Geometry and physics: Though lovers be lost love shall not
To paraphrase a certain Polish mathematician: “the most important ideas in mathematics come from physics”. While there is no reason why mathematics — as mathematics — should come from physics, there is some deep connection between mathematics and our understanding of the Universe. Wigner in 1960 in his famous “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” article, noticed how the mathematical structure of physical theories can lead to new physical insight. And of course, physical insight can lead to new mathematics. By physics, I will mean the construction of mathematical models of natural phenomena and the comparing of the predictions of these models against nature.
From the very nature of physics it is clear that there is at least some superficial relation with mathematics. After all, physics uses mathematics. However, physics is not mathematics in the sense that mathematical constructions in physics should have (maybe not directly) some meaning. There must be some relation of the mathematics to a physical law. In mathematics, there is no such constraint that any of it have any meaning beyond what it mathematically means. It is a complete mystery as to why nature seems generally amenable to being understood in terms of abstract mathematics.
The deep interconnection between mathematics and physics seems especially true when focusing in on geometry: literally geometry means `Earth measurement’. At the most basic level, geometry is the study of spaces, which are understood as collections of points, together with a notion of points being `close to each other or not’, and usually with some further mathematical structures on them, such as a notion of the distance between to near by points. But this is definition in terms of points is not enough to cover the modern usage of `geometry’. So, what is geometry and where does it come from? Moreover, what has the study of spaces got to do with physics?
The first work on synthetic geometry is the book Elements written Euclid of Alexandria (c.325–265 BC). In this book an axiomatic approach to plane geometry, so parallel lines on flat surfaces etc., is established. For example, the internal angles of a triangle on the plane always add up to 180 degrees. However, curves, circles and spheres had been known about since antiquity. Solid geometry — the study of three dimensional objects — was needed as soon as humans started to imagine buildings such as domes and pyramids. In addition to this, the heavenly sky can be imagined as the inner surface of a dome speckled with stars — at least as we see it, and ancient astronomers saw it!
Methods of calculating the volume of simple regular three dimensional objects were developed. For example, the ancient Egyptians knew how to calculate the volume of pyramids and chambers therein: they were the mummy of all modern mathematicians! Archimedes (287–212 BC) in his `eureka’ moment realised that one could deduce the volume of three dimensional irregular objects based on the amount of water they displaced. However, Archimedes was unable to actually calculate volumes in any generality.
In another direction, Apollonius of Perga (c.262–190 BC) showed that the regular curves — circles; ellipses; parabola; and hyperbola — can be formed by cutting the cone, hence conic sections. Amazingly, in Newtonian gravity (circa 1686) the orbits of the two massive bodies are described by conic sections. This is part of the unifying power of mathematics: the mathematics involved in cutting cones is exactly the mathematics needed to describe orbits, for example the path of the Moon around the Earth! These mathematical coincidences are abound.
The most important mathematical works on conic sections — as far as our story goes — are that of Descartes (1596–1650) and Fermat (1601–1665), who in the 17th century brought algebra in to the game. Conic sections can be described by algebraic equations via coordinates — analytic and algebraic geometry were born! \par
The use coordinates (eg. x and x on the plane) opens up the use of calculus in geometry. Newton’s differential and integral calculus allows for methods of calculating gradients of curves, areas under curves, the volumes of objects etc. — calculus today is a common method of torturing undergraduate students! Differential geometry was born … or at least the seeds of the theory were planted by Newton (1642–1727) and Leibniz (1646–1716). One should not forget that much of Newton’s inspiration in developing calculus comes from his work on classical mechanics: so the mathematical description of the motion of massive bodies.
Curved surfaces – such as the sphere – represent non-Euclidean geometries. Lines drawn on them violate the axioms of Euclid’s plane geometry: this was seen as a real problem by mathematicians. It was Eugenio Beltrami (1835–1899) who showed that hyperbolic geometry is consistent: this is the geometry of surfaces of constant curvature for which the internal angles of a triangle add up to less that 180 degrees. Similar results were obtained for spherical geometries, so geometries of constant curvature for which the internal angles of triangles add up to more that 180 degrees.
Bernhard Riemann (1826–1866) in his PhD thesis extended the work of Beltrami to surfaces that have non-uniform curvature, and to higher dimensions. The work of Riemann allowed algebra and calculus to be applied to spaces known as smooth manifolds, i.e., spaces such that every `small piece’ of them looks like a `small piece’ of the n-dimensional plane for some integer n. One should keep in mind the relation between a globe and a map: any small piece of the globe can be represented on a sheet of paper as a map, and points on the globe are then represented by two numbers, the coordinates with respect to the given map. The notion of a smooth manifold underpins Einstein’s special and general relativity, as well as Maxwell’s theory of electromagnetism, Yang–Mills theories and classical mechanics: even thermodynamics has a geometric formulation!
It is worth saying a little more about Einstein’s general relativity (1916). This theory is a theory of gravity, and to date it is the most accurate theory of gravity we have. Moreover four dimensional smooth manifolds are central to the theory. Einstein took the earlier idea that space and time should be unified into space-time seriously, we have one time coordinate and three space coordinates. Einstein then told us that gravity is not your typical force, but rather it really is due to the local shape of space-time! The mathematical theory of curved smooth manifolds is vital to our understanding of gravity and the Universe as a whole, and vice versa, physics has been the impetus for many mathematical works on curved smooth manifolds.
There is a duality between a space and the algebra of functions on that space ( i.e., maps from that space to the real or complex numbers). Loosely, if you know the algebra of functions on a space, then you know the space. The algebra of functions on a classical space is commutative: the order of pointwise multiplication does not matter. We can imagine a more general notion of a `space’ by considering any algebra — not necessarily commutative — as the algebra of functions on some `space’. The phase space of quantum mechanics, that is the `space’ of positions and momenta of a quantum particle, is a noncommutative geometry.
A quantum theory of gravity could be some kind of noncommutative geometry: both string theory and loop quantum gravity suggest noncommutativity of space-time at some level — both the loopers and p-braners agree on this! Trying to make sense of physics at the smallest scales pushes what we mean by geometry well beyond our original understanding. Noncommutative geometries are in general not set theoretical objects, i.e., they do not consist of a collection of points — it is all rather pointless!
There is a kind of `halfway house’ between classical and quantum geometry: here I refer to supermanifolds as defined by Berezin and Leites in 1976. Without details and being very loose, a supermanifold is a `manifold-like object’ which comes with some coordinates that commute with all the coordinates, and some coordinates that anticommute amongst themselves. By anticommute we mean that they pick up a minus sign when we exchange the order they appear in expressions. In particular we have some coordinates that square to zero!
Supermanifolds play the role of manifolds when, for example, fermions such as the electron are present in the theory. If we want to develop a `classical’ theory of fermions then we must employ objects that anticommute: one can justify this using the Pauli exclusion principle — no more than one fermion can be in a given quantum state, while for bosons there is no such restriction. Heuristically, one can say that bosons like to be together, while fermions are rather more like hermits.
Supermanifolds offer a conceptual and geometric way to treat bosons and fermions on equal footing: supermanifolds are the geometry of supersymmetry. In short, supersymmetry is an operation that allows us to `rotate’ a boson into a fermion and vice versa. It turns out that this is not just a neat way of unifying bosons and fermions, but theories that posses supersymmetry can have remarkable mathematical and phenomenological properties — we await CERN’s confirmation that nature uses supersymmetry!
Another amazing link between geometry and physics can be found in mirror symmetry which relates pairs of particular manifolds called Calabi-Yau manifolds. Superstring theory is 10 dimensional, yet our physical world appears 4 dimensional — one time and three space dimensions. To overcome this discrepancy one can postulate that 6 of these dimensions is `scrunched up tightly’, and all we see is four dimensions on all but the very smallest scales. These compactifications as they are known, are Calabi-Yau manifolds, and different compactifications in general lead to different physics. However, it was noticed in the late 1980s by Dixon, Lerche, Vafa, and Warner that two different versions of superstring theory (type IIA and IIB) can be compactified on two different Calabi-Yau manifolds, yet lead to the same physics. In this case the two Calabi-Yau manifolds are said to be mirror duals, and the symmetry between the physics is known as mirror symmetry. This pairing of Calabi-Yau manifolds is now an active area of mathematical research with much effort devoted to carefully understanding the intuitive physics based picture.
In conclusion, not only has geometry been essential in developing physical theories, but these theories then push our understanding of geometry and lead to new mathematics. I have only touched upon a tiny part of this interrelation. There are a great number of other things I could have described and new links are being uncovered all the time. What will future mathematicians understand by the term `geometry’ is anyone’s guess. However, I am sure it will be closely related to our understanding of the physical Universe.
]]>It was suggested to me on Avvo.com that I report you to authorities for “illegal” activity in relation to the place of employment that I worked for you at. I, however, am wise enough to be aware that the current mobocratic system is corrupt and for the most part not worth my while to deal with. It’s extremely inefficient, in my experience.
However, it would appear that I’m the, if not only, person on Earth who can walk about 9.9 miles along some piece of Earth, Terra, what-have-you. As such, I am considering walking to Toni’s of Winnebago and paying you a visit in a manner that I hope will not cause me to use a 2-dimensional nature of adversarial conflict toward you.
I believe I am reasonable in what I do and that you’re schizoaffective behavior is unacceptable. As such, I believe it’s reasonable to suggest to people close to you that you’re thrown in a white-padded cell for quite a while.
Yes, I could write a letter. However, I am under the interpretation that you will not respond. However, I am extremely curious as to how you will behave after I personally walk there and pay you a visit.
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Ok, here is something that annoys me about employer’s and their websites.
This is from pci pharma services:
“How did you hear about this position?
Please tell us how you heard about this position by selecting a source from this list.”
Ok…. uhhhhhh. Subvocalization aside, I don’t think I “heard” about the position. I mean, I was on indeed.com, saw an ad for on pci pharma job (packer), went to the pci (I think) website and saw that it wasn’t being listed. I went to look around at the pci website for other jobs, found something of interested, and “decided” to apply. Now, did I “hear” about it?
I mean, you’re asking me to apply for a job that involves analysis, right?
Business Development Estimation Manager
So, I’ve got a bachelor’s degree. And you’re a pharma company. I find me degree reasonably relative to what you’re hustling, as I’m a scientist with a bachelor’s in science. I also have experience running my own business.
But, wow… just, “How did you hear about this job?”
Know what I think? I think I ought to short your website, your company, and etc.. for just negligence. In other words, I ought to ignore you and have your assets and company diffuse to me rather than deal with your ignorance.
What? You think I’m a paranoid schizophrenic and “hear” voices?
*roll my eyes*
Uptight? No, I’m a hustler, baby. Yeeahhhh.
I’m a male, but I like this song, so I’mma push it: https://www.youtube.com/watch?v=h5kHpmXOoc8
Nurul Adhwa Abdul Rahman, Siti Hanna Muharram, Oduola Abiola*
Pengiran Anak Puteri Rashidah Sa’adatul Bolkiah Institute of Health Sciences, Universiti Brunei Darussalam.
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I reason with something like the Bitcoin project, the SHA-2 protocol (as an auditor) would eventually lay the smackdown on them. So, I find it interesting and odd that I come across erroneous and contradicting information, such as what I experienced on Thermo Fisher. I have issues recalling how to spell the company name at times.
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I’ve used the alias Cyberman, this is something different.
]]>Funny enough, the website even said that it forgot my e-mail address login information. And I’m all like, “Alright, I’ll play your bullshit game.”
Then it spits out that somehow I’m in the system. Obscure, really.
A lot of this bullshit reminds me of some guy from MIT that I read/learned about that was calling himself Cyberman. The dude would document stuff with camera’s, etc… People did not like him. As I’m on a library computer, as an individual with the first name “Michael” supposedly tossed my laptop due to an alleged bedbug infestation, I am lacking in the whole documentation aspect of things and telling Thermo Fisher that they suck. Otherwise, I reason I’ll just hack and crack into the website.
No, I don’t think I forgot my password. Actually, I’d like to see the damn website tell me what it thinks my password is rather than allege that I’ve typed the incorrect password. Some website actually give you the option of showing the password you’ve typed in. AKA, I’m think for some obscure reason the Thermo Fisher website is engaged in illusion/negligence.
]]>Agnes Brown’s grandson Bono asked his grandmother about Einstein’s theory of relativity. The joke is that this is far to complicated for an old Irish Granny to answer.
This made me think. Einstein first published on special relativity in 1905 and the field equations for general relativity were published in 1916. So humanity has had knowledge of Einstein’s relativity for 100 years now, yet Mrs Brown was unable to say anything!
Not that we should expect everyone to have a detailed mathematical knowledge of Einstein’s relativity, but should just about everyone be able to say something?
Special relativity
Special relativity is based on two postulates – that then lead to a consistent mathematical description in terms of differential geometry, but we should ignore that for now – that can be paraphrased as follows
Even more pedagogically, all observers that are not experiencing a net force are `equally as good’ as far as determining the laws of physics are concerned, and on top of that, light waves travel at a fixed speed as measured by these privileged observers.
General relativity
This is a bit harder to paraphrase, but basically we have three key features
Again, all this can be made mathematically precise.
Are we expecting too much?
I think it is too much to expect any real knowledge of Einstein’s relativity from the general public. So, we should not ask for this, but only a vague idea of the ideas from most people.
From my own perspective it is all differential geometry
]]>I have a contribution with Janusz Grabowski, Katarzyna Grabowska and Paweł Urbański entitled New developments in geometric mechanics.
Gennadi Sardanashvily – passed away on the September 1, 2016 – also has a contribution in the proceedings. I did not know Sardanashvily well, but our few interactions told me he was a nice guy. I am sure the community will miss him.
In better news, my wife Gemma had a portrait of Janusz Grabowski published in the proceedings!
]]>1.Infinite list of binary sequences
2.About the Power set of ℕ
3.Frame of Natural Infinity
4.List of numbers smaller than 1
a.Creation of the numbers
b.Denseness of R..
c.Completeness of R..
d.Real numbers in [0,1[
5.About Cantor’s first proof
6.About the diagonal argument
7.Conclusion
Please read the article at
PDF Lists of binary sequences and uncountability
http://pengkuanonmaths.blogspot.com/2016/11/lists-of-binary-sequences-and.html
or Word https://www.academia.edu/30072323/Lists_of_binary_sequences_and_uncountability
In the paper we show that Graded bundles (cf. [2]), which are a particular kind of graded manifold (cf. [3]), can be `fully linearised’ or `polarised’. That is, given any graded bundle of degree k, we can associate with it in a functorial way a k-fold vector bundle – we call this the full linearisation functor. In the paper [1], we fully characterise this functor. Hopefully, this notion will prove fruitful in applications as k-fold vector bundles are nice objects that that various equivalent ways of describing them.
