Sum-free sets and Ramsey numbers

In this note we obtain new lower bounds for the Ramsey numbers R(5, 5) and R(5, 6). The method is based on computational results of partitioning the integers into sum-free sets. We obtain R(5, 5)?42 and R(5, 6)?53.     

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.     

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}.     

Some Results on Cross-Positive Matrices

Let K be a cone in Rⁿ, K1 its dual cone. An n×n matrix A is called cross-positive on K if and only if for all y?K, z?K1 such that (z, y) = 0 we have (z, Ay)?0. In this short note new equivalent conditions for matrices cross-positive on K will be given in terms of the partial ordering in Rⁿ induced by the cone K.     

Optimal bounds for ordinal comparison maps

In this note, we answer the following question: Given two recursively presented well orderingsR, S of subsets of the natural numbers, what is the complexity of an ordinal comparison mapκ takingR isomorphically onto an initial segment ofS (or vice versa)?     

Existence of derivations on near-rings

We obtain conditions on (R,+) which force that the zero map is the only derivation on a zero-symmetric near-ring R. Throughout the paper we construct several new examples of near-rings which are not rings admitting non-zero derivations, non-zero (σ, σ)-derivations and non-zero (1, σ)-derivations.     

Derivations on Lie ideals in semiprime rings

We prove that ifR is a semiprime 2-torsion free ring with a derivationd andU a Lie ideal ofR such thatd ² (U)=0, thend(U)?Z(R), the center ofR.     

A Method of Representing Rough Sets System Determined by Quasi Orders

In this paper, we study about the ordered structure of rough sets determined by a quasi order. A characterization theorem for rough sets of an approximation space (U, R) based on a quasi order R is given in Nagarajan and Umadevi (2010). Then using the characterization of rough sets determined by a quasi order, its rough sets system is represented by a new construction. This construction is generalized and abstracted into a new method of constructing Kleene based algebraic structures from dually isomorphic distributive lattices. Then by using different varieties of distributive lattices, we obtain various Kleene based algebraic structures. By this construction, we give various algebraic structures to the rough sets system determined by a quasi order R.     

The inertia of a Hermitian pencil

Let A and B be Hermitian matrices and P = λA + μB where (λ,μ)?$\text{R}$². Using parametric dependence of the eigenvalues, we study the inertia of P under variation of (λ,μ) and under small Hermitian perturbations. In particular, we give semicontinuous dependence results for the set of (λ,μ) where inertia of P is discontinuous.     

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.     

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.     

The topological dimension of limits of vertex replacements

Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R(G). By iterating this procedure, a sequence of graphs {Rn(G)} is obtained. When each graph in this sequence is normalized to have diameter one, the resulting sequence may converge in the Gromov-Hausdorff metric. In this paper, we compute the topological dimension of limit spaces of normalized sequences of iterated vertex replacements involving more than one replacement graph. We also give examples of vertex replacement rules that yield fractals.     

Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.     

Analytical solutions of a coupled nonlinear system arising in a flow between stretching disks

In this paper, we consider the axi-symmetric flow between two infinite stretching disks. By using a similarity transformation, we reduce the governing Navier-Stokes equations to a system of nonlinear ordinary differential equations. We first obtain analytical solutions via a four-term perturbation method for small and large values of the Reynolds number R. Also, we apply the Homotopy Analysis Method (which may be used for all values of R) to obtain analytical solutions. These solutions converge over a larger range of values of the Reynolds number than the perturbation solutions. Our results agree well with the numerical results of Fang and Zhang [22]. Furthermore, we obtain the analytical solutions valid for moderate values of R by use of Homotopy Analysis.     

Subgroups of Lie type groups containing a unipotent radical

In this paper we show, that if G is a Lie type group and R a proper subgroup of G containing some unipotent radical, then R has a nilpotent normal subgroup generated by long root subgroups and R is contained in a proper parabolic subgroup of G. We also obtain some consequences of this result.     

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;     
   
   

Hardy-Sobolev spaces on product domains and application

We introduce the product Hardy-Sobolev space H1,p(Rn×Rm) and obtain its characterization in terms of atomic decomposition. As an application, we give an endpoint version of multi-parameter div-curl lemma.     

The existence of k-convex hypersurface with prescribed mean curvature

Using the strong maximum principle, we obtain a constant rank theorem for the k-convex solutions of semilinear elliptic partial differential equations. As an application we obtain an existence theorem of k-convex starshaped hypersurface with prescribed mean curvature in R ⁿ⁺¹.     

Estimates for harmonic conjugate functions and Riesz-system

In this paper we generalize a theorem of Hardy and Littlewood concerning the growth of conjugate harmonic functions. Moreover, we obtain an estimate for Riesz systems in the upper half-space R ₊ ⁿ⁺¹ of R ⁿ⁺¹.