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In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.

During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.

Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha $$ , there exist any $$n \in \mathbb {Z}^+$$ such that $$\tau (n) = \alpha $$ . A series of recent papers excludes many integers as possible values of the $$\tau $$ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $$\tau (n)$$ . We synthesize these results and methods to prove that if $$0< \left| \alpha \right| < 100$$ and $$\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$$ , then $$\tau (n) \ne \alpha $$ for all $$n > 1$$ . Moreover, if $$\alpha \in T$$ and $$\tau (n) = \alpha $$ , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $$\left| \tau (n) \right| > 100$$ for all $$n > 2$$ .

In this paper, we extend compactness theorems of Cheeger, Gromov, Taylor, and Sprouse to the Bakry–Emery Ricci tensor and generalized quasi-Einstein tensors. Our results generalize previous results obtained by Yun and Wan.

In this paper, the arithmetic-geometric mean inequalities of indefinite type are discussed. We show that for a J-selfadjoint matrix A satisfying $$I \ge ^J A$$ and $${\mathrm{sp}}(A) \subseteq [1, \infty ),$$ the inequality $$\begin{aligned} \frac{I + A}{2} \le ^J \sqrt{A} \end{aligned}$$ holds, and the reverse does for A with $$I \ge ^J A$$ and $${\mathrm{sp}}(A) \subseteq [0, 1]$$ . We also prove that for J-positive invertible operators A, B acting on a Hilbert space of arbitrary dimension, the inequality $$\begin{aligned} \frac{A + B}{2} \ge ^J A \sharp ^J B \end{aligned}$$ holds, where $$A \sharp ^J B:= J \bigl ( (JA) \sharp (JB) \bigr )$$ . Several examples involving Pauli matrices are provided to illustrate the main results.

The Liouville–von Neumann equation describes the change in the density matrix with time. Interestingly, this equation was recently regarded as a wave equation for wave functions but not as an equation for density functions. This setting leads to an extended form of the Schrodinger wave equation governing the motion of a quantum particle. In this paper, we obtain the integrability of the wave propagator arising from the Liouville–von Neumann equation in this setting.

In this paper, we use the Fourier–Hermite functionals to extend the structure of the Cameron–Martin translation theorem on an abstract Wiener space $$(H,B,\nu )$$ . The directional function in our translation theorem may not be in the Cameron–Martin subspace H of B. We then proceed to obtain an explicit formula for our general translation theorem.

Let A be a positive bounded linear operator acting on a complex Hilbert space $${\mathcal {H}}$$ . Our aim in this paper is to prove some A-numerical radius inequalities of bounded linear operators acting on $${\mathcal {H}}$$ when an additional semi-inner product structure induced by A is considered. In particular, an alternative proof of a recent result proved in Moslehian et al. (Linear Algebra Appl 591:299–321 2020) is given.

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, we prove the Besse conjecture for compact manifolds with pinched Weyl curvature. Moreover, we shall conclude that such a conjecture is true if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

As an application of the curve shortening flow, this paper will show an inequality for the maximum curvature of a smooth simple closed curve on surfaces.

It is known that under the Dirichlet product, the set of arithmetic functions in several variables becomes a unique factorization domain. A. Zaharescu and M. Zaki proved an analog of the ABC conjecture in this ring and showed that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$). It is also known that under the Cauchy product, the set of arithmetic functions becomes a unique factorization domain. In this paper, we consider the ring of arithmetic functions in several variables under the Cauchy product and prove an analog of the ABC conjecture in this ring to show that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$).

The Nekrasov–Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The first conjecture is a refined Nekrasov–Okounkov formula involving hooks with trivial legs. We give a proof of the conjecture. The second conjecture is on properties of the roots of the underlying D’Arcais polynomials. We give a counterexample and present a new conjecture. The third conjecture is on the unimodality of the coefficients of the involved polynomials. We confirm the conjecture up to the polynomial degree 1000.

Let $$S^p$$ be the space of holomorphic functions whose derivative lies in the classical Hardy space $$H^p$$ over the unit disk. We prove in this paper that the composition operator $$C_\varphi $$ with $$\varphi $$ an automorphism is hypercyclic on $$S^p$$ , $$0

In this paper, our purpose is to give a characterization of a sequentially Cohen–Macaulay module, which was introduced by Stanley (Combinatorics and Commutative Algebra, 2nd edn, Birkhauser, Boston, 1996), in terms of its index of reducibility of parameter ideals, which was given by Noether in 1921 (Math Ann 83:24–66, 1921). This applies in particular to characterizing the Gorensteinness, Cohen–Macaulayness of local rings in terms of eventually the index of reducibility for parameter ideals.

