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1.
In [1], V.I. Arnold conjectured “the matrix Euler congruence” for any integer matrix A, prime p, and natural number n. He proved it for p ≤ 5, n ≤ 4. In fact the conjecture immediately follows from a result of C.J. Smyth [5]. We give a simple proof of this result and discuss a related conjecture of Arnold concerning some congruences for multinomial coefficients.  相似文献   

2.
This paper contains a proof of a conjecture of Braverman concerning Laumon quasiflag spaces. We consider the generating function Z(m), whose coefficients are the integrals of the equivariant Chern polynomial (with variable m) of the tangent bundles of the Laumon spaces. We prove Braverman’s conjecture, which states that Z(m) coincides with the eigenfunction of the Calogero-Sutherland hamiltonian, up to a simple factor which we specify. This conjecture was inspired by the work of Nekrasov in the affine [^( \mathfrak sl)]n\widehat{ {\mathfrak {sl}}}_{n} setting, where a similar conjecture is still open.  相似文献   

3.
We obtain two results concerning the Feichtinger conjecture for systems of normalized reproducing kernels in the model subspace K Θ=H 2⊖ΘH 2 of the Hardy space H 2, where Θ is an inner function. First, we verify the Feichtinger conjecture for the kernels [(k)\tilde]ln=kln/||kln||\tilde{k}_{\lambda_{n}}=k_{\lambda_{n}}/\|k_{\lambda _{n}}\| under the assumption that sup  n |Θ(λ n )|<1. Second, we prove the Feichtinger conjecture in the case where Θ is a one-component inner function, meaning that the set {z:|Θ(z)|<ε} is connected for some ε∈(0,1).  相似文献   

4.
In this paper we prove the Gromov–Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on \mathbbRn{\mathbb{R}^{n}}, and improve bounds on the number n(d, k) in the analogous conjecture for odd degrees d (this case is known as the Birch theorem) and complex polynomials.  相似文献   

5.
Let ℝ n be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ℝ n of radius $ \sqrt {n/4} $ \sqrt {n/4} contains a point of ∧. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.  相似文献   

6.
We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of : an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.  相似文献   

7.
We prove—for sufficiently large n—the following conjecture of Faudree and Schelp:
, for the three-color Ramsey numbers of paths on n vertices. * The second author was supported in part by OTKA Grants T038198 and T046234. † Research supported in part by the National Science Foundation under Grant No. DMS-0456401.  相似文献   

8.
Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality holds for α,β > − 1 and n ≥ 1, θ ∈ (0, π), where are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where .   相似文献   

9.
The well known Zarankiewicz' conjecture is said that the crossing number of the complete bipartite graph Km,n (m≤ n) is Z(m,n), where Z(m,n)=\lfloor\frac{m}{2}\rfloor\lfloor\frac{m-1}{2}\rfloor\lfloor\frac{n}{2}\rfloor$\lfloor\frac{n-1}{2}\rfloor$ (for any real number x, $\lfloor x\rfloor$ denotes the maximal integer no more than x). Presently, Zarankiewicz' conjecture is proved true only for the case m≤ 6. In this article, the authors prove that if Zarankiewicz' conjecture holds for m≤9, then the crossing number of the complete tripartite graph K1,8,n is $Z(9, n)+ 12\lfloor\frac{n}{2}\rfloor$.  相似文献   

10.
Remarks on the Extremal Functions for the Moser-Trudinger Inequality   总被引:1,自引:0,他引:1  
We will show in this paper that if A is very close to 1, then I(M,λ,m) =supu∈H0^1,n(m),∫m|△↓u|^ndV=1∫Ω(e^αn|u|^n/(n-1)-λm∑k=1|αnun/(n-1)|k/k!)dV can be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" (Comm. Pure. Appl. Math., 55, 135-152, 2002).  相似文献   

11.
We compute the largest dimension of the Abelian Lie subalgebras contained in the Lie algebra of n×n strictly upper triangular matrices, where n ∈ ℕ \ {1}. We do this by proving a conjecture, which we previously advanced, about this dimension. We introduce an algorithm and use it first to study the two simplest particular cases and then to study the general case. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 419–429, September, 2007.  相似文献   

