tag:blogger.com,1999:blog-922111353248289998Sun, 04 Oct 2026 21:37:02 +0000Definitions and PropositionsInformation TablesExploiting an Open-Source Office SuiteThe Bias PlanetTo Develop UNO Extensions (LibreOffice Extensions or Apache OpenOffice Extensions)Let Me Understand C++Let Me Understand the Python Programming LanguageJava TipsSchool Mathematics from Higher ViewpointsTo Disentangle Confusing Terms or DiscoursesLet Me Understand GradleLet Me Understand the Java Programming LanguageLet Me Understand C#Let Me Understand GitProjects Build SystemsGradle TipsHow to Use UNO (Handle LibreOffice or Apache OpenOffice Documents) in External Java ProgramsNotes About Using UNO in Basic MacrosUNO Dispatch CommandsT.B.P.https://thebiasplanet.blogspot.com/noreply@blogger.com (Unknown)Blogger2525125tag:blogger.com,1999:blog-922111353248289998.post-288298474128464048Sun, 04 Oct 2026 12:59:31 +00002026-10-04T21:59:31.456+09:00Definitions and Propositions2038: For Module over Division Ring with Finite Basis, There Is No Basis That Has Larger Cardinality <The previous article in this series | The table of contents of this series | description/proof of that for module over division ring with finite basis, there is no basis that has larger cardinality Topics About: module The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3: Proof Starting https://thebiasplanet.blogspot.com/2026/10/formoduleoverdivisionringwithfinitebasisthereisnobasisthathaslargercardinality.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6163779093311907467Sun, 04 Oct 2026 12:58:08 +00002026-10-04T21:58:08.395+09:00Definitions and Propositions2037: For Module over Division Ring Basis, Replacing Element by Linear Combination of Elements with Nonzero Coefficient for Element Forms Basis <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for module over division ring basis, replacing element by linear combination of elements with nonzero coefficient for element forms basis Topics About: module The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/10/formoduleoverdivisionringbasisreplacingelementbylinearcombinationofelementswithnonzerocoefficientforelementformsbasis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-425442174388875627Sun, 04 Oct 2026 12:56:01 +00002026-10-04T21:56:01.505+09:00Definitions and Propositions2036: For Topological Group, Subgroup as Topological Subspace Is Topological Group <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for topological group, subgroup as topological subspace is topological group Topics About: group About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/10/fortopologicalgroupsubgroupastopologicalsubspaceistopologicalgroup.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3314858685855525570Sun, 04 Oct 2026 12:54:36 +00002026-10-04T21:54:36.425+09:00Definitions and Propositions2035: For Group Topological Space, if There Is 'Groups - Homomorphisms' Isomorphism Homeomorphism from Space onto Topological Group, Space Is Topological Group <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for group topological space, if there is 'groups - homomorphisms' isomorphism homeomorphism from space onto topological group, space is topological group Topics About: group About: topological space The table of contents https://thebiasplanet.blogspot.com/2026/10/forgrouptopologicalspaceifthereisgroupshomomorphismsisomorphismhomeomorphismfromspaceontotopologicalgroupspaceistopologicalgroup.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3317321061431211868Sun, 04 Oct 2026 12:53:12 +00002026-10-04T21:53:12.855+09:00Definitions and Propositions2034: 'Rings - Homomorphisms' Isomorphism Is Restricted to Multiplicative 'Groups - Homomorphisms' Isomorphism <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that 'rings - homomorphisms' isomorphism is restricted to multiplicative 'groups - homomorphisms' isomorphism Topics About: ring About: group The table of contents of this article Starting Context Target Context Orientation https://thebiasplanet.blogspot.com/2026/10/ringshomomorphismsisomorphismisrestrictedtomultiplicativegroupshomomorphismsisomorphism.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1692495367296539128Sun, 04 Oct 2026 12:51:48 +00002026-10-04T21:51:48.920+09:00Definitions and Propositions2033: For Ring, Subset of Invertible Elements Forms Multiplicative Group <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for ring, subset of invertible elements forms multiplicative group Topics About: ring About: group The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: https://thebiasplanet.blogspot.com/2026/10/forringsubsetofinvertibleelementsformsmultiplicativegroup.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5274406007343765131Sun, 04 Oct 2026 12:50:25 +00002026-10-04T21:50:25.260+09:00Definitions and Propositions2032: For Ring of \(n \times n\) Field Matrices, if Element Multiplied by Another Element from Left or Right Is \(1\), Another Element Is Inverse <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for ring of \(n \times n\) field matrices, if element multiplied by another element from left or right is \(1\), another element is inverse Topics About: ring The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/10/forringofntimesnfieldmatricesifelementmultipliedbyanotherelementfromleftorrightis1anotherelementisinverse.