tag:blogger.com,1999:blog-922111353248289998Sun, 25 Oct 2026 10:54:06 +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)Blogger2572125tag:blogger.com,1999:blog-922111353248289998.post-411699619043319485Sun, 25 Oct 2026 10:54:06 +00002026-10-25T19:54:06.259+09:00Definitions and Propositions2085: For Topological Group Action, Valid Restriction of Action w.r.t. Subgroup Topological Subspace and Topological Subspace Is Topological Group Action <The previous article in this series | The table of contents of this series | description/proof of that for topological group action, valid restriction of action w.r.t. subgroup topological subspace and topological subspace is topological group action Topics About: group About: topological space The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/10/fortopologicalgroupactionvalidrestrictionofactionwrtsubgrouptopologicalsubspaceandtopologicalsubspaceistopologicalgroupaction.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8103056679554927588Sun, 25 Oct 2026 10:52:44 +00002026-10-25T19:52:44.485+09:00Definitions and Propositions2084: Transpose of Unitary Matrix Is Unitary <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that transpose of unitary matrix is unitary Topics About: matrices space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof Starting Context The reader knowshttps://thebiasplanet.blogspot.com/2026/10/transposeofunitarymatrixisunitary.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5485182475898088185Sun, 25 Oct 2026 10:51:24 +00002026-10-25T19:51:24.025+09:00Definitions and Propositions2083: For Complex Matrix, Hermitian Conjugate of Transpose of Matrix Is Transpose of Hermitian Conjugate of Matrix <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for complex matrix, Hermitian conjugate of transpose of matrix is transpose of Hermitian conjugate of matrix Topics About: matrices space The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/10/forcomplexmatrixhermitianconjugateoftransposeofmatrixistransposeofhermitianconjugateofmatrix.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3033077659365896163Sun, 25 Oct 2026 10:49:56 +00002026-10-25T19:49:56.697+09:00Definitions and Propositions2082: 'Rings - Homomorphisms' Isomorphism from Ring of \(n \times n\) Quaternion Matrices onto Subring of \(2 n \times 2 n\) Complex Matrices Maps Symplectic Matrix to Unitary Matrix <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 from ring of \(n \times n\) quaternion matrices onto subring of \(2 n \times 2 n\) complex matrices maps symplectic matrix to unitary matrix Topics About: ring The table of contents of this article https://thebiasplanet.blogspot.com/2026/10/ringshomomorphismsisomorphismfromringofntimesnquaternionmatricesontosubringof2ntimes2ncomplexmatricesmapssymplecticmatrixtounitarymatrix.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6867836587514922490Sun, 25 Oct 2026 10:48:31 +00002026-10-25T19:48:31.144+09:00Definitions and Propositions2081: 'Rings - Homomorphisms' Isomorphism from Quaternions Division Associative Algebra onto Subring of \(2 \times 2\) Complex Matrices Ring Maps Quaternion Conjugate to Hermitian Conjugate <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 from quaternions division associative algebra onto subring of \(2 \times 2\) complex matrices ring maps quaternion conjugate to Hermitian conjugate Topics About: ring The table of contents of this https://thebiasplanet.blogspot.com/2026/10/ringshomomorphismsisomorphismfromquaternionsdivisionassociativealgebraontosubringof2times2complexmatricesringmapsquaternionconjugatetohermitianconjugate.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1102242234444918050Sun, 25 Oct 2026 10:47:07 +00002026-10-25T19:47:07.740+09:00Definitions and Propositions2080: For Module over Division Ring with Finite Basis, Linearly Independent Subset Can Be Supplemented with Some Elements of Basis to Be 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 with finite basis, linearly independent subset can be supplemented with some elements of basis to be basis Topics About: module The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/10/formoduleoverdivisionringwithfinitebasislinearlyindependentsubsetcanbesupplementedwithsomeelementsofbasistobebasis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2551751878343116423Sun, 25 Oct 2026 10:45:22 +00002026-10-25T19:45:22.849+09:00Definitions and Propositions2079: Gram-Schmidt Orthonormalization of Countable Subset of Quaternion Euclidean Module <The previous article in this series | The table of contents of this series | The next article in this series> definition of Gram-Schmidt orthonormalization of countable subset of quaternion Euclidean module Topics About: module The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting https://thebiasplanet.blogspot.com/2026/10/gramschmidtorthonormalizationofcountablesubsetofquaternioneuclideanmodule.