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By "continuous vector space", I mean that the identity operator can be resolved for a complete orthonormal basis \( |\phi(x) \rangle \) for all \( x \) such that \( \langle \phi(x), \phi(x') \rangle = \delta(x - x') \) as \( \hat{1} = \int |\phi(x)\rangle\langle \phi(x)|~\mathrm{d}x \). If an operator \( \hat{A} \) has continuous matrix elements \( A(x, x') = \langle \phi(x), \hat{A}\phi(x') \rangle \), then it is easy to see that the conditions for its trace \( \operatorname{trace}(\hat{A}) = \int A(x, x)~\mathrm{d}x \) to be finite are that the integral must converge, so the "function" \( A(x, x) \) must asymptotically approach 0 strictly faster than \( 1/x \) as \( |x| \to \infty \) and must at most have singularities at finite points \( x_{0} \) that diverge strictly slower than \( 1/|x - x_{0}| \). This can be seen as the continuum limit of a sum over the diagonal. However, the determinant is harder to express in this way because it involves products over diagonals & subdiagonals that are harder to express in a continuum space.</p><p>For this post, I will only consider Hermitian positive-definite operators. The conditions that I will list for which the determinant exists for such operators are sufficient for the determinant to exist, but I am not convinced that they are necessary. If such operators have an eigenvalue decomposition \( \hat{A} = \int a(x) |\phi(x)\rangle\langle \phi(x)|~\mathrm{d}x \) where the vectors \( \{ |\phi(x) \rangle \} \) form a complete orthonormal basis and the eigenvalues satisfy \( a(x) > 0 \) for all \( x \), then one can make use of the identity \( \ln(\det(\hat{A})) = \operatorname{trace}(\ln(\hat{A})) \) to say that \( \ln(\det(\hat{A})) = \int \ln(a(x))~\mathrm{d}x \). For the right-hand side to converge, then \( \ln(a(x)) \) must asymptotically approach 0 with \( x \) as \( |x| \to \infty \) strictly faster than \( 1/x \), which means that \( a(x) \) must asymptotically 1 with \( x \) as \( |x| \to \infty \) strictly faster than \( \exp(1/x) \) (which is <i>not</i> the same as \( e^{-x} \)), and \( \ln(a(x)) \) can at most have singularities at finite points \( x_{0} \) that diverge strictly slower than \( 1/|x - x_{0}| \), which means that \( a(x) \) must either diverge to \( \infty \) strictly slower than \( \exp(1/|x - x_{0}|) \) or drop to 0 strictly slower than \( \exp(-1/|x - x_{0}|) \). For example, \( a(x) = \exp(1/(x^{2} + x_{0}^{2})) \) fits the bill; note that this is <i>not</i> the same as the Gaussian kernel \( \exp(-(x^{2} + x_{0}^{2})) \). Intuitively, this condition makes sense, because for a finite-dimensional diagonal matrix as the dimension becomes arbitrarily large, the diagonal elements must mostly be exactly or very close to 1 for the determinant to not grow arbitrarily large with the dimension.<br /></p><p>In finite-dimensional vector spaces, it is also easy to compute the determinants of triangular matrices simply as the products of the diagonal elements. (This is why the determinant is most often computed by an algorithm like first computing the LU decomposition and then taking the product of the diagonal elements of the upper-triangular matrix, which for an \( N \times N \) matrix involves \( O(N^{3}) \) operations, as opposed to the Leibniz formula involving every permutation which involves \( O(N!N) \) operations.) In infinite-dimensional vector spaces, a matrix that is triangular in a countable basis can have the determinant computed similarly as in finite-dimensional vector spaces; if an operator \( \hat{A} \) in that basis has elements \( A_{ij} \), then using the definition \( \ln(|\det(\hat{A})|) = \prod_{i} \ln(|A_{ii}|) \), the determinant converges as long as the diagonal elements \( |A_{ii}| \) are mostly exactly or very close to 1, specifically such that as \( |i| \to \infty \), \( \ln(|A_{ii}|) \) decays to 0 strictly faster than \( 1/i \). (Note that \( i \) is an integer index written in slanted font, not the imaginary unit \( \operatorname{i} \) written in upright font.) However, I am not sure how to generalize this to operators that are expressed as triangular matrices in continuous bases.<br /></p>http://dasublogbyprashanth.blogspot.com/2024/05/finite-determinants-of-linear-operators.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-7805462786473208475Mon, 01 Apr 2024 15:44:00 +00002024-04-01T11:44:54.229-04:00calculusclassmathematicsphysicsquantum electrodynamicsquantum mechanicssciencestatistical mechanicsTransitioning from microscopic to macroscopic and quantum to classical regimes<p>I recently read two things that were of interest to me having previously worked in physics. One was an article in <i>The New Yorker</i> magazine [<a href="https://www.newyorker.com/culture/annals-of-inquiry/how-much-of-the-world-is-it-possible-to-model">LINK</a>], in which the author does a good job of going over the successes of mathematical modeling in the physical sciences and contrasting this with the limitations of mathematical modeling in public health (showing, for example, how many models of the spread of contagions fail when governments & societies take fast & drastic collective actions to limit the spread), the failures of mathematical models in social sciences where the outputs of those models can create feedback loops with public sentiment (for example in political polling), and the way that many people who use machine learning models in different domains expect the fancy curve-fitting of those models to represent fundamental understanding when that might not really be so. The other was a journal article published in Physical Review Letters [<a href="https://doi.org/10.1103/PhysRevLett.132.030202">LINK</a>] about how it can be possible to test the extent to which a massive (as opposed to massless) object which exhibits the dynamics of a simple harmonic oscillator and prepared in a quantum coherent state can be tested for deviations from classical behavior using a protocol that does not depend on the mass of the object (although I question this given that the protocol depends on timed measurements that depend on the frequency of oscillation, and in many physics contexts the frequency does depend on the mass as \( \omega = \sqrt{k/m}\), but this is somewhat of a quibble). These two things got me to think about something that I realized I never got out of many years of formal undergraduate & graduate education in physics. This can be illustrated with the following example.<br /></p><p>In introductory physics classes that focus on Newtonian mechanics, a prototypical problem involves a block, modeled as a point mass, sliding (with or without friction) down a fixed triangular incline in the constant gravitational field of the Earth. In the context of those classes, instructors will be careful to note that this is merely a model, and corrections could come from the inclusion of the variation of the Earth's gravitational field & surface curvature, the technical possibility of moving the triangular incline (which must be much more massive than the block in question), the shape of the block, variations in the touching surfaces, air resistance, et cetera. In later classes, instructors may point out corrections due to special relativity (i.e. the speed of light) and general relativity (as it relates to the Earth's gravitational field).</p><p>However, in later classes about quantum mechanics & statistical mechanics, instructors explain how different the models are from models of Newtonian mechanics at human scales, but they often promise that appropriate treatments of aggregates of microscopic constituents can consistently recover results from Newtonian mechanics, yet this promise is almost never fulfilled. In particular, wavefunctions that describe pure states of single microscopic particles are quite far removed from the simple dynamical variables describing blocks on inclined planes, although statistical mechanics can probabilistically describe the solid states of the block & inclined plane as well as the gaseous state of the surrounding air, it is not usually extended to describe the dynamics of the block sliding down the inclined plane. For example, if a block sliding down a fixed inclined plane of horizontal angle \( \theta \) in a uniform gravitational field is described as having equations of motion \( m\ddot{x} = mg\sin(\theta) \) where the \( x \)-axis is defined as pointing downward parallel to the slope of the inclined plane for increasing \( x \) and the \( y \)-axis points outward in the normal direction from the inclined plane, then I wish to see corrections of the form \( m\ddot{\vec{x}} = \sum_{\mu = 0}^{\infty} \sum_{\nu = 0}^{\infty} \hbar^{\mu} k_{\mathrm{B}}^{\nu} \vec{f}^{(\mu, \nu)} \) where the lowest-order term is \( \vec{f}^{(0, 0)} = mg\sin(\theta)\vec{e}_{x} \). I have never seen these sorts of quantum or statistical corrections to Newtonian equations of motion in simple (in the context of Newtonian mechanics) systems. Similarly, it is rare to see how quantum or statistical mechanical systems can, in appropriate limits, reproduce classical systems; I can only think of the quantum coherent state of the simple harmonic oscillator as well as how the Moyal bracket in the phase space formulation of quantum mechanics reduces to lowest order in \( \hbar \) to the Poisson bracket, and in the latter case, intuitive construction of the quantum phase space quasiprobability function is made more difficult (compared to construction of a classical phase space probability density function, as I did in a post [<a href="https://dasublogbyprashanth.blogspot.com/2020/07/classical-phase-space-densities-for-one.html">LINK</a>] from a few years ago) by the fact that unlike the classical phase space probability density function, the quantum phase space quasiprobability function cannot be arbitrarily localized in phase space, it can take on negative values for certain wavefunctions, it is compressible in phase space with respect to its own evolution over time, and it is not obvious how it should look for a system of many particles constituting a macroscopic object like a block (in contrast to a classical phase space probability density function, which for such a system could just be a product of Dirac delta functions localizing each microscopic constituent to a point in phase space).</p>These considerations reminded me of a discussion I had last year with friends from college, who also did course 8 (physics) with me. We came to a consensus that while people who do not become physics majors should, as usual, get exposure to Newtonian physics and the basics of electricity & magnetism, people who become physics majors should have a curriculum over 3-4 years that exhibits a sensible conceptual progression. In particular, after seeing Newtonian mechanics, such students should then be exposed to Lagrangian & Hamiltonian formulations of classical mechanics. The Lagrangian formulation of classical mechanics should then be used to develop intuitions about mechanical waves, which in turn can lead to introductions to classical field theory and development of classical electromagnetic theory as a rich example of a classical field theory. (I would also personally recommend using the introduction of mechanical waves to introduce the linear algebraic treatment of waves and then reintroduce the linear algebraic treatment of waves into the treatment of linear classical field theories in general & linear classical electromagnetic theory in particular.) The Hamiltonian formulation of classical mechanics should then be used to develop intuitions about probability distributions in classical mechanics, which in turn can be used to develop intuitions about statistical mechanics. Optionally, at this point, the Hamiltonian formulation of classical mechanics can also be used to develop intuitions about nonlinear dynamics & chaos theory, but while this is good for the broader education of physics students, it is less immediately relevant for the introduction of quantum theory to come soon after (because quantum mechanics is linear). Finally, only after these things happen should quantum theory be introduced, such that there are clear connections of the wavefunction formulation of quantum mechanics to mechanical waves, the phase space formulation of quantum mechanics to classical phase space probability distributions, and the linear algebraic framework of quantum mechanics to linear algebraic treatments of classical field theories (including linear classical electromagnetic theory); this will ensure that students understand how ideas like superposition, interference, rotation through a Hilbert space, statistical uncertainty, and related ideas are not unique to quantum mechanics (which is unfortunately too often a consequence of the way quantum mechanics is typically introduced in undergraduate curricula, at least in the US). We also came to a consensus that in each course, there should be clear explanations of what prototypical systems are analytically solvable, what prototypical systems are not analytically solvable, and why (in each case).<br />http://dasublogbyprashanth.blogspot.com/2024/04/transitioning-from-microscopic-to.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-330502932149100853Fri, 01 Mar 2024 16:34:00 +00002024-03-01T11:34:17.871-05:00changeclimateFOLLOW-UPscienceProgression of Winter Storms across the Contiguous US<p>This winter has featured many winter storms over the contiguous US that have swept from the west coast to the east coast. In previous posts, I have discussed basic intuitions for why different climates occur in different regions [<a href="https://dasublogbyprashanth.blogspot.com/2023/04/my-rough-intuition-of-climate.html">LINK</a>], my assessment of the deficiencies of the Trewartha climate classification system [<a href="https://dasublogbyprashanth.blogspot.com/2023/05/follow-up-my-rough-intuition-of-climate.html">LINK</a>], what I would change about the Trewartha climate classification system [<a href="https://dasublogbyprashanth.blogspot.com/2023/07/more-on-climate-categorization.html">LINK</a>], how my proposed changes to the Trewartha climate classification system can be applied to understand what climates occur where in middle latitudes [<a href="https://dasublogbyprashanth.blogspot.com/2023/10/where-different-climate-types-do-or-do.html">LINK</a>], why popular understanding of the effects of the Gulf Stream over the Atlantic Ocean on the climate of Europe is incorrect in many ways [<a href="https://dasublogbyprashanth.blogspot.com/2023/12/myth-of-effects-of-gulf-stream-on.html">LINK</a>], and why different climates occur in coastal locations on different coasts at different latitudes [<a href="https://dasublogbyprashanth.blogspot.com/2024/01/variations-of-coastal-monsoon-climates.html">LINK</a>]. These posts have suggested, among other things, that many winter storms on the east coast of the US would come from warm moist air from over the Gulf of Mexico or mild moist air from over the Atlantic Ocean colliding with cold dry air over the continent, but these collisions would be somewhat more sporadic because the prevailing westerlies, which would have dumped moisture primarily over the west coast, would be weak & dry by the time they reach the east coast. Thus, it is somewhat surprising to me that these winter storms seem to be driven by the prevailing westerlies over the continent. The following is my attempt to intuitively explain, based only on sea-/surface-level temperatures, air pressures, and air flows, why this happens. <b>Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.