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We prove a log-free zero density estimate for automorphic L -functions defined over a number field k. This work generalizes and sharpens the method of pseudo-characters and the large sieve used earlier by Kowalski and Michel. As applications, we demonstrate for a particular family of number fields of degree n over k (for any n) that an effective Chebotarev density theorem and a bound on ℓ -torsion in class groups hold for almost all fields in the family. [ABSTRACT FROM AUTHOR]

AbstractThis paper is devoted to develop the concepts of semi-c-periodicity, c-uniform recurrence and almost automorphy for functions defined on vertical strips in the complex plane, where c is a non-zero complex number. As an extension of the study performed for functions defined on the real axis, this work aims to investigate the main properties of these classes of functions and establish their connections with the more known class of c-almost periodic functions defined on vertical strips. In fact, we resolve an open problem which was raised in 2020 for the real case. Additionally, we also use this approach to introduce an asymptotic version of these new classes of functions. [ABSTRACT FROM AUTHOR]

We consider solutions of some two-dimensional fourth-order equation with a biharmonic operator and exponential nonlinearity of a counterpart of the classical Gauss–Bieberbach–Rademacher second-order equation, which was previously inspected by many authors in connection with problems of the geometry of surfaces with negative Gaussian curvature, rarefied gas dynamics, and the theory of automorphic functions. We obtain some conditions for the absence of a solution in a disk of sufficiently large radius and show that global solutions on the plane can exist only if the coefficient of nonlinearity decays at infinity at the rate at least . Otherwise the mean value of the solution on a circle of radius would tend to with exponential rate as . The Pokhozhaev–Mitidieri nonlinear capacity method, based on the choice of appropriate cutoff test functions, proves the impossibility of the existence of such global solution. Also, for the solutions in , periodic in all but variables, the absence of global solutions is obtained by similar methods when the nonlinearity coefficient decays at rate slower than . [ABSTRACT FROM AUTHOR]

In this paper, we are going to obtain some normal superchar- acter theories of a group of order 6n with the presentation special cases. We will also prove that the automorphic supercharacter theories of this group can be computed with the other methods. [ABSTRACT FROM AUTHOR]

This work is devoted to the consideration of the behavior of conformal quasi-harmonics that characterize the behavior of vortex domains within the framework of the conformal mapping of the Vaitkevichius-Volk function. A conformal analysis of the automorphic function of this conformal mapping was carried out and the pattern of existence of the connectivity of the function with the core of the lemniscate strip was revealed. The mechanism of formation of isogonal potential and isothermal vortex fields of function (1) on a circle of a Riemannian two-sheet plane was studied. A new characteristic is given to the complex values of the divergence and shear of these fields. [ABSTRACT FROM AUTHOR]

In this paper, we consider generalized Möbius functions associated with two types of L -functions: Rankin–Selberg L -functions of symmetric powers of distinct holomorphic cusp forms and L -functions derived from Maass cusp forms. We show that these generalized Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak's conjecture holds for these two types of Möbius functions. [ABSTRACT FROM AUTHOR]

This paper presents the conditions under which a coded digital communication system's performance may be improved using better signal sets design matched to groups. This improvement is directly related to the determination of the uniformization region of a hyperelliptic curve by using a Fuchsian differential equation and, consequently, the determination of the Fuchsian group and subgroup generators. The hyperelliptic curves of interest are given by y 2 = z n ± 1 , where n ∈ I N and n = 2 g + 1 or n = 2 g + 2 , and satisfy the following conditions: all the roots are distinct, they are symmetrically arranged (regular polygon) in the Poincaré disk, the juxtaposition of two such polygons gives the uniformization region. These conditions imply that the uniformization regions are regular polygons and that the associate Fuchsian group leads to regular tessellations from which arithmetic Fuchsian groups may be found. This work aims to answer and establish the conditions for the following questions: Can Fuchsian subgroups corresponding to two distinct hyperelliptic curves with the same degree and genus be isomorphic in D 2 as well as in H 2 ? What are the conditions under which an isometry acting on isomorphic Fuchsian subgroups in D 2 remain isomorphic in H 2 ? To show the isomorphism, we use the concept of conjugation. The first question's positive response is achieved for hyperelliptic curves given by y 2 = z n ± 1 . Finally, from the results established in D 2 and H 2 , if the Fuchsian subgroups are isomorphic by conjugation in D 2 , they remain isomorphic in H 2 by the action of an appropriate isometry as established in Theorem 3 and its generalization in Theorem 4. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to the contribution to giving the deep understanding for diffusion equations associated with the entire functions of order one in number theory. [ABSTRACT FROM AUTHOR]