Graded Bundles
Graded bundles are particular examples of polynomial bundles: that is we have a fibre bundle whose are $latex \mathbb{R}^{N}$ and the admissible changes of local coordinates are polynomial. A little more specifically, a graded bundle $F$, is a polynomial bundle for which the base coordinates are assigned a weight of zero, while the fibre coordinates are assigned a weight in $latex \mathbb{N} \setminus 0$. Moreover we require that admissible changes of local coordinates respect the weight. The degree of a graded bundle is the highest weight that we assign to the fibre coordinates.
Any graded bundle admits a series of affine fibrations
$latex F = F_k \rightarrow F_{k-1} \rightarrow \cdots \rightarrow F_{1} \rightarrow F_{0} =M$,
which is locally given by projecting out the higher weight coordinates.
For example, a graded bundle of degree 2 admits local coordinates $latex (x, y ,z)$ of weight 0,1, and 2 respectively. Changes of coordinates are then, `symbolically’
$latex x’ = x'(x)$,
$latex y’ = y T(x)$,
$latex z’ = z G(x) + \frac{1}{2} y y H(x)$,
which clearly preserve the weight.
We then have a series of fibrations
$latex F_2 \rightarrow F_1 \rightarrow M$,
given (locally) by
$latex (x,y,z) \mapsto (x,y) \mapsto (x)$.
Linearisation
The basic idea of the full linearisation is quite simple – I won’t go into details here. Recall the notion of polarisation of a homogeneous polynomial. The idea is that one adjoins new variables in order to produce a multi-linear form from a homogeneous polynomial. The original polynomial can be recovered by examining the diagonal.
As graded bundles are polynomial bundles, and the changes of local coordinates respect the weight, we too can apply this idea to fully linearise a graded bundle. That is, we can enlarge the manifold by including more and more coordinates in the correct way as to linearise the changes of coordinates. In this way we obtain a k-fold vector bundle, and the original graded bundle, which we take to be of degree k.
So, how do we decide on these extra coordinates? The method is to differentiate, reduce and project. That is we should apply the tangent functor as many times as is needed and then look for a substructure thereof. So, let us look at the degree 2 case, which is simple enough to see what is going on. In particular we only need to differentiate once, but you can quickly convince yourself that for higher degrees we just repeat the procedure.
The tangent bundle $latex T F_2$ – which we consider the tangent bundle as a double graded bundle – admits local coordinates
$latex (\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)} ,\; \underbrace{z}_{(2,0)} \; \underbrace{\dot{x}}_{(0,1)}, \; \underbrace{\dot{y}}_{(1,1)} ,\; \underbrace{\dot{z}}_{(2,1)})$
The changes of coordinates for the ‘dotted’ coordinates are inherited from the changes of coordinates on $latex F_2$,
$latex \dot{x}’ = \dot{x}\frac{\partial x’}{\partial x}$,
$latex \dot{y}’ = \dot{y}T(x) + y \dot{x} \frac{\partial T}{\partial x}$,
$latex \dot{z}’ = \dot{z}G(x) + z \dot{x}\frac{\partial G}{\partial x} + y \dot{y}H(x) + \frac{1}{2}y y \dot{x}\frac{\partial H}{\partial x}$.
Thus we have differentiated.
Clearly we can restrict to the vertical bundle while still respecting the assignment of weights – one inherited from $latex F_2$ and the other comes from the vector bundle structure of a tangent bundle. In fact, what we need to do is shift the first weight by minus the second weight. Technically, this means that we no longer are dealing with graded bundles, the coordinate $latex \dot{x}$ will be of bi-weight (-1,1). However, the amazing thing here is that we can set this coordinate to zero – as we should do when looking at the vertical bundle – and remain in the category of graded bundles. That is, not only is setting $latex \dot{x}=0$ well-defined, you see this from the coordinate transformations; but also this keeps us in the right category. We have preformed a reduction of the (shifted) tangent bundle.
Thus we arrive at a double graded bundle $latex VF_2$ which admits local coordinates
$latex (\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)} ,\; \underbrace{z}_{(2,0)}, \; \underbrace{\dot{y}}_{(0,1)} ,\; \underbrace{\dot{z}}_{(1,1)})$,
and the obvious admissible changes thereof.
Now, observe that we have the degree of $latex z$ as (2,0), which is the coordinate with the highest first component of the bi-weight. Thus, as we have the structure of a graded bundle, we can project to a graded bundle of one lower degree $latex \pi : VF_2 \rightarrow l(F_2)$. The resulting double vector bundle is what we will call the linearisation of $latex F_2$.
So we have constructed a manifold with coordinates
$latex (\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)}, \; \underbrace{\dot{y}}_{(0,1)} ,\; \underbrace{\dot{z}}_{(1,1)})$,
with changes of coordinates
$latex x’ = x'(x)$,
$latex y’ = y T(x)$
$latex \dot{y}’ = \dot{y}T(x)$,
$latex \dot{z}’ = \dot{z}G(x) + y \dot{y}H(x)$.
Then, by comparison with the changes of local coordinates on $latex F_2$ you see that we have a canonical embedding of the original graded bundle in its linearisation as a ‘diagonal’
$latex \iota : F_2 \rightarrow l(F_2)$,
by setting $latex \dot{y} = y$ and $latex \dot{z} = 2 z$.
References
[1] Andrew James Bruce, Janusz Grabowski and Mikołaj Rotkiewicz, Polarisation of Graded Bundles, SIGMA 12 (2016), 106, 30 pages.
[2] Janusz Grabowski and Mikołaj Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36.
[3] Th.Th. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 131-168.
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HISTRUCT — Workshop on higher structures
When: 13–16 December 2016
Where: University of Luxembourg-campus Kirchberg, Luxembourg, LUXEMBOURG
Website: http://math.uni.lu/leibniz/
Aim and scope
The purpose of this workshop is to bring together mathematicians working on Leibniz algebras and other higher structures.
Confirmed speakers include:
Olivier ELCHINGER (University of Luxembourg)
Yaël FRÉGIER (Université d’Artois)
Xevi GUITART (Universitat de Barcelona)
Honglei LANG (Max Planck Institute for Mathematics)
Camille LAURENT-GENGOUX (University of Lorraine)
Zhangju LIU (Peking University)
Mykola MATVIICHUK (University of Toronto)
Sergei MERKULOV (University of Luxembourg)
Norbert PONCIN (University of Luxembourg)
Florian SCHÄTZ (University of Luxembourg)
Martin SCHLICHENMAIER (University of Luxembourg)
Boris SHOIKET (Antwerp University)
Mathieu STIENON (Pennsylvania State University, USA)
Ping XU (Pennsylvania State University, USA)
Registration : http://math.uni.lu/leibniz/reg.html
The deadline for registration is the 2nd of December 2016.
Research Project
– This conference is funded in the frame of the OPEN Scheme of the Fonds National de la Recherche Luxembourg (FNR) with the project QUANTMOD O13/5707106 and
– Partial funding by the Mathematics Research Unit is acknowledged.
Please feel free to circulate this announcement around you!
The organizers:
Martin Schlichenmaier (Luxembourg)
Ping Xu (Penn State, USA)
Olivier Elchinger (Luxembourg)
Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean?
Please read the article at
PDF Continuity and uncountability
http://pengkuanonmaths.blogspot.com/2016/09/continuity-and-uncountability.html
or Word https://www.academia.edu/28750869/Continuity_and_uncountability
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Twenty four members of the Department of Mathematics at the University of Leicester – the great majority of the members of the department – have been informed that their post is at risk of redundancy, and will have to reapply for their positions by the end of September. Only eighteen of those applying will be re-appointed (and some of those have been changed to purely teaching positions). This is supposedly because of a financial crisis at the University, though the union disputes this claim. It should be noted that there is no formal tenure in the UK, but such mass redundancies are highly unusual.
You can add your name to the online petition against this unusual
attempt at:
http://www2.le.ac.uk/institution/unions/ucu/news/no-redundancies-no-confidence
In this case it would be helpful to mention in the comments section that your signature is in support of the Mathematics Department (the petition is for the whole University, but apparently only the Math Dept has been formally notified of the redundancies at this stage).
You can also write directly to:
Professor Paul Boyle
President and Vice-Chancellor
University of Leicester
University Road,
Leicester, LE1 7RH,
United Kingdom
In the `classical language’, a Courant algebroid is a vector bundle, whose sections come equipped with a bracket – bilinear map – together with an anchor map and a nondegenerate symmetric bilinear form that satisfy some compatibility conditions. The bracket on the space of sections is not a Lie bracket, but rather a non-skewsymmetric bracket that satisfies the Jacobi identity in Loday-Leibniz form. This bracket is usually called the Courant–Dorfman bracket.
A pre-Courant algebroid can be thought of as a Courant algebroid but without the Jacobi identity on the Courant–Dorfman pre-bracket.
It has long be known, due to Roytenberg [1], that Courant algebroids are `really’ symplectic Lie 2-algebroids. That is, we have an N-manifold of degree 2 (a supermanifold with a particular additional grading), equipped with a nondegenerate Poisson bracket of degree -2 and a homological vector field of degree 1 that is Hamiltonian. The brackets of Courant algebroid can then be recovered using the derived bracket formalism and the bilinear form is encoded in the symplectic structure.
Pre-Courant algebroids in the superlanguage
So, do we have a similar understanding of pre-Courant algebroids? The answer is yes…
First back to Courant algebroids. As stated above, they can be encoded in a Hamiltonian vector field – and so they can be encoded in a Grassmann odd Hamiltonian of degree/weight 3, which we denote as $latex \Theta$. The fact that the Hamiltonian vector field is homological (Grassmann odd and squares to zero) is equivalent to
$latex \{ \Theta, \Theta \} =0 $.
This condition encodes all the compatibility conditions between the bracket and the anchor map (a particular vector bundle map to the tangent bundle). More than that, this condition also encodes the Jacobi identity for the bracket. Thus, we need a weaker condition that is not too weak – we only want to lose the Jacobi identity and keep the other conditions. It turns out that we require
$latex \{\{ \Theta, \Theta \}, f\} =0 $,
for all weight zero functions f, if we want to encode a pre-Courant algebroid in exactly the same way as we do a Courant algebroid. In the preprint we define what we call symplectic almost Lie 2-algebroids in this way and show how they correspond to pre-Courant algebroids.
Does this help any?
This change in starting position simplifies many basic facts about pre-Courant algebroids – just as it does with Courant algebroids. In particular, the notion a Dirac structures as a particular Lagrangian submanifolds is quite clear.
In the preprint was also show that including a compatible N-grading is quite simple when one uses the language of homogeneity structures [2]. One should also consult [3,4] where the notion of weighted Lie groupoids and weighted Lie algebroids are explored. As an example VB-Courant algebroids – Courant algebroids with a compatible vector bundle structure – are natural examples of weighted (pre-)Courant algebroids. This change of postion to `graded super bundles’ with some additional structures allows for a very neat understanding of weighted Dirac structure and in particular VB-Dirac structures. This framework simplifes the understanding of many thing.
Conclusion
The bottom line seems to be that Courant algebroids are `really’ sympelectic Lie 2-algebroids and pre-Courant algebroids are really symplectic almost Lie 2-algebroids.
References
[1] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in: Quantization, Poisson brackets and beyond (Manchester, 2001), 169–185, Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002.
[2] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21–36.
[3] A.J. Bruce, K. Grabowska & J. Grabowski, Graded bundles in the category of Lie groupoids, SIGMA 11 (2015), 090.
[4] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101
(2016), 71–99.
Here is a very interesting look at the split-brain experiments by CGP Grey: https://www.youtube.com/watch?v=wfYbgdo8e-8
]]>In this paper we take the point of view that Jacobi geometry is best understood as homogeneous Poisson geometry – that is Poisson geometry on principle $latex \mathbb{R}^{\times}$-bundles. Every line bundle over a manifold can be understood in terms of such a principle bundle.
The same holds try when we pass to supermanifolds. With this in mind Alfonso and I more-or-less just replace Poisson with higher or homotopy Poisson. This allows us to neatly define an $latex L_{\infty}$-algebra on the space of sections of an even line bundle in the categeory of supermanifolds. This algebra is the higher/homotopy generalisation of Kirillov’s local Lie algebra on the space of sections of a line bundle.
We show that the basic theorems from Kirillov’s local Lie algebras or Jacobi bundles all passes to this higher case.
Refrences
[1] Andrew James Bruce & Alfonso Giuseppe Tortorella, Kirillov structures up to homotopy, Differential Geometry and its Applications Volume 48, October 2016, Pages 72–86.
In this work we take the notion of a curve on a supermanifold to be an S-curve, which is an ‘element’ of the mapping supermanifold Hom(R,M) [1]. This mapping supermanifold is a generalised supermanifold and so it is a functor from the (opposite) category of supermanifolds to sets. Each ‘element’ needs to be ‘probed’ by a supermanifold, and so S-curves are ‘curves’ that are parameterised by all supermanifolds. Or maybe better to say that an S-curve is a family of functors paramaterised by time. At any given time and a given supermanifold S, we have a morphism of supermanifolds S → M. That is, an S-curve tracks out the S-points of M.
With this robust notion of a curve, we go on to define what we mean by an autonomous ordinary differential equation on a supermaifold, and more importantly what we mean by a solution. This seems to have been a notion not at all clearly defined in the existing literature. For us, a differential equation is a sub-structure of the tangent bundle of the said supermanifold, and solutions are S-curves on the supermanifold for which their tangent prolongation sit inside the differential equation. This is very close to the classical notions, but now we use S-points and not just the topological points.
We then take these notion and apply them to supermechanical systems given in terms of a Lagrangian. We use Tulczyjew’s geometric approach to Lagrangian mechanics, and really we only modify the notion of a curve and not the underlying geometry of Tulczyjew’s approach [2]. In doing so, we have a well defined notion of the phase dynamics, the Euler-Lagrange equations and solutions thereof for mechanical systems on supermanifolds. We present a few nice example, includinh Witten’s N=2 supersymmetric model [3] and geodesics on a super-sphere.
The importance of this work is not so much in the equations we present, these can be derived using formal variations. The point is we give some proper mathematical understanding of solutions to the equations.
References
[1] Andrew James Bruce, On curves and jets of curves on supermanifolds, Archivum Mathematicum, vol. 50 (2014), issue 2, pp. 115-130.
[2] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincare Sect. A (N.S.), 27(1):101–114, 1977.
[3] Edward Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. , B188:513, 1981.
The preprint outlines much of our recent work on graded bundles (a nice kind of graded manifold) and their linearisation (as a functor to k-fold vector bundles), as well as the notions of weighted Lie groupoids and algebroids, including the Lie theory.
One key observation that must be made is that there are many examples of graded bundles that appear in the existing literature, it is just that they are not recognised as such and their graded structure is not really exploited. The canonical example here are the higher order tangent bundles which are well studied from the perspective of higher order mechanics.
Anyway, if anyone want to get a quick overview of some of the ideas behind my work, then I direct them to this preprint. If you are interested in the applications to mechanics, then I suggest [2] as well as references therein.