</b></p><h2 style="text-align: left;">Why this happens in North America</h2><p>This happens in North America mainly because of the arrangement of landmasses & seas/oceans. In the winter half of the year in North America, the subtropical ridge is strongest around 30 degrees in latitude (north of the equator) to the west of the continents of North America in the Pacific Ocean & of Africa in the Atlantic Ocean. Prevailing westerlies generated by the subtropical ridge over the Pacific Ocean bring moisture to the west coast of the US and turn clockwise due to the Coriolis force, meaning that around the time the prevailing westerlies reach the Rocky Mountains, they may have turned more toward the Gulf of Mexico, though this is not guaranteed to happen every time. In doing so, the prevailing westerlies, by this point colder & drier, can pick up warm moist air from the Gulf of Mexico. This clockwise turn by the Coriolis force is reversed within the Gulf of Mexico by southerly winds coming from air coming clockwise off of the subtropical ridge over the Atlantic Ocean, so this newly warmed & moistened air turns toward the east coast of the US, bringing moisture there before moving east & turning clockwise (again due to the Coriolis force) over the Atlantic Ocean toward Europe. This is how the subtropical ridge can function like a conveyor belt of moisture. Essentially, the continent of North America & the Atlantic Ocean are both narrow enough (with respect to the ranges of longitudes), and the Gulf of Mexico with warm water is favorably placed, to ensure that this can happen. That said, the prevailing westerlies will not always turn clockwise enough to go over the Gulf of Mexico and then counterclockwise enough to go over the east coast of the US, which is why the prevailing westerlies are more likely to bring moisture to the west coast of the US but only sporadically do so for the east coast of the US.</p><p>I should clarify that the storms that sweep across the contiguous US are often localized highly mobile systems of low pressure. They internally turn counterclockwise, but the motion of the centers of these storms is affected by the aforementioned prevailing westerlies coming from the subtropical ridges over the eastern Pacific Ocean & Atlantic Ocean in the northern hemisphere.<br /></p><h2 style="text-align: left;">Why this does not happen in other continents</h2><p>This does not happen in other continents because of unfavorable arrangements of landmasses & seas/oceans. I will give details for each continent in turn.</p><h3 style="text-align: left;">Eurasia</h3><p>In the northern hemisphere, Eurasia & the Pacific Ocean are much wider (with respect to the range of longitudes) than North America & the Atlantic Ocean, so the conveyor belt effect is lost there; this point is amplified by the much stronger system of high pressure forming due to the settling of cold dry air over the continent in the winter half of the year. Additionally, the Indian Ocean (which would supply warm moist air) is not far enough from the equator and there are too many mountains in between for the Indian Ocean to function analogously to the Gulf of Mexico.</p><h3 style="text-align: left;">South America</h3><p>The east coast of South America in the middle latitudes would refer to the east coast of Argentina. There is no major body of water immediately to the north (toward the equator) of Argentina analogous to the Gulf of Mexico, so although the subtropical ridge over the Atlantic Ocean to the west of South Africa is somewhat close by, the prevailing westerlies are largely dry by the time they reach Argentina and have no way of replenishing moisture & warmth before reaching the east coast.</p><h3 style="text-align: left;">Africa</h3><p>In the southern hemisphere, Africa does not extend much into the middle latitudes. Thus, this issue is moot there.</p><h3 style="text-align: left;">Oceania</h3><p>Oceania does not extend much into the middle latitudes and is surrounded by much more water, keeping the temperatures more moderate anyway (so there is less opportunity for big temperature contrasts between land & water to form, which would lead to stronger winter storms). Additionally, the Pacific Ocean in the southern hemisphere is much wider (with respect to the range of longitudes) than the Atlantic Ocean in the northern hemisphere, so the conveyor belt effect is lost there.</p>http://dasublogbyprashanth.blogspot.com/2024/03/progression-of-winter-storms-across.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-1064789301039958919Fri, 02 Feb 2024 16:29:00 +00002024-02-02T11:29:17.698-05:00conferenceDavisdisabilitypoliticspostpresentationsocial policytransportationMy time at the TRB 2024 Annual MeetingLast month, I attended the TRB 2024 Annual Meeting. The conference, which was held in DC, was a lot of fun. The graduate student researcher working with me was able to present our work as a poster. Additionally, although I didn't meet as many people whom I had not met before, this was a good opportunity for me to deepen connections with people with whom I had connected more briefly in past conferences in person or remotely. In particular, these included people working like me at the intersection of transportation & disability as well as people working in the autonomous vehicle industry. Strengthening connections with people in the former group was especially important to me because of the relevance of my work, how few of us there are in the US, and being able to feel like I am part of a group of like-minded academic researchers (given that I lead so much of my work essentially alone in the context of academic researchers). Strengthening connections with people in the latter group was important to me because of recent turbulence in the space of autonomous vehicles and the need for consistent pushes for inclusion of people with disabilities as users of such vehicles.<br />http://dasublogbyprashanth.blogspot.com/2024/02/my-time-at-trb-2024-annual-meeting.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-2999912643016071717Mon, 01 Jan 2024 14:52:00 +00002024-01-01T09:52:52.495-05:00changeclimateFOLLOW-UPscienceVariations of Coastal Monsoon Climates with Latitude<p>I have learned about different aspects of the Earth's climate and shared what I've learned over 5 posts in 2022, including basic intuitions for why different climates occur in different regions [<a href="https://dasublogbyprashanth.blogspot.com/2023/04/my-rough-intuition-of-climate.html">LINK</a>], my assessment of the deficiencies of the Trewartha climate classification system [<a href="https://dasublogbyprashanth.blogspot.com/2023/05/follow-up-my-rough-intuition-of-climate.html">LINK</a>], what I would change about the Trewartha climate classification system [<a href="https://dasublogbyprashanth.blogspot.com/2023/07/more-on-climate-categorization.html">LINK</a>], how my proposed changes to the Trewartha climate classification system can be applied to understand what climates occur where in middle latitudes [<a href="https://dasublogbyprashanth.blogspot.com/2023/10/where-different-climate-types-do-or-do.html">LINK</a>], and why popular understanding of the effects of the Gulf Stream over the Atlantic Ocean on the climate of Europe is incorrect in many ways [<a href="https://dasublogbyprashanth.blogspot.com/2023/12/myth-of-effects-of-gulf-stream-on.html">LINK</a>]. Ultimately, my learning about different aspects of the climates of the world was done with the personal aim of understanding why cities on opposite coasts of the US at the same latitudes have such different climates, with those on the west coast having characteristically mild to hot arid rainless summers & cool (but not cold) rainy winters and those on the east coast having typically warm or hot humid rainy summers & cool or cold slightly drier but still rainy or snowy winters. I did learn about that to a great extent, but as I learned more, I started to question whether my previous intuitions (from when I started learning about different climates) were correct. Follow the jump to see more and the resolution to this problem. <b>Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.</b><br></p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2024/01/variations-of-coastal-monsoon-climates.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2024/01/variations-of-coastal-monsoon-climates.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-7466288905781000940Sat, 02 Dec 2023 18:08:00 +00002023-12-02T13:08:51.089-05:00changeclimateFOLLOW-UPscienceMyth of the Effects of the Gulf Stream on the Climate of Europe<p>Recently, I happened to come across articles online [<a href="https://www.americanscientist.org/article/the-source-of-europes-mild-climate">LINK</a>] clarifying that there are some competing explanations for why the climate of Europe immediately to the east of the Atlantic Ocean is milder in the winter than the climate of North America at similar latitudes immediately to the west of the Atlantic Ocean but that the Gulf Stream in the Atlantic Ocean only plays a minimal role. It got me to think whether I have unwittingly repeated the myth of the importance of the Gulf Stream for the climate of Europe in recent blog posts like my most recent one about climate types [<a href="https://dasublogbyprashanth.blogspot.com/2023/10/where-different-climate-types-do-or-do.html">LINK</a>]. Having gone through that blog post, I can say more confidently that I did not repeat that myth with respect to the big picture of Europe's climate, but there may have been certain aspects of Europe's climate (especially in eastern Europe) for which I overstated the effect of the Gulf Stream, so I want to set the record straight in an effort to not spread known misinformation or myths as if they were facts. Follow the jump to see more details. <b>Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.</b></p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2023/12/myth-of-effects-of-gulf-stream-on.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2023/12/myth-of-effects-of-gulf-stream-on.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-8996487395635871902Wed, 01 Nov 2023 15:22:00 +00002023-11-01T11:22:06.827-04:00calculusmathematicsphysicsscienceContravariant and Covariant Objects in Matrix Notation<p>For many years when and since I was in college, I wondered whether it might be possible to consistently represent contravariant & covariant objects using vector & matrix notation. In particular, when I learned about the idea of covariant representations of [invariant] vectors being duals to contravariant representations of [invariant] vectors, meaning that if a contravariant representation of a [invariant] vector can be seen as a column vector, then a covariant representation of a [invariant] vector can be seen as a row vector, I wondered how it would be possible to represent the fully covariant metric tensor as a metric tensor if it multiplies a contravariant representation of a [invariant] vector (i.e. a column vector) to yield a covariant representation of a [invariant] vector (i.e. a row vector), especially as traditionally in linear algebra, a matrix acting on a column vector yields another column vector (while transposition, though linear in the sense of respecting addition and scalar multiplication, cannot be represented simply as the action of another matrix). At various points, I've wondered if this means that fully contravariant or fully covariant representations of multi-index tensors should be represented as columns of columns or rows of rows, and I've tried to play around with these ideas more. This post is not the first to explore such ideas even online, as I came across notes online by Viktor T. Toth [<a href="https://www.vttoth.com/CMS/physics-notes/139-on-tensors-and-their-matrix-representations">LINK</a>], but this post is my attempt to flesh out these ideas further. Follow the jump to see more. Throughout this post, I will work with the notation of 2 spatial indices, in which the fully covariant representation of the metric tensor \( g_{ij} = \vec{e}_{i} \cdot \vec{e}_{j} \) might not be Euclidean, where indices will use English letters \( i, j, k, \ldots \in \{1, 2\} \), where superscripts do not imply exponents, and where multiple superscripts do not imply single numbers (for example, \( g_{12} \) is the fully covariant component of the metric tensor with first index 1 and second index 2, not the covariant component at index 12 of a single-index tensor (vector)); extensions to spacetime (where the convention is to use indices labeled by Greek letters) and in particular to 3 spatial + 1 temporal dimensions are trivial. Additionally, Einstein summation will be assumed, and all tensors (including vectors & scalars) are assumed to be real-valued. Finally, I will do my best to ensure that when indices are raised or lowered, the ordering of indices is clear (as examples, distinguishing \( T^{i}_{\, j} \) from \( T_{i}^{\, j} \) instead of ambiguously using \( T^{i}_{j} \) or \( T^{j}_{i} \)), but this will depend on the quality of LaTeX rendering in this post.<br></p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2023/11/contravariant-and-covariant-objects-in.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2023/11/contravariant-and-covariant-objects-in.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-778649482013545138Thu, 12 Oct 2023 16:40:00 +00002023-10-12T12:40:35.062-04:00changeclimateFOLLOW-UPscienceWhere Different Climate Types Do or Do Not Occur in Middle Latitudes<p>As a follow-up to a recent post [<a href="https://dasublogbyprashanth.blogspot.com/2023/07/more-on-climate-categorization.html">LINK</a>], I've been able to somewhat comprehensively catalogue & categorize climates of different population centers roughly in the middle latitudes (23-67 degrees, which are outside of both the tropics and the polar circles) in each continent to understand why certain climate types occur in certain continents and not others. This post explains that further. <b>Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.</b></p><p>For the rest of this post, I will use the modification of the Trewartha categorization that I explained in the recent post. Each climate label in this categorization has four letters, with the first being uppercase and the remaining 3 being lowercase.</p><h2 style="text-align: left;">Climate categorization definitions <br></h2><h3 style="text-align: left;">First letter<br></h3><p>The first letter can be A, B, C, D, E, or F. Climate types have the first letter F (polar) if if the mean temperature of the hottest month is less than 10 degrees Celsius. Climate types with the first letter B (semi-arid or arid) are defined based on precipitation thresholds regardless of mean temperatures each month; this will be explained soon. If a climate type does not meet a precipitation threshold for the first letter B, then the first letter is A (tropical) if the mean temperature of the coldest month is at least 18 degrees Celsius, C (subtropical) if the mean temperature of the coldest month is less than 18 degrees Celsius but 8-12 months in the year have a mean temperature of at least 10 degrees Celsius, D (intermediate) if 4-7 months in the year have a mean temperature of at least
10 degrees Celsius (which means the mean temperature of the remaining 5-8 months, including the coldest month, must be less than 10 degrees Celsius), or E (subpolar) if 1-3 months in the year have a mean temperature of at least
10 degrees Celsius (which means the mean temperature of the remaining 9 to 11 months, including the coldest month, must be less than 10 degrees