The novel concept called neutrosophic set was launched to take care of indeterminate factors in real-life data. The hybrid model of neutrosophic set and soft set has been widely studied in different areas of algebra, especially in associative structures such as fields, groups, rings, and modules. In this current paper, the novel concept is further introduce to a non-associative structure termed Q-neutrosophic soft quasigroup (Q-NSG) and investigate its different algebraic properties of the quasigroups. We shown the conditions for the sets of a-level cut of Q-NSG to be subquasigroups, the condition for each set of subquasigroups of a quasigroup to be Q-level cut neutrosophic soft subquasigroup were established. It was shown that Q-NSG obeys alternative property and flexible law. In addition, We defined Q-neutrosophic soft loop and investigate some of its characteristics. In particular, it was shown that Q-neutrosophic soft loop obeys inverse, weak inverse and cross inverse properties. We established the condition for a Q-neutrosophic soft loop to obey antiautomorphic inverse, semi-automorphic inverse and super anti-automorphic inverse properties. The necessary and sufficient condition for Q-neutrosophic soft set under a loop (G, ◦,/, \) to be a Q-neutrosophic soft loop was also established. [ABSTRACT FROM AUTHOR]

Let |$\lambda _{\phi }(n)$| be the Fourier coefficients of a Hecke holomorphic or Hecke–Maass cusp form on |$\textrm{SL}_{2}(\mathbb Z)$| and |$f$| be any multiplicative function that satisfies two mild hypotheses. We establish a nontrivial upper bound for the correlation |$\sum _{n \leq X}f(n)\lambda _{\phi }(n+h)$| uniformly in |$0<|h|\leq X$|⁠. As applications, we consider some special cases, including |$f(n)=\lambda _{\pi }(n), \,\mu (n)\lambda _{\pi }(n)$| and any divisor-bounded multiplicative function. Here, |$\lambda _{\pi }(n)$| denotes the |$n$| -th Dirichlet coefficient of a |$\textrm{GL}_{m}$| automorphic |$L$| -function |$L(s,\pi)$| for an irreducible cuspidal automorphic representation |$\pi $|⁠ , and |$\mu (n)$| denotes the Möbius function. In particular, nontrivial savings are achieved for shifted convolution problems on |$\textrm{GL}_{m}\times \textrm{GL}_{2}\, (m\geq 4)$| and Hypothesis C of Iwaniec–Luo–Sarnak for the first time. [ABSTRACT FROM AUTHOR]

Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields.We establish various properties of these correspondences regarding functoriality for cyclic base change: For Z/pZ-extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For Z/pZ-extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along Z/pZ-extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann- Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from modular representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod p spherical Hecke algebras, in a joint appendix with Gus Lonergan. [ABSTRACT FROM AUTHOR]

We relate the study of Landau–Siegel zeros to the ranks of Jacobians J0(q)$J_0(q)$ of modular curves for large primes q$q$. By a conjecture of Brumer–Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level q$q$ have analytic rank ⩽1$\leqslant 1$. We show that either Landau–Siegel zeros do not exist, or that, for wide ranges of q$q$, almost all such newforms have analytic rank ⩽2$\leqslant 2$. In particular, in wide ranges, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes q$q$ in a wide range, we show that the rank of J0(q)$J_0(q)$ is asymptotically equal to the rank predicted by the Brumer–Murty conjecture. [ABSTRACT FROM AUTHOR]

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights k1 and k2 for the full modular group Γ = SL(2, Z), respectively. Denote by λf (n) and λg(n) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we consider the summatory function ... for x ≥ 2, where a, b ∈ Z and i, j ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

