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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]

We consider gradient fields on Z d for potentials V that can be expressed as e - V (x) = p e - q x 2 2 + (1 - p) e - x 2 2 . This representation allows us to associate a random conductance type model to the gradient fields with zero tilt. We investigate this random conductance model and prove correlation inequalities, duality properties, and uniqueness of the Gibbs measure in certain regimes. We then show that there is a close relation between Gibbs measures of the random conductance model and gradient Gibbs measures with zero tilt for the potential V. Based on these results we can give a new proof for the non-uniqueness of ergodic zero-tilt gradient Gibbs measures in dimension 2. In contrast to the first proof of this result we rely on planar duality and do not use reflection positivity. Moreover, we show uniqueness of ergodic zero tilt gradient Gibbs measures for almost all values of p and q and, in dimension d ≥ 4 , for q close to one or for p (1 - p) sufficiently small. [ABSTRACT FROM AUTHOR]

Motivated by the theory of Painlevé equations and associated hierarchies, we study nonautonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed into a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stäckel systems from 14. [ABSTRACT FROM AUTHOR]

Let K be an Sn-field. For a nonidentity conjugacy class C, define NK,C to be the smallest prime p such that Frobp ∈ C. By using the observation that NK,C is interpreted as the first prime sign change of the Dirichlet coefficients of automorphic L-functions, we improve the known bound on NK,C for n = 3,4,5. (For n = 5, we need to assume the strong Artin conjecture.) [ABSTRACT FROM AUTHOR]

The existence and uniqueness of a global positive solution is proven for the system of stochastic differential equations describing a nonautonomous stochastic predator-prey model with a modified version of the Leslie-Gower term and Holling-type II functional response disturbed by white noise, centered and noncentered Poisson noises. Sufficient conditions are obtained for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, weak persistence in the mean and extinction of a solution to the considered system. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of μ‐pseudo almost automorphic solution for a delayed non‐autonomous partial functional differential equation in the exponential dichotomic case, where the nonlinear operator F satisfies the φ‐Lipschitz condition and φ belongs to some admissible spaces. We further prove the existence of an invariant stable manifold around the μ‐pseudo almost automorphic solution in that case. An application is given to illustrate our theory. [ABSTRACT FROM AUTHOR]

Given |$\boldsymbol{F}$| a number field with ring of integers |${\mathcal{O}}_{\boldsymbol{F}}$| and |${\mathfrak{p}},{\mathfrak{q}}$| two coprime and squarefree ideals of |${\mathcal{O}}_{\boldsymbol{F}}$|⁠ , we prove a reciprocity relation for the 1st moment of the triple product |$L$| -functions |$L(\pi \otimes \pi _1\otimes \pi _2,\frac{1}{2})$| twisted by |$\lambda _{\pi }({\mathfrak{p}})$|⁠ , where |$\pi _1$| and |$\pi _2$| are fixed unitary automorphic representations of |$\textrm{PGL}_2({\mathbb{A}}_{\boldsymbol{F}})$| with |$\pi _1$| cuspidal and |$\pi$| runs through unitary automorphic representations of conductor dividing |${\mathfrak{q}}$|⁠. The method uses adelic integral representations of |$L$| -functions and the symmetric identity is established for a particular period. Finally, the integral period is connected to the moment via Parseval formula. [ABSTRACT FROM AUTHOR]

We prove the multiplicity formula for the automorphic discrete spectrum of the metaplectic group |$\textrm{Mp}_4$| of rank |$2$|⁠. [ABSTRACT FROM AUTHOR]

We define the space of nearly holomorphic automorphic forms on a connected reductive group G over Q such that the homogeneous space G (R) 1 / K ∞ ∘ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's study, there are the classification of indecomposable (g , K ∞) -modules which occur in the space of nearly holomorphic elliptic modular forms and Siegel modular forms of degree 2. This paper studies global representations of the adele group G (A Q) which occur in the space of nearly holomorphic Hilbert modular forms. In the case of elliptic modular forms, the result of this paper is an adelization of Pitale, Saha and Schmidt's result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of an a -pseudo complementation x ↦ x ∗ a on an Almost Distributive Lattice A for an arbitrary fixed element a and any x of A and discuss several properties of this. Here, we obtain a bijection correspondence between the a -pseudo complementations on A and maximal elements of A. We prove that the set A ∗ a = { x ∗ a : a ∈ A } is a Boolean algebra which is independent(upto isomorphism) of the a -pseudo complementation x ↦ x ∗ a. [ABSTRACT FROM AUTHOR]

The aim of this article is to study neutral type Clifford-valued high-order Hopfield neural networks with mixed delays and D operator. New criteria are established for the existence, uniqueness and global exponential stability of (μ , ν) - pseudo almost automorphic solutions of the considered model via Banach's fixed point principle and differential inequality techniques. An example is given to show the effectiveness of the main new criteria. [ABSTRACT FROM AUTHOR]

