成果名称:Shapley's Conjecture on the Cores of Abstract Market Games主要作者:曹志刚,秦承忠,杨晓光奖项类别:著作论文奖获奖等级:二等奖获奖论文《Shapley's Conjecture on the Cores of Abstract Market Games》发表于博...成果名称:Shapley's Conjecture on the Cores of Abstract Market Games主要作者:曹志刚,秦承忠,杨晓光奖项类别:著作论文奖获奖等级:二等奖获奖论文《Shapley's Conjecture on the Cores of Abstract Market Games》发表于博弈论领域顶级期刊《Games and Economic Behavior》2018年第2期。论文研究成果初步解决了诺贝尔经济学奖获得者罗伊德·沙普利(Lloyd S. Shapley)提出的抽象市场博弈核非空的猜想。展开更多
We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through developm...We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through development of a rigorous quantum complexity operator formalism, we show that mirror pairs must have equivalent complexity spectra and that the SYZ fibration naturally preserves these computational invariants while implementing the required geometric transformations. Our proof proceeds by first establishing a precise mathematical framework connecting quantum complexity with geometric structures, then demonstrating that the special Lagrangian torus fibration preserves computational complexity at both local and global levels, and finally proving that this preservation necessarily implies the geometric correspondences required by the SYZ conjecture. This approach not only resolves the conjecture but reveals deeper insights about the relationship between computation and geometry in string theory. We introduce new complexity-based invariants for studying mirror symmetry and demonstrate how our framework extends naturally to related geometric structures.展开更多
We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show tha...We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show that its unique nonnegative fixed point is globally attractive.As application,our results not only improve many known ones,but also solve several“Open Problems and Conjectures”given by Professors Ladas and Camouzis,et al.展开更多
In this paper,a formula is given. The formula gives the number of prime number solutions of the indefinite equation p 1+p 2=2n , and based on it, an equivalent proposition to the conjecture of Goldbach is obtained.
The well-known Yau＇s uniformization conjecture states that any complete noncompact Kahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal ...The well-known Yau＇s uniformization conjecture states that any complete noncompact Kahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed, by G. Liu in [23]. In the first part, we will give a survey on thc progress. In the second part, we will consider Yau＇s conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number Cn1 is an essential condition to solve Yau＇s conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, Cn1 is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau＇s uniformization conjecture on Kahler manifolds with minimal volume growth.展开更多
A new structure with the special property that instantaneous state and catas-trophes is imposed to ordinary birth-death processes is considered. Kendall's conjecture forthe processes is proved to be right.
Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function &...Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function <img alt="" src="Edit_8fcdfff5-6b95-42a4-8f47-2cabe2723dfc.bmp" />, <img alt="" src="Edit_6ce3a4bd-4c68-49e5-aabe-dec3e904e282.bmp" />, <img alt="" src="Edit_29ea252e-a81e-4b21-a41c-09209c780bb2.bmp" /> by geometric analysis, which has the symmetry: v=0 if β=0, and basic expression <img alt="" src="Edit_bc7a883f-312d-44fd-bcdd-00f25c92f80a.bmp" />. We show that |u| is single peak in each root-interval <img alt="" src="Edit_d7ca54c7-4866-4419-a4bd-cbb808b365af.bmp" /> of u for fixed β ∈(0,1/2]. Using the slope u<sub>t</sub>, we prove that v has opposite signs at two end-points of I<sub>j</sub>. There surely exists an inner point such that , so {|u|,|v|/β} form a local peak-valley structure, and have positive lower bound <img alt="" src="Edit_bac1a5f6-673e-49b6-892c-5adff0141376.bmp" /> in I<sub>j</sub>. Because each t must lie in some I<sub>j</sub>, then ||ξ|| > 0 is valid for any t (i.e. RH is true). Using the positivity <img alt="" src="Edit_83c3d2cf-aa7e-4aba-89f5-0eb44659918a.bmp" /> of Lagarias (1999), we show the strict monotone <img alt="" src="Edit_87eb4e9e-bc7b-43e3-b316-5dcf0efaf0d5.bmp" /> for β > β<sub>0</sub> ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.展开更多
Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequalit...Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty＇s conjectured projection inequality and the Lp-Petty projection inequality are given, respectively.展开更多
In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product P;×P;of two strongly geometrically bounded symplectic manifolds under some conditions with P;. In particula...In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product P;×P;of two strongly geometrically bounded symplectic manifolds under some conditions with P;. In particular, if N is a closed manifold or a noncompact manifold of finite topological type, our result implies that the Weinstein conjecture in CP;×T*N holds.展开更多
This paper proves Riemann conjecture (RH), <em>i.e</em>., that all the zeros in critical region of Riemann <span style="white-space:nowrap;"><em><span style="white-space:nowra...This paper proves Riemann conjecture (RH), <em>i.e</em>., that all the zeros in critical region of Riemann <span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><em>ξ</em><span style="white-space:normal;"> </span></span></em></span>-function lie on symmetric line <span style="white-space:nowrap;"><em>σ</em></span> =1/2 . Its proof is based on two important properties: the symmetry and alternative oscillation for <span style="white-space:nowrap;"><em><em>ξ</em><span style="white-space:normal;"> </span></em>=<em> u </em>+<em> iv</em></span> . Denote <img src="Edit_317839cd-bad0-44d8-b081-c473bcb336f1.png" width="170" height="15" alt="" />. Riemann proved that u is real and <em>v</em> <span style="white-space:nowrap;">≡ </span>0 for <span style="white-space:nowrap;"><em><span style="white-space:nowrap;">β</span></em> =0</span> (the symmetry). We prove that the zeros of u and v for <em>β</em> <span style="white-space:nowrap;">> 0</span> are alternative, so <span style="white-space:nowrap;"><em>u</em> (<em>t</em>,0)</span> is the single peak. A geometric model was proposed. <img src="Edit_27688061-de42-4bce-ad80-6fb3dd1e3d4b.png" width="85" height="27" alt="" /> is called the root-interval of <em>u </em>(<em>t</em>,<em style="white-space:normal;">β</em>) , if |<span style="white-space:nowrap;"><em>u</em>| <em>> </em>0</span> is inside <em>I</em><sub><em>j</em> </sub>and <span style="white-space:nowrap;"><em>u</em> = 0</span> is at its two ends. If |<em>u</em> (<em style="white-space:normal;">t</em><span style="white-space:normal;">,</span><em style="white-space:normal;">β</em>)| has only one peak on each <em style="white-space:normal;">I</em><sub style="white-space:normal;"><em>j</em></sub>, which is called the single peak, else called multiple peaks (it will be proved that the multiple peaks do not exist). The important expressions of u and v for <em style="white-space:normal;">β</em><span style="white-space:normal;"> </span>> 0 were derived. By <img src="Edit_b6369c2e-6a6d-4e1a-8a75-00d743cecaf1.png" width="240" height="28" alt="" />, the peak <em style="white-space:normal;">u </em><span style="white-space:normal;">(</span><em style="white-space:normal;">t</em><span style="white-space:normal;">,</span><em style="white-space:normal;">β</em><span style="white-space:normal;">)</span> will develop toward its convex direction. Besides, <em style="white-space:normal;">u<sub>t</sub> </em><span style="white-space:normal;">(</span><em style="white-space:normal;">t</em><span style="white-space:normal;">,</span><em style="white-space:normal;">β</em><span style="white-space:normal;">)</span> has opposite signs at two ends <em>t</em> = <em>t<sub>j</sub></em><sub> </sub>, <em>t<sub>j+1</sub></em> of <em>I<sub>j </sub></em>, <img src="Edit_be3f0d63-1d24-4165-ac2c-141c9a47d1c8.png" width="145" height="28" alt="" /> also does, then there exists some inner point <span style="white-space:nowrap;"><em>t</em>′</span> such that <span style="white-space:nowrap;"><em>v</em><em></em> (<em>t′</em>,<em>β</em>) = 0</span>. Therefore {|<em>u</em>|,|<em>v</em>|/<em>β</em>} in <em>I<sub>j</sub></em><sub> </sub>form a peak-valley structure such that <img src="Edit_70bb530a-662f-464a-b3c8-4d5625fbf679.png" width="180" height="22" alt="" /> has positive lower bound independent of <em>t</em> <span style="white-space:nowrap;">∈ </span><em>I<sub>j</sub></em><sub> </sub>(<em>i.e</em>. RH holds in <em style="white-space:normal;">I<sub>j</sub></em><sub style="white-space:normal;"> </sub>). As <em style="white-space:normal;">u </em><span style="white-space:normal;">(</span><em style="white-space:normal;">t</em><span style="white-space:normal;">,</span><em style="white-space:normal;">β</em><span style="white-space:normal;">)</span> does not have the finite condensation point (unless <span style="white-space:nowrap;"><em>u</em> = <em>cons</em><em>t</em>.