We provide a concrete characterization of the poly-analytic planar automorphic functions, a special class of non analytic planar automorphic functions with respect to the Appell–Humbert automorphy factor, arising as images of the holomorphic ones by means of the creation differential operator. This is closely connected to the spectral theory of the magnetic Laplacian on the complex plane. [ABSTRACT FROM AUTHOR]

A half-isomorphism f : G → K between multiplicative systems G and K is a bijection from G onto K such that f (a b) ∈ { f (a) f (b) , f (b) f (a) } for any a , b ∈ G . A half-isomorphism is trivial when it is either an isomorphism or an anti-isomorphism. Consider the class of automorphic loops such that the equation x ⋅ (x ⋅ y) = (y ⋅ x) ⋅ x is equivalent to x ⋅ y = y ⋅ x . Here we show that this class of loops includes automorphic loops of odd order and uniquely 2-divisible. Furthermore, we prove that every half-isomorphism between loops in that class is trivial. [ABSTRACT FROM AUTHOR]

Let n⩾1$n \geqslant 1$ be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature (1,2n−1)$(1, 2n-1)$. [ABSTRACT FROM AUTHOR]

In this paper, we study the functorial descent from self-contragredient cuspidal automorphic representations π of GL7(A) with LS (s; π; ^ ³ / having a pole at s=1 to the split exceptional group G2(A), using Fourier coefficients associated to two nilpotent orbits of E7. We show that one descent module is generic, and under suitable local conditions, it is cuspidal and π is a weak functorial lift of each of its irreducible summands. This establishes the first functorial descent involving the exotic exterior cube L-function. However, we show that the other descent module supports not only the nondegenerate Whittaker–Fourier integral on G2(A) but also every degenerate Whittaker– Fourier integral. Thus it is generic, but not cuspidal. [ABSTRACT FROM AUTHOR]

In this paper, we study the functorial descent from self-contragredient cuspidal automorphic representations π of GL7(A) with LS (s; π; ^ ³ / having a pole at s=1 to the split exceptional group G2(A), using Fourier coefficients associated to two nilpotent orbits of E7. We show that one descent module is generic, and under suitable local conditions, it is cuspidal and π is a weak functorial lift of each of its irreducible summands. This establishes the first functorial descent involving the exotic exterior cube L-function. However, we show that the other descent module supports not only the nondegenerate Whittaker–Fourier integral on G2(A) but also every degenerate Whittaker– Fourier integral. Thus it is generic, but not cuspidal. [ABSTRACT FROM AUTHOR]

In this paper, we consider recurrent behavior of bounded solutions for a functional integro-differential equation arising from heat conduction in materials with memory. Prior to the main results, we give a new version of composite theorem on measure pseudo almost automorphic functions involved in delay. Based on recently obtained results on the uniform exponential stability as well as contraction mapping principle, we prove some existence and uniqueness theorems on the recurrence of bounded mild solutions for the addressed equations with infinite delay. Finally, we finish this paper with an example on partial integro-differential equation which frequently comes to light in the study of heat conduction. [ABSTRACT FROM AUTHOR]

In this paper, we analyze various classes of Stepanov-p-almost periodic functions and Stepanov-p-almost automorphic functions ( p > 0 ). The class of Stepanov-p-almost periodic (automorphic) functions in norm ( p > 0 ) is also introduced and analyzed. Some structural results for the introduced classes of functions are clarified. We also provide several important theoretical examples, useful remarks and some new applications of Stepanov-p-almost periodic type functions to the abstract (impulsive) first-order differential inclusions and the abstract (impulsive) fractional differential inclusions. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study generalizations of the Caginalp phase-field system based on a thermomechanical theory involving two temperatures and a nonlinear coupling. In particular, we prove well-posedness results. More precisely, the existence of a pullback attractor for a nonautonomous parabolic of type Cahn-Hilliard phase-field system. The pullback attractor is a compact set, invariant with respect to the cocycle and which attracts the solutions in the neighborhood of minus infinity, consequently the attractor pullback (or attractor retrograde) exhibits a infinite fractal dimension. [ABSTRACT FROM AUTHOR]

