A Proof of KÜhnel’s Conjecture for n ⩾ k2 + 3k Eric Sparla

In this note we show that an Upper Bound Conjecture made by Kühnel for combinatorial 2k-manifolds holds for fixed k if its number of vertices is at least n ? k² + 3k. Together with known results this provides a simple proof of the conjecture for k = 1 and k = 2.     

A solution to de Groot’s absolute cone conjecture

A compactum X is an ‘absolute cone’ if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n≤3 and false for n≥5. For n=4, the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.     

A maximum degree theorem for diameter-2-critical graphs

A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ?n ²/4? and that the extremal graphs are the complete bipartite graphs K _?n/2?,?n/2?. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n ₀ where n ₀ is a tower of 2’s of height about 10¹⁴. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.     

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel n-Unipotent Conjecture and the Vanishing n  -Massey Conjecture for n≥3$n\ge 3$ were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of p-maximal extensions are free pro-p-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish various filtration results for free pro-p-groups, provide examples of pro-p-groups which do not have the kernel n-unipotent property, compare various Zassenhaus filtrations with the descending p-central series and establish new type of automatic Galois realization.     

On the Isomorphism Conjecture in algebraic K-theory

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RΓ, where Γ is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and arbitrary coefficient rings R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RΓ in terms of the K-theory of R and the homology of the group.     

A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture(see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional(n≥3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau number theoretic conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5≤y 17, compared with the result obtained by Ennola.     

The Anderson-Badawi conjecture for commutative algebras over infinite fields

In [1], Anderson and Badawi conjecture that every n-absorbing ideal of a commutative ring is strongly n-absorbing. In this article we prove their conjecture in certain cases (in particular this is the case for commutative algebras over an infinite field). We also show that an affirmative answer to another conjecture in [1] implies the Anderson-Badawi Conjecture.     

On a generalization of the Evans Conjecture

The Evans Conjecture states that a partial Latin square of order n with at most n-1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order n with at most n-1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist.     

P<Subscript>4k-1</Subscript>-factorization of complete bipartite graphs

Let K_m,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A P_v-factorization of K_m,n is a set of edge-disjoint P_v-factors of K_m,n which partition the set of edges of K_m,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of P_v-factorization of K_m,n. When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P_4k-1-factorization of K_m,n is (1) (2k - 1)m ⩽ 2kn, (2) (2k - 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k - 1), (4) (4k - 1)mn/[2(2k - 1)(m + n)] is an integer.     

Remarks on Harada’s conjecture

An open conjecture by Harada from 1981 gives an easy characterization of the p-blocks of a finite group in terms of the ordinary character table. Kiyota and Okuyama have shown that the conjecture holds for p-solvable groups. In the present work we extend this result using a criterion on the decomposition matrix. In this way we prove Harada’s Conjecture for several new families of defect groups and for all blocks of sporadic simple groups. In the second part of the paper we present a dual approach to Harada’s Conjecture.     

A method for proving polynomial enumeration formulas

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by “explaining” their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to prove a new refinement of the Bender-Knuth (ex-)Conjecture, which easily implies the Bender-Knuth (ex-)Conjecture itself. This is probably the most elementary way to prove this result currently known. Furthermore we adapt our method to q-polynomials, which allows us to derive generating function results as well. Finally we use this method to give another proof for the enumeration of semistandard tableaux of a fixed shape which differs from our proof of the Bender-Knuth (ex-)Conjecture in that it is a multivariate application of our method.     