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2078979003880920881Sun, 25 Oct 2026 10:43:58 +00002026-10-25T19:43:58.649+09:00Definitions and Propositions2078: Quaternion Euclidean Module <The previous article in this series | The table of contents of this series | The next article in this series> definition of quaternion Euclidean module Topics About: module 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 definition of quaternions https://thebiasplanet.blogspot.com/2026/10/quaternioneuclideanmodule.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5643207837274441Sun, 25 Oct 2026 10:42:34 +00002026-10-25T19:42:34.632+09:00Definitions and Propositions2077: For \(n \times n\) Unitary Matrix, Each of Set of Columns and Set of Rows Is Orthonormal Basis of \(n\)-Dimensional Complex Euclidean Vectors Space with Complex Euclidean Inner Product <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(n \times n\) unitary matrix, each of set of columns and set of rows is orthonormal basis of \(n\)-dimensional complex Euclidean vectors space with complex Euclidean inner product Topics About: matrices space About: vectors https://thebiasplanet.blogspot.com/2026/10/forntimesnunitarymatrixeachofsetofcolumnsandsetofrowsisorthonormalbasisofndimensionalcomplexeuclideanvectorsspacewithcomplexeuclideaninnerproduct.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8484219352323105981Sun, 25 Oct 2026 10:41:14 +00002026-10-25T19:41:14.824+09:00Definitions and Propositions2076: For \(n \times n\) Quaternion Matrix, iff for Each \(n\)-Column-or-Row Element, Matrix Multiplied by Element and Quaternion-Conjugate-Transpose of Element Is Element Multiplied by Quaternion-Conjugate-Transpose of Element, Matrix Is Identity <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(n \times n\) quaternion matrix, iff for each \(n\)-column-or-row element, matrix multiplied by element and quaternion-conjugate-transpose of element is element multiplied by quaternion-conjugate-transpose of element, matrix is identity Topics https://thebiasplanet.blogspot.com/2026/10/forntimesnquaternionmatrixiffforeachncolumnorrowelementmatrixmultipliedbyelementandquaternionconjugatetransposeofelementiselementmultipliedbyquaternionconjugatetransposeofelementmatrixisidentity.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-164274070040473602Sun, 25 Oct 2026 10:39:48 +00002026-10-25T19:39:48.293+09:00Definitions and Propositions2075: For Quaternion, iff Sum of Quaternion Multiplied by \(i\) from Left and Right, Sum of Quaternion Multiplied by \(j\) from Left and Right, and Sum of Quaternion Multiplied by \(k\) from Left and Right Are \(0\), Quaternion Is \(0\) <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for quaternion, iff sum of quaternion multiplied by \(i\) from left and right, sum of quaternion multiplied by \(j\) from left and right, and sum of quaternion multiplied by \(k\) from left and right are \(0\), quaternion is \(0\) Topics About: https://thebiasplanet.blogspot.com/2026/10/forquaternioniffsumofquaternionmultipliedbyifromleftandrightsumofquaternionmultipliedbyjfromleftandrightandsumofquaternionmultipliedbykfromleftandrightare0quaternionis0.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2370857044912476581Sun, 25 Oct 2026 10:38:30 +00002026-10-25T19:38:30.125+09:00Definitions and Propositions2074: For \(n \times n\) Real Matrix, iff for Each \(n\)-Column-or-Row Vector, Matrix Multiplied by Vector and Transpose of Vector Is Vector Multiplied by Transpose of Vector, Matrix Is with Diagonals \(1\) and Non-Diagonals Antisymmetric <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(n \times n\) real matrix, iff for each \(n\)-column-or-row vector, matrix multiplied by vector and transpose of vector is vector multiplied by transpose of vector, matrix is with diagonals \(1\) and non-diagonals antisymmetric Topics About:https://thebiasplanet.blogspot.com/2026/10/forntimesnrealmatrixiffforeachncolumnorrowvectormatrixmultipliedbyvectorandtransposeofvectorisvectormultipliedbytransposeofvectormatrixiswithdiagonals1andnondiagonalsantisymmetric.