References
[1] Introduction to graded bundles, Andrew J. Bruce, K. Grabowska, J. Grabowski, arXiv:1605.03296 [math.DG]
[2] New developments in geometric mechanics, A. J. Bruce, K. Grabowska, J. Grabowski, P. Urbanski, arXiv:1510.00296 [math-ph].
]]>Most exciting is that my graduate student made a documentary! Check it out!
New paper 1: Genetic diversity in RNA virus populations drives adaptation to spatially mixed host environments
We just had a super exciting paper accepted too! stay tuned for an update on it.
]]>The mathematics of special relativity
It is more-or-less true that Einstein’s original works on special relativity do not really use any highbrow mathematics. In a standard undergraduate introduction to the subject no more than linear algebra is really used: vector spaces, matrices and quadratic forms.
So, as linear algebra is well-founded, one is not going to find some internal inconsistencies in special relativity.
Moreover, today we understand special relativity to be based on the geometry of Minkowski space-time. Basically, this is Euclidean with an awkward minus sign in the metric. Thus, special relativity, from a geometric perspective, is as well-founded as any thing in differential geometry.
So one is not going to mathematically prove that special relativity is wrong in any mathematical sense.
On to physics…
However, the theory of special relativity is falsifiable in the sense of Popper. That is, taking into account the domain of validity (ie., just the situations you expect the theory to work), experimental accuracy, statistical errors etc. one can compare the theoretical predictions with what is measured in experiments. If the predictions match the theory well, up to some pre-described level, then the theory is said to be ‘good’. Otherwise the theory is ‘bad’ and not considered to be a viable description of nature.
In this sense, using not much more that linear algebra one could in principle calculate something within special relativity that does not agree well with nature (being careful with the domain of validity etc). Thus, one can in principle show that special relativity is not a ‘good’ theory by finding some mismatch between the theory and observations. This must be the case if we want to consider special relativity as a scientific theory.
Is special relativity ‘good’ or ‘bad’?
Today we have no evidence, direct or indirect, to suggest that special relativity is not a viable description of nature (as ever taking into account the domain of validity). For example, the standard model of particle physics has at its heart special relativity. So far we have had great agreement with theory and experiment, the electromagnetic sector is extremely well tested. This tells us that special relativity is ‘good’.
Even the more strange predictions like time dilation are realised. For example the difference in the life-time of muons as measured at rest and at high speed via cosmic rays agrees very well with the predictions of special relativity.
Including gravity into the mix produces general relativity. However, we know that on small enough scales general relativity reduces to special relativity. Any evidence that general relativity is a ‘good’ theory also indirectly tells us that special relativity is ‘good’. Apart from all the other tests, I offer the discovery of gravitational waves as evidence that general relativity is ‘good’ and thus special relativity is also ‘good’.
The clause
The important thing to remember is that the domain of validity is vital in deciding if a theory is ‘good’ or ‘bad’. We know that physics depends on the scales at which you observe, so we in no way would expect special relativity be a viable description across all scales. For example, when gravity comes into play we have to consider general relativity.
On the very smallest length scales, outside of what we can probe, we expect the nature of space-time to be modified to take into account quantum mechanics. Thus, at these smallest length scales we would not expect the description of space-time using special relativity to be a very accurate one. So, no one is claiming that special relativity, nor general relativity is the final say on the structure of space and time. All we are claiming is that we do have ‘good’ theories by the widely accepted definition.
Are all claims that relativity is wrong bogus?
Well, one would have to examine all claims carefully to answer that…
However, in my experience most objections to special relativity are based on either philosophical grounds or misinterpreting the calculations. Neither of these are enough to claim that Einstein was completely wrong in regards to relativity.
]]>Please read the article at
PDF Cardinality of the set of decimal numbers http://pengkuanonmaths.blogspot.com/2016/03/cardinality-of-set-of-decimal-numbers.html
or Word https://www.academia.edu/23155464/Cardinality_of_the_set_of_decimal_numbers
]]>The endgame is that nobody really knows what they are doing. Nobody really knows how to judge a case, but there are rules and procedures. But I want to talk into consideration something of the utmost importance that’s found in just about any legal case: The concept of blame.
The concept of blame can be taken a few ways:
I’d like to take stance three with this whole issue. Thus, I post that people should “blame God” or simply not blame, as the universe simply works out the way it does. It reasoning appears circular, but the universe appears to be circular itself. But what I want to get at is this: Stop blaming people. It doesn’t make any sense. No one has any control over their actions. Furthermore, believing that anyone can affect the outcome of any behavior through any judicial technique or methodology (therapeutic justice, restorative justice, etc..) is a delusion. So, the endgame is to get people to stop blaming people.
And I think society is the way it is right now with a lot of people not understanding much law, science, and philosophy. So, my proposal is this: Stop making new laws. That’s right. Immediately, stop making new laws. And from there, start creating more lenient sentencing guidelines.
One issue with laws is that the creation of laws really outpace how fast people can understand them, know about them, read them, and adapt to them in society. And if we want to consider the learning process and adaption an illusion, then fine. You might say that I’m engaging in similar behavior as the judges might in case 2, which is saying there is determinism and treating it like free-will. However, I have not stated that I believe anyone can actually implement this system, “cause” it to be implemented,” or “willed” into existence. I’m suggesting that it occur. That’s all I’m doing: Suggesting.
What do I think would be the “effect” or ideal “effect” from this situation?
People stop blaming other people. Easy. The world starts to release itself from a schizoaffective mindframe about reality. The world becomes more realistic. The fabric of society becomes realized for the laws that are and are not in place. Things become what I call an “anarchic adventure,” whereby individuals have to “accept” there is a “risk” that shit can hit the fan from their “actions.”
I mean, the Internet is a great example. Pre- year 2000, it was anarcho-commnuist. It was decent. People just did whatever based on their views. There wasn’t government control so much. It was communities going about their own business. There wasn’t “harssment” law and the such. If someone thought you were pestering another, you could get banned without the potential of legal ramification. And if you were using the Internet, you had an understanding that people might start crap with you on the Internet without legal ramifications, so you “took that risk” of having a bad day from someone talking shit to you.
No new laws.
Genecks for President.
]]>Please read the article at
PDF Prime numbers and irrational numbers
http://pengkuanonmaths.blogspot.com/2016/02/prime-numbers-and-irrational-numbers.html
or Word https://www.academia.edu/22457358/Prime_numbers_and_irrational_numbers
]]>The year 2020 had been a wild roller coaster of a year; world wide nuclear war had just barely been averted as political and religious violence came to a head in a land that had seen millennia of conflict.
The synergy of constant political upheaval, religious conflict and ethnic violence had finally been given access to the planets most destructive weapons technology. The horrific outcome had very nearly been preordained.
Nuclear warheads had been detonated over six targets. Two other warheads did not reach a target. Most of them exploded over high tech cities in the Middle East, one had exploded accidentally while it was being loaded into an airplane and one detonating in a remote corner of China after being launched by North Korea. A mistake certainly, but China had not been amused.
China, despite the insane ranting of the North Korean government, had invaded and took over North Korea in just a few days with very little fighting. The North Korean military had folded almost immediately, the military leaders of the DPRK had long grown weary of the frequent purges, executions, wide spread hunger and despotism of Kim Jong-un. The military, using the crisis as an opportunity, had surrendered almost without resistance. Now, much to the world’s surprise, the Chinese were negotiating a unification of the Koreas under the banner of South Korea.
Many unlikely things had happened this year. After the dust had more or less settled in the Middle East, people began to understand the destruction that had been wrought, even in that limited war was unacceptable. The countries involved had centuries of conflict and hate to motivate them. But, the horrors of modern cities obliterated, billions of dollars of high tech engineering marvels, millions of lives swept away like feathers in a hurricane was heady motivation. Enough motivation to force a change in thinking from the inherently violent, often inhuman conquest in the name of religion and politics mind set toward a state that was at least tolerant of each other if not all out cooperation.
Israel was the country that lost part of a city and an airbase to the accidental detonation of a nuke as it was being loaded on an aircraft, supposed to be impossible but this year was full of impossible events that didn’t seen to know they were impossible. As a couple of destroyed mega cities, built by oil money, were being mourned and clean up was being planned something truly improbable if not outright impossible had occurred.
On December 19th as many people were trying to calm back down and live a bare two kilometers from the international space station, an alien space craft seemed to simply pop into existence. No one on board the space station had actually seen it appear. An automated camera system had been looking almost directly at the spot where the alien craft had suddenly materialized. A two kilometer long space craft that looked amazingly like a famous science fiction space craft, in the video the space craft had appeared but what looked like an after image had receded into the distance. Two people on the ground had been looking at the space station with small telescopes noticed this as well. In the coming months why this after image had appeared to recede into the distance was a popular point of discussion among both scientists and laymen alike.
Sadly this was one of the few things that could be discussed about the alien craft. It hung in low earth orbit for several weeks as humanity tried desperately to communicate with the aliens.
The spacecraft, now called Star Ship by nearly everyone, could be seen with the naked eye and after seven weeks odd little metallic balls started to appear around the planet. The initial metallic spheres were about a meter in diameter. These large globes were seen disgorging smaller balls about the size of golf balls. These smaller spheres gave birth to even smaller BB sized specks. Floating through the air these metallic balls showed up nearly everywhere on the planet as days rolled by. Giving one more oddity to be debated fruitlessly.
In the North Pacific an American Navel Task Force was on routine maneuvers when an object appeared on the radar approaching fast. Before planes could be scrambled or any measures taken the elongated diamond shaped craft was hovering, then landing on the deck of the aircraft carrier Gerald R. Ford.
The military had been given orders not to interfere with any of the strange spheres for fear of provoking some sort of destructive response. This craft was obviously not one of the spheres but the aircraft’s speed was such that no response was in actuality possible The craft sat on the deck as people scrambled to do something, anything. No one in fact, had time to do much as the craft off loaded a large metallic box and subsequently rose and speed away into the air over the empty ocean.
Waves of patterns and colors passed over the strange looking bodies of the aliens. Waving their antenna which were amazingly like moth antenna and clicking popping several pairs of forelimbs together with a release of pheromones completed the communication they were sharing while they watched a holographic representation of the vicinity of the box as it was passed into the innards of the Aircraft Carrier:
“Ghastly diminutive creatures don’t you think?”
“Yes” answered his companion with a similar display of colors, patterns, clicking noises and chemicals. “Yes they are pitiful looking creatures, so simple, yet primitive in a peculiar manner notwithstanding in sentient life forms!”
“Agreed” replied his fellow, “Do you want to bet on how long it takes for them to figure out what the box does?”
“Despite their appearance they do seem to have a propensity for solving puzzles, I’d say it won’t take more than a few of their planetary orbits”
“I am not so sure, the rules of contact do state they must figure out how to communicate with us with no help. Giving them a communications terminal is stretching the letter of the law well past what should be its’ breaking point!”
His partner hesitated, patterns of colors flowed over his lower half and his communication limbs clacked in a nervous manner. “Face it if we are going to make a profit off annexing this planet into the Empire we need to stretch the rules as far as possible!” He topped off his reply with a powerful burst of chemicals that made his friend draw back slightly.
“Look at the image! They are taking the Cbox into their moving colony; make sure we are recording this! This will be our first look into one of their nests!”
“Not nests! The data we have mined from their communications net indicates they do not nest in groups like most social beings do, they are individuals, not a collective!”
His partner hesitated as his mind collected the right scents to form an answer. “The data does suggest they are not a collective mind. But I resolve to wait for conformation of that idea before I make any assumptions. It’s difficult to see how such a simple being could be intelligent and social with out some collective thought processes.”
“I agree it would be a rare thing but the beings we subjugated in grid 1086 x 232 x 16,108 were equivalent. And they were individuals and not much more advanced biologically than these… humans…”
The second alien hesitated as his mutualistic symbiotes crawled over his body slowing down from their normal near blurring motions as they sensed his discomfort in the discussion. The small creatures consisted of several different species of arthropod appearing animals that assist in the bodily functions of the aliens. Transferring waste and food in and out of the body through special orifices, clearing the tracheal tubes, even moving food from one part of the digestive system to another. As the symbiotes calmed down and resumed their tasks the other aliens symbiotes relaxed from their defensive posture, in an argument the symbiotes of the two creatures fight each other and the damage, if any, done depends on the severity of the disagreement. Generally only a few of the little animals would harm each other but in extreme cases the damage could be extensive…
Both aliens assumed a more relaxed stance as the question was mulled over finally resulting in agreement. “ These humans may indeed be unique but all intelligences are unique in some manner, these are just a bit more so than most”
With a quick succession of pops and ticks followed by a spray of pheromones the discussion was resumed:
“We stand to gain a considerable amount of wealth from these humans if we can exploit them before anyone else happens upon our operation. The exotic foodstuffs alone will be worth several more resuscitations for our entire group. The biology on this planet is amazingly compatible, fully 10% of the empire can make use of them as a food source and if we are able to use them as workers, the profits make me giddy, the Empire will have to grant us full ownership of these simple beings!”
His partner hesitated, “we will have to cull them ruthlessly, I can’t see how more than 5% of the current population will be necessary for operating the farms and processing factories, we will need much more land than we need workers…”
“How soon until you can write up a proposal to the Throne? We have to get out ahead of this to avoid and competition! So far every animal we have tested will be desirable as food for one or more of the races of the Empire!”
“I have almost completed a formal prospectus, two maybe three standard turns should be enough time to complete my estimates.”
Turning back to the hologram being displayed in the middle of the room they both watched as the sailors on the Aircraft carrier moved the Cbox into an open area as the people stared almost dumbfounded at the apparatus.
“How much time do we have?”
“About 1000 of their planets orbit will be plenty and as short lived as they are none of them will realize what is happening until they are in too deep to stop it!”
“What about their weapons, they do have nuclear weapons you know!”
Yes but by the time they realize they need them we will have eliminated the threat!”
]]>A few of our blogs:
I think Carl Jung is dismissed too much.
I think there are two pivotal point in human history:
With #1, it has to do with this issue: Is there a truth or (T)ruth worth pursuing?
With #2, it has to do with this issue: Is there a truth or (T)ruth worth pursuing?
And you could say it boils down to the following: Is there an “intent” behind all of reality?
Now, culpability is a pain in the ass. I think deterministic beings are incapable of intent, because intent is something that requires free-will. It’s a philosophical notion. Wants, desires, etc.. They’re illusions.
I’d like to think, however, if there is some being that had or has free-will, then it was or is culpable, thus had intent. With that said, everything occurring around us has a “reason” for occurring.
Philosophers of old wanted to know if there was a Truth to be known, much like saying there is a God to be known. Carl Jung was touching on whether or not there is meaning to anything in reality, whether or not there is intent behind space-time events and their occurrence.
And I’d like to bring up the simulation argument, whereby reality is a virtual simulation. Under that premise, you can start arguing there is intent.
A video game designer often has various intents with designing a game. In general, with modern society the way it is, the “intent” is to entertain, thus make money for the company by getting people to buy something entertaining. But I like to take things a little further and look at the details of the game, what was behind each of the aspects in the game.