Celsius). These conditions are the same as in the Trewartha categorization.<br></p><p>The precipitation threshold for climates with the first letter B is \( H = 10(T - 10) + 300S \), where \( T \) is the mean annual temperature in degrees Celsius and \( S \) is the fraction (between 0 and 1) of yearly precipitation that occurs in the summer half of the year (inclusively between April through September in the northern hemisphere, or October through March in the southern hemisphere). These conditions are the same as in the Trewartha categorization.</p><p>If the mean temperature of the hottest month is less than 10 degrees Celsius, then the climate type automatically has the first letter F (polar). This holds even if the climate type would otherwise qualify for the first letter B (arid or semi-arid), because close enough to the pole, the air is too cold to hold much moisture anyway, and features of the vegetation are more influenced by the coldness than the dryness per se; more precisely, as a climate becomes colder, less moisture from the ground is lost to evapotranspiration, so the amount of precipitation needed per year to avoid a climate type with the first letter B (arid or semi-arid) is lessened anyway. If a climate type does not qualify for the first letter F (polar) but the yearly precipitation is \( P \leq 2H \), then the climate type has the first letter B (arid or semi-arid); otherwise, the first letter must be A (tropical), C (subtropical), D (intermediate), or E (subpolar) depending on the mean temperatures of the hottest and coldest months and the number of months with mean temperatures of at least 10 degrees Celsius. These conditions are the same as in the Trewartha categorization.</p><h3 style="text-align: left;">Second letter</h3><p>The second letter depends on the first letter. If the first letter is B (arid or semi-arid), then the second letter denotes whether the climate is either semi-arid or arid. A semi-arid (steppe) climate, with the second letter being 's', has \( H < P \leq 2H \). An arid (desert) climate, with the second letter being 'w', has \( P \leq H \). These conditions are the same as in the Trewartha categorization.</p><p>If the first letter is A (tropical), then the second letter denotes whether the climate is a tropical rainforest climate or a tropical wet-and-dry climate. If at least 10 months each have at least 60 millimeters of precipitation, then the second letter is 'r' (tropical rainforest climate). Otherwise, the climate is a tropical wet-and-dry climate; the second letter is 'w' if the dry season is during the winter half of the year or 's' if the dry season is during the summer half of the year. These conditions are the same as in the Trewartha categorization.</p><p>If the first letter is F (polar), then the second letter denotes whether the climate is a tundra climate or an ice cap climate. If the mean temperature of the hottest month is at least 0 degrees Celsius but below 10 degrees Celsius, then the second letter is 't' (tundra climate); otherwise, as every month has a mean temperature below 0 degrees Celsius (implying permanent ice where water is present), the second letter is 'i' (ice cap climate). These conditions are the same as in the Trewartha categorization.</p><p>If the first letter is C (subtropical), D (intermediate), or E (subpolar), then the second letter denotes whether the climate has a dry summer or generally uniform precipitation through the year, as dry summers indicate vulnerability to droughts, wildfires, and related natural disasters. This is the main way that my modification differs from the original Trewartha categorization. If all of the following conditions hold, namely that the driest month is in the summer half of the year, the wettest month is in the winter half of the year, the wettest month has at least 3 times the mean precipitation as the driest month, and the summer half of the year has at least 3 months where the mean precipitation is at most 40 millimeters (including the driest month, by definition), then the second letter is 's', indicating a dry summer. Otherwise, the second letter is 'f', indicating a humid summer. Unlike the Köppen categorization, neither the Trewartha categorization nor my modification to it allow for the second letter to be 'w', which would indicate dry winters, when the first letter is C (subtropical), D (intermediate), or E (subpolar), and this is for two related reasons. First, there is no particular climactic or ecological feature unique to places with dry winters, as the dryness corresponds to the time of the year with the least amount of sunlight and the lowest temperatures; this is unlike when the second letter is 's' (dry summer), because dryness in the summer allows for temperatures to become arbitrarily high in the absence of precipitation (even if average temperatures are somewhat more moderate, as may happen when moisture comes in other forms like fog), which can easily lead to wildfires as is characteristic of places that have climate types with the second letter 's' (dry summer). Second, the threshold \( H \) for precipitation for a climate type to have the first letter B (arid or semi-arid) is defined to depend not only on the average temperature for the year but also on the percentage of precipitation in the summer half of the year, because evapotranspiration rates increase as the temperature increases. This means that for two places that have the same average temperature for the year, the one that has a greater percentage of precipitation occurring in the summer half of the year will experience more evapotranspiration because the temperatures in that half of the year are higher, so the climate type there is more likely to have the first letter B (arid or semi-arid) under the Trewartha categorization or my modification of it even if the Köppen categorization would make the first letter C (subtropical) or D (continental) with the second letter 'w' (dry winter), because that becomes the more salient feature of such a climate; if the climate type doesn't have the first letter B (arid or semi-arid), then there is less of a salient difference in the climates & ecologies of areas with climate types with the first letter C (subtropical), D (intermediate), or E (subpolar) and the second letter 'f' (humid summer) whether the winter is dry or not.<br></p><h3 style="text-align: left;">Third and fourth letters</h3><p>The third and fourth letters are more needed in my modification of the Trewartha categorization for comparison of different climates to make sense, but the actual letters are the same (although at or below 0 degrees Celsius, I may have shifted things by 0.1 degree Celsius). In particular, the third letter indicates the mean temperature of the hottest month and the fourth letter indicates the mean temperature of the coldest month. Both the third and fourth letters come from the following set of letters. These letters are 'i' for temperatures of at least 35 degrees Celsius, 'h' for temperatures of at least 28 degrees Celsius but less than 35 degrees Celsius, 'a' for temperatures of at least 22.2 degrees Celsius but less than 28 degrees Celsius, 'b' for temperatures of at least 18 degrees Celsius but less than 22.2 degrees Celsius, 'l' for temperatures of at least 10 degrees Celsius but less than 18 degrees Celsius, 'k' for temperatures of at least 0 degrees Celsius but less than 10 degrees Celsius, 'o' for temperatures of at least -10 degrees Celsius but less than 0 degrees Celsius, 'c' for temperatures of at least -25 degrees Celsius but less than -10 degrees Celsius, 'd' for temperatures of at least -40 degrees Celsius but less than -25 degrees Celsius, and 'e' for temperatures less than -40 degrees Celsius. Thus, when I speak of the temperature-indicative third or fourth letters being higher or lower when comparing two climate types, such statements refer to this temperature scale.</p><h2 style="text-align: left;">Effects of mountains</h2><p>Frequently, when considering transitions between climate types, I will refer to mountains lying in some direction relative to an area with a climate type and not further discuss the climate types on or across those mountains. Mountains have their own, typically polar-like, climate types and significantly break up continuity between otherwise geographically adjacent climate types in a given continent. In particular, as I discussed in a previous post [<a href="https://dasublogbyprashanth.blogspot.com/2023/04/my-rough-intuition-of-climate.html">LINK</a>], a mountain range that lies roughly along a line of longitude (meridian) creates a significant rain shadow that will depend on the direction of the prevailing winds; at more tropical latitudes, the prevailing winds are the trade winds going from east to west, so areas east of a mountain will get much more precipitation than areas west of a mountain, while at middle latitudes, the prevailing winds are the prevailing westerlies going from west to east, so areas west of a mountain will get much more precipitation than areas east of a mountain. A mountain range that lies roughly along a line of latitude usually will not create a significant rain shadow unless there is a specific warm ocean current driving wind from the equator to a pole roughly along a line of longitude (meridian), but it will block warm air going from the equator toward a pole and cold air going from a pole toward the equator; thus, it is more likely to create sharper transitions in temperature profiles (third & fourth letters in the climate type), and if this affects the position of the subtropical ridge especially around the west coast of a continent, then it can further create sharper transitions between precipitation profiles based on whether summers are dry.<br></p><p>Follow the jump to see further discussion of actual climate type occurrences. I will focus mostly on climates with the first letter being C (subtropical), D (intermediate), or E (subpolar), as those are the most common in the middle latitudes; there will be some discussion of climates with the first letter being B (arid or semi-arid), as there are many areas in middle latitudes that have semi-arid or arid climates, and there will be brief discussion of climates with the first letter being A (tropical) or F (polar), as those are rare outside of the tropical or polar regions respectively. I should note that this post contains two large biases in sampling. First, I have only considered population centers that are clear on Google Maps. Therefore, some of these climates may actually be more widespread in area than they look based only on where people live. Second, as I'm most familiar with North America, I may have picked more small or mid-sized cities in North America compared to other continents. Therefore, some of these climates may actually be more widespread in other continents than this post may seem to suggest.<br></p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2023/10/where-different-climate-types-do-or-do.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2023/10/where-different-climate-types-do-or-do.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-5187355806607841632Mon, 18 Sep 2023 15:00:00 +00002023-09-18T11:00:00.149-04:00ethicsindiaMuslimracial profilingreligionsurveillanceterroristUSwarfareStand-Up Comedy and Emotional Resonance<p>I recently read an article in <i>The New Yorker</i> magazine [<a href="https://www.newyorker.com/news/annals-of-communications/hasan-minhajs-emotional-truths">LINK</a>] about how the stand-up comedian Hasan Minhaj significantly exaggerated or conflated stories in his recent big stand-up comedy routines. In particular, these stories were about instances of racism or Islamophobia, including being the victim of police brutality, being part of a mosque that was infiltrated by an FBI agent, and being sent a mysterious powder that led to his child's hospitalization, that either didn't happen at all or were significantly exaggerated. As someone who has liked his work in the past and who could identify to some degree with his stand-up comedy material based on experiences as the child of immigrants from India, I found these allegations quite troubling, yet I also found myself struggling to articulate exactly why I found these allegations to be so troubling. This post is my attempt, in the current zeitgeist (as this is a very new story and new details could soon arise that would make this post irrelevant or incorrect), to make sense of these things. Follow the jump to see more.</p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2023/09/stand-up-comedy-and-emotional-resonance.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2023/09/stand-up-comedy-and-emotional-resonance.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-6662871559494495186Sun, 06 Aug 2023 20:10:00 +00002023-08-06T16:10:58.753-04:00Movie ReviewOppenheimerMovie Review: Oppenheimer<p>I should note that the last movie review on this blog was almost exactly 12 years ago, when I had watched the movie <i>Source Code</i> [<a href="https://dasublogbyprashanth.blogspot.com/2011/08/movie-review-source-code.html">LINK</a>]; going back to that post let me cringe a little again at my writing style as a college student. In any case, although I have watched many movies since then but haven't felt compelled to review them for this blog, I felt a little more compelled to do so after recently watching the movie <i>Oppenheimer</i> in IMAX (though not 170 mm IMAX), because of the historical & scientific significance as well as the hype around its release. That movie is essentially a dramatized adaptation of the book <i>American Prometheus</i> by Kai Bird & Martin J. Sherwin (which I haven't yet read), covering the life of J. Robert Oppenheimer during his career as a physicist & developer of nuclear weapons for the US and particularly focusing on his involvement in the Manhattan Project & his subsequently being stripped of a security clearance.</p><p>There were a few things that I liked about the movie. I understand that Edward Teller was ostracized by the scientific community by giving testimony that would further bolster investigators turning scientific disputes & personal friction between Teller & Oppenheimer into a reason for claiming Oppenheimer to be a national security risk, but considering that Teller's further ostracism came more when he further dug into developing nuclear arsenals & using nuclear weapons in absurd ways that signaled a weird lust for nuclear explosions, I appreciated that the movie stuck with Teller's role in the Manhattan Project (without letting later views of Teller color his portrayal during the time of the Manhattan Project) and made explicit his real-life testimony praising Oppenheimer's integrity & ultimate loyalty to the US (as opposed to other countries). I also appreciated how the movie made clear that arguments against the use of nuclear weapons after the actual bombing of Japan could be seen as facile or hypocritical when compared to similar arguments before the initial test in Los Alamos. In particular, Oppenheimer initially rationalized concerns about the US having access to the destructive power of nuclear weapons by recognizing the far greater threat to humanity of Nazi Germany getting & using such weapons first, so later claims of being disgusted by their use need to be shaped with a lot more nuance than Oppenheimer actually provided. Additionally, as I have read most of the Bhagavadgītā, I could see that Oppenheimer quoting Kṛṣṇa's line (repeating the translation that Oppenheimer used) "I am become Death, the shatterer of worlds" is arguably a misunderstanding of the philosophical implication, considering that Truman essentially had to correct Oppenheimer in the same way that Kṛṣṇa had to correct Arjuna: the US (with the president, at that time Truman, as the symbolic executor), like Kṛṣṇa, was the entity with the will to destroy, while Oppenheimer/Arjuna was the human instrument and the nuclear/celestial weapons were the insentient instruments. I wonder if more people will recognize this and thus not blindly praise Oppenheimer just for quoting the Bhagavadgītā.</p><p>There has been a lot of controversy, especially in India and also among Hindus outside of South Asia, about the depiction of a Sanskrit copy of the Bhagavadgītā during a sex scene being sacrilegious. I'm not religious, and I knew of Oppenheimer's fascination with Hindu mysticism, so I initially gave the director the benefit of the doubt that it may perhaps reflect some combined mystical view of sex & spirituality by Oppenheimer in real life, especially given that reactions about these things tend to be much harsher in India than in the US. Now that I have watched that scene, I can say that the presence of a Sanskrit copy of the Bhagavadgītā added nothing to the sex scene or to the understanding of Oppenheimer's life and was probably not something that happened in real life, so it seems to be in gratuitously bad taste. Moreover, I felt like the scenes where Oppenheimer used Kṛṣṇa's aforementioned line, including but not limited to the sex scene, made it feel cheap & unnecessary; in particular, using it first in the sex scene robbed it of the gravitas that it could have had when portraying the nuclear test explosion.