The classical Hecke operators act on the space of polyharmonic Maass forms, but the space of higher depth is not spanned by Hecke eigenforms, and therefore has no bases consisting of Hecke eigenforms. Alternatively, we give a basis consisting of generalized Hecke eigenforms, not ordinary eigenforms, characterize generalized Hecke eigenspaces, and give the trace formula on the space of polyharmonic Maass forms.
Assume (Ω , 𝒜, ℙ) is a probability space, (X,ρ ) is a complete and separable metric space with the σ –algebra ℬ of all its Borel subsets and f: X ×Ω→ X is measurable for ℬ⊗𝒜 and such that ∫ _Ωρ (f(x, ω ), f(z, ω ) ) ℙ(dω ) ≤β (ρ (x, z) ) for x, z ∈ X with a concave β : [0,∞ ) → [0,∞ ) satisfying ∑ _n=1^∞β ^n(t)<∞ for t ∈ (0,∞ ) , and ∫ _Ωρ (f(x_0, ω ), x_0 ) ℙ(d ω ) <∞ for an x_0 ∈ X . We consider the sequence of iterates of f defined on X ×Ω ^ℕ by f^0(x, ω ) = x and f^n(x, ω ) = f (f^n-1(x, ω ), ω _n ) for n ∈ℕ , its weak limit π ^f and the problem of the almost sure convergence lim _n →∞1/n∑ _k=1^n ψ∘ f^k(x,· )=∫ _Xψ dπ ^f for x ∈ X and Lipschitzian ψ :X →ℝ .
Within the framework of the theory of semigroups of operators, we consider a local singular perturbation of a one-dimensional Laplace operator. The Feller semigroups we study describe Brownian motions on an interval with killing that is more and more concentrated around a chosen point. As a result, the description of the limit semigroup involves a Robin-type boundary condition that reflects the killing mechanism that applies to this point and to this point only. As an application we provide a semigroup-theoretical approach to calculating the distribution of the local time for Brownian motion when it exits an interval.
In this paper, we study the Dirichlet problem for Monge-Ampère type equations for p-plurisubharmonic functions on Riemannian manifolds. A priori estimates up to second order derivatives of solutions are established. The existence of a solution then follows by the continuity method.
Let Q_ν(a,b) be the generalized Marcum Q-function and R_ν(a,b) be the generalized Marcum function of the second kind of order ν >0 . In this paper our aim is to present some new results related to the monotonicity and bounds on the ratio of the generalized Marcum functions of the first and second kinds Q_ν(a,b) /R_ν(a,b). We also present new closed form series and integral representations for R_ν(a,b) and its higher order derivatives. The results on the ratio of the generalized Marcum functions of the first and second kinds are based on some earlier established results on Q_ν(a,b) and R_ν(a,b), and also on properties of modified Bessel functions of the first and second kind. Moreover, we show that the generalized Marcum function of the second kind is related to a distribution, which is not only infinitely divisible, but belongs also to the classes of self-decomposable distributions, generalized gamma convolutions and hyperbolically completely monotone densities.
Employing the creative microscoping method and the Chinese remainder theorem for polynomials, we give a parametric q-supercongruence and two Dwork-type q-supercongruences. As a corollary we deduce that, for primes p≡ 14 and r⩾ 1 , ∑ _k=0^p^r-1(6k+1)(1/2)_k(1/4)_k^2/k!^34^k ≡ p ∑ _k=0^p^r-1-1(6k+1)(1/2)_k(1/4)_k^2/k!^34^k p^3r, and a similar result for primes p≡ 34 and r⩾ 2 , ∑ _k=0^p^r-1(6k+1)(1/2)_k(1/4)_k^2 4^k /k!^3≡ p^2 ∑ _k=0^p^r-2-1(6k+1)(1/2)_k(1/4)_k^2 4^k /k!^3p^3r-2, where (x)_0=1 and (x)_k=x(x+1)⋯ (x+k-1) for k⩾ 1 .
After having analysed and defined the “surface of a translation-like triangle" in each non-constant curvature Thurston geometry [7], i.e. in S^2×R, H^2×R, SL_2R, Nil, Sol , we extend the famous Menelaus and Ceva theorems for translation triangles in the mentioned spaces by generalizing the concept of simple ratio. Our method makes possible to transfer further classical Euclidean theorems and notions to the above Thurston geometries. In our work we will use the projective models of Thurston geometries introduced by Molnár in [10].
Almost abelian Lie groups are those admitting a normal subgroup which is commutative and of codimension one. We determine all Lie algebras of almost abelian Lie groups admitting a left-invariant Lorentzian metric of vanishing curvature. For each of those algebras, we completely classify the set of inequivalent flat Lorentzian products.