In this paper we consider the following quasilinear Schrodinger–Poisson system in a bounded domain in $${\mathbb {R}}^{2}$$ : $$\begin{aligned} \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &{}\ \text{ in } \Omega , \\ -\Delta \phi - \varepsilon ^{4}\Delta _4 \phi = u^{2} &{} \ \text{ in } \Omega ,\\ u=\phi =0 &{} \ \text{ on } \partial \Omega \end{array} \right. \end{aligned}$$ depending on the parameter $$\varepsilon >0$$ . The nonlinearity f is assumed to have critical exponential growth. We first prove existence of nontrivial solutions $$(u_{\varepsilon }, \phi _{\varepsilon })$$ and then we show that as $$\varepsilon \rightarrow 0^{+}$$ , these solutions converge to a nontrivial solution of the associated Schrodinger–Poisson system, that is, by making $$\varepsilon =0$$ in the system above.

In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite., Comment: 9 pages, revised version, main results unchanged, Archiv der Mathematik, 2018

In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for these two zeta functions.

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let $$D: = {\sum\limits_{i = 1}^n {\frac{\partial }{{\partial x_{i} }}e_{i}}}$$ be the Euclidean Dirac operator in the n-dimensional flat space $$\mathbb{R}^{n},\;{\mathbf{E}}: = {\sum\limits_{i = 1}^n {x_{i} \frac{\partial }{{\partial x_{i} }}}}$$ the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system $$\left[ {D - \lambda - (1 + \alpha )\frac{{\text{x}}} {{\text{|x|}}^{2}}{\mathbf{E}}} \right]f = 0$$ in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials.

Cuckovic and Paudyal recently characterized the lattice of invariant subspaces of the shift plus a complex Volterra operator on the Hilbert space \(H^2\) on the unit disk. Motivated by the idea of Ong, in this paper, we give a complete characterization of the lattice of invariant subspaces of the shift operator plus a positive integer multiple of the Volterra operator on Hardy spaces \(H^p\), which essentially extends their works to the more general cases when \(1\le p

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by $$S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}$$ . In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space $$\mathcal {L}^2[0,2\pi ]$$ is a simple point set $$\{0\}$$ .

This paper studies the asymptotic behaviour of the powers $$C_\varphi ^n$$ of a composition operator $$C_\varphi $$ on certain spaces of holomorphic functions defined on the right half plane $$\mathbb {C}_+$$ . It is shown that for composition operators on the Hardy spaces and the standard weighted Bergman spaces, if the inducing map $$\varphi $$ is not of parabolic type, then either the powers $$C_\varphi ^n$$ converge uniformly only to 0 or they do not converge even strongly.

The known (explicit) examples of Riemann surfaces not definable over their field of moduli are not real whose field of moduli is a subfield of the reals. In this paper we provide explicit examples of real Riemann surfaces which cannot be defined over the field of moduli., In this new version, Eslam Badr has been added as a coauthor and the examples provided in the previous version has been generalised

The paper presents a simple new approach to the problem of computing Fourier transforms of SO(d)-finite measures on the unit sphere in the euclidean space. Representing such measures as restrictions of homogeneous polynomials we use the canonical decomposition of homogeneous polynomials together with the plane wave expansion to derive a formula expressing such transforms under two forms, one of which was established previously by F. J. Gonzalez Vieli. We showthat equivalence of these two forms is related to a certain multi-step recurrence relation for Bessel functions, which encompasses several classical identities satisfied by Bessel functions. We show it leads further to a certain periodicity relation for the Hankel transform, related to the Bochner- Coifman periodicity relation for the Fourier transform. The purported novelty of this approach rests on the systematic use of the detailed form of the canonical decomposition of homogeneous polynomials, which replaces the more traditional approach based on integral identities related to the Funk-Hecke theorem. In fact, in the companion paper the present authors were able to deduce this way a fairly general expansion theorem for zonal functions, which includes the plane wave expansion used here as a special case.