</p><p>Overall, perhaps because I had some familiarity with the historical events, I felt like the director tried too hard to make an ultimately simply story about the life of a complicated person seem more complicated (as a story) & visually engaging than necessary. It is perhaps damning to the movie that I felt that despite having seen trailers where the cast of the movie encouraged people to watch it in an IMAX movie theater, I felt that I could have enjoyed it equally on a small screen in an airplane. As an example, I could see that the director was in many scenes trying to visually depict the turmoil in & tortured state of Oppenheimer's mind, but the effects often felt too overwrought with crazy pictures & loud sounds. I thus would only recommend it to people who may then be inspired to read the book (as I myself have yet to do).</p><p>On another note, there was a scene with a graph on a chalkboard for one of Oppenheimer's lectures showing a single particle tunneling quantum mechanically through a flat barrier in 1 dimension, but the wavefunction was so badly drawn that it didn't seem to show exponential suppression in space in the region of the barrier. When I saw that scene, I immediately thought that if I were a TA for a class in which he was a student and he had submitted that as part of a homework assignment, I would have deducted points.<br /></p>http://dasublogbyprashanth.blogspot.com/2023/08/movie-review-oppenheimer.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-2936607837227038641Sat, 01 Jul 2023 13:21:00 +00002023-07-01T09:21:00.475-04:00changeclimateFOLLOW-UPscienceMore on Climate Categorization<p>This post is essentially a follow-up to a recent post [<a href="https://dasublogbyprashanth.blogspot.com/2023/05/follow-up-my-rough-intuition-of-climate.html">LINK</a>] about the Köppen & Trewartha categorizations of climates; that post was in turn a follow-up to a recent post [<a href="https://dasublogbyprashanth.blogspot.com/2023/04/my-rough-intuition-of-climate.html">LINK</a>] about my intuitions of various climates. This post will discuss, more systematically & in more detail, the climates that are impossible or not typically observed in the Trewartha categorization even when consistently using the third & fourth letters to specify the hottest & coldest mean monthly temperatures respectively, the pros & cons of the Trewartha categorization, and a proposal that I thought of to address some of the cons of the Trewartha categorization. <b>Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.</b> Follow the jump to see more.</p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2023/07/more-on-climate-categorization.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2023/07/more-on-climate-categorization.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-1574468016959209086Sun, 11 Jun 2023 18:46:00 +00002023-06-11T14:46:41.750-04:00Book ReviewFermat's Last Theoremgame theoryHow Not to Be WrongJordan Ellenbergmathematicsphilosophypoliticsreligionstatistical mechanicsBook Review: "How Not to Be Wrong" by Jordan Ellenberg<p>I've recently read the book <i>How Not to Be Wrong</i> by Jordan Ellenberg. As the author states in the introduction, it is an exposition of simple yet profound ideas in mathematics, meant for laypeople. Topics include nonlinear phenomena (in opposition to naïve linear extrapolation), probability, Bayesian reasoning, and statistical testing of hypotheses. All chapters refer to many examples in politics, economics, and everyday life to make the concepts easier for laypeople to digest.</p><p>I found the book to be fairly easy to follow. I can't say that I learned much in terms of concepts, as these are all concepts that I've come across one way or another in school, college, graduate school, or my work now, though I did appreciate the discussion of how conspiracy theorists like to add hypotheses after the fact to make a conspiracy theory harder to fully disprove, how the fact that random fluctuations in many phenomena observed over time are time-reversal invariant implies that the phenomenon of regression toward the mean is also time-reversal invariant in a probabilistic sense, and the intuitive explanations of common causes & common effects in leading to correlations between random variables that are otherwise not causally connected. Additionally, I felt like this book did a better job than the book <i>Algorithms to Live By</i> by Brian Christian & Tom Griffiths (which I have reviewed on this blog before [<a href="https://dasublogbyprashanth.blogspot.com/2022/05/book-review-algorithms-to-live-by-by.html">LINK</a>]) in having some structure in the progression from one chapter to the next and in using topics from earlier chapters in later chapters even though this book, unlike that book, didn't pretend to have a unified message. My only quibbles are the claim that the impossibility of accurately running the fundamental equations describing atmospheric & oceanic dynamics for more than 2 weeks implies impossibility in forecasting through other methods (like machine learning models looking for patterns in weather effects & progression) and the fact that the chapter connecting ideas from probability, geometry, and signal processing (particularly around error correction) took me a fair bit of effort to follow (unlike the other chapters, which tells me that laypeople will likely struggle with that chapter much more). Additionally, I think readers should be aware that the author often makes reference to sports that are mostly popular in the US and to US politics and that the author at a few points espouses more liberal or progressive political views (though I think such espousal is not gratuitous but is done in a way that fits well with broader discussions of assumptions underlying mathematical, political, and legal judgments). Overall, I think the author has done a good job of fulfilling the goal of communicating these ideas to a lay audience, so I recommend this book to anyone who might be interested in these ideas.<br /></p>http://dasublogbyprashanth.blogspot.com/2023/06/book-review-how-not-to-be-wrong-by.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-9037995433658278336Sat, 20 May 2023 21:31:00 +00002023-05-21T02:18:22.697-04:00changeclimateFOLLOW-UPscienceFOLLOW-UP: My Rough Intuition of Climate, Especially in the US<p>The previous post in this blog [<a href="https://dasublogbyprashanth.blogspot.com/2023/04/my-rough-intuition-of-climate.html">LINK</a>] went over my rough intuition of climate, primarily in middle latitudes like those of the US. Most of the broad categories that I described were largely aligned with the Köppen climate classification system (henceforth called the Köppen categorization). However, there is a more recent categorization known as the Trewartha climate classification system (henceforth called the Trewartha categorization) that is supposed to be more representative of middle latitudes like those of the US. Essentially, tropical, desert, and semi-arid climates, as well as polar and ice cap climates, are defined in the same ways between the two categorizations. The differences lie in the definitions of subtropical, continental, and subpolar oceanic/subarctic climates. One benefit of the Trewartha categorization is that it clearly separates
boreal/subpolar climates from other oceanic and continental climates,
whereas the Köppen categorization uses subcategories that could be a little more confusing. However, the definitions of subtropical, oceanic, and continental climates in the Trewartha categorization seem less justifiable to me. Follow the jump to see more details. <b>Again, I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.</b></p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2023/05/follow-up-my-rough-intuition-of-climate.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2023/05/follow-up-my-rough-intuition-of-climate.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-7699225903679515802Sat, 01 Apr 2023 15:38:00 +00002023-04-01T11:38:25.937-04:00changeclimatescienceMy Rough Intuition of Climate, Especially in the US<p>For a long time, I had wondered why the climates of San Francisco, Sacramento, and Los Angeles are so different from those respectively of Richmond, DC, and Atlanta. I had read a few articles on Wikipedia on occasion, so I got a sense that it has to do in part with different ocean currents; this made sense to me, as I had become very comfortable (growing up in the DC area) with the warm waters at beaches along the East Coast in the summer, and I was always surprised by the comparatively much colder waters at beaches along the West Coast whenever I'd visit California even in the summer. I knew though that this wasn't the whole story, and I was surprised to see, for example, that even in South America, South Africa, Western Europe versus East Asia, and Australia, there were very similar contrasts in climates between cities along west versus east coasts in the middle latitudes. This made me more curious about the reasons for these similarities, so I recently went down a rabbit hole of Wikipedia articles to learn more and form an intuition about why different places have different kinds of climates. Follow the jump to see my explanation. <b>I am not a trained climatologist or meteorologist; I can't guarantee that this information is accurate, and I can only say that my intuitions seem through my limited understanding to align with superficial aspects of more detailed explanations.</b> I'm just putting this out there in case this intuition is helpful to anyone else <b>as a starting point</b> to learn more (and I recognize that an incorrect initial intuition could hurt rather than help when trying to learn more).</p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2023/04/my-rough-intuition-of-climate.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2023/04/my-rough-intuition-of-climate.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-8304123550227744028Wed, 01 Mar 2023 16:22:00 +00002023-03-03T11:26:16.754-05:00agricultural companyBook ReviewcommunicationcultureEuropeevolutionhistoryindiamisrepresentationnative americanphilosophyreligionSapiensscienceYuval Noah HarariBook Review: "Sapiens" by Yuval Noah Harari<p>I've recently read the book <i>Sapiens</i> by Yuval Noah Harari; this was highly recommended to me 7 years ago by a friend in graduate school with whom I had enthusiastic discussions about the material in the book, but I hadn't gotten a chance to read the book until now. This book is supposed to be a history of humans, going from an evolutionary perspective for the first 2 million years since the genus Homo became distinct and then getting into the developments of language, collective myths, agriculture, urbanization, and industry. Even my summary of the flow of the book contains my opinion about its progression (though I should note that the "parts" that I speak of overlap with but aren't identical to the 4 parts that formally divide the book): the beginning part of the book seems to be a serious discussion of evolution, language, and the advent of agriculture, the middle part tries to be serious but has more inconsistencies that I find problematic, and the last part seems more clearly to be more like a "pop-history" type of book with less rigorous speculation (so I read it with a lighter heart even if the author didn't intend it that way and I therefore heavily discounted it in my overall opinion of the book); furthermore, the parts about scientific development since 1500 can be understood more clearly in other books.<br /></p><p>There were several things that I learned from the book and several ways in which the book forced me to consider a different perspective. These are as follows, in no particular order. First, I learned that the further development of language to be able to convey detailed information, gossip, abstract ideas, and fictions happened 70,000 years ago and coincided with humans becoming dominant in the food chain and in their spread across the world. Second, I had always uncritically believed in the advent of agriculture and later urbanization as a good thing, to the extent that I've recently sometimes wondered (without being particularly informed about history & sociology) whether clashes between the Mongol invaders & Hindu natives (in contrast to earlier arrivals in India of urban Muslim traders from Mesopotamia) as well as between the expectations of Western government & the reality of the House of Saud represent clashes between urbanized cultures that have developed to a great degree versus nomadic cultures that have endured much harsher conditions and only wish to plunder the cities for wealth without care for more refined aspects of urban cultures; this book forced me to consider that individuals within nomadic tribes had much more varied diets & activities within each day, that the first few millennia of the transition to agriculture may have led to a lot of suffering compared to what came immediately before, and that the domestication of wheat can be reinterpreted as a mutalistic domestication of wheat & humans. Third, I learned that empires might be defined only by the number of culturally distinct tribes under their yokes and the flexibility of their borders in expansion, not by population or area per se. Fourth, while I had some familiarity with how capitalism & European imperialism fueled each other and with the use of scientifically-inspired racism as a justification for European colonization, I didn't have a good sense for how these all tied together until I read this book, especially in the context of scientific voyages only being funded if the scientists could tag along with naval officers ordered to colonize the lands they would reach. Fifth, I appreciated the distinction between ancient & medieval empires which grew in predictable ways by absorbing naval territories versus early modern European empires which grew unpredictably across the world with long-distance seafaring. Sixth, I appreciated the explicit call to attention about how liberal humanistic political philosophies, which profess to be atheistic in themselves & multireligious in the sense of tolerance, cast freedom & political empowerment in terms of a special nature of individuals that is drawn directly from Christian notions of creation & individual souls (though the concepts of creation & individual souls aren't unique to Christianity among traditional religions); this is something that I've pondered before but have typically glossed over, so I appreciated being challenged in this way.</p><p>There were a few points that I was happy to see in the sense of agreeing with those worldviews. These include the ideas of collective myths (not only in traditional religions but in the systems of trust that underlie monetary systems & democracies), historical predictions leading to self-fulfilling or self-negating prophecies, the existence of hierarchies of some form in almost all societies larger than about 150 people (as I've wondered, for example, if a solution to the problem of inflation coming from an immediate cash payout to everyone in a universal basic income plan would be to sprinkle it randomly upon different people at different times to ensure that the economic system doesn't stray too far from its previous state and can better respond to bigger numbers of people getting such payouts later even if that creates an effective hierarchy between those who get such payouts at a given time and those who don't), and the ways that even cultures free of external pressures can develop internal contradictions that in turn can lead to continued development of the culture as a unified entity or a split of the culture into multiple descendants. On a lighter note, I also enjoyed seeing the author, in the otherwise problematic speculation about science, point out that science fiction can only rarely, if ever, attempt to describe what would truly be alien experiences to humans, and that most science fiction stories ultimately revolve around myths & social conflicts that in one form or another have been recognizable for millennia, which leads me to the conclusion that there is no reason beyond snobbery to claim that <i>Star Trek</i> is science fiction while <i>Star Wars</i> supposedly is not (because if science fiction is defined as only portraying truly alien experiences in encounters with new technology or new intelligent species that aren't just thinly-veiled allegories for known interactions among human groups, there may only be a few books, movies, and TV series that may be called "science fiction", perhaps including <i>Black Mirror</i> or <i>2001: A Space Odyssey</i>, and those works are rare probably exactly because readers or viewers would find them less relatable).