Dehn Invariant and Scissors Congruence of Flexible Polyhedra

We prove that the Dehn invariant of any flexible polyhedron in n-dimensional Euclidean space, where n ≥ 3, is constant during the flexion. For n = 3 and 4 this implies that any flexible polyhedron remains scissors congruent to itself during the flexion. This proves the Strong Bellows Conjecture posed by R. Connelly in 1979. It was believed that this conjecture was disproved by V. Alexandrov and R. Connelly in 2009. However, we find an error in their counterexample. Further, we show that the Dehn invariant of a flexible polyhedron in the n-dimensional sphere or n-dimensional Lobachevsky space, where n ≥ 3, is constant during the flexion whenever this polyhedron satisfies the usual Bellows Conjecture, i.e., whenever its volume is constant during every flexion of it. Using previous results of the first named author, we deduce that the Dehn invariant is constant during the flexion for every bounded flexible polyhedron in odd-dimensional Lobachevsky space and for every flexible polyhedron with sufficiently small edge lengths in any space of constant curvature of dimension at least 3.     

The steiner ratio conjecture is true for five points

The long-standing conjecture of Gilbert and Pollak states that for any set of n given points in the Euclidean plane, the ratio of the length of a Steiner minimal tree and the length of a minimal (spanning) tree is at least $\text{3}\text{2}$. This conjecture was shown to be true for n = 3 by Gilbert and Pollak, and for n = 4 by Pollak. Recently, Du, Yao and Hwang used a different approach to give a shorter proof for n = 4. In this paper we continue this approach to prove the conjecture for n = 5. Such results for small n are useful in obtaining bounds for the ratio of the two lengths in the general case.     

Isomorphism Conjecture for homotopy K-theory and groups acting on trees

We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell-Jones Conjecture in algebraic K-theory.     

K- and L-theory of group rings over GL n (Z)

We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for GL _n (Z).     

On Thompson's Conjecture of A 10

In this short article, we prove the Thompson's Conjecture is true for the alternating group A ₁₀, where A ₁₀ has exactly one connected prime graph component.     

Induced lines in Hales-Jewett cubes

A line in d[n] is a set {x(1),…,x(n)} of n elements of d[n] such that for each 1?i?d, the sequence is either strictly increasing from 1 to n, or strictly decreasing from n to 1, or constant. How many lines can a set S⊆d[n] of a given size contain?One of our aims in this paper is to give a counterexample to the Ratio Conjecture of Patashnik, which states that the greatest average degree is attained when S=d[n]. Our other main aim is to prove the result (which would have been strongly suggested by the Ratio Conjecture) that the number of lines contained in S is at most |S|^2−ε for some ε>0.We also prove similar results for combinatorial, or Hales-Jewett, lines, i.e. lines such that only strictly increasing or constant sequences are allowed.     

Blocks with Defect Group \boldsymbol{Q_{2^n}\times C_{2^m}} and \boldsymbol{SD_{2^n}\times C_{2^m}}

We determine the numerical invariants of blocks with defect group $Q_{2^n}\times C_{2^m}$ and $SD_{2^n}\times C_{2^m}$ , where $Q_{2^n}$ denotes a quaternion group of order 2 ⁿ , $C_{2^m}$ denotes a cyclic group of order 2 ^m , and $SD_{2^n}$ denotes a semidihedral group of order 2 ⁿ . This generalizes Olsson’s results for m?=?0. As a consequence, we prove Brauer’s k(B)-Conjecture, Olsson’s Conjecture, Brauer’s Height-Zero Conjecture, the Alperin–McKay Conjecture, Alperin’s Weight Conjecture and Robinson’s Ordinary Weight Conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper follows (and uses) (Sambale, J Pure Appl Algebra 216:119–125, 2012; Proc Amer Math Soc, 2012).     

Groupes de lie nilpotents à structure affine invariante à gauche

We study the Filiform Lie Groups admitting a left invariant affine structure. For this class of Lie Groups we prove the Auslander Conjecture whenever the dimensionn of the group is odd and we show that it is false forn even andn>2.     

Chair-Free Berge Graphs Are Perfect

A graph G is called Berge if neither G nor its complement contains a chordless cycle with an odd number of nodes. The famous Berge’s Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd}, eb. We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect Graph Conjecture is true for chair-free graphs.