The novelties of this paper is the study of existence and uniqueness results for a class of incommensurate fractional differential equation systems with general multiorders using the weighted infinity norm with respect to the classical Mittag-Leffler function via the contraction mapping principle. The lack of stability results in the Ulam–Hyers sense on fractional multidimensional differential equations motivates us to generalize our theory to stability analysis based on fixed point approach.

The minus order on sets of matrices over a field and bounded linear operators on arbitrary Hilbert spaces has been an area of intense research. Recently, this partial order has been studied on regular modules by the present authors. We investigate lattice properties of the minus order on regular modules and present various characterizations of the minus order as an application.

Let $\mathrm{\Phi }:[0,+\infty )\to [0,+\infty ]$ be an arbitrary Young function and $K$ be a compact set in ${ℝ}^n$ having (O)-property. Then there exists a constant independent of $f$ such that

$$\parallel D^{\alpha }f{\parallel }_{(\mathrm{\Phi })}\le C_{K,\mathrm{\Phi }}\underset{z\in K}{\text{sup}}\left|z^{\alpha }\right|\parallel f{\parallel }_{{\mathcal{ℒ}}_{K,3}}$$

for all $\alpha \in {\mathbb{Z}}_{+}^n$ and $f\in {\mathcal{ℒ}}_{K,3}$, where ${\mathcal{ℒ}}_{K,3}=\{f\in {\mathcal{𝒮}}^{\prime }({ℝ}^n):\text{supp}\widehat{f}\subset K,D^{(3,3,\dots ,3)}\widehat{f}\in C({ℝ}^n)\}$, $\parallel f{\parallel }_{{\mathcal{ℒ}}_{K,3}}=\parallel D^{(3,3,\dots ,3)}\widehat{f}{\parallel }_{\infty }$, $\widehat{f}$ is the Fourier transform of $f$ and $\parallel \cdot {\parallel }_{(\mathrm{\Phi })}$ is the Luxemburg norm. As an application, a new Paley–Wiener theorem is given.

Let $C^{\ast }(E)$ be the graph ${\text{C}}^{\ast }$-algebra of a row-finite graph $E$. We give a complete description of the vertex sets of the gauge-invariant regular ideals of $C^{\ast }(E)$. It is shown that when $E$ satisfies condition (L), the regular ideals $C^{\ast }(E)$ are a class of gauge-invariant ideals which preserve condition (L) under quotients. That is, we show that if $E$ satisfies condition (L) then a regular ideal $J⊴C^{\ast }(E)$ is necessarily gauge-invariant. Further, if $J⊴C^{\ast }(E)$ is a regular ideal, it is shown that $C^{\ast }(E)∕J\simeq C^{\ast }(F)$, where $F$ satisfies condition (L).

We derive a series representation of the generalized Stieltjes constants which arise in the Laurent series expansion of partial zeta function at the point $s=1$. In the process, we introduce a generalized gamma function and deduce its properties such as functional equation, Weierstrass product and reflection formulas along the lines of the study of a generalized gamma function introduced by Dilcher in 1994. These properties are used to obtain a series representation for the $k$-th derivative of Dirichlet series with periodic coefficients at the point $s=1$. Another application involves evaluation of a class of infinite products of which a special case is an identity of Ramanujan.

We introduced certain Toeplitz-type operator associated to the pseudodifferential operator and obtained the boundedness of the Toeplitz-type operator from Lebesgue space to Orlicz space by using some sharp inequalities.

We prove the existence of a positive periodic solution for a nonlinear differential equation with attractive-repulsive singularities by using a Green’s function and a fixed point theorem in cones.

We investigate the existence of the periodic solutions of a quasilinear impulsive differential equation with alternately advanced and retarded arguments of generalized type, in short IDEPCAG. By using some fixed point theorems and some new analysis techniques, sufficient conditions are obtained for the existence and uniqueness of periodic solutions of these IDEPCAG systems. A new IDEPCAG’s Gronwall-type lemma is proved. Some examples concerning impulsive biological models such as Lasota–Ważewska model, Nicholson’s blowflies and logistic models are treated.

