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User:Jianing Song
- Which of 222Rn and 222Fr has lower energy?
- Which of the state and the state of 214At has lower energy? (levels of 214At)
- Which of 247Cm and the state of 247Bk has lower energy? (levels of 247Bk)
- Which of 259Fm and 259Md has lower energy?
Note the error margins! These questions do not have decisive answers by now.
Important: Please kindly contact me by jianing.song@polytechnique.edu. My old email address sjn1508@163.com is no longer in use.
Replacing and in linear algebra
Traditionally, various topics on linear algebra assume that the base field of study is either or . But wait, we put too many restrictions on ourselves!
Definition. An ordered field is called archimedean if the integers are unbounded. An ordered field is called an Euclidean field if every positive element is a square. A field is called quadratically closed if every element is a square. A field is called real-closed if is not a square , and is algebraically closed.
We note that a field can be turned into an Euclidean field if and only if is not a square in , and is quadratically closed.
Theory on inner product: can be replaced by any archimedean Euclidean field. Technically inner products can be defined on every ordered field, but in order to allow Gram–Schmidt process we need that every positive element is a square, so it's better if the base field is Euclidean. Moreover, we want that a norm satisfying parallelogram law is induced by an inner product, so we want to be dense in the base field, which means that it's best if the base field is also archimedean. (If we do not care about norms, of course an Euclidean field is enough.) I didn't figure out how we could replace . I presume that it can be replaced by an arbitrary quadratically closed field with characteristic 0 and cardinality , but must such a field have an archimedean Euclidean subfield of index 2? (Anyway, must a quadratically closed field with characteristic 0 have an Euclidean subfield of index 2? Of course this is true for an algebraically closed field with characteristic 0: see the next paragraph.)
Theory on orthogonal/unitary diagonalization: can be replaced by any real-closed field, on which every symmetric matrix is diagonalizable. In fact real-closedness is a sufficient but not necessary condition (see here), but for the sake of simplicity we will assume it. can be replaced by any algebraically closed field with characteristic 0, because such a field must have a subfield of index 2 (which must be real closed; see here), so an automorphism of order 2 which looks like the ordinary complex conjugation.
Theory on binary forms: can be replaced by any Euclidean field, which is precisely one of the two kinds of fields on which the Sylvester law of inertia holds (see here). This means that every symmetric matrix on an Euclidean field is congruent to a diagonal matrix consisting of 1s, (-1)'s, and 0s, and the occurence of each number is independent of the congruence factor. can be replaced by any quadratically closed field (the other kind of fields on which the Sylvester law of inertia is valid; see the link aformentioned in this paragraph): every symmetric matrix on such a field is is congruent to a diagonal matrix consisting of 1s and 0s, where the number of 1s is equal to the rank of the matrix.
Character values of representations of finite group
Let be a field. What is the set of all numbers of the form , where is a representation of a finite group in , and is an element in ?
Lemma. Let be an algebraically closed field, and be the set of elements in that can be expressed as a sum of roots of unity. If is of characteristic , then , where is the prime field of and is the algebraic closure of within . If is of characteristic 0, then , where is the prime field of , is the extension of generated by all roots of unity, and is the set of algebraic integers in ,
Proof. The inclusion is clear. For the case characteristic , every element in is automatically a root of unity, so lies in . For characteristic 0, each element in lies in some , so it is a sum of roots of unity (possibly with repetitions).
Since is of finite order, is also of finite order, so the eigenvalues of in (an algebraic closure of ) must be roots of unity, which means that a character value must be an element in which can be expressed as a sum of roots of unity in .
If is of characteristic , let be the prime field of , then the lemma above tells us that a character value lies in , which is the maximal algebraic subextension of within . Since every element in is itself a root of unity, every element in is automatically a character value of a representation of dimension 1 of a finite cyclic group in .
If is of characteristic 0, let be the prime field of , then the lemma above tells us that a character value lies in (note that is the maximal abelian subextension of within ). It remains to show that if an element in which can be expressed as a sum of roots of unity in , then it must occur as a character value. ...
Factoring rational primes on the quadratic number field with discriminant
- D = -376. Decomposing: A191056; remaining inert: A191086.
- D = -344. Decomposing: A191051; remaining inert: A191083.
- D = -312. Decomposing: A191047; remaining inert: A191080.