We prove the multiplicity formula for the automorphic discrete spectrum of the metaplectic group |$\textrm{Mp}_4$| of rank |$2$|⁠. [ABSTRACT FROM AUTHOR]

The definition of Stepanov-like almost automorphic functions on time scales had been proposed in the literature, but at least one result was incorrect, which involved Bochner transform. In our work, we give the Bochner definition of Stepanov-like almost automorphic functions on time scales, and prove that a function is Stepanov-like almost automorphic if and only if it satisfies Bochner definition of Stepanov-like almost automorphic function on time scales. The Bochner definition of Stepanov-like almost automorphic functions on time scales corrects the faulty result, and perfects the definition of Stepanov-like almost automorphic functions. As applications, we discuss the almost automorphy of a certain dynamic equation and some cellular neural networks with delays on time scales. [ABSTRACT FROM AUTHOR]

This paper investigates the equivalence between p‐mean Poisson μ‐pseudo almost automorphic and Poisson asymptotically almost automorphic stochastic processes outside one μ‐ergodic zero set and then obtains the completeness of the space of Poisson μ‐pseudo almost automorphic stochastic processes without the Lipschitz condition. These properties established enrich the knowledge of Poisson μ‐pseudo almost automorphic stochastic processes. Meanwhile, a more significant point is that without the Amerio semiseparation condition, we establish the Amerio‐type theorem that all the Lp‐bounded mild solutions are μ‐pseudo almost automorphic in distribution for a class of nonlinear Sobolev‐type stochastic differential equations with Markov switching and jumps. In this sense, we partly contribute to a better understanding of the stochastic analysis between Lp‐bounded mild solutions and μ‐pseudo almost automorphic mild solutions in distribution for nonlinear SDEs. Finally, we give an example to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

We prove effective forms of the Sato–Tate conjecture for holomorphic cuspidal newforms which improve on the author's previous work (solo and joint with Lemke Oliver). We also prove an effective form of the joint Sato–Tate distribution for two twist-inequivalent newforms. Our results are unconditional because of recent work of Newton and Thorne. [ABSTRACT FROM AUTHOR]

We study the argument of L(s, f) associated with holomorphic cusp form in both weight aspect and t-aspect. We prove that for −1 ≤ σ ≤ 2 and t ≥ 4, argL(s, f) ≪ log kt, where s = σ + it. Assuming the generalized Riemann hypothesis (GRH), we have arg L(s, f) ≪ log kt/log log kt for σ ≥ 1/2 and t ≥ 4. [ABSTRACT FROM AUTHOR]

We show that the finite part of the adjoint |$L$| -function (including contributions from all non-archimedean places, including ramified places) is holomorphic in |${\textrm{Re}}(s) \ge 1/2$| for a cuspidal automorphic representation of |${\textrm{GL}}_3$| over a number field. This improves the main result of [ 21 ]. We obtain more general results for twisted adjoint |$L$| -functions of both |${\textrm{GL}}_3$| and quasisplit unitary groups. For unitary groups, we explicate the relationship between poles of twisted adjoint |$L$| -functions, endoscopy, and the structure of the stable base change lifting. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the classes of (ω,c)-pseudo almost periodic functions and (ω,c)-pseudo almost automorphic functions. These collections include (ω,c)-pseudo periodic functions, pseudo almost periodic functions and their automorphic analogues. We present an application to the abstract semilinear first-order Cauchy inclusions in Banach spaces. [ABSTRACT FROM AUTHOR]

In this article we show the existence of positive Stepanov-like almost automorphic solutions for systems of nonlinear delay integral equations. To do this, we apply the well-known Guo-Krasnosel'skii fixed point theorem for cone expansion and compression. [ABSTRACT FROM AUTHOR]

We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$. A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$. This generalises a result of Vogan [Vog91]. We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G. Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20]. Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models. [ABSTRACT FROM AUTHOR]

Let m be an integer greater than three and l be an odd prime. In this paper we prove that at least one of the following groups: PΩ±2m(Fls), PSO±2m(Fls), PO2m±(Fls), or PGO±2m(Fls) is a Galois group of Q for infinitely many integers s > 0. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen, and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of GL2m(AQ). [ABSTRACT FROM AUTHOR]

Recently, the problem of bounding the sup norms of L2-normalized cuspidal automorphic newforms ϕ on GL2 in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ϕ∥∞ ≪ N1/4 + ϵ, at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ϕ∥∞ ≪ N1/2 + ϵ in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case. [ABSTRACT FROM AUTHOR]

