We construct some families of full diversity rotated $D_n$-lattices via $\mathbb{Z}$-modules for any $n\ge 3$. We show that $\mathbb{Z}$-modules known in previous works to obtain rotated ${\mathbb{Z}}^n$-lattices with $n$ an odd integer are ideals and we find a sufficient condition for such ideals to be principal ideals. We also present bounds and formulas for the minimum product distance of ${\mathbb{Z}}^n$ and $D_n$ restricted to some conditions.

We use a symmetric mountain pass lemma of Kajikiya to prove the existence of infinitely many weak solutions for the Schrödinger $\mathrm{\Phi }$-Laplace equation

$${\left(-\mathrm{\Delta }\right)}_{\mathrm{\Phi }}u+V\left(x\right)\phi \left(u\right)=\xi \left(x\right)f\left(u\right)\text{in }{ℝ}^d,$$

where $\mathrm{\Phi }\left(t\right)={\int }_0^t\phi \left(s\right)ds$ is an $N$-function, ${\mathrm{\Delta }}_{\mathrm{\Phi }}$ is the $\mathrm{\Phi }$-Laplacian operator, $V:{ℝ}^d\to ℝ$ is a continuous function, $\xi$ is a function with sign-changing on ${ℝ}^d$ and the nonlinearity $f$ is sublinear as $\left|u\right|\to \infty$. During the study of our problem, we deal with a new compact embedding theorem for the Orlicz–Sobolev spaces.

We also study the existence and multiplicity of solutions to the general fractional $\mathrm{\Phi }$-Laplacian equations of Kirchhoff type

$$\left\{\begin{array}{cc}M\left({\int }_{{ℝ}^{2d}}\mathrm{\Phi }\left(\frac{u\left(x\right)-u\left(y\right)}{K\left(|x-y|\right)}\right)\frac{dxdy}{N\left(|x-y|\right)}\right){\left(-\mathrm{\Delta }\right)}_{\mathrm{\Phi }}^{K,N}u=f\left(x,u\right)\hfill & \text{ in }\mathrm{\Omega },\hfill \\ & \\ u=0\hfill & \text{ in }{ℝ}^d\setminus \mathrm{\Omega },\hfill \end{array}\right.$$

where $\mathrm{\Omega }$ is an open bounded subset of ${ℝ}^d$ with smooth boundary $\partial \mathrm{\Omega }$, $d>2$, and $M:{ℝ}_0^{+}\to {ℝ}^{+}$ is a continuous function and $f:\mathrm{\Omega }\times ℝ\to ℝ$ is a Carathéodory function. The proofs rely essentially on the fountain theorem and the genus theory.

If we consider $G=\left(V,E\right)$ to be a connected graph, with $v$ $\in$ $V$ and $e=uw$ $\in$ $E$, then $d_G\left(e,v\right)=min\left\{d_G\left(u,v\right),d_G\left(w,v\right)\right\}$ has been defined as the distance between a vertex $v$ and an edge $e$. The cardinality of the smallest subset $S\subseteq V$ which can assign a unique distance vector to every edge of $G$ is referred to as edge metric dimension (EMD) given by $edim\left(G\right)$. A $k$ multiwheel graph $W_{1,n,k}$ is composed of $k$ cycles $C_n$ along with a central vertex $x$ such that $x$ is adjacent to each of the vertices of $C_1$ and the corresponding vertices of the two consecutive cycles $C_i$ and $C_{i+1}$ are also adjacent for all $1\le i\le \left(k-1\right)$. This article discusses the EMD of the double wheel graph and extends the results to the general $k$ multiwheel graph.

We determine explicitly the $2$-radical subgroups and their normalizers for the group ${Sp}_4\left(q\right)$, where $q$ is odd. We then show that the corresponding simple group ${PSp}_4\left(q\right)$ satisfies the inductive blockwise Alperin weight conditions for the prime $2$ and odd primes dividing $q^2-1$. When combined with existing literature, this completes the verification that ${PSp}_4\left(q\right)$ satisfies the conditions for all primes and all choices of $q$.

