Let u$u$ be a supertemperature on an open set E$E$, and let v$v$ be a related temperature on an open subset D$D$ of E$E$. For example, v$v$ could be the greatest thermic minorant of u$u$ on D$D$, if it exists. Putting w=u$w=u$ on E\setminus D$E\setminus D$ and w=v$w=v$ on D$D$, we investigate whether w$w$, or its lower semicontinuous smoothing, is a supertemperature on E$E$. We also give a representation of the greatest thermic minorant on E$E$, if it exists, in terms of PWB solutions on an expanding sequence of open subsets of E$E$ with union E$E$.  In addition, in the case of a nonnegative supertemperature, we prove inequalities that relate reductions to Dirichlet solutions. We also prove that the value of any reduction at a given time depends only on earlier times.

We determine the limiting distribution of the distance from the root of a tree to any nearest endnode of the tree (other than the root) for certain families of rooted trees.

We describe measurable Hilbert sheaves as Hilbert space objects in a sheaf category constructed from a measure space. These are quite useful for the interpretation of the direct integral of Hilbert spaces as an indexed functor. We set up a framework to put this and similar constructions of operator theory on an indexed categorical footing.

A balanced incomplete block design or BIBD is defined as an arrangement of v objects in b blocks containing k objects all different, so that there are r blocks containing a given object and λ blocks containing any two given objects.

A regular double p-algebra L satisfying (i) ∩(xn(+*); n < ω) for every 1 ≠ x ∈ L and (ii) L is not subdirectly irreducible, is constructed. The construction is purely topological and the desired result is obtained via the known Priestly duality. The notion of an auxiliary regular double p-algebra is introduced and the algebras having this property are characterized.

A simple technique for obtaining bounds in terms of means and variances for the expectations of certain functions of random variables in a given class is examined. The bounds given are sharp in the sense that they are attainable by at least one random variable in the class. This technique is applied to obtain bounds for moment generating functions, the coefficient of skewness and parameters associated with branching processes. In particular an improved lower bound for the Malthusian parameter in an age-dependent branching process is derived.

In [3], Ligh proved that every distributively generated Boolean near-ring is a ring, and he gave an example to which the above fact can not be extended. That is, let G be an additive group and let the multiplication on G be defined by xy = y for all x, y in G. Ligh called this Boolean near-ring G a general Boolean near-ring. near-ring. Then in [4], Ligh called Ra β-near-ring if for each x in R, x2 = x and xyz = yxz for all x, y, z in R, and he proved that the structure of a β-near-ring is “very close” to that of a usual Boolean ring. We note that general Boolean near-rings and Boolean semirings as defined in [5] are β-near-rings. The purpose of this paper is to generalize the structure theorem on β-near-rings given by Ligh in [4] to a broader class of near-rings.

In this paper we study the size of the value set of the Carmichael λ-functions.

Let E be a given field and G some (not necessarily the total) group of automorphisms of E. We introduce a topology in G by saying that a net of elements Tα in G converges to T in G if for each x∈E, Tα(x) coincides with Tα(x) ultimately. We shall call this the Krull topology on G.

We attempt to discuss a new circle problem. Let \unicode[STIX]{x1D701}(s)$\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function \sum _{n=1}^{\infty }n^{-s}$\sum _{n=1}^{\infty }n^{-s}$ (\text{Re}\,s>1$\text{Re}\,s>1$) and L(s,\unicode[STIX]{x1D712}_{4})$L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet L$L$-function \sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ (\text{Re}\,s>1$\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function R_{(1,1)}(n)$R_{(1,1)}(n)$ by the coefficient of the Dirichlet series \unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$(\text{Re}\,s>1)$(\text{Re}\,s>1)$. This is an analogue of r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for \sum _{n\leq x}r(n)$\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for \sum _{n\leq x}R_{(1,1)}(n)$\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

An elementary proof is given of a theorem of Castro and Lazer that the degree of the gradient of a coercive functional on a large ball of Rn is one.

We study commutators in pseudo-orthogonal groups O2nR (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO2nR. We estimate the number of commutators, c(O2nR) and c(GO2nR), needed to represent every element in the commutator subgroup. We show that c(O2nR) ≤ 4 if R satisfies the ∧-stable condition and either n ≥ 3 or n = 2 and 1 is the sum of two units in R, and that c(GO2nR) ≤ 3 when the involution is trivial and ∧ = R∈. We also show that c(O2nR) ≤ 3 and c(GO2nR) ≤ 2 for the ordinary orthogonal group O2nR over a commutative ring R of absolute stable rank 1 where either n ≥ 3 or n = 2 and 1 is the sum of two units in R.

A right R-module AR will be said to be right projectively torsion-free (AR is PTF) if for every a ∈ A, there exist subsets {a1, a2, …, an,} ⊆ A and {x1, x2, …, xn} ⊆ R such that a = Σni = 1 aixi and for all x ∈ R, if ax = 0 then xix = 0 for all 1 ≦ i ≦ n.

A semigroup is a nonvoid Hausdorff space together with a continuous associative operation. A semiring is a nonvoid Hausdorff space together with a couple of continuous associative operations, one of which (usually denoted as multiplication) distributes over the other (usually denoted as addition). If R is a semiring then an R-semimodule is a semigroup M under addition together with a continuous operation R × M → M which satisfies the associativity and distributivity conditions usually stipulated in the instance of an R-module. It is purpose of this paper to establish for semimodules certain propositions proved by Kaplansky [4], Pearson [8], Selden [9], Beidleman-Cox [1] and others.

We prove that an operator system is (min, ess)-nuclear if its C^{\ast }$C^{\ast }$ -envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by Farenick et al. [‘Operator systems from discrete groups’, Comm. Math. Phys.329(1) (2014), 207–238] is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete).

Let Fp be the residue field modulo a prime number p. The mappings of Fp, into itself are viewed as functions in one variable over Fp. When the mapping is onto, the function is a permutation.

A well-known theorem of Hardy and Littlewood gives a three-term asymptotic formula, counting the lattice points inside an expanding, right triangle. In this paper a generalisation of their theorem is presented. Also an analytic method is developed which enables one to interpret the coefficients in the formula. These methods are combined to give a generalisation of a “heightcounting” formula of Györy and Pethö which itself was a generalisation of a theorem of Lang.

Let K be an arbitrary field of characteristic 2, F a free group of countably infinite rank. We construct a finitely generated fully invariant subgroup U in F such that the relatively free group F/U satisfies the maximal condition on fully invariant subgroups but the group algebra K (F/U) does not satisfy the maximal condition on fully invariant ideals. This solves a problem posed by Plotkin and Vovsi. Using the developed techniques we also construct the first example of a non-finitely based (nilpotent of class 2)-by-(nilpotent of class 2) variety whose Abelian-by-(nilpotent of class at most 2) groups form a hereditarily finitely based subvariety.

V. Krishnamurthy has shown that on a finite set X all topologies can be mapped into a certain set of matrices of zeros and ones. In this paper it is shown that all lattices, algebras and rings on a finite set X can be mapped onto particular sets of matrices of zeros and ones.

Perturbation expansions are sought for the flow variables associated with the diffraction of a plane weak shock wave around convex-angled corners in a polytropic, inviscid, thermally-nonconducting gas. Lighthill's method of strained co-ordinates [4] produces a uniformly valid expansion for most of the diffracted front, while the remainder of this front is treated by a modification of the shock-ray theory of Whitham [6]. The solutions from these approaches are patched just inside the ‘shadow’ region yielding a plausible description of the entire diffracted shock front.