In this paper we consider the PDE system of vanishing normal projection of the Laplacian for $C^2$ maps $u:{ℝ}^n\supseteq \mathrm{\Omega }\to {ℝ}^N$:

$${\left[\left[Du\right]\right]}^{\perp }\mathrm{\Delta }u=0\text{ in }\mathrm{\Omega }.$$

This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the $p$-Laplace system for all $p\in \left[2,\infty \right]$. For $p=\infty$, the $\infty$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial calculus of variations in $L^{\infty }$. Herein we show that the image of a solution $u$ is piecewise affine if either the rank of $Du$ is equal to one or $n=2$ and $u$ has additively separated form. As a consequence we obtain corresponding flatness results for $p$-Harmonic maps for $p\in \left[2,\infty \right]$.

Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its optimality properties and goodness of fit. Further, we present a notion of MDS on infinite metric measure spaces that generalizes these optimality properties. As a consequence we can study the MDS embeddings of the geodesic circle $S^1$ into ${ℝ}^m$ for all $m$, and ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into ${ℝ}^m$. Finally, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space $X$, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of $X$?

We discuss some of the mathematical properties of the fractional derivative defined by means of Fourier transforms. We first consider its action on the set of test functions $\mathcal{𝒮}\left(ℝ\right)$, and then we extend it to its dual set, ${\mathcal{𝒮}}^{\prime }\left(ℝ\right)$, the set of tempered distributions, provided they satisfy some mild conditions. We discuss some examples, and we show how our definition can be used in a quantum mechanical context.

We study the asymptotic distribution of the linearly normalized extremes under finite mixture models. Moreover, the asymptotic distributions of the generally normalized central and intermediate order statistics under finite mixture models are studied.

We study the existence of weak positive solutions for a new class of Kirchhoff parabolic systems in bounded domains with multiple parameters taking into account the symmetry conditions and the right-hand side defined as a multiplication of two separate functions. Our results are natural extensions of previous results in the field, which used the same method for some classical elliptic equations.

We first define weighted area balance function. Then we prove two identities for the integrable functions involving weighted area balance. Moreover, using these equalities, we obtain some inequalities for mappings of bounded variation and for Lipschitzian functions, respectively.

It is known that there exist classes of sets of positive integers which provide arbitrary given values as lower and upper logarithmic and asymptotic densities. The only restriction is the inequality between these densities. We generalize this result for a sequence of weighted densities, i.e., for countably infinitely many weighted densities.

We investigate the meromorphic solutions of two types of nonlinear differential equations of the form

$$\begin{array}{lllllll}\hfill b{f}^n+a{f}^{n-1}{f}^{\prime }+Q_d\left(f\right)& =u\left(z\right)e^{v\left(z\right)},& \hfill n\ge d+2,& & \hfill & & \hfill \\ \hfill f^n+f^{n-1}{f}^{\prime }+Q_d\left(f\right)& =P_1\left(z\right)e^{{\alpha }_1}+P_2\left(z\right)e^{{\alpha }_2},& \hfill n\ge d+3,& & \hfill & & \hfill \end{array}$$

where $a,b$ are constants with $\left(a,b\right)\ne \left(0,0\right)$ and $n,d$ are positive integers, $u,P_1,P_2$ are nonzero rational functions, $v,{\alpha }_1,{\alpha }_2$ are nonconstant polynomials, and $Q_d\left(f\right)$ denotes a differential polynomial in $f$ with rational functions as its coefficients. Our results improve some recent related results.