<br /></p><p>There were a few specific stories that I liked reading. One was of the Chinese seafarer Zheng He, as it shows that Chinese seafaring technology was as advanced as European seafaring technology around 1500 but China simply didn't have the same ambition to conquer faraway lands through seafaring. The other was of how the accompaniment of Hernán Cortés by Aztec people carrying burning incense sticks near him convinced him that the "primitive" Aztecs were treating him as a deity but was actually because he had terrible body odor due to bad hygiene, as it is a funny story, it shows that the Aztecs, immediately upon encountering Europeans, figured out what other contemporary peoples of Asia & Africa had known for many centuries (leading those peoples to set up quarantine areas for European visitors at ports), namely that Europeans had bad hygiene at that time, and it shows how the self-delusion of European winners of such conflicts (in this case Cortés believing that he was being treated as a deity so the Aztecs must have been "primitive") could persist in "official" historical narratives for many centuries.</p><p>Beyond the problematic historiography (especially ignoring the way that so many consequential scientific discoveries were made in Europe individually by people who were independently wealthy while also not clearly explaining which technological discoveries were systematically funded & used by governments, though those parts could be fixed with better writing) and excessively serious-sounding speculation about science in the last few chapters & sprinkled elsewhere in the book, there were three major points of disagreement that I had with the author, in the sense that I believe that these points strike at the fundamental arguments of the book. These are as follows.</p><p>First, the author makes a big deal throughout the book about how the global unity in understanding of political, economic, and other norms that has emerged in the last 500 years is unprecedented in all prior years of human existence. My counterargument is that this argument depends too much on the specific way that previous interactions between cultures went or on the fact that certain cultures happened to not interact. It will be based primarily on [Native] American and European cultures before and around the time of their first contact in the middle of the second millennium, as the author makes a big deal about how American tribes were among the groups that were totally isolated from the continuum of groups across Africa, Asia, and Europe (with Australian tribes being among the others). As the author argues, the lack of contact before may well have been because of a combination of technological limitations along with limitations in cultural ambition. However, in a counterfactual situation where, for example, English people looking to start local democratic governments met on a truly equal footing with Iroquois people who embodied the spirit of democracy in their local confederated governments, there is no specific reason to believe that they would have been talking past each other; the author's conception of "global unity" as a phenomenon that developed in the last 500 years with no precedent seems to depend too strongly on peoples having met or being aware of each other's existence and not enough on actual similarities between each other's cultures. Additionally, the example of Hernán Cortés and the incense sticks, along with the example (not in the book) of the origin of the ethnic slur "Indian giver" from a deliberate misunderstanding of Native Americans' attempts to barter with Europeans as gifts that were then demanded to be returned, shows that the author's view of the establishment of "global unity" depended strongly on the actual course of history (in this case Europeans deliberately ignoring what Native Americans were telling them) and not on the broader cultural similarities already present. This dependence on the actual course of history makes this a hindsight-based account that the author supposedly disclaims, making the author rather hypocritical.</p><p>Second, in later parts of the book, when the author discusses the reasons for European armies so easily conquering peoples in faraway lands, the author puts a lot of stock in the idea that success was due to the European drive for exploration of the unknown, even before that commitment to exploration started bearing systematic fruit in the forms of scientific discovery or technological advancement; conversely, the author briefly mentions and otherwise glosses over the role of more effective forms of social organization & discipline in those armies. This seems to completely undermine previous chapters in the book that so clearly emphasized the ways that communication of collective myths could lead to new forms of social organization. Perhaps this seeming contradiction can be resolved by interpreting the "drive to explore the unknown" as extending to European armies systematically developing new ways, including new forms of organization of their own armies, of dealing with unknown peoples whom they wish to conquer. However, this seems like a stupid semantic difference and again seems like the author is engaging excessively in analysis from narrow hindsight, contrary to the author's own stated claims. (<b>UPDATE</b>: A related point is about how the author implies that the drive for European colonists to learn about unknown cultures & explore how to systematically conquer unknown peoples led them to use what they learned about these cultures to systematically deepen existing divisions or create new divisions. I could agree that Europeans were the first to do this so systematically or so tightly coupled to the seemingly more noble goal of learning things that weren't known to them. However, I cannot agree with the idea that Europeans were the first to exploit & inflame divisions or engage in proxy wars. as ancient Egyptian kingdoms were known to have done this to the Assyrian Empire. Perhaps this can be forgiven if it turns out that this book was published before we knew about how the ancient Egyptians fomented rebellions, civil wars, and proxy wars in the Assyrian Empire, but in any case, the author's seeming unwillingness to directly assert or refute the idea that European colonists' "drive to explore" specifically included exploration of how to organize themselves better & exploit other people's weaknesses more effectively to conquer those other peoples is a much bigger problem with this book.) Moreover, the author does not attempt to explain why peoples were so consistently conquered at all by Europeans from the perspective of those conquered peoples other than simply stating the claim that those peoples couldn't imagine that their knowledge could be incomplete. I think I could do a better job than the author by imagining a counterfactual situation, using the example of Hernán Cortés encountering the Aztecs: even if the Aztecs were similarly driven as the Spanish by exploration of the unknown and had expanded their empire that way before the Spanish landed in America as historically happened, the only way from the perspective of social dynamics that I can see the Aztecs successfully repelling the invasion is by using their knowledge of dealing with unknown peoples to see through Cortés's lies into his true intentions and organize accordingly, yet there is no guarantee that knowing what to do when attempting to conquer unknown peoples would lead an empire to develop knowledge of what to do at the receiving end of a conquest attempt. Finally, in the specific cases of Europeans interacting with Native Americans, the author in a few places briefly acknowledges the role of infectious disease (used by Europeans sometimes accidentally and other times, as in the case of pox blankets in the 1763 Native American siege of the British-occupied Fort Pitt, intentionally) but otherwise glosses over this in favor of explanations based on exploration of the unknown. Yet, as the examples of Cortés as well as the slur "Indian giver" point out, it is quite plausible that Europeans, seeing how easily Native Americans were wiped out by disease, used this as propaganda to better organize themselves and portray Native Americans as weak (independent of specific technology or ideals about exploring the unknown), and I think it is irresponsible for the author to ignore this obvious possibility.</p><p>Third, there is a whole chapter about the history of traditional religions, including animistic, polytheistic, monotheistic, and atheistic religions. My problem is that the author makes too broad claims about religious trends even though there are so few surviving or [relatively] recently extinguished major traditional religions; the sample sizes are so small as to make the claims unconvincing. The author acknowledges similar problems in other contexts elsewhere but not in that chapter.</p><p>Beyond these issues, I noted several more minor issues at various points in the book. Although some of these issues personally offended me, I still categorize them as minor because I think that deleting the offending passages from the book would not significantly reduce support for or otherwise qualitatively change the main arguments of the book. These are as follows, in no particular order. Even if some of these points raise questions that have no clear answer, I think the author was irresponsible in not addressing the existence of these questions and clearly stating the lack of a clear answer.<br /></p><p>First, the author claims that when big social orders are sustained through collective myths, those collective myths require genuine belief from members of the elite too. Recent news about how Fox News executives & star hosts privately disbelieved claims that the 2020 US presidential election was rigged but knowingly pushed such claims in public just to boost TV ratings & stock prices. On the one hand, perhaps it isn't fair to pin this on the author as this news is much more recent than the publication of the book. On the other hand, I would be curious to see how the author would react to this news now; if the author reacts by claiming to be correct because the degree of true belief among the elite was "always destined to wane at some point" or for some similar reason purely in hindsight, then that tells me that the author's approach is worthless because it would be unfalsifiable.</p><p>Second, the author does such a consistently bad job with the history of India that I have to wonder if the author's research about that specific topic consisted exclusively of books written by British colonizers to portray India to their own benefit. Problems include claims that Indo-Aryans "invaded" (as that word is usually used to imply a systematic movement of an army to bring forth a violent clash, yet there is no historical evidence for such singular violent clashes between ancient Central Asian migrants and South Asian natives), the treatment of caste in the Vedas (as even people who aren't apologists for Brahmins or deniers of the history of caste will recognize a lot of subtlety in the way the Vedas used terms associated now with caste, especially as those castes didn't exist in Indian society until after Vedic times), the treatment of caste in general (conflating jati & varna to claim that the "original" 4 varnas over time split into thousands of jatis, when the reality is much more complicated & less clear), the claim that Brahmins could have "learned" from the KKK how to enforce caste divisions (as Brahmins, especially in South India, were already brutally effective in enforcing caste divisions long before the KKK existed), and the claim that India had no national consciousness before the British Empire (which undermines the author's own prior acknowledgment of the Gupta & Maurya Empires as empires by the author's own definition). Another problematic statement by the author that I am willing to forgive given when the book was published (before the political rise of right-wing Hindu nationalism in India in the last 10 years) is the rhetorical question about whether right-wing Hindu nationalists would do away with all symbols of the Mughal Empire, include those of beauty like the Taj Mahal; the author clearly implied that they wouldn't dare to do so, but that view now seems laughably quaint. Finally, a statement by the author about religion in a broader context yields problems in the context of India. In particular, the author claims that polytheistic kings didn't try to convert conquered peoples or make them destroy temples to their own deities, but historical conflicts between Saiva & Vaisnava kingdoms in South India over religion suggest otherwise; perhaps this point can also be forgiven as a rare exception to the rule.</p><p>Third, the author's definition of an empire in terms of flexibly expanding borders still leads contemporary readers to imagine borders that strictly control the flow of people across them, which I think is historically misleading given the ease with which people could pass across. Even now, people from Mexico & countries in Central America freely pass through the international border between California & Baja California as seasonal migrant workers although the US border is otherwise very strictly controlled.</p><p>Fourth, the author makes claims about the positive cultural developments by empires, but those claims seem incomplete. Additionally, the author tries to distinguish Cyrus of Persia's claim that the empire would benefit all people from the more limited ambitions of Assyrian emperors, but this distinction is not clear at all.</p><p>Fifth, the author claims that interest (in the sense of a guaranteed geometric return on an investment) requires the existence of a currency that is not useful for any other reason. I disagree in principle because livestock and crops, which have historically been used as currencies or items of barter, have the potential to multiply over time. That said, this may be a moot point if there is no evidence for societies having charged interest directly on livestock or crops as forms of currency.</p><p>Sixth, the explanations of why some peoples did not develop agriculture until much later contact by faraway urbanized peoples seems incomplete. I agree with the idea that some peoples settled in areas where plants & animals simply could not be domesticated; the capability to be domesticated by humans is rare among species. However, this does not explain why many native peoples of America and Australia never developed agriculture even though they lived on grasslands that later turned out to support agriculture very easily; in the case of Australia, the author's omission is especially troubling given that the author explains how humans who moved to Australia 45,000 years ago had no problem with destroying the forests that were already there & replacing them with grasslands.<br /></p><p>Seventh, near the end of the book, the author distinguishes ecological destruction from resource scarcity as the more likely cause of future human suffering or extinction. This seems to undermine an earlier chapter in which the author acknowledges that problems with allocating resources to maintain a certain standard of living under certain norms, even if the resources themselves are technically abundant, are more likely to lead to conflict. I wish the author had dealt with this more carefully.</p><p>Eighth, the author writes about the seemingly inexorable trend toward globalization and the way that world war has come to seem implausible since World War II. The failure to predict both the retreat from globalization especially since 2014 as well as the Russian military invasion of Ukraine in 2022 are forgivable given when the book was published. However, I'm more troubled that the author's claims about the implausibility of world war are phrased in ways that are unfalsifiable, as the author can always claim either a different definition of "world war" or a finite time period of validity (since the last world war) that the author would not have previously clearly stated.