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-9026266061917880664Sun, 25 Oct 2026 10:37:07 +00002026-10-25T19:37:07.515+09:00Definitions and Propositions2073: For \(n \times n\) Complex Matrix, iff for Each \(n\)-Column-or-Row Vector, Matrix Multiplied by Vector and Hermitian Conjugate of Vector Is Vector Multiplied by Hermitian Conjugate of Vector, Matrix Is Identity <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(n \times n\) complex matrix, iff for each \(n\)-column-or-row vector, matrix multiplied by vector and Hermitian conjugate of vector is vector multiplied by Hermitian conjugate of vector, matrix is identity Topics About: matrices space https://thebiasplanet.blogspot.com/2026/10/forntimesncomplexmatrixiffforeachncolumnorrowvectormatrixmultipliedbyvectorandhermitianconjugateofvectorisvectormultipliedbyhermitianconjugateofvectormatrixisidentity.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2918086544636100000Sun, 25 Oct 2026 10:35:38 +00002026-10-25T19:35:38.597+09:00Definitions and Propositions2072: Operation of \(n \times n\) Quaternion Matrix on Quaternion Euclidean Topological Space Corresponds to Operation of Corresponding Complex Matrix <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that operation of \(n \times n\) quaternion matrix on quaternion euclidean topological space corresponds to operation of corresponding complex matrix Topics About: topological space The table of contents of this article Starting Context https://thebiasplanet.blogspot.com/2026/10/operationofntimesnquaternionmatrixonquaternioneuclideantopologicalspacecorrespondstooperationofcorrespondingcomplexmatrix.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4824072254582645956Sun, 25 Oct 2026 10:34:06 +00002026-10-25T19:34:06.025+09:00Definitions and Propositions2071: For Complex-Euclidean-Normed Complex Euclidean Vectors Space, Norm of Element Is Norm of Corresponding Element of Euclidean-Normed Euclidean Vectors Space <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for complex-Euclidean-normed complex Euclidean vectors space, norm of element is norm of corresponding element of Euclidean-normed Euclidean vectors space Topics About: vectors space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/10/forcomplexeuclideannormedcomplexeuclideanvectorsspacenormofelementisnormofcorrespondingelementofeuclideannormedeuclideanvectorsspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3128423336346588864Sun, 25 Oct 2026 10:32:43 +00002026-10-25T19:32:43.080+09:00Definitions and Propositions2070: Quaternion-Conjugate-Transpose of Product of Quaternion Matrices Is Product of Quaternion-Conjugate-Transposes of Constituents in Reverse Order <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that quaternion-conjugate-transpose of product of quaternion matrices is product of quaternion-conjugate-transposes of constituents in reverse order Topics About: matrices space The table of contents of this article Starting Context https://thebiasplanet.blogspot.com/2026/10/quaternionconjugatetransposeofproductofquaternionmatricesisproductofquaternionconjugatetransposesofconstituentsinreverseorder.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7434793801823628997Sun, 18 Oct 2026 12:18:37 +00002026-10-25T19:31:12.300+09:00Definitions and Propositions2069: For Stabilizer of Point w.r.t. Compact Topological Group Right Action on Hausdorff Topological Space, Canonical Map from Right Cosets of Stabilizer Quotient Topological Space into Orbit of Point Topological Subspace Is Homeomorphism <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for stabilizer of point w.r.t. compact topological group right action on Hausdorff topological space, canonical map from right cosets of stabilizer quotient topological space into orbit of point topological subspace is homeomorphism Topics About:https://thebiasplanet.blogspot.com/2026/10/forstabilizerofpointwrtcompacttopologicalgrouprightactiononhausdorfftopologicalspacecanonicalmapfromrightcosetsofstabilizerquotienttopologicalspaceintoorbitofpointtopologicalsubspaceishomeomorphism.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5084510407573348377Sun, 18 Oct 2026 12:16:42 +00002026-10-18T21:16:42.230+09:00Definitions and Propositions2068: For Stabilizer of Point w.r.t. Compact Topological Group Left Action on Hausdorff Topological Space, Canonical Map from Left Cosets of Stabilizer Quotient Topological Space into Orbit of Point Topological Subspace Is Homeomorphism <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for stabilizer of point w.r.t. compact topological group left action on Hausdorff topological space, canonical map from left cosets of stabilizer quotient topological space into orbit of point topological subspace is homeomorphism Topics About: https://thebiasplanet.blogspot.com/2026/10/forstabilizerofpointwrtcompacttopologicalgroupleftactiononhausdorfftopologicalspacecanonicalmapfromleftcosetsofstabilizerquotienttopologicalspaceintoorbitofpointtopologicalsubspaceishomeomorphism.