We prove, under some conditions on the domains, that the adjoint of the sum of two unbounded operators is the sum of their adjoints in both Hilbert and Banach space settings. A similar result about the closure of operators is also proved. Some interesting consequences and examples “spice up” the paper.

We determine an explicit cell decomposition of the wonderful compactification of a semisimple algebraic group. To do this we first identify the $B\,\times \,B$-orbits using the generalized Bruhat decomposition of a reductive monoid. From there we show how each cell is made up from $B\,\times \,B$-orbits.

A counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension 3 with larger Dilworth number than Sperner number. The Dilworth number of $A\left[ {Z}/{pZ}\;\,\oplus \,{Z}/{pZ}\; \right]$ is computed when $A$ is an unramified principal Artin local ring.

Let $G$ be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net $\left( {{f}_{\alpha }} \right)$ of positive, normalized functions in ${{L}_{1}}\left( G \right)$ such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.

It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalities

We prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.

We study here the smoothness of solutions of the Dirichlet problem for elliptic equations in a region G with a piece-wise smooth boundary. The smoothness of the solution given depends on the smoothness of the coefficients of the equation, the boundary, the boundary function and the values of the angles on the boundary and the values of the coefficients of the second derivatives at the corners.

By an r-graph, we mean a finite set V of elements called vertices and a collection of some of the r-subsets of V called edges with the property that each vertex is incident with at least one edge. An A-chromatic r-graph is an r-graph all of whose edges are coloured A.

Theorem. Let G1, …, Gt denote r-graphs. There exists a nonempty class of r-graphs such that for each if the edges of G are painted arbitrarily in t colours A1, …, At, then for at least one i in {1, …, t}, G has an Ai-chromatic r-subgraph which is isomorphic to Gi.

We use the general factorisation theorems of Grothendieck, Nikishin and Maurey to characterise coefficient multipliers between Bergman spaces and into the Nevanlinna class.

Various weighted ${{l}_{p}}$-norm inequalities in convolutions are derived by a simple and general principle whose ${{l}_{2}}$ version was obtained by using the theory of reproducing kernels. Applications to the Riemann zeta function and a difference equation are also considered.

J.H. van Lint has recently shown [1] that if A(n, m) denotes the n number of representations of m in the form , where εk = 0 or 1 for -n ≤ k ≤ n then

(1)

Using this result, the fact that A(n, m) is a non-increasing function of |m|, and a simple recurrence relation for A(n, m) we derive the following extension of (1):

(2)

where [0 (n)] is any integral valued function m(n) = 0(n).

Let n and k be positive integers, k≥3. Denote by ϕ(n, k) the least positive integer such that if F is any family of more than ϕ(n, k) sets, each set with n elements, then some k members of F have pairwise the same intersection. In this paper we obtain a new asymptotic upper bound for ϕ(n, k), k fixed, n approaching infinity.

Consider the second order superlinear dynamic equation

$$(*)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{x}^{\Delta \Delta }}(t)+p(t)f(x(\sigma (t)))=0$$

where $p\,\in \,C(\mathbb{T},\,\mathbb{R})$, $\mathbb{T}$ is a time scale, $f\,:\,\mathbb{R}\,\to \,\mathbb{R}$ is continuously differentiable and satisfies ${{f}^{'}}(x)>0$, and $x\,f\,(x)\,>\,0$ for $x\,\ne \,0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which includes the nonlinear function $f(x)\,=\,{{x}^{\alpha }}$ with $\alpha \,>\,1$, commonly known as the Emden–Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for arbitrarily large values of $t$. In addition to extending the result of Kiguradze for $\left( * \right)$ in the real case $\mathbb{T}\,=\,\mathbb{R}$, we obtain analogues in the difference equation and $q$-difference equation cases.

A strong version of Levine′s decomposition of continuity leads to the result that a closed graph weakly continuous function into a rim-compact space is continuous. This result implies a closed graph theorem: every almost continuous closed graph function into a strongly locally compact space is continuous. An open problem of Shwu-Yeng T. Lin and Y.-F. Lin asks if every almost continuous closed graph function from a Baire space to a second countable space is necessarily continuous. This question is answered in the negative by an example.

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.

We find a lower bound on the number of imaginary quadratic extensions of the function field ${{\mathbb{F}}_{q}}\left( T \right)$ whose class groups have an element of a fixed order.

More precisely, let $q\,\ge \,5$ be a power of an odd prime and let $g$ be a fixed positive integer $\ge \,3$. There are $\gg \,{{q}^{\ell \left( \frac{1}{2}+\frac{1}{g} \right)}}$ polynomials $D\,\in \,{{\mathbb{F}}_{q}}\left[ T \right]$ with $\deg \left( D \right)\,\le \,\ell $ such that the class groups of the quadratic extensions ${{\mathbb{F}}_{q}}\left( T,\,\sqrt{D} \right)$ have an element of order $g$.

We provide more characterizations of varieties with a weak difference term and of neutral varieties. We prove that a variety has a (weak) difference term (is neutral) with respect to the $\text{TC}$-commutator iff it has a (weak) difference term (is neutral) with respect to the linear commutator. We show that a variety $V$ is congruence meet semi-distributive iff $V$ is neutral, iff ${{M}_{3}}$ is not a sublattice of Con $\mathbf{A}$, for $\mathbf{A}\in V$, iff there is a positive integer $n$ such that $V{{\vDash }_{Con}}\alpha (\beta \,o\,\gamma )\le \alpha {{\beta }_{n}}$.

We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces $X$ with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space $\left( \mathcal{P}\left( X \right),\,{{W}_{2}} \right)$ satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.

If R is a ring and I is a right ideal of R then I is called faithful if R - I is a faithful right R-module, i.e. if { r ∊ R: Rr⊆ I} = (0). I is called irreducible [ 1 ] provided that if J1 and J2 are right ideals such that J1 ∩ J2 = I, then J1 or J2 = I. Let N(I){ r ∊ R: rI⊆ I} and [ I: a ] = { r ∊ R: ar⊆ I} for a ∊ R. We write (a)r for [(0): a ].

A combinatorial proof is given for Uchimura’s identity

As a corollary to this proof we derive a formula for the sum of the nth powers of the divisors of m in terms of partitions of m.

It is known that every Toeplitz matrix $T$ enjoys a circulant and skew circulant splitting (denoted $\text{CSCS}$) i.e., $T=C-S$ a circulant matrix and $S$ a skew circulant matrix. Based on the variant of such a splitting (also referred to as $\text{CSCS}$), we first develop classical $\text{CSCS}$ iterative methods and then introduce shifted $\text{CSCS}$ iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical $\text{CSCS}$ iterative methods work slightly better than the Gauss–Seidel $(\text{GS})$ iterative methods if the $\text{CSCS}$ is convergent, and that there is always a constant $\alpha $ such that the shifted $\text{CSCS}$ iteration converges much faster than the Gauss–Seidel iteration, no matter whether the $\text{CSCS}$ itself is convergent or not.