</span>), any finite t surely falls in some <em style="white-space:normal;">I<sub>j</sub></em><sub style="white-space:normal;"> </sub>, then <img src="Edit_166a9981-aac8-476b-a29a-496763297b35.png" width="50" height="23" alt="" /> holds for any t (RH is proved). Our previous paper “Local geometric proof of Riemann conjecture” (APM, V.10:8, 2020) has two defects, this paper has amended these defects and given a complete proof of RH.展开更多
This paper does not claim to prove the Goldbach conjecture, but it does provide a new way of proof (LiKe sequence);And in detailed introduces the proof process of this method: by indirect transformation, Goldbach conj...This paper does not claim to prove the Goldbach conjecture, but it does provide a new way of proof (LiKe sequence);And in detailed introduces the proof process of this method: by indirect transformation, Goldbach conjecture is transformed to prove that, for any odd prime sequence (3, 5, 7, <span style="font-size:12px;white-space:nowrap;">…</span>, <em>P<sub>n</sub></em>), there must have no LiKe sequence when the terms must be less than 3 <span style="font-size:12px;white-space:nowrap;">×</span> <em>P<sub>n</sub></em>. This method only studies prime numbers and corresponding composite numbers, replaced the relationship between even numbers and indeterminate prime numbers. In order to illustrate the importance of the idea of transforming the addition problem into the multiplication problem, we take the twin prime conjecture as an example and know there must exist twin primes in the interval [3<em>P<sub>n</sub></em>, <span><em>P</em></span><sup>2</sup><sub style="margin-left:-8px;"><em>n</em></sub>]. This idea is very important for the study of Goldbach conjecture and twin prime conjecture. It’s worth further study.展开更多
The stability conjecture of Cauchy horizons of black holes suggested by Helliwell and Konkowski is used to investigate the 1+1-dimensional(2D)black holes under perturbations of infalling null dust and of both infallin...The stability conjecture of Cauchy horizons of black holes suggested by Helliwell and Konkowski is used to investigate the 1+1-dimensional(2D)black holes under perturbations of infalling null dust and of both infalling and outgoing null dust.The result given by this conjecture agrees with that from mass inflation scenario for 2D charged dilaton black hole.For 2D black holes,we show that the Cauchy horizons are unstable and the corresponding singularities exist.展开更多
In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C-1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved...In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C-1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved still holding for it. As a consequence some results on Weinstein conjecture are generalized to C-1-smooth hypersurface of contact type.展开更多
Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t −iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product ex...Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t −iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product expression , where z<sub>j</sub> are all roots of ξ(z), including complex roots. He proposed Riemann conjecture (RC): All roots of ξ(z) are real. As the product expression can only be used as a tool of contradiction, we prove RC by contradiction. To avoid the zeros of ξ(1/2 + it), define a subset . We have basic estimate , on L (R). One can construct by all real roots t<sub>j</sub> of ξ(t). If ξ has no complex roots, then w(t) = G(s)ξ(s) for s = 1/2 + it. If the product expression has a complex root z'=t' −iα, where 0 a ≤ 1/2, R' = |z′| > 10, then ξ(z) has four complex roots ±(t′ ± iα), and should contain fourth order factor p(z), i.e. ξ(z) = w(z)p(z). But p(z) can not be contained in ξ(s), as we have on L(R) and p(t) ≥ 0.5(t/R)<sup>4</sup> . As a result, we can rewrite ξ(t) = w(t)p(t) =G(s)ξ(s)p(t) on and get This contradicts the basic estimate. Therefore ξ(z) has no complex roots and RC holds.展开更多