Given a finite set $A\subset ℝ\setminus \left\{0\right\}$, define

$$\begin{array}{cccc}\hfill A\cdot A& =\left\{a_i\cdot a_j\mid a_i,a_j\in A\right\},\hfill & \hfill A∕A& =\left\{a_i∕a_j\mid a_i,a_j\in A\right\},\hfill \\ \hfill A+A& =\left\{a_i+a_j\mid a_i,a_j\in A\right\},\hfill & \hfill A-A& =\left\{a_i-a_j\mid a_i,a_j\in A\right\}.\hfill \end{array}$$

The set $A$ is said to be MPTQ (more product than quotient) if $|A\cdot A|>|A∕A|$ and MSTD (more sum than difference) if $|A+A|>|A-A|$. Since multiplication and addition are commutative while division and subtraction are not, it is natural to think that MPTQ and MSTD sets are very rare. However, they do exist. This paper first shows an efficient search for MPTQ subsets of $\left\{1,2,\dots ,n\right\}$ and proves that as $n\to \infty$, the proportion of MPTQ subsets approaches $0$. Next, we prove that MPTQ sets of positive numbers must have at least $8$ elements, while MPTQ sets of both negative and positive numbers must have at least $5$ elements. Finally, we investigate several sequences that do not have MPTQ subsets.

Define $F\left(q\right):={\sum }_{n=-\infty }^{\infty }{\left(-1\right)}^{\delta n}\left(an+b\right)q^{\left(c{n}^2+dn\right)∕2}$, which includes Ramanujan’s theta function as a special case. We establish a dissection identity for this function, and use it to derive congruence properties for the coefficients of $F\left(q\right)$. As an application we deduce several infinite families of congruences for $\ell$-regular partitions and $\ell$-regular bipartitions. In addition, we give a new proof of Ramanujan’s congruence for the unrestricted partition function modulo $5$.

We prove the global (in time) existence of small data solutions from energy spaces based on $L^q$ spaces, $q\in \left(1,\infty \right)$, to the Cauchy problem for a weakly coupled system of semilinear viscoelastic damped $\sigma$-evolution equations, where we consider nonlinearity terms with powers $p_1,p_2>1$ and any ${\sigma }_1,{\sigma }_2\ge 1$ in the comparison between two single equations. To do this, by mixing additional $L^m$ regularity for the data on the basis of $L^q$-$L^q$ estimates, with $q\in \left(1,\infty \right)$ and $m\in \left[1,q\right)$, we apply $\left(L^m\cap L^q\right)$-$L^q$ estimates for solutions to the corresponding linear Cauchy problems to treat semilinear problems. In addition, two different strategies allowing no loss of decay and some loss of decay combined with the flexible choice of admissible parameters ${\sigma }_1$, ${\sigma }_2$, $m$ and $q$ bring some benefits to relax the restrictions on the admissible exponents $p_1,p_2$.

Let $\mathcal{𝒜}$ be a prime $\ast$-algebra and suppose that $\mathrm{\Phi }$ preserves triple $\ast$-Jordan derivation on $\mathcal{𝒜}$, that is, for every $A,B\in \mathcal{𝒜}$,

$$\mathrm{\Phi }\left(A◇B◇C\right)=\mathrm{\Phi }\left(A\right)◇B◇C+A◇\mathrm{\Phi }\left(B\right)◇C+A◇B◇\mathrm{\Phi }\left(C\right),$$

where $A◇B=AB+B{A}^{\ast }$. Then $\mathrm{\Phi }$ is additive. Moreover, if $\mathrm{\Phi }\left(\alpha I\right)$ is selfadjoint for $\alpha \in \left\{1,i\right\}$, then $\mathrm{\Phi }$ is a $\ast$-derivation.

We evaluate the multiple zeta value $\zeta \left(1,{\left\{2\right\}}^{n+1}\right)$ or its dual $\zeta \left({\left\{2\right\}}^n,3\right)$. When $n$ is even, along with stuffle relations already available, it is enough to evaluate all multiple zeta values of the form $\zeta \left({\left\{2\right\}}^a,3,{\left\{2\right\}}^b\right)$ with $a+b=n$. Furthermore, we obtain a decomposition for $2\zeta \left(2n+3\right)$ as

$${\zeta }^{\star }\left(3,{\left\{2\right\}}^n\right)+\zeta \left({\left\{2\right\}}^n,3\right)+\sum _{r=1}^n\sum _{\left|\alpha \right|=n+1}\zeta \left(1,2{\alpha }_1,2{\alpha }_2,\dots ,2{\alpha }_r\right),$$

which also can be used to evaluate $\zeta \left({\left\{2\right\}}^n,3\right)$ when $n$ is even.

