An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is ...An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.展开更多
This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An...This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.展开更多
In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The m...In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.展开更多
Riemann proved three results: analytically continue ζ(s) over the whole complex plane s =σ + it with a pole s =1;(Theorem A) functional equation ξ(t) = G(s<sub>0</sub>)ζ (s<sub>0</sub>), s&...Riemann proved three results: analytically continue ζ(s) over the whole complex plane s =σ + it with a pole s =1;(Theorem A) functional equation ξ(t) = G(s<sub>0</sub>)ζ (s<sub>0</sub>), s<sub>0</sub> =1/2 + it and (Theorem B) product expression ξ<sub>1</sub>(t) by all roots of ξ(t). He stated Riemann conjecture (RC): All roots of ξ (t) are real. We find a mistake of Riemann: he used the same notation ξ(t) in two theorems. Theorem B must contain complex roots;it conflicts with RC. Thus theorem B can only be used by contradiction. Our research can be completed on s<sub>0</sub> =1/2 + it. Using all real roots r<sub>k</sub><sub> </sub>and (true) complex roots z<sub>j</sub> = t<sub>j</sub> + ia<sub>j</sub> of ξ (z), define product expressions w(t), w(0) =ξ(0) and Q(t) > 0, Q(0) =1 respectively, so ξ<sub>1</sub>(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s<sub>0</sub>)| =ω} for small ω > 0. If ξ(t) has complex roots, then ω =ωQ(t) on L(ω). Finally in a large interval of the first module |z<sub>1</sub>|>>1, we can find many points t ∈ L(ω) to make Q(t) . This contraction proves RC. In addition, Riemann hypothesis (RH) ζ for also holds, but it cannot be proved by ζ.展开更多
The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inf...The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inference restricted to the odd prime number set only. Based on this redefinition, one can describe an empirical-heuristic proof of the Collatz conjecture.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
In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C1-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 C1-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 C1-smooth hypersurface of contact type.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 paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product P_1 ×P_2 of two strongly geometrically bounded symplectic manifolds under some conditions with P_1. In part...In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product P_1 ×P_2 of two strongly geometrically bounded symplectic manifolds under some conditions with P_1. 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^2×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
Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> and using analysis method likely are two incor-rect guides. Actually, a unique hope may study...Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> 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 <i>β</i>=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 <i>u</i> for fixed <em>β</em> ∈(0,1/2]. Using the slope u<sub>t</sub>, we prove that <i>v</i> has opposite signs at two end-points of I<sub>j</sub>. There surely exists an inner point such that , so {|u|,|v|/<em>β</em>} 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 <i>t</i> must lie in some I<sub>j</sub>, then ||<em>ξ</em>|| > 0 is valid for any <i>t</i> (<i>i.e.</i> 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 <i>β</i> > <i>β</i><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”.</i>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 , some examples are given to show that the maximal number of admissible preference orderings is greater than 2n-1 when n is greater than 3. And the recurrence formula of local maximal number is presented.
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
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
In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest e...In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian(introduced by Kigami) attains its maximum and minimum on the boundary.展开更多
Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The aut...Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent, it is both sufficient and necessary that there exist three numbers X0 and X1 < X2 such that limsup(FN(X2) - FN(X1)) > 0 holds, and Lo = N→ ∞ lim FN(X0) exists. Moreover, the authors point out that they can also obtain the same result in the weakened case of lim inf P(Xn = 0) > 0.展开更多
Treating the cosmological constant as a dynamical variable,we investigate the thermodynamics and weak cosmic censorship conjecture(WCCC)of a charged AdS black hole(BH)in the Rastall gravity.We determine the energy mom...Treating the cosmological constant as a dynamical variable,we investigate the thermodynamics and weak cosmic censorship conjecture(WCCC)of a charged AdS black hole(BH)in the Rastall gravity.We determine the energy momentum relation of charged fermion at the horizon of the BH using the Dirac equation.Based on this relation,it is shown that the first law of thermodynamics still holds as a fermion is absorbed by the BH.However,the entropy of both the extremal and near-extremal BH decreases in the irreversible process,which means that the second law of thermodynamics is violated.Furthermore,we verify the validity of the WCCC by the minimum values of the metric function h(r)at its final state.For the extremal charged AdS BH in the Rastall gravity,we find that the WCCC is always valid since the BH is extreme.While for the case of near-extremal BH,we find that the WCCC could be violable in the extended phase space(EPS),depending on the value of the parameters of the BH and their variations.展开更多
1 IntroductionFor an n×n matrix A which is an inverse M-matrix,M.Neumann in [1]conjecturedthat the Hadamard product A·A is an inverse of an M-matrix.They have checked hisconjecture without failure on Ultrame...1 IntroductionFor an n×n matrix A which is an inverse M-matrix,M.Neumann in [1]conjecturedthat the Hadamard product A·A is an inverse of an M-matrix.They have checked hisconjecture without failure on Ultrametric matrices and inverse of MMA-matrices,Uni-pathicmatrices and the Willongby inverse M-matrices.Bo-Ying Wang et al.in[2]haveinvestigated Triangular inverse M-matrices which are closed under the Hadamard multipli-cation.Lu Linzheng,Sun Weiwei and Li Wen in[3]presented a more general展开更多
文摘An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.