In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth col...In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, thus we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of <span style="white-space:nowrap;">σ(<em>n</em>)</span> , where <img src="Edit_dad9267d-3c54-455b-b30e-63819c207e54.png" width="300" height="117" alt="" /> Then, with the help of the infinite-row-four-column-matrix, we continue to use the recursive method to prove this conjecture strictly.展开更多
In this paper,we study the thermodynamics and the weak cosmic censorship conjecture of the nonlinear electrodynamics black hole under the scattering of a charged complex scalar field.According to the energy and charge...In this paper,we study the thermodynamics and the weak cosmic censorship conjecture of the nonlinear electrodynamics black hole under the scattering of a charged complex scalar field.According to the energy and charge fluxes of the scalar field,the variations of this black hole’s energy and charge can be calculated during an infinitesimal time interval.With scalar field scattering,the variation of the black hole is calculated in the extended and normal phase spaces.In the normal phase space,the cosmological constant and the normalization parameter are fixed,and the first and second laws of thermodynamics can also be satisfied.In the extended phase space,the cosmological constant and the normalization parameter are considered as thermodynamic variables,and the first law of thermodynamics is valid,but the second law of thermodynamics is not valid.Furthermore,the weak cosmic censorship conjecture is both valid in the extended and normal phase spaces.展开更多
Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which im...Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which implies a large tensor-to-scalar ratio is strongly in tension with recent observational results. In this paper, we explore the possibility the swampland conjectures could be compatible with single-field inflationary scenarios if the effects due to the quantum theory of gravity are considered. We show that the quantum gravitational effects due to the nonlinear dispersion relation provides significant modifications on the amplitude of both the scalar and tensor perturbation spectra. Such modifications could be either raise or reduce the perturbation spectra depending on the values of the parameters in the nonlinear terms of the dispersion relations. Therefore, these effects can reduce the tensor-to-scalar ratio to a smaller value, which helps to relax the tension between the swampland conjecture and observational data.展开更多
In this paper, we consider some problems involving Strong Lemoine Conjecture in additive number theory. Based on Dusart's inequality and Rosser-Schoenfeld's inequality, we obtain several new results and give a...In this paper, we consider some problems involving Strong Lemoine Conjecture in additive number theory. Based on Dusart's inequality and Rosser-Schoenfeld's inequality, we obtain several new results and give an equivalent form of Strong Lemoine Conjecture.展开更多
Terai presented the following conjecture: Ifa2 + b2 = c2 vith a, b, c ∈ N, gcd (a, b, c) = 1 and a even, then the diophantine equation x2 + bm = cn has the only peitive integral solution (x, m, n ) = (a,2, 2). In thi...Terai presented the following conjecture: Ifa2 + b2 = c2 vith a, b, c ∈ N, gcd (a, b, c) = 1 and a even, then the diophantine equation x2 + bm = cn has the only peitive integral solution (x, m, n ) = (a,2, 2). In this paper we prove that if c is a prime power, b 1 (mod 8) and b 1 (mod 16) if b2 + 1 = 2c, then Terai’s conjecture holds.展开更多
Erdosa and Sós conjectured in 1963 that if G is a graph o ofof ordeq >1/2p(k - 1), then G contains every tree of size k. It is shown in this paper that the conjecture is true if the complement G of G contains ...Erdosa and Sós conjectured in 1963 that if G is a graph o ofof ordeq >1/2p(k - 1), then G contains every tree of size k. It is shown in this paper that the conjecture is true if the complement G of G contains no a copy of K3 as an induced subgraph of G.展开更多