<br /></p><p>Overall, I think this is still an interesting book, though I wasn't incredibly impressed by it (unlike, for example, my friend from graduate school). I'd recommend it with the caveats discussed above. In any case, as this was among the first books to go on the reading list (for books unrelated to my work) that I made for myself in graduate school, I'm glad to have finally read it.<br /></p>http://dasublogbyprashanth.blogspot.com/2023/03/book-review-sapiens-by-yuval-noah-harari.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-5345177680148060907Wed, 01 Feb 2023 13:36:00 +00002023-02-01T08:36:53.702-05:00conferenceDavisdisabilitypoliticspostpresentationsocial policytransportationMy Time at the 2023 TRB Annual Meeting<p>Last month, I attended the 2023 TRB Annual Meeting. I meant to write this post immediately afterwards, but I became busy with work and with preparing for international travel which I started soon after that and finished just yesterday. Thus, for the first time in the history of this blog, there was a month with no new post. I'll try my best going forward to continue posting at least once each month, but sometimes, these issues can come up.</p><p>The conference, which was held in DC, was a lot of fun. The graduate student researcher working with me was able to present our work as a poster. Additionally, it was my first experience attending an external conference (i.e. one not hosted by UC Davis) since changing fields to get into transportation policy research, so it was an excellent opportunity for professional networking. Because I had previously attended the APS March Meeting for 3 years (2017-2019), I was prepared for a conference of a similar size and scope; in particular, I intentionally avoided overloading my schedule with presentation sessions, made time to take breaks, and made time to meet people informally. Additionally, as transportation is a much more policy-oriented field than physics and is not confined to academia, I made sure to attend TRB committee meetings, different organizations' receptions in the evenings, and other events technically outside of the conference itself. I certainly professionally got what I wanted out of it, and I look forward to attending again in the future.<br /></p>http://dasublogbyprashanth.blogspot.com/2023/02/my-time-at-2023-trb-annual-meeting.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-5934516851612614059Fri, 02 Dec 2022 16:12:00 +00002022-12-02T11:12:49.594-05:00calculusmathematicsphysicsscienceFundamental Theorem of Calculus for Functionals<p>I happened to think more about the idea of recovering a functional by somehow integrating its functional derivative. In the process, I realized that certain ideas that I would have to consider make this post a natural follow-up to a recent post [<a href="https://dasublogbyprashanth.blogspot.com/2022/11/mapping-scalars-to-functions.html">LINK</a>] about mapping scalars to functions. This will become clear later in this post.</p><p>For a single variable, a function \( f(x) \) has an antiderivative \( F(x) \) such that \( f(x) = \frac{\mathrm{d}F}{\mathrm{d}x} \). One statement of the fundamental theorem of calculus is that this implies that \[ \int_{a}^{b} f(x)~\mathrm{d}x = F(b) - F(a) \] for these functions. In turn, this means \( F(x) \) can be extracted directly from \( f(x) \) through \[ F(x) = \int_{x_{0}}^{x} f(x')~\mathrm{d}x' \] in which \( x_{0} \) is chosen such that \( F(x_{0}) = 0 \).</p><p>For multiple variables, a conservative vector field \( \mathbf{f}(\mathbf{x}) \) in which \( \mathbf{f} \) must have the same number of components as \( \mathbf{x} \) can be said to have a scalar antiderivative \( F(\mathbf{x}) \) in the sense that \( \mathbf{f} \) is the gradient of \( F \), meaning \( \mathbf{f}(\mathbf{x}) = \nabla F(\mathbf{x}) \); more precisely, \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) = \frac{\partial F}{\partial x_{i}} \) for all \( i \in \{1, 2, \ldots, N \} \). (Note that if \( \mathbf{f} \) is not conservative, then it by definition cannot be written as the gradient of a scalar function! This is an important point to which I will return later in this post.) In such a case, a line integral (which, as I will emphasize again later in this post, is distinct from a functional path integral) from vector point \( \mathbf{a} \) to vector point \( \mathbf{b} \) of \( \mathbf{f} \) can be computed as \( \int \mathbf{f}(\mathbf{x}) \cdot \mathrm{d}\mathbf{x} = F(\mathbf{b}) - F(\mathbf{a}) \); more precisely, this equality holds along any contour, so if a contour is defined as \( \mathbf{x}(s) \) for \( s \in [0, 1] \), no matter what \( \mathbf{x}(s) \) actually is, as long as \( \mathbf{x}(0) = \mathbf{a} \) and \( \mathbf{x}(1) = \mathbf{b} \) hold, then \[ \sum_{i = 1}^{N} \int_{0}^{1} f_{i}(x_{1}(s), x_{2}(s), \ldots, x_{N}(s)) \frac{\mathrm{d}x_{i}}{\mathrm{d}s} \mathrm{d}s = F(\mathbf{b}) - F(\mathbf{a}) \] must also hold. This therefore suggests that \( F(\mathbf{x}) \) can be extracted from \( \mathbf{f}(\mathbf{x}) \) by relabeling \( \mathbf{x}(s) \to \mathbf{x}'(s) \), \( \mathbf{a} \) to a point such that \( F(\mathbf{a}) = 0 \), and \( \mathbf{b} \to \mathbf{x} \). Once again, if \( \mathbf{f}(\mathbf{x}) \) is not conservative, then it cannot be written as the gradient of a scalar field \( F \), and the integral \( \sum_{i = 1}^{N} \int_{0}^{1} f_{i}(x_{1}(s), x_{2}(s), \ldots, x_{N}(s)) \frac{\mathrm{d}x_{i}}{\mathrm{d}s} \mathrm{d}s \) will depend on the specific choice of \( \mathbf{x}(s) \), not just the endpoints \( \mathbf{a} \) and \( \mathbf{b} \).</p><p>For continuous functions, the generalization of a vector \( \mathbf{x} \), or more precisely \( x_{i} \) for \( i \in \{1, 2, \ldots, N\} \), is a function \( x(t) \) where \( t \) is a continuous dummy index or parameter analogous to the discrete index \( i \). This means the generalization of a scalar field \( F(\mathbf{x}) \) is the scalar functional \( F[x] \). What is the generalization of a vector field \( \mathbf{f}(\mathbf{x}) \)? To be precise, a vector field is a collection of functions \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) \) for all \( i \in \{1, 2, \ldots, N \} \). This suggests that its generalization should be a function of \( t \) and must somehow depend on \( x(t) \) as well. It is tempting therefore to write this as \( f(t, x(t)) \) for all \( t \). However, although this is a valid subset of the generalization, it is not the whole generalization, because vector fields of the form \( f_{i}(x_{i}) \) are collections of single-variable functions that do not fully capture all vector fields of the form \( f_{i}(x_{1}, x_{2}, \ldots, x_{N}) \) for all \( i \in \{1, 2, \ldots, N \} \). As a specific example, for \( N = 2 \), the vector field with components \( f_{1}(x_{1}, x_{2}) = (x_{1} - x_{2})^{2} \) and \( f_{2}(x_{1}, x_{2}) = (x_{1} + x_{2})^{3} \) cannot be written as just \( f_{1}(x_{1}) \) and \( f_{2}(x_{2}) \), as \( f_{1} \) depends on \( x_{2} \) and \( f_{2} \) depends on \( x_{1} \) as well. Similarly, in the generalization, one could imagine a function of the form \( f = \frac{x(t)}{x(t - t_{0})} \mathrm{exp}(-(t - t_{0})^{2}) \); in this case, it is not correct to write it as \( f(t, x(t)) \) because the dependence of \( f \) on \( x \) at a given dummy index value \( t \) comes through not only \( x(t) \) but also \( x(t - t_{0}) \) for some fixed parameter \( t_{0} \). Additionally, the function may depend not only on \( x \) per se but also on derivatives \( \frac{\mathrm{d}^{n} x}{\mathrm{d}t^{n}} \); the case of the first derivative \( \frac{\mathrm{d}x}{\mathrm{d}t} = \lim_{t_{0} \to 0} \frac{x(t) - x(t - t_{0})}{t_{0}} \) illustrates the connection to the aforementioned example. Therefore, the most generic way to write such a function is effectively as a functional \( f[x; t] \) with a dummy index \( t \). The example \( f = \frac{x(t)}{x(t - t_{0})} \mathrm{exp}(-(t - t_{0})^{2}) \) can be formalized as \( f[t, x] = \int_{-\infty}^{\infty} \frac{x(t')}{x(t' - t_{0})} \mathrm{exp}(-(t' - t_{0})^{2}) \delta(t - t')~\mathrm{d}t' \) where the dummy index \( t' \) is the integration variable while the dummy index \( t \) is free. (For \( N = 3 \), the condition of a vector field being conservative is often written as \( \nabla \times \mathbf{f}(\mathbf{x}) = 0 \). I have not used that condition in this post because the curl operator does not easily generalize to \( N \neq 3 \).)<br /></p><p>If a functional \( f[x; t] \) is conservative, then there exists a functional \( F[x] \) (with no free dummy index) such that \( f \) is the functional derivative \( f[x; t] = \frac{\delta F}{\delta x(t)} \). Comparing the notation between scalar fields and functionals, \( \sum_{i} A_{i} \to \int A(t)~\mathrm{d}t \) and \( \mathrm{d}x_{i} \to \delta x(t) \), in which \( \delta x(t) \) is a small variation in a function \( x \) specifically at the index value \( t \) and nowhere else. This suggests a generalization of the fundamental theorem of calculus to functionals as follows. If \( a(t) \) and \( b(t) \) are fixed functions, then \( \int_{-\infty}^{\infty} \int f[x; t]~\delta x(t)~\mathrm{d}t = F[b] - F[a] \). More precisely, a path from the function \( a(t) \) to the function \( b(t) \) at every index value \( t \) can be parameterized by \( s \in [0, 1] \) by the map \( s \to x(t, s) \) which is a function of \( t \) for each \( s \) such that \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \); this is why I linked this post to the most recent post on this blog. With this in mind, the fundamental theorem of calculus becomes \[ \int_{-\infty}^{\infty} \int_{0}^{1} f[x(s); t] \frac{\partial x}{\partial s}~\mathrm{d}s~\mathrm{d}t = F[b] - F[a] \] where, in the integrand, the argument \( x \) in \( f \) has the parameter \( s \) explicit but the dummy index \( t \) implicit; the point is that this equality holds regardless of the specific parameterization \( x(t, s) \) as long as \( x \) at the endpoints of \( s \) satisfies \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \). This also means that \( F[x] \) can be recovered if \( b(t) = x(t) \) and \( a(t) \) is chosen such that \( F[a] = 0 \), in which case \[ F[x] = \int_{-\infty}^{\infty} \int_{0}^{1} f[x'(s); t]~\frac{\partial x'}{\partial s}~\mathrm{d}s~\mathrm{d}t \] (where \( x(t, s) \) has been renamed to \( x'(t, s) \) to avoid confusion with \( x(t) \)). If \( f[x; t] \) is not conservative, then there is no functional \( F[x] \) whose functional derivative with respect to \( x(t) \) would yield \( f[x; t] \); in that case, with \( x(t, 0) = a(t) \) and \( x(t, 1) = b(t) \), the integral \( \int_{-\infty}^{\infty} \int_{0}^{1} f[x(s); t] \frac{\partial x}{\partial s}~\mathrm{d}s~\mathrm{d}t \) does depend on the specific choice of parameterization \( x(t, s) \) with respect to \( s \) and not just on the functions \( a(t) \) and \( b(t) \) at the endpoints of \( s \).</p><p>As an example, consider from a previous post [<a href="https://dasublogbyprashanth.blogspot.com/2022/03/how-to-tell-whether-functional-is.html">LINK</a>] the nonrelativistic Newtonian action \[ S[x] = \int_{-\infty}^{\infty} \left(\frac{m}{2} \left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^{2} + F_{0} x(t) \right)~\mathrm{d}t \] for a particle under the influence of a uniform force \( F_{0} \) (which may vanish). The first functional derivative is \[ f[x; t] = \frac{\delta S}{\delta x(t)} = F_{0} - m\frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}} \] and its vanishing would yield the usual equation of motion. The action itself vanishes for \( x(t) = 0 \), which will be helpful when using the fundamental theorem of calculus to recover the action from the equation of motion. In particular, one can parameterize \( x'(t, s) = sx(t) \) such that \( x'(t, 0) = 0 \) and \( x'(t, 1) = x(t) \). This gives the integral \( \int_{0}^{1} \left(F_{0} - ms\frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}}\right)x(t)~\mathrm{d}s = F_{0} x(t) - \frac{m}{2} x(t) \frac{\mathrm{d}^{2} x}{\mathrm{d}t^{2}} \). This is then integrated over all \( t \), so the first term is identical to the corresponding term in the definition of \( S[x] \), and the second term becomes the same as the corresponding term in the definition of \( S[x] \) after integrating over \( t \) by parts and setting the boundary conditions that \( x(t) \to 0 \) for \( |t| \to \infty \). (Other boundary conditions may require more care.) In any case, the parameterization \( x'(t, s) = sx(t) \) is not the only choice that could fulfill the boundary conditions; the salient point is that any parameterization fulfilling the boundary conditions would yield the correct action \( S[x] \).<br /></p><p>I considered that example because I wondered whether any special formulas need to be considered if \( f[x; t] \) depends explicitly on first or second derivatives of \( x(t) \), as might be the case in nonrelativistic Newtonian mechanics. That example shows that no special formulas are needed because even if the Lagrangian explicitly depends on the velocity \( \frac{\mathrm{d}x}{\mathrm{d}t} \), the action \( S \) only explicitly depends as a functional on \( x(t) \), so proper application of functional differentiation and regular integration by parts will ensure proper accounting of each piece.</p><p>This post has been about the fundamental theorem of calculus saying that the 1-dimensional integral of a function in \( N \) dimensions along a contour, if that function is conservative, is equal to the difference between the two endpoints of its scalar antiderivative. This generalizes easily to infinite dimensions and continuous functions instead of finite-dimensional vectors. There is another fundamental theorem of calculus saying that the \( N \)-dimensional integral in a finite volume of the scalar divergence of an \( N \)-dimensional vector function, if that volume has a closed orientable surface, is equal to the \( N - 1 \)-dimensional integral of the inner product of that function with the normal vector (of unit 2-norm) at every point on the surface across the whole surface, meaning \[ \int_{V} \sum_{i = 1}^{N} \frac{\partial f_{i}}{\partial x_{i}}~\mathrm{d}V = \oint_{\partial V} \sum_{i = 1}^{N} f_{i}(x_{1}, x_{2}, \ldots, x_{N}) n_{i}(x_{1}, x_{2}, \ldots, x_{N})~\mathrm{d}S \] where \( \sum_{i = 1}^{N} |n_{i}(x_{1}, x_{2}, \ldots, x_{N})|^{2} = 1 \) for every \( \mathbf{x} \). From a purely formal perspective, this could generalize to something like \( \int_{V} \int_{-\infty}^{\infty} \frac{\delta f[x; t]}{\delta x(t)}~\mathrm{d}t~\mathcal{D}x = \oint_{\partial V} \int_{-\infty}^{\infty} f[x; t]n[x; t]~\mathrm{d}t~\mathcal{D}x \) having generalized \( \frac{\partial}{\partial x_{i}} \to \frac{\delta}{\delta x(t)} \), \( \prod_{i} \mathrm{d}x_{i} \to \mathcal{D}x \), and \( n_{i}(\mathbf{x}) \to n[x; t] \) where \( n[x; t] \) is normalized such that \( \int_{-\infty}^{\infty} |n[x; t]|^{2}~\mathrm{d}t = 1 \) for all \( x(t) \) on the surface. However, this formalism may be hard to further develop because the space has infinite dimensions. Even when working in a countable basis, it might not be possible to characterize an orientable surface enclosing a volume in an infinite-dimensional space; the surface is also infinite-dimensional. While the choice of basis is arbitrary, things become even less intuitive when choosing to work in an uncountable basis.</p>http://dasublogbyprashanth.blogspot.com/2022/12/fundamental-theorem-of-calculus-for.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-4054743065572645467Tue, 01 Nov 2022 13:12:00 +00002022-11-01T09:12:45.062-04:00calculusmathematicsphysicsscienceMapping Scalars to Functions<p>In just over a year, I've written three posts for this blog about functionals, specifically about their application to probability theory [<a href="https://dasublogbyprashanth.blogspot.com/2021/10/functionals-in-probability-and-bayesian.html">LINK</a>], finding their stationary points [<a href="https://dasublogbyprashanth.blogspot.com/2022/03/how-to-tell-whether-functional-is.html">LINK</a>], and the use of their stationary points in classical mechanics [<a href="https://dasublogbyprashanth.blogspot.com/2022/04/follow-up-how-to-tell-whether.html">LINK</a>]. As a reminder, a functional is an object that maps a space of functions to a space of numbers. This got me thinking about what the reverse, namely an object that maps a space of numbers to a space of functions, looks like. To be clear, this is not the same as an ordinary function which, as an element in a space of functions, maps a space of numbers to a space of numbers.</p><p>As I thought about it more, I realized that this is a bit easier to understand and therefore more commonly encountered than a functional. An extremely glib way to describe such an object is a function of multiple variables. However, it may be more enlightening to describe this in further detail to avoid potentially deceptive images that may arise from that glib description.</p><p>In the discrete case, the matrix elements \( A_{ij} \) can be described as a map from integers to vectors, in which an integer \( j \) is associated with a vector whose elements indexed by an integer \( i \) are \( A_{ij} \). This is the essential idea behind seeing the columns of the matrix with elements \( A_{ij} \) as a collection of vectors. Formally, this maps \( i \to (j \to A_{ij}) \) where the map \( j \to A_{ij} \) defines a vector indexed by the free variable \( i \).</p><p>Similarly, in the continuous case, the function elements \( f(x, y) \) can be described as a map from numbers to functions, in which a number \( y \) is associated with a function whose