And I like to look at things in the game, such as when people question if there is a God. Because it’s not people questioning if there is a God: It’s “God” having programmed them to question if there is a “God,” which seems kind of like a silly thing to do.
Imagine a game programmer making an NPC question in the game, upon approaching and interacting with the NPC, if there is a programmer. So, let’s saying I’m playing a game and I come across an NPC. I press “X” or whatever controller button and it says, “I wonder if there is a programmer that created me.”
Well, certainly there was. And it was the programmer that “caused” the NPC to state such. And if you look around the “real” world, there is a lot of the same going on. People question if there is a God.
I take it in two possible ways:
Now, of curious note is whether or not I can engage in “self-reflection” in order to say, “That’s not funny.”
Self-reflection could just as well be an illusion. Thus, any chance of self-reflection being a delusion of my mind. But in my believed self-awareness, I look at a person questioning God’s existence as something pre-planned since the dawn of existence. And my own observation of this individual as also pre-planned since the beginning of time. Thus, I even further question, “What’s the point?”
The problem I have is that Descartes has no legitimate reason to believe his senses are correct. The phrase “I think; therefore I am” is flawed. Along a time-space spectrum, his thinking occurs along various points in times as does his identity. As a Buddhist concept, the self is changing, one flame on a candle being transferred to another candle. So, he is never the same self as he is contemplating his own existence: He is simply deceived as to having a whole “self.”
Thus, the demon, metaphorically, is his ignorance. However, I can take this a step further and actually bring a demon into existence, as much as bringing a God into existence. The idea that God is the Truth is an issue, because if God is everything, then God is not only the Truth but also The Illusion. And, sure enough, God eludes us all, thus being a masterful illusionist preventing its existence being known.
But I think the illusion is the demon, the ignorance. Synonymous.
So, with that, I’d like to put forth that reality may as well be an illusion. It’s both an illusion and the truth. The truth is that reality is an illusion.
]]>Now, someone might ask, “What do you mean does most of the world have schizoaffective disorder?”
We have a world where people believe in free will, use words that imply having free will at some point in time, and the such: People believe they somehow are “free” and able to escape determinism, eternalism, etc.. the such. Now, if we take certain philosophical paradigms as true, then yeah, the majority of the world is straight-up bonkers. I’m not sure we can say they are legally insane. We “can’t,” if we say we don’t have free-will. Does it qualify as legal insanity? Are the necessary criteria met? Well, again, that’s like asking if I “can” even cause a criteria set to occur.
So, there ends up being a breakdown of language. Deconstructive nihilism.
If I were to empathize, though, legal insanity tends to be thinking people are something other than they really are: You think you see a bear rather than actually seeing a human. Being delusional means you believe something that is not true. And the sticky issue becomes, “What is true?” “How do you define Truth?” “How do you define what is true?”
So, I would posit that it’s true that free will does not exist. So, that kills morality, the idea of responsibility.
I guess if there is no responsibility, one might posit there is no “meaning” to life, as one is not responsible to accomplish anything in one’s lifetime.
]]>First off, I’ve not completely examined the Benjamin Libet experiments, the published papers, and so forth. Ok, that’s fine. I could go down to a medical library or the public library, request the papers electronically, and start reading them. It’s all hearsay, anyway, so what’s it matter?
Regardless, were I to consider the validity of the papers and experimental results, then a person might find anything I have to say worth something. That doesn’t mean the whole “free won’t” thing is not crap. Because I highly suspect that it’s exactly that: Bullshit. Authors in Scientific American and Psychology Today more than likely undergo cognitive dissonance and end up contradicting themselves.
Here’s someone’s definition of “free won’t,”
We have free will to abort an action. So, we may better think of volitional action in this case not as free will, but as “free won’t.” We can stop an action initiated by our brain nonconsciously.
– source: https://www.psychologytoday.com/blog/dont-delay/201106/free-wont-it-may-be-all-we-have-or-need
Ok, so there are issues. Once again, there is, as cited and sourced, the pushing forward of the premise by an author that there is “free will.” Ok, so the author in that source is like, “Well, there isn’t free will. Well, ok, there is free will, but I’m going to change its definition to make it exist: it’s now called free won’t. And, sure, I contradicted myself and changed what free will is called rather than changing it’s definition, but I don’t think most people will notice.”
One is the linguistic use of the word “can,” which brings in the philosophy of language. I want to write an article or blog entry about how I believe most of the world has schizoaffective disorder. When people use words, such as “try,” “can,” “will,” or “want,” then they are using a set of words that are part of a “free will lexicon.”
Imagine a world where people didn’t use words or language that implied that somehow they have free will and are able to “change” reality as it is.
]]>
Please read the article at
PDF On Cantor’s first proof of uncountability
http://pengkuanonmaths.blogspot.com/2016/02/on-cantors-first-proof-of-uncountability.html
or Word https://www.academia.edu/22104462/On_Cantors_first_proof_of_uncountability
HarvardX: PH207x Health in Numbers: Quantitative Methods in Clinical & Public Health Research
Available on edX.
What it teaches:
Harvard’s introductory course to biostatistics, which is statistics that focuses on public health models and analysis, and epidemiology, which is an investigation of the causes of diseases and conditions and their correlations. Epidemiology uses various case studies as frameworks to set up valid analysis for biostatistics.
Harvard originally taught these subjects in two different courses, and this particular course is a fusion. The course also teaches how to use the
statistical software package Stata for assistance in your analyses. It also introduces guest lecturers who lead the class in critiquing published case papers.
PH207x was planned as a 12 week course. If you already have a knowledge of statistics, you can probably fly through it. Most of the online homework questions refer to basic calculations and concepts.
Input[1]:
Basic fluency of algebra. Computer literacy.
Output:
Ability to understand basic statistics [2].
Ability to critique papers on epidemiological studies.
Ability to assist in biostatistics and epidemiology data analysis.
Ability to use basic Stata analysis.
Mixers:
Principles of Biostatistics (co-written by one of the lecturers, reviewed earlier in this blog)
Stata software package. Version 12 was used in the class, 14 was used for this review. The differences are mostly trivial[3].
Why You Should Take This Course:
First, if you have no prior knowledge of statistics, learning the subject is very informative. Statistics allows you to account for variability and probability, and to make predictions. The field of statistics gives you a set of tools which can be just as interesting and valuable as those acquired from calculus. Second, you will be able to call bullshit on many claims given by people not just on statistics, but on health and nutrition as well. You’ll know what makes a valid analysis and when to ask for evidence. Third, it was fun to see what a proper statistical program can do.
Do I recommend it?
Yes. The two instructors are well-versed and communicate the material of their lectures very well. The textbook is lucid and not overly formal, which is fine for applied mathematics. I had no problem understanding the material. For a very small investment: algebra and computer literacy, you may acquire many useful rewards. Most especially, statistical analysis is a tool that isn’t just limited to public health, but a wide array of sciences, and I highly recommend learning it.
[1] What skills you need to properly process the course.
[2] The course will bring you up to linear and logistic regression, then recommends that you take another course in regression to progress.
[3] A calculator was relocated, and pseudo-random seeds will not be the same.
The Polish group
The Virgo-POLGRAW group, lead by Prof. Andrzej Królak at IMPAN.
The Welsh group
The Cardiff Gravitational Physics Group, and within that the Data Innovation Institute lead by Prof Bernard F Schutz.
]]>
More importantly is looking at things I tried covering some time ago: Einstein’s theories of relativity and the philosophical influence it has. If we’re in a block universe, static as can be, the ideas of repeatability and testability are ideas that require free-will to be accomplished. I think reality ends up becoming a psychological experience, whereby the existential crisis is resolved by the perpetuation of insanity and delusion. Falsification as an idea is an axiological one, whereby in a physicalist reality, there can be no right nor wrong.
So, science dies. The human race ends up in turmoil due to not being able to find any decent resolves of answering the why and hows of reality came to be. Perhaps Gödel’s incompleteness theorem is what drives life into insanity or seeking some reclusive state of mind to never question reality again, thus preventing existential issues. Personally, I’ve considered the reason we can never fully understand reality is because “God” or the universe didn’t/doesn’t want us to: If you understand how things came to be, maybe you could destroy them. In terms of a pain/suffering perspective of life, ending suffering means ending reality: Absolute prevention.
Those are long-term things. Short-term things involve the evolution of STEM. I see transhumanists coming into being if politicians don’t corrupt everything with their greed. With transhumanists, I see society becoming a Type III civilization, looking for other life-forms. I don’t think that will resolve much. Issues such as parallel dimensions, the multiverse, etc.. may come into fruition or at least be observed. The ability to make or observe such may come into fruition.
I see the ideal transhumanist discovering the emotional connectivity with reality, thus being able to manipulate reality with the mind. But at no time do I see the resolve for philosophical problems. The grandfather paradox (time traveler’s paradox) and other issues stick around. Reality becomes a one-track situation. One could only hope for a deus ex machine situation, whereby an individual sits in the driver-seat of God, becoming master of reality in order to change things, thus paradoxes not being an issue.
I think neuroscience is going to head in the direction where people walk around like Dr. Gero from Dragon Ball Z with their brains in robot bodies. It resolves a lot of issues, thus enabling focus on regeneration and restoration to decaying brain parts. I like to think neural Darwinism due to the mechanisms that exist now would beat out any engineering “attempt” to make something better in the next million years. If that hypothesis is wrong, something could be engineered.
With the brain isolated and the body no longer an issue, people will have the opportunity to live long lives and focus on philosophical issues and understanding the “physical” nature of reality. From there, they can get their Star Trek or Stargate on. That doesn’t necessitate anything will be resolved, though. The expansion of the universe will be an issue. If something can be engineered to bounce back thing, great. Punch wormholes, manipulate reality, travel the cosmos, look for other life.
I guess that’s the say science eventually hits a brick wall: The philosophical issues remain. Sure, it’d be great if a scientific description of God could occur. And God might be best described as “the first cause” or “what caused the universe to occur,” if you want to claim that’s quantum fluctuations, some obscure physical phenomenon, etc.. It’s a definitional issue.
Scientifically, I see God as an infinite-dimensional being, thus allowing it to have free-will. It may or may not be logical.
I see the transhumanists taking over for the most part. But with the issues of not seeing time travelers or aliens, it makes a person question if individuals in the “future” are extremely capitalistic or secretive: Selfish vs. reclusive. I think the latin description of science being knowledge is one thing: It’s another to take a perspective that science is what a person uses to accomplish a desire. There are those who engage in science to understand and describe reality as it is. There are others who believe science can be used to understand, describe, and manipulate reality. The last part might seem like a feat of engineering, perhaps at least requiring free-will if not a break in reality: Arguable, any attempt could be the delusional acts of an individual desiring something more from reality.
Some transhumanist schools have leaned toward hedonism, which I suppose is to battle the existential crisis. My subscribed school is truth-seeking, thus is more interested in what exists and does not exist in reality. It may be that “truth” is never found. I’d like to think that if any transhumanist figured out reality, he or she would have found the off-switch, thus putting the universe to an end. My continued existence in reality, as I type this, appears to be evidence that confirms no transhumanist has done such.
An evidential interpretation of my place in reality convinces me that God is not dead. God being reality, nature, all of existence, etc.. God is not dead. As long as God can be defined, God is not dead. As long as there is a definition for God, God is not dead. That doesn’t necessarily mean that one can bring forth a definition in “court,” which might be the only place a definition matters. But the paradoxical relationship between there not being evidence of a God yet still existing seems to only bring forth a solipsistic view of reality, which might be the unfortunate situation of reality.
It could be that reality is simply the mind of God, everything in reality all pre-planned. So, I would say, ideally, science would find a way to kill God. But I also think it’s interesting that nature, reality, and the universe has even allowed at least one individual to think about doing such. But that doesn’t mean it will be possible: And eventually, it appears the death of reality never occurs. It becomes questionable, on a philosophical level, why the universe never dies, doesn’t want to die, or at least doesn’t end. It’s just not part of reality for reality to die.
The thing missing from science is free-will, which science ends up not doing anyone any “good,” which is an axiological ideal requiring free-will in the first place. That may or may not be reasonable, but I think matters of axiological necessitate free-will to exist in the first place.
If you look at medicine, you see the issue. I think the issue with making a stem-cell treatment for AIDs started to show the issue. People tried replicating the stem-cell treatment by breaking bones, something like that, and entering in stem cells that won’t get infected by HIV, and it just wasn’t successful. It could be the discovery wasn’t correct. It could be that free-will is necessary to implement the treatment. A treatment is never conducted. That would require free-will. Doctors would need free-will to treat a patient. Otherwise, it’s just the flow of reality. It ends up appearing a huge situation of chemotaxis. A chemical soup looking for a chemical treatment to alter its chemical makeup. There’s a philosophical topic that deals with that: It goes along the lines that if someone is going to heal, they’ll heal as part of what’s in the great design of reality.
Science ends up going so far as it can be implemented for engineering purposes. It only goes so far for truth-seeking purposes. And it goes only so far to resolve philosophical topics. Once it hits the glass-ceiling, it’s done for.
I think you look around, no time travelers or aliens: It hits the brick wall in the future. The engineering uses seem to be moot. Otherwise, we’d have time travelers around talking about the scientific methodology they used to come back, the science behind “time-space” that their time machine or machine uses.
I don’t think that means a spiritual enlightenment occurs: I’ve come across various spiritualists who argue that being alive is about “loving” others, so I’d imagine that any transcendent beings would come back to our timeline to describe how we can transcend ourselves: Kind of like Morpheus in the Matrix movie helping out Neo to seek the truth. I grew up around a person who believed/(s) in the ascended master theology.
I think the utilitarians and various transhumanists are looking into futures and considering hedonism to be the only thing worth doing. Science is a descriptive technique. Engineering a creative one. If either is an art, either may as well by a sophism.
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Imagine God is free-will and all that can be. Each timeline and universe that is created is a “choice” that God makes. And essentially, God is attempting to figure out how to find another God.
Each extension is a time-line or universe. They end up being deterministic. They can, however, go back into the pool of what God is. Each time an extension occurs, it’s influenced by what is outside of it, thus causing God to change. With that said, one might say, “What’s the point,” and I think my answer is that “God” is learning. God has free-will, makes universes, but finds itself “imperfect” despite being perfect. It’s attempting to figure out if there is more to its existence than simply being.
My hypothesis is that God is looking for another God.
I like to postulate that God is insane. Let’s say schizophrenia or insanity is a complete break from logic, thus being able to step out of the realm of logic. Something about Mental State “x” enables a break from logic. That might mean breaking free from constraints. If God is all that can be, logically so, then logically there can’t be another God. But being God, if God can break logic, then God would be looking for another God. It wouldn’t be logical. So, it’s questionable what to make of it, even if it were possible. Logically, not so, but then the question becomes what kind of “system” or “methodology” is being used to accomplish such.