文摘成果名称:Shapley's Conjecture on the Cores of Abstract Market Games主要作者:曹志刚,秦承忠,杨晓光奖项类别:著作论文奖获奖等级:二等奖获奖论文《Shapley's Conjecture on the Cores of Abstract Market Games》发表于博弈论领域顶级期刊《Games and Economic Behavior》2018年第2期。论文研究成果初步解决了诺贝尔经济学奖获得者罗伊德·沙普利(Lloyd S. Shapley)提出的抽象市场博弈核非空的猜想。
文摘We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through development of a rigorous quantum complexity operator formalism, we show that mirror pairs must have equivalent complexity spectra and that the SYZ fibration naturally preserves these computational invariants while implementing the required geometric transformations. Our proof proceeds by first establishing a precise mathematical framework connecting quantum complexity with geometric structures, then demonstrating that the special Lagrangian torus fibration preserves computational complexity at both local and global levels, and finally proving that this preservation necessarily implies the geometric correspondences required by the SYZ conjecture. This approach not only resolves the conjecture but reveals deeper insights about the relationship between computation and geometry in string theory. We introduce new complexity-based invariants for studying mirror symmetry and demonstrate how our framework extends naturally to related geometric structures.
基金Supported by the National Natural Science Foundation of China(61473340)Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province(F703108L02)。
文摘We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show that its unique nonnegative fixed point is globally attractive.As application,our results not only improve many known ones,but also solve several“Open Problems and Conjectures”given by Professors Ladas and Camouzis,et al.
文摘In this paper,a formula is given. The formula gives the number of prime number solutions of the indefinite equation p 1+p 2=2n , and based on it, an equivalent proposition to the conjecture of Goldbach is obtained.
文摘The well-known Yau＇s uniformization conjecture states that any complete noncompact Kahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed, by G. Liu in [23]. In the first part, we will give a survey on thc progress. In the second part, we will consider Yau＇s conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number Cn1 is an essential condition to solve Yau＇s conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, Cn1 is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau＇s uniformization conjecture on Kahler manifolds with minimal volume growth.
基金Supported by the Guangxi Science Foundation(0339071)
文摘A new structure with the special property that instantaneous state and catas-trophes is imposed to ordinary birth-death processes is considered. Kendall's conjecture forthe processes is proved to be right.
文摘Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function <img alt="" src="Edit_8fcdfff5-6b95-42a4-8f47-2cabe2723dfc.bmp" />, <img alt="" src="Edit_6ce3a4bd-4c68-49e5-aabe-dec3e904e282.bmp" />, <img alt="" src="Edit_29ea252e-a81e-4b21-a41c-09209c780bb2.bmp" /> by geometric analysis, which has the symmetry: v=0 if β=0, and basic expression <img alt="" src="Edit_bc7a883f-312d-44fd-bcdd-00f25c92f80a.bmp" />. We show that |u| is single peak in each root-interval <img alt="" src="Edit_d7ca54c7-4866-4419-a4bd-cbb808b365af.bmp" /> of u for fixed β ∈(0,1/2]. Using the slope u<sub>t</sub>, we prove that v has opposite signs at two end-points of I<sub>j</sub>. There surely exists an inner point such that , so {|u|,|v|/β} form a local peak-valley structure, and have positive lower bound <img alt="" src="Edit_bac1a5f6-673e-49b6-892c-5adff0141376.bmp" /> in I<sub>j</sub>. Because each t must lie in some I<sub>j</sub>, then ||ξ|| > 0 is valid for any t (i.e. RH is true). Using the positivity <img alt="" src="Edit_83c3d2cf-aa7e-4aba-89f5-0eb44659918a.bmp" /> of Lagarias (1999), we show the strict monotone <img alt="" src="Edit_87eb4e9e-bc7b-43e3-b316-5dcf0efaf0d5.bmp" /> for β > β<sub>0</sub> ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.