In this paper we investigate the behavior of the Hilbert transform and the Cauchy transform. It is well known, that for absolut integrable functions the Hilbert transform and the Cauchy transform is finite almost everywhere. In this paper it is shown, that for each set \( E\subset [-\pi ,\pi ) \) with Lebesgue measure zero there exists a continuous function such that the Hilbert transform and the Cauchy transform of this function is infinite for all points of the set E. So for continuous functions the Hilbert transform and the Cauchy transform have a similar divergence behavior as for absolute integrable functions.

In this paper we give two counterexamples to the closedness of the sum of two sectorial operators with commuting resolvents. In the first example the operators are defined on an L p-space, with \(1 \le p \neq 2 \le \infty \), and one of them admits bounded imaginary powers. The second example is concerned with operators defined on a Hilbert valued L p-space; one acts on L p and admits bounded imaginary powers as the other acts on the Hilbert space. In the last section of the paper we show that the two partial derivations on \(L^2 ({\Bbb R}^2;X)\) admit a so-called bounded joint functional calculus if and only if X is a UMD Banach space with property \((\alpha )\) (geometric property introduced by G. Pisier).

The first main theorem of this paper asserts that any $$(\sigma , \tau )$$ -derivation d, under certain conditions, either is a $$\sigma $$ -derivation or is a scalar multiple of ( $$\sigma - \tau $$ ), i.e. $$d = \lambda (\sigma - \tau )$$ for some $$\lambda \in \mathbb {C} \backslash \{0\}$$ . By using this characterization, we achieve a result concerning the automatic continuity of $$(\sigma , \tau $$ )-derivations on Banach algebras which reads as follows. Let $$\mathcal {A}$$ be a unital, commutative, semi-simple Banach algebra, and let $$\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}$$ be two distinct endomorphisms such that $$\varphi \sigma (\mathbf e )$$ and $$\varphi \tau (\mathbf e )$$ are non-zero complex numbers for all $$\varphi \in \Phi _\mathcal {A}$$ . If $$d : \mathcal {A} \rightarrow \mathcal {A}$$ is a $$(\sigma , \tau )$$ -derivation such that $$\varphi d$$ is a non-zero linear functional for every $$\varphi \in \Phi _\mathcal {A}$$ , then d is automatically continuous. As another objective of this research, we prove that if $$\mathfrak {M}$$ is a commutative von Neumann algebra and $$\sigma :\mathfrak {M} \rightarrow \mathfrak {M}$$ is an endomorphism, then every Jordan $$\sigma $$ -derivation $$d:\mathfrak {M} \rightarrow \mathfrak {M}$$ is identically zero.

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension $$n\ge 3$$ with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of $${\mathbb {S}}^n$$ under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).

In a 1988 paper, Cowen found a formula expressing the adjoint of any linear fractional composition operator on the Hardy space as a product of Toeplitz operators and another linear fractional composition operator. In this paper, we use Cowen's adjoint formula to give a unitary equivalence relating composition operators on different weighted Hardy spaces. This result is then applied to some composition operators on the Sa spaces. We find the spectrum of any linear fractional composition operator whose symbol has exactly one fixed point of multiplicity one on the unit circle.

We shall work in a universal class ~ of right near-rings, (that is, every hornomorphic image and every ideal of a near-ring in ~J is again in lIJ), and we shall assume that the universal class ~J is closed under certain near-ring constructions which will be specified later on. Radical and semisimple classes of near-rings are meant in the sense of Kurosh and Amitsur. There are two kinds of trivial multiplications for near-rings: the zero-multiplication and the constant multiplication. I1 is a natural requirement that a radical of near-tings should contain all near-rings with trivial multiplication belonging to the considered universal class. It is the purpose of the presem paper to characterize the semisimple classes of such near-ring radicals. A radical class P, is said to be hypernilpo~em, if it contains all nilpotent near-rings (cf. [3] Proposition 2.11. We may call a radical class N hyperconstant, if IR contains all constant near-rings of lJ. For recent developments in the radical theory of near-rings the excellent survey paper [8] can be consulted. In the sequel N O and N c will stand for the zero-near-ring and the constant near-ring, respectively, built on the additive group N + In our considerations Veldsman's near-ring construction [6] will play a decisive role: for any additive group N = N-, let us define an addition and a multiplication on the cartesian product N x N x N as follows

