Analysis and Mathematical Physics最新文献

Let m be a nonnegative integer, and let (nge 2^{m+1}+1.) In this paper, we consider the higher order Schrödinger type operator ({mathcal {H}}_{2^m}=(-Delta )^{2^m}+V^{2^m} ) on ({mathbb {R}}^n,) and establish the (L^p({mathbb {R}}^n)) boundedness of Riesz transforms (nabla ^j {mathcal {H}}_{2^m}^{-frac{j}{2^{m+1}}} (j=1,2,cdot cdot cdot ,2^{m+1}-1)) and their commutators. Here, V is a nonnegative potential belonging to both the reverse Hölder class (RH_s) for (s ge frac{n}{2}), and the Gaussian class associated with ((-Delta )^{2^m}).

Every positive multilinear map between (C^*)-algebras is separately weak(^*)-continuous. We show that the joint weak(^*)-continuity is equivalent to the joint weak(^*)-continuity of the multiplications of the (C^*)-algebras under consideration. We study the behavior of general tracial positive maps on properly infinite von Neumann algebras and by applying the Aron–Berner extension of multilinear maps, we establish that under some mild conditions every tracial positive multilinear map between general (C^*)-algebras enjoys a decomposition (Phi =varphi _2 circ varphi _1), in which (varphi _1) is a tracial positive linear map with the commutative range and (varphi _2) is a tracial completely positive map with the commutative domain. As an immediate consequence, tracial positive multilinear maps are completely positive. Furthermore, we prove that if the domain of a general tracial completely positive map (Phi ) between (C^*)-algebra is a von Neumann algebra, then (Phi ) has a similar decomposition. As an application, we investigate the generalized variance and covariance in quantum mechanics for arbitrary positive maps. Among others, an uncertainty relation inequality for commuting observables in a composite physical system is presented.

In this paper, we start from the periodic geodesic of generalized Hermite operators, and analyze their geometric characteristics and analytical properties. For the quantitative study of periodic solutions to the Schrödinger operators with non-polynomial potentials, we systematically discuss the corresponding Hamilton system, and use the harmonic balance method (HBM) and the modified harmonic balance method (mHBM) to approximate and estimate the periodic solution in high accuracy.

In this paper, we consider boundary extensions of two classes of mappings between metric measure spaces. These two mapping classes extend in particular the well-studied geometric mappings such as quasiregular mappings with integrable Jacobian determinant and mappings of exponentially integrable distortion with integrable Jacobian determinant. Our main results extend the corresponding results of Äkkinen and Guo (Ann. Mat. Pure. Appl. 2017) to the setting of metric measure spaces.

In this paper, we give sufficient conditions for functions defined on the space (L^{p}(mathbb {R},dmu )), (1<ple 2), providing the weighted integrability of their Cherednik-Opdam transforms. These results generalize a famous Titchmarsh’s theorem and Younis’ theorem for functions from Lipschitz classes.

If ({mathcal {S}}) denotes the class of all univalent functions in the open unit disk ({mathbb {D}}:=left{ zin {mathbb {C}}:|z|<1right} ) with the form (f(z)=z+sum nolimits _{n=2}^{infty }a_{n}z^n), then the logarithmic coefficients (gamma _{n}) of (fin {mathcal {S}}) are defined by

The logarithmic coefficients were brought to the forefront by I.M. Milin in the 1960’s as a method of calculating the coefficients (a_{n}) for (fin {mathcal {S}}). He concerned himself with logarithmic coefficients and their role in the theory of univalent functions, while in 1965 Bazilevič also pointed out that the logarithmic coefficients are crucial in problems concerning the coefficients of univalent functions. In this paper we estimate the bounds for the logarithmic coefficients (|gamma _{n}(f)|) when f belongs to the class ({mathcal {B}}(alpha ,beta )) of Bazilevič function of type ((alpha ,beta )).

In [23], Koepf proved that for a function (f(xi )=xi +sum limits _{m=2}^infty a_mxi ^m) in the class of normalized close-to-convex functions in the unit disk,

In this paper, considering the zero of order (i.e., the mapping (f(x)-x) has zero of order (k+1) at the point (x=0)), we generalize the above classical result and establish the modified Fekete-Szegö functional for s subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.

In this paper we characterise the pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where these two metrics are not equivalent. We further establish a suitable version of the pointwise kernel lower bound of the Dunkl Riesz transform via the Euclidean metric only. Then we show that the lower bound of commutator of the Dunkl Riesz transform is with respect to the BMO space associated with the Euclidean metric, and that the upper bound is respect to the BMO space associated with the Dunkl metric. Moreover, the compactness and the two types of VMO are also addressed.

In this article, we consider the following class of stochastic partial differential equations (SPDEs):

with fully locally monotone coefficients in a Gelfand triplet (mathbb {V}subset mathbb {H}subset mathbb {V}^*), where the mappings

are measurable, (text {L}_2(mathbb {U},mathbb {H})) is the space of all Hilbert-Schmidt operators from (mathbb {U}rightarrow mathbb {H}), (text {W}) is a (mathbb {U})-cylindrical Wiener process and (widetilde{pi }) is a compensated time homogeneous Poisson random measure. This class of SPDEs covers various fluid dynamic models and also includes quasi-linear SPDEs, the convection-diffusion equation, the Cahn-Hilliard equation, and the two-dimensional liquid crystal model. Under certain generic assumptions of (text {A},text {B}) and (gamma ), using the classical Faedo–Galekin technique, a compactness method and a version of Skorokhod’s representation theorem, we prove the existence of a probabilistic weak solution as well as pathwise uniqueness of solution. We use the classical Yamada-Watanabe theorem to obtain the existence of a unique probabilistic strong solution. Furthermore, we establish a result on the continuous dependence of the solutions on the initial data. Finally, we allow both diffusion coefficient (text {B}(t,cdot )) and jump noise coefficient (gamma (t,cdot ,z)) to depend on both (mathbb {H})-norm and (mathbb {V})-norm, which implies that both the coefficients could also depend on the gradient of solution. Under some assumptions on the growth coefficient corresponding to the (mathbb {V})-norm, we establish the global solvability results also.

We study Hill’s differential equation with potential expressed by elliptic functions which arises in some problems of physics and mathematics. Analytical method can be applied to study the local properties of the potential in asymptotic regions of the parameter space. The locations of the saddle points of the potential are determined, the locations of turning points can be determined too when they are close to a saddle point. Combined with the quadratic differential associated with the differential equation, these local data give a qualitative explanation for the asymptotic eigensolutions obtained recently. A relevant topic is about the generalisation of Floquet theorem for ODE with doubly-periodic elliptic function coefficient which bears some new features compared to the case of ODE with real valued singly-periodic coefficient. Beyond the local asymptotic regions, global properties of the elliptic potential are studied using numerical method.