In this paper, we use the 11-point biplane and its automorphisms in $L_2(11)$ to label and study the Livingstone graph ($\mathrm{\Gamma }$) and $J_1$, with an aim of using the simplest methods possible. We detail the action of $J_1$ on $\mathrm{\Gamma }$, along with the adjacencies and coadjacencies (vertices at maximum distance) in $\mathrm{\Gamma }$. In the last section, we use this apparatus to describe the generation of subgroups of the form $2^3:7:3$ and an elegant substructure of $\mathrm{\Gamma }$ fixed by a maximal subgroup of $J_1$ isomorphic to $19:6$.

We find the exact book thickness of all nilpotent graphs associated to commutative rings with nonzero identity whose genus is at most one. Also, we investigate the book thickness of nilpotent graphs from local rings and some product rings.

We consider the simplicity of the $C^{\ast }$-algebra associated to an arbitrary weakly left-resolving labeled space $(E,\mathcal{ℒ},\mathcal{ℰ})$, where $\mathcal{ℰ}$ is the smallest nondegenerate accommodating set. We classify all gauge-invariant ideals of $C^{\ast }(E,\mathcal{ℒ},\mathcal{ℰ})$ and characterize minimality of $(E,\mathcal{ℒ},\mathcal{ℰ})$ in terms of ideal structure of $C^{\ast }(E,\mathcal{ℒ},\mathcal{ℰ})$. Using these results, we prove that $C^{\ast }(E,\mathcal{ℒ},\mathcal{ℰ})$ is simple if and only if $(E,\mathcal{ℒ},\mathcal{ℰ})$ is strongly cofinal and satisfies condition $(\text{L})$, and for any $A\in \mathcal{ℰ}\setminus \{\varnothing \}$ and $B\in \mathcal{ℰ}$, there is $C\in {\mathcal{ℰ}}_{\text{reg}}$ such that $B\setminus C\in \mathcal{\mathscr{H}}(A)$, and if and only if $(E,\mathcal{ℒ},\mathcal{ℰ})$ is minimal and satisfies condition $(\text{L})$, and if and only if $(E,\mathcal{ℒ},\mathcal{ℰ})$ is minimal and satisfies condition $(\text{K})$.

Let $r$ be any positive integer. In this paper, we revisit explicit and recurrence formulas satisfied by the Bernoulli numbers $B_n^{(r)}$ of higher order $r$. By using the unsigned Stirling numbers, we give them new forms and a clearer description. The second part of this study consists of providing and proving analogues of Kummer congruences and a von Staudt–Clausen theorem for the numbers $B_n^{(r)}∕(r\left(\genfrac{}{}{0.0pt}{}{n}{r}\right))$, as well as investigating their numerators. Moreover, we study the $p$-integrality of these numbers. In the third part of this paper, we construct a new family of Eisenstein series $E_k^{(r)}(z)$ whose constant terms are the numbers $B_n^{(r)}∕(r\left(\genfrac{}{}{0.0pt}{}{n}{r}\right))$. We prove a congruence for these new Eisenstein series. This generalizes the classical von Staudt–Clausen’s and Kummer’s congruences of Eisenstein series. Our study leads to several applications in algebraic number theory.

In this paper we prove the following result. Let $n\ge 3$ be some fixed integer, let $X$ be a real or complex Banach space, let $\mathcal{ℒ}(X)$ be the algebra of all bounded linear operators on $X$, and let $\mathcal{𝒜}(X)\subseteq \mathcal{ℒ}(X)$ be a standard operator algebra. Suppose there exists a linear mapping $D:\mathcal{𝒜}(X)\to \mathcal{ℒ}(X)$ satisfying the relation $${2^{n - 2}}D({A^n}) = \sum\nolimits_{i = 0}^{n - 2} {\left( {\matrix{ {n - 2} i } } \right)} {A^i}D({A^2}){A^{n - 2 - i}} + ({2^{n - 2}} - 1)(D(A){A^{n - 1}} + {A^{n - 1}}D(A)) + \sum\nolimits_{i = 1}^{n - 2} {\left( {\sum\nolimits_{k = 2}^i {({2^{k - 1}} - 1)\left( {\matrix{ {n - k - 2} {i - k} } } \right)} + \sum\nolimits_{k = 2}^{n - 1 - i} {({2^{k - 1}} - 1)\left( {\matrix{ {n - k - 2} {n - i - k - 1} } } \right)} } \right)} {A^i}D(A){A^{n - 1 - i}}$$ for all $A\in \mathcal{𝒜}(X)$. In this case $D$ is of the form $D(A)=[B,A]$ for all $A\in \mathcal{𝒜}(X)$, and some fixed $B\in \mathcal{ℒ}(X)$. In particular, $D$ is continuous. This result is related to a classical result of Chernoff.