We show that the finite part of the adjoint |$L$| -function (including contributions from all non-archimedean places, including ramified places) is holomorphic in |${\textrm{Re}}(s) \ge 1/2$| for a cuspidal automorphic representation of |${\textrm{GL}}_3$| over a number field. This improves the main result of [ 21 ]. We obtain more general results for twisted adjoint |$L$| -functions of both |${\textrm{GL}}_3$| and quasisplit unitary groups. For unitary groups, we explicate the relationship between poles of twisted adjoint |$L$| -functions, endoscopy, and the structure of the stable base change lifting. [ABSTRACT FROM AUTHOR]

Let m be an integer greater than three and l be an odd prime. In this paper we prove that at least one of the following groups: PΩ±2m(Fls), PSO±2m(Fls), PO2m±(Fls), or PGO±2m(Fls) is a Galois group of Q for infinitely many integers s > 0. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen, and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of GL2m(AQ). [ABSTRACT FROM AUTHOR]

We propose a class of Clifford-valued neutral-type fuzzy cellular neural network models with leakage delays on time scales, give the definition and basic properties of Clifford-valued almost automorphic functions on time scales, and prove a composition theorem of Clifford-valued almost automorphic functions on time scales, and on this basis, we study the existence and global exponential stability of almost automorphic solutions of this class of neural networks. The research in this paper can unify the research of real-valued neural networks, complex-valued neural networks and quaternion-valued neural networks, as well as the research of continuous-time neural networks and discrete-time neural networks. Finally, an example illustrates the feasibility of the results obtained. [ABSTRACT FROM AUTHOR]

The principal aim of this article is to attach and study $p$ -adic $L$ -functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$ -adic $L$ -functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$ -adic $L$ -functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$ -function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi)$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$ -power conductor. [ABSTRACT FROM AUTHOR]

We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line. [ABSTRACT FROM AUTHOR]

We estimate the growth of cuspidal cohomology of G L 2 (A Q). Quantitatively, we provide bounds on the total number of normalised eigenforms of Hecke operators which are obtained by automorphic induction from Hecke characters of imaginary quadratic fields grows as level structure varies. We further investigate how much of cuspidal cohomology of G L 3 (A Q) is obtained by symmetric square transfer from G L 2 (A Q) when the level structure corresponds to power of an odd prime. [ABSTRACT FROM AUTHOR]

We consider the nonrelativistic limit of a semilinear field equation in a homogeneous and isotropic space, where the scale function of the space is constructed based on the Einstein equations.Furthermore,we examine the Cauchy problem for the limit equation and showthe existence of global and blowup solutions in Sobolev spaces. Finally,we study the effects of spatial variance on the problem and remark on some dissipative and antidissipative properties of the limit equation. [ABSTRACT FROM AUTHOR]

We study modulo p reduction of compatible systems of p -adic representations of the absolute Galois group of Q , arising from an algebraic automorphic representation. In particular, we prove that there is a field extension of Q with the Galois group G 2 (p) , ramified at 5 and p only, for a set of primes p of density one. [ABSTRACT FROM AUTHOR]

We present the basics of the local theory, which arises from global Rankin-Selberg integrals, attached to pairs of irreducible globally generic cuspidal automorphic representations of the quasi-split unitary group U 2 n + 1 and Res E / F GL m , for a quadratic extension of number fields E / F , when m > n. [ABSTRACT FROM AUTHOR]

Abstract—We give a solution to the Kolmogorov problem on uniqueness of probability solutions to a parabolic Fokker–Planck–Kolmogorov equation. [ABSTRACT FROM AUTHOR]

Automorphic loops or A -loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. Given a Lie ring (Q , + , [. ,. ]) we can define an operation (∗) such that (Q , ∗) is an A -loop. We call it Lie automorphic loop. 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. It was shown by [W. R. Scott, Half-homomorphisms of groups, Proc. Amer. Math. Soc.8 (1957) 1141–1144] that if G is a group then f is either an isomorphism or an anti-isomorphism. This was used to prove that a finite group is determined by its group determinant. Here, we show that every half-automorphism of a Lie automorphic loop of odd order is either an automorphism or an anti-automorphism. [ABSTRACT FROM AUTHOR]

Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of $K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such $K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds' conjecture. [ABSTRACT FROM AUTHOR]

Let p , q be different primes and suppose that the principal p - and the principal q -block of a finite group have only one irreducible complex character in common, namely the trivial one. We conjecture that this condition implies the existence of a nilpotent Hall { p , q } -subgroup and prove that a minimal counter-example must be an almost simple group where pq divides the order of its simple nonabelian normal subgroup. As an immediate consequence we obtain that the conjecture holds true for p -solvable or q -solvable groups. Furthermore, we prove the conjecture in case 2 ∈ { p , q } using the classification theorem of finite simple groups. Finally, we consider the situation that the intersection of an arbitrary p -block with an arbitrary q -block contains only one irreducible character. [ABSTRACT FROM AUTHOR]