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5872908810539678013Sun, 18 Oct 2026 12:14:41 +00002026-10-18T21:14:41.117+09:00Definitions and Propositions2067: For Stabilizer of Point w.r.t. Topological Group Right Action, Canonical Map from Right Cosets of Stabilizer Quotient Topological Space into Orbit of Point Topological Subspace Is Continuous Bijection <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for stabilizer of point w.r.t. topological group right action, canonical map from right cosets of stabilizer quotient topological space into orbit of point topological subspace is continuous bijection Topics About: group About: https://thebiasplanet.blogspot.com/2026/10/forstabilizerofpointwrttopologicalgrouprightactioncanonicalmapfromrightcosetsofstabilizerquotienttopologicalspaceintoorbitofpointtopologicalsubspaceiscontinuousbijection.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4720372695475706939Sun, 18 Oct 2026 12:13:21 +00002026-10-18T21:13:21.881+09:00Definitions and Propositions2066: For Stabilizer of Point w.r.t. Topological Group Left Action, Canonical Map from Left Cosets of Stabilizer Quotient Topological Space into Orbit of Point Topological Subspace Is Continuous Bijection <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for stabilizer of point w.r.t. topological group left action, canonical map from left cosets of stabilizer quotient topological space into orbit of point topological subspace is continuous bijection Topics About: group About: https://thebiasplanet.blogspot.com/2026/10/forstabilizerofpointwrttopologicalgroupleftactioncanonicalmapfromleftcosetsofstabilizerquotienttopologicalspaceintoorbitofpointtopologicalsubspaceiscontinuousbijection.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4942942392039964470Sun, 18 Oct 2026 12:11:54 +00002026-10-18T21:11:54.308+09:00Definitions and Propositions2065: For Stabilizer of Point w.r.t. Group Right Action, Canonical Map from Right Cosets of Stabilizer Quotient Set into Orbit of Point Is Bijection <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for stabilizer of point w.r.t. group right action, canonical map from right cosets of stabilizer quotient set into orbit of point is bijection Topics About: group The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/10/forstabilizerofpointwrtgrouprightactioncanonicalmapfromrightcosetsofstabilizerquotientsetintoorbitofpointisbijection.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6285049877735058630Sun, 18 Oct 2026 12:10:20 +00002026-10-18T21:10:20.144+09:00Definitions and Propositions2064: For Stabilizer of Point w.r.t. Group Left Action, Canonical Map from Left Cosets of Stabilizer Quotient Set into Orbit of Point Is Bijection <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for stabilizer of point w.r.t. group left action, canonical map from left cosets of stabilizer quotient set into orbit of point is bijection Topics About: group The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/10/forstabilizerofpointwrtgroupleftactioncanonicalmapfromleftcosetsofstabilizerquotientsetintoorbitofpointisbijection.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7616701235856961477Sun, 18 Oct 2026 12:08:55 +00002026-10-18T21:08:55.781+09:00Definitions and Propositions2063: Effective Group Right Action <The previous article in this series | The table of contents of this series | The next article in this series> definition of effective group right action Topics About: group 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 definition of stabilizer of https://thebiasplanet.blogspot.com/2026/10/effectivegrouprightaction.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5376593461868330478Sun, 18 Oct 2026 12:07:36 +00002026-10-18T21:07:36.641+09:00Definitions and Propositions2062: Effective Group Left Action <The previous article in this series | The table of contents of this series | The next article in this series> definition of effective group left action Topics About: group 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 definition of stabilizer of https://thebiasplanet.blogspot.com/2026/10/effectivegroupleftaction.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6688647951943478117Sun, 18 Oct 2026 12:06:17 +00002026-10-18T21:06:17.725+09:00Definitions and Propositions2061: Free Group Right Action <The previous article in this series | The table of contents of this series | The next article in this series> definition of free group right action Topics About: group The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description Starting Context The reader knows a definition of stabilizer of point with https://thebiasplanet.blogspot.com/2026/10/freegrouprightaction.htmlnoreply@blogger.com (Unknown)