New upper bound formulas with parameters for Ramsey numbers

In this paper, we obtain some new results R(5,12)?848, R(5,14)?1461, etc., and we obtain new upper bound formulas for Ramsey numbers with parameters.     

On the Brauer group and factoriality of normal domains

Let B(R) be the Brauer group of the integrally closed noetherian domain R with quotient field K. We reexamine the proof that B(R) → R(K) is monic for R regular from the point of view of factoriality of R and its extensions. For R local with maximal ideal m, henselization R^h and divisor class group Cl(R) we embed ker $\text{{B(R) → B(K)?B(}\text{R}\text{M}\text{)}}$ into $\text{Cl(R}{}^{\mathrm{h}}\text{)}\text{Cl(R)}$. This is applied to obtain examples of non-regular geometric local domains R for which B(R) → B(K) is monic.     

Lower bounds on covering codes via partition matrices

Let Kq(n,R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σq(n,s;r) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound Kq(n,n−2) and σq(n,s;s−2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on Kq(n,n−2) and to the new bound Kq(n,n−2)?3q−2n+2, improving the first iff 5?n<q?2n−4. We determine Kq(q,q−2)=q−2+σ₂(q,2;0) if q?10. Moreover, we obtain the new powerful recursive bound Kq+1(n+1,R+1)?min{2(q+1),Kq(n,R)+1}.     

Further results on the covering radius of small codes

The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K(t,b,R). In the paper, necessary and sufficient conditions for K(t,b,R)=M are given for M=6 and 7 by proving that there exist exactly three families of optimal codes with six codewords and two families of optimal codes with seven codewords. The cases M?5 were settled in an earlier study by the same authors. For binary codes, it is proved that K(0,2b+4,b)?9 for b?1. For ternary codes, it is shown that K(3t+2,0,2t)=9 for t?2. New upper bounds obtained include K(3t+4,0,2t)?36 for t?2. Thus, we have K(13,0,6)?36 (instead of 45, the previous best known upper bound).     

Automorphisms with unipotent values

In this paper we obtain the following result: ifR is a prime ring andg is an automorphism ofR such that (g(x)?x) ⁿ=0 for allx∈R, wheren≥1 is a fixed integer, theng=1.     

Convergence of a subgradient method for computing the bound norm of matrices

Let φ and ψ be any norms on $\text{R}$^m and $\text{R}$ⁿ respectively. We study a subgradient method for computing the associated bound norm S_φψ(A) = sup{φ(Ax), ψ(x)?1} (a nonconvex optimization problem). It is proved that homodual method converges when one of the norms φ and ψ is polyhedral.     

Existence of non-abelian representations of the near hexagon <Emphasis Type="BoldItalic">Q(5,2)⊗Q(5,2)</Emphasis>

In [5], a new combinatorial model with four types of points and nine types of lines of the slim dense near hexagon Q(5,2)?Q(5,2) was provided and it was then used to construct a non-abelain representation of Q(5,2)?Q(5,2) in the extraspecial 2-group \(2_{-}^{1+18}\). In this paper, we give a direct proof for the existence of a non-abelian representation of Q(5,2)?Q(5,2) in \(2_{-}^{1+18}\).     

On a generalization of Minkowski's convex body theorem

Given a lattice $\text{Λ ? R}{}^{\mathrm{n}}$ and a bounded function g(x), x ∈ Rⁿ, vanishing outside of a bounded set, the functions ?(x)$\text{g}\text{?}\text{(x)?}$max_u∈Λg(u +x), ?(x)?Σ_{u∈Λ g(u +x)}, and ?₊(x)?Σ_u∈Λ max_{v∈Λ min {g(v + x); g(u + v + x)}} are defined and periodic mod Λ on Rⁿ. In the paper we prove that ?(x) + ?₊(x) ? 2?(x) ≥ ?(x) + h?₊(x) ? 2?(x) holds for all x ∈ Rⁿ, where h(x) is any “truncation” of g by a constant c ≥ 0, i.e., any function of the form h(x)?g(x) if g(x) ≤ c and h(x)?c if g(x) > c. This inequality easily implies some known estimations in the geometry of numbers due to Rado [1] and Cassels [2]. Moreover, some sharper and more general results are also derived from it. In the paper another inequality of a similar type is also proved.     