For an $r$-discrete Hausdorff groupoid $\mathcal{𝒢}$ and an inverse semigroup $S$ of slices of $\mathcal{𝒢}$ there is an isomorphism between $\mathcal{𝒢}$-equivariant $KK$-theory and compatible $S$-equivariant $KK$-theory. We use it to define descent homomorphisms for $S$, and indicate a Baum–Connes map for inverse semigroups. Also findings by Khoshkam and Skandalis for crossed products by inverse semigroups are reflected in $KK$-theory.

We derive a closed formula for the number of quasiplatonic topological actions of the cyclic group on surfaces of genus two or greater. The formula implies that the number of quasiplatonic cyclic group actions is roughly one-sixth the number of regular dessins d’enfants with a cyclic group of automorphisms.

Given a positive integer $n$, the factorial base representation of $n$ is given by $n={\sum }_{i=1}^k{a}_i\cdot i!$, where $a_k\ne 0$ and $0\le a_i\le i$ for all $1\le i\le k$. For $e\ge 1$, we define $S_{e,!}:{\mathbb{Z}}_{\ge 0}\to {\mathbb{Z}}_{\ge 0}$ by $S_{e,!}\left(0\right)=0$ and $S_{e,!}\left(n\right)={\sum }_{i=1}^n{a}_i^e$, for $n\ne 0$. For $\ell \ge 0$, we let $S_{e,!}^{\ell }\left(n\right)$ denote the $\ell$-th iteration of $S_{e,!}$, while $S_{e,!}^0\left(n\right)=n$. If $p\in {\mathbb{Z}}^{+}$ satisfies $S_{e,!}\left(p\right)=p$, then we say that $p$ is an $e$-power factoradic fixed point of $S_{e,!}$. Moreover, given $x\in {\mathbb{Z}}^{+}$, if $p$ is an $e$-power factoradic fixed point and if there exists $\ell \in {\mathbb{Z}}_{\ge 0}$ such that $S_{e,!}^{\ell }\left(x\right)=p$, then we say that $x$ is an $e$-power factoradic $p$-happy number. Note an integer $n$ is said to be an $e$-power factoradic happy number if it is an $e$-power factoradic $1$-happy number. We prove that all positive integers are $1$-power factoradic happy and, for $2\le e\le 4$, we prove the existence of arbitrarily long sequences of $e$-power factoradic $p$-happy numbers. A curious result establishes that for any $e\ge 2$, the $e$-power factoradic fixed points of $S_{e,!}$ that are greater than $1$ always appear in sets of consecutive pairs. Our last contribution provides the smallest sequences of $m$ consecutive $e$-power factoradic happy numbers for $2\le e\le 5$, for some values of $m$.

We express some general type of infinite series such as

$$\sum _{n=1}^{\infty }\frac{F\left(H_n^{\left(m\right)}\left(z\right),H_n^{\left(2m\right)}\left(z\right),\dots ,H_n^{\left(\ell m\right)}\left(z\right)\right)}{{\left(n+z\right)}^{s_1}{\left(n+1+z\right)}^{s_2}\cdots {\left(n+k-1+z\right)}^{s_k}},$$

where $F\left(x_1,\dots ,x_{\ell }\right)\in \mathbb{Q}\left[x_1,\dots ,x_{\ell }\right]$, $H_n^{\left(m\right)}\left(z\right)={\sum }_{j=1}^{n}1∕{\left(j+z\right)}^m$, $z\in \left(-1,0\right]$, and $s_1,\dots ,s_k$ are nonnegative integers with $s_1+\cdots +s_k\ge 2$, as a linear combination of multiple Hurwitz zeta functions and some special values of $H_n^{\left(m\right)}\left(z\right)$.

