“Polynomials” with degree as an ordered monoid say $\mathrm{\Delta }$ are constructed along with corresponding schemes and line bundles $\mathcal{𝒪}\left(d\right)$, $d\in \mathrm{\Delta }$. The cohomology of these line bundles is then computed using Čech complex and new proof of zero cohomology of affine schemes is given. The last section of the paper applies the theory developed for the construction and computation of affine cohomology of perfectoid Tate algebras.

A representation of the solutions for the Darboux problem of partial fractional differential equations with Caputo derivative in the linear case is given. The Ulam–Hyers–Rassias Mittag–Leffler stability of the solutions for the Darboux problem of partial fractional differential equations is investigated. An illustrative example is presented to validate our results.

We study the orbits under the natural action of a permutation group $G\le {\text{S}}_n$ on the powerset $\mathcal{𝒫}\left(\left\{1,\dots ,n\right\}\right)$. The permutation groups having exactly $n+1$ orbits on the powerset can be characterized as set-transitive groups and were fully classified by Beaumont and Peterson in 1955. In this paper, we establish a general method that allows one to classify the permutation groups with $n+r$ set-orbits for a given $r$, and apply it to integers $2\le r\le 15$ with the help of GAP.

We are concerned with the third-order linear differential equation $\left(q\left(qy\prime \right)\prime \right)\prime +py\prime +ry=0$ defined on an interval $\left[t_0,\infty \right)$. We derive asymptotic solutions for cases characterized by $p$ being “small” and “large”, respectively. Our main tool is a result by Eastham on the asymptotic solution of linear differential systems.

We present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let $P$, $Q$, $R_0$ and $R_1$ be fixed integers, and let $R={\left(R_n\right)}_{n\ge 0}$ be the recurrence sequence defined by $R_{n+2}=P{R}_{n+1}-Q{R}_n$ for all $n\ge 0$. Under some conditions on the parameters, we determine a rational nontrivial divisor for $L_{k,n}:=lcm\left(R_k,R_{k+1},\dots ,R_n\right)$, for all positive integers $n$ and $k$, such that $n\ge k$. As consequences, we derive nontrivial effective lower bounds for $L_{k,n}$, and we establish an asymptotic formula for $log\left(L_{n,n+m}\right)$, where $m$ is a fixed positive integer. Denoting by ${\left(F_n\right)}_n$ the usual Fibonacci sequence, we prove, for example, that for any $m\ge 1$, we have

$$loglcm\left(F_n,F_{n+1},\dots ,F_{n+m}\right)\sim n\left(m+1\right)log\mathrm{\Phi }\left(\text{ as }n\to +\infty \right),$$

where $\mathrm{\Phi }$ denotes the golden ratio. We conclude the paper with some interesting identities and properties regarding the least common multiple of Lucas sequences.

We define the manifold of configurations to be the quotient set of $k$ points in Euclidean space identified under congruence, and prove that compact subsets of ${ℝ}^d$, $d\ge 2$, of large Hausdorff dimension have a non-null set of configurations in them. Our method simplifies previous work and achieves a better dimensional threshold.

We consider the scaling behavior of the range and $p$-multiple range, that is the number of points visited and the number of points visited exactly $p\ge 1$ times, of a simple random walk on ${\mathbb{Z}}^d$, for dimensions $d\ge 2$, up to time of exit from a domain $D_N$ of the form $D_N=ND$, where $D\subset {ℝ}^d$, as $N\uparrow \infty$. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case $D$ is a cube in $d\ge 3$, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in $d\ge 2$, both weakly converge to proportional exit times of Brownian motion from $D$, and that the corresponding limit moments are “polyharmonic”, solving a hierarchy of Poisson equations.

We give a geometric criterion for Dirichlet $L$-functions associated to cyclic characters over the rational function field ${\mathbb{𝔽}}_q\left(t\right)$ to vanish at the central point $s=\frac{1}{2}$. The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over ${\mathbb{𝔽}}_q$. Using this geometric criterion, we obtain a lower bound on the number of cubic characters over ${\mathbb{𝔽}}_q\left(t\right)$ whose $L$-functions vanish at the central point where $q=p^{4n}$ for any rational prime $p\equiv 2mod3$. We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the $L$-functions of Dirichlet characters of other orders.