基金Project supported by the National Natural Science Foundation of China(No.10671117)the Key Science Research Foundation of Education Department of Hubei Province of China(No.2003A005).
文摘Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty＇s conjectured projection inequality and the Lp-Petty projection inequality are given, respectively.
文摘In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product P;×P;of two strongly geometrically bounded symplectic manifolds under some conditions with P;. In particular, if N is a closed manifold or a noncompact manifold of finite topological type, our result implies that the Weinstein conjecture in CP;×T*N holds.
文摘This paper proves Riemann conjecture (RH), <em>i.e</em>., that all the zeros in critical region of Riemann <span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><em>ξ</em><span style="white-space:normal;"> </span></span></em></span>-function lie on symmetric line <span style="white-space:nowrap;"><em>σ</em></span> =1/2 . Its proof is based on two important properties: the symmetry and alternative oscillation for <span style="white-space:nowrap;"><em><em>ξ</em><span style="white-space:normal;"> </span></em>=<em> u </em>+<em> iv</em></span> . Denote <img src="Edit_317839cd-bad0-44d8-b081-c473bcb336f1.png" width="170" height="15" alt="" />. Riemann proved that u is real and <em>v</em> <span style="white-space:nowrap;">≡ </span>0 for <span style="white-space:nowrap;"><em><span style="white-space:nowrap;">β</span></em> =0</span> (the symmetry). We prove that the zeros of u and v for <em>β</em> <span style="white-space:nowrap;">> 0</span> are alternative, so <span style="white-space:nowrap;"><em>u</em> (<em>t</em>,0)</span> is the single peak. A geometric model was proposed. <img src="Edit_27688061-de42-4bce-ad80-6fb3dd1e3d4b.png" width="85" height="27" alt="" /> is called the root-interval of <em>u </em>(<em>t</em>,<em style="white-space:normal;">β</em>) , if |<span style="white-space:nowrap;"><em>u</em>| <em>> </em>0</span> is inside <em>I</em><sub><em>j</em> </sub>and <span style="white-space:nowrap;"><em>u</em> = 0</span> is at its two ends. If |<em>u</em> (<em style="white-space:normal;">t</em><span style="white-space:normal;">,</span><em style="white-space:normal;">β</em>)| has only one peak on each <em style="white-space:normal;">I</em><sub style="white-space:normal;"><em>j</em></sub>, which is called the single peak, else called multiple peaks (it will be proved that the multiple peaks do not exist). The important expressions of u and v for <em style="white-space:normal;">β</em><span style="white-space:normal;"> </span>> 0 were derived. By <img src="Edit_b6369c2e-6a6d-4e1a-8a75-00d743cecaf1.png" width="240" height="28" alt="" />, the peak <em style="white-space:normal;">u </em><span style="white-space:normal;">(</span><em style="white-space:normal;">t</em><span style="white-space:normal;">,</span><em style="white-space:normal;">β</em><span style="white-space:normal;">)</span> will develop toward its convex direction. Besides, <em style="white-space:normal;">u<sub>t</sub> </em><span style="white-space:normal;">(</span><em style="white-space:normal;">t</em><span style="white-space:normal;">,</span><em style="white-space:normal;">β</em><span style="white-space:normal;">)</span> has opposite signs at two ends <em>t</em> = <em>t<sub>j</sub></em><sub> </sub>, <em>t<sub>j+1</sub></em> of <em>I<sub>j </sub></em>, <img src="Edit_be3f0d63-1d24-4165-ac2c-141c9a47d1c8.png" width="145" height="28" alt="" /> also does, then there exists some inner point <span style="white-space:nowrap;"><em>t</em>′</span> such that <span style="white-space:nowrap;"><em>v</em><em></em> (<em>t′</em>,<em>β</em>) = 0</span>. Therefore {|<em>u</em>|,|<em>v</em>|/<em>β</em>} in <em>I<sub>j</sub></em><sub> </sub>form a peak-valley structure such that <img src="Edit_70bb530a-662f-464a-b3c8-4d5625fbf679.png" width="180" height="22" alt="" /> has positive lower bound independent of <em>t</em> <span style="white-space:nowrap;">∈ </span><em>I<sub>j</sub></em><sub> </sub>(<em>i.e</em>. RH holds in <em style="white-space:normal;">I<sub>j</sub></em><sub style="white-space:normal;"> </sub>). As <em style="white-space:normal;">u </em><span style="white-space:normal;">(</span><em style="white-space:normal;">t</em><span style="white-space:normal;">,</span><em style="white-space:normal;">β</em><span style="white-space:normal;">)</span> does not have the finite condensation point (unless <span style="white-space:nowrap;"><em>u</em> = <em>cons</em><em>t</em>.