Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ...Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .展开更多
To shed some light on the John-Nirenberg space,the authors of this article introduce the John-Nirenberg-Q space via congruent cubes,JNQp,qα(Rn),which,when p=∞and q=2,coincides with the space Qα(Rn)introduced by Ess...To shed some light on the John-Nirenberg space,the authors of this article introduce the John-Nirenberg-Q space via congruent cubes,JNQp,qα(Rn),which,when p=∞and q=2,coincides with the space Qα(Rn)introduced by Essen,Janson,Peng and Xiao in[Indiana Univ Math J,2000,49(2):575-615].Moreover,the authors show that,for some particular indices,JNQp,qα(Rn)coincides with the congruent John-Nirenberg space,or that the(fractional)Sobolev space is continuously embedded into JNQp,qα(Rn).Furthermore,the authors characterize JNQp,qα(Rn)via mean oscillations,and then use this characterization to study the dyadic counterparts.Also,the authors obtain some properties of composition operators on such spaces.The main novelties of this article are twofold:establishing a general equivalence principle for a kind of’almost increasing’set function that is here introduced,and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.展开更多
Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an int...Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.展开更多
As one of important members of refractory materials,tungsten phosphide(WP)holds great potential for fundamental study and industrial applications in many fields of science and technology,due to its excellent propertie...As one of important members of refractory materials,tungsten phosphide(WP)holds great potential for fundamental study and industrial applications in many fields of science and technology,due to its excellent properties such as superconductivity and as-predicted topological band structure.However,synthesis of high-quality WP crystals is still a challenge by using tradition synthetic methods,because the synthesis temperature for growing its large crystals is very stringently required to be as high as 3000℃,which is far beyond the temperature capability of most laboratory-based devices for crystal growth.In addition,high temperature often induces the decomposition of metal phosphides,leading to off-stoichiometric samples based on which the materials'intrinsic properties cannot be explored.In this work,we report a high-pressure synthesis of single-crystal WP through a direct crystallization from cooling the congruent W-P melts at 5 GPa and^3200℃.In combination of x-ray diffraction,electron microscope,and thermal analysis,the crystal structure,morphology,and stability of recovered sample are well investigated.The final product is phase-pure and nearly stoichiometric WP in a single-crystal form with a large grain size,in excess of one millimeter,thus making it feasible to implement most experimental measurements,especially,for the case where a large crystal is required.Success in synthesis of high-quality WP crystals at high pressure can offer great opportunities for determining their intrinsic properties and also making more efforts to study the family of transition-metal phosphides.展开更多
For two triangles to be congruent,SAS theorem requires two sides and the included angle of the first triangle to be congruent to the corresponding two sides and included angle of the second triangle.If the congruent a...For two triangles to be congruent,SAS theorem requires two sides and the included angle of the first triangle to be congruent to the corresponding two sides and included angle of the second triangle.If the congruent angles are not between the corresponding congruent sides,then such triangles could be different.It turns out that it is possible to describe four cases in which triangles are congruent even though congruent angles.For two triangles to be congruent,SAS theorem requires two sides and the included angle of the first triangle to be congruent to the corresponding two sides and included angle of the second triangle.If the congruent angles are not between the corresponding congruent sides,then such triangles could be different.It turns out that it is possible to describe four cases in which triangles are congruent even though congruent angles are not between the corresponding congruent sides.Such a theorem could be named,for example,SSA theorem.Many texts state that two triangles cannot be shown to be congruent if the condition of SSA exists.However,the author describes cases in which such triangles could be proven congruent with the SSA theorem.An immediate consequence of this new understanding is the necessity of revising many problems and answers in high school and college-level texts related to congruent triangles.are not between the corresponding congruent sides.Such a theorem could be named,for example,SSA theorem.An immediate consequence of this new understanding is the necessity of revising many problems and answers in high school and college-level texts related to congruent triangles.展开更多
We give some sufficient conditions for non-congruent numbers in terms of the Monsky matrices.Many known criteria for non-congruent numbers can be viewed as special cases of our results.