文摘This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.
文摘In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.
文摘Riemann proved three results: analytically continue ζ(s) over the whole complex plane s =σ + it with a pole s =1;(Theorem A) functional equation ξ(t) = G(s<sub>0</sub>)ζ (s<sub>0</sub>), s<sub>0</sub> =1/2 + it and (Theorem B) product expression ξ<sub>1</sub>(t) by all roots of ξ(t). He stated Riemann conjecture (RC): All roots of ξ (t) are real. We find a mistake of Riemann: he used the same notation ξ(t) in two theorems. Theorem B must contain complex roots;it conflicts with RC. Thus theorem B can only be used by contradiction. Our research can be completed on s<sub>0</sub> =1/2 + it. Using all real roots r<sub>k</sub><sub> </sub>and (true) complex roots z<sub>j</sub> = t<sub>j</sub> + ia<sub>j</sub> of ξ (z), define product expressions w(t), w(0) =ξ(0) and Q(t) > 0, Q(0) =1 respectively, so ξ<sub>1</sub>(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s<sub>0</sub>)| =ω} for small ω > 0. If ξ(t) has complex roots, then ω =ωQ(t) on L(ω). Finally in a large interval of the first module |z<sub>1</sub>|>>1, we can find many points t ∈ L(ω) to make Q(t) . This contraction proves RC. In addition, Riemann hypothesis (RH) ζ for also holds, but it cannot be proved by ζ.
文摘The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inference restricted to the odd prime number set only. Based on this redefinition, one can describe an empirical-heuristic proof of the Collatz conjecture.
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.
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 C1-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 C1-smooth hypersurface of contact type.
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.
文摘In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product P_1 ×P_2 of two strongly geometrically bounded symplectic manifolds under some conditions with P_1. 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^2×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Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> 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 <i>β</i>=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 <i>u</i> for fixed <em>β</em> ∈(0,1/2]. Using the slope u<sub>t</sub>, we prove that <i>v</i> has opposite signs at two end-points of I<sub>j</sub>. There surely exists an inner point such that , so {|u|,|v|/<em>β</em>} 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 <i>t</i> must lie in some I<sub>j</sub>, then ||<em>ξ</em>|| > 0 is valid for any <i>t</i> (<i>i.e.</i> 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 <i>β</i> > <i>β</i><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”.</i>
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.
文摘In this paper , some examples are given to show that the maximal number of admissible preference orderings is greater than 2n-1 when n is greater than 3. And the recurrence formula of local maximal number is presented.
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.
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.
基金supported in part by NSFC grants Nos.11271327, 11771391
文摘In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian(introduced by Kigami) attains its maximum and minimum on the boundary.
基金Supported by National Natural Science Foundation of China
文摘Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent, it is both sufficient and necessary that there exist three numbers X0 and X1 < X2 such that limsup(FN(X2) - FN(X1)) > 0 holds, and Lo = N→ ∞ lim FN(X0) exists. Moreover, the authors point out that they can also obtain the same result in the weakened case of lim inf P(Xn = 0) > 0.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11875095 and 11903025)the Basic Research Project of Science and Technology Committee of Chongqing(Grant No.cstc2018jcyjA2480)the Fundamental Research Funds of ChinaWest Normal University(Grant No.18Q062).
文摘Treating the cosmological constant as a dynamical variable,we investigate the thermodynamics and weak cosmic censorship conjecture(WCCC)of a charged AdS black hole(BH)in the Rastall gravity.We determine the energy momentum relation of charged fermion at the horizon of the BH using the Dirac equation.Based on this relation,it is shown that the first law of thermodynamics still holds as a fermion is absorbed by the BH.However,the entropy of both the extremal and near-extremal BH decreases in the irreversible process,which means that the second law of thermodynamics is violated.Furthermore,we verify the validity of the WCCC by the minimum values of the metric function h(r)at its final state.For the extremal charged AdS BH in the Rastall gravity,we find that the WCCC is always valid since the BH is extreme.While for the case of near-extremal BH,we find that the WCCC could be violable in the extended phase space(EPS),depending on the value of the parameters of the BH and their variations.
文摘1 IntroductionFor an n×n matrix A which is an inverse M-matrix,M.Neumann in [1]conjecturedthat the Hadamard product A·A is an inverse of an M-matrix.They have checked hisconjecture without failure on Ultrametric matrices and inverse of MMA-matrices,Uni-pathicmatrices and the Willongby inverse M-matrices.Bo-Ying Wang et al.in[2]haveinvestigated Triangular inverse M-matrices which are closed under the Hadamard multipli-cation.Lu Linzheng,Sun Weiwei and Li Wen in[3]presented a more general