</span>), any finite t surely falls in some <em style="white-space:normal;">I<sub>j</sub></em><sub style="white-space:normal;"> </sub>, then <img src="Edit_166a9981-aac8-476b-a29a-496763297b35.png" width="50" height="23" alt="" /> holds for any t (RH is proved). Our previous paper “Local geometric proof of Riemann conjecture” (APM, V.10:8, 2020) has two defects, this paper has amended these defects and given a complete proof of RH.
文摘This paper does not claim to prove the Goldbach conjecture, but it does provide a new way of proof (LiKe sequence);And in detailed introduces the proof process of this method: by indirect transformation, Goldbach conjecture is transformed to prove that, for any odd prime sequence (3, 5, 7, <span style="font-size:12px;white-space:nowrap;">…</span>, <em>P<sub>n</sub></em>), there must have no LiKe sequence when the terms must be less than 3 <span style="font-size:12px;white-space:nowrap;">×</span> <em>P<sub>n</sub></em>. This method only studies prime numbers and corresponding composite numbers, replaced the relationship between even numbers and indeterminate prime numbers. In order to illustrate the importance of the idea of transforming the addition problem into the multiplication problem, we take the twin prime conjecture as an example and know there must exist twin primes in the interval [3<em>P<sub>n</sub></em>, <span><em>P</em></span><sup>2</sup><sub style="margin-left:-8px;"><em>n</em></sub>]. This idea is very important for the study of Goldbach conjecture and twin prime conjecture. It’s worth further study.
文摘The stability conjecture of Cauchy horizons of black holes suggested by Helliwell and Konkowski is used to investigate the 1+1-dimensional(2D)black holes under perturbations of infalling null dust and of both infalling and outgoing null dust.The result given by this conjecture agrees with that from mass inflation scenario for 2D charged dilaton black hole.For 2D black holes,we show that the Cauchy horizons are unstable and the corresponding singularities exist.
基金Supported by the NNSF of China(19971045) the MCF of Chinese University
文摘In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C-1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved still holding for it. As a consequence some results on Weinstein conjecture are generalized to C-1-smooth hypersurface of contact type.
文摘Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t −iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product expression , where z<sub>j</sub> are all roots of ξ(z), including complex roots. He proposed Riemann conjecture (RC): All roots of ξ(z) are real. As the product expression can only be used as a tool of contradiction, we prove RC by contradiction. To avoid the zeros of ξ(1/2 + it), define a subset . We have basic estimate , on L (R). One can construct by all real roots t<sub>j</sub> of ξ(t). If ξ has no complex roots, then w(t) = G(s)ξ(s) for s = 1/2 + it. If the product expression has a complex root z'=t' −iα, where 0 a ≤ 1/2, R' = |z′| > 10, then ξ(z) has four complex roots ±(t′ ± iα), and should contain fourth order factor p(z), i.e. ξ(z) = w(z)p(z). But p(z) can not be contained in ξ(s), as we have on L(R) and p(t) ≥ 0.5(t/R)<sup>4</sup> . As a result, we can rewrite ξ(t) = w(t)p(t) =G(s)ξ(s)p(t) on and get This contradicts the basic estimate. Therefore ξ(z) has no complex roots and RC holds.