We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and (S)n of the elliptic curve En: y2 = x(x - n)(x - 2n) and its dual curve (E)n: y2 =x3 + 6nx2 + n2x have the smallest size: Sn = ...We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and (S)n of the elliptic curve En: y2 = x(x - n)(x - 2n) and its dual curve (E)n: y2 =x3 + 6nx2 + n2x have the smallest size: Sn = {1}, (S)n = {1, 2, n, 2n}. It is well known that for such integer n, the rank of group En(Q) of the rational points on En is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves En with rank zero and such series of integers n are non-congruent numbers.展开更多
Let p ∈ [1, ∞), q ∈ [1, ∞), α∈ R, and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg(1961) and the space B introduced by Bourgain et al.(2015), we introduce a special Joh...Let p ∈ [1, ∞), q ∈ [1, ∞), α∈ R, and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg(1961) and the space B introduced by Bourgain et al.(2015), we introduce a special John-Nirenberg-Campanato space JN^(con)_((p,q,s)) over R^(n) or a given cube of R;with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as p →∞ is just the Campanato space which coincides with the space BMO(the space of functions with bounded mean oscillations)when α = 0. Moreover, a vanishing subspace of this new space is introduced, and its equivalent characterization is established as well, which is a counterpart of the known characterization for the classical space VMO(the space of functions with vanishing mean oscillations) over R^(n) or a given cube of R^(n) with finite side length.Furthermore, some VMO-H^(1)-BMO-type results for this new space are also obtained, which are based on the aforementioned vanishing subspaces and the Hardy-type space defined via congruent cubes in this article. The geometrical properties of both the Euclidean space via its dyadic system and congruent cubes play a key role in the proofs of all these results.展开更多
The development of a formal semantics for a given programming language can proceed in several stages. At each stage we give an alternative semantic definition of the language, and each definition embodies successively...The development of a formal semantics for a given programming language can proceed in several stages. At each stage we give an alternative semantic definition of the language, and each definition embodies successively more and more implementation details. Then we formulate and prove at each stage the congruence conditions between successive definitions in the sequence. This paper presents two formal semantics for Prolog with “cut” and shows the congruence condition between them.展开更多
In this paper,we use the 2-descent method to find a series of odd non-congruent numbers≡1(mod 8)whose prime factors are≡1(mod 4)such that the congruent elliptic curves have second lowest Selmer groups,which include ...In this paper,we use the 2-descent method to find a series of odd non-congruent numbers≡1(mod 8)whose prime factors are≡1(mod 4)such that the congruent elliptic curves have second lowest Selmer groups,which include Li and Tian’s result as special cases.展开更多
We study 2-primary parts ⅢX(E^((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E^((n)): y^2= x^3-n^2x, n ∈Q~×/Q^(×2). Previous results focused on finding sufficient conditions for ⅢX...We study 2-primary parts ⅢX(E^((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E^((n)): y^2= x^3-n^2x, n ∈Q~×/Q^(×2). Previous results focused on finding sufficient conditions for ⅢX(E^((n))/Q)[2~∞]trivial or isomorphic to(Z/2Z)~2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E^((n))is isomorphic to(Z/2Z)～2 and the Mordell-Weil rank of E^((n)) is zero,provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for ⅢX(E^((n))/Q)[2~∞]■(Z/2Z)^(2k), where k≥2.展开更多
Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime ...Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.展开更多
文摘Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .
基金partially supported by the National Natural Science Foundation of China(12122102 and 11871100)the National Key Research and Development Program of China(2020YFA0712900)。
文摘To shed some light on the John-Nirenberg space,the authors of this article introduce the John-Nirenberg-Q space via congruent cubes,JNQp,qα(Rn),which,when p=∞and q=2,coincides with the space Qα(Rn)introduced by Essen,Janson,Peng and Xiao in[Indiana Univ Math J,2000,49(2):575-615].Moreover,the authors show that,for some particular indices,JNQp,qα(Rn)coincides with the congruent John-Nirenberg space,or that the(fractional)Sobolev space is continuously embedded into JNQp,qα(Rn).Furthermore,the authors characterize JNQp,qα(Rn)via mean oscillations,and then use this characterization to study the dyadic counterparts.Also,the authors obtain some properties of composition operators on such spaces.The main novelties of this article are twofold:establishing a general equivalence principle for a kind of’almost increasing’set function that is here introduced,and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.
文摘Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.
基金the National Key Research and Development Program of China(Grant Nos.2016YFA0401503 and 2018YFA0305700)the National Natural Science Foundation of China(Grant No.11575288)+4 种基金the Youth Innovation Promotion Association of Chinese Academy of Sciences(Grant No.2016006)the Key Research Platforms and Research Projects of Universities in Guangdong Province,China(Grant No.2018KZDXM062)the Guangdong Innovative&Entrepreneurial Research Team Program,China(Grant No.2016ZT06C279)the Shenzhen Peacock Plan,China(Grant No.KQTD2016053019134356)the Shenzhen Development&Reform Commission Foundation for Novel Nano-Material Sciences,China,the Research Platform for Crystal Growth&Thin-Film Preparation at SUST,China,and the Shenzhen Development and Reform Commission Foundation for Shenzhen Engineering Research Center for Frontier Materials Synthesis at High Pressure,China.