On questions related to saturated chains of primes in polynomial rings

We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is R〈X₁,…,X_n〉 residually Jaffard (resp., totally Jaffard) when R(X₁,…,X_n) is ? We construct a three-dimensional local ring R such that R(X₁,…,X_n) is totally Jaffard (and hence, residually Jaffard) whereas R〈X₁,…,X_n〉 is not residually Jaffard (and hence, not totally Jaffard).     

Slight improvements of the Singleton bound

The problem of determining A_q(n,d), the maximum cardinality of a q-ary code of length n with minimum distance at least d, is considered in some cases where corresponding MDS codes do not exist. Slight improvements of the Singleton bound are given, including A_q(q+2,q)?q³-3 if q is odd, A₅(7,5)?5³-4 and A₁₆(18,15)?18⁴-4.     

Derivations in prime rings

LetR be a prime ring andD a nonzero derivation ofR. If one of the four conditions holds inR, thenR is commutative:

1. (i)
   X ²D(X)?D(X)X²∈Z(R), CharR≠2;     
   
   

Sure wins, separating probabilities and the representation of linear functionals

We discuss conditions under which a convex cone K⊂RΩ admits a finitely additive probability m such that supk∈Km(k)?0. Based on these, we characterise those linear functionals that are representable as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions.     

Global strong solution to the Thirring model in critical space

We obtain a strong solution in charge critical space L²(R) of the Thirring system and Federbusch equations in one space dimension by using solution representation of the models. The uniqueness is obtained for the solution Ψ∈L^∞([0,T];L²(R)∩L⁴(R)). A decay of local charge and asymptotic behavior of the field can be shown directly.     

Compact Sobolev imbeddings on finite measure spaces

We provide conditions on a finite measure μ on $\text{R}$ⁿ which insure that the imbeddings W^{k, p}($\text{R}$ⁿdμ)?L^p($\text{R}$ⁿdμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by $\text{dμ = e}{}^{\mathrm{?¦x¦\alpha }}\text{dx,}\text{where}\text{α > 1}$.     

Algorithms for finding connected separators between antipodal points

A set (or a collection of sets) contained in the Euclidean space Rm is symmetric if it is invariant under the antipodal map. Given a symmetric unicoherent polyhedron X (like an n-dimensional cube or a sphere) and an odd real function f defined on vertices of a certain symmetric triangulation of X, we algorithmically construct a connected symmetric separator of X by choosing a subcollection of the triangulation. Each element of the subcollection contains the vertices v and u such that f(v)f(u)?0.     

On mappings related to the gradient of the conformal radius

We establish a criterion for the gradient ?R(D, z) of the conformal radius of a convex domain D to be conformal: the boundary ?D must be a circle. We obtain estimates for the coefficients K(r) for the K(r)-quasiconformal mappings ?R(D, z), D(r) ? D, 0 < r < 1, and supplement the results of Avkhadiev and Wirths concerning the structure of the boundary under diffeomorphic mappings of the domain D.     

Bounds for sorting by prefix reversal

For a permutation σ of the integers from 1 to n, let ?(σ) be the smallest number of prefix reversals that will transform σ to the identity permutation, and let ?(n) be the largest such ?(σ) for all σ in (the symmetric group) S_n. We show that $\text{?(n)?}\text{(5n+5)}\text{3}$, and that $\text{?(n)?}\text{17n}\text{16}$ for n a multiple of 16. If, furthermore, each integer is required to participate in an even number of reversed prefixes, the corresponding function g(n) is shown to obey $\text{3n}\text{2?1}\text{?g(n)?2n+3}$.     

A new proof of Ramanujan’s modular equation relating R(q) with R(q 5)

We give a new proof of Ramanujan’s modular identity relating R(q) with R(q ⁵), where R(q) is the famous Rogers–Ramanujan continued fraction. Our formulation is stronger than those of preceding authors; in particular, we give for the first time identities for the expressions appearing in the numerator and the denominator of Ramanujan’s identity. A related identity for R(q) that has partition-theoretic connections is also proved.