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The main aim of this work is to study a second-order nonlinear differential equation with mixed delay and time-varying coefficients. By employing the fixed-point theorem and some properties of the doubly measure pseudo almost automorphic functions, we prove the existence, uniqueness and global exponential stability of doubly measure pseudo-almost automorphic solution to the second-order delay differential equation. Our approach generalizes the classical results on weighted pseudo almost automorphic functions. Finally, a numerical example is provided to illustrate the effectiveness and feasibility of the obtained results. [ABSTRACT FROM AUTHOR]

Let F be a number field, and n\geq 3 be an integer. In this paper we give an effective procedure which (1) determines whether two given quadratic lattices on F^n are isometric or not, and (2) produces an invertible linear transformation realizing the isometry provided the two given lattices are isometric. A key ingredient in our approach is a search bound for the equivalence of two given quadratic forms over number fields which we prove using methods from algebraic groups, homogeneous dynamics and spectral theory of automorphic forms. [ABSTRACT FROM AUTHOR]

We investigate the Cohen-Grosberg differential equations with mixed delays and time-varying coefficient: Several useful results on the functional space of such functions like completeness and composition theorems are established. By using the fixed-point theorem and some properties of the doubly measure pseudo almost automorphic functions, a set of sufficient criteria are established to ensure the existence, uniqueness and global exponential stability of a (μ, ν)-pseudo almost automorphic solution. The theory of this work generalizes the classical results on weighted pseudo almost automorphic functions. Finally, a numerical example is provided to illustrate the validity of the proposed theoretical results. [ABSTRACT FROM AUTHOR]

Let A be the ring of adeles of a number field F. Given a self-dual irreducible, automorphic, cuspidal representation τ of GL n (A) , with a trivial central character, we construct its full inverse image under the weak Langlands functorial lift from the appropriate split classical group G. We do this by a new automorphic descent method, namely the double descent. This method is derived from the recent generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan [CFGK17] , which represent the standard L -functions for G × GL n. Our results are valid also for double covers of symplectic groups. [ABSTRACT FROM AUTHOR]

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on Γ 0 (N) in the case where N is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O'Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of "square root cancelation", while in another case we find more cancelation. [ABSTRACT FROM AUTHOR]

In this paper, using the subvariant functional method due to Favard [Favard J. Sur les équations différentielles linéaires á coefficients presque-périodiques. ActaMathematica. 1928;51(1):31–81.], we prove the existence and uniqueness of compact almost automorphic solutions for a class of semilinear evolution equations in Banach spaces provided the existence of at least one bounded solution on the right half line. More specifically, we improve the assumptions in [Cieutat P, Ezzinbi K. Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities. J Funct Anal. 2011;260(9):2598–2634.], we show that the almost automorphy of the coefficients in a weaker sense (Stepanov almost automorphy of order 1 ≤ p < ∞) is enough to obtain solutions that are almost automorphic in a strong sense (Bochner almost automorphy). For that purpose we distinguish two cases, p = 1 and p > 1. The main difficulty in this work, is to prove the existence of at least one solution with relatively compact range while the forcing term is not necessarily bounded. Moreover, we propose to study a large class of reaction-diffusion problems with unbounded forcing terms. [ABSTRACT FROM AUTHOR]

We study p -adic families of cohomological automorphic forms for GL (2) over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. We show how to computationally determine when this is not the case and establish concrete examples of families interpolating only finitely many Bianchi modular forms. [ABSTRACT FROM AUTHOR]

The paper is dedicated to the study of problem of Poisson stability (in particular, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, Levitan almost periodicity, pseudo-periodicity, almost recurrence in the sense of Bebutov, recurrence in the sense of Birkhoff, pseudo recurrence and Poisson stability) and asymptotically Poisson stability of motions of monotone non-autonomous difference equations which have a strong monotone first integral. We solve this problem in the framework of general non-autonomous dynamical systems with discrete time. [ABSTRACT FROM AUTHOR]

