By applying a Mawhin continuation theorem of coincidence degree theory, we establish sufficient conditions for the existence of a periodic solution for a class of impulsive neutral differential equations. The procedure adopted in this work makes use of a nonimpulsive associated equation in order to overcome the difficulties resulting from the moments of impulse effects.

Linear Riemann–Liouville fractional differential equations with impulses are studied in the case of scalar equations and the case of systems. Both cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of impulse. Two types of initial conditions and impulsive conditions are applied to set up initial value problems for fractional differential equations with impulses. Explicit formulas for the solutions are obtained. The Mittag-Leffler function and the matrix generalization of the fractional exponential function are applied.

We prove an existence result for solutions to a class of unilateral problems for the nonlinear elliptic equation whose prototype is $-div\left(|\nabla u{|}^{p-2}\nabla u\right)+b\left(x\right)|\nabla u{|}^{\lambda }=f-divF\text{in }\mathrm{\Omega }$, where $\mathrm{\Omega }$ is a bounded open set of ${ℝ}^N$, $N\ge 2$, $1<p<N$, $0\le \lambda \le p-1$, $b\left(x\right)$ belongs to the Lorentz space $L^{N,1}\left(\mathrm{\Omega }\right),f\in L^1\left(\mathrm{\Omega }\right)$ and $F\in {\left(L^{p^{\prime }}\left(\mathrm{\Omega }\right)\right)}^N$, $p^{\prime }=p∕\left(p-1\right)$.

Classical parabolic Riesz and parabolic Bessel type potentials are interpreted as negative fractional powers of the differential operators $\left(-△+\partial ∕\partial t\right)$ and $\left(I-△+\partial ∕\partial t\right)$. Here, $△$ is the Laplacian and $I$ is the identity operator. We introduce some generalizations of these potentials, namely, we define the family of operators $A_{\beta ,𝜃}^{\alpha }={\left(𝜃I+{\left(-△\right)}^{\beta ∕2}+\partial ∕\partial t\right)}^{-\alpha }$ for $𝜃\ge 0$ and $\alpha ,\beta >0$, and investigate its behavior in the framework of $L_p\left({ℝ}^{n+1}\right)$-spaces.

For any prime power $p^n$ we obtain estimates for the quantity ${\mathrm{\Gamma }}^{\ast }\left(k,p^n\right)$, the smallest positive integer $s$ such that for any integers $a_i$ with $p\nmid a_i$, and integer $a$, the congruence

$$a_1{x}_1^k+\cdots +a_s{x}_s^k\equiv a\left(mod{p}^n\right)$$

is solvable in integers $x_i$, with $p\nmid x_i$, $1\le i\le n$. Let $k=p^e{k}_1$ with $k_1|p-1$, $e\le n-1$ and $t_1=\left(p-1\right)∕k_1$. For $p$ odd and $t_1>2$ we obtain

$${\mathrm{\Gamma }}^{\ast }\left(k,p^n\right)\le \lceil \frac{p}{p-1}k\rceil \text{ and}{\mathrm{\Gamma }}^{\ast }\left(k,p^n\right)\le {\left(Clog2k_1\right)}^{3e+3}{k}^{2∕\varphi \left(t_1\right)}$$

for some constant $C$. Several other estimates are given for ${\mathrm{\Gamma }}^{\ast }\left(k,p^n\right)$ as well as for the related quantities $\mathrm{\Gamma }\left(k,p^n\right)$, for primitive representations, and ${\mathrm{\Gamma }}_0\left(k,p^n\right)$, for any representation.

In the 1980s, Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then, this has been extended by works of Cimasoni, Florens, Mellor, Melvin, Conway, Toffoli, Friedl, and others to compute other link invariants. Informally, a C-complex is a union of surfaces which are allowed to intersect each other in clasps. We study the minimal number of clasps amongst all C-complexes bounded by a fixed link $L$. This measure of complexity is related to the number of crossing changes needed to reduce $L$ to a boundary link. We prove that if $L$ is a 2-component link with nonzero linking number, then the linking number determines the minimal number of clasps amongst all C-complexes. In the case of 3-component links, the triple linking number provides an additional lower bound on the number of clasps in a C-complex.

Recently, Carlini, Catalisano, Guardo and Van Tuyl introduced a new construction using the Hadamard product to present star configurations of codimension $c$ of ${\mathbb{P}}^n$ and which they called Hadamard star configurations. We introduce a more general type of Hadamard star configuration; any star configuration constructed by our approach is called a weak Hadamard star configuration. We classify weak Hadamard star configurations, and in the case $c=n$, we investigate the existence of a (weak) Hadamard star configuration which is apolar to the generic homogeneous polynomials of degree $d$.