The singular Kirchhoff-type $p$-Laplacian problem

is considered, where $\mathrm{\Omega }\subset {ℝ}^N\left(N\ge 3\right)$ is a bounded domain, $a,b\ge 0$ with

$p^{\ast }=pN∕\left(N-p\right)$ is the critical Sobolev exponent, and the coefficient $f$ is in $L^{p^{\ast }∕\left(p^{\ast }+\gamma -1\right)}\left(\mathrm{\Omega }\right)$ with . By the variational method and some analysis techniques, a uniqueness result for a class of singular $p$-Kirchhoff type problems is obtained.

In this note we study the lineability and algebrability problems for certain classes of sequence spaces and series. In particular, we extend some algebrability results to the context of $p$-adic fields. We also prove that the subset of nonconvergent sequences of ${\ell }_{\infty }\left(ℝ\right)$ whose set of accumulation points is not a Cantor-type set is strongly $\mathfrak{𝔠}$-algebrable.

We apply the discrete multiparameter local Calderón identity and Littlewood–Paley–Stein theory with weights to carry out the weighted multiparameter local Hardy spaces $h_w^p$ of arbitrary $k$ number of parameters $\left(k\ge 3\right)$. We will derive the $\left(h_w^p,h_w^p\right)$ and $\left(h_w^p,L_w^p\right)$ boundedness for multiparameter inhomogeneous singular integral operators with slightly weaker assumptions on the kernel.

We provide a short proof on the change-of-basis coefficients from the Specht basis to the Kazhdan–Lusztig basis, using Kazhdan–Lusztig theory for the parabolic Hecke algebra.

We investigate the parabolic Allen-Cahn equation

$$\frac{\partial u}{\partial t}=\mathrm{\Delta }u+\left(1-u^2\right)u$$

under general geometric flow on complete noncompact manifolds. We first prove a local gradient estimate for the positive solution $u$ to this equation, then we obtain a Harnack inequality as an application.

It is well known that increasing functions do not preserve operator order in general; nor do decreasing functions reverse operator order. However, operator monotone increasing or operator monotone decreasing functions do. In this article, we employ a convex approach to discuss operator order preserving or conversing functions. As an easy consequence of more general results, we find nonnegative constants $\gamma$ and $\psi$ such that $A\le B$ implies

$$f\left(B\right)\le f\left(A\right)+\gamma {\mathbf{1}}_{\mathcal{\mathscr{H}}}\text{ and}f\left(A\right)\le f\left(B\right)+\psi {\mathbf{1}}_{\mathcal{\mathscr{H}}},$$

for the self-adjoint operators $A$, $B$ on a Hilbert space $\mathcal{\mathscr{H}}$ with identity operator ${\mathbf{1}}_{\mathcal{\mathscr{H}}}$ and for the convex function $f$ whose domain contains the spectra of both $A$ and $B$.

The connection of these results to the existing literature will be discussed and the significance will be emphasized by some examples.

We focus on the problems of sign changes and nonvanishing of Fourier coefficients of holomorphic cusp forms. Let ${\lambda }_f\left(n\right)$, ${\lambda }_g\left(n\right)$ and ${\lambda }_h\left(n\right)$ be the $n$-th Fourier coefficients of primitive holomorphic cusp forms $f$, $g$ and $h$ for $SL\left(2,\mathbb{Z}\right)$, respectively. Firstly, we study the behavior of the signs of the sequences $\left\{{\lambda }_f\left(n^j\right)\right\}$ with $j\ge 3$, $\left\{{\lambda }_f\left(n^i\right){\lambda }_g\left(n^j\right)\right\}$ with $i\ge 1,j\ge 2$, and $\left\{{\lambda }_f\left(n\right){\lambda }_g\left(n\right){\lambda }_h\left(n\right)\right\}$ in short intervals, and present some quantitative results for the number of sign changes for $n\le x$. Here we improve and generalize previous results. Moreover, we investigate simultaneous nonvanishing of $\left\{{\lambda }_f\left(n^j\right){\lambda }_g\left(n^j\right){\lambda }_h\left(n^j\right)\right\}$ with $j\ge 1$ in short intervals.