Let $\alpha $ and $\beta $ be bounded measurable functions on the unit circle T. The singular integral operator $S_{\alpha ,\,\beta }$ is defined by $S_{\alpha ,\,\beta } f = \alpha Pf + \beta Qf(f \in L^2 (T))$ where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of $S_{\alpha ,\,\beta }$ was calculated in general, using $\alpha ,\beta $ and $\alpha \bar {\beta } + H^\infty $ where $H^\infty $ is a Hardy space in $L^\infty (T).$ In this paper, the essential norm $\Vert S_{\alpha ,\,\beta } \Vert _e$ of $S_{\alpha ,\,\beta }$ is calculated in general, using $\alpha \bar {\beta } + H^\infty + C$ where C is a set of all continuous functions on T. Hence if $\alpha \bar {\beta }$ is in $H^\infty + C$ then $\Vert S_{\alpha ,\,\beta } \Vert _e = \max (\Vert \alpha \Vert _\infty , \Vert \beta \Vert _\infty ).$ This gives a known result when $\alpha , \beta $ are in C.

In this paper, we obtain existence and orbital stability results for the planar Schrodinger-Poisson system. Our results are based on the Gagliardo-Nirenberg inequality, the concentration-compactness principle, and the extremum principle in critical point theory.

In this paper, we give a method of constructing conformal mappings defined in the unit disk which can fix arbitrarily many points on the unit circle.

The main goal of this paper is the presentation of an elementary analytic technique which enables the evaluation of the so-called restricted sum formulas involving multiple zeta values with even arguments, i.e. $$E(2c,K):=\sum_{\substack{\sum_{j=1}^{K}c_{j}=c\\{c}_{j}\in\mathbb{N}}} \zeta(2c_1,\ldots ,2c_K),$$ where c and K are arbitrary positive integers with $${c\ge K}$$ . Though the young and general theory of the multiple Riemann zeta function with a rich application potential may be rather complicated, our contribution makes the evaluation of the term E(2c,K) intelligible to a broad mathematical audience.

We consider a linear non-autonomous evolutionary Cauchy problem \begin{equation} \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time depending sesquilinear form $a(t,.,.)$ on a Hilbert space $H$ with constant domain $V.$ Recently a result on $L^2$-maximal regularity in $H,$ i.e., for each given $f\in L^2(0,T,H)$ and $u_0 \in V$ the problem above has a unique solution $u\in L^2(0,T,V)\cap H^1(0,T,H),$ is proved in [10] under the assumption that $a$ is symmetric and of bounded variation. The aim of this paper is to prove that the solutions of an approximate non-autonomous Cauchy problem in which $a$ is symmetric and piecewise affine are closed to the solutions of that governed by symmetric and of bounded variation form. In particular, this provide an alternative proof of the result in [10] on $L^2$-maximal regularity in $H.$, 12 pages

In this paper, we study properties of the topological space of composition operators acting on the space \({\mathcal{H}^{\infty}}\) of Dirichlet series. Especially, we show that there are two compact composition operators which are not in the same path component on \({\mathcal{H}^{\infty}}\). This is in sharp contrast with the classical case where all compact composition operators on \({H^{\infty}}\) of one variable or several variables lie in the same path component.

The paper deals with bounded non-selfadjoint operators having Hilbert–Schmidt imaginary Hermitian components. A perturbation bound for invariant subspaces is established. Our results can be considered as a particular generalization of the well-known Davis–Kahan sin θ-theorem for selfadjoint operators.

Let \({\mathbb{F}}\) be a finite field. The function field analog of Hermite’s theorem says that there are at most finitely many finite separable extensions of \({\mathbb{F}(T)}\) inside a fixed separable closure of \({\mathbb{F}(T)}\) whose discriminant divisors have bounded degree. In this paper we give a field theoretic proof of this result, inspired by a lemma of Faltings for comparing semisimple \({\ell }\)-adic Galois representations.

