We classify completely three-generator finite p-groups G such that Ф(G)≤Z(G)and|G′|≤p2.This paper is a part of the classification of finite p-groups with a minimal non-abelian subgroup of index p,and solve partly ...We classify completely three-generator finite p-groups G such that Ф(G)≤Z(G)and|G′|≤p2.This paper is a part of the classification of finite p-groups with a minimal non-abelian subgroup of index p,and solve partly a problem proposed by Berkovich.展开更多
This paper finishes the classification of three-generator finite p-groups G such that Φ(G) Z(G).This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p, and partly sol...This paper finishes the classification of three-generator finite p-groups G such that Φ(G) Z(G).This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p, and partly solves a problem proposed by Berkovich(2008).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.
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展开更多
In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.
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
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 D = p1p2 …pm, where p1,p2, ……,pm are distinct rational primes with p1 ≡p2 ≡3（mod 8）, pi =1（mod 8）（3 ≤ i ≤ m）, and m is any positive integer. In this paper, we give a simple combinatorial criterion fo...Let D = p1p2 …pm, where p1,p2, ……,pm are distinct rational primes with p1 ≡p2 ≡3（mod 8）, pi =1（mod 8）（3 ≤ i ≤ m）, and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the complex L-function of the congruent elliptic curve ED2 ： y^2 = x^3- D^2x at s = 1, divided by the period ω defined below, to be exactly divisible by 2^2m-2, the second lowest 2-power with respect to the number of the Gaussian prime factors of D. As a corollary, we obtain a new series of non-congruent numbers whose prime factors can be arbitrarily many. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer.展开更多
Huntington’s disease(HD)is a genetic neurodegenerative disorder that affects not only the motor but also the cognitive domain.In particular,cognitive symptoms such as impaired executive skills and deficits in recogni...Huntington’s disease(HD)is a genetic neurodegenerative disorder that affects not only the motor but also the cognitive domain.In particular,cognitive symptoms such as impaired executive skills and deficits in recognizing other individuals’mental state may emerge many years before the motor symptoms.This study was aimed at testing two cognitive hypotheses suggested by previous research with a new Stroop task created for the purpose:1)the impairment of emotion recognition in HD is moderated by the emotions’valence,and 2)inhibitory control is impaired in HD.Forty manifest and 20 pre-manifest HD patients and their age-and gender-matched controls completed both the traditional“Stroop Color and Word Test”(SCWT)and the newly created“Stroop Emotion Recognition under Word Interference Task”(SERWIT),which consist in 120 photographs of sad,calm,or happy faces with either congruent or incongruent word interference.On the SERWIT,impaired emotion recognition in manifest HD was moderated by emotion type,with deficits being larger in recognizing sadness and calmness than in recognizing happiness,but it was not moderated by stimulus congruency.On the SCWT,six different interference scores yielded as many different patterns of group effects.Overall our results corroborate the hypothesis that impaired emotion recognition in HD is moderated by the emotions’valence,but do not provide evidence for the hypothesis that inhibitory control is impaired in HD.Further research is needed to learn more about the psychological mechanisms underlying the moderating effect of emotional valence on impaired emotion recognition in HD,and to corroborate the hypothesis that the inhibitory processes involved in Stroop tasks are not impaired in HD.Looking beyond this study,the SERWIT promises to make important contributions to disentangling the cognitive and the psychomotor aspects of neurological disorders.The research was approved by the Ethics Committee of the“Istituto Leonarda Vaccari”,Rome on January 24,2018.展开更多
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the uni...In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on ?by using the rotation group [3] [4]. It will be proved that the group acts on elements of ?in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation ?in terms of matrix operations using the operator and the Hadamard Product;this construction is consistent with the group operation defined in the first article.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11371232)Natural Science Foundation of Shanxi Province (Grant Nos. 2012011001-3 and 2013011001-1)
文摘We classify completely three-generator finite p-groups G such that Ф(G)≤Z(G)and|G′|≤p2.This paper is a part of the classification of finite p-groups with a minimal non-abelian subgroup of index p,and solve partly a problem proposed by Berkovich.
基金supported by National Natural Science Foundation of China(Grant No.11371232)Natural Science Foundation of Shanxi Province(Grant Nos.2012011001-3 and 2013011001-1)
文摘This paper finishes the classification of three-generator finite p-groups G such that Φ(G) Z(G).This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p, and partly solves a problem proposed by Berkovich(2008).
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.
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
文摘In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.
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.
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.
文摘Let D = p1p2 …pm, where p1,p2, ……,pm are distinct rational primes with p1 ≡p2 ≡3（mod 8）, pi =1（mod 8）（3 ≤ i ≤ m）, and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the complex L-function of the congruent elliptic curve ED2 ： y^2 = x^3- D^2x at s = 1, divided by the period ω defined below, to be exactly divisible by 2^2m-2, the second lowest 2-power with respect to the number of the Gaussian prime factors of D. As a corollary, we obtain a new series of non-congruent numbers whose prime factors can be arbitrarily many. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer.
基金“Fondazione Cattolica Assicurazione” for funding the observational research of LIRH Foundation (www. lirh.it)
文摘Huntington’s disease(HD)is a genetic neurodegenerative disorder that affects not only the motor but also the cognitive domain.In particular,cognitive symptoms such as impaired executive skills and deficits in recognizing other individuals’mental state may emerge many years before the motor symptoms.This study was aimed at testing two cognitive hypotheses suggested by previous research with a new Stroop task created for the purpose:1)the impairment of emotion recognition in HD is moderated by the emotions’valence,and 2)inhibitory control is impaired in HD.Forty manifest and 20 pre-manifest HD patients and their age-and gender-matched controls completed both the traditional“Stroop Color and Word Test”(SCWT)and the newly created“Stroop Emotion Recognition under Word Interference Task”(SERWIT),which consist in 120 photographs of sad,calm,or happy faces with either congruent or incongruent word interference.On the SERWIT,impaired emotion recognition in manifest HD was moderated by emotion type,with deficits being larger in recognizing sadness and calmness than in recognizing happiness,but it was not moderated by stimulus congruency.On the SCWT,six different interference scores yielded as many different patterns of group effects.Overall our results corroborate the hypothesis that impaired emotion recognition in HD is moderated by the emotions’valence,but do not provide evidence for the hypothesis that inhibitory control is impaired in HD.Further research is needed to learn more about the psychological mechanisms underlying the moderating effect of emotional valence on impaired emotion recognition in HD,and to corroborate the hypothesis that the inhibitory processes involved in Stroop tasks are not impaired in HD.Looking beyond this study,the SERWIT promises to make important contributions to disentangling the cognitive and the psychomotor aspects of neurological disorders.The research was approved by the Ethics Committee of the“Istituto Leonarda Vaccari”,Rome on January 24,2018.
文摘In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on ?by using the rotation group [3] [4]. It will be proved that the group acts on elements of ?in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation ?in terms of matrix operations using the operator and the Hadamard Product;this construction is consistent with the group operation defined in the first article.
