Analysis and Mathematical Physics最新文献

Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler–Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.

We consider the Schrödinger operator on the quantum graph whose edges connect the points of ({{mathbb {Z}}}). The numbers of the edges connecting two consecutive points n and (n+1) are read along the orbits of a shift of finite type. We prove that the Lyapunov exponent is potitive for energies E that do not belong to a discrete subset of ([0,infty )). The number of points E of this subset in ([(pi (j-1))^2, (pi j)^2]) is the same for all (jin {{mathbb {N}}}).

In this paper, we investigate the existence of multiple weak solutions for a Schrödinger-Kirchhoff type elliptic system involving nonlocal ((alpha _1(cdot ), ldots , alpha _N(cdot )))-Laplacian operator. The system is modeled as follows:

We apply the three critical points theorem to establish sufficient conditions for the existence of at least three weak solutions under appropriate assumptions on the system’s parameters and nonlinearity terms. This work extends the analysis of elliptic systems involving variable exponent spaces and nonlocal operators, offering novel insights into their mathematical structure and solution properties.

We find two Lax representations for the reduced magnetohydrodynamics equations (rmhd) and construct a local variational Poisson structure (a Hamiltonian operator) for them. Its inverse defines a nonlocal symplectic structure for the same equations. We describe the action of both operators on the second-order cosymmetries and on the infinitesimal contact symmetries of rmhd, respectively. The reduction of rmhd by the symmetry of shifts along the z-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics (imhd). Applied to the Lax representations and the variational Poisson structure of rmhd, the reduction provides analogous constructions for imhd.

In this work we introduce a family of conformal flows generalizing the classical Yamabe flow. We prove that for a large class of such flows long-time existence holds, and the arguments are in fact simpler than in the classical case. Moreover, we establish convergence for the case of negative scalar curvature and expect a similar statement for the positive and the flat cases as well.

Let X be a ball quasi-Banach function space, (alpha in mathbb {R}) and (qin (0,infty )). In this article, the authors first introduce the Herz-type Hardy space (mathcal {Hdot{K}}_{X}^{alpha ,,q}({mathbb {R}}^n)), which is defined via the non-tangential grand maximal function. Under some mild assumptions on X, the authors establish the atomic decompositions of (mathcal {Hdot{K}}_{X}^{alpha ,,q}({mathbb {R}}^n)). As an application, the authors obtain the boundedness of certain sublinear operators from (mathcal {Hdot{K}}_{X}^{alpha ,,q}({mathbb {R}}^n)) to (mathcal {dot{K}}_{X}^{alpha ,,q}({mathbb {R}}^n)), where (mathcal {dot{K}}_{X}^{alpha ,,q}({mathbb {R}}^n)) denotes the Herz-type space associated with ball quasi-Banach function space X. Finally, the authors apply these results to three concrete function spaces: Herz-type Hardy spaces with variable exponent, mixed Herz-Hardy spaces and Orlicz-Herz Hardy spaces, which belong to the family of Herz-type Hardy spaces associated with ball quasi-Banach function spaces.

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.