I think the answer is that God is attempting to break its own ignorance. That’s like saying, God thinks its the only God yet ignorant of how it would be possible for there to be another: Thus, it’s engaging in a method to break its ignorance of what would be necessary to find another God. You might say, then, that God is not God: And that might be the logical answer. But if the system is not dependent on logic, then logic is thrown out the window.
]]>Well, let me throw out an idea: everything that could exist does exist. Infinite universes seems to create the issue that if there is a person that can cause something to occur, there is another that would cancel out that possibility, thus making it feel as though there is one universe.
Imagine the following:
There exists a person in a universe that cause A to occur. There exists a person in a universe that can prevent A to occur. Their existences cancel out, thus giving the appearance there is no multiverse. But then you might make the logical argument that if a person can travel to this universe, then he or she cannot be prevented.
So, with the infinite universe idea, there MUST be a person in an outside universe who wants to let me know that he or she is (1) from another universe, (2) prove such, (3) do such right now. I looked around me after typing such, and encountered no such being: One might take such to be evidence of absence that there are not infinite universes, thus the multiverse idea is not practical, at least for there to be infinite universes.
Part of me, then has been considering what if there is an aspect of reality, however, that does exist. Let’s call this the realm of the “imaginary.” It exists. It’s a dimension of it’s own. And it’s “imaginary,” whereby it’s like a Platonic realm where crossover is possible.
I’ll throw a bone and give an example.
In Dragon Ball Z, Goku learns to go faster than light after he learns instant transmission. As such, he has become a master of reality in his dimension.
You might think that’s questionable. Let’s take Goku and throw him in the hypercube shown in the Cube 2 movie. Goku would use his brute strength and instant transmission, and he would more than likely get out of there real fast. You might posit that a 5th dimensional being would be able to see the maze for what it is, but Goku uses brute strength (using FTL technology: instant transmission) and gets out after using a brute force hacking methodology of using every combination of getting out of there.
Now let’s take that knowledge a step further. Goku exists inside his dimension. Interestingly, using his FTL knowledge, he figured out his reality is like a hypercube. He found a way out. Making things more interesting, it possible that the Goku that comes back to Earth from space is actually a damn lie. Everything you see Goku doing on television is a damn lie. The cell games: A damn lie. Fighting buu: A damn lie. The only Goku you need to concern yourself is the Goku that occurs once he learns FTL technology.
If we argue that being able to master FTL technology provides an individual to live an infinite if not close to infinite amount of time, that’s like saying Goku spent an infinite amount of time in his reality, coming to understand the aspects of reality that he was bound in, and eventually finding an exit door. And with that, he might have come to realize there are people observing him on the television screen. And once he learned that, he created a huge number of illusions to prevent anyone from following him, catching up to him, and taking him down. After Goku learned FTL technology, he soon enough found out how to travel to another dimension.
I guess if anyone follows Flash or The Flash comics, something similar happens with The Flash. The thing is, however, a person argues that these individuals are of “fantasy.” Well, so is the number “one.” But we don’t generally attribute characteristics to the number one, such as the ability to engage in FTL travel.
So it comes to the next question, why would an individual deceive others as to him or herself being a multiverse being, that there being a multiverse, and teaching others how to go into the multiverse. I think the answer is that there is something axiological about it. If one uses empathy, one could consider that amount of heroic nature an individual must possess to be able to involve him or herself with such matters. A level of social order or innature knowledge of social order would be necessary. This seems to contradict with multiverse beings that are discussed in comics: I don’t know if the Beyonder would contradict such.
A person might ask what gave me these ideas. I’d have to say it was the curious nature of observing some animes, thinking to myself, “Wouldn’t it be peculiar if x, y, z occurred in the anime?” and then such happening shortly soon enough in the episode I was watching. It was at the least worthy of raising an eyebrow, that such was either a coincidence or something.
But I do think there is one other thing worthy of mention here. Something that leads me to believe if not question the multiverse. It’s something that has caused me to question if we live in something like Valhalla. It’s “new evidence.” It’s a concept in court.
New evidence is anything that wasn’t presented when it could have been presented as evidence. But what if a God or Goddess your worship is someone or something that can carry you through to victory in a court case. If somehow you lose the case, does that mean your God or Goddess was falsified? Well, under total falsification, you might consider that in no way will your God or Goddess present itself in reality again to save your arse from potentially harmful issues in court. However, there have been interesting things going on in reality: Loki, Thor, Jesus, Moses, etc… find their ways into our media still. Gods and Goddesses of old have found their ways into our reality, coming back from the “dead” to persist. And with that, I think that’s about the only way to argue that, yes, Thor exists.
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So, you might be questioning, how would Goku be able to see on the other side of the television?
It’s my hypothesis as to what a 5th dimensional being would be like. As humans, I posit that we’re 4-dimensional beings: x, y, z, and time. But a 5th dimensional being can go through “versions.” Versions occur throughout time.
A 5th dimensional being can observe the versions. Let’s say reality were to destroy itself and re-create itself many times over. A 5th dimensional being would be able to flip through the versions like a book with pages, each page being a different version of reality, and inject itself into the different versions if so desired. Thus, it could be argued that the 5th dimension is an “alternate reality.”
Thus, a being that can access the 5th dimension or a 5th dimension can access an alternate reality. Goku having FTL technology gives him the maze dilemma, whereby he eventually discovers the 5th dimension or an alternate reality. However, the amount of time for him to accomplish such is infinite or almost infinite rather than instantaneous.
A 4th dimensional being that can use FTL technology would spend infinite or almost infinite time to access an alternate reality. A 5th dimensional being could instantaneously access an alternate reality. Perhaps the Trunks timeline is evidence that Goku played with time-space, or maybe not: I’d have to analyze that more. The Trunks timeline has time-space issues that anyone with time-space knowledge would find odd. Does Buu, nonetheless, kill Trunks? Shouldn’t future Trunks have encountered Buu? Etc. etc..
The other side of the television would be an alternate reality, our reality: A 5th dimension to Goku.
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And one other thing I’d like to add to this. I have a hypothesis as to how Goku would have obtained FTL technology. It’s based on a hypothesis as how to create a “reasonable person,” which the reasonable person concept is actually somewhat a legal concept. However, I tried to mesh ideas of laws and science together. In law, there is something called “novus actus interveniens,” which means a new act intervening: A new act will intervene and remove an individual’s negligence.
Let’s imagine I create a building with a tunnel. On the outside is a door. After you walk in the door, you live and die a large number of times, maybe infinite. After you die, you come back to life. The only way you’ll get out is by an act of “God.” Thus, eventually, Goku walks out of the door he went in. The question is, “How did that happen?” The answer is “novus actus interveniens,” which means a reasonable person intervened and removed Goku’s negligence. My hypothesis is that the aliens Goku met told him to walk into a 4-dimensional box that he dies in, resets itself, and brings him back to the version he was in when he first walked in: Thus, a system for creating a 5th dimensional being or allowing FTL technology to be learned. It could be argued that Goku became a 5th dimensional being once he learned FTL technology, but that’s debatable.
Imagine something like Kami’s time chamber, except a person spends their life there until they die. And after they die, someone on the outside resets things to the point where the person first went in. If anything were to change inside the time chamber, then it must be outside the control of the person who was resetting the time chamber, thus an intervening act by a being that can access at least 5-dimensional space.
One of my current obscure ideas is that reality has things that deceive us, forms of knowledge that we’re aiming for to achieve. And these things “deceive us,” but if we ourself were to become deceptive, then we would be able to obtain those realms of knowledge. If court is where fantasy meets reality and fantasy is synonymous with illusion, the imaginary, and/or death, then the more times we die, the closer we get to death, which technically is an illusion in a reality where we keep coming back to life: Thus the more we die the more we see the illusions that are in reality, because we ourselves become illusionists.
It’s a speculation.
But let’s go further.
Let’s say I meet a woman named Morgana. She was a cavewoman, typical barbaric woman trying to avoid the sabertooth tiger. But she died… a lot. And as she died more and more, she got closer to death, because she increasingly experienced death. And death could be synonymous with illusion. She never really “died,” because in her reality the universe re-creates itself and she comes back to life in a “block” universe no memory of her former lives. But as she dies more and more, she eventually learns about the illusion of reality surrounding her. And she somehow herself becomes part of the illusion. And with that, she becomes an illusionist. She courted life-and-death so many times, her experiences piled up: Something more reasonable than her intervened to remove her of her negligence and unreasonableness. And she sprung up to be a Goddess.
But those who know lore, you might think to yourself, “How come none of us have seen this Morgana?” And I think the answer goes back to something in my last post: Competing forces nullifying the others influences and existence.
It’s a nice thought, but I’m not sure its realistic.
This post is more about the multiverse. I think the Dragon Ball Z example is good, because it shows someone with a technology that might be able to get someone into our reality. If you can go faster than light and you have infinite time to do something, you might be able to get into this dimension.
The 5th dimension described as “versions” is a good analogy. But I think the term “version” is not adequate. But a 5th-dimensional being or being that can access the 5th-dimension can access parallel objects, objects that don’t touch each other. The individual finds or creates the bridge.
Cube 2: Hypercube is a decent movie if you haven’t watched it. Other 5th-dimensional beings might be observed in the movie Shocker, which is a horror film. Freddy Krueger might be considered a 5th-dimensional being.
]]>If you haven’t seen a time traveler or alien yet, you’re doomed. If no one has seen a time traveler or alien yet, we’re all doomed.
The lack of time travelers and aliens seems to create a significant evidential aspect to further the existential crisis. Part of me wants to ask, “How long would it take a reasonable person to discover alien life?”
Maybe I’ve come across a time traveler. Maybe I’ve come across an alien. And, sure, I probably shouldn’t have been doing some “unethical” things at the time. But I attest that my encountering of them didn’t seem to solve any of the more complex philosophical issues that we humans already deal with. But, I think if I learned something, then it’s that they want to be able to figure out how to break paradoxes, especially the time traveler’s paradox.
With aliens, I think to myself that there may be biological life out there in the cosmos, but it might not be “intelligent.” As a neuroscientist, I well enough understand intelligence to be relative. Thus, my issue is more with whether or not these biological entities can grasp philosophical conundrums, such as the meaning of existence, what is “truth,” and whether or not there can be any feasible objectives to pursue while in existence.
So, with aliens, it could well enough be that there are evolved lifeforms but they aren’t much more communicative than cockroaches, germs, or armadillos. Their life styles don’t require that they look much further than around them rather than above them. Looking back at my education, humans tended to have looked at the sky, questioning the greater aspects of reality and whether or not there was something “out there,” but given that the universe has lifeforms that don’t question, interrogate, or find concerning what is “out there,” there would never have been a necessitate aspect to move beyond their planet.
Sure, it would be interesting for there to be other life forms, but more interesting is if their knowledge of metaphysics and reality has surpassed our own, finding answers to what we find to be philosophical matters that are unsolvable.
No time travelers, no aliens… it seems like no hope. It starts to create evidence of absence, whereby I question if the human race is doomed. Sure, well enough, there might be aliens and time travelers moving around in reality, competing against each other for possession of reality itself. As I once explained to someone, if we could build a time machine and understand reality quite well, more than likely, there would be what I call “the race to zero.”
Imagine you could inject yourself into the beginning of time. All you have to do is move some particles around in the right way, and you’ve completely manipulated time and space and the way reality will unfold. It becomes an issue of control. It might make an individual feel God-like, but not necessarily make someone God. It would, nonetheless, change reality, thus possibly change the futures of societies and cultures from coming into existence. And any aliens or time travelers aware of such a person “wanting” to do such might compete against the individual to “prevent” such from occurring.
So, perhaps there are aliens and time travelers, but they have more important issues to attend to, more important matters that secure their existence in reality.
I think the Fermi paradox is a serious matter. Even if there was other intelligent life, I’m sure they would be considered with issues, such as the existential crisis. They might come across matters of religion. Perhaps it could be posited that God simply wants people to believe whatever they want, thus to find fulfillment in life: But then, one might insist such a philosophy would necessitate free-will.
Personally, I would think if time travelers or aliens existed, they would be concerned with resolving matters of metaphysics and reality.
]]>I’m starting to hypothesize various aspects about reality not necessarily being what they are. That’s due to me taking various perspectives of “law” and reality and experimenting with them. In a lot of ways, what has peaked my interest is the concept of “court,” which I consider to be a Platonic form.
Neuroscience meeting law is similar to philosophy meeting law. Where I see the line is between the conscious experience and the funneling of a “moral” system into reality, whereby a moral system is the entrance of some kind of “free will” ideal into a reality where there may or may not be free will. What I’ve found is that I consider “court” to be a concept similar to a Platonic form, so I will set forward a definition of court here:
Court is where fantasy meets reality. To a purist, one might argue that there can be no such thing as fantasy, thus all things are reality. Thus, I can further argue and posit the term “fantasy” and the term “imaginary.”
Fantasy is theoretical, hypothetical, imaginative, untouchable, and yet existing. It’s observed through a glass window to exist but never touchable. It’s tangible aspects are questionable, but it’s observable.
For anyone who has ever studied law, law is like a religion. However, as the same time, it’s something that exists in this reality. And with law, there is a God, which is the reasonable person. I think the bigger aspect of all of this is furthering defining what “court” is: Court is the mind of God.
Nonetheless, there is something I’d like to speculate on. Interestingly, everything that exists outside of court is “hearsay.” Everything that exists outside of it is hearsay. That’s like saying it’s black magic, rumor, etc.. If you look at it, then, every court hearing that has ever existed is fraudulent. By the standards of man, any word that comes out of any person’s mouth is hearsay in the court of law. The word “dog,” it meaning, etc… are all things that come from outside of court.
Court is an illusion. As the mind of God, it knows the truth, the reasonable answer of resolve, and so forth. Nonetheless, it’s deceptive. These are my “scientific” perspectives on Court. Court, nonetheless, has entered itself into reality. It exists as a mechanism of adversarial dispute and resolve. It’s meant to be a truth-seeking mechanism, as science is meant to be a truth-seeking if not a mechanism to discover aspects of reality and describe them to form knowledge.
The significant difference appears to be in objective: One to understand the social fabric of reality, the other to understand the physical fabric of reality. Law vs. science. The entrance of psychologism, however, seems to cause the lines between both to blur if we consider knowledge or what is “truth” to be a psychological consensus occurrence: Mobocratic truth, Kuhnian.
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The other day Bill Clinton attacked Sanders’s proposal for a single-payer health plan as unfeasible and a “recipe for gridlock.” But these days, nothing of any significance is politically feasible and every bold idea is a recipe for gridlock. This election is about changing the parameters of what’s feasible and ending the choke hold of big money on our political system. In other words, it’s about power – whether the very wealthy who now have it will keep it, or whether average Americans will get some as well.
See Robert Reich:The Most Pragmatic Way to Fix American Democracy
Update Feb 14,2016:
Government Gridlock has involved taxes, a shutdown, a sequester, and the debt ceiling. Now it’s looks like it’s also going to include a Supreme Court appointment.