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7599244610915632451Sun, 04 Oct 2026 12:48:52 +00002026-10-04T21:48:52.521+09:00Definitions and Propositions2031: For Ring, if Element Has Inverse and Element Multiplied by Another Element from Left or Right Is \(1\), Another Element Is Inverse <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for ring, if element has inverse and element multiplied by another element from left or right is \(1\), another element is inverse Topics About: ring The table of contents of this article Starting Context Target Context Orientation https://thebiasplanet.blogspot.com/2026/10/forringifelementhasinverseandelementmultipliedbyanotherelementfromleftorrightis1anotherelementisinverse.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2497460028504851647Sun, 04 Oct 2026 12:47:14 +00002026-10-04T21:47:14.102+09:00Definitions and Propositions2030: For \(1\)-Dimensional Euclidean Topological Space with Canonical Linear-Ordering and Subset, if Infimum of Subset Exists out of Subset, Infimum Is Accumulation Point of Subset, and if Supremum of Subset Exists out of Subset, Supremum Is Accumulation Point of Subset <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(1\)-dimensional Euclidean topological space with canonical linear-ordering and subset, if infimum of subset exists out of subset, infimum is accumulation point of subset, and if supremum of subset exists out of subset, supremum is accumulation https://thebiasplanet.blogspot.com/2026/10/for1dimensionaleuclideantopologicalspacewithcanonicallinearorderingandsubsetifinfimumofsubsetexistsoutofsubsetinfimumisaccumulationpointofsubsetandifsupremumofsubsetexistsoutofsubsetsupremumisaccumulationpointofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5011464237766438266Sun, 04 Oct 2026 12:45:00 +00002026-10-04T21:45:00.465+09:00Definitions and Propositions2029: For Partially-Ordered Set and Subset, if Infimum of Subset Exists in Subset, Infimum Is Minimum of Subset and if Supremum of Subset Exists in Subset, Supremum Is Maximum of Subset <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for partially-ordered set and subset, if infimum of subset exists in subset, infimum is minimum of subset and if supremum of subset exists in subset, supremum is maximum of subset Topics About: set The table of contents of this https://thebiasplanet.blogspot.com/2026/10/forpartiallyorderedsetandsubsetifinfimumofsubsetexistsinsubsetinfimumisminimumofsubsetandifsupremumofsubsetexistsinsubsetsupremumismaximumofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4274021220670803615Sun, 04 Oct 2026 12:43:38 +00002026-10-04T21:43:38.733+09:00Definitions and Propositions2028: For Topological Space, Subset Is Closed iff Closure of Subset Is Subset <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for topological space, subset is closed iff closure of subset is subset Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof https://thebiasplanet.blogspot.com/2026/10/fortopologicalspacesubsetisclosediffclosureofsubsetissubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3194648209956156021Sun, 04 Oct 2026 12:41:58 +00002026-10-04T21:41:58.619+09:00Definitions and Propositions2027: Conjugate of Unitary Matrix by Invertible Matrix Is Not Necessarily Unitary <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that conjugate of unitary matrix by invertible matrix is not necessarily unitary Topics About: matrices space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3:https://thebiasplanet.blogspot.com/2026/10/conjugateofunitarymatrixbyinvertiblematrixisnotnecessarilyunitary.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1666707056532914883Sun, 04 Oct 2026 12:40:35 +00002026-10-04T21:40:35.981+09:00Definitions and Propositions2026: \(m \times n\) Quaternion Matrices Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of \(m \times n\) quaternion matrices topological space Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting Context The reader https://thebiasplanet.blogspot.com/2026/10/mtimesnquaternionmatricestopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2921042741830213277Sun, 04 Oct 2026 12:39:06 +00002026-10-04T21:39:06.501+09:00Definitions and Propositions2025: Quaternion Euclidean Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of quaternion Euclidean topological space Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting Context The reader knows a https://thebiasplanet.blogspot.com/2026/10/quaternioneuclideantopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1255967479672956313Sun, 04 Oct 2026 12:37:46 +00002026-10-04T21:37:46.909+09:00Definitions and Propositions2024: \(m \times n\) Complex Matrices Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of \(m \times n\) complex matrices topological space Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting Context The reader https://thebiasplanet.blogspot.com/2026/10/mtimesncomplexmatricestopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5305675098013550547Sun, 04 Oct 2026 12:36:25 +00002026-10-04T21:36:25.727+09:00Definitions and Propositions2023: \(m \times n\) Real Matrices Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of \(m \times n\) real matrices topological space Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting Context The reader knows ahttps://thebiasplanet.blogspot.com/2026/10/mtimesnrealmatricestopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4849874182060277611Sun, 04 Oct 2026 12:35:02 +00002026-10-04T21:35:02.071+09:00Definitions and Propositions2022: Component of Symplectic Matrix Has Absolute Value Equal to or Smaller than \(1\) <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that component of symplectic matrix has absolute value equal to or smaller than \(1\) Topics About: matrices space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: https://thebiasplanet.blogspot.com/2026/10/componentofsymplecticmatrixhasabsolutevalueequaltoorsmallerthan1.