- D = -280. Decomposing: A191043; remaining inert: A191078.
- D = -248. Decomposing: A191040; remaining inert: A191076.
- D = -184. Decomposing: A191032; remaining inert: A191071.
- D = -163. Decomposing: A296921; remaining inert: A296915; not remaining inert: A257362.
- D = -152. Decomposing: A191028; remaining inert: A191069.
- D = -120. Decomposing: A191023; remaining inert: A191066.
- D = -95. Decomposing: A191057; remaining inert: A191087.
- D = -91. Decomposing: A191054; remaining inert: A191085.
- D = -88. Decomposing: A191020; remaining inert: A191064.
- D = -87. Decomposing: A191052; remaining inert: A191084.
- D = -83. Decomposing: A191050; remaining inert: A191082.
- D = -79. Decomposing: A191048; remaining inert: A191081.
- D = -71. Decomposing: A191044; remaining inert: A191079.
- D = -68. Decomposing: A296929; remaining inert: A296930; not remaining inert: A296931.
- D = -67. Decomposing: A191041; remaining inert: A191077; not remaining inert: A106933.
- D = -59. Decomposing: A191038; remaining inert: A191075.
- D = -56. Decomposing: A191017; remaining inert: A191061; not decomposing: A274504.
- D = -55. Decomposing: A191036; remaining inert: A191074.
- D = -52. Decomposing: A296926; remaining inert: A296927; not remaining inert: A296928 U {2}.
- D = -51. Decomposing: A191034; remaining inert: A191073.
- D = -47. Decomposing: A191033; remaining inert: A191072.
- D = -43. Decomposing: A191031; remaining inert: A184902; not remaining inert: A106891.
- D = -40. Decomposing: A155488; remaining inert: A296925; not remaining inert: A293859.
- D = -39. Decomposing: A191029; remaining inert: A191070.
- D = -35. Decomposing: A191026; remaining inert: A191068.
- D = -31. Decomposing: A191024; remaining inert: A191067.
- D = -24. Decomposing: A157437; remaining inert: A191059; not remaining inert: A296924.
- D = -23. Decomposing: A191021; remaining inert: A191065; not remaining inert: A296932.
- D = -20. Decomposing: A139513; remaining inert: A003626; not remaining inert: A240920 = A296922 U {2}; not decomposing: A296923 U {5}.
- D = -19. Decomposing: A191019; remaining inert: A191063; not remaining inert: A106863.
- D = -15. Decomposing: A191018; remaining inert: A191062.
- D = -11. Decomposing: A296920; remaining inert: A191060; not remaining inert: A056874.
- D = -8. Decomposing: A033200; remaining inert: A003628; not remaining inert: A033203; not decomposing: A045355.
- D = -7. Decomposing: A045386; remaining inert: A003625; not remaining inert: A045373; not decomposing: A045399.
- D = -4. Decomposing: A002144; remaining inert: A002145; not remaining inert: A002313; not decomposing: A045326.
- D = -3. Decomposing: A002476; remaining inert: A003627 = A007528 U {2}; not remaining inert: A007645; not decomposing: A045309 = A045410 U {2}.
- D = 5. Decomposing: A045468 = A064739 \ {2}; remaining inert: A003631 = A097957 U {2}; not remaining inert: A038872; not decomposing: A042993.
- D = 8. Decomposing: A001132 = A097958 \ {3}; remaining inert: A003629; not remaining inert: A038873; not decomposing: A042999.
- D = 12. Decomposing: A097933; remaining inert: A003630; not remaining inert: A038874 = A296933 U {2}; not decomposing: A038875 U {3}.
- D = 13. Decomposing: A296937; remaining inert: A038884; not remaining inert: A038883; not decomposing: A120330.
- D = 17. Decomposing: A296938; remaining inert: A038890; not remaining inert: A038889.
- D = 21. Remaining inert: A038894; not remaining inert: A038893.
- D = 24. Decomposing: A097934; remaining inert: A038877; not remaining inert: A038876.
- D = 28. Decomposing: A296934; remaining inert: A003632; not remaining inert: A038878.
- D = 29. Decomposing: A191022; remaining inert: A038902; not remaining inert: A038901.
- D = 33. Remaining inert: A038908; not remaining inert: A038907.
- D = 37. Decomposing: A191027; remaining inert: A038914; not remaining inert: A038913.