文摘In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, thus we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of <span style="white-space:nowrap;">σ(<em>n</em>)</span> , where <img src="Edit_dad9267d-3c54-455b-b30e-63819c207e54.png" width="300" height="117" alt="" /> Then, with the help of the infinite-row-four-column-matrix, we continue to use the recursive method to prove this conjecture strictly.
基金supported in part by NSFC(Grant No.11375121,11747171,11747302 and 11847305)Natural Science Foundation of Chengdu University of TCM(Grants No.ZRYY1729 and ZRQN1656)+2 种基金Discipline Talent Promotion Program of Xinglin Scholars(Grant No.QNXZ2018050)The Key Fund Project for Education Department of Sichuan(Grant No.18ZA0173)Sichuan University Students Platform for Innovation and Entrepreneurship Training Program(Grant No.C2019104639).
文摘In this paper,we study the thermodynamics and the weak cosmic censorship conjecture of the nonlinear electrodynamics black hole under the scattering of a charged complex scalar field.According to the energy and charge fluxes of the scalar field,the variations of this black hole’s energy and charge can be calculated during an infinitesimal time interval.With scalar field scattering,the variation of the black hole is calculated in the extended and normal phase spaces.In the normal phase space,the cosmological constant and the normalization parameter are fixed,and the first and second laws of thermodynamics can also be satisfied.In the extended phase space,the cosmological constant and the normalization parameter are considered as thermodynamic variables,and the first law of thermodynamics is valid,but the second law of thermodynamics is not valid.Furthermore,the weak cosmic censorship conjecture is both valid in the extended and normal phase spaces.
基金Supported by National Natural Science Foundation of China under Grant No.11675143the Fundamental Research for the Provincial Universities of Zhejiang in China under Grant No.RF-A2019015
文摘Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which implies a large tensor-to-scalar ratio is strongly in tension with recent observational results. In this paper, we explore the possibility the swampland conjectures could be compatible with single-field inflationary scenarios if the effects due to the quantum theory of gravity are considered. We show that the quantum gravitational effects due to the nonlinear dispersion relation provides significant modifications on the amplitude of both the scalar and tensor perturbation spectra. Such modifications could be either raise or reduce the perturbation spectra depending on the values of the parameters in the nonlinear terms of the dispersion relations. Therefore, these effects can reduce the tensor-to-scalar ratio to a smaller value, which helps to relax the tension between the swampland conjecture and observational data.
基金Supported by the National Natural Science Foundation of China(11401050)Scientific Research Innovation Team Project Affiliated to Yangtze Normal University(2016XJTD01)
文摘In this paper, we consider some problems involving Strong Lemoine Conjecture in additive number theory. Based on Dusart's inequality and Rosser-Schoenfeld's inequality, we obtain several new results and give an equivalent form of Strong Lemoine Conjecture.
文摘Terai presented the following conjecture: Ifa2 + b2 = c2 vith a, b, c ∈ N, gcd (a, b, c) = 1 and a even, then the diophantine equation x2 + bm = cn has the only peitive integral solution (x, m, n ) = (a,2, 2). In this paper we prove that if c is a prime power, b 1 (mod 8) and b 1 (mod 16) if b2 + 1 = 2c, then Terai’s conjecture holds.
文摘Erdosa and Sós conjectured in 1963 that if G is a graph o ofof ordeq >1/2p(k - 1), then G contains every tree of size k. It is shown in this paper that the conjecture is true if the complement G of G contains no a copy of K3 as an induced subgraph of G.
