Analysis and Mathematical Physics最新文献

This paper investigates a class of time scales for which the forward jump function is given by (sigma (t)=qt+h), where q, and h are constants. This framework allows us to treat the standard, h-, q-, and (q, h)-derivatives simultaneously as special cases of the delta derivative. We establish a key connection between the nth delta derivative and specific nth divided difference, which serves as the foundation for generalizing several classical results from q-calculus to the broader context of (q, h)-calculus. In the second part of the paper, we present explicit formulas for the nth delta derivative of a quotient of two functions, extending familiar results from classical calculus. As an application, we use the obtained results to study the (q, h)-analogs of the power and exponential functions, yielding explicit expressions for the nth derivatives of their reciprocals and leading to a novel q-binomial identity.

We establish local energy decay for damped magnetic wave equations on stationary, asymptotically flat space-times subject to the geometric control condition. More specifically, we allow for the addition of time-independent magnetic and scalar potentials, which negatively affect energy coercivity and may add in unwieldy spectral effects. By asserting the non-existence of eigenvalues in the lower half-plane and resonances on the real line, we are able to apply spectral theory from the work of Metcalfe, Sterbenz, and Tataru and combine with a generalization of prior work by the present author to extend the latter work and establish local energy decay, under one additional symmetry hypothesis. Namely, we assume that the damping term is the dominant principal term in the skew-adjoint part of the damped wave operator within the region where the metric perturbation from that of Minkowski space is permitted to be large. We also obtain an energy dichotomy if we do not prohibit non-zero real resonances. In order to make the structure of the argument more cohesive, we contextualize the present work within the requisite existing theory.

It is proven that Schrödinger-type problem (w'_t=text {i}mathfrak {A} w), (w(0)=f), ({(t>0)}) in the Gaussian Hilbert space (L^2_mathbb {C}(X,mathcal {B},gamma )) has the unique solution ({e}^{text {i}tmathfrak {A}}f=frac{1}{sqrt{4pi t}}{mathop {mathbb {E}}}_xe ^{-frac{1}{4t}Vert xVert _X^2}mathcal {W}_{text {i}x}f), where the semigroup ({e}^{text {i}tmathfrak {A}}) is irreducible-intertwined via Weyl pairs (left{ mathcal {W}_{text {i}x}:xin Xright} ) with the shift and multiplication coordinate groups on the space (mathcal {H}^2_mathbb {C}) of Hilbert-Schmidt analytic functionals on ({Hoplus text {i}H}). The expectation ({mathop {mathbb {E}}}f={int f,dgamma }) is defined by Gaussian measure (gamma ) on a real separable Banach space X, using Gross’s theory of an abstract Wiener space (jmath :Hlooparrowright X) with the reproducing Hilbert space H. It is established the explicit formula for Hamiltonian (mathfrak {A}) in the form of a closure of sums ({sum [mathfrak {h}_2(phi _j)+mathbb {1}_j]}) with the 2nd-degree Hermite polynomial (mathfrak {h}_2) from Gaussian variables (phi _j) and number operators (mathbb {1}_j) generated by the basis ((mathfrak {e}_j)subset H) in the probability space ((X,mathcal {B},gamma )) with Borel’s field (mathcal {B}) created by (jmath ). The Jackson inequalities with explicit constants for best approximations of (mathfrak {A}) are established.

Let (mathcal {M}) be a von Neumann algebra equipped with a normal semifinite faithful trace (tau ). Let (mathcal {H}_p(mathbb {R}^d,,mathcal {M})) denote the operator-valued Hardy space with (1le p<infty ), which is first studied by T. Mei [Mem. Amer. Math. Soc. 188 (2007), vi+64 pp; MR2327840]. In this paper, the authors mainly establish some new square function characterizations of operator-valued Hardy space (mathcal {H}_p(mathbb {R}^d,,mathcal {M})) for all (1le p<infty ), which can describe the predual spaces of noncommutative BMO spaces.

In this paper we derive new symmetry and new expression for 6j-symbols of the unitary principal series representations of the (SL(2,mathbb {C})) group. This allowed us to derive for them the analogue of the Regge symmetry.

Let X be a compact strictly pseudoconvex embeddable CR manifold and let A be the Toeplitz operator on X associated with a Reeb vector field ({mathcal {T}}in {mathscr {C}}^infty (X,TX)). Consider the operator (chi _k(A)) defined by the functional calculus of A, where (chi ) is a smooth function with compact support in the positive real line and (chi _k(lambda ):=chi (k^{-1}lambda )). It was established recently that (chi _k(A)(x,y)) admits a full asymptotic expansion in k when (k) becomes large. The second coefficient of the expansion plays an important role in the further studies of CR geometry. In this work, we calculate the second coefficient of the expansion.

Recent studies on linear positive operators have led to the investigation of approximation properties of Szász–Mirkyan operators related to the modified Bessel function of order 0. In this paper, we analyse the asymptotic behavior of these operators, convergence theorems, Voronovskaya and Grüss-Voronovskaya type results. A comparative assessment with classical Szász–Mirakyan operators is also presented. These results may have an impact on wide branches of knowledge, such as probability theory, statistics, physical chemistry, optics, and computer science, especially signal processing.

For a compact Riemannian manifold (M, g) with boundary (partial M), the Dirichlet-to-Neumann operator (Lambda _g:C^infty (partial M)longrightarrow C^infty (partial M)) is defined by (Lambda _gf=left. frac{partial u}{partial nu }right| _{partial M}), where (nu ) is the unit outer normal vector to the boundary and u is the solution to the Dirichlet problem (Delta _gu=0, u|_{partial M}=f). Let (g_partial ) be the Riemannian metric on (partial M) induced by g. The Calderón problem is posed as follows: To what extent is (M, g) determined by the data ((partial M,g_partial ,Lambda _g))? We prove the uniqueness theorem: A compact connected two-dimensional Riemannian manifold (M, g) with non-empty boundary is determined by the data ((partial M,g_partial ,Lambda _g)) uniquely up to conformal equivalence.

In this paper, we investigate the nonexistence of solutions of certain nonlinear elliptic equations, focusing on solutions that are stable or stable outside a compact set, potentially unbounded, and sign-changing. Our primary methods include integral estimates, Pohozaev-type identity and the monotonicity formula. Our classification approaches as a sharp result, specifically, in the subcritical case (i.e, (1< p < frac{n+4}{n-4})), we establish the existence of a mountain pass solution with a Morse index of 1 in the subspace of (H^2 cap H_0^1(Omega )) that exhibits cylindrical symmetry.

In this paper, we first introduced two time scales based on the interval [a, b] and ( mathbb {Z} ). Then, by using one of these time scale and substitutions rules, we prove a new version of discrete Hermite-Hadamard inequality for discrete convex functions. Moreover, we investigate the fractional version of this inequality involving fractional delta and nabla sums.