elements indexed by a number \( x \) are \( f(x, y) \). Formally, this maps \( x \to (y \to f(x, y)) \) where the map \( y \to f(x, y) \) defines a function indexed
by the free variable \( x \). These ideas are foundational to the development of more abstract notions of functions, like lambda calculus.<br /></p>http://dasublogbyprashanth.blogspot.com/2022/11/mapping-scalars-to-functions.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-4951752779578776841Sun, 02 Oct 2022 05:34:00 +00002022-10-02T01:34:38.076-04:00bancultureDRMebooklicense feeracial profilingrepublicanTechnological Restrictions on E-Books and Culture Wars on Books in 2022<p>Despite the long title, this post will be fairly short. This blog used to publish a lot more often (I had a lot more free time in high school & college) and focus a lot more on issues related to free software, free culture, and things like that, yet even after looking through posts on this blog from its early years (which, aligning with the stereotype of an adult looking through essays written in high school, made me cringe at the quality of writing even if I agreed with some of the basic opinions), I actually couldn't find any posts specifically about the effects of so-called digital rights management (DRM) on E-books.</p><p>In any case, I was motivated to write this because I recently listened to an episode [<a href="https://www.youtube.com/watch?v=i2iiyU-z5E4">LINK</a>] of a podcast associated with The Daily Show in which the guests discussed recent instances of conservative politicians in the US preventing public schools & libraries from teaching or carrying books that offend those politicians' cultural sensibilities. I distinctly remember reading in high school & college about warnings of the consequences of putting DRM on E-books, including making it easier to ban such books. At that time, I and many others felt it would be ridiculous for politically motivated book bans to take effect in the US especially given respect for the First Amendment to the US Constitution. Leaving aside whether such book bans from public schools & libraries technically violate that amendment if such bans don't go beyond those domains, it is disheartening to see a direct example of censorship so closely connected to technological restrictions that are politically motivated (not due to fundamental technical limitations). It will be interesting to see whether authors of banned books encourage or tolerate people scanning & sharing unauthorized PDF files of the books for free; this wouldn't be unprecedented, given that the huge markup of textbooks in the US compared to other countries has led many textbook authors to encourage students to buy cheaper editions from other countries.<br /></p>http://dasublogbyprashanth.blogspot.com/2022/10/technological-restrictions-on-e-books.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-6525314255859203444Sat, 17 Sep 2022 22:20:00 +00002022-09-17T18:20:23.777-04:00Book ReviewElizabeth HintonethnicityFrom the War on Poverty to the War on CrimehistoryjailingMore Than Just RacepoliceUSviolationBook Review: "From the War on Poverty to the War on Crime" by Elizabeth Hinton<p>I've recently read the book <i>From the War on Poverty to the War on Crime</i> by Elizabeth Hinton. This book is a history of the progression through the titular subjects in the US, starting with the Kennedy presidency and ending with the Reagan presidency. It shows how while some problems associated with the war on poverty did come from good but conflicting intentions when implementing social welfare programs, many more problems came from halfhearted implementation of social welfare programs with the intent & through the lens of fighting crime leading ultimately to replacement of those programs with more explicit expansions of policing to fight crime especially in response to high-profile riots in large cities in the 1960s & 1970s.</p><p>The introduction makes clear that the war on crime started with the presidency of Lyndon Johnson and arguably even earlier with the Kennedy administration's efforts to combat juvenile delinquency, so the association of the war on crime primarily with the presidencies Nixon, Ford, and Reagan (as Carter is often left out of popular discussions about this despite being equally responsible in these ways) is because those presidents (and Carter) cut funding for social welfare programs that Johnson initially saw as integral to the success of programs combating crime but began to back away from by the end of his presidency, increased funding for policing, and shifted focus to arrests & imprisonments as ways to prevent future crimes. The author discusses how law enforcement agencies started to develop biased metrics for crime & collect data in biased ways to justify racist theories about the supposedly inherent pathologies of black Americans in cities, even as some politicians at that time wondered if law enforcement agencies should be collecting crime statistics given the conflict of interest. The author emphasizes the bipartisan white American political consensus about crime after 1960 to show that it wasn't just Reagan or other Republicans in the 1980s who focused on punishment through veiled racism. The author also discusses how black American community leaders in & after the 1960s wanted to partner with law enforcement agencies to develop effective strategies together to deal with local problems at the root of local crimes, but conservative politicians (from both parties) deliberately moved away from such partnerships toward the federal block grant funding model that would incentivize states to conduct law enforcement in the most heavy-handed & punitive way possible especially in urban black neighborhoods, which led many black Americans to stop trusting law enforcement. My only criticism specifically about the introduction (that doesn't have to do with the rest of the book) is that the author's language about conditional probabilities is quite sloppy, which is problematic in the context of discussions about biases in data collection & statistical analysis by law enforcement agencies about crime.</p><p>The rest of the book simply goes through the history in detail. In the introduction, I wasn't sure who the target audience of the book was supposed to be given frequent references to gaps in academic literature in the main text, but the narrative became more clear through the rest of the book.</p><p>There are a few points that I credit the book for. These are as follows.</p><p>First, the author repeats points effectively to reinforce the narrative. This makes the narrative easy to follow, and the narrative is clearly well-sourced. <br /></p><p>Second, I didn't know that the war on crime dated back to the Kennedy administration. I can claim to have learned that from this book.</p><p>Third, I didn't know that close federal cooperation directly with local governments also dated back to the Kennedy administration at the latest. I can claim to have learned that from this book. This is an issue that has been on my mind for a while in the context of empowering cities whose political views oppose those of their state governments which want to disempower them (because while there is a clear federal relationship among the federal, state, and tribal governments in the US Constitution, the US Constitution doesn't govern states' internal affairs, and many states treat their constituent cities as fully subordinate to state governments in all matters).</p><p>Fourth, in my view, the author correctly recognizes that policies to combat crime should be evaluated for effectiveness several years afterwards, as there are no quick fixes. The author therefore evaluates whether different parts of the war on crime had effects on crime 10-20 years later instead of just a year later, as the latter would have been a political cheap shot.</p><p>Fifth, I appreciate the author's recognition that the US made the same mistakes in both military & social welfare strategies domestically as in invasions of other countries. The author makes clear that many politicians at that time recognized this in the context of wars in Southeast Asia. This adds another dimension to arguments from the book <i>How Everything Became War and the Military Became Everything</i> by Rosa Brooks (which I have reviewed on this blog [<a href="https://dasublogbyprashanth.blogspot.com/2017/03/book-review-how-everything-became-war.html">LINK</a>]) which, from what I remember, focused a little more on more recent invasions of Afghanistan & Iraq.</p><p>However, there are many more points in the book which I find problematic. These are as follows.</p><p>First, at several points, the author claims that news media & politicians often overstated the prevalence of violent crime, but the author's claims that federal law enforcement statistics about crime were less biased before the 1960s are undercut by the author's acknowledgment that such statistics were collected much more sparsely and in a more ad hoc way before the 1960s. Even leaving aside this point, the author doesn't (in the main text, leaving out the endnotes) name or cite enough specific sources to effectively argue against the dominant narrative of that time, which the book simply restates while feebly arguing against, or intuitively explain why certain crime statistics may initially look alarming but may actually imply rarity of such incidents in practice for an individual. For example, it could be argued that a certain homicide rate given in homicides per million people per year could look big at first but could be argued to imply that an individual has a very low chance of being killed by another person on a given day. (I don't know enough about crime statistics to know what specific number could plausibly fit this description.)<br /></p><p>Second, at several points, the author basically wags a finger against what the federal government did, but in some cases where solutions are longer-term and therefore more obvious like investments in unleaded plumbing, better street lighting, or better sidewalks, the author barely identifies these, and in other cases where issues like imminent or continuing riots are spiraling out of control, the author doesn't convincingly argue in favor of specific alternatives except in perhaps 1 or 2 isolated cases. Part of the problem, as becomes clear in the epilogue, may be that the federal government prematurely dismissed such alternatives before seriously trying them, but then the author should have spent more space arguing for such programs on their own potential specific merits instead of giving most of the space to the arguments of contemporary politicians that, by dominating the narrative, may unintentionally seem more convincing than the author may have wanted.</p><p>Third, the author argues at some points that poor black Americans should have been empowered from the bottom-up but at other points that they should have gotten similar top-down federal assistance as poor white Americans (which did happen in the war on poverty, albeit at much less monetary amounts per person). This seems incoherent, and the author makes no attempt to explain why these views are compatible with each other.</p><p>Fourth, the author ignores the issues of continued slum clearance, urban highway construction, and the dynamics of white flight from urban cores in much of the narrative. I've read in many other places how critical these concerns were in the context of urban crime, so it is surprising to see no mention of these concerns in the book.</p><p>Fifth, the author seems to unduly dismiss the challenges that the Carter administration faced in rebuilding damaged urban neighborhoods in the face of high interest rates & high inflation in the 1970s. Perhaps the argument would have been stronger if the author could have found examples of the Carter administration spending scarce resources on less dire issues.</p><p>Sixth, in the introduction, the author claims that the war on crime specifically wasn't a reincarnation of the Jim Crow era, but later in the book, the author at many points implies & comes close to explicitly saying exactly that. This seems inconsistent, though to be fair, it was obvious to me that the author would ultimately argue that the war on crime was related to the Jim Crow era, so this inconsistency only threw me off in the introduction.</p><p>Seventh, I found it interesting that the author used scare quotes around the term "evil empire" and called the Cold War "Reagan's Cold War". It could be argued that the latter was a more specific reference to how the Reagan administration waged the Cold War in the 1980s as a smaller part of the broader conflict over decades, but based on the author's other stated & implied views through the book, I see it more likely as evidence of the author having far left-wing sympathies, because the latter term in the broader context of the author's views through the book sounds like the author believes the Reagan administration was too bellicose toward the USSR and was too taken by American propaganda over decades to admit that some parts of Soviet propaganda especially about race relations could be true (which is debatable in the context of internal affairs in the USSR).<br /></p><p>Eighth, the author seems to argue that the implementation of the ban of handguns but not shotguns was racist because it failed to separate bans on weapons (which should mostly be about destroying those weapons and removing sources of weapons production, though perhaps some further consequences could be appropriate for repeat offenders) from the harsh punishment of offenders. I agree with this in the context of having excessive punishments and in the sense that, even now, it is clear in the US that white Americans are much more often than black Americans given the benefit of the doubt with respect to usage of firearms in self-defense or the possession of firearms per the Second Amendment to the US Constitution. Moreover, it is worth remembering that while courts of law in the US didn't start to systematically recognize an individual right to bear arms until the 2000s (many decades after the 1960s), that doesn't mean that everyone was prosecuted equally for bearing firearms before the 2000s; it is much more plausible that there were racial disparities in enforcement of such laws. However, I think the author doesn't do a good job of acknowledging how the much greater difficulty in using a shotgun to commit violent crimes in urban settings compared to using a handgun for the same purpose makes a handgun ban more sensible without such a ban (separate from the consequences to humans who violate such bans) necessarily being racist per se.