Please read the article at
PDF On the uncountability of the power set of ℕ
http://pengkuanonmaths.blogspot.com/2016/02/on-uncountability-of-power-set-of.html
or Word https://www.academia.edu/21601620/On_the_uncountability_of_the_power_set_of_N
Paraphrasing what he said:
Surely all mathematics was worked out and finalised years ago?
I think he was willing to accept that there are still some classical open problems, but essentially he thought that mathematics was now ‘done and dusted’.
Of course this cannot be the case, as evidence I offer all the preprints that appear on the arXiv everyday. Mathematics departments are not full of people who just teach linear algebra and calculus to engineering students! I also submit that my boss Prof. Grabowski would be wondering what I am doing day in day out!
But why would he think mathematics research is over?
High School Mathematics
I think this belief stems from mathematics teaching in schools. Let me explain…
Let us start with physics and science in general. Students and the public at large know that scientists are working on open problems and discovering new things. For example we hear about new materials (eg. graphene); we know that the likes of Hawking are wrestling with the theory of black holes; we see images of all kinds of things in observational cosmology; we hear about medical scientists working on cancer cures; biologist discovering new species can make the news; CERN discovered the higgs boson…
High school students are aware that science is far from over and the syllabus for A-level physics is periodically updated to reflect some of these new discoveries.
But what about mathematics?
Linear algebra first emerged in 1693 with the work of Leibniz. By about 1900 all the main ingredients were know, so vectors have a modern treatment by 1900. This is all quite dated, but some open questions remain (for example in relation to quantum information theory).
Quadratic functions were solved by Euclid (circa 300 BC) and ‘the formula’ was known to Brahmagupta by 628 AD.
Calculus the foundations are from the 17th century in the works of Newton and Leibniz.
Plane geometry goes back to 300 BC and Euclid. Coordinate geometry is due to Descartes in the 1600’s.
Probability theory has it origins in Cardano’s work in the 16th century. Fermat and Pascal in the 17th century also made fundamental advances here.
Logarithms and exponentials in their modern form is due to Euler in the 18th century.
Trigonometry has roots going back to the Greek mathematicians from the 3rd century BC. Islamic mathematicians by the 10th century were using all six trigonometric functions.
So in sort, much of the typical pure mathematics syllabus at advanced level in high school is quite old. This I think, together with the ‘unchanging’ nature of mathematics (once proven a statement is always true) leads to the idea that it is all done already and nothing new can be discovered.
It also take from my friends question that he understood that the applications of mathematics are important and that plenty of work in applied mathematics is going on, for example in computational approaches to chemical dynamics. However, the ideas that mathematics as mathematics is finished remained.
For me personally, these applications of mathematics can lead to new structures in mathematics and this is worth studying. Indeed much of my professional work is in studying geometries inspired by applications in physics, particularly mechanics and field theory.
What can be done?
The ‘unchanging’ nature of mathematics is hard to get around. In science some new evidence could come to light and change our views. Indeed the scientific method is an integral part of teaching physics at advanced level in the UK. This ‘flexibility’ of science to adapt is important in student understanding of the philosophy of science.
So, we could try to promote new discoveries in mathematics to the general public, including high school students. The problem is that the background needed to understand the questions, let alone have any idea about the solutions prevents wide public engagement. Astronomers are lucky, we have all seen stars in the sky and can admire nice pictures!
Trying to start at a much higher level of mathematics would be futile, given the prerequisites that are needed. Moreover, most students will not become researchers in mathematics and will only need to be comfortable applying basic mathematics to their later field of study and work.
In short I have no idea how to promote the idea that mathematics research is not over, but please take my word it is not over!
]]>Please read the article at
PDF Hidden assumption of the diagonal argument http://pengkuanonmaths.blogspot.com/2016/01/hidden-assumption-of-diagonal-argument.html
or Word https://www.academia.edu/20805963/Hidden_assumption_of_the_diagonal_argument
So here’s a question: how old is life insurance?
I shan’t lead you on: 2600 years, give or take. The Romans and Greeks had guilds which a person would pay into, and they would take care of funeral expenses and stipends for the family of the deceased. The concept of insurance in general goes back to the Code of Hammurabi, which gave a form of maritime insurance to those who take a loan and lose a ship at sea. If you think about it, the idea of property insurance is just a redistribution of risk from taking out a loan. Loans are a result of an individual or organization having a surfeit of money and wishing to invest. Money itself is a shorthand for transfer of goods and labor, which comes from a settled, agricultural society[1]. So property insurance itself isn’t that complicated, it just naturally arises from money. Life insurance is a little more complex, but it too naturally arises from property insurance.
[1] I neglected to mention trade as a factor, which is a whole topic in itself: measuring investment against the risk of maritime loss. I do not exaggerate when I say that trade is perhaps one of the hugest factors in world history, and is often hidden as a motivation in high school textbooks.
]]>All you have to do is follow this link.
]]>That’s if you make the deadline. The King is quite cross with you if you don’t make it. One of those little speeches goes (from memory).
“There you are, always in the middle of things, never quite finishing anything!!!
Blogs are like being in the middle of things. So rarely do they start from a proper beginning, with a proper introduction. At least, perhaps the good ones do. People like to talk about themselves, which is often Not Very Interesting. I’d rather talk about other things and inform the reader prior to when they need information about how I view things or operate. Here and there I’ll leave clues perhaps to who I am, but its really not that important.
So this first post is a review of the book Introduction into Biostatistics by Marcello Pagano and Kimberlee Gauvreau. This is the second edition. I suppose I should inform you[1] that I have a habit of studying textbooks and courses. There is an excellent resource for courses called EdX, with top level university courses. Introduction to Biostatistics (henceforth abbreviated to ItB) is the textbook for the
course, which is the first course of the archives.
I prefer to go through the entire textbook and do all of the problems before taking a course. Hermione would be proud. I tried reading the textbook and doing all of the problems with the course, but some instructors jump around the textbook a bit, which is fine for them but distracting for me.
I do not have a medical or statistics background. I do have a Baccalaureates in Electrical Engineering. So my mathematics background is multivariable calculus, ordinary differential equations, probability, discrete math (which I enjoyed the most), and some linear algebra. Probability is taught to EE’s as a component to separating noise in radio communications, but we never quite got to that part in the course, which makes me cross. We also never got around to using ODEs in electronics as well. *grumble emoji would go here*
Prior to reading this textbook, I knew what means, medians, and modes were, but I couldn’t rightfully tell you what a p value was. I am absolutely delighted by this textbook, which serves as an introduction to a large set of new and fascinating tools. The book is composed of 22 chapters, each ranging from 20 to 40 pages long, and ends with 10 to 20 questions. I never had a problem understanding the material, and only towards the end, on the topic of linear regression, is when the text started to gloss a bit over information, which is fine for an introductory textbook. Linear regression is also where I started leaning on software packages for analysis, since performing linear regression on a data set of 100 entries seemed unnecessarily tedious. For most of the book I was fine using online web pages for making graphs and lengthy calculations, though I tried to do some things by hand for understanding’s sake.
Introduction into Biostatistics uses real information and has many references and sources, which is a bonus. The real data made me feel like I was doing actual work and not playing with made up numbers. I did not have access to an answer key (or even half a key as most textbooks provide), but at no point did I feel lost. I might have an unresolved error here or there, but the nature of the problems was that I felt I could always answer them with the information given in the chapter. The book expects you to use either Minitab or Stata with it and does not teach you how to use these packages, so if you’re not computer savvy, you may have some confusion.
Overall I am very happy with what I learned and the material wasn’t difficult at all. I consider statistics to be a very welcome addition to my toolbox of problem solving. I look forward to taking the EdX course, which combines biostatistics with epidemiology and Pagano is one of the instructors of the course.
[1] I’m thinking of a name for the reader, Gentle Reader is taken by Miss Manners. Cecil Adams of The Straight Dope used the Teeming Millions to refer to his readership. I’ll think about an original name, but let’s keep the ball rolling. Suggestions are welcome.
]]>Please read the article at
PDF Which infinity for irrational numbers? http://pengkuanonmaths.blogspot.com/2016/01/which-infinity-for-irrational-numbers.html
or Word https://www.academia.edu/20147272/Which_infinity_for_irrational_numbers
1. Rational numbers are discrete
2. Real numbers are continuous
3. Collectively exhaustive and mutually exclusive events
4. Continuum hypothesis
5. Cardinality of discontinuous subsets of real numbers
Please read the article at
PDF Continuous set and continuum hypothesis
http://pengkuanonmaths.blogspot.com/2015/12/continuous-set-and-continuum-hypothesis.html
or
Word https://www.academia.edu/19589645/Continuous_set_and_continuum_hypothesis
We start with the problem of passing from a particular ‘species’ of graded manifold, known as graded bundles [1]. Graded bundles are non-negatively graded (purely even) manifolds for which the grading is associated with a smooth action of the multiplicative monoid of reals. Such graded manifolds have a well defined structure, nice topological properties and a well defined differential calculus. For these reason we decided that this special class of graded manifold should be the starting place.
Moreover, any vector bundle structure can be encoded in a regular action of the monoid of multiplicative reals. A graded bundle is a ‘vector bundle’ for which we relax the condition of being regular. As everyone knows, the parity reversion functor takes a vector bundle (the total space of) and produces a linearly fibred supermanifold. This functor just declares the fibre coordinates of the vector bundle (in the category of smooth manifolds) to be Grassmann odd. Importantly, one can ‘undo’ this superisation by once again shifting the Grassmann parity of the fibre coordinates. Thus, the parity reversion functor acting on purely even vector bundles is an inconvertible functor and we establish a categorical equivalence between vector bundles and linearly fibred supermanifolds.
Passing to graded bundles
However, such a direct functor cannot exist for graded bundles. Graded bundles are not ‘linear objects’, the changes of non-zero weight local coordinates are polynomial. Simply declaring some coordinates to be Grassmann odd is not going to produce an invertible functor: we have nilpotent elements and now terms that are skew-symmetric which by contraction with symmetric terms in the transformation laws will vanish. In short, some information about the changes of local coordinates is going to be lost when we superise by brute force. We do obtain a functor that takes a graded bundle and produces a supermanifold, but we cannot go back!
Any meaningful ‘superisation’ of a graded bundle must be in terms of an invertible functor and allow us to establish a categorical equivalence between the category of graded bundles and some subcategory of the category of supermanifolds (or some other ‘super-objects’).
Our solution to this conundrum is a two stage plan of attack: first fully linearise and then superise.
Full linearisation
First we fully linearise a graded bundle by repeated application of the linearisation functor [2]. In this way we get a functor that takes a graded bundle of degree k and produces a k-fold vector bundle. In the paper we characterise this functor and make several interesting observations, especially in relation to the degree two case.
The basic idea of the full linearisation is that we polarise the polynomial changes of local coordinates. That is, we add more and more local coordinates in such a way as to fully linearise the changes of coordinates. We do this by repeated application of the tangent functor and substructures thereof. We also have an inverse procedure of diagonalisation, which allows us to ‘undo’ the full linearsation.
As a k-fold vector bundle is ‘multi-linear’ we can superise it!
Standard superisation
Following Voronov [3], we can apply the standard parity reversion functor to a k-fold vector bundle in each ‘direction’ and obtain a supermanifold. Thus, by fully linearising a graded bundle and then application of the parity reversion functor in each ‘direction’ we obtain a supermanifold.
However, this procedure is not really unique: one obtains different functors depending on which order each parity reversion functor is applied. These different functor are of course related by a natural transformation, so there is no deep problem here. However, when we consider just vector bundles the parity reversion functor works perfectly and we have no ambiguities in our choice of functor. This suggest that we can do something better for k-fold vector bundles and our superisation of graded bundles.
Higher supermanifolds
Instead of using standard supermanifolds we can employ $latex \mathbb{Z}_{2}^{k}$-supermanifolds [4]. It is known from [4] that these ‘higher supermanifolds’ offer a neat way to superise k-fold vector bundles without any ambiguities. Thus, in our paper we apply this higher superisation to the lineariastion of a graded bundle.
In short, we can in a functorial and invertible way associate a $latex \mathbb{Z}_{2}^{k}$-supermanifold with a graded bundle answering our opening question.
References
[1] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21–36.
[2] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439 [math-ph], (2014).
[3] Th.Th. Voronov, Q-manifolds and Mackenzie theory, Comm. Math. Phys. 315 (2012), no. 2, 279-310.
[4] T. Covolo, J. Grabowski & N. Poncin, $latex \mathbb{Z}_{2}^{n}$-Supergeometry I: Manifolds and Morphisms, arXiv:1408.2755[math.DG], (2014).
]]>Please read the article at
Cardinality of the set of binary-expressed real numbers
PDF http://pengkuanonmaths.blogspot.com/2015/12/cardinality-of-set-of-binary-expressed.html
or
Word https://www.academia.edu/19403597/Cardinality_of_the_set_of_binary-expressed_real_numbers
Abstract: The trypanosome Trypanosoma brucei gambiense (Tbg) is a cause of human African trypanosomiasis (HAT) endemic to many parts of sub-Saharan Africa. The disease is almost invariably fatal if untreated and there is no vaccine, which makes monitoring and managing drug resistance highly relevant. A recent study of HAT cases from the Democratic Republic of the Congo (DRC) reported a high incidence of relapses in patients treated with melarsoprol. Of the 19 Tbg strains isolated from patients enrolled in this study, four pairs were obtained from the same patient before treatment and after relapse. We used whole genome sequencing to investigate whether these patients were infected with a new strain, or if the original strain had regrown to pathogenic levels. Clustering analysis of 5,938 single nucleotide polymorphisms (SNPs) supports the hypothesis of regrowth of the original strain, as we found that strains isolated before and after treatment from the same patient were more similar to each other than to other isolates. We also identified 23 novel genes that could affect melarsoprol sensitivity, representing a promising new set of targets for future functional studies. This work exemplifies the utility of using evolutionary approaches to provide novel insights and tools for disease control.
]]>In this paper we define weighed Lie groupoids as Lie groupoids with a compatible action of the multiplicative monoid of reals. Such actions are known as homogeneity structures [1]. Compatibility means that the action for any ‘time’ acts as a morphism of Lie groupoids. These actions encode a non-negative integer grading on the Lie groupoid compatible with the groupoid structure, and so we have a kind of ‘graded Lie groupoid’. Importantly, weighted Lie groupoids form a nice generalisation of VB-groupoids (VB = Vector Bundle), which can be defined as a Lie groupoids with regular homogeneity structures [2].
Based on our earlier work [3], in which we similarly define weighed Lie algebroids, we present the basics of weighted Lie theory. In particular we show that weighted Lie algebroids and weighted Lie groupoids are related by more-or-less standard Lie theory: we just need to use Lie II to integrate the action of the homogeneity structure on the weighted Lie algebroid.