We show the existence of insensitizing controls and study null controllability with constraints for the Burgers equation. We solve these problems by first considering the linearized problem, then, by the Kakutani fixed point theorem, we show that the same results hold for the Burgers equation. The main difficulty in this paper is dealing with the nonlinear term in the Burgers equation.

Let $f$ and $\tilde{f}$ be entire functions of order less than two, and $\mathrm{\Omega }=\left\{z\in ℂ:\left|z\right|=1\right\}$. Let $i_{in}\left(f\right)$ and $i_{out}\left(f\right)$ denote the numbers of the zeros of $f$ taken with their multiplicities located inside and outside $\mathrm{\Omega }$, respectively. Besides, $i_{out}\left(f\right)$ can be infinite. We consider the following problem: how “close” should $f$ and $\tilde{f}$ be in order to provide the equalities $i_{in}\left(\tilde{f}\right)=i_{in}\left(f\right)$ and $i_{out}\left(\tilde{f}\right)=i_{out}\left(f\right)$? If for $f$ we have the lower bound on the boundary, that problem sometimes can be solved by the Rouché theorem, but the calculation of such a bound is often a hard task. We do not require the lower bounds. We restrict ourselves by functions of order no more than two. Our results are new even for polynomials.

We present several enumerative identities involving third-order recursive sequences. In particular, we prove a dual of such identities. We also give a combinatorial interpretation of these sequences through weighted tilings and use it to find bijective proofs of presented relations.

We study the $n$-th-order nonlinear dynamic equations

$$x^{\left[n\right]}\left(t\right)+p\left(t\right){\varphi }_{{\alpha }_{n-1}}\left[{\left(x^{\left[n-2\right]}\left(t\right)\right)}^{\mathrm{\Delta }\sigma }\right]+q\left(t\right){\varphi }_{\gamma }\left(x\left(g\left(t\right)\right)\right)=0$$

on an unbounded time scale $\mathbb{𝕋}$, where $n\ge 2$ and for $i=1,\dots ,n-1$

$$x^{\left[i\right]}\left(t\right):=r_i\left(t\right){\varphi }_{{\alpha }_i}\left[{\left(x^{\left[i-1\right]}\left(t\right)\right)}^{\mathrm{\Delta }}\right],$$

with $r_n={\alpha }_n=1$ and $x^{\left[0\right]}=x$; here the constants ${\alpha }_i$ and the functions $r_i$, $i=1,\dots ,n-1$, are positive and $p$, $q$ are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when $g$ is not (delta) differentiable and the forward jump operator $\sigma$ and $g$ do not commute.

We give a new proof of the Cohen type theorem for $S$-Noetherian rings and a slight generalization of the Eakin–Nagata type theorem for $S$-Noetherian rings.

We consider the representative coordinates of S. Bergman and give some applications to the study of holomorphic mappings.

We set up the selection conditions for time series ${\left\{t_k\right\}}_{k=1}^{\infty }$ which converge to 0 as $k\to \infty$ and for which the solutions of a class of generalized Schrödinger equations pointwise converge almost everywhere to their initial data in $H^s\left({ℝ}^n\right)$ for $s>0$.

We investigate the properties of solutions to operator Riccati equations in connection with Bergman spaces. We characterize some necessary conditions for the solvability of the Riccati equation

$$XAX+XB-CX-D=0$$

on the set $\tau$ of all Toeplitz operators on the Bergman spaces $A_{\alpha }^2\left(\mathbb{𝔻}\right)$. This extends the results of Karaev on Hardy space.

We establish some new oscillation criteria of first-order $p$-Laplacian nonlinear dynamic equations on time-scales. We also discuss the Kamenev and Philos-type oscillation criteria. Consequently, we apply these techniques to improve and extend the results in the literature. Further, our results are demonstrated through some interesting examples.

We prove that certain twisted flag varieties carry Ulrich bundles.