</p><p>Ninth, at several points, the author seems to go beyond merely describing large-scale riots led by black Americans as a predictable consequence of oppression to being an apologist for such riots. I do at least acknowledge the merit in describing such predictable consequences from a sociological perspective. Also, I admit that it took me a while to understand the author's point that the problem with claiming in the 1960s & later that black Americans are culturally pathological is that such claims were coming from white American politicians who wanted to claim that such cultural pathology was the root of poverty & crime (and not the other way around). Finally, while I still believe that any human culture can have pathological elements in which some elements come from a history of being oppressed & therefore traumatized while other elements may be evidence of being a privileged or oppressive group (so there is no such thing as being a purely "good" or "oppressed" group or a purely "bad" or "oppressive" group), I can understand the author's desire to call white American racism as the dominant group in the US something closer to "cultural pathology" and the cultural pathologies that may result directly from centuries of oppression "trauma" as qualitatively distinct things within the context of US history. Having said those things, I don't think the author did a good enough job of acknowledging that even if some ideal form of reparations for these harms could be formulated & implemented overnight, the lasting effects of these traumas could continue to have negative consequences for different localities & the US as a whole for many decades, and especially now with the US having so many residents & citizens who come from other places & don't identify as white or black, I'm not sure how many Americans would have the patience to wait decades for those things to resolve even if they are much more understanding & supportive of the need for reparations than most white Americans would have been in the 1960s (as the author shows how white Americans in the 1960s wanted quick fixes to the eruptions of violence in response to oppression, which led to further oppression through brutal crackdowns by law enforcement).</p><p>Overall, I still recommend this book because the basic historical narrative is well-written & well-sourced; I do feel like I learned a little bit and I had a lot to think about. I just think that readers should be aware of the author's biases as I've discussed above.<br /></p>http://dasublogbyprashanth.blogspot.com/2022/09/book-review-from-war-on-poverty-to-war.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-4493465719258343201Sun, 07 Aug 2022 15:38:00 +00002022-08-07T13:06:05.851-04:00desktop effectsdolphingwenviewKDEKDE ActivitieskolourpaintokularUnixoid Reviewvirtual desktopReview: KDE neon 5.25<p>It has been a long time since I've reviewed a Linux distribution on this blog; the last one was of Linux Mint 19 "Tara" from 4 years ago [<a href="https://dasublogbyprashanth.blogspot.com/2018/07/review-linux-mint-19-tara-mate-xfce.html">LINK</a>]. In a more recent post about problems that I had with a scanner that required me to install Linux Mint 20 "Ulyana" MATE because the existing operating system was damaged beyond repair [<a href="https://dasublogbyprashanth.blogspot.com/2020/09/ongoing-problems-connecting-canon.html">LINK</a>], I explained that I had come to trust the consistency & stability of Linux Mint enough and liked it enough that, in conjunction with the lack of novelty in Linux distributions compared to 10 years prior, I no longer felt motivated to do such reviews. Thus, it may seem strange that I should do a review like this now. In truth, the motivation wasn't hugely compelling, but I thought it might make for a nice post on this blog as I didn't have much else in mind. I thought of checking out a showcase of KDE, namely KDE neon, because it had been a long time since I tried KDE and I was getting a little concerned that the odd artifacts I was starting to see in Linux Mint 20 "Ulyana" MATE when hovering over right-click menus might be the tip of an iceberg of problems. While the latter concern has thankfully not come to pass even after several months of experiencing these more minor issues, I figured it might be nice to see what KDE is like now.</p><p></p><p></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB5F2InMHrnwfPscij7Y3wlF2mHBS2eI2VorlyL4tlVduJCVeo_bTbNj4FbSsoX4Vt0GHjqGlH6samRGFUeHCyqKy7UZh3fyCnrV0G8skRCHiRPvZVP3fJ1MhoVOcTmVhP9prL4nQmQvyYWPi1OZwsMcM6JXoSP3VM7q2k4GUAKkoyPdWDKBJOb3Qz/s1920/Screenshot_20220807_021510.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1080" data-original-width="1920" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB5F2InMHrnwfPscij7Y3wlF2mHBS2eI2VorlyL4tlVduJCVeo_bTbNj4FbSsoX4Vt0GHjqGlH6samRGFUeHCyqKy7UZh3fyCnrV0G8skRCHiRPvZVP3fJ1MhoVOcTmVhP9prL4nQmQvyYWPi1OZwsMcM6JXoSP3VM7q2k4GUAKkoyPdWDKBJOb3Qz/s320/Screenshot_20220807_021510.png" width="320"></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Default desktop, before changes<br></td></tr></tbody></table>This review will be a bit different from past reviews. In particular, in past reviews, I took the perspective of a newbie to Linux trying to do ordinary tasks, whereas the purpose of this review is to see whether I can replicate the look & feel of my desktop in Linux Mint 20 "Ulyana" MATE. Thus, I will focus mostly on changing the desktop and on using the default KDE applications; I will not focus on the presence or absence of other applications or on other parts of the live USB environment. Follow the jump to see what it is like.<p></p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2022/08/review-kde-neon-525.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2022/08/review-kde-neon-525.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-2793458649876694442Sun, 10 Jul 2022 16:02:00 +00002022-07-10T12:02:38.841-04:00cryptographyFOLLOW-UPlinuxprivacysurveillancetech companyFOLLOW-UP: Some Recent Troubles with pCloud and Google Chat<p>Almost exactly 11 months ago, I wrote a post [<a href="https://dasublogbyprashanth.blogspot.com/2021/08/some-recent-troubles-with-pcloud-and.html">LINK</a>] about problems I was experiencing with pCloud and Google Chat. I don't have any updates about Google Chat (or Google Meet), but I do have an update regarding pCloud. In the previous post, I noted that for files & folders protected by standard encryption (as opposed to stronger zero-knowledge encryption, for which this problem doesn't exist in pCloud), some files & folders require multiple attempts to transfer; I also speculated that pCloud may have been secretly deleting files & folders. After having spent more time using pCloud, I've been able to verify that my files & folders protected by standard encryption have transferred properly, and I think I've figured out why they initially seemed to require multiple attempts to transfer.</p><p>As I understand, pCloud creates a temporary folder, essentially like a cache, on the local hard drive, to transfer files before they are uploaded to pCloud. The process of transferring from the local cache to pCloud is limited by upload speeds, which are quite slow (as I mentioned in the previous post). Additionally, once the pCloud program is closed and the remote drive is unmounted, file transfer stops, so many folders & files that the user might think were transferred might not have been transferred. A good way to verify this is to leave the desktop application for pCloud open, monitor how many files as well as what total amount of data still remain to be transferred, and only close the application when all folders & files have been transferred; I like to think of it as a practice similar to leaving a torrent open for uploading after it has finished downloading.<br /></p>http://dasublogbyprashanth.blogspot.com/2022/07/follow-up-some-recent-troubles-with.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-6786466082991490736Wed, 01 Jun 2022 13:21:00 +00002022-06-01T09:21:18.190-04:00mathematicsphysicsquantum electrodynamicsNonlocality and Infinite LDOS in Lossy Media<p>While I have written many posts on this blog about various topics in physics or math unrelated to my graduate work as well as posts promoting papers from my graduate work, it is rare that I've written direct technical posts about my graduate work. It is even more unusual that I should be doing so 2 years after leaving physics as a career. However, I felt compelled to do so after meeting again with my PhD advisor (a day before the Princeton University 2020 Commencement, which was held in person after a delay of 2 years due to this pandemic), as we had a conversation about the problem of infinite local density of states (LDOS) in a lossy medium.</p><p>Essentially, the idea is the following. Working in the frequency domain, the electric field produced by a polarization density in any EM environment is \( E_{i}(\omega, \vec{x}) = \int G_{ij}(\omega, \vec{x}, \vec{x}')P_{j}(\omega, \vec{x}')~\mathrm{d}^{3} x' \) which can be written in bra-ket notation (dispensing with the explicit dependence on frequency) as \( |\vec{E}\rangle = \hat{G}|\vec{P}\rangle \). The LDOS is proportional to the power radiated by a point dipole and can be written as \( \mathrm{LDOS}(\omega, \vec{x}) \propto \sum_{i} \mathrm{Im}(G_{ii}(\omega, \vec{x}, \vec{x})) \). This power should be finite as long as the power put into the dipole to keep it oscillating forever at a given frequency \( \omega \) is finite. However, there appears to be a paradox in that if the position \( \vec{x} \) corresponds to a point embedded in a local lossy medium, the LDOS diverges there.</p><p>I wondered if an intuitive explanation could be that loss should properly imply the existence of energy leaving the system by traveling out of its boundaries, so the idea of a medium that is local everywhere (in the sense that the susceptibility operator takes the form \( \chi_{ij}(\omega, \vec{x}, \vec{x}') = \chi_{ij}(\omega, \vec{x})\delta^{3} (\vec{x} - \vec{x}') \) at all positions) and is lossy at every point in its domain may not be well-posed as energy is somehow disappearing "into" the system instead of leaving it. Then, I wondered if the problem may actually be with locality and whether a nonlocal description of the susceptibility could help. This is where my graduate work could come in. Follow the jump to see a very technical sketch of how this might work (as I won't work out all of the details myself).</p><span></span><a href="http://dasublogbyprashanth.blogspot.com/2022/06/nonlocality-and-infinite-ldos-in-lossy.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2022/06/nonlocality-and-infinite-ldos-in-lossy.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-5294118625568170914Sun, 01 May 2022 16:13:00 +00002022-05-01T12:13:33.691-04:00Algorithms to Live ByBook ReviewBrian Christianethicsgame theorymathematicsminimumscienceTom GriffithsBook Review: "Algorithms to Live By" by Brian Christian & Tom Griffiths<p>I've recently read the book <i>Algorithms to Live By</i> by Brian Christian & Tom Griffiths. This book shows how many problems & heuristics in computer science can be applied to explain or improve human decision-making. Each chapter focuses on a certain class of problems or issues. Such classes include the optimal stopping problem, the multi-armed bandit problem, searching & sorting, task scheduling, Bayesian inference, overfitting data, constraint relaxation, random stimulus, communication protocols, and social interaction. Additionally, most chapters try to show how results from computer science can either improve or justify certain human behaviors.</p><p>This book was frustrating for me to read. If it had fully met my expectation that it would show, in a unified & consistent way, how these computer science problems apply to human behavior and connect to each other, I would be singing its praises. If it had completely failed, I'd be happy to rhetorically trash this book. Instead, I found that each chapter would be a great vignette on its own, and each chapter showed the great potential of what the book could have been, but the book failed to live up to that potential. First, there was very little connection among the chapters, and any acknowledgment that the authors did make of such connections was almost always superficial instead of deeply insightful. For example, the respective chapters about the optimal stopping problem, caches, and overfitting each could have been so much better with greater discussion about the connection to social pressure & game theory, yet those topics were discussed only in the last chapter, which I think was a mistake. Second, only in the concluding section did the authors make clear that they wanted to either improve or justify human behavior with each class of problems or issues. This because clear over the course of reading the book, yet there was very little guidance in each chapter about whether improvement versus justification would be the goal. Perhaps the worst offender was the chapter about constraint relaxation, as there was little connection to human behavior in a way that would be obvious to lay readers. These problems meant that reading the last numbered chapter (about game theory) and the conclusion felt simultaneously wonderful for finally seeing these concepts discussed clearly and maddening for knowing that the book could have been so much better if these ideas had been more consistently executed through the book.</p><p>There are two other minor criticisms I have of the book too. First, the chapter about overfitting seems to use the word "overfitting" to mean too many different things, which is ironic and undermines any clarity that the discussion could have provided. Second, the chapter about randomized algorithms attempts to make a tenuous connection between randomized algorithms used in computer science and the way that random mental stimuli can produce very creative responses in people, but it never makes clear whether the latter result is true at an individual level or only holds statistically for large populations.<br /></p><p>Overall, I think the author's goals were laudable and that each chapter is interesting to read in isolation. However, other readers may be disappointed, as I was, in the way that the authors fail to synthesize many of the ideas across chapters in a smooth & unified manner. Thus, I would advise that readers who may be interested in these topics go into this book with lower expectations.<br /></p>http://dasublogbyprashanth.blogspot.com/2022/05/book-review-algorithms-to-live-by-by.htmlnoreply@blogger.com (PV)tag:blogger.com,1999:blog-9037508989778638319.post-1403045745727570716Mon, 04 Apr 2022 15:21:00 +00002022-04-04T19:29:55.021-04:00FOLLOW-UPmathematicsminimumphysicsquantum electrodynamicsquantum mechanicsscienceFOLLOW-UP: How to Tell Whether a Functional is Extremized <p>This post is a follow-up to an earlier post (<a href="https://dasublogbyprashanth.blogspot.com/2022/03/how-to-tell-whether-functional-is.html">link here</a>) about how to tell whether a stationary point of a functional is a maximum, minimum, or saddle point. In particular, as I thought about it more, I realized that using the analogy to discrete vectors could help when formulating a more general expression for the second derivative of the nonrelativistic classical action for a single degree of freedom (i.e. the corresponding Hessian operator). Additionally, I thought of a few other examples of actions whose Hessian operators are positive-definite. Finally, I've thought more about how to express these equations for systems with multiple degrees of freedom (DOFs) as well as for fields and about how these ideas connect to the path integral formulation of quantum mechanics. Follow the jump to see more</p><a href="http://dasublogbyprashanth.blogspot.com/2022/04/follow-up-how-to-tell-whether.html#more">Read more »</a>http://dasublogbyprashanth.blogspot.com/2022/04/follow-up-how-to-tell-whether.htmlnoreply@blogger.com (PV)