We prove a new (conditional) result toward the subconvexity problem for certain automorphic L-functions for GL2 × GL3. This follows from the computation of new SL2-period integrals associated with newforms f and g of even weight and odd squarefree level. The same computations also lead to a central value formula for degree six complex L-series of the form L(f Ad(g), s), extending previous work in Pal and de Vera-Piquero [Documenta Math. 24 (2019), 1935-2036]. [ABSTRACT FROM AUTHOR]

Groups of F-type were introduced in [B. Fine and G. Rosenberger, Generalizing Algebraic Properties of Fuchsian Groups, London Math. Soc. Lecture Note Ser., 159 (1991) 124{147.] as a natural algebraic generalization of Fuchsian groups. They can be considered as the analogs of cyclically pinched one-relator groups where torsion is allowed. Using the methods In [B. Fine. M. Kreuzer and G. Rosenberger, Faithful Real Representations of Cyclically Pinched One-Relator Groups, Int. J. Group Theory, 3 (2014) 1{8.] we prove that any hyperbolic group of F-type has a faithful representation in PSL(2;R). From this we also obtain that a cyclically pinched one-relator group has a faithful real representation if and only if it is hyperbolic. We further survey the many nice properties of groups of F-type. [ABSTRACT FROM AUTHOR]

In this work we consider the class of nonlocal non-autonomous evolution problems in a bounded smooth domain... Under appropriate assumptions we study the asymptotic behavior of the evolution process, generated by this problem in a suitable Banach space. We prove results on existence, uniqueness and smoothness of the solutions and on the existence of pullback attractor for the evolution process. We also prove a continuous dependence of the evolution process with respect to the external stimuli function present in the model. Furthermore, using the continuous dependence of the evolution process, we prove the upper semicontinuity of pullback attractors with respect to the external stimuli function. We finish this article with a small discussion about the model and about a biological interpretation of the result on the continuous dependence of neuronal activity with respect to the external stimuli function. [ABSTRACT FROM AUTHOR]

In this work, the existence and uniqueness of random attractors of a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation driven by colored noise with a nonlinear diffusion term is established. We comment that compared to white noise, the colored noise is much easier to handle in examining the pathwise dynamics of stochastic systems. Additionally, we prove the attractors of the random fractional Ginzburg-Landau system driven by a linear multiplicative colored noise converge to those of the corresponding stochastic system driven by a linear multiplicative white noise. [ABSTRACT FROM AUTHOR]

It is shown by Fera that there exists uncountably many exceptional points for cocompact Fuchsian groups. We generalize this result to the case that Fuchsian groups are of finite coarea. [ABSTRACT FROM AUTHOR]

The breakdown voltage, VBR, of a superjunction whose pillar parameters, namely doping, length, and width, are optimized to yield the least specific ON-resistance, RONSP, for the specified VBR, falls drastically for even small charge imbalances. Hence, a structure fabricated with these target parameter values often has a VBR much lower than specified due to the charge imbalance caused by random process variations. We give an analytical solution for alternate target parameter values which yield the specified VBR in spite of process variations, with minimum sacrifice in RONSP. The solution is obtained by the method of Lagrange multipliers and a simple equation for the breakdown field versus charge imbalance. It is validated by TCAD simulation. It can either be approximated to a closed-form or can yield the target parameter values accurately with just a few iterations. Apart from eliminating the tedious iterations of prior design methods, our solution provides two new insights: when compared with an optimum balanced device, a device optimized considering typical charge imbalance has significantly lower pillar doping but almost the same pillar length and width; the breakdown field of a balanced device is a power law function of the pillar length alone. Although illustrated for 4H-SiC structures with VBR of 1–10 kV, our solution should work for any semiconductor after material specific changes. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence and uniqueness theorem and a comparison theorem for a class of anticipated mean-field backward stochastic differential equations with jumps. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence and uniqueness theorem and a comparison theorem for a class of anticipated mean-field backward stochastic differential equations with jumps. [ABSTRACT FROM AUTHOR]

This paper is concerned with existence, uniqueness, and almost sure exponential stability of solutions to nonlinear stochastic system with Markovian switching and Lévy noises. Firstly, the existence and uniqueness of solutions to the system is studied. Then, the almost sure exponential stability of the system is derived. Finally, an example is presented to illustrate the results. [ABSTRACT FROM AUTHOR]

Let G be a connected split reductive group over a finite field Fq and X a smooth projective geometrically connected curve over Fq. The ℓ-adic cohomology of stacks of G-shtukas is a generalization of the space of automorphic forms with compact support over the function field of X. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue. [ABSTRACT FROM AUTHOR]