In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated with even integer weight vector-valued automorphic forms of non-cocompact Fuchsian groups of the first kind. In the end, their L -functions and exponential sums have been discussed. [ABSTRACT FROM AUTHOR]

Summary: While Sherwin J. Carlquist (1930–2021) did not originate the concept of long-distance dispersal and its role in evolution — a major pillar in Darwin's theory (1859) — he almost single-handedly turned research on dispersal to insular habitats into an empirical and experimental research area. This contribution explains how and why this occurred based on Carlquist's own papers and personal account, and provides a brief assessment of the historical context of his research on long-distance dispersal. I end on a personal note; in 1981, when I was a graduate student, Carlquist participated in a symposium on 'Dispersal and Distribution' in Hamburg, and the paper he gave there on intercontinental dispersal greatly influenced my own work. [ABSTRACT FROM AUTHOR]

Let |$G$| be a reductive group over a local field |$F$| of characteristic zero. Let |$X$| be a |$G$| -space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on |$X$|⁠ , considered as a representation of |$G$|⁠. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying our theorems to symmetric pairs |$(G,H)$| we show that there exists an infinite-dimensional |$H$| -distinguished representation of |$G$| if and only if the real reductive group corresponding to the pair |$(G,H)$| is non-compact. For quasi-split |$G$| we also extend to the Archimedean case the theorem of Prasad stating that there exists a generic |$H$| -distinguished representation of |$G$| if and only if the real reductive group corresponding to the pair |$(G,H)$| is quasi-split. In the non-Archimedean case our result also gives rather sharp bounds on the wave-front sets of distinguished representations. Finally, we deduce a corollary on vanishing of period integrals of automorphic forms with certain Whittaker supports. This corollary, when combined with the restrictions on the Whittaker support of cuspidal automorphic representations proven by Gomez–Gourevitch–Sahi, implies many of the vanishing results on periods of automorphic forms proved by Ash–Ginzburg–Rallis. [ABSTRACT FROM AUTHOR]

In this paper, we establish a one-to-one correspondence between conjugacy classes of any Hecke group and irreducible systems of poles of rational period functions for automorphic integrals on the same group. We use this correspondence to construct irreducible systems of poles and to count poles. We characterize Hecke-conjugation and Hecke-symmetry for poles of rational period functions in terms of the transpose of matrices in conjugacy classes. We construct new rational period functions and families of rational period functions. [ABSTRACT FROM AUTHOR]

Recently in Dvurečenskij and Zahiri (J Appl Log IfCoLog J Log Appl 8:2365–2399, 2021b, J Appl Log IfCoLog J Log Appl 8:2401–2433, 2021c), new algebras, called weak pseudo EMV-algebras (wPEMV-algebras in short), were introduced. The authors do not assume the existence of a top element —they generalize MV-algebras, pseudo MV-algebras, and pseudo EMV-algebras. A g-state is defined on a unital wPEMV-algebra M as a mapping from M into the positive half-line of reals such that it preserves a partial addition + , and in a fixed strong unit, it takes the value 1. They form a Bauer simplex, and extremal points are exactly g-states whose kernel is a maximal and normal ideal. We show that extremal g-states generate all g-states, and it can happen that in some unital wPEMV-algebra, even commutative, there is no g-state. We present some conditions for existence of g-states and establish an integral representation of g-states. In addition, we give a topological characterization of the spaces of g-states and extremal g-states, respectively. Moreover, discrete g-states are investigated. [ABSTRACT FROM AUTHOR]

Let F (s) = ∑ n = 1 ∞ a n n s be a Dirichlet series in the axiomatically defined class # . The class # is known to contain the extended Selberg class # , as well as all the L -functions of automorphic forms on GL n / K , where K is a number field. Let d be the degree of F (s). We show that ∑ n < X | a n | = Ω (X 1 2 + 1 2 d ) , and hence, that the abscissa of absolute convergence of σ a of F (s) must satisfy σ a ≥ 1 / 2 + 1 / 2 d. [ABSTRACT FROM AUTHOR]

