$K$-frames, as a new generalization of frames, have important applications, especially in sampling theory, to help us to reconstruct elements from a range of a bounded linear operator $K$ in a separable Hilbert space. In this paper, we focus on the reconstruction formulae to characterize all $K$-duals of a given $K$-frame. Also, we give several approaches for constructing $K$-frames.

Requirements under which the Daugavet equation and the alternative Daugavet equation hold for pairs of nonlinear maps between Banach spaces are analyzed. A geometric description is given in terms of nonlinear slices. Some local versions of these properties are also introduced and studied, as well as tests for checking whether the required conditions are satisfied in relevant cases.

This paper examines a class of Kirchhoff nonlocal operators involving concave-convex nonlinearities and sign-changing weight functions. With the aid of the Nehari manifold, the existence of multiple nontrivial nonnegative solutions is obtained.

We investigate the number of parts modulo~$m$ of $m$-ary partitions of a positive integer~$n$. We prove that the number of parts is equidistributed modulo~$m$ on a special subset of $m$-ary partitions. As consequences, we explain when the number of parts is equidistributed modulo~$m$ on the entire set of partitions, and we provide an alternate proof of a recent result of Andrews, Fraenkel and Sellers regarding the number of $m$-ary partitions modulo~$m$.

To a semi-cosimplicial object (SCO) in a category, we associate a system of partial shifts on the inductive limit. We show how to produce an SCO from an action of the infinite braid monoid $\mathbb{B} ^+_\infty $ and provide examples. In categories of (noncommutative) probability spaces, SCOs correspond to spreadable sequences of random variables; hence, SCOs can be considered as the algebraic structure underlying spreadability.

Let $R$ be an integral domain and $I$ a non\-zero ideal of $R$. An ideal $J\subseteq I$ is a $t$-reduction of~$I$ if $(JI^{n})_{t}=(I^{n+1})_{t}$ for some integer $n\geq 0$. An ele\-ment $x\in R$ is $t$-integral over $I$ if there is an equation $x^{n}+a_{1}x^{n-1}+ \cdots +a_{n-1}x+a_{n}=0$ with $a_{i}\in (I^{i})_{t}$ for $i=1,\ldots ,n$. The set of all elements that are $t$-integral over $I$ is called the $t$-integral closure of $I$. This paper investigates the $t$-reductions and $t$-integral closure of ideals. Our objective is to establish satisfactory $t$-analogues of well known results in the literature, on the integral closure of ideals and~its corr\-el\-ation with reductions, namely, Section 2 identifies basic properties of $t$-reductions of ideals and features explicit examples discriminating between the notions of reduction~and $t$-reduction. Section~3 investigates the concept of $t$-integral closure of ideals, including its correlation with $t$-reductions. Section~4 studies the persistence and contraction of $t$-integral closure of id\-eals under ring homomorphisms. Throughout the paper, the main results are illustrated with original examples.

In this paper, we study some algebraic properties of the spanning simplicial complex $\Delta _s(G)$ associated to a multigraph~$G$. It is proved that, for any finite multi\-graph~$G$, $\Delta _s(G)$ is a pure vertex decomposable simplicial complex and therefore shellable and Cohen-Macaulay. As a consequence, we deduce that, for any multigraph~$G$, the quotient ring $R/I_c(G)$ is Cohen-Macaulay, where \[ I_c(G)=(x_{i_1} \cdots x_{i_k} \mid \{x_{i_1},\ldots , x_{i_k}\}\qquad \qquad \qquad \] \[ \qquad \qquad \qquad \mbox {is the edge set of a cycle in~$G$}). \] Also, some homological invariants of the Stanley-Reisner ring of $\Delta _s(G)$, such as projective dimension and regularity, are investigated.