文摘As one of important members of refractory materials,tungsten phosphide(WP)holds great potential for fundamental study and industrial applications in many fields of science and technology,due to its excellent properties such as superconductivity and as-predicted topological band structure.However,synthesis of high-quality WP crystals is still a challenge by using tradition synthetic methods,because the synthesis temperature for growing its large crystals is very stringently required to be as high as 3000℃,which is far beyond the temperature capability of most laboratory-based devices for crystal growth.In addition,high temperature often induces the decomposition of metal phosphides,leading to off-stoichiometric samples based on which the materials'intrinsic properties cannot be explored.In this work,we report a high-pressure synthesis of single-crystal WP through a direct crystallization from cooling the congruent W-P melts at 5 GPa and^3200℃.In combination of x-ray diffraction,electron microscope,and thermal analysis,the crystal structure,morphology,and stability of recovered sample are well investigated.The final product is phase-pure and nearly stoichiometric WP in a single-crystal form with a large grain size,in excess of one millimeter,thus making it feasible to implement most experimental measurements,especially,for the case where a large crystal is required.Success in synthesis of high-quality WP crystals at high pressure can offer great opportunities for determining their intrinsic properties and also making more efforts to study the family of transition-metal phosphides.
文摘For two triangles to be congruent,SAS theorem requires two sides and the included angle of the first triangle to be congruent to the corresponding two sides and included angle of the second triangle.If the congruent angles are not between the corresponding congruent sides,then such triangles could be different.It turns out that it is possible to describe four cases in which triangles are congruent even though congruent angles.For two triangles to be congruent,SAS theorem requires two sides and the included angle of the first triangle to be congruent to the corresponding two sides and included angle of the second triangle.If the congruent angles are not between the corresponding congruent sides,then such triangles could be different.It turns out that it is possible to describe four cases in which triangles are congruent even though congruent angles are not between the corresponding congruent sides.Such a theorem could be named,for example,SSA theorem.Many texts state that two triangles cannot be shown to be congruent if the condition of SSA exists.However,the author describes cases in which such triangles could be proven congruent with the SSA theorem.An immediate consequence of this new understanding is the necessity of revising many problems and answers in high school and college-level texts related to congruent triangles.are not between the corresponding congruent sides.Such a theorem could be named,for example,SSA theorem.An immediate consequence of this new understanding is the necessity of revising many problems and answers in high school and college-level texts related to congruent triangles.
基金supported by NSFC(Nos.12231009,11971224,12071209).
文摘We give some sufficient conditions for non-congruent numbers in terms of the Monsky matrices.Many known criteria for non-congruent numbers can be viewed as special cases of our results.
基金This work was supported by the National Scientific Research Project 973 of China(Grant No.2004 CB 3180004)the National Natural Science Foundation of China(Grant No.60433050).
文摘We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and (S)n of the elliptic curve En: y2 = x(x - n)(x - 2n) and its dual curve (E)n: y2 =x3 + 6nx2 + n2x have the smallest size: Sn = {1}, (S)n = {1, 2, n, 2n}. It is well known that for such integer n, the rank of group En(Q) of the rational points on En is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves En with rank zero and such series of integers n are non-congruent numbers.
基金supported by National Natural Science Foundation of China(Grant Nos.11971058,12071197 and 11871100)。
文摘Let p ∈ [1, ∞), q ∈ [1, ∞), α∈ R, and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg(1961) and the space B introduced by Bourgain et al.(2015), we introduce a special John-Nirenberg-Campanato space JN^(con)_((p,q,s)) over R^(n) or a given cube of R;with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as p →∞ is just the Campanato space which coincides with the space BMO(the space of functions with bounded mean oscillations)when α = 0. Moreover, a vanishing subspace of this new space is introduced, and its equivalent characterization is established as well, which is a counterpart of the known characterization for the classical space VMO(the space of functions with vanishing mean oscillations) over R^(n) or a given cube of R^(n) with finite side length.Furthermore, some VMO-H^(1)-BMO-type results for this new space are also obtained, which are based on the aforementioned vanishing subspaces and the Hardy-type space defined via congruent cubes in this article. The geometrical properties of both the Euclidean space via its dyadic system and congruent cubes play a key role in the proofs of all these results.
文摘The development of a formal semantics for a given programming language can proceed in several stages. At each stage we give an alternative semantic definition of the language, and each definition embodies successively more and more implementation details. Then we formulate and prove at each stage the congruence conditions between successive definitions in the sequence. This paper presents two formal semantics for Prolog with “cut” and shows the congruence condition between them.
基金supported by National Natural Science Foundation of China(Grant No.11171317)National Key Basic Research Program of China(Grant No.2013CB834202)
文摘In this paper,we use the 2-descent method to find a series of odd non-congruent numbers≡1(mod 8)whose prime factors are≡1(mod 4)such that the congruent elliptic curves have second lowest Selmer groups,which include Li and Tian’s result as special cases.
文摘We study 2-primary parts ⅢX(E^((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E^((n)): y^2= x^3-n^2x, n ∈Q~×/Q^(×2). Previous results focused on finding sufficient conditions for ⅢX(E^((n))/Q)[2~∞]trivial or isomorphic to(Z/2Z)~2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E^((n))is isomorphic to(Z/2Z)～2 and the Mordell-Weil rank of E^((n)) is zero,provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for ⅢX(E^((n))/Q)[2~∞]■(Z/2Z)^(2k), where k≥2.
基金supported by National Natural Science Foundation of China (Grant No. 11501541)
文摘Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.
