We provide bounds on the average distance between two points uniformly and independently chosen from a compact convex subset of the s-dimensional Euclidean space.

Let ℳ be a regular map of genus g>1 and X be the underlying Riemann surface. A reflection of ℳ fixes some simple closed curves on X, which we call mirrors. Each mirror passes through at least two of the geometric points (vertices, face-centers and edge-centers) of ℳ. In this paper we study the surfaces which contain mirrors passing through just two geometric points, and show that only Wiman surfaces have this property.

In this paper, we study the mean square asymptotic stability of a generalized half-linear neutral stochastic differential equation with variable delays applying fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton, Zhang and Luo. Two examples are given to illustrate our results.

Some classical examples in vector integration due to Phillips, Hagler and Talagrand are revisited from the point of view of the Birkhoff and McShane integrals.

In this article, we define lattice graphs (which generalise ultragraphs) as well as their Cuntz–Krieger families and C*-algebras. We will give a thorough study in the special case of lattice atomic graphs.

Modular forms for a discrete subgroup Γ of SL(2,ℝ) can be identified with holomorphic sections of line bundles over the modular curve U corresponding to Γ, and quasimodular forms generalize modular forms. We construct vector bundles over U whose sections can be identified with quasimodular forms for Γ.

Let a,b,c be relatively prime positive integers such that a2+b2=c2 with b even. In 1956 Jeśmanowicz conjectured that the equation ax+by=cz has no solution other than (x,y,z)=(2,2,2) in positive integers. Most of the known results of this conjecture were proved under the assumption that 4 exactly divides b. The main results of this paper include the case where 8 divides b. One of our results treats the case where a has no prime factor congruent to 1 modulo 4, which can be regarded as a relevant analogue of results due to Deng and Cohen concerning the prime factors of b. Furthermore, we examine parities of the three variables x,y,z, and give new triples a,b,c such that the conjecture holds for the case where b is divisible by 8. In particular, to prove our results, we shall show an important result which asserts that if x,y,z are all even, then x/2,y/2,z/2 are all odd. Our methods are based on elementary congruence and several strong results on generalized Fermat equations given by Darmon and Merel.

A radical α has the Amitsur property if α(A[x])=(α(A[x])∩A)[x] for all rings A. For rings R⊆S with the same unity, we call S a finite centralizing extension of R if there exist b1,b2,…,bt∈S such that S=b1R+b2R+⋯+btR and bir=rbi for all r∈R and i=1,2,…,t. A radical α is FCE-friendly if α(S)∩R⊆α(R) for any finite centralizing extension S of a ring R. We show that if α is a supernilpotent radical whose semisimple class contains the ring ℤ of all integers and α is FCE-friendly, then α has the Amitsur property. In this way the Amitsur property of many well-known radicals such as the prime radical, the Jacobson radical, the Brown–McCoy radical, the antisimple radical and the Behrens radical can be established. Moreover, applying this condition, we will show that the upper radical 𝒰(*k) generated by the essential cover *k of the class * of all *-rings has the Amitsur property and 𝒰(*k)(A[x])=𝒰(*k)(A)[x], where a semiprime ring R is called a *-ring if the factor ring R/I is prime radical for every nonzero ideal I of R. The importance of *-rings stems from the fact that a *-ring A is Jacobson semisimple if and only if A is a primitive ring.

We give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and nonperiodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.

It is shown that a rational map of degree at least 2 admits a meromorphic invariant line field if and only if it is conformally conjugate to either an integral Lattès map, a power map, or a Chebyshev polynomial.

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.

In this paper, we generalize a result of Bennett and Lutzer and give a condition under which a continuously Urysohn space must have a one-parameter continuous separating family.

A characterization of Banach spaces possessing the weak Radon–Nikodým property is given in terms of finitely additive interval functions. Due to that characterization several Banach space valued set functions that are only finitely additive can be represented as integrals.

Determining meaningful lower bounds on the supremal strict p-negative type of classes of finite metric spaces is a difficult nonlinear problem. In this paper we use an elementary approach to obtain the following result: given a finite metric space (X,d) there is a constant ζ>0, dependent only on n=∣X∣ and the scaled diameter 𝔇=(diamX)/min{d(x,y)∣x⁄=y} of X (which we may assume is >1), such that (X,d) has p-negative type for all p∈[0,ζ] and strict p-negative type for all p∈[0,ζ). In fact, we obtain A consideration of basic examples shows that our value of ζ is optimal provided that 𝔇≤2. In other words, for each 𝔇∈(1,2] and natural number n≥3, there exists an n-point metric space of scaled diameter 𝔇 whose supremal strict p-negative type is exactly ζ. The results of this paper hold more generally for all finite semi-metric spaces since the triangle inequality is not used in any of the proofs. Moreover, ζ is always optimal in the case of finite semi-metric spaces.

Let p be a prime. We say that a transitive action of a group L on a set Ω is p-sub-regular if there exist x,y∈Ω such that 〈Lx,Ly〉=L and LYx≅ℤp, where Y =yLx is the orbit of y under Lx. Our main result is that if Γ is a G-arc-transitive graph and the permutation group induced by the action of Gv on Γ(v) is p-sub-regular, then the order of a G-arc-stabilizer is equal to ps−1 where s≤7, s≠6, and moreover, if p=2, then s≤5. This generalizes a classical result of Tutte on cubic arc-transitive graphs as well as some more recent results. We also give a characterization of p-sub-regular actions in terms of arc-regular actions on digraphs and discuss some interesting examples of small degree.

In a wide variety of related problems, a flexible but inextensible membrane is filled to capacity with incompressible fluid. In all such cases, the resulting shape satisfies a set of three simultaneous partial differential equations. At present there are no known solutions of any generality, and the main aim of this paper is to formulate the partial differential equations with a view to stimulating further interest in this important class of problems.

We discuss a Conley index calculation which is of importance in population models with large interaction. In particular, we prove that a certain Conley index is trivial.