The main point here is that we not only naturally generalise ‘VB-objects’, we simplify working with them. In particular, VB-objects require that the homogeneity structure be regular as this encodes a vector bundle structure [4]. The nice, but somewhat technical results of Bursztyn, Cabrera and del Hoyo [2] rely on showing that regularity of the homogeneity structure is preserved under ‘differentiation’ and ‘integration’. That is, when you pass from a groupoid to an algebroid and vice versa. Differentiation is no problem here, but integration is a much tougher question.
However, if we now consider VB-objects as sitting inside the larger category of weighted-objects then we can forget about the preservation of regularity during integration and simply check after that regularity is preserved. Bursztyn et al forced themselves to work in a smaller and not so nice category. We showed that working in this larger category of weighted-objects can simplify working with VB-objects.
Along side this, we show that there are plenty of nice and natural examples of weighted Lie groupoids. For example, the higher order tangent bundle of a Lie groupoid is a weighted Lie groupoid. This and other examples convince us that weighted Lie groupoids are important objects and that there is plenty of work to do.
References
[1] Grabowski J., Rotkiewicz M., Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36, arXiv:1102.0180.
[2] Bursztyn H., Cabrera A., del Hoyo M., Vector bundles over Lie groupoids and algebroids, arXiv:1410.5135.
[3] Bruce A.J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439.
[4] Grabowski J., Rotkiewicz M., Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305, math.DG/0702772.
]]>Please read the article at
A 1.95 m long solenoid exerting Aharonov–Bohm force on a coil
http://pengkuanem.blogspot.com/2015/10/a-195-m-long-solenoid-exerting.html
or
https://www.academia.edu/17214485/A_1.95_m_long_solenoid_exerting_Aharonov_Bohm_force_on_a_coil
A Congressional report produced in 1946 contained a section, a minority report that censured President Roosevelt as bearing responsibility for the attack on Pearl Harbor. Some of the arguments presented there seem to echo or elaborate the point that Donald Trump was attempting to make when he cast blame on President Bush for the September 11 attacks. The relevant section from the Pearl Harbor report is contained below:
INVESTIGATION OF THE PEARL HARBOR ATTACK
REPORT OF THE JOINT COMMITTEE ON THE INVESTIGATION
OF THE PEARL HARBOR ATTACK
CONGRESS OF THE UNITED STATES
See CONTENTS OF THE MINORITY PEARL HARBOR REPORT
The President of the United States was responsible for the failure
to enforce continuous, efficient, and appropriate cooperation among the
Secretary of War, the Secretary of the Navy, the Chief of Staff,
and the Chief of Naval Operations, in evaluating information and
dispatching clear and positive orders to the Hawaiian commanders as
events indicated the growing imminence of war; for the Constitution and
laws of the United States vested in the President full power, as Chief
Executive and Commander in Chief, to compel such cooperation and vested
this power in him alone with a view to establishing his responsibility
to the people of the United States.
As to the power, and therefore of necessity, the responsibility of the
President in relation to the chain of events leading to the catastrophe
at Pearl Harbor, there can be no doubt. The terms of the Constitution
and the laws in this respect are clear beyond all cavil.
The Constitution vests in the President the whole and indivisible
executive power subject to provisions for the approval of appointments
and treaties by the Senate.
The President, by and with the advice and consent of the Senate,
appoints high officers, civil and military.
He is Chief Magistrate in all civil affairs, including those related to
the maintenance and operation of the Military and Naval Establishments.
Under the law he conducts all diplomatic negotiations on behalf the
United States, assigning to his appointee, the Secretary of State, such
duties connected therewith as he sees fit, always subject to his own
instructions and authorizations.
Under the Constitution the President is Commander in Chief of the armed
forces of the United States, and with the approval of the Senate he
appoints all high military and naval officers. He assigns them to their
duties in his discretion except in the case of the Chief Staff and Chief
of Naval Operations-these appointments must approved by the Senate.
And why did the framers of the Constitution vest these immense powers in
one magistrate-not in a directory or a single official checked by a
council, as was proposed in the Convention of 1787?
The answer to this question is to be found in No. 70 of The
Federalist. The purpose of establishing a single rather than a plural
executive was to assure “energy in the Executive,” “a due dependence the
people,” and “a due responsibility.” A plural Executive, it is there
argued, “tends to deprive the people of the two greatest securities they
can have for the faithful exercise of any delegated power, first, the
restraints of public opinion; and, secondly, the opportunity of
discovering with facility and clearness the misconduct persons they
trust.”
The acts of Congress providing for the organization, operations, powers,
and duties of the Military Establishments under the President
particularized the powers and duties of the President in relation them;
in brief, they empowered him to issue orders and instructions the civil
Secretaries and also directly to the Chief of Staff and the Chief of
Naval Operations.
Such are the terms of the Constitution and the laws relative to the
Chief Executive.
From March 4, 1933, to December 7, 1941, Franklin D. Roosevelt was
President and Commander in Chief of the armed forces of the United
States and in him was vested all Executive powers under the Constitution
and the laws.
See President Roosevelt’s failure to enforce cooperation between high military authorities in Washington
The title is ‘New developments in geometric mechanics’. As well as myself the authors are K. Grabowska, J. Grabowski and P. Urbanski. We present a 16 page overview of our collective recent work in geometric mechanics. A little more specifically the main theme of the contribution is our application of graded bundles to geometric mechanics in the spirit of Tulczyjew.
For more details, consult the arXiv version and the original literature cited therein.
]]>However, I am very pleased that a Japanese group, Tsuguhiko Asakawa, Hisayoshi Muraki and Satoshi Watamura [2] found my work interesting and cited my work on Lie algebroid sigma models [1].
I placed my preprint on the arXiv on June 25th and the first version of their preprint was placed on the arXiv on Aug 24th. This is a record for me (excluding self-citations that nobody counts).
I don’t always check my citation very regularly and the automatic notifications are not always very reliable. Anyway…
The Japanese group constructed a gravity theory on a Poisson manifold equipped with a Riemannian metric. They do this in the context of Poisson generalised geometry and use the Lie algebroid of a Poisson manifold. Fascinating stuff.
References
[1] Andrew James Bruce, Killing sections and sigma models with Lie algebroid targets, arXiv:1506.07738 [math.DG].
[2] Tsuguhiko Asakawa , Hisayoshi Muraki and Satoshi Watamura, Gravity theory on Poisson manifold with R-flux, arXiv:1508.05706 [hep-th].
]]>Any chance you could make an expository post on Lie Theory for those of us who only known some abstract algebra and calculus? The topic seems very inaccessible otherwise, but I hear Lie Groups and Lie Algebras mentioned regularly.
As your friendly neighbourhood mathematician I will try to oblige.
Disclaimer What I do is give an informal overview and not worry too much about details and proper proofs. Proofs you can find in textbooks. Rather I want to present the ideas and sketch some constructions.
I will build this account up over the period of a few weeks.
Rough Plan
The things I would like to cover are the following.
There maybe some changes here as the work develops.
I will also include some simple exercises for those that are interested. I will post solutions at the end.
Part 0: Introduction
Anybody who reads anything about modern physics will encounter the terms ‘Lie group’ and ‘Lie algebra’. Lie theory is all about the relation between these two structures.
A Lie group is a group that also has a smooth manifold structure, importantly the group operations are compatible with this smooth structure. Groups represent transformations and symmetries of mathematical objects. Lie groups are the mathematical framework for studying continuous symmetries of mathematical objects. Thus, Lie groups are fundamental in geometry and theoretical physics.
Now, every Lie group has associated with it a Lie algebra, whose vector space structure is the tangent space of the Lie group at the identity element. The Lie algebra describes the local structure of the group. Informally one can think of the Lie algebra as describing the elements of the Lie group that are ‘very close to the identity element’.
The theory of Lie groups and Lie algebras was initiated by Sophus Lie, and hence the nomenclature. Lie’s motivation was to extend Galois theory, which proved useful in the study of algebraic equations, to cope with continuous symmetries of differential equations. Lie laid down much of the basic theory of continuous symmetry groups.
The plan is with these notes is to sketch the relation between Lie groups and Lie algebras. I will stick to the finite dimensional case for this first look.
Part I: Abstract Lie algebras
Let us start with a completely algebraic set-up. Informally, a Lie algebra is a vector space with a non-associative product, known as a ‘bracket’ that satisfies some nice properties. We will only consider algebras over the reals or complex here, though everything will generalise to more arbitrary fields (with some minor modifications if necessary).
Definition
A Lie algebra is a vector space $latex \mathfrak{g}$ together with a bilinear operation $latex [\bullet,\bullet]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}$, that satisfies the following conditions
$latex [x,[y,z]] + [z,[x,y]] +[y,[z,x]]=0$
for all $latex x,y, z \in \mathfrak{g}$.
Note that Lie algebras are non-associative. Thinking of the bracket as a form of multiplication we see that the Jacobi identity is related to the ‘associator’ which is non-zero in general
$latex [x,[y,z]] -[[x,y],z]= [x,[y,z]] + [z,[x,y]] = [[z,x],y] \neq 0$.
The Jacobi identity can also be written in ‘Loday form’
$latex [x,[y,z]] = [[x,y],z] + [y,[x,z]]$,
which means that the operator $latex Ad_{x}:= [x, \bullet]$ satisfies the Leibniz rule, the so called adjoint operator is a derivation. Note that this form of the Jacobi identity has this interpretation even if the bracket is not skewsymmetric. In fact such bracket algebras are well studied and are usually called “Loday” or “Leibniz-Loday” algebras.
The dimension of a Lie algebra is defined to be the dimension of the underlying vector space. Elements of a Lie algebra are said to generate that Lie algebra if they form the smallest subalgebra that contains these elements is the Lie algebra itself.
Example Any vector space equipped with a vanishing bracket $latex [x,y]=0$, is a Lie algebra. We call any Lie algebra with a vanishing bracket an abelian Lie algebra.
Example The (real) vector space of all n×n skew-hermitian matrices together with the standard commutator is Lie algebra. This Lie algebra is denoted $latex \mathfrak{u}(n)$.
Example The Heisenberg algebra is the Lie algebra generated by three elements x,y,z and the Lie brackets are defined as
$latex [x,y] =z$, $latex [x,z] =0$ and $latex [y,z] =0$.
Given a set of generators $latex \{T_{a}\}$ we can define the Lie algebra in terms of its structure constants. As the Lie bracket of any pair of generators must be a linear combination of the generators we have
$latex [T_{a}, T_{b}] = C^{c}_{ab}\: T_{c}$,
and so the Lie algebra is determined by the structure constants $latex C^{c}_{ab}$.
Exercise How many one dimensional Lie algebras are there up to isomorphisms?
Exercise There are exactly two Lie algebras of dimension two over the real numbers, up to isomorphism. Can you write these down in terms of generators?
Exercise What conditions do the structure constants need to satisfy in order to have a Lie algebra? (Hint: think about the two defining conditions of a Lie algebra)
People study Lie algebras in their own right, but historically they arose from the study of Lie groups. From my own perspective, it is the fact that Lie algebras are ‘infinitesimal Lie groups’ that makes them interesting and useful. In the next section I will move on to groups and in particular Lie groups.
Part II: Lie groups
Before we move on to Lie groups, let us recall the notion of a group. Generically, one thinks of groups as encoding transformations and symmetries of mathematical objects, so they arise all across mathematics.
Definition
A group is a set $latex G$ together with a binary operation $latex \circ: G \times G \rightarrow G$ that satisfies the following axioms
It can be shown that the identity element $latex e$ is unique. There is only one identity element. Note we have said noting about commutativity. Generally $latex a\circ b$ is not the same as $latex b\circ a$. Groups for which these two expression are always equal are called abelian groups.
Example The set of integers $latex \mathbb{Z}$ together with standard addition form an abelian group. The identity element is zero and the inverse of any element is $latex a^{-1} = {-}a $.
Exercise Does the set of real numbers $latex \mathbb{R}$ equipped with standard addition form a group? Does the set of real numbers with standard multiplication form a group?
Example A symmetric group a set consists of permutations on the given set; ie. bijective maps from the set to itself. The product is just composition of the permutations as functions. The identity element is just the identity function from the set to itself. The inverse of an element is just the inverse as a function.
Example Probably the simplest non-abelian group is the rotation group $latex SO(3)$. This group consists of all rotations about the origin of three-dimensional Euclidean space and the composition is just standard composition of linear maps. Because all linear transformations can be represented by matrices (once a basis has been chosen) the group $latex SO(3)$ can be represented by the set of orthogonal 3×3 matrices and standard matrix multiplication. This group is non-abelian as the order of which rotations are composed matters.
Now, Lie groups are both groups and smooth manifolds at the same time. Before we make this statement a bit more precise I should say a few words about manifolds…
For an informal overview of the idea of manifolds you can consult an earlier post I made here. I will assume everyone had read this, or is at least familiar with the basic idea. I will review the minimum needed to define a Lie group.
A manifold is a ‘space’ that is locally similar to $latex \mathbb{R}^{n}$ for some n. A smooth manifold is a refinement of that notion to allow us to do calculus. Any manifold can be described by a collection of charts, also known as an atlas.
An atlas on a topological space $latex X$ (say) is a collection of pairs $latex \{(U_{\alpha},\phi_{\alpha})\} $ called charts, where the $latex U_{\alpha}$ are open sets that cover the topological space, such that
$latex \phi_{\alpha}: U_{\alpha} \rightarrow \mathbb{R}^{n},$
is a homomorphism of $latex U_{\alpha}$ onto an open subset of $latex \mathbb{R}^{n}$. Loosley this means that locally we can ways think about cutting our topological space up into small pieces of the real linear space.
The transition maps are defined as
$latex \phi_{\alpha \beta}:= \phi_{\beta} \circ \phi^{-1}_{\alpha}|_{\phi_{\alpha}(U_{\alpha} \cap U_{\beta})}: \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \rightarrow \phi_{\beta}(U_{\alpha} \cap U_{\beta}).$
Any topological space with an atlas is a topological manifold. Loosley, the transition maps allow you to sew together the local patches by telling you what happens on the overlap of such patches.
We will be interested smooth manifolds, that is we insist that the transition maps be infinitely differentiable in the standard sense. Because we can describe everything locally on a smooth manifold in terms of smooth transition functions and local patches of $latex \mathbb{R}$ we can extend all our knowledge of standard multi-variable calculus to smooth manifolds.
In particular we know what a smooth map between two smooth manifolds is. As topological spaces a map between smooth manifolds is a continuous map. To define it as ‘smooth’ we compose the function with a chart on our source and target manifolds and as we know what smoothness means for map from $latex \mathbb{R}^{n}$ to say $latex \mathbb{R}^{m}$ we can accordingly define smoothness for maps between smooth manifolds.
Exercise Fill in details for the above paragraph.
We can now state what a Lie group is…
Definition A Lie group $latex G$ is a smooth manifold that also carries a group structure whose product and inversion operations are smooth maps.