A function (meromorphic or harmonic) from the hyperbolic disk of the complex plane to the Riemann sphere is normal if its dilatation is bounded. In this paper, normal families of meromorphic functions is studied in view of sharing values by their differential monomials. Moreover, several properties of normal harmonic mappings are also studied in view of certain general settings. [ABSTRACT FROM AUTHOR]

Let F be a non-Archimedean local field and G be (the F-points of) a connected reductive group defined over F. Fix U_0 to be the unipotent radical of a minimal parabolic subgroup P_0 of G, and \psi :U_0\rightarrow \mathbb {C}^\times be a non-degenerate character of U_0. Let P=MU\supseteq P_0 be a standard parabolic subgroup of G so that the restriction \psi _M of \psi to M\cap U_0 is non-degenerate. We denote by \mathcal {W}(G,\psi) the space of smooth \psi-Whittaker functions on G and by \mathcal {W}_c(G,\psi) its G-stable subspace consisting of functions with compact support modulo U_0. In this situation Bushnell and Henniart identified \mathcal {W}_c(M,\psi _M^{-1}) to the Jacquet module of \mathcal {W}_c(G,\psi ^{-1}) with respect to P^- (Bushnell and Henniart [Amer. J. Math. 125 (2003), pp. 513–547]). On the other hand Delorme defined a constant term map from \mathcal {W}(G,\psi) to \mathcal {W}(M,\psi _M) which descends to the Jacquet module of \mathcal {W}(G,\psi) with respect to P (Delorme [Trans. Amer. Math. Soc. 362 (2010), pp. 933–955]). We show (as we surprisingly could not find a proof of this statement in the literature) that the descent of Delorme's constant term map is the dual map of the isomorphism of Bushnell and Henniart, in particular the constant term map is surjective. We also show that the constant term map coincides on admissible submodules of \mathcal {W}(G,\psi) with the inflation of the "germ map" defined by Lapid and Mao [Represent. Theory 13 (2009), pp. 63–81] following earlier works of Casselman and Shalika [Compositio Math. 41 (1980), pp. 207–231]. From these results we derive a simple proof of a slight generalization of a theorem of Delorme and Sakellaridis–Venkatesh ([Ast'erisque 396 (2017), pp. viii+360] for quasi-split G) on irreducible discrete series with a generalized Whittaker model to the setting of admissible representations with a central character under the split component of G, and similar statements in the cuspidal case (also generalizing a result of Delorme) and in the tempered case. We also show that the germ map of Lapid and Mao is injective, answering one of their questions. Finally using a result of Vignéras [ Contributions to automorphic forms, geometry, and number theory , Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 773–801] and recent results of Dat, Helm, Kurinczuk, and Moss [ Finiteness for hecke algebras of p-adic groups , arXiv: 2203.04929 , 2022], we show in the context \ell-adic representations that the asymptotic expansion of Lapid and Mao can be chosen to be integral for functions in integral G-submodules of \mathcal {W}(\pi,\psi) of finite length. [ABSTRACT FROM AUTHOR]

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let |$\Gamma $| be a finite index subgroup of |$\textrm {SL}(2,\mathbb Z)$| with an action on a complex simple Lie algebra |$\mathfrak g$|⁠ , which can be extended to |$\textrm {SL}(2,{\mathbb {C}})$|⁠. We show that the Lie algebra of the corresponding |$\mathfrak {g}$| -valued modular forms is isomorphic to the extension of |$\mathfrak {g}$| over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups |$\Gamma (N), \, N\leq 6$|⁠ , is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and uniqueness of solutions of linear and semilinear second–order equations involving time scales. To obtain such results, we make use of exponential dichotomy and fixed point results. Also, we present some examples and applications to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, to begin with, we introduce the concept of Besicovitch almost periodic stochastic processes in distribution sense and study the relationship between it and the concept of Besicovitch almost periodic stochastic processes in p-th mean sense. In addition, we take a class of Clifford-valued stochastic neural networks with time-varying delays as an example to investigate the existence and uniqueness of Besicovitch almost periodic solutions in distribution sense of this class of neural networks by using Banach fixed point theorem and a variant of Gronwall lemma. Moreover, we study the global exponential stability of this unique Besicovitch almost periodic solution in distribution sense by using inequality techniques. Finally, we give an example to illustrate our results. The results of this paper are completely new. [ABSTRACT FROM AUTHOR]

