The Overfull Conjecture and the Conformability Conjecture

In this paper we show that under some fairly general conditions the Overfull Conjecture about the chromatic index of a graph G implies the Conformability Conjecture about the total chromatic number of G. We also show that if G has even order and high maximum degree, then G is conformable unless the deficiency is very small.     

The Atiyah Conjecture and $$L^2$$-cohomology computations

If G is a discrete group with bounded torsion, the Strong Atiyah Conjecture predicts that the \(L^2\)-Betti numbers of any G-space are rational, with denominators determined by the orders of the torsion subgroups. We prove the conjecture for some new examples of groups, including the class of virtually sparse special groups. We then show that the conjecture can be used to compute the \(L^2\)-cohomology of certain Coxeter groups that have highly symmetric nerves.     

The Gaussian Moments Conjecture and the Jacobian Conjecture

We first propose what we call the Gaussian Moments Conjecture. We then show that the Jacobian Conjecture follows from the Gaussian Moments Conjecture. Note that the the Gaussian Moments Conjecture is a special case of [11, Conjecture 3.2]. The latter conjecture was referred to as the Moment Vanishing Conjecture in [7, Conjecture A] and the Integral Conjecture in [6, Conjecture 3.1] (for the one-dimensional case). We also give a counter-example to show that [11, Conjecture 3.2] fails in general for polynomials in more than two variables.     

The Standard Map and a Probabilistic Conjecture

Statistical properties of trajectories of a class of dynamical systems, which includes the standard map, are studied. In particular a well known conjecture is stated and its natural probabilistic analogue is defined in the framework of ergodic theory of random transformations. The main result is a proof of a probabilistic analogue of this conjecture.     

A Positivity Property of a Quantum Anharmonic Oscillator Suggested by the BMV Conjecture

In this work an observation concerning a positivity property of the quantum anharmonic oscillator is made. This positivity property is suggested by the BMV conjecture.     

一个关于(k;g)-笼的猜想的证明

(k;g) -笼是指具有围长 g的 k-正则图中那些顶点数最小的图 .文 [2 ]中有下面的猜想 :设 G为一个 ( k;g) -笼 ,则它的每一个 g-圈 C是不可分离的 ( nonseparating) ,也就是说 ,对 G中任意的 g-圈 C,G- C仍是连通的 .对于偶数 g,[2 ]已给出了此猜想的证明 .本文中 ,证明 :对于奇数 g,此猜想也是正确的 .     

The relation between the Baum-Connes Conjecture and the Trace Conjecture

We prove a version of the L ²-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ⊂λ ^G ⊂ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K ₀(C ^* _r (G))→ℝ takes values in λ ^G . The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [15]. Oblatum 10-IV-2001 & 18-X-2001?Published online: 15 April 2002     

The Honeycomb Conjecture

This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Received August 30, 1999, and in revised form April 17, 2000. Online publication September 22, 2000.     

The Cameron-Erdos Conjecture

A subset A of the integers is said to be sum-free if there donot exist elements x, y, z A with x + y = z. It is shown thatthe number of sum-free subsets of {1,...,N} is O(2^N/2), confirminga well-known conjecture of Cameron and Erds. 2000 MathematicsSubject Classification 11B75.     

The Cohen--Lusk Conjecture

The question of Cohen and Lusk about the partial gluing of an orbit under a map of a free Z _p-space to ⁿ is answered in part.     

关于Katai的一个猜想的证明

本文证明了：如果实值加性函数ｆ（ｎ）满足条件‖ｆ（ｎ＋１）－ｆ（ｎ）‖＝ｏ（１），（ｎ→∞）这里‖‖表示一实数与最近的整数的距离，则一定有某常数ｃ使ｆ（ｎ）一ｃｌｏｇｎ为整数值加性函数．这证实了ＫａｔａｉⅠ的一个猜想．     

The Local Stark Conjecture at a Real Place

A refinement of the rank 1 Abelian Stark conjecture has been formulated by B.Gross. This conjecture predicts some -adic analytic nature of a modification of the Stark unit. The conjecture makes perfect sense even when is an Archimedean place. Here we consider the conjecture when is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the conjecture for CM extensions; here the original Stark conjecture is uninteresting, but the refined conjecture is nontrivial. In more generality, we show that, under mild hypotheses, if the subgroup of the Galois group generated by complex conjugations has less than full rank, then the refined conjecture implies that the Stark unit should be a square. This phenomenon has been discovered by Dummit and Hayes in a particular type of situation. We show that it should hold in much greater generality.     

The Analytic Caratheodory Conjecture

The aim of this article is to provide the reader with a real possibility of becoming confident that the index of an isolated umbilic point of an analytic surface is never greater than one. For a surface homeomorphic to a sphere, this means in particular that on the surface there necessarily exist at least two umbilic points as it was conjectured by Caratheodory.     

The Two Hyperplane Conjecture

We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar′e inequalities, Harnack inequalities, and NTA(non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.     

The Proof of a Conjecture of Additive Number Theory

The aim of this paper is to show that for any n∈N, n>3, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having the same parity (see the text for the definition of the “length” of a natural number). Also we will show that for any n∈N, n>2, n≠5, 10, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having different parities. We will prove also that for any prime p≡7(mod 8) there exist a, b∈N* such that p=a²+b, the “length” of b being an even number.     

The Hexagonal Honeycomb Conjecture

It is conjectured that the planar hexagonal honeycomb provides the least-perimeter way to enclose and separate infinitely many regions of unit area. Various natural formulations of the question are not known to be equivalent. We prove existence for two formulations. Many questions remain open.

     

On a Conjecture of Finotti

We prove a conjecture of Luis Finotti about cubic polynomials of one variable in characteristic p. He checked it by computer for primes p < 890 and uses it to define and study the minimal degree lift of the generic point of an ordinary elliptic curve in characteristic p to the canonical lift mod p ³ of the curve. Received: 5 June 2002     

On a Conjecture of Kemnitz

A classic theorem of Erdis, Ginzburg and Ziv states that in a sequence of 2n-1 integers there is a subsequence of length n whose sum is divisble by n. This result has led to several extensions and generalizations. A multi-dimensional problem from this line of research is the following. Let Z_nZ_n stand for the additive group of integers modulo n. Let s(n, d) denote the smallest integer s such that in any sequence of s elements from Z_n^dZ_n^d (the direct sum of d copies of Z_nZ_n) there is a subsequence of length n whose sum is 0 in Z_n^dZ_n^d. Kemnitz conjectured that s(n, 2) = 4n - 3. In this note we prove that s(p,2) ￡ 4p - 2s(p,2) \le 4p - 2 holds for every prime p. This implies that the value of s(p, 2) is either 4p-3 or 4p-2. For an arbitrary positive integer n it follows that s(n, 2) ￡ (41/10)ns(n, 2) \le (41/10)n. The proof uses an algebraic approach.     

The Gold Partition Conjecture

We present the Gold Partition Conjecture which immediately implies the – Conjecture and tight upper bound for sorting. We prove the Gold Partition Conjecture for posets of width two, semiorders and posets containing at most elements. We prove that the fraction of partial orders on an -element set satisfying our conjecture converges to when approaches infinity. We discuss properties of a hypothetical counterexample.