We consider a class of space-time fractional stochastic partial differential equation on a bounded domain with Lévy noise. We prove that the second moment of the solution $u\left(t,x\right)$ can not decay exponentially and for $\beta \in \left(0,\frac{1}{2}\right]$, ${sup}_{x\in D}\mathbb{𝔼}|u\left(t,x\right){|}^2$ grows exponentially fast for large $t$. When $\beta \in \left(\frac{1}{2},1\right)$, there is some phase transition of the second moment growth, depending on the noise level $\lambda$.

By Cartan’s version of Nevanlinna’s theory, we prove the following result: Let $m$ and $n$ be two positive integers satisfying $n\ge m+2$ and $m\ge 1$, let $p$ be a polynomial such that $p\not\equiv 0$, let $\eta$ be a finite complex number such that $\eta \ne 0$ and ${\mathrm{\Delta }}_{\eta }f\not\equiv 0$, let ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m$ be $m$ distinct finite nonzero complex numbers, and let $H_j$ be either an exponential polynomial of degree less than $q$ or an ordinary polynomial in $z$ such that $H_j\not\equiv 0$ for $1\le j\le m$. Suppose that $f$ is a nonconstant meromorphic solution of the difference equation

$$f^n\left(z\right)+p\left(z\right){\mathrm{\Delta }}_{\eta }f\left(z\right)=H_1\left(z\right)e^{{\alpha }_1{z}^q}+H_2\left(z\right)e^{{\alpha }_2{z}^q}+\cdots +H_m\left(z\right)e^{{\alpha }_m{z}^q}.$$

Then $m\ge 2$ and $f$ is reduced to a transcendental entire function such that its order $\rho \left(f\right)=\infty$ or its order satisfies $\rho \left(f\right)=q$ with $m=2$, while $f$ can be either expressed as

$$f\left(z\right)=A_1\left(z\right)e^{{\alpha }_1{z}^q},\text{ with }{A}_1\left(z\right)=\frac{H_{1}f}{p{\mathrm{\Delta }}_{\eta }f}\text{and }n{\alpha }_1-{\alpha }_2=0,$$

or

$$f\left(z\right)=A_2\left(z\right)e^{{\alpha }_2{z}^q},\text{ with }{A}_2\left(z\right)=\frac{H_{2}f}{p{\mathrm{\Delta }}_{\eta }f}\text{and }n{\alpha }_2-{\alpha }_1=0,$$

where $A_1$ and $A_2$ are entire function such that their characteristic functions satisfy

$$T\left(r,A_j\right)=o\left(T\left(r,f\right)\right),1\le j\le 2,$$

as $r\notin E$ and $r\to \infty$. Here, $E\subset \left(0,+\infty \right)$ is a subset of finite linear measure.

An example is provided to show that the assumption $n\ge m+2$, in a sense, is the best possible.

This result improves and extends the corresponding results from Yang–Laine (2010), Liu–Lü–Shen–Yang (2014), and Latreuch (2017).

The generalized Abel equation (GAE) with constant coefficients, ${\gamma }_1{}_a{D}_x^{-\sigma }u+{\gamma }_2{}_x{D}_b^{-\sigma }u=f$, , has renewed interest since it has appeared as a component in double-sided fractional diffusion equations. Motivated by direct applications in modeling, we seek the necessary and sufficient conditions for interchanging the order of differentiation and fractional integration in GAEs. Put another way, assume functions $u,v,f$ lie in Hölder spaces that admit integrable singularities at the endpoints and

$${\gamma }_1{}_a{D}_x^{-\sigma }u+{\gamma }_2{}_x{D}_b^{-\sigma }u=f\text{ and}{\gamma }_1{}_a{D}_x^{-\sigma }v+{\gamma }_2{}_x{D}_b^{-\sigma }v=Df,$$

then we prove that $Du=v$ if and only if $u\left(a\right)=u\left(b\right)=0$.

We present interrelations between several topological properties of, for example, spaces with a point-countable base, spaces with a $\sigma$-hereditarily closure-preserving $k$-network, spaces with a base of countable order, spaces with a point-regular base, Nagata-spaces, $c$-semi-stratifiable spaces, $\gamma$-spaces, or semi-metrizable spaces of $X$ and the hyperspaces of nonempty compact subsets and finite subsets with the Vietoris topology.

This paper is concerned with the asymptotic behavior of solutions of a new class of the nonlinear rational difference equations. We establish new criteria for stability (local and global) and boundedness of solutions of the studied equation. Moreover, we investigate the periodic character (periodic two and three) of the solutions of these equations, by using new techniques. Finally, we give some interesting examples in order to verify our results.