Given a field $k$ of characteristic~$0$ and~an indeterminate $T$, the main topic of this paper is the con\-struction of specializations of any given finite extension of $k(T)$ of degree~$n$ that are degree~$n$ field extensions of~$k$ with specified local behavior at any given finite set of primes of~$k$. First, we give a full non-Galois analog of a result with a ramified-type conclusion from a preceding paper, and next we prove a unifying statement which combines our results and previous work devoted to the unramified part of the problem in the case where $k$ is a number field.

Let $P_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved, among other results, that, for every sufficiently large, even integer $N$ satisfying the congruence condition $N \not \equiv 2\pmod 3$, the equation \[ N=x^2+p^2+p_1^3+p_2^4+p_3^4+p_4^4 \] is solvable with $x$ being a $P_{5}$ and the other variable primes. This result constitutes an enhancement upon that of Vaughan \cite {s12} and Mu \cite {s9}.

This paper gives a description of various recent results, which construct monomial ideals with a given minimal free resolution. We show that these results are all instances of coordinatizing a finite atomic lattice, as found in~\cite {mapes}. Subsequently, we explain how, in some of these cases \cite {Faridi, Floystad1} where questions still remain, this point of view can be applied. We also prove an equivalence for trees between the notion of \textit {maximal} defined in~\cite {Floystad1} and the notion of being maximal in a Betti stratum.

In this paper, we extend the Li coefficients for automorphic $L$-functions and the Li criterion for the Riemann hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips for automorphic $L$-functions inside the critical strip. Next, we give an arithmetical and asymptotical formula for these coefficients. Finally, we show that there exists an entire function of exponential type that interpolates the extended Li coefficients (or the $\tau $-Li coefficients) at integer values. The results of this paper arise from ideas of the author~\cite {15}, Freitas~\cite {8}, Lagarias~\cite {10} and Odz$\breve {a}$k and Smajlovi$\grave {c}$~\cite {17}.

In this paper, an eigenvalue problem for a regular fourth order ordinary differential equation is considered, where one of the boundary conditions linearly depends upon the eigenvalue parameter. The first four terms in the asymptotic expansion of the eigenvalues are derived.

For $f \in \mathbb{Q} [x]$, we say that a rational prime~$p$ is a prime divisor of $f$ if $p$ divides the numerator of $f(n)$ for some integer $n$. Let $\mathcal{P} (f)$ denote the set of prime divisors of~$f$. We present an elementary proof of the following theo\-rem, which generalizes results of Bauer and Brauer: fix a nonzero integer~$g$. Suppose that $f(x) \in \mathbb{Q} [x]$ is a nonconstant polynomial having a root in $\mathbb{Q} _p$ for every prime $p$ dividing $g$, and having a root in $\mathbb{R} $ if $g \lt 0$. Let $m$ be a positive integer coprime to~$g$, and let~$H$ be a subgroup of $(\mathbb{Z} /m\mathbb{Z} )^{\times }$ not containing $g\bmod {m}$. Then there are infinitely many primes $p \in \mathcal{P} (f)$ with $p\bmod {m} \notin H$.

We study the notion of affine ringed space, see its meaning in topological, differentiable and algebro-geometric contexts and show how to reduce the affineness of a ringed space to that of a ringed finite space. Then, we characterize schematic finite spaces and affine schematic spaces in terms of combinatorial data. Finally, we prove Serre's criterion of affineness for schematic finite spaces. This yields, in particular, Serre's criterion of affineness on schemes.

Let $\mathcal {A}$ be a factor von Neumann algebra, with unit $ I $, which contains a nontrivial projection $ P_{1} $, and let $\psi :\mathcal {A} \rightarrow \mathcal {A}$ be a surjective map that satisfies one of the two conditions: $\psi (A)\psi (P) + \lambda \psi (P)\psi (A) = AP + \lambda PA$ and $\psi (A)\psi (P) + \lambda \psi (P)\psi (A)^{\ast } = AP + \lambda PA^{\ast }$ for all $A \in \mathcal {A}$ and $P \in \lbrace P_{1}, I - P_{1}\rbrace $ and $\lambda \in \lbrace -1, 1\rbrace $. Then, we determine the concrete form of $\psi $.