We prove a formula for the second order subdifferential of the sum of two lower semi continuous functions in finite dimensions. This formula yields an Alexandrov type theorem for continuous functions. We derive from this uniqueness results of viscosity solutions of second order Hamilton-Jacobi equations and singlevaluedness of the associated Hamilton-Jacobi operators. We also provide conterexamples in infinite dimensional Hilbert spaces.

Let p be an odd prime with p > 3. In this paper we give all positive integer solutions (x, y, m, n) of the equation x2 + p2m = yn, gcd (x, y) = 1, n > 2 satisfying 2 | n of 2 ∤ n and p ≢ (−1)(p−1)/2(mod 4n.

An elementary new, simple and purely analytic proof of Laguerre's theorem about the zeros of the polar derivative of a polynomial P(z) with respect to the point α, is given. The proof is based on a lemma which is also of independent interest.

In this paper we introduce 𝒯-noncosingular modules. Rings for which all right modules are𝒯-noncosingular are shown to be precisely those for which every simple right module is injective. Moreover, for any ring R we show that the right R-module R is 𝒯-noncosingular precisely when R has zero Jacobson radical. We also study the 𝒯-noncosingular condition in association with (strongly) FI-lifting modules.

We prove a blow-up formula for Morse–Novikov cohomology on a compact locally conformal Kähler manifold.

Let G be a finite group. It is shown that G is 2-closed if and only if

(a) every 2-block of G has full defect, and

(b) every Sylow 2-intersection is centralized by a Sylow 2-subgroup of G.

As a consequence it is shown that G is a TI-group if and only if every 2-block of G has either full defect or defect zero and (b) holds. This result and a theorem of Kwok yield complete characterizations of finite groups with certain relations being satisfied by every nonprincipal irreducible character.

A new method is found for solving the general Dirichlet problem for two non-overlapping spheres of different radius. The expression for the external potential involves hypergeometric functions and is obtained from an infinite set of linear equations. In essence the method makes use of the fact that (I–A)–1 = I + A + A2 + A3 + …, where A belongs to a certain class of infinite matrices and I is the unit matrix.

We classify unit speed curves contained in a real space form of arbitrary dimension Nm(c), whose mean curvature vector is proper for the Laplacian. Then we use these results to classify Hopf cylinders of S3 and semi-Riemannian Hopf cylinders of with proper mean curvature function.

Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm.

Recently, Talagrand [Israel J. Math. 44 (1983), 317–321] constructed a Banach space E having the Dunford-Pettis property such that C([0, 1], E) fails to have the Dunford-Pettis property. So he answered negatively a question which was posed some years ago.

We prove in this paper that for a large class of compacts K (the scattered compacts), C(K, E) has either the Dunford-Pettis property, or the reciprocal Dunford-Pettis property, or the Dieudonné property, or property V if and only if E has the same property.

Also some properties of the operators defined on C(K, E) are studied.

In terms of a new notion, called prebalancedness, those extensions of a Butler group by a Butler group which are again Butler groups are characterised. It is shown that the prebalanced extensions of torsion-free abelian groups form a divisible subgroup in the group of extensions.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite 2-group. If $G$ is of coclass 2 or $(G,Z(G))$ is a Camina pair, then $G$ admits a noninner automorphism of order 2 or 4 leaving the Frattini subgroup elementwise fixed.

In this paper, we first discuss some basic properties of semipreinvex functions. We then show that the ratio of semipreinvex functions is semipreinvex, which extends earlier results by Khan and Hanson [6] and Craven and Mond [3]. Finally, saddle point optimality criteria are developed for a multiobjective fractional programming problem under semipreinvexity conditions.

We show that for every integer k ≥ 3 and every prime p ≤ 5 there is a group with soluble length k and order .

We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

In this paper, we prove that there exist three finite sets Sj (j = 1, 2, 3) such that any two non-constant meromorphic functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1,2,3 must be identical. As a particular case of the above result, we obtain that there exist two finite sets Sj (j = 1, 2) such that any two non-constant entire functions f and g satisfying Ej(Sj) = Eg(Sj) for j = 1, 2 must be identical, which answers a question posed by Gross.

We study the long-standing problem of the existence of non-Berwaldian Landsberg spaces from the perspective of conformal transformations. We calculate the Berwald and Landsberg tensors in terms of the T-tensor and show that there are Landsberg spaces with nonvanishing T-tensor. We give a necessary condition for a Landsberg space to be Berwaldian. We find conditions under which the Landsberg spaces cannot be Berwaldian and give examples of ($y$-local) non-Berwaldian Landsberg spaces.

Roughly speaking, we show that a Banach space X has the fixed point property for nonexpansive mappings whenever X has the WORTH property and the unit sphere of X does not contain a triangle with sides of length larger than 2.

Let ${ \mathbb{F} }_{q} $ be the finite field of characteristic $p$ containing $q= {p}^{r} $ elements and $f(x)= a{x}^{n} + {x}^{m} $, a binomial with coefficients in this field. If some conditions on the greatest common divisor of $n- m$ and $q- 1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x)= a{x}^{n} + {x}^{m} $ permutes ${ \mathbb{F} }_{p} $, where $n\gt m\gt 0$ and $a\in { \mathbb{F} }_{p}^{\ast } $, then $p- 1\leq (d- 1)d$, where $d= \gcd (n- m, p- 1)$, and that this bound of $p$, in terms of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of ${ \mathbb{F} }_{q} $ from a permutation binomial over ${ \mathbb{F} }_{q} $.

We construct a semigroup S with an epimorphically embedded proper subband U. The band U furnishes an example of a regular semigroup which is not saturated, thus answering a question posed by Hall, Semigroup Forum (to appear).