We consider the Prym variety $P\left(\tilde{C}∕C\right)$ associated to a Galois coverings of curves $f:\tilde{C}\to C$ branched at $r$ points. We discuss some properties and equivalent definitions and then consider the Prym map $\mathcal{𝒫}=\mathcal{𝒫}\left(G,g,r\right):R\left(G,g,r\right)\to A_{p,\delta }$ with $\delta$ the type of the polarization. For Galois coverings whose Galois group is abelian and metabelian (nonabelian) we show that the differential of this map at certain points is injective. We also consider the Abel-Prym map $u:\tilde{C}\to P\left(\tilde{C}∕C\right)$ and prove some results for its injectivity. In particular, we show that in contrast to the classical and cyclic case, the behavior of this map here is more complicated. The theories of abelian and metabelian Galois coverings play a substantial role in our analysis and have been used extensively throughout the paper.

We prove a Liouville-type theorem for the entire solutions of $\mathrm{\Delta }u\ge f\left(u\right)$, complementing a classical result by Osserman (1957).

In this work we give sufficient conditions for hyponormality of Toeplitz operators on weighted Bergman spaces when the analytic part of the symbol is a monomial and the conjugate part is a polynomial. We also adapt a function theoretic method, due to P. Ahern and Ž. Čučković (1996), to a class of Toeplitz operators on weighted Bergman spaces to get an extension of the necessary condition for hyponormality.

Let $E$ be an elliptic curve over $\mathbb{Q}$ defined by $y^2=a{x}^3+b{x}^2+cx+d$. We say a sequence of rational points $\left(x_i,y_i\right)\in E\left(\mathbb{Q}\right)$, $i=0,1,\dots ,\ell$, forms a sequence of consecutive $n$-th powers on $E$ of length $\ell$ whenever the sequence of $x$-coordinates, $x_i$, $i=0,1,\dots ,\ell$, consists of consecutive powers of degree $n$ in the form $x_i={\left(g+i\right)}^n$, for some rational $g$. Applying the known Mestre’s theorem, for an arbitrary natural number $n\ge 2$, we produce a one-parameter family of elliptic curves over $\mathbb{Q}$ which contains an $8$-term sequence of consecutive $n$-th powers. Furthermore, we show that for $n=2$ and $3$ the associated families of elliptic curves are of generic rank $\ge 6$ and $7$, respectively. We also provide an explicit set of linearly independent points for those families. Finally, according to our limited trial conducted for $4\le n\le 50$, we discovered that the generic rank of the corresponding families is $\ge 7$. We guess that this holds for all $n\ge 4$; however, we are not able to prove it at this time.

We give an explicit formula for the number of subgroups of the modular group of a given index that are genus zero and torsion-free, and a formula for their conjugacy classes. We do so by exhibiting a correspondence between these groups and the trivalent maps on a sphere. We focus on the particular case of the subgroups of index 18 which have some interesting geometric properties.

We first establish a new Hilbert-type inequality involving the kernel including both the homogeneous and the nonhomogeneous cases, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of Leibniz integral rule and the rational fraction expansion of trigonometric functions, we derive the representation of the best possible constant factor by the logarithmic function and the higher derivative of trigonometric functions, which leads to a few special and interesting inequalities. Furthermore, We also demonstrate that our newly obtained inequality is a unified generalization of some classical Hilbert-type inequalities.

In this paper, we consider the existence and asymptotic behavior of ground state sign-changing solutions for the following nonlinear Schrödinger–Poisson–Kirchhoff system:

Under some mild assumptions, by using a series of constructive Nehari manifolds and a quantitative deformation lemma, we prove that the Schrödinger–Poisson–Kirchhoff system possesses at least one ground state sign-changing solution $u_b$ for all . Moreover, for any sequence $\left\{b_n\right\}\to 0^{+}$ $\left(n\to \infty \right)$, there exists a subsequence $\left\{b_{n_k}\right\}$, such that $\left\{u_{b_{n_k}}\right\}$ converges to $u_0$, where $u_0$ is a sign-changing solution of the equation $-a△u+V\left(x\right)u=\lambda f\left(x\right)u+g\left(x,u\right)$, $x\in {ℝ}^3$.