We introduce the generalized weighted Morrey spaces and the generalized weighted weak Morrey spaces on RD-spaces. As applications, the boundedness of the Hardy–Littlewood maximal operator and the Calderón–Zygmund operator is established.

In this article we generalize the main results of [7] and [12]. More specifically, we show that there are branching systems (which induce representations of the graph $C^{\ast }\left(E\right)$) associated to each row-countable graph $E$. For row-countable graphs, we characterize the Condition $\left(L\right)$ via branching systems. Moreover, we show that each permutative representation by operators in Hilbert spaces is unitarily equivalent to one induced by a branching system, even the spaces being not separable. Furthermore, under some hypothesis on the graph, we show that each representation of the graph $C^{\ast }$-algebra is permutative.

Let $\alpha$ be a nonzero algebraic integer of degree $d$, all of whose conjugates ${\alpha }_1=\alpha ,{\alpha }_2,\dots ,{\alpha }_d$ lie in a sector $|argz|\le 𝜃$, $0<𝜃\le 90^{\circ }$. We define the S-measure of $\alpha$ by $S\left(\alpha \right)={\sum }_{i=1}^d\left|{\alpha }_i\right|$ and the absolute $S$-measure of $\alpha$ by $\text{ s}\left(\alpha \right)=S\left(\alpha \right)∕d$. We compute the greatest lower bound $c\left(𝜃\right)$ of $\text{ s}\left(\alpha \right)$ for $\alpha$ belonging to twelve subintervals of $\left(0,𝜃\right)$. Among these subintervals, three are complete. These computations use the principle of explicit auxiliary functions and our recursive algorithm.

We consider the problem of existence of the extreme points and support points of harmonic $\alpha$-Bloch mappings and little harmonic $\alpha$-Bloch mappings. First, we prove a necessary condition for a little harmonic $\alpha$-Bloch mapping to be an extreme point of the unit ball of the normalized little harmonic $\alpha$-Bloch spaces in the unit disk $\mathbb{𝔻}$, and we also show that a harmonic $\alpha$-Bloch unit-valued set consists of several simple closed pairwise disjoint analytic curves and several isolated points. Then we show that a harmonic $\alpha$-Bloch mapping $f$ is a support point of the unit ball of the normalized harmonic $\alpha$-Bloch spaces in $\mathbb{𝔻}$ if and only if the $\alpha$-Bloch unit-valued set of $f$ is not empty. We also give a characterization for the support points of the unit ball of the harmonic $\alpha$-Bloch spaces in $\mathbb{𝔻}$.

We investigate a more general nonlinear biharmonic equation

$${\mathrm{\Delta }}^{2}u-\beta {\mathrm{\Delta }}_{p}u+V_{\lambda }\left(x\right)u=f\left(x,u\right)\text{ in }{ℝ}^N,$$

where ${\mathrm{\Delta }}^2:=\mathrm{\Delta }\left(\mathrm{\Delta }\right)$ is the biharmonic operator, $N\ge 1$, $\lambda >0$ and $\beta \in ℝ$ are parameters, ${\mathrm{\Delta }}_{p}u=div\left(|\nabla u{|}^{p-2}\nabla u\right)$ with $p\ge 2$. Differently from previous works on biharmonic problems, we replace Laplacian with $p$-Laplacian, and suppose that $V\left(x\right)=\lambda a\left(x\right)-b\left(x\right)$ with $\lambda >0$ and $b\left(x\right)$ can be singular at the origin, in particular we allow $\beta$ to be a real number. Under suitable conditions on $V_{\lambda }\left(x\right)$ and $f\left(x,u\right)$, the multiplicity of solutions is obtained for $\lambda >0$ sufficiently large. Our analysis is based on variational methods as well as the Gagliardo–Nirenberg inequality.

