A generalised dual to a non-differentiable programming problem is given and duality established under general convexity and invexity conditions. A second order dual is also given and duality theorems proved under generalised second order invexity condition.

Let G be a finite group. We prove that if the set of p-regular conjugacy class sizes of G has exactly two elements, then G has Abelian p-complement or G=PQ×A, with P∈Sylp(G), Q∈Sylq(G) and A Abelian.

It is well-known that a boolean ring is commutative. In this note we show that a distributively generated boolean near-ring is multiplicatively commutative, and therefore a ring. This is accomplished by using subdirect sum representations of near-rings.

Let 〈𝒫〉⊂N be a multiplicative subsemigroup of the natural numbers N={1,2,3,…} generated by an arbitrary set 𝒫 of primes (finite or infinite). We give an elementary proof that the partial sums ∑ n∈〈𝒫〉:n≤x(μ(n))/n are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to ∏ p∈𝒫(1−(1/p)) (the case where 𝒫 is all the primes is a well-known observation of Landau). Interestingly, this convergence holds even in the presence of nontrivial zeros and poles of the associated zeta function ζ𝒫(s)≔∏ p∈𝒫(1−(1/ps))−1 on the line {Re(s)=1}. As equivalent forms of the first inequality, we have ∣∑ n≤x:(n,P)=1(μ(n))/n∣≤1, ∣∑ n∣N:n≤x(μ(n))/n∣≤1, and ∣∑ n≤x(μ(mn))/n∣≤1 for all m,x,N,P≥1.

Let q be a prime power, Fq a field with q elements, f ∈ Fq[x] a polynomial of degree n ≥ 1, V(f) = #f(Fq) the number of different values f(α) of f, with α ∈ Fq, and p = q – V(f). It is shown that either ρ = 0 or 4n4 > q or 2pn > q. Hence, if q is “large” and f is not a permutation polynomial, then either n or ρ is “large”.

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.

An associative ring is called strongly right (left) bounded if every nonzero right (left) ideal contains a nonzero ideal. We prove that if R is a strongly right bounded finite ring with unity and the order |R| of R has no factors of the form p5, then R is strongly left bounded. This answers a question of Birkenmeier and Tucci.

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and S is completely semisimple of a type that includes countable free inverse semigroups.

While the separable quotient problem is famously open for Banach spaces, in the broader context of barrelled spaces we give negative solutions. Obversely, the study of pseudocompact $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ and Warner bounded $X$ allows us to expand Rosenthal’s positive solution for Banach spaces of the form $ C_{c}(X) $ to barrelled spaces of the same form, and see that strong duals of arbitrary $C_{c}(X) $ spaces admit separable quotients.

The article examines the connection between the concepts of isoclinism and covering groups for finite groups. The main result is that all covering groups for a given finite group are mutually isoclinic. The converse is false.

Techniques from martingale theory are used to obtain the Skorokhod embedding of reverse martingales in Brownian motion. This result is then used to obtain a functional law of the iterated logarithm for reverse martingales.

We consider a nonlinear hemivariational inequality with the p-Laplacian at resonance. Using an extension of the nonsmooth mountain pass theorem of Chang, which makes use of the Cerami compactness condition, we prove the existence of a nontrivial solution. Our existence results here extends a recent theorem on resonant hemivariational inequalities, by the authors in 1999.

We discuss here the boundedness of the fractional integral operator Iα and its generalized version on generalized nonhomogeneous Morrey spaces. To prove the boundedness of Iα, we employ the boundedness of the so-called maximal fractional integral operator Ia,κ*. In addition, we prove an Olsen-type inequality, which is analogous to that in the case of homogeneous type.

We characterise some star sign pattern matrices and linear tree sign pattern matrices that allow a nilpotent matrix.

In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.

In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl. 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X.

Strong solvability and uniqueness in the Sobolev space W2, q(Ω), q > n, are proved for the oblique derivative problem

assuming the coefficients of the quasilinear elliptic operator to be Carathéodory functions, aij ∈ VMO∩L∞ with respect to x, and b to grow at most quadratically with respect to the gradient.

We develop a discrete version of the weak quadratic estimates for operators of type w explained by Cowling, Doust, McIntosh and Yagi, and show that analogous theorems hold. The method is direct and can be generalised to the case of finding necessary and sufficient conditions for an operator T to have a bounded functional calculus on a domain which touches σ(T) nontangentially at several points. For operators on Lp, 1 < p < ∞, it follows that T has a bounded functional calculus if and only if T satisfies discrete quadratic estimates. Using this, one easily obtains Albrecht's extension to a joint functional calculus for several commuting operators. In Hilbert space the methods show that an operator with a bounded functional calculus has a uniformly bounded matricial functional calculus.

The basic idea is to take a dyadic decomposition of the boundary of a sector Sv. Then on the kth ingerval consider an orthonormal sequence of polynomials . For h ∈ H∞(Sν), estimates for the uniform norm of h on a smaller sector Sμ are obtained from the coefficients akj = (h, ek, j). These estimates are then used to prove the theorems.

In this paper, sufficient conditions for oscillations of the first order neutral differential equation with variable coefficients

are obtained, where c, τ, σ and µ are positive constants, p, q ∈ C ([t0, ∞), R+).

The paper provides irrationality measures for certain values of binomial functions and definite integrals of some rational functions. The results are obtained using Jacobi type polynomials and divisibility considerations of their coefficients.