We obtain a necessary and sufficient condition for the existence of wavelet frames. We define and study the synthesis and analysis operators associated with wavelet frames. We discuss some applications of operator value (OPV) frames in the theory of wavelet frames. Also, we discuss the minimal property of wavelet frame coefficients and study the property of over completeness of wavelet frames. Various characterizations of wavelet frame, Riesz wavelet basis and orthonormal wavelet basis are given. Further, dual wavelet frames are discussed and a characterization of dual wavelet frames is given. Finally, we give a characterization of a pair of biorthogonal Riesz bases.

Galanopoulos et al. proved that the mean Lipschitz condition for $f$ in the classical Bergman space is characterized by the growth of the area integral mean of its derivative as well as by the growth of the norm of the difference between $f$ and the dilated function of $f$. We prove that functions in the weighted Bergman space with admissible Békollé weights also have the same property. Furthermore we investigate the Bloch and Zygmund-type spaces for admissible weight.

This paper concerns the coincidence of the socle of a semiprimitive ring (or just a reduced one, in some cases) with the intersection of all essential prime, essential minimal prime, or essential maximal ideals of the ring. More precisely, we prove first that the socle of a reduced ring coincides with the intersection of all essential minimal prime ideals if and only if every minimal prime (or, equivalently, every prime) ideal is either essential or it is a direct summand which is also a maximal ideal. Next, we show that the socle of a semiprimitive ring $R$ is equal to the intersection of all essential maximal ideals of $R$ (i.e., ${Soc}_{\mathsf{max}}\left(R\right)$) if and only if the set of isolated points of $Max\left(R\right)$ with the Zariski topology contains no infinite basic open set. Whenever $R$ is a semiprimitive c.a.c. ring, we prove that for every essential ideal $I$ of $R$ containing ${Soc}_{\mathsf{max}}\left(R\right)$, $I∕{Soc}_{\mathsf{max}}\left(R\right)$ is essential in $R∕{Soc}_{\mathsf{max}}\left(R\right)$ if and only if the set of isolated points of $Max\left(R\right)$ is finite. We apply this result to rings of continuous real-valued functions on a topological space.

We study the fractional Schrödinger–Poisson system with critical Sobolev exponent

$$\left\{\begin{array}{cc}{\left(-\mathrm{\Delta }\right)}^{s}u+V\left(x\right)u+\varphi u=f\left(x,u\right)+K\left(x\right)|u{|}^{2_s^{\ast }-2}u\hfill & \text{ in }{ℝ}^3,\hfill \\ {\left(-\mathrm{\Delta }\right)}^t\varphi =u^2\hfill & \text{ in }{ℝ}^3,\hfill \end{array}\right.$$

where ${\left(-\mathrm{\Delta }\right)}^{\alpha }$ denotes the fractional Laplacian of order $\alpha =s,t\in \left(0,1\right)$; $V\left(x\right)$, $f\left(x,u\right)$ and $K\left(x\right)$ are $1$-periodic in the $x$-variables; $2_s^{\ast }=6∕\left(3-2s\right)$ is the fractional critical Sobolev exponent in dimension $3$. Under some weaker conditions on $f$, we prove the existence of ground state solutions for such a system via the mountain pass theorem in combination with the concentration-compactness principle. Our results are new even for $s=t=1$.

We discuss the existence and uniqueness of monotone positive solutions for a class of higher-order nonlinear fractional differential equations infinite-point boundary value problems (for short, BVPs) on cones. The existence and uniqueness of solutions are obtained via applying the properties of the Green function and a fixed point theorem. Our analysis is based on the operator equation $T\omega +S\omega =\omega$ on an ordered Banach space. Finally, a example is given to illustrate our results.

We study the induced Ekedahl–Oort stratification on the moduli of curves of genus 4 in positive characteristic by computing the de Rham cohomology of curves. For moduli space of hyperelliptic curves of genus 4 in characteristic 3, we determine the dimension and reducibility of Ekedahl–Oort strata. For moduli space of curves of genus 4 in odd characteristic, we show the existence of certain Ekedahl–Oort strata and discuss the existence of superspecial curves.