We introduce the class of split regular BiHom-Leibniz color algebras as the natural generalization of split regular Hom-Leibniz algebras. By developing techniques of connections of roots for this kind of algebra, we show that such a split regular BiHom-Leibniz color algebra $\mathcal{ℒ}$ is of the form $\mathcal{ℒ}=\mathcal{𝒰}\oplus {\sum }_{\left[\alpha \right]\in \mathrm{\Pi }∕\sim }{I}_{\left[\alpha \right]}$, with $\mathcal{𝒰}$ a subspace of the abelian subalgebra $\mathcal{\mathscr{H}}$ and any $I_{\left[\alpha \right]}$, a well-described ideal of $\mathcal{ℒ}$, satisfying $\left[I_{\left[\alpha \right]},I_{\left[\beta \right]}\right]=0$ if $\left[\alpha \right]\ne \left[\beta \right]$. Under certain conditions, in the case of $\mathcal{ℒ}$ being of maximal length, the simplicity and the primeness of the algebra is characterized.

We define and study the generalized Drazin inverse of bounded and closed linear relations on Banach spaces. Then, we provide necessary and sufficient conditions under which a linear relation be generalized Drazin invertible.

For the Cauchy problem of the one-dimensional free surface equation of shallow water waves, we obtain local well-posedness of the free surface equation of shallow water waves in Sobolev spaces. We also derive a wave-breaking mechanism for strong solutions.

Consider the diophantine equation $\left(3^{x_1}-1\right)\left(3^{x_2}-1\right)=\left(5^{y_1}-1\right)\left(5^{y_2}-1\right)$ in positive integers $x_1\le x_2$ and $y_1\le y_2$. Each side of the equation is a product of two terms of a given binary recurrence. We prove that the only solution to the title equation is $\left(x_1,x_2,y_1,y_2\right)=\left(1,2,1,1\right)$. The main novelty of our result is that we allow products of two terms on both sides.

Based on the method of contractive functions, the existence of the time-dependent global attractor of the plate equation with nonlinear damping and linear memory is proved.

It is well known that the expected number of real zeros of a random cosine polynomial $V_n\left(x\right)={\sum }_{j=0}^n{a}_{j}cos\left(jx\right)$, $x\in \left(0,2\pi \right)$, with the $a_j$ being standard Gaussian i.i.d. random variables, is asymptotically $2n∕\sqrt{3}$. On the other hand, some of the previous works on the random cosine polynomials with dependent coefficients show that such polynomials have at least $2n∕\sqrt{3}$ expected real zeros lying in one period. We investigate two classes of random cosine polynomials with pairwise equal blocks of coefficients. First, we prove that a random cosine polynomial with the blocks of coefficients being of a fixed length and satisfying $A_{2j+1}=A_{2j}$ possesses the same expected real zeros as the classical case. Afterwards, we study a case containing only two equal blocks of coefficients, and show that in this case significantly more real zeros should be expected compared with those of the classical case.

We consider alternative orders of summation for the conditionally convergent series defining the weight-2 Eisenstein series $G_2$ and the Weierstrass $\wp$-function. The resulting sums differ from the standard ones by a residual term that can be thought of as a function of the shapes with respect to which we sum. We compute this residual function explicitly and give some examples. The results generalize the well-known quasimodularity relationship between $G_2$ and its series summed in the reverse order.

We give an extension of the operator Bellman inequality. This result is estimated via the Kantorovich constant.

In this paper, we obtain two identities related to mock theta functions ${\mathcal{ℱ}}_2\left(q\right)$ and $\varphi \left(q\right)$ by employing the $2$-fold Bailey lemma. And, we also deduce an identity as mock theta function $\nu \left(q\right)$ by using Bailey’s lemma.

We study the Cauchy problem of the nonhomogeneous Bénard system in the whole two-dimensional (2D) space, where the density is allowed to vanish initially. We prove that there exists a unique local strong solution. To compensate for the lack of integrability of the velocity in the whole space, a careful space weight is imposed on the initial density, which cannot decay too slowly in the far field.