Given a directed graph $E$ and a labeling $\mathcal{ℒ}$, one forms the labeled graph $C^{\ast }$-algebra by taking a weakly left-resolving labeled space $\left(E,\mathcal{ℒ},\mathcal{ℬ}\right)$ and considering a universal generating family of partial isometries and projections. We work on ideals for a labeled graph $C^{\ast }$-algebra when the graph contains sinks. Using some of the tools we build, we compute $C^{\ast }\left(E,\mathcal{ℒ},\mathcal{ℬ}\right)$ when $E$ is a finite graph.

We prove the existence of mild solutions for a class of conformable fractional differential equations with nonlocal conditions. The main results are based on semigroup theory combined with the Schaefer fixed point theorem. Moreover, we give an example to illustrate the applicability of our results.

We classify the canonical threefold singularities that allow an effective two-torus action. This extends classification results of Mori on terminal threefold singularities and of Ishida and Iwashita on toric canonical threefold singularities. Our classification relies on lattice point emptiness of certain polytopes with rational vertices. We show that in dimension two, such polytopes are sporadic or are given by Farey sequences. We finally present the Cox ring iteration tree of the classified singularities.

The Newton–Girard Formula allows one to write any elementary symmetric polynomial as a sum of products of power sum symmetric polynomials and elementary symmetric polynomials of lesser degree. It has numerous applications. We have generalized this identity by replacing the elementary symmetric polynomials with monomial symmetric polynomials. Our formula has an application in the field of Lie algebras.

Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $J$ (resp. $J^{\prime }$) be an ideal of $B$ (resp. $C$) such that $f^{-1}\left(J\right)=g^{-1}\left(J^{\prime }\right)$. We investigate the transfer of the notions of (A)-rings and strong (A)-rings to the bi-amalgamation of $A$ with $\left(B,C\right)$ along $\left(J,J^{\prime }\right)$ with respect to $\left(f,g\right)$, denoted by $A{\bowtie }^{f,g}\left(J,J^{\prime }\right)$, introduced and studied by Kabbaj, Louartiti and Tamekkante in 2013. Our results generate new original examples of rings satisfying these properties.

We derive explicit formulas for the expected cover time and cover time distribution for Markov-generated binary patterns of length two, including the special case of sequences of independent observations. We include links to output and coding for Mathematica and R simulations and calculations as appropriate.

For a connected graph $G$, the reduced second Zagreb index is defined as

$$R{M}_2=R{M}_2\left(G\right)=\sum _{uv\in E\left(G\right)}\left(d_G\left(u\right)-1\right)\left(d_G\left(v\right)-1\right),$$

where $d_G\left(u\right)$ and $d_G\left(v\right)$ denote the degrees of vertices $u$ and $v$ in graph $G$, respectively. We determine the extremal graphs among all connected graphs of order $n$ and with $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)-m$ edges with respect to $R{M}_2$, and obtain the first ten largest $R{M}_2$ for connected graphs of order $n$. Moreover, we characterize the sharp upper bound on $R{M}_2$ for all connected bipartite graphs with given connectivity $k$. Finally, the extremal graphs with respect to $R{M}_2$ are determined in all connected bipartite graphs with matching number $q$.

We define some metrics on the set of all bounded normal composition operators in $L^2\left(\mathrm{\Sigma }\right)$, and show that these metrics are equivalent with the metric induced by the usual operator norm.

We derive a dimensionally reduced model for a thin film prestrained with a given incompatible Riemannian metric:

$$G^h\left(x^{\prime },x_3\right)=I_3+2h^{\gamma }S\left(x^{\prime }\right)+2h^{\gamma ∕2}{x}_{3}B\left(x^{\prime }\right)+\text{ higher order terms},\gamma >2,$$

where $0<h\ll 1$ is the thickness of the film. The problem is studied rigorously by using a variational approach and establishing the $\mathrm{\Gamma }$-convergence of the non-Euclidean version of the nonlinear elasticity functional. It is shown that the residual nonlinear elastic energy scales as $O\left(h^{\gamma +2}\right)$ as $h\to 0$.

