A similar characterization, as the Gelfand-Kolmogoroff theorem for the maximal ideals in $C(X)$, is given for the maximal ideals of $C_c(X)$. It is observed that the $z_c$-ideals in $C_c(X)$ are contractions of the $z$-ideals of $C(X)$. Using this, it turns out that maximal ideals (respectively, prime $z_c$-ideals) of $C_c(X)$ are precisely the contractions of maximal ideals (respectively, prime $z$-ideals) of $C(X)$, as well. Maximal ideals of $C^*_c(X)$ are also characterized, and two representations are given. We reveal some more useful basic properties of $C_c(X)$. In particular, we observe that, for any space $X$, $C_c(X)$ and $C^*_c(X)$ are always clean rings. It is also shown that $\beta _0X$, the Banaschewski compactification of a zero-dimensional space $X$, is homeomorphic with the structure spaces of $C_c(X)$, $C^F(X)$, $C_c(\beta _0X)$, as well as with that of $C(\beta _0 X)$. $F_c$-spaces are characterized, the spaces $X$ for which $C_c(X)_P$, the localization of $C_c(X)$ at prime ideals $P$, are uniform (or equivalently are integral domain). We observe that $X$ is an $F_c$-space if and only if $\beta _0X$ has this property. In the class of strongly zero-dimensional spaces, we show that $F_c$-spaces and $F$-spaces coincide. It is observed that, if either $C_c(X)$ or $C^*_c(X)$ is ax Bezout ring, then $X$ is an $F_c$-space. Finally, $C_c(X)$ and $C^*_c(X)$ are contrasted with regards to being an absolutely Bezout ring. Consequently, it is observed that the ideals in $C_c(X)$ are convex if and only if they are absolutely convex if and only if $C_c(X)$ and $C^*_c(X)$ are both unitarily absolute Bezout rings.

A well-known result of Beukers on the generalized Ramanujan-Nagell equation has, at its heart, a lower bound on the quantity $|x^2-2^n|$. In this paper, we derive an inequality of the shape $|x^3-2^n| \geq x^{4/3}$, valid provided $x^3 \neq 2^n$ and $(x,n) \neq (5,7)$, and then discuss its implications for a variety of Diophantine problems.

Stuart proved that the Cesaro matrix $C_1$ cannot sum almost every subsequence of a bounded divergent sequence $x$. At the end of the paper, he remarked, ``It seems likely that this proposition could be generalized for any regular matrix, but we do not have a proof of this.'' In this note, we confirm Stuart's conjecture, and we extend it to the more general case of divergent sequences $x$.

We prove the existence of infinitely many solutions for $p$-biharmonic problems in a bounded, smooth domain $\Omega $ with concave-convex nonlinearities dependent upon a parameter $\lambda $ and a positive continuous function $f\colon \overline {\Omega }\to \mathbb {R}$. We simultaneously handle critical case problems with both Navier and Dirichlet boundary conditions by applying the Ljusternik-Schnirelmann method. The multiplicity of solutions is obtained when $\lambda $ is small enough. In the case of Navier boundary conditions, all solutions are positive, and a regularity result is proved.

Exponential polynomials as solutions of differential equations with constant coefficients are widely used for approximation purposes. Recently, mixed spaces containing algebraic, trigonometric and exponential functions have been extensively considered for design purposes. The analysis of these spaces leads to constructions that can be reduced to Hermite interpolation problems. In this paper, we focus on spaces generated by algebraic polynomials, hyperbolic sine and hyperbolic cosine. We present classical interpolation formulae, such as Newton and Aitken-Neville formulae and a suggestion of implementation. We explore another technique, expressing the Hermite interpolant in terms of polynomial interpolants and derive practical error bounds for the hyperbolic interpolant.

${\mathcal{A}}'(\kappa)$ asserts the existence of pairwise almost compatible finite-to-one functions $A\to \omega$ for each countable subset $A$ of $\kappa$. The existence of winning $2$-Markov strategies in several infinite-length games, including the Menger game on the one-point Lindelofication $\kappa^\dagger$ of $\kappa$, are guaranteed by ${\mathcal{A}}'(\kappa)$. ${\mathcal{A}}'(\kappa)$ is implied by the existence of cofinal Kurepa families of size $\kappa$, and thus, holds for all cardinals less than $\aleph _\omega$. It is consistent that ${\mathcal{A}}'({\aleph _\omega })$ fails; however, there must always be a winning $2$-Markov strategy for the second player in the Menger game on $\omega_\omega^\dagger$.

The aim of this paper is to establish symmetry and monotonicity of solutions to the equation involving fractional Laplacians of higher order. For this purpose, we first reduce the equation into a system via the composition of lower fractional Laplacians and then obtain symmetry and monotonicity of solutions to the system by applying the method of moving planes.

In this paper, we consider the algebra of WCE operators on $L^p$-spaces, and we investigate some al\-ge\-braic properties of it. For instance, we show that the set of normal WCE operators is a unital finite Von Neumann algebra, and we obtain the spectral measure of a normal WCE operator on $L^2(\mathcal {F})$. Then, we specify the form of projections in the Von Neumann algebra of normal WCE operators, and we obtain that, if the underlying measure space is purely atomic, then all projections are minimal. In the non-atomic case, there is no minimal projection. Also, we give a non-commutative operator algebra on which the spectral map is subadditive and submultiplicative. As a consequence, we obtain that the set of quasinilpotents is an ideal, and we get a relation between quasinilpotents and commutators. Moreover, we give some sufficient conditions for an algebra of WCE operators to be triangularizable, and consequently, that its quotient space over its quasinilpotents is commutative.