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1396372811319122835Sun, 27 Sep 2026 12:49:58 +00002026-10-04T21:33:43.786+09:00Definitions and Propositions2021: \(n \times n\) Symplectic Group <The previous article in this series | The table of contents of this series | The next article in this series> definition of \(n \times n\) symplectic group Topics About: group About: matrices space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting Context The reader https://thebiasplanet.blogspot.com/2026/09/ntimesnsymplecticgroup.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1720282753961694194Sun, 27 Sep 2026 12:48:41 +00002026-09-27T21:48:41.938+09:00Definitions and Propositions2020: Special Linear Group of Finite-Dimensional Vectors Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of special linear group of finite-dimensional vectors space Topics About: vectors space About: group The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note https://thebiasplanet.blogspot.com/2026/09/speciallineargroupoffinitedimensionalvectorsspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-964139372188900304Sun, 27 Sep 2026 12:47:21 +00002026-09-27T21:47:21.937+09:00Definitions and Propositions2019: For Group, Nonempty Subset Is Subgroup iff It Is Closed Under Operation and Inversion <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for group, nonempty subset is subgroup iff it is closed under operation and inversion Topics About: group The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3https://thebiasplanet.blogspot.com/2026/09/forgroupnonemptysubsetissubgroupiffitisclosedunderoperationandinversion.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4535290136552398254Sun, 27 Sep 2026 12:45:55 +00002026-09-27T21:45:55.463+09:00Definitions and Propositions2018: For 'Modules - Linear Morphisms' Isomorphism, Linearly Independent Subset or Basis of Domain Is Mapped to Linearly Independent Subset or Basis <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for 'modules - linear morphisms' isomorphism, linearly independent subset or basis of domain is mapped to linearly independent subset or basis Topics About: module The table of contents of this article Starting Context Target Contexthttps://thebiasplanet.blogspot.com/2026/09/formoduleslinearmorphismsisomorphismlinearlyindependentsubsetorbasisofdomainismappedtolinearlyindependentsubsetorbasis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5192263062576318868Sun, 27 Sep 2026 12:44:30 +00002026-09-27T21:44:30.409+09:00Definitions and Propositions2017: For Module with Finite Basis, Module Is 'Modules - Linear Morphisms' Isomorphic to Components Module <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for module with finite basis, module is 'modules - linear morphisms' isomorphic to components module Topics About: module The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/09/formodulewithfinitebasismoduleismoduleslinearmorphismsisomorphictocomponentsmodule.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6830312281291860250Sun, 27 Sep 2026 12:43:07 +00002026-09-27T21:43:07.643+09:00Definitions and Propositions2016: Module Linearly Independent Subsets or Bases Do Not Necessarily Have Some Properties of Vectors Space Linearly Independent Subsets or Bases <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that module linearly independent subsets or bases do not necessarily have some properties of vectors space linearly independent subsets or bases Topics About: module The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/09/modulelinearlyindependentsubsetsorbasesdonotnecessarilyhavesomepropertiesofvectorsspacelinearlyindependentsubsetsorbases.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5015204047053528687Sun, 27 Sep 2026 12:41:48 +00002026-09-27T21:41:48.805+09:00Definitions and Propositions2015: For Module, Basis Cannot Be Supplemented with Element to Keep Linearly Independent <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for module, basis cannot be supplemented with element to keep linearly independent Topics About: module The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3: https://thebiasplanet.blogspot.com/2026/09/formodulebasiscannotbesupplementedwithelementtokeeplinearlyindependent.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8231396549923249005Sun, 27 Sep 2026 12:40:09 +00002026-09-27T21:40:09.749+09:00Definitions and Propositions2014: For Module over Division Ring, Nonzero Element Nonzero-Scalar Multiplied Is Nonzero <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for module over division ring, nonzero element nonzero-scalar multiplied is nonzero Topics About: module The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3:https://thebiasplanet.blogspot.com/2026/09/formoduleoverdivisionringnonzeroelementnonzeroscalarmultipliedisnonzero.htmlnoreply@blogger.com (Unknown)