12.
A nonlinear deformation is conjectured for the reduction of the 3rd KP flow on the subspace of skew-symmetric operators, and the conjecture is proved for the linearized flow. As a by-product, we find a peculiar (nonquantum) polynomial deformation of the numbers , where B m ’s are the Bernoulli numbers. General open questions and generalizations are also discussed. The conjecture is extended to all the flows, and its linearized version is proved. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 101–116, 2006.  相似文献   

13.
We consider the Weil elliptic curve E/ℚ and let be its canonical L-series. Admitting the Birch-Swinnerton-Dyer conjecture and fixing the curve E, a criterion is given for the finiteness of the group ED(ℚ) for twisted elliptic curves ED, defined by the condition
  相似文献   

14.
Let G be a finite group and $$\psi (G)=\sum _{g\in {G}}{o(g)}$$. There are some results about the relation between $$\psi (G)$$ and the structure of G. For instance, it is proved that if G is a group of order n and $$\psi (G)>\dfrac{211}{1617}\psi (C_n)$$, then G is solvable. Herzog et al. in (J Algebra 511:215–226, 2018) put forward the following conjecture: Conjecture. If G is a non-solvable group of order n, then $$\begin{aligned} {\psi (G)}\,{\le }\,{{\dfrac{211}{1617}}{\psi (C_n)}}, \end{aligned}$$with equality if and only if $$G \cong A_5$$. In particular, this inequality holds for all non-Abelian simple groups. In this paper, we prove a modified version of Herzog’s Conjecture.  相似文献   

15.
16.
The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.   相似文献   

17.
Let K denote the middle third Cantor set and . Given a real, positive function ψ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |xp/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K—an assertion attributed to K. Mahler. Explicit examples of irrational numbers satisfying Mahler’s assertion are also given. Dedicated to Maurice Dodson on his retirement—finally!  相似文献   

18.
Additive functions on translation quivers have played an important role in the representation theory of finite-dimensional algebras, the most prominent ones are the hammock functions introduced by S.?Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type $\mathbb{A}_{n}, \mathbb{D}_{n}, \mathbb{E}_{6}, \mathbb {E}_{7}, \mathbb{E}_{8}$ are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case $\mathbb{A}_{n}$ .  相似文献   

19.
Let X be an algebraic curve over \({\mathbb {Q}}\) and \({t\in {\mathbb {Q}}(X)}\) a non-constant rational function such that \({{\mathbb {Q}}(X)\ne {\mathbb {Q}}(t)}\). For every \({ n \in {\mathbb {Z}}}\) pick \({P_ n \in X(\bar{{\mathbb {Q}}})}\) such that \({t(P_n)=n}\). We conjecture that, for large N, among the number fields \({\mathbb {Q}}(P_1), \ldots , {\mathbb {Q}}(P_N)\) there are at least cN distinct. We prove this conjecture in the special case when \(\bar{{\mathbb {Q}}}(X)/\bar{{\mathbb {Q}}}(t)\) is an abelian field extension and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a more famous conjecture of Schinzel.  相似文献   

20.
早在20世纪50年代,Zarankiewicz 猜想完全2-部图K_{m,n}(m\leq n)的交叉数为\lfloor\frac{m}{2}\rfloor\times \lfloor\frac{m-1}{2}\rfloor\times\lfloor\frac{n}{2}\rfloor\times\lfloor\frac{n-1}{2}\rfloor (对任意实数x,\lfloor x\rfloor表示不超过x的最大整数). 目前这一猜想的正确性只证明了当m\leq6时成立. 假定著名的Zarankiewicz的猜想对m=7的情形成立,确定了6-轮W_{6}与星S_{n}的笛卡尔积图的交叉是 cr(W_{6}\times S_{n})=9\lfloor\frac{n}{2}\rfloor\times\lfloor\frac{n-1}{2}\rfloor+2n+5\lfloor\frac{n}{2}\rfloor.  相似文献   

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