We consider the maximal regularity problem for non-autonomous evolution equations \begin{equation} \left\{ \begin{array}{rcl} u'(t) + A(t)\,u(t) &=& f(t), \ t \in (0, ��] u(0)&=&u_0. \end{array} \right. \end{equation} Each operator $A(t)$ is associated with a sesquilinear form $\mathfrak{a}(t)$ on a Hilbert space $H$. We assume that these forms all have the same domain $V$. It is proved in \cite{HO14} that if the forms have some regularity with respect to $t$ (e.g., piecewise $��$-H��lder continuous for some $��> 1/2$) then the above problem has maximal $L_p$--regularity for all $u_0 $ in the real-interpolation space $(H, D(A(0)))_{1-1/p,p}$. In this paper we prove that the regularity required there can be improved for a class of sesquilinear forms. The forms considered here are such that the difference $\mathfrak{a}(t;\cdot,\cdot) - \mathfrak{a}(s; \cdot,\cdot)$ is continuous on a larger space than the common domain $V$. We give three examples which illustrate our results., arXiv admin note: text overlap with arXiv:1402.1136

Let \({\Omega \subset \mathbb{R}^n}\) be a convex set. If \({u: \Omega \rightarrow \mathbb{R}}\) has mean 0, then we have the classical Poincare inequality $$\|u \|_{L^p} \leq c_p {\rm diam}(\Omega) \| \nabla u \|_{L^p}$$ with sharp constants \({c_2 = 1/\pi}\) (Payne and Weinberger, 1960) and c1 = 1/2 (Acosta and Duran, 2005) independent of the dimension. The sharp constants cp for 1 < p < 2 have recently been found by Valtorta (2012) and Ferone, Nitsch and Trombetti (2012). The purpose of this short paper is to prove a much stronger inequality in the endpoint L1: we combine results of Cianchi and Kannan, Lovasz and Simonovits to show that $$\left\|u\right\|_{L^{1}(\Omega)}\leq \frac{2}{\log{2}} M_{}(\Omega)\left\|\nabla u\right\|_{L^{1}(\Omega)},$$ where \({M(\Omega)}\) is the average distance between a point in \({\Omega}\) and the center of gravity of \({\Omega}\) . If \({\Omega}\) is a regular simplex in \({\mathbb{R}^n}\) , this yields an improvement by a factor of \({\sim \sqrt{n}}\) .

In this paper, we prove the Hardy’s and Cowling–Price’s uncertainty principles for the generalized Fourier transform associated to a Cherednik type operator on the real line.

The aim of this paper is to present a formula for the Gaussian curvature of an immersed surface in the Berger sphere \({\mathbb{S}_{\kappa,\tau}^3}\) which involves the contact angle \({\beta}\) . This allows us to conclude that, in the case of \({\kappa-4\tau^2 > 0}\) , the connected CMC compact surface M in this Berger sphere with sign-preserving contact angle must be a Hopf torus.

In this paper we prove a generalized version of a joint discrete universality theorem on the approximation of a collection of analytic functions by discrete shifts of Dirichlet L-functions.

We introduce a new class of unbounded model subdomains of $\mathbb{C}^2$ for the $\Box_b$ problem. Unlike previous finite type models, these domains need not be bounded by algebraic varieties. In this paper we obtain precise global estimates for the Carnot-Caratheodory metric induced on the boundary of such domains by the real and imaginary parts of the CR vector field.

Let $${\mathbb{F}_q}$$ be the finite field of characteristic p > 2 with q elements. Carlitz proposed the problem of finding an explicit formula for the number of solutions to the equation $$(x_1+ x_2+\cdots+x_n)^2=a\, x_1x_2\cdots x_n,$$ where $${a\in \mathbb{F}_q^*}$$ and n ≥ 3. By using the augmented degree matrix and Gauss sums, we consider the generalizations of the above equation and partially solve Carlitz’s problem. Moreover, the technique developed in this paper may be applied to other equations of the form $${h_1^\lambda=h_2}$$ with $${h_1, h_2 \in \mathbb{F}_q[x_1,\ldots,x_n]}$$ and $${\lambda \in \mathbb{N}}$$ .

In this paper we prove that Morita equivalences under particular conditions are compatible with affine cellular structures.

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry–Emery–Ricci curvature if the space of weighted L2 harmonic one-forms is non-trivial, then the weighted volume of the manifold is finite and the universal cover of the manifold splits isometrically as the product of the real line with a hypersurface.

A positive integer n is called a square-full number if p2 divides n whenever p is a prime divisor of n. In this paper we study the distribution of square-full numbers in arithmetic progressions by using the properties of Riemann zeta functions and Dirichlet L-functions.