In this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed $p$-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel $\text{I}$. Goldberg for the case of 1-forms. As first examples of application we obtain new identities on locally conformally Kähler manifolds and quasi-Sasakian manifolds. Moreover, we prove that under suitable conditions a certain subalgebra of differential forms in a compact manifold is quasi-isomorphic as a $\text{CDGA}$ to the full de Rham algebra.

This paper studies the uncertainty principle for spherical $h$ -harmonic expansions on the unit sphere of ${{\mathbb{R}}^{d}}$ associated with a weight function invariant under a general finite reflection group, which is in full analogy with the classical Heisenberg inequality. Our proof is motivated by a new decomposition of the Dunkl–Laplace–Beltrami operator on the weighted sphere.

We study the distribution of the discrete spectrumof the Schrödinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$.

Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincaré– Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

We consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiffusive type. Using variational methods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies.

A subset $E$ of a discrete abelian group is called $\varepsilon$-Kronecker if all $E$-functions of modulus one can be approximated to within ϵ by characters. $E$ is called a Sidon set if all bounded $E$-functions can be interpolated by the Fourier transform of measures on the dual group. As $\varepsilon$-Kronecker sets with $\varepsilon \,<\,2$ possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.

An incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a ‘hole’. If there are v points, a hole of size $w$ , and all (other) block sizes equal $k$ , this is denoted $\text{IPBD}\left( \left( v;w \right),\,k \right)$ . In addition to congruence restrictions on $v$ and $w$ , there is also a necessary inequality: $v\,>\,\left( k\,-\,1 \right)w$ . This article establishes two main existence results for $\text{IPBD}\left( \left( v;w \right),\,k \right)$ : one in which $w$ is fixed and $v$ is large, and the other in the case $v>\,\left( k-1+\varepsilon \right)w$ when $w$ is large (depending on $\varepsilon$ ). Several possible generalizations of the problemare also discussed.

We study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial, and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields.

We propose a version of the classical Artin approximation that allows us to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a Nash equation by a Nash solution in a compatible way with a given Nash change of variables. This result is closely related to the so-called nested Artin approximation and becomes false in the analytic setting. We provide local and global versions of this approximation in real and complex geometry together with an application to the Right-Left equivalence of Nash maps.

In this paper we study the zero sets of harmonic functions on open sets in ${{\mathbb{R}}^{N}}$ and holomorphic functions on open sets in ${{\mathbb{C}}^{N}}$. We show that the non-extendability of such zero sets is a generic phenomenon.

A simple proof is given for the fact that for $m$ a non-negative integer, a function $f\,\in \,{{C}^{(m)}}\,(\mathbb{R})$, and an arbitrary positive continuous function $\in$, there is an entire function $g$ such that $\left| {{g}^{(i)}}(x)\,-\,{{f}^{(i)}}(x) \right|\,<\,\in (x)$, for all $x\,\in \,\mathbb{R}$ and for each $i\,=\,0,\,1\,.\,.\,.\,,\,m$. We also consider the situation where $\mathbb{R}$ is replaced by an open interval.

In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated with the scaling function for multiresolution analysis of ${{L}^{2}}\left( K \right)$. We use probability and martingale methods to provide such a characterization.

A dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation, the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary differential equation.

We study some geometric properties related to the set

$${{\Pi }_{X}}\,:=\left\{ \left( x,\,{{x}^{*}} \right)\,\in \,{{\text{S}}_{X}}\,\times \,{{\text{S}}_{{{X}^{*}}}}\,:\,{{x}^{*}}\left( x \right)\,=\,1 \right\}$$

obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element $\left( h,\,k \right)\,\in \,H\,{{\oplus }_{2}}\,H$ to ${{\Pi }_{H}}$ for $H$ a Hilbert space.

We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz–Krieger uniqueness theorem for graph algebras by means of branching systems.

We show that the product rank of the $3\,\times \,3$ determinant ${{\det }_{3}}$ is $5$ , and the product rank of the $3\,\times \,3$ permanent $\text{per}{{\text{m}}_{3}}$ is $4$ . As a corollary, we obtain that the tensor rank of ${{\det }_{3}}$ is $5$ and the tensor rank of $\text{per}{{\text{m}}_{3}}$ is $4$ . We show moreover that the border product rank of $\text{per}{{\text{m}}_{3}}$ is larger than $n$ for any $n\,\ge \,3$ .

The energy of a type $\text{II}$ superconductor submitted to an external magnetic field of intensity close to the second critical field is given by the celebrated Abrikosov energy. If the external magnetic field is comparable to and below the second critical field, the energy is given by a reference function obtained as a special (thermodynamic) limit of a non-linear energy. In this note, we give a new formula for this reference energy. In particular, we obtain it as a special limit of a linear energy defined over configurations normalized in the ${{L}^{4}}$-norm.

In this paper, we study uniform perturbations of von Neumann subalgebras of a von Neumann algebra. Let $M$ and $N$ be von Neumann subalgebras of a von Neumann algebra with finite probabilistic index in the sense of Pimsner and Popa. If $M$ and $N$ are sufficiently close, then $M$ and $N$ are unitarily equivalent. The implementing unitary can be chosen as being close to the identity.

The Osgood–Carathéodory theorem asserts that conformal mappings between Jordan domains extend to homeomorphisms between their closures. For multiply-connected domains on Riemann surfaces, similar results can be reduced to the simply-connected case, but we find it simpler to deduce such results using a direct analogue of the Carathéodory reflection principle.

In this paper, the authors characterize second-order Sobolev spaces ${{W}^{2,p}}({{\mathbb{R}}^{n}})$ , with $p\,\in \,[2,\,\infty )$ and $n\,\in \,\mathbb{N}\,\text{or}\,p\,\in \,(1,\,2)\,\text{and}\,n\,\in \,\left\{ 1,\,2,\,3 \right\}$ , via the Lusin area function and the Littlewood–Paley $g_{\text{ }\!\!\lambda\!\!\text{ }}^{*}$ -function in terms of ball means.