The $k$-graphs in the sense of Kumjian and Pask~\cite {KP} are discrete Conduch\'{e} fibrations over the monoid~$\mathbb {N}^k$, satisfying a finiteness condition. We examine the generalization of this construction to discrete Conduch\'{e} fibrations with the same finiteness condition and a lifting property for completions of cospans to commutative squares, over any category satisfying a strong version of the right Ore condition, including all categories with pullbacks and right Ore categories in which all morphisms are monic.

Let $B(R)\subset \mathbb{R}^n $, $n\in \mathbb{N} $, $n\geq 2$, be an open ball. By a result from~\cite {AdT}, the Moser functional with the borderline exponent from the Moser-Trudinger inequality fails to be sequentially weakly continuous on the set of radial functions from the unit ball in $W_0^{1,n}(B(R))$, only in the exceptional case of sequences acting like a~concentrating Moser sequence.

We extend this result into the Lorentz-Sobolev space $W_0^1L^{n,q}(B(R))$, with $q\in (1,n]$, equipped with the norm $$ ||\nabla u||_{n,q}:= ||t^{1/n-1/q}|\nabla u|^*(t)||_{L^q((0,|B(R)|))}. $$ We also consider the case of a nontrivial weak limit and the corresponding Moser functional with the borderline exponent from the concentration-compactness alternative.

We introduce, for any set $S$, the concept of a $\mathfrak {K}$-family between two Hilbert $C^*$-modules over two $C^*$-algebras, for a given completely positive definite (CPD-) kernel $\mathfrak {K}$ over $S$ between those $C^*$-algebras, and we obtain a factorization theorem for such $\mathfrak {K}$-families. If $\mathfrak {K}$ is a CPD-kernel and $E$ is a full Hilbert $C^*$-module, then any $\mathfrak {K}$-family which is covariant with respect to a dynamical system $(G,\eta ,E)$ on $E$, extends to a $\widetilde {\mathfrak {K}}$-family on the crossed product $E \times _\eta G$, where $\widetilde {\mathfrak {K}}$ is a CPD-kernel. Several characterizations of $\mathfrak {K}$-families, under the assumption that ${E}$ is full, are obtained, and covariant versions of these results are also given. One of these characterizations says that such $\mathfrak {K}$-families extend as CPD-kernels, between associated (extended) linking algebras, whose $(2,2)$-corner is a homomorphism and vice versa. We discuss a dilation theory of CPD-kernels in relation to $\mathfrak {K}$-families.

Let $m$ be a square-free integer, $m\equiv 2,3\pmod 4$. We show that the number field $K=\mathbb{Q} (i,\sqrt [4]{m})$ is non-monogene, that is, it does not admit any power integral bases of type $\{1,\alpha ,\ldots ,\alpha ^7\}$. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using congruence considerations only.

Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields which is applicable for parametric families of number fields. We calculate the index of elements as polynomials dependent upon the parameter, factor these polynomials, and consider systems of congruences according to the factors.

We discuss the notions of circumradius, inradius, diameter and minimum width in generalized Minkowski spaces (that is, with respect to gauges), i.e., we measure the ``size'' of a given convex set in a finite-dimensional real vector space with respect to another convex set. This is done via formulating some kind of containment problem incorporating homothetic bodies of the latter set or strips bounded by parallel supporting hyperplanes thereof. This paper can be seen as a theoretical starting point for studying metric problems of convex sets in generalized Minkowski spaces.

We discuss a problem of Dixmier \cite [Problem 10.10.11]{D} on continuous fields of postliminal \st -algebras and the greatest liminal ideals of the fibers.

We complete the demonstration of source algebra equivalences between spin blocks of families of covering groups $\{\widetilde {S}_n\}$ and $\{\widetilde {A}_n\}$ of symmetric and alternating groups, for pairs of blocks at the ends of maximal strings. These equivalences remain within the family of groups if cores of the two blocks have the same parity and cross over from one family to the other if the cores are of opposite parity. This demonstrates Kessar and Schaps' crossover conjecture for the easier case of extremal points of maximal strings. We use this result to give an improved bound for the highest degree necessary in order to get representatives of all Morita equivalence classes of spin blocks for a given weight.

In this paper, we investigate shared value problems related to an entire function $f(z)$ of hyper-order less than one and its linear difference polynomial $L(f)=\sum _{i=1}^{k}a_{i}f(z+c_{i})$, where $a_{i}, c_{i}\in \mathbb {C}$. We give sufficient conditions in terms of weighted value sharing and truncated deficiencies, which imply that $L(f)\equiv f$.

We introduce martingale weighted two-parameter Lorentz spaces and establish atomic decomposition theorems. As an application of atomic decomposition we obtain a sufficient condition for sublinear operators defined on martingale weighted Lorentz spaces to be bounded. Moreover, some interpolation properties with a function parameter of those spaces are obtained.

It is shown that, if the processes $B$ and $f(B)$ are both Brownian motions (without a random time change), then $f$ must be an affine function. As a by-product of the proof it is shown that the only functions which are solutions to both the Laplace equation and the eikonal equation are affine.

In this article, we establish the existence of solutions of a system of integro-differential equations arising in population dynamics in the case of anomalous diffusion. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without the Fredholm property in unbounded domains are used.

This paper deals with a class of nonlinear problems $$ -(r^{n-1}|u'|^{p-2}u')'+r^{n-1}q(r)|u|^{p-2}u =r^{n-1}w(r)f(u) $$ in $(0,1)$, where $1\leq n\lt p\lt \infty $ and $'={d}/{dr}$. We study the existence of nodal solutions to this nonautonomous system. We give necessary and sufficient conditions for the existence of sign-changing solutions and also observe an application related to the case of multi-point boundary conditions. Methods used here are energy function control, shooting arguments and Pr\"{u}fer-type substitutions.

This note gives a characterization of matrix structures for multipliers of a stable $C^*$-algebra $A\otimes \mathcal{K} $ with any $C^*$-algebra $A$. We represent elements in $(A\otimes \mathcal{K} )''$, $QM(A\otimes \mathcal{K} )$ and $M(A\otimes \mathcal{K} )$ as infinite matrices over certain $C^*$-algebras, respectively. These results generalize the related work of Brown, Lin and Zhang in this area.

Let $\mathcal {T}$ be a triangulated category with triangulation $\Delta $, $\xi \subseteq \Delta $ a proper class of triangles and $\mathcal {C}$ an additive full subcategory of $\mathcal {T}$. We provide a method for constructing a proper $\mathcal {C}(\xi )$-resolution (respectively, coproper $\mathcal {C}(\xi )$-coresolution) of one term in a triangle in $\xi $ from those of the other two terms. By using this construction, we show the stability of the Gorenstein category $\mathcal {GC}(\xi )$ in triangulated categories. Some applications are given.