We prove the modularity of most reducible, odd representations ρ ¯ : Γ Q → GL 2 (k) with k a finite field of characteristic an odd prime p. This is an analogue of Serre's celebrated modularity conjecture (which concerned irreducible, odd representations ρ ¯ : Γ Q → GL 2 (k) ) for reducible, odd representations. Our proof lifts ρ ¯ to an irreducible geometric p-adic representation ρ which is known to arise from a newform by results of Skinner–Wiles and Pan. We likewise prove automorphy of many reducible representations ρ ¯ : Γ F → GL n (k) when F is a global function field of characteristic different from p, by establishing a p-adic lifting theorem and invoking the work of L. Lafforgue. Crucially, in both cases we show that the actual representation ρ ¯ , rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation. Our main theorem establishes a geometric lifting result for mod p representations ρ ¯ : Γ F → G (k) of Galois groups of global fields F, valued in reductive groups G(k), and assumed to be odd when F is a number field. Thus we find that lifting theorems, combined with automorphy lifting results pioneered by Wiles in the number field case and the results in the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod p Galois representations both in reducible and irreducible cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze the notions of -almost periodicity and Stepanov -almost periodicity for functions with values in complex Banach spaces. In order to do that, we use the recently introduced notions of lower and upper (Banach) -densities. We also analyze uniformly recurrent functions, generalized almost automorphic functions and apply our results in the qualitative analysis of solutions of inhomogeneous abstract integro-differential inclusions. We present plenty of illustrative examples, results of independent interest, questions and unsolved problems. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the link between evolution algebras and (pseudo)digraphs. We study some theoretical properties about this association and determine the properties of the (pseudo)digraphs associated with each type of evolution algebras. We also analyze the isomorphism classes for each configuration associated with these algebras providing a new method to classify them, and we compare our results with the current classifications of two‐ and three‐dimensional evolution algebras. In order to complement the theoretical study, we have designed and performed the implementation of an algorithm, which constructs and draws the (pseudo)digraph associated with a given evolution algebra and another procedure to study the solvability of a given evolution algebra. [ABSTRACT FROM AUTHOR]

In this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension E/F at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla--Rapoport conjecture for exotic smooth Rapoport-Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomology of integral models of unitary Shimura varieties with Drinfeld level structures. [ABSTRACT FROM AUTHOR]

Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional (2D) hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in an experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps correspond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact. [ABSTRACT FROM AUTHOR]

We define piecewise continuous almost automorphic (p.c.a.a.) functions in the manners of Bochner, Bohr and Levitan, respectively, to describe almost automorphic motions in impulsive systems, and prove that with certain prefixed possible discontinuities they are equivalent to quasi-uniformly continuous Stepanov almost automorphic ones. Spatially almost automorphic sets on the line, which serve as suitable objects containing discontinuities of p.c.a.a. functions, are characterized in the manners of Bochner, Bohr and Levitan, respectively, and shown to be equivalent. Two Favard's theorems are established to illuminate the importance and convenience of p.c.a.a. functions in the study of almost periodically forced impulsive systems. [ABSTRACT FROM AUTHOR]

Let \pi be a unitary cuspidal automorphic representation of \mathrm {GL}_n over a number field, and let \widetilde {\pi } be contragredient to \pi. We prove effective upper and lower bounds of the correct order in the short interval prime number theorem for the Rankin–Selberg L-function L(s,\pi \times \widetilde {\pi }), extending the work of Hoheisel and Linnik. Along the way, we prove for the first time that L(s,\pi \times \widetilde {\pi }) has an unconditional standard zero-free region apart from a possible Landau–Siegel zero. [ABSTRACT FROM AUTHOR]

We review and reformulate old and prove new results about the triad , which provides a universal generalization of the classical automorphic triad. The leading P or p in the universal setting stands for piecewise, and the group PPSL 2 (Z) plays at once the role of universal modular group, universal mapping class group, Thompson group T and Ptolemy group. In particular, we construct and study new framed holographic coordinates on the universal Teichmüller space and its symmetry group PPSL 2 (R) , the group of piecewise PSL 2 (R) homeomorphisms of the circle with finitely many pieces, which is dense in the group of orientation-preserving homeomorphisms of the circle. We produce a new basis of its Lie algebra p p s l 2 (R) and compute the structure constants of the Lie bracket in this basis. We define a central extension of p p s l 2 (R) and compare it with the Weil-Petersson form. Finally, we construct a PPSL 2 (Z) -invariant 1-form on the universal Teichmüller space formally as the Maurer-Cartan form of p p s l 2 (R) , which suggests the full program for developing the theory of automorphic functions for the universal triad which is analogous, as much as possible, to the classical triad. In the last section we discuss the representation theory of the Lie algebra p p s l 2 (R) and then pursue the universal analogy for the invariant 1-form E 2 (z) d z , which gives rise to the spin 1 representation of p s l 2 (R) extended by the trivial representation. We conjecture that the corresponding automorphic representation of p p s l 2 (R) yields the bosonic CFT 2 . Relaxing the automorphic condition from PSL 2 (Z) to its commutant allows the increase of the space of 1-forms six-fold additively in the classical case and twelve-fold multiplicatively in our universal case. This leads to our ultimate conjecture that we can realize the Monster CFT 2 via the automorphic representation for the universal triad. This conjecture is also bolstered by the links of both the universal Teichmüller and the Monster CFT 2 theories to the three-dimensional quantum gravity. [ABSTRACT FROM AUTHOR]

Let Π be a cohomological irreducible cuspidal automorphic representation of GL 2 (A F) with central character ω Π over a totally real number field F. In this paper, we prove the algebraicity of the near central non-critical value of the symmetric fourth L -function of Π twisted by ω Π − 2. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of Π and the top degree Whittaker period of the Gelbart–Jacquet lift Sym 2 Π of Π. [ABSTRACT FROM AUTHOR]

We construct an Euler system for Galois representations associated to cohomological cuspidal automorphic representations of GSp4, using the pushforwards of Eisenstein classes for GL2 x GL2. [ABSTRACT FROM AUTHOR]

In this paper we give a description of the structure of compact global attractor (Levinson center) for monotone Bohr/Levitan almost periodic dynamical system x' = f(t; x) (*) with the strictly monotone first integral. It is shown that Levinson center of equation (*) consists of the Bohr/Levitan almost periodic (respectively, almost automorphic, recurrent or Poisson stable) solutions. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We also give some applications of theses results to different classes of differential/difference equations. [ABSTRACT FROM AUTHOR]

We focus on the presence of almost automorphy in strongly monotone skew‐product semiflows on Banach spaces. Under the C1$C^1$‐smoothness assumption, it is shown that any linearly stable minimal set must be almost automorphic. This extends the celebrated result of Shen and Yi [Mem. Amer. Math. Soc. 136(1998), No. 647] for the classical C1,α$C^{1,\alpha }$‐smooth systems. Based on this, one can reduce the regularity of the almost periodically forced differential equations and obtain the almost automorphic phenomena in a wider range. [ABSTRACT FROM AUTHOR]

We construct Hida and Coleman theories for the degree 0 and 1 cohomology of automorphic line bundles on the modular curve, and we define a p-adic duality pairing between the theories in degree 0 and 1. [ABSTRACT FROM AUTHOR]

In this paper, we show that for almost every m > 0 , the solution to the semi-linear Klein–Gordon equation associated with the harmonic oscillator, with small initial data, exists over a longer time interval than the one given by local existence theory, using the normal form method. A similar result for the quadratic wave equation is also obtained. [ABSTRACT FROM AUTHOR]

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), whereG=Sp4(R) andK is itsmaximal compact subgroup. In particular, we determineK-types and composition series, andwrite down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowestweight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegelmodular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we areable toprove astructure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight detl symm with respect to an arbitrary congruence subgroup of Sp4(Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight detl' symm' with (l',m') varying over a certain set. The structure theorem for the space of all modular forms is similar, except thatwemay nowhave an additional component coming from certain nearly holomorphic forms of weight det³ symm' that cannot be obtained from holomorphic forms. As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction. [ABSTRACT FROM AUTHOR]

This paper is devoted to exploring the translation invariance and convolution invariance of doubly weighted pseudo almost automorphic stochastic processes with impulses on time scales. Based on these results, the existence and uniqueness of the doubly weighted pseudo almost automorphic solutions to a class of stochastic nonlinear impulsive equations on time scales are obtained by taking advantage of a new approach, which enrich the dynamics of doubly weighted pseudo almost automorphic stochastic processes. Finally, an example is researched to illustrate our conclusions. [ABSTRACT FROM AUTHOR]

In this manuscript, we studied a class of delayed Fuzzy Genetic Regulatory Networks (FGRNs) with Stepanov-like weighted pseudo almost automorphic coefficients. New criteria for the existence, uniqueness and global exponential stability of its weighted pseudo almost automorphic solution are established. Our approach is based on Banach fixed point theorem and novel analysis techniques. Moreover, a numerical example is given to illustrate the validity of the obtained results. [ABSTRACT FROM AUTHOR]

The τ -Keiper/Li coefficients attached to a function F are closely related to its zero-free regions. However, the absence of a closed formula for calculating these coefficients makes them challenging to use. Motivated by Voros approach, we introduce the discretized τ -Keiper/Li coefficients. A finite sum representation derived for these coefficients is useful for numerical calculations. Representation in terms of zeros of the corresponding function is basis for analytic considerations. We prove that the violation of τ / 2 -generalized Riemann hypothesis implies oscillations of corresponding discretized τ -Li coefficients with power-growing amplitudes. Results are obtained for the class S ♯ ♭ (σ 0 , σ 1) , which contains the Selberg class, the class of all automorphic L-functions, the Rankin–Selberg L-functions, as well as products of suitable shifts of those functions. [ABSTRACT FROM AUTHOR]

Let G be a reductive group over a number field F, which is split at a finite place 𝔭 of F, and let π be a cuspidal automorphic representation of G, which is cohomological with respect to the trivial coefficient system and Steinberg at 𝔭. We use the cohomology of 𝔭-arithmetic subgroups of G to attach automorphic ℒ -invariants to π. This generalizes a construction of Darmon (respectively Spieß), who considered the case G = GL2 over the rationals (respectively over a totally real number field). These ℒ-invariants depend a priori on a choice of degree of cohomology, in which the representation π occurs. We show that they are independent of this choice provided that the π-isotypic part of cohomology is cyclic over Venkatesh's derived Hecke algebra. Further, we show that automorphic ℒ-invariants can be detected by completed cohomology. Combined with a local-global compatibility result of Ding it follows that for certain representations of definite unitary groups the automorphic ℒ-invariants are equal to the Fontaine–Mazur ℒ-invariants of the associated Galois representation. [ABSTRACT FROM AUTHOR]

In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields. [ABSTRACT FROM AUTHOR]

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen. [ABSTRACT FROM AUTHOR]

This article's main objective is to establish the measure pseudo-almost automorphic solution of an integro-differential equation with impulses. We develop the existence results based on the Banach contraction principle mapping and Krasnoselskii and Krasnoselskii–Schaefer type fixed point theorems. Finally, some examples are given to illustrate the significance of our theoretical findings. [ABSTRACT FROM AUTHOR]

In this article, we investigated a class of genetic regulatory networks (GRNs) with time-varying delays and Stepanov-like weighted pseudo-almost automorphic coefficients. By Banach fixed point theorem, we firstly established the existence results of the weighted pseudo-almost automorphic solutions for the proposed GRNs. Secondly, by using suitable Lyapunov functional and mathematical analysis skills, new criteria are derived for ensuring the global exponential stability of the solutions. Finally, a numerical example and remarks are provided to substantiate the efficiency of obtained results. The derived results of this article are new and complement many earlier works and the ideas of this work can be applied to investigate other similar systems. [ABSTRACT FROM AUTHOR]

Let k / F p denote a finite field. For any split connected reductive group G/W(k) and a CM number field F, we deform nearly ordinary Galois representations ρ ¯ : Gal (F ¯ / F) → G (k) to analytic families X ρ ¯ of Galois representations Γ F → G (Q ¯ p) lifting ρ ¯ such that the set of points of X ρ ¯ which are geometric (in the sense of the Fontaine–Mazur conjecture) with parallel Hodge–Tate weights is contained in an analytic subvariety of X ρ ¯ with positive codimension. Thus, the set of points in X ρ ¯ which could (conjecturally) be associated to automorphic forms is sparse. This generalizes a result of Calegari and Mazur for F / Q quadratic imaginary and G = GL 2 . The sparsity of automorphic points for a CM field F contrasts with the situation when F is a totally real field, where automorphic points are often provably dense. [ABSTRACT FROM AUTHOR]

In this paper, we establish a very flexible and explicit Voronoï summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic L -functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural p -adic analogue to the well-studied t -aspect. [ABSTRACT FROM AUTHOR]

The cohomology H*(Γ, E) of a torsion-free arithmetic subgroup Γ of the special linear ℚ-group 𝖦 = SLn may be interpreted in terms of the automorphic spectrum of Γ. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes {𝖯} of associate proper parabolic ℚ-subgroups of 𝖦 {\mathsf{G}}. Each summand H{P}*(Γ, E) is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in {𝖯}. The cohomology H*{P}(Γ, E) vanishes above the degree given by the cohomological dimension cd(Γ) = 1/2n(n − 1). We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes {𝖯} for which the corresponding summand H{𝖯}cd (Γ) (Γ, E) vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span Hcd(Γ){𝖰}(Γ,C). Finally, in the case of a principal congruence subgroup Γ(q), q = pν > 5, p ≥ 3 a prime, we give lower bounds for the size of these spaces. In addition, for certain associate classes {𝖰 }, there is a precise formula for their dimension. [ABSTRACT FROM AUTHOR]

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We discuss mock automorphic forms from the point of view of representation theory, that is, mock automorphic forms obtained from weak harmonic Maaß forms giving rise to nontrivial -cohomology. We consider the possibility of replacing the 'holomorphic' condition with a "cohomological" one when generalizing to general reductive groups. Such a candidate for replacement allows for growing Fourier coefficients, in contrast to automorphic forms under the Miatello-Wallach conjecture. In the second part, we provide an overview of the connection with BPS black hole counts as a physical motivation for studying mock automorphic forms. [ABSTRACT FROM AUTHOR]

This article deals with the concept of measure pseudo almost automorphic functions with impulsive effects. Then, based on systematic study associated with impulsive condition, the existence and uniqueness theorem is established. As far as we know, this is the first article to analyze such type of solutions with impulses. The Krasnoselskii's fixed point theorem is considered as a working tool to prove the main results. Also, an example is given to illustrate the significance of our findings. [ABSTRACT FROM AUTHOR]

This article deals with idea of piecewise stepanov-like almost automorphic solution to fractional Fredholm-Volterra integro differential equation with impulsive condition. First we demonstrate existence of such solution by using fixed point theorem and analyse the stability of the solution. An example is provided to outline the thought developed on this work. [ABSTRACT FROM AUTHOR]

Let G be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of G, subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of \operatorname {Bun}_G. It is independent of the work of Lafforgue on the global Langlands correspondence. [ABSTRACT FROM AUTHOR]

In this paper, a recurrent neural network with mixed delays which plays an important role is considered. We are concerned with the existence, uniqueness and global exponential stability of the doubly measure pseudo almost automorphic solutions. First, we establish results that are interesting on the functional space of such functions like composition theorem. Second, by employing the fixed-point theorem and some properties of the measure pseudo almost automorphic functions, some sufficient conditions for the existence, uniqueness and global exponential stability of the doubly measure pseudo almost automorphic solutions have been established. Our results obtained in this paper are new. Finally, two commonly used numerical examples are given to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let p be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L‐function of the base‐change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch–Kato conjecture predicts that this should force the existence of non‐trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Salazar and the second author to construct certain p‐adic Abel–Jacobi maps, which we use to propose a construction of such classes via Stark–Heegner cycles. This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms. [ABSTRACT FROM AUTHOR]

In this paper, we propose the concept of a weighted pseudo δ-almost automorphic function under the matched space for time scales and we present some properties. Also, we obtain sufficient conditions for the existence of weighted pseudo δ-almost automorphic mild solutions to a class of semilinear dynamic equations under the matched spaces for time scales. [ABSTRACT FROM AUTHOR]

We construct p -adic L -functions associated with triples of finite slope p -adic families of quaternionic automorphic eigenforms over totally real fields on Shimura curves. These results generalize a previous construction, joint work with D.Barrera, performed in the ordinary setting. [ABSTRACT FROM AUTHOR]

In this article we investigate the global existence as well as long-term dynamics for a wide class of lattice plate equations on the entire integer set with nonlinear damping driven by infinite-dimensional nonlinear noise. The well-posedness of the system is established for a class of nonlinear drift functions of polynomial growth of arbitrary order as well as locally Lipschitz continuous diffusion functions depending on time. Both existence and uniqueness of weak pullback mean random attractors are established for the non-autonomous system when the growth rate of the drift function is almost linear. In addition, the existence of invariant measures for the autonomous system is also established in ℓ 2 × ℓ 2 when the growth rate of the drift function is superlinear. The main difficulty of deriving the tightness of a family of distribution laws of the solutions is surmounted in light of the idea of uniform tail-estimates on the solutions developed by Wang (Phys D 128:41–52, 1999). [ABSTRACT FROM AUTHOR]

Under consideration is the class of nonlinear systems of nonautonomous differential equations of neutral type with a variable delay that can be unbounded. Using a Lyapunov–Krasovskii functional, we establish some estimates of solutions that allow us to conclude whether the solutions are stable. In the case of exponential and asymptotic stability, we estimate the attraction domains and the rate of stabilization of solutions at infinity. [ABSTRACT FROM AUTHOR]

We study the dynamical properties of irregular model sets and show that the translation action on their hull always admits an infinite independence set. The dynamics can therefore not be tame and the topological sequence entropy is strictly positive. Extending the proof to a more general setting, we further obtain that tame implies regular for almost automorphic group actions on compact spaces. In the converse direction, we show that even in the restrictive case of Euclidean cut and project schemes irregular model sets may be uniquely ergodic and have zero topological entropy. This provides negative answers to questions by Schlottmann and Moody in the Euclidean setting. [ABSTRACT FROM AUTHOR]

Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F0. When the residue characteristic of F0 is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U(2,1)(F/ F0). [ABSTRACT FROM AUTHOR]

We prove several important results concerning existence and uniqueness of pseudo almost automorphic (paa) solutions with measure for integro-differential equations with reflection. We use the properties of almost automorphic functions with measure and the Banach fixed point theorem, and we discuss two linear and nonlinear cases. We conclude with an example and some observations. [ABSTRACT FROM AUTHOR]