We extend, using an elementary method, results of Banach (1924), Fan (1952), and Sanders (1961), which concern a finite collection ${\{f_i:A_i\to A_{i+1}\}}_{i=1}^n$ of mappings with $A_{n+1}=A_1$ which is decomposable as $f_i(B_i)=A_{i+1}\setminus B_{i+1}$, where $B_i\subseteq A_i$ for all $i$ and $B_{n+1}=B_1$. Our theorem determines when such a collection is decomposable. We also show that such a set $B_1$ is unique up to an addition of a certain set, which was conjectured by Sanders.

We merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety.

Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of rational points of scheme theoretic tropicalizations, including a characterization of the respective topologies. Our second main result shows that the Giansiracusa bend relations can be derived by a natural construction from the scheme theoretic tropicalization.

In this article, we consider the bilinear operator $T$ satisfying that there exists a positive constant $C_{(T)}$, depending on $T$, such that, for any measurable functions $f$ and $g$ with compact support, $t\in ℝ$ with , and $x\in {ℝ}^n$ with $0\notin \text{supp}(f(x-t\cdot ))\cap \text{supp}(g(x-\cdot ))$,

$$\left|T(f,g)(x)\right|\le C_{(T)}{\displaystyle {\int }_{{ℝ}^n}\frac{\left|f(x-ty)g(x-y)\right|}{|y{|}^n}dy}.$$

We investigate the boundedness of $T$ on the vanishing generalized Morrey spaces $V_0{L}^{p,\phi }({ℝ}^n)$ and $V_{\infty }{L}^{p,\phi }({ℝ}^n)$, and the boundedness of the subbilinear maximal operator $\mathcal{ℳ}$ on the vanishing generalized Morrey space $V^{(\ast )}{L}^{p,\phi }({ℝ}^n)$, and their applications to some classical (sub)bilinear operators in harmonic analysis. As a byproduct, we also show that $T$ is bounded on generalized Morrey spaces $L^{p,\phi }({ℝ}^n)$. Some typical examples for the main results of this paper are also included.

Real rank zero was a reoccurring, though implicit, theme in the original Brown–Douglas–Fillmore theory. Motivated by this and other considerations, we prove that a large class of (not necessarily simple) purely infinite corona algebras has real rank zero.

Let $f$ and $g$ be two meromorphic mappings of ${ℂ}^m$ into ${\mathbb{P}}^n(ℂ)$ and let $a_1,\dots ,a_{2n+2}$ be $2n+2$ moving hyperplanes which are slow with respect to $f$ and $g$. We will show that if $f$ and $g$ have the same inverse images for all $a_i(1\le i\le 2n+2)$ with multiplicities counted to level $l_i$ such that

where $q=\left(\genfrac{}{}{0.0pt}{}{2n+2}{n+1}\right)$, then the map $f\times g$ into ${\mathbb{P}}^n(ℂ)\times {\mathbb{P}}^n(ℂ)$ must be algebraically degenerate over the field ${\mathcal{ℛ}}_{{\{a_i\}}_{i=1}^{2n+2}}$. Our result extends and improves the previous result in this topic.

Let $I$ be a square-free monomial ideal in a polynomial ring $R=K[x_1,\dots ,x_n]$ over a field $K$, $\mathfrak{𝔪}=(x_1,\dots ,x_n)$ be the graded maximal ideal of $R$, and $\{u_1,\dots ,u_{{\beta }_1(I)}\}$ be a maximal independent set of minimal generators of $I$ such that $\mathfrak{𝔪}\setminus x_i\notin \text{Ass}(R∕{(I\setminus x_i)}^t)$ for all $x_i\mid {\prod }_{i=1}^{{\beta }_1(I)}{u}_i$ and some positive integer $t$, where $I\setminus x_i$ denotes the deletion of $I$ at $x_i$ and ${\beta }_1(I)$ denotes the maximum cardinality of an independent set in $I$. We prove that if $\mathfrak{𝔪}\in \text{Ass}(R∕I^t)$, then $t\ge {\beta }_1(I)+1$. As an application, we verify that under certain conditions, every unmixed Kőnig ideal is normally torsion-free, and so has the strong persistence property. In addition, we show that every square-free transversal polymatroidal ideal is normally torsion-free. Next, we state some results on the corner elements of monomial ideals. In particular, we prove that if $I$ is a monomial ideal in a polynomial ring $R=K[x_1,\dots ,x_n]$ over a field $K$ and $z$ is an $I^t$ corner element for some positive integer $t$ such that $\mathfrak{𝔪}\setminus x_i\notin \text{Ass}{(I\setminus x_i)}^t$ for some $1\le i\le n$, then $x_i$ divides $z$.

This note deals with vectorial local minimizers of some integral functionals related to nonlinear elasticity theory. Under some structural assumptions, we derive that each component of the local minimizers lies in the De Giorgi class, thus the minimizers are locally Hölder continuous. We emphasize that no convexity is assumed. On the other hand, a special dependence on minors, taken from the Jacobian matrix, plays an important role in the proof.

We are interested in studying the Cauchy problem for a weakly coupled system of semilinear $\sigma$-evolution equations with frictional damping. The main purpose of this paper is two-fold. We would like to not only prove the global (in time) existence of small data energy solutions but also indicate the blow-up result for Sobolev solutions when $\sigma$ is assumed to be any fractional number. Moreover, some further generalizations will be also discussed in the present paper.

Let $\mathcal{𝒜}$ be a prime $\ast$-algebra and let $\xi \in ℂ\setminus \{0,\pm 1\}$. We prove that a mapping $\varphi :\mathcal{𝒜}\to \mathcal{𝒜}$ satisfies

$$\varphi (A{◇}_{\xi }B{◇}_{\xi }C)=\varphi (A){◇}_{\xi }B{◇}_{\xi }C+A{◇}_{\xi }\varphi (B){◇}_{\xi }C+A{◇}_{\xi }B{◇}_{\xi }\varphi (C)$$

for all $A,B,C\in \mathcal{𝒜}$ if and only if $\varphi$ is an additive $\ast$-derivation and $\varphi (\xi A)=\xi \varphi (A)$ for all $A\in \mathcal{𝒜}$.

We consider the weighted Lichnerowicz equation

$${△}_{f}u+c{u}^{\sigma }=0$$

on smooth metric measure spaces, where $c\ge 0,\sigma \le 1$ are real constants. A local gradient estimate for positive solutions to this equation is derived, and as an application we give a corresponding Harnack inequality.

Suppose and . Let $\mathcal{ℬ}{\mathcal{\mathscr{H}}}_{p,\alpha }$ denote the logarithmic Hardy–Bloch type space of those functions $f$ which are analytic in the unit disk $\mathbb{𝔻}$ such that

In this paper, we mainly obtain the relation between the logarithmic Hardy–Bloch type space $\mathcal{ℬ}{\mathcal{\mathscr{H}}}_{p,\alpha }$ and the Hardy space $H^p$ (or the Dirichlet space ${\mathcal{𝒟}}_{p-1}^p$). We also give the characterization of lacunary series on $\mathcal{ℬ}{\mathcal{\mathscr{H}}}_{p,\alpha }$ when .

Let $\pi :X\to Y$ be an $n$-dimensional iterated Kodaira fibration with fiber of genus $g$ and injective monodromy. In [6], it is proved that we can pass to a finite index subgroup of ${\pi }_1(X)$ to get the base space of an ($n+1$)-dimensional iterated Kodaira fibration with injective monodromy, and they asked about bounding the index of such a group. We provide a bound on this index.