That is both
$latex \mu : G \times G \rightarrow G$
$latex (x,y) \mapsto \mu(x,y) = x\cdot y$
and
$latex inv : G \rightarrow G$
$latex x \mapsto x^{-1}$
are smooth maps.
Examples to follow…
]]>Of course you are all wondering what a Lie system is. Well, basically a Lie system is a systems of first-order ordinary differential equations whose general solution can be written in terms of a finite family of particular solutions and a superposition rule. There is a rich geometric theory here and many motivating examples that arise from physics.
]]>Please read the article at
Disc magnet parallel action experiment
http://pengkuanem.blogspot.com/2015/06/disc-magnet-parallel-action-experiment.html
or Disc magnet parallel action experiment (video included)
https://www.academia.edu/13033082/Disc_magnet_parallel_action_experiment_video_included_
The magnetic field of the earth is uniform on its surface. The resultant Lorentz force a uniform magnetic field exerts on a coil of any shape is zero. The torque perpendicular to a flat coil is also zero. So, a current carrying coil in the magnetic field of the earth should stay immobile. However, my experiment shows that the test coil rotates in its plane. See the video of this experiment: http://youtu.be/JKMG8jY1RRg
Please read the article at
Earth’s magnetic field and parallel action
http://pengkuanem.blogspot.com/2015/06/earths-magnetic-field-and-parallel.html
or Earth’s magnetic field and parallel action (video included)
https://www.academia.edu/12926460/Earth_s_magnetic_field_and_parallel_action_video_included_
Please read the article at
Solenoid parallel action experiment
http://pengkuanem.blogspot.com/2015/06/solenoid-parallel-action-experiment.html
or Solenoid parallel action experiment with video included
https://www.academia.edu/12722675/Solenoid_parallel_action_experiment_with_video_included
However, the magnetic field outside a solenoid is zero and cannot act any force on a current. Moreover, there cannot be force parallel to current for classical theory.
So, If I do this experiment, will the coil rotate or not?
Please read the article at
Q: Parallel action with a solenoid
http://pengkuanem.blogspot.com/2015/05/q-parallel-action-with-solenoid.html
or
https://www.academia.edu/12426677/Q_Parallel_action_with_a_solenoid
Most of the posts I found myself writing were about silly mistakes that I made in the laboratory. Sometimes these mistakes were pretty funny and I thought it could help people by sharing them. But then I had a crisis of confidence: do I really want to admit to the world that sometimes I am completely and utterly stupid?
On a similar vein, I have quite a wacky sense of humour and sometimes I like to write about really silly subjects (see: the iguana post). Writing in a more light-hearted way was a welcome relief from serious science, but I worried that I wouldn’t be taken seriously by being quite oddball on my blog. I think I would be really embarrassed if someone who didn’t realise how nutty I am really came across my blog.
Sometimes I found myself writing about what I was working on and wondered about how much detail it’s safe to go in to prior to publication. Another issue is that I am quite private with my online identity and struggled with how wise it would be to put my name and face to what I was writing.
I would be interested to get feedback on what I’ve written, so please feel free to leave a comment.
]]>Please read the article at
Aharonov–Bohm effect in CRT experiment
http://pengkuanem.blogspot.com/2015/04/aharonovbohm-effect-in-crt-experiment.html
or
https://www.academia.edu/12052541/Aharonov_Bohm_effect_in_CRT_experiment_Video_included_
I’m very excited to be starting my own lab, moving to California and teaching at UC Merced.
My lab will be focused on evolution in viral and bacterial systems – I am recruiting postdocs, graduate students and undergraduate interns for fall semester 2015! My website is here: https://sistromlab.wordpress.com/
Looking forward to keeping SFN updated on how our research develops.
]]>Please read the article at
Non-loop induced voltage problem
http://pengkuanem.blogspot.com/2015/03/non-loop-induced-voltage-problem.html
or
https://www.academia.edu/11701041/Non-loop_induced_voltage_problem
Please read the article at
Induced conductor net problem
http://pengkuanem.blogspot.com/2015/03/induced-conductor-net-problem.html
or
https://www.academia.edu/11470407/Induced_conductor_net_problem
Please read the article at
Coil and resistor induction paradox pdf or word
http://pengkuanem.blogspot.com/2015/02/coil-and-resistor-induction-paradox.html
or
https://www.academia.edu/11113117/Coil_and_resistor_induction_paradox
I’m here to not apologize for my recent absence. I started blogging a little over 7 years ago; tried it, liked it, and kept going. I had no really clear plan other than I’d do it as long as I was enjoying it.
Lately, I’m not enjoying it. (I think the technical term for it is “being in a funk”). So I’m taking a break. (It’s not you, it’s me. Continue to see other blogs. We were never exclusive)
This isn’t the first time in a funk — motivation wanes and waxes. Each time I eventually found something interesting to blog about, and posting links filled the gaps. Even as things slowed down over time — I went from multiple posts pretty much every day to eventually one post on most weekdays. In the past several years I had a pick-me-up this time of year in the form of the Science Online conference. Getting to see friends and meeting new people was a lot of fun, and there were always new things discussed in the sessions that rejuvenated me. But the organization folded last year, which is a long story, but the result is that there is no conference (which should have been happening this very week). No salvation there. It’s gotten tiresome slogging through feeds and lists looking for interesting things, and my normal fallback — ranting about bad physics reporting — just feels like it’s all been done.
There are bloggers out there who have done this longer, but I think most have not (heavily skewed by a large infant mortality rate). I expect I’ll return — someday — but if and when that happens there are no guarantees as to the frequency of my posting. We will see.
]]>“There is no question but that Hitler belongs in the category of the truly mystic medicine man…since the time of Mohammed nothing like it has been seen in this world. This markedly mystic characteristic of Hitler is what makes him do things which seem to us illogical, inexplicable, curious and unreasonable….Don’t you know that if you choose one hundred of the most intelligent people in the world and get them all together, they are a stupid mob? Ten thousand of them together would have the collective intelligence of an alligator…. In a crowd, the qualities which everybody possesses multiply, pile up, and become the dominant characteristics of the whole crowd. Not everybody has virtues, but everybody has the low animal instincts, the basic primitive caveman suggestibility, the suspicions and vicious traits of the savage age. The result is that when you get a nation of many millions of people, it is not even human. It is a lizard or a crocodile or a wolf.”
~Carl Jung interview with H.R. Knickerbocker in Cosmopolitan [1938] See: C.G. Jung Speaks; Pages 115-135.
]]>Three!
]]>All of these are electromagnetic waves and they all travel at the same speed (the speed of light). However, they have different interactions with matter. If you are inside, your mobile phone can still get data from a cell tower since these radio waves pass through most walls. Can you see through the walls? No. Visible light does not pass through most walls. X-rays mostly go through your skin, but you can’t see (with visible light) through skin – that would just be weird.
Technically the interaction with light and matter depends on the frequency of light – but since frequency and wavelength are related, we can just talk about the wavelength.
Not sure of the wisdom of advertising that there are two jerks working there.
]]>Real scientific controversies play out in the scientific literature, through papers drawing on many other sources of data.
…
Phony controversies tend to play out in the media, through press releases, stump speeches, and polemical writing reshared via social media.
Somewhat related: something I wrote a while back. Each step along the way of doing the science increases your confidence, but ultimately what you need in any scientific finding is confirmation of a result.
]]>Once the weight of experimental result hits a certain critical mass, the expectations swing away from needing data to confirm a theory to needing exceptional data to disprove it.
]]>Understanding my limits and being willing to acknowledge them is, in fact, one of my strengths. I don’t think it should be pathologized alongside the very real problem of “impostor syndrome”.
In fact, it is the opposite behavior—the belief that you can do anything, including things you are blatantly not qualified for or straight up lying about—should be pathologized.
]]>The Harriss spiral is constructed from rectangles in the ratio of the plastic number (1.3247…), in a similar way to how a Fibonacci spiral is created from rectangles in the related golden ratio (1.6180…). These plastic rectangles can be split into two smaller plastic rectangles, leaving a square. Recursively splitting the rectangles, and drawing curves in the squares gives this fractal spiral.
]]>When you say we should work harder, I hear two things: 1) we aren’t working hard, and 2) we don’t think we have to. Professors seem like an easy target. We have good job security, we’re paid well, we often come from privileged backgrounds. We appear to have little to do but teach a class for a few hours a week, and we have extended vacations. It’s easy to see us as cloistered in the Ivory Tower, without much experience with the “real world” and the concerns of average folks.
The picture I’ve painted for you is incomplete, though.
]]>The whiteness of newly fallen snow is, of course, one of its primary defining characteristics, so it’s tempting to just say that, you know, that’s the way snow is. But it’s actually a pretty good question, because snow is really just frozen water, and frozen water tends to be transparent
A teaser. For the setup, go to the link.
…
In each case the rolling coin has made one complete rotation. But the red arc at the top is half the length of the red line at the bottom. Why?
I have a more physics-y than a formal math-y explanation of why, which I will post soon.
————
OK, here’s my answer.
In the rolling case, all you have is rotation. On rotation gives you 2*pi, so it rolls one circumference.
But in the other case you have rotation and revolution (spin and also orbital motion). Going halfway around the coin gives you an equal contribution of each, so the amount of spin only requires pi rotation, and it rolls half of the circumference. If the coin’s point of contact never changed, it would still do a rotation over the course of its revolution. If the orientation stayed fixed, the point of contact would make a complete trip around the coin.
A related example of this is the moon. If viewed from an external inertial frame (where the distant stars appear to be fixed), the moon rotates around the earth every ~4 weeks. But since it’s tidally locked and always has the same part facing the earth, it also rotates once about its axis.
]]>]]>Gav shows you how insanely quick the inside of a DSLR camera moves when it takes a picture, by filming it at 10,000 fps.
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Camera filmed is a Canon 7D.
This video is a good demonstration of how a rolling shutter works.
Shot with a Phantom Flex at 10,000fps
Why 50 million smart meters still haven’t fixed America’s energy habits
The upshot: Right now, smart meters aren’t waking Americans up and making them conscious of their energy use — because they aren’t being paired with what behavioral research shows us is needed for that to happen.
This is the story of why the smart meter revolution has, thus far, fallen short — and how we can better use one of the most pivotal innovations in the electricity sphere to save energy, cut greenhouse gas emissions and save a lot of money.
I can vouch for the notion of immediate feedback being an important component to changing behavior — something that’s discussed in the article. My new-ish car tells me my instantaneous gas efficiency and reminds me of things that I know but would not necessarily be thinking about, such as how wasteful it is to romp on the gas when speeding up, or how hitting the brakes means you are bleeding away your kinetic energy as heat. So it’s modified how I drive — smaller accelerations. Less gas when speeding up and coasting to slow down, when it’s appropriate to do so. So I can see how this would work for home energy use, too.
]]>The explosion, say Pavel Jungwirth and his collaborators at the Czech Academy of Sciences in Prague, is not merely a consequence of the ignition of the hydrogen gas that the alkali metals release from water. That may happen eventually, but it begins as something far stranger: a rapid exodus of electrons followed by explosion of the metal driven by electrical repulsion.
Neat. It’s not the hydrogen reacting with air that causes the alkali to explode. That reaction doesn’t cause more surface area to be created as the reaction unfolds, so it can’t “accelerate”
Video in the link, including slow-motion views of the explosion.
]]>]]>There was also another controversy raging at the time, concerning the nature of light. It was known that light travelled through space with a finite speed, rather than leaping instantaneously from its source to our eyes.
But no-one knew, a century-and-a-half ago, what light was actually made of.
Most physicists agreed it travelled through space as a wave but they didn’t know what these light waves were made of, and they didn’t know how they got from one place to another. Maxwell was about to solve all these mysteries.
A couple of science stories that don’t quite count as science because the results have never been duplicated.
]]>As the video warns, magnets like these are potentially very dangerous.
]]>I was a tad confused at first, because to me “kinetic sculpture” refers to those systems where balls are continually moving about on various tracks in a closed system. But these are rather nice, too.
]]>]]>Having two pictures of the exact same lightning bolt lets you do something pretty amazing; reconstruct its path in 3D. In this case because the precise location and elevation of the photographers isn’t known this is slightly more art than science, but it is still fun!
Fascinating 3D-Printed Fibonacci Zoetrope Sculptures
These 3d-printed zoetrope sculptures were designed by John Edmark, and they only animate when filmed under a strobe light or with the help of a camera with an extremely short shutter speed.
… just like any other object would. Maybe it’s just me, but this sounds like the author is implying this is special to this particular class of structures — it’s not. That’s just how the strobe effect works.
Marvellous rube goldberg mechanical lightswitch covers
These are wonderful. But having a few gears doesn’t turn it in to a Rube Goldberg device; it’s not just a matter of being slightly more complex than it needs to be — in this case, mostly by adding one layer of complexity. There are no chain reactions and no diversity of mechanism, two hallmarks of such devices.
]]>Please read the article at
Magnetism and dark matter http://pengkuanem.blogspot.com/2015/01/magnetism-and-dark-matter.html
or
https://www.academia.edu/10183169/Magnetism_and_dark_matter
“I have a friend — or had a friend, now dead — Abdus Salam, a very devout Muslim, who was trying to bring science into the universities in the Gulf states and he told me that he had a terrible time because, although they were very receptive to technology, they felt that science would be a corrosive to religious belief, and they were worried about it… and damn it, I think they were right. It is corrosive of religious belief, and it’s a good thing too.”
“There are those whose views about religion are not very different from my own, but who nevertheless feel that we should try to damp down the conflict, that we should compromise it. … I respect their views and I understand their motives, and I don’t condemn them, but I’m not having it. To me, the conflict between science and religion is more important than these issues of science education or even environmentalism. I think the world needs to wake up from its long nightmare of religious belief; and anything that we scientists can do to weaken the hold of religion should be done, and may in fact be our greatest contribution to civilization.”
Today these religious fanatics are murdering satirical cartoonists. I would not be surprised if in the future they turned their attentions to attacking intelligent atheists who express themselves as eloquently as Steven Weinberg does.
]]>What Feynman is presenting in the video is an atheist’s version of the story of Adam and Eve and the Tree of Knowledge. Feynman’s banana is analogous to the fruit of the Tree of Knowledge, which is more typically depicted as an apple.
And as an atheist, Feynman rejects the version of man’s origins presented in Genesis, but rather considers man to be a close relative and a descendant of the apes. Plucking a banana from its tree is like obtaining one additional bit of understanding of “the ultimate laws of physics”.
]]>“Feynman once said, ‘Science is imagination in a straitjacket.’ It is ironic that in the case of quantum mechanics, the people without the straitjackets are generally the nuts.”
–Lawrence M. Krauss
Here is link to a video clip where Feynman makes his science as imagination in a tight straitjacket assertion: http://youtu.be/ysYEAC4z66c
Here are two shots I took of a street performer entertaining spectators by struggling to get out of a straitjacket while balancing on a large ball. It’s interesting that some of the spectators watching him have their arms crossed as if they themselves were wearing straitjackets: