Fundamental solutions to differential operators lead to integral operators providing integral representation formulas for solutions to related differential equations. Proper modifications of the fundamental solutions result in integral operators which are related to certain boundary value problems. For complex partial differential operators of arbitrary order in the plane, fundamental solutions are achievable by properly integrating the Cauchy kernel. Particular such complex model differential operators are the polyanalytic and the polyharmonic operators. A hierarchy of integral operators is available for these model operators leading to polyanalytic Cauchy–Schwarz and to polyharmonic Green, Neumann, Robin, and hybrid Green integral operators. The theory is supplemented here by constructing a new polyanalytic Pompeiu (Pompeiu–Vekua) integral operator of any order adjusted to (iterated) Neumann boundary conditions.

This paper extends the concept of a $B$-happy number, for $B\ge 2$, from the positive rational integers, ${\mathbb{Z}}^{+}$, to the Gaussian integers, $\mathbb{Z}[i]$. We investigate the fixed points and cycles of the Gaussian $B$-happy functions, determining them for small values of $B$ and providing a method for computing them for any $B\ge 2$. We discuss heights of Gaussian $B$-happy numbers, proving results concerning the smallest Gaussian $B$-happy numbers of certain heights, and we prove conditions for the existence and nonexistence of arbitrarily long arithmetic sequences of Gaussian $B$-happy numbers. Finally, we consider an alternative definition of Gaussian happy numbers using expansions in base $-1+i$.

In this paper we uncover some fundamental relationships involving the weights one needs to calculate the $n$-th difference of a function on an arbitrary discrete time scale. One of several interesting results we obtain is a formula that allows one to calculate the value of any desired finite difference coefficient directly from the graininess function for the time scale under consideration. This foundational work beautifully combines analysis, algebra, and combinatorics to obtain this and other interesting results. Some of the other interesting results include (i) for a fixed $n$, the $n$-th finite difference coefficients sum to 0 and (ii) there are nice and useful generating functions that encode various sequences of finite difference coefficients. Throughout we show the results coincide with well-known results in the special cases where the time scales are either quantum time scales or time scales with constant graininess.

We systematically study variational inequalities for hypersingular integral operators. More precisely, we show the variational inequalities for the families of truncated hypersingular integrals with variable kernels, which are defined by

where $\alpha \ge 0$ and the kernel $\mathrm{\Omega }$ belongs to $L^{\infty }({ℝ}^n)\times L^1({\mathbb{𝕊}}^{n-1})$. We first prove that the variation of the hypersingular integral with variable kernel is bounded from the Sobolev space ${\dot{L}}_{\alpha }^2$ to the Lebesgue space $L^2$ when $\mathrm{\Omega }\in L^{\infty }({ℝ}^n)\times L^q({\mathbb{𝕊}}^{n-1})$ for and satisfies some cancellation condition in its second variable. The result is sharp in the sense that the $({\dot{L}}_{\alpha }^2,L^2)$ boundedness of ${\mathcal{𝒯}}_{\alpha }$ fails if $q\le 2(n-1)∕(n+2\alpha )$. After strengthening the smoothness of $\mathrm{\Omega }(x,z^{\prime })$ in its second variable, we give the weighted boundedness of the variation of the hypersingular integrals with smooth variable kernels from ${\dot{L}}_{\alpha }^p(w)$ to $L^p(w)$ for and $w\in A_p$. Finally, we extend the result to the Sobolev–Morrey spaces.

We construct infinitely many new families of integers, each containing infinitely many even noncongruent numbers. These noncongruent numbers are obtained as products of an arbitrary number of primes in certain patterns.

Let $n\ge 2$ denote an integer, and let $l\in \{1,\dots ,n-1\}$. A family of $(l+1)$-point boundary value problems for an $n$-th order ordinary differential equation is studied and sufficient conditions for existence of solutions are obtained. In particular, it is shown that a type of global uniqueness of solutions implies global existence of solutions. Then explicit conditions in terms of monotonicity of the nonlinear term are obtained to imply the global uniqueness of solutions.

We study the minimal regularity required on the datum to guarantee the existence of classical $C^1$ solutions to the inhomogeneous Cauchy–Riemann equations on planar domains.

We consider trinomials of the form $z^n+a{z}^m+b$, where are relatively prime integers and $a$ and $b$ are nonzero complex numbers. Typically (but not exclusively), $a$ will be an integer with $\left|a\right|\ge 2$, while $b=\pm 1$. Our main results cover the irreducibility, Mahler measure and house of such trinomials.

Let $H$ be a hypergroup with left Haar measure and let $L^1(H)$ be the complex Lebesgue space associated with it. Let $L^{\infty }(H)$ be the set of all locally measurable functions that are bounded except on a locally null set, modulo functions that are zero locally a.e. It is a standard device to embed $L^{\infty }(H)$ into $\mathcal{ℬ}(L^1(H),L^{\infty }(H))$. We denote the strong operator topology and the weak operator topology on $L^{\infty }(H)$ by ${\tau }_{\text{so}}$ and ${\tau }_{\text{wo}}$. Unlike the uniform norm and weak topologies on $L^{\infty }(H)$, they depend essentially on the hypergroup structure of $H$. We derive that the ${\tau }_{\text{so}}$-topology is always different from the weak${}^{\ast }$-topology whenever $H$ is infinite. We can conclude that for a compact hypergroup $H$, $L^1(H)$ is the dual of $(L^{\infty }(H),{\tau }_{\text{so}})$. The properties of ${\tau }_{\text{so}}$ and ${\tau }_{\text{wo}}$ are then studied further and we pay attention to the ${\tau }_{\text{wo}}$-almost periodic elements of $L^{\infty }(H)$. Finally we give some further results about bounded linear operators which are ${\tau }_{\text{so}}$-${\tau }_{\text{so}}$-continuous.

In this paper, we will prove the existence of multiple solutions for the fractional Schrödinger–Poisson system with critical Sobolev exponent setting in the whole space ${ℝ}^3$. We achieve our goal by making use of variational methods and the minimax method.

We study bounds of the rank of CM-types of CM-fields, and using the bounds, we show that all CM-types of Galois CM-fields with cyclic or dicyclic Galois groups of order $2^{k}p$ with a prime $p$ are nondegenerate. We also show the nondegeneracy of CM-types for certain non-Galois CM-fields. As a consequence, the Hodge conjecture is true for abelian varieties with complex multiplication by such CM-fields.

For a group $G$, embedded in its group of permutations $B=\text{Perm}(G)$ via the left regular representation $\lambda :G\to B$, the normalizer of $\lambda (G)$ in $B$ is $\text{Hol}(G)$, the holomorph of $G$. The set $\mathcal{\mathscr{H}}(G)$ of those regular $N\le \text{Hol}(G)$ such that $N\cong G$ and ${\text{Norm}}_B(N)=\text{Hol}(G)$ is keyed to the structure of the so-called multiple holomorph of $G$, $\text{NHol}(\text{G})={\text{Norm}}_B(\text{Hol}(G))$, in that $\mathcal{\mathscr{H}}(G)$ is the set of conjugates of $\lambda (G)$ by $\text{NHol}(G)$. We wish to generalize this by considering a certain set $\mathcal{𝒬}(G)$ consisting of regular subgroups $M\le \text{Hol}(G)$, where $M\cong G$, that contains $\mathcal{\mathscr{H}}(G)$ with the property that its members mutually normalize each other. This set will generally give rise to a group $Q\text{Hol}(G)$ which we will call the quasi-holomorph of $G$, where the orbit of $\lambda (G)$ under $Q\text{Hol}(G)$ is $\mathcal{𝒬}(G)$. The multiple holomorph is a group extension of $\text{Hol}(G)$ and the quasi-holomorph will contain $\text{NHol}(\text{G})$, but, when larger than $\text{NHol}(\text{G})$, is frequently a Zappa–Szép product with the holomorph.

We confirm two conjectures of Lassak on the area of reduced spherical polygons. The area of every reduced spherical nonregular $n$-gon is less than that of the regular spherical $n$-gon of the same thickness. Moreover, the area of every reduced spherical polygon is less than that of the regular spherical odd-gons of the same thickness and whose number of vertices tends to infinity.

Let $T_1$ and $T_2$ be Calderón–Zygmund operators, and let $T_{1,b}$ denote the commutator of $T_1$ and the BMO function $b$. By establishing bisublinear sparse dominations, we obtain the quantitative weighted bounds for composite operators $T_1{T}_2$ and $T_{1,b}{T}_2$ on $L^p(X,\omega )$, with $\omega \in A_p(X)$.

We construct a family of biharmonic mappings $W_{\alpha ,\beta }$, which arises from analytic functions and has two parameters. Some sufficient conditions for $W_{\alpha ,\beta }$ to be sense-preserving and univalent are explored. The radii of full convexity and starlikeness of $W_{\alpha ,\beta }$ are determined. Some sufficient conditions for biharmonic mappings to be fully starlike and fully convex are obtained. Many related previous results are generalized.

We present different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations, we prove several new identities and simplify the proofs of several known identities. Some open problems are discussed towards the end of the paper.

The state or configuration space for $r$ vertices on a complete bipartite graph $K_{m,n}$ is a $\text{CAT}(0)$ cube complex, which is sometimes interpreted as a parameter space for $r$ robots moving on a $K_{m,n}$. We combine an analysis of the topology of links of vertices in this complex, the description of a hidden symmetry among the parameters, and known results from the literature to explicitly compute the exact degree to which the universal covers of these complexes are connected at infinity.

The Ohno relation is a well known relation in the theory of multiple zeta values. Recently, Seki and Yamamoto introduced a connector method and gave its succinct proof. On the other hand, Igarashi obtained the generalization of the Ohno relation for parametrized multiple series. In this paper, we give new connectors to simplify his proof.

Let $\mathcal{𝒜}$ be a Banach algebra and let $J$ be a closed ideal of $\mathcal{𝒜}$ such that $\phi {|}_J\ne 0$ for some nonzero character $\phi$ on $\mathcal{𝒜}$. We obtain some relations between the existence of compact and weakly compact multipliers on $J$ and on $\mathcal{𝒜}$ in some sense. Then we apply these results to hypergroup algebra $L^1(K)$ when $K$ is a locally compact hypergroup. In particular, for a closed ideal $J$ in $L^1(K)$ we prove that $K$ is compact if and only if there is $f\in J$ such that ${\phi }_1(f)\ne 0$ and the multiplication operator ${\lambda }_f:g\mapsto g\ast f$ is weakly compact on $J$. Using this, we study Arens regularity of $J$ whenever it has a bounded left approximate identity. Finally, we apply these results on some abstract Segal algebras with respect to the $L^1(K)$.

Peakon traveling wave solutions on the circle are derived for the cubic $ab$-family of equations, which includes both the Fokas–Olver–Rosenau–Qiao (FORQ) and Novikov (NE) equations. For $a\ne 0$, it is proved that there exists an initial data in the Sobolev space $H^s$, , with nonunique solutions on the circle. We construct a two-peakon solution with an asymmetric peakon–antipeakon initial profile that collides at a finite time. At the time of collision, the two-peakon solution reduces to a single peakon solution, which will complete the proof of nonuniqueness.

In a remark on page 80 of his classical book 250 Problems in Elementary Number Theory, Sierpiński stated that it was not known if the equation $x∕y+y∕z+z∕x=4$ has solutions in positive integers. Bondarenko (Investigation of a class of Diophantine equations, Ukraïn. Mat. Zh. 52:6 (2000), 831–836) gave a negative answer to Sierpiński’s remark by showing that the equation $x∕y+y∕z+z∕x=4k^2$ does not have solutions in positive integers if $3\nmid k$. However, Garaev (Diophantine equations of the third degree, Tr. Mat. Inst. Steklova 218 (1997), 99–108) had already proved that the equation $x^3+y^3+z^3=nxyz$ has no positive integer solutions if $n=4k$, $n=8k-1$, or $n=2^{2m+1}(2k-1)+3$, where $m,k\in {\mathbb{Z}}^{+}$, which Bondarenko’s result is a consequence of. In this paper, we shall partially extend Garaev’s result by showing that the equation $x∕y+y∕z+m\cdot (z∕x)=nm$ does not have solutions in positive integers if $m$ is odd and $4\mid n$ or $8\mid n+1$. Our method is different from Garaev’s method and has been successfully applied to several situations.

We prove a global mixed-norm gradient estimate in the framework of anisotropic Lorentz spaces to general elliptic equations of $p$-growth under weak regularity data. It is assumed that the principle coefficients are small BMO seminorm in $x$-variable, and the boundary of underlying domain is locally $C^{0,1}$-smooth with a small Lipschitz constant.

We characterize the modules of infinite projective dimension over the endomorphism algebras of Oppermann–Thomas cluster-tilting objects $X$ in $(n+2)$-angulated categories $(\mathcal{𝒞},{\mathrm{\Sigma }}^n,\mathrm{\Theta })$. For an indecomposable object $M$ of $\mathcal{𝒞}$, we define in this article the ideal $I_M$ of ${End}_{\mathcal{𝒞}}({\mathrm{\Sigma }}^{n}X)$ given by all endomorphisms that factor through $addM$, and show that the ${End}_{\mathcal{𝒞}}(X)$-module ${\text{Hom}}_{\mathcal{𝒞}}(X,M)$ has infinite projective dimension precisely when $I_M$ is nonzero. As an application, we generalize a recent result by Beaudet–Brüstle–Todorov for cluster-tilted algebras.