In this paper, the generalized Miranda theorem is applied for second-order systems of differential equations with one boundary condition given by Riemann-Stieltjes integral \[ x'' = f(t,x,x'), \quad x(0) = 0, \ x'(1) = \int _0^1 x(s) \, dg(s),\] where $f : [0,1]\times \mathbb{R} ^k\times \mathbb{R} ^k \to \mathbb{R} ^k$ is continuous and $g : [0,1] \to \mathbb{R} ^k$ has bounded variation. Under suitable assumptions upon $f$ and $g$ we prove the existence of solutions to such posed problem.

In this paper, we prove that the space of two parameter matrix-valued BMO functions can be char\-ac\-terized by considering iterated commutators with the Hilbert transform. Specifically, we prove that $$ \| B \|_{BMO} \!\lesssim \! \| [[M_B, H_1],H_2] \|_{L^2(\mathbb{R} ^2;\mathbb{C} ^d) \rightarrow L^2(\mathbb{R} ^2;\mathbb{C} ^d)} \!\lesssim \! \| B \|_{BMO}. $$ The upper estimate relies on Petermichl's representation of the Hilbert transform as an average of dyadic shifts and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's proof for the scalar case.

Let $R$ be a Noetherian ring, $N$ a finitely generated $R$-module and $I$ an ideal of $R$. It is shown that the sequences $Ass _R R/(I^n)_a^{(N)}$, $Ass _R (I^n)_a^{(N)}/ (I^{n+1})^{(N)}_a$ and $Ass _R (I^n)_a^{(N)}/ (I^n)_a$, $n= 1,2, \ldots $, of associated prime ideals, are increasing and ultimately constant for large $n$. Moreover, it is shown that, if $S$ is a multiplicatively closed subset of $R$, then the topologies defined by $(I^n)_a^{(N)}$ and $S((I^n)_a^{(N)})$, $n\geq 1$, are equivalent if and only if $S$ is disjoint from the quintasymptotic primes of $I$. By using this, we also show that, if $(R, \mathfrak {m})$ is local and $N$ is quasi-unmixed, then the local cohomology module $H^{\dim N}_I(N)$ vanishes if and only if there exists a multiplicatively closed subset $S$ of $R$ such that $\mathfrak {m} \cap S \neq \emptyset $ and the topologies induced by $(I^n)_a^{(N)}$ and $S((I^n)_a^{(N)})$, $n\geq 1$, are equivalent.

In this paper, a closed-form expression for counting all SAWs, irrespective of length, but restricted to the finite lattice strip $\{ -a,\ldots ,0,\ldots ,b\}\times \{0,1\}$, shall be obtained in terms of the non-negative integer parameters $a$ and $b$. In addition, the argument used to prove this result will be extended to establish an enumerating formula for counting all SAWs, irrespective of length, but restricted to the half-finite lattice strip of width two $\{ 0,1,\ldots ,n\}\times \{ 0,1,2\}$, in terms of $n$.

A real $n\times n$ matrix $A$ is called an expansive dilation if all of its eigenvalues $\lambda $ satisfy $|\lambda |\!\!>\!\!1$. Let $\varphi : \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function. The aim of this article is to find an appropriate general space which includes the weak Hardy space of Fefferman and Soria, the weighted weak Hardy space of Quek and Yang}, the anisotropic weak Hardy space of Ding and Lan, the Musielak-Orlicz Hardy space of Ky and the anisotropic Hardy space of Musielak-Orlicz type of Li, Yang and Yuan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $m\in \mathbb {N}$ and obtain some new real-variable characterizations of $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ in terms of the radial, the non-tangential and the tangential maximal functions via a new monotone convergence theorem adapted to the weak anisotropic Musielak-Orlicz space $L^{\varphi , \infty }({\mathbb {R}}^n)$. These maximal function characterizations generalize the known results on the anisotropic weak Hardy space $H^{p, \infty }_A({\mathbb {R}}^n)$ with $p\in (0, 1]$ and are new even for their weighted variants or weak Orlicz-Hardy variants. As an application, the authors show the boundedness of a class of multilinear operators formed by the anisotropic Calderon-Zygmund operators from product weighted Lebesgue space to $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $\varphi (x,t):=t^p\omega (x)$ and $\omega \in \mathbb {A}_1(A)$, which is a weighted and non-isotropic extension of Grafakos.

In this work, a representation of functionals and a necessary and sufficient condition for uniformly non-square points of Orlicz-Bochner sequence spaces endowed with the Orlicz norm are given.

Generalized frames (in short, $g$-frames) are a natural generalization of standard frames in separable Hilbert spaces. Motivated by the concept of weaving frames in separable Hilbert spaces by Bemrose, Casazza, Grochenig, Lammers and Lynch in the context of distributed signal processing, we study weaving properties of $g$-frames. Firstly, we present necessary and sufficient con\-ditions for weaving $g$-frames in Hilbert spaces. We extend some results of \cite Bemrose, Casazza, Grochenig, Lammers and Lynch, and Casazza and Lynch regarding conversion of standard weaving frames to $g$-weaving frames. Some Paley-Wiener type perturbation results for weaving $g$-frames are obtained. Finally, we give necessary and sufficient conditions for weaving $g$-Riesz bases.

The conditions on a Banach space $E$ under which the algebra $\mathcal {K}(E)$ of compact operators on $E$ is right flat or homologically unital are investigated. These homological properties are related to factorization in the algebra, and, it is shown that, for $\mathcal {K}(E)$, they are closely associated with the approximation property for $E$. The class of spaces $E$ such that $\mathcal {K}(E)$ is known to be right flat and homologically unital is extended to include spaces which do not have the bounded compact approximation property.