- D = 40. Decomposing: A097955; remaining inert: A038880; not remaining inert: A038879.
- D = 41. Decomposing: A191030; remaining inert: A038920; not remaining inert: A038919.
- D = 44. Decomposing: A296935; remaining inert: A296936; not remaining inert: A038881 U {2}; not decomposing: A038882 U {11}.
- D = 53. Decomposing: A191035; remaining inert: A038932; not remaining inert: A038931.
- D = 56. Remaining inert: A038886; not remaining inert: A038885.
- D = 57. Remaining inert: A038936; not remaining inert: A038935.
- D = 60. Decomposing: A097956; remaining inert: A038888; not remaining inert: A038887.
- D = 61. Decomposing: A191039; remaining inert: A038942; not remaining inert: A038941.
- D = 65. Remaining inert: A038946; not remaining inert: A038945.
- D = 69. Decomposing: A191042; remaining inert: A038952; not remaining inert: A038951.
- D = 73. Decomposing: A191045; remaining inert: A038958; not remaining inert: A038957.
- D = 76. Decomposing: A297175; remaining inert: A297176 = A038892 \ {2}; not remaining inert: A038891 U {2}.
- D = 77. Remaining inert: A038962; not remaining inert: A038961.
- D = 85. Remaining inert: A038972; not remaining inert: A038971.
- D = 88. Remaining inert: A038896; not remaining inert: A038895.
- D = 89. Decomposing: A191053; remaining inert: A038978; not remaining inert: A038977.
- D = 92. Decomposing: A297177; remaining inert: A038898; not remaining inert: A038897.
- D = 93. Decomposing: A191055; remaining inert: A038982; not remaining inert: A038981.
- D = 97. Decomposing: A191058; remaining inert: A038988; not remaining inert: A038987.
- D = 104. Remaining inert: A038900; not remaining inert: A038899.
- D = 120. Decomposing: A097959; remaining inert: A038904; not remaining inert: A038903.
- D = 124. Remaining inert: A038906; not remaining inert: A038905.
- D = 136. Decomposing: A191025; remaining inert: A038910; not remaining inert: A038909.
- D = 140. Remaining inert: A038912 \ {2}; not remaining inert: A038911 U {2}.
- D = 152. Remaining inert: A038916; not remaining inert: A038915.
- D = 156. Remaining inert: A038918; not remaining inert: A038917.
- D = 168. Remaining inert: A038922; not remaining inert: A038921.
- D = 172. Remaining inert: A038924 \ {2}; not remaining inert: A038923 U {2}.
- D = 184. Remaining inert: A038926; not remaining inert: A038925.
- D = 188. Remaining inert: A038928; not remaining inert: A038927.
- D = 204. Remaining inert: A038930 \ {2}; not remaining inert: A038929 U {2}.
- D = 220. Remaining inert: A038934; not remaining inert: A038933.
- D = 232. Decomposing: A191037; remaining inert: A038938; not remaining inert: A038937.
- D = 236. Remaining inert: A038940 \ {2}; not remaining inert: A038939 U {2}.
- D = 248. Remaining inert: A038944; not remaining inert: A038943.
- D = 264. Remaining inert: A038948; not remaining inert: A038947.
- D = 268. Remaining inert: A038950 \ {2}; not remaining inert: A038949 U {2}.
- D = 280. Remaining inert: A038954; not remaining inert: A038953.
- D = 284. Remaining inert: A038956; not remaining inert: A038955.
- D = 296. Decomposing: A191046; remaining inert: A038960; not remaining inert: A038959.
- D = 312. Remaining inert: A038964; not remaining inert: A038963.
- D = 316. Remaining inert: A038966; not remaining inert: A038965.
- D = 328. Decomposing: A191049; remaining inert: A038968; not remaining inert: A038967.
- D = 332. Remaining inert: A038970 \ {2}; not remaining inert: A038969 U {2}.
- D = 344. Remaining inert: A038974; not remaining inert: A038973.
- D = 348. Remaining inert: A038976; not remaining inert: A038975 U {2}.
- D = 364. Remaining inert: A038980 \ {2}; not remaining inert: A038979 U {2}.
- D = 376. Remaining inert: A038984; not remaining inert: A038983.
- D = 380. Remaining inert: A038986; not remaining inert: A038985.