Weyl almost automorphy is a natural generalization of Bochner almost automorphy and Stepanov almost automorphy. However, the space composed of Weyl almost automorphic functions is not a Banach space. Therefore, the results of the existence of Weyl almost automorphic solutions of differential equations are few, and the results of the existence of Weyl almost automorphic solutions of difference equations are rare. Since the study of dynamic equations on time scales can unify the study of differential equations and difference equations. Therefore, in this paper, we first propose a concept of Weyl almost automorphic functions on time scales and then take the Clifford‐valued shunt inhibitory cellular neural networks with time‐varying delays on time scales as an example of dynamic equations on time scales to study the existence and global exponential stability of their Weyl almost automorphic solutions. We also give a numerical example to illustrate the feasibility of our results. [ABSTRACT FROM AUTHOR]

This paper’s main motivation is to study the notion of weighted pseudo almost automorphic (WPAA) functions and establish the existence results of piecewise continuous mild solution of fractional order integro-differential equation with instantaneous impulses. The usual WPAA functions may not work since the solution of impulsive differential equations may not be continuous. Thus in order to give a broader spectrum, we introduce this concept. We establish main results by using the Banach contraction mapping principle and Sadovskii’s fixed point theorem. An example is shown to exhibit our analytic findings. [ABSTRACT FROM AUTHOR]

This article establishes a reduction principle for partial functional differential equation without compactness of the semigroup generated by the linear part. Under conditions more general than the compactness of the C0-semigroup generated by the linear part, we establish the quasi-compactness of the C0-semigroup associated to the linear part of the partial functional differential equation. This result allows as to construct a reduced system that is posed by an ordinary differential equation posed in a finite dimensional space. Through this result we study the existence of almost automorphic and almost periodic solutions for partial functional differential equations. For illustration, we study a transport model. [ABSTRACT FROM AUTHOR]

We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard-Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank 7 motive whose Galois group is of type G2 in the cohomology of Siegel modular varieties of genus 3. Under some additional local hypotheses we also show automorphic multiplicity 1 as well as meromorphic continuation of the spin L-functions. [ABSTRACT FROM AUTHOR]

We define a set of holomorphic functions in terms of the Hauptmodul of a quotient Riemann surface and prove that these functions are holomorphic on the upper half-plane. It is also shown that these functions are automorphic forms of weight k with respect to a Fuchsian group. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the existence and uniqueness of μ-pseudo almost periodic (resp. μ-pseudo compact almost automorphic) weak solutions for the following partial functional differential inclusion: x ′ (t) + A x (t) ∋ f (t , x t) f o r t ∈ R , where A : H ⟶ 2 H is a strongly maximal monotone operator on a real Hilbert space H , f : R × C ⟶ H is a Stepanov μ-pseudo almost periodic (resp. μ-pseudo compact almost automorphic) function of class r, C = C ([ − r , 0 ] , H) is the Banach space of all continuous functions from [ − r , 0 ] to H and the history function x t is defined by x t (θ) = x (t + θ) for θ ∈ [ − r , 0 ]. Two examples are given for parabolic and subdifferential systems. [ABSTRACT FROM AUTHOR]

We give constraints on the existence of (χ , b) -factors in the global A -parameter of a genuine cuspidal automorphic representation σ of a metaplectic group in terms of the invariant, lowest occurrence index, of theta lifts to odd orthogonal groups. We also give a refined result that relates the invariant, first occurrence index, to non-vanishing of period integrals of residues of Eisenstein series associated to the cuspidal datum χ ⊗ σ. This complements our previous results for symplectic groups. [ABSTRACT FROM AUTHOR]