The adèle ring ${\mathbb{𝔸}}_K$ of a global field $K$ is a locally compact, metrizable topological ring which is complete with respect to any invariant metric on ${\mathbb{𝔸}}_K$. For a fixed global field $F$ and a possibly infinite algebraic extension $E∕F$, there is a natural partial ordering on $\left\{{\mathbb{𝔸}}_K:F\subseteq K\subseteq E\right\}$. Therefore, we may form the direct limit

$${\mathbb{𝔸}}_E=\underrightarrow{lim}{\mathbb{𝔸}}_K,$$

which provides one possible generalization of adèle rings to arbitrary algebraic extensions $E∕F$. In the case where $E∕F$ is Galois, we define an alternate generalization of the adèles, denoted by ${\overline{\mathbb{𝕍}}}_E$, to be a certain metrizable topological ring of continuous functions on the set of places of $E$. We show that ${\overline{\mathbb{𝕍}}}_E$ is isomorphic to the completion of ${\mathbb{𝔸}}_E$ with respect to any invariant metric and use this isomorphism to establish several topological properties of ${\mathbb{𝔸}}_E$.

We establish two identities involving the Gamma function and Bernoulli polynomials, namely

$$\sum _{k\le x}\frac{1}{k^s}\sum _{j=1}^{k^s}log\mathrm{\Gamma }\left(\frac{j}{k^s}\right)\sum _{d|k,d^s|j}f\ast \mu \left(d\right)\text{ and}\sum _{k\le x}\frac{1}{k^s}\sum _{j=0}^{k^s-1}{B}_m\sum _{d|k,d^s|j}f\ast \mu \left(d\right)$$

with any fixed integer $s>1$ and any arithmetical function $f$. We give asymptotic formulas for them with various multiplicative functions $f$. We also consider several formulas of Dirichlet series associated with the above identities. This paper is a continuation of an earlier work of the authors.

By the theory of elliptic curves, we show that given a convex angle $𝜃$, there exist, except for finitely many exceptions, infinitely many pairs of rational $𝜃$-triangle and $\omega$-parallelogram with areas and perimeters in fixed proportions $\left(\alpha ,\beta \right)$ respectively, satisfying that $sin\omega$ is a previously fixed rational multiple of $sin𝜃$, where $\alpha$ and $\beta$ are positive rational numbers.

We investigate the uniqueness and boundedness of solutions and finite-time stability for nonlinear fractional $h$-difference systems with “maxima” using the generalized fractional $h$-Gronwall inequality. We give explicit bounds on solutions for this fractional $h$-difference system with “maxima”. Finally, an example is given to illustrate one of our main results.

For $q\in \left(0,1\right)$, let $B_q$ denote the limit $q$-Bernstein operator. The distance between $B_q$ and $B_r$ for distinct $q$ and $r$ in the operator norm on $C\left[0,1\right]$ is estimated, and it is proved that $1\le \parallel B_q-B_r\parallel \le 2$, where both of the equalities can be attained. Furthermore, the distance depends on whether or not $r$ and $q$ are rational powers of each other. For example, if $r^j\ne q^m$ for all $j,m\in \mathbb{N}$, then $\parallel B_q-B_r\parallel =2$, and if $r=q^m$ for some $m\in \mathbb{N}$, then $\parallel B_q-B_r\parallel =2\left(m-1\right)∕m$.

We introduce a family of squarefree monomial ideals associated to finite simple graphs, whose monomial generators correspond to closed neighborhoods of vertices of the underlying graph. Any such ideal is called the closed neighborhood ideal of the graph. We study some algebraic invariants of these ideals like Castelnuovo–Mumford regularity and projective dimension and present some combinatorial descriptions for these invariants in terms of graph invariants.

We introduce a natural extension of time weighted entropy of homeomorphism on locally compact separable metrizable spaces, and some new properties are given.

Palindromic numbers are positive integers that remain unchanged when their decimal digits are reversed. We characterize palindromic numbers whose squares are also palindromic. We use this to determine the number of $n$-digit palindromic numbers whose squares are palindromic, and the number of palindromic numbers whose squares are palindromic and which are not greater than a fixed positive integer.

We consider the Cauchy problem of the inhomogeneous pseudoparabolic equation $u_t-k\mathrm{\Delta }{u}_t=\mathrm{\Delta }u+|u{|}^p+\omega \left(x\right)$ in ${ℝ}^n$ with nonnegative initial value $u_0\left(x\right)$, where $k>0$ and $p>1$. When the equation is homogeneous, i.e., $\omega \equiv 0$, it was shown by Cao, Yin and Wang (2009) that the critical Fujita exponent $p_c$ is given by $p_c=1+2∕n$, which means for $p\in \left(1,p_c\right)$, the distribution of the initial data has no effect on the blow-up phenomena; for $p\in \left(p_c,\infty \right)$, the distribution of the initial data does have effect on the blow-up phenomena. We study the effect of the inhomogeneous term $\omega \left(x\right)$ on the critical Fujita exponent $p_c$, and we show $p_c=\infty$ if $n=1,2$, and $p_c=1+2∕\left(n-2\right)$ if $n\ge 3$.