In this paper, we study non-degenerate almost complex surfaces in SL(2,R)& times;SL(2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL(2,\mathbb {R}) imes SL(2,\mathbb {R})$$\end{document} for which the almost product structure P preserves the tangent bundle or maps the tangent bundle into the normal bundle. When P preserves the tangent bundle, the non-degenerate almost complex surface is totally geodesic. When P maps the tangent bundle into the normal bundle, the non-degenerate almost complex surface is determined by a spacelike surface immersed in the Minkowski 3-space R13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}_1<<^>>3$$\end{document} with constant mean curvature -23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- frac{2}{\sqrt{3}}$$\end{document}. We also find a flat non-degenerate almost complex surface in this category.
Let (X,d,μ ) be a metric measure space equipped with a doubling measure and supporting a p -Poincaré inequality, where 1< p < ∞ . For a function u ∈ N^1,p(X) , we establish a sufficient condition for fine continuity at a point in terms of the convergence of a Wiener-type integral involving its minimal p -weak upper gradient. This condition holds outside a set of zero q -capacity for each 1< q < p , and hence yields fine continuity q -quasieverywhere.
We fix attention on a nonlinear partial differential Cauchy problem with polynomial coefficients in complex time and space paired with time acting Mahler transforms. A bounded holomorphic solution to the problem is achieved near the origin through a convergent power series in the space variable with Taylor coefficients being Laplace transforms in time. Each Taylor coefficient is shown to carry a formal Puiseux series as asymptotic expansion in time that comprise a finite number of ramifications. The resulting exponential series of these Puiseux expansions in the space variable turn out to formally solve the given Cauchy problem and represent generically a so-called Hahn series relatively to time with rational support.
The aim of the present paper is to establish multiple-term refinements and reverses of the real power form for convex and log-convex functions, thereby exploring their applications to inequalities for scalar and operator means, as well as for unitarily invariant and numerical radius norms. The results of the paper provide a comprehensive and systematic overview of refined and interpolated inequalities; moreover, they cover or refine most of the known results in the literature.
This work proves the energy scattering for an inter-critical non-linear biharmonic Schödinger equation with an unbounded inhomogeneous term. The proof follows the method of Dodson-Murphy (Proc. Am. Math. Soc. 145, no. 11 (2017), 485904867), based on variance type identities and the scattering criteria of Tao (Dyn. Partial. Differ. Equ. 1 (2004), no. 1, 1–48). The challenge is to overcome the presence of an unbounded inhomogeneous term. This note naturally complements (Calc. Var. 60, 113 (2021)).
Row-action methods provide a powerful framework for large-scale linear problems by iteratively enforcing constraints through low-dimensional projections. While highly successful for over-determined linear systems, their potential for matrix equations remains underexplored. This paper extends the row-action paradigm to the matrix equation AXB = F by introducing the Alternating Randomized Block Kaczmarz (ARBK) method. ARBK operates by alternately applying randomized block Kaczmarz iterations to two interrelated linear systems with multiple right-hand sides, effectively decomposing the matrix problem into a sequence of manageable row-action steps. We establish a tight linear convergence rate for ARBK, with an explicit bound that quantifies its performance. Numerical experiments demonstrate that our method not only validates the theory but also achieves competitive performance on large-scale data fitting tasks, highlighting the efficacy of row-action methods in this extended setting.
Maligranda introduced the p-angular distance in 2006, and Rooin proposed the skew p-angular distance in 2018. This paper introduces the Maligranda-Rooin constant, a new geometric constant in Banach spaces to compare these two distances. We denote the Maligranda-Rooin constant as ℳℛ_p(𝒳) . First, the bounds of ℳℛ_p(𝒳) are derived. Next, it is shown that a normed linear space is an inner product space if and only if ℳℛ_p(𝒳)=1 . Furthermore, an equivalent form of this new constant is established. Finally, the relationships between ℳℛ_p(𝒳) and uniform non-squareness, uniform convexity, uniform smoothness, as well as uniform normal structure are investigated.
Employing the method of “creative microscoping” introduced by the first author and Wadim Zudilin, we deduce two q-supercongruences from a quadratic transformation by Rahman. They may be deemed generalizations of two q-congruences implicitly contained in a paper by Liu and Wang. By specializing q→ 1 in one of our q-supercongruences, we obtain the result: for primes p≡ 38 , ∑ _k=0^p-1(6k+1)(1/4)_k (3/8)_k^2 (1/2)_k^3/(1)_k^4 (3/4)_k^2≡ 0 p^4, where (a)_0=1 and (a)_n=a(a+1)⋯ (a+n-1) for n⩾ 1 . Meanwhile, by specializing q→ -1 in the same q-supercongruence, we get the result: for primes p≡ 34 , ∑ _k=0^p-1(6k+1)(1/2)_k^3(1/4)_k/k!^4 4^k≡ 0p^4, which was conjectured by Bing He and first confirmed by Chuanan Wei.
We present several inequalities for sums involving the modulus of the sine function. One of our results states that for all integers n≥ 1 and real numbers x, we have ∑ _k=1^n (n-k+1)(n-k+2)|sin (kx)| ≤ C (n+1)^3 with the best possible constant factor C=1/432 (9+√(41) ) √(14+6√(41))=0.25814... .