Let Λ π (n) denote the n th coefficient in the Dirichlet series expansion of the logarithmic derivative of L (s , π) associated with an automorphic irreducible cuspidal representation of GL m over Q . In this paper, for all α of irrational type 1 lying in the interval [ 0 , 1 ] , we investigate the best possible estimate for the sum ∑ n ≤ x Λ π (n) e (n α) under a certain assumption. And we consider the metric result on the exponential sum involving automorphic L -functions without any assumptions. Let Λ (n) be the von Mangoldt function. Then as an application, for ε > 0 and all 0 < α < 1 in a set of full Lebesgue measure (depending on π ), we obtain ∑ n ≤ x Λ (n) λ π (n) e (n α) = O (x 5 6 + ε) . [ABSTRACT FROM AUTHOR]

Let D be a definite quaternion algebra over Q and O an Eichler order in D of square-free level. We study distribution of the toric periods of algebraic modular forms of level O . We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on O , we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld's conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic L-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients { a (n) } n of weight 3 2 modular forms, where n ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them. [ABSTRACT FROM AUTHOR]

In this paper, we consider quaternion-valued fuzzy recurrent neural networks with time-varying delays on time scales. Different from the previous literature, we use a direct method to obtain our theoretical results to avoid decomposing the model into real-valued or complexvalued systems. Then, we obtain some sufficient conditions on the existence, uniqueness, and Sp-global exponential stability of weighted Stepanov-like pseudo almost periodic solution on time scales of the considered model by applying inequality analysis techniques on time scales, a fixed point theorem, and composition theorem, and by constructing an appropriate Lyapunov function. At the end of this work, we give a numerical example and simulations to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze several new classes of multi-dimensional almost automorphic functions which generalize the classical one of Bochner. We develop the basic theory for the introduced classes, investigating the themes like composition principles, convolution invariance and the invariance under the actions of convolution products. We present several illustrative examples and applications to the abstract Volterra integro-differential equations and partial differential equations, providing also a mini appendix about almost automorphic functions on semi-topological groups. [ABSTRACT FROM AUTHOR]

We consider a class of neutral type Clifford-valued cellular neural networks with discrete delays and infinitely distributed delays. Unlike most previous studies on Clifford-valued neural networks, we assume that the self feedback connection weights of the networks are Clifford numbers rather than real numbers. In order to study the existence of $ (\mu, \nu) $-pseudo compact almost automorphic solutions of the networks, we prove a composition theorem of $ (\mu, \nu) $-pseudo compact almost automorphic functions with varying deviating arguments. Based on this composition theorem and the fixed point theorem, we establish the existence and the uniqueness of $ (\mu, \nu) $-pseudo compact almost automorphic solutions of the networks. Then, we investigate the global exponential stability of the solution by employing differential inequality techniques. Finally, we give an example to illustrate our theoretical finding. Our results obtained in this paper are completely new, even when the considered networks are degenerated into real-valued, complex-valued or quaternion-valued networks. [ABSTRACT FROM AUTHOR]

We prove that a cuspidal automorphic representation of |$\mathrm{GL}(3)$| over any number field is determined by the quadratic twists of its central value. In the case π is not a Gelbart–Jacquet lift, the result is conditional on the analytic behavior of a certain Euler product. We deduce the nonvanishing of infinitely many quadratic twists of central values. This generalizes a result of Chinta and Diaconu that was valid only over Q and explored only for Gelbart–Jacquet lifts. [ABSTRACT FROM AUTHOR]

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We prove a Galois-equivariant algebraicity result for the ratios of successive critical values of |$L$| -functions for |${\textrm GL}(n)/F,$| where |$F$| is a totally imaginary quadratic extension of a totally real number field |$F^+$|⁠. The proof uses (1) results of Arthur and Clozel on automorphic induction from |${\textrm GL}(n)/F$| to |${\textrm GL}(2n)/F^+$|⁠ , (2) results of my work with Harder on ratios of critical values for |$L$| -functions of |${\textrm GL}(2n)/F^+$|⁠ , and (3) period relations amongst various automorphic and cohomological periods for |${\textrm GL}(2n)/F^+$| using my work with Shahidi. The reciprocity law inherent in the algebraicity result is exactly as predicted by Deligne's conjecture on the special values of motivic |$L$| -functions. [ABSTRACT FROM AUTHOR]