Type Theorem Research Articles

Cameron–Martin type theorem for a class of non-Gaussian measures

. In this article, we study the quasi-translation-invariant property of a class of non-Gaussian measures. These measures are associated with the family of generalized grey Brownian motions. We identify the Cameron–Martin space and derive the explicit Radon-Nikodym density in terms of the Wiener integral with respect to the fractional Brownian motion. Moreover, we show an integration by parts formula for the derivative operator in the directions of the Cameron–Martin space. As a consequence, we derive the closability of both the derivative and the corresponding gradient operators.

Future worldwide coronavirus disease 2019 epidemic predictions by Gaidai multivariate risk evaluation method.

Accurate estimation of pandemic likelihood in every US state of interest and at any time. Coronavirus disease 2019 (COVID-19) is an infectious illness with a high potential for global dissemination and low rates of fatality and morbidity, placing some strains on national public health systems. This research intends to benchmark a novel technique, that enables hazard assessment, based on available clinical data, and dynamically observed patient numbers while taking into account pertinent territorial and temporal mapping. Multicentre, population-based, and biostatistical strategies have been utilized to process raw/unfiltered medical survey data. The expansion of extreme value statistics from the univariate to the bivariate situation meets with numerous challenges. First, the univariate extreme value types theorem cannot be directly extended to the bivariate (2D) case,-not to mention challenges with system dimensionality higher than 2D. Assessing outbreak risks of future outbreaks in any nation/region of interest. Existing bio-statistical approaches do not always have the benefits of effectively handling large regional dimensionality and cross-correlation between various regional observations. These methods deal with temporal observations of multi-regional phenomena. Apply contemporary, novel statistical/reliability techniques directly to raw/unfiltered clinical data. The current study outlines a novel bio-system hazard assessment technique that is particularly suited for multi-regional environmental, bio, and public health systems, observed over a representative period. With the use of the Gaidai multivariate hazard assessment approach, epidemic outbreak spatiotemporal risks may be properly assessed. Based on raw/unfiltered clinical survey data, the Gaidai multivariate hazard assessment approach may be applied to a variety of public health applications. The study's primary finding was an assessment of the risks of epidemic outbreaks, along with a matching confidence range. Future global COVID-19/severe acute respiratory syndrome coronavirus 2 (SARS-COV2) epidemic risks have been examined in the current study; however, COVID-19/SARS-COV2 infection transmission mechanisms have not been discussed.

Low-energy theorem revisited and OPE in massless QCD

We revisit a low-energy theorem (LET) of NSVZ type in SU(N) QCD with Nf massless quarks derived in [1] by implementing it in dimensional regularization. The LET relates n-point correlators in the l.h.s. to n + 1-point correlators with the extra insertion of TrF2 at zero momentum in the r.h.s. We demonstrate that, for 2-point correlators of an operator O in the l.h.s., the LET implies that, in general, the integrated 3-point correlator in the r.h.s. needs in perturbation theory an infinite additive renormalization in addition to the multiplicative one. We relate the above counterterm to a corresponding divergent contact term in a certain coefficient of the OPE of TrF2 with O in the momentum representation, thus extending to any operator O an independent argument that first appeared for O = TrF2 in [2]. Finally, we demonstrate that in the asymptotically free phase of QCD the aforementioned counterterm in the LET is actually finite nonperturbatively after resummation to all perturbative orders. We also briefly recall the implications of the LET in the gauge-invariant framework of dimensional regularization for the perturbative and nonperturbative renormalization in large-N QCD. The implications of the LET inside and above the conformal window of SU(N) QCD with Nf massless quarks will appear in a forthcoming paper.

Titchmarsh’s-type theorem for two-sided quaternion Fourier transform and sharp Hausdorff–Young inequality for quaternion linear canonical transform

In this work, we first introduce the two-sided quaternion Fourier transform and demonstrate its essential properties. We generalize Titchmarsh’s-type theorem in the framework of the two-sided quaternion Fourier transform. Based on the interaction between the quaternion Fourier transform and quaternion linear canonical transform we explore sharp Hausdorff–Young inequality for the quaternion linear canonical transform. The obtained result can be considered as a generalized version of sharp Hausdorff–Young inequality for the two-dimensional quaternion Fourier transformation in the literature.

Viscosity solutions to the inhomogeneous reaction–diffusion equation involving the infinity Laplacian

In this paper, we study the inhomogeneous reaction–diffusion equation involving the infinity Laplacian: [Formula: see text] where the continuous function [Formula: see text] satisfies [Formula: see text] a positive function [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text]. Such a model permits existence of solutions with dead core zones, i.e. a priori unknown regions where non-negative solutions vanish identically. For [Formula: see text] and the non-positive inhomogeneous term [Formula: see text] we establish the existence, uniqueness and stability of the viscosity solution of the corresponding continuous Dirichlet problem. Under additional structure conditions on [Formula: see text] and [Formula: see text] we obtain the optimal [Formula: see text] regularity across the free boundary [Formula: see text] Moreover, we establish the porosity of the free boundary and Liouville type theorem for entire solutions. Finally, we prove that the dead core vanishes in the limit case [Formula: see text]

Liouville type theorems for generalized P-harmonic maps

Some theorems of Liouville type are given for such P-harmonic maps when target manifold have conformal vector field or convex function or have non-positive sectional curvature.

A Whitney Type Theorem for Surfaces: Characterising Graphs with Locally Planar Embeddings

AbstractGiven a graph G and a parameter r, we define the r-local matroid of G to be the matroid generated by its cycles of length at most r. Extending Whitney’s abstract planar duality theorem from 1932, we prove that for every r the r-local matroid of G is co-graphic if and only if G admits a certain type of embedding in a surface, which we call r-planar embedding. The maximum value of r such that a graph G admits an r-planar embedding is closely related to face-width, and such embeddings for this maximum value of r are quite often embeddings of minimum genus. Unlike minimum genus embeddings, these r-planar embeddings can be computed in polynomial time. This provides the first systematic and polynomially computable method to construct for every graph G a surface so that G embeds in that surface in an optimal way (phrased in our notion of r-planarity).

Fully nonlinear dead-core systems

We study fully nonlinear dead-core systems coupled with strong absorption terms. We discover a chain reaction, exploiting properties of an equation along the system and obtain higher sharp regularity across the free boundary. Additionally, we prove geometric measure estimates and obtain coincidence of the free boundaries. We also derive Liouville type theorems for entire solutions. These results are new even for linear systems.

On Liouville type results for the stationary MHD in R3

This paper is concerned with the Liouville type theorems for the steady incompressible magnetohydrodynamics (MHD) equations. We establish that the solution to the steady MHD equations is identically zero under the integrability assumptions on (v, b). We show that, in particular, a combination of a strong integrability condition on the velocity of a fluid and a weak integrability condition on the magnetic field gives a sufficient condition on the Liouville type theorems. Furthermore, we show that the combination of the growth condition of the potential for the fluid velocity and the integrability condition for the magnetic field leads to the triviality of the solution.

Llarull type theorems on complete manifolds with positive scalar curvature

In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold ( M n , g ) (M^{n},g) with scalar curvature R g ≥ 6 R_{g}\geq 6 admits a non-zero degree and 1 1 -Lipschitz map to ( S 3 × T n − 3 , g S 3 + g T n − 3 ) (\mathbb {S}^{3}\times \mathbb {T}^{n-3},g_{\mathbb {S}^{3}}+g_{\mathbb {T}^{n-3}}) , for 4 ≤ n ≤ 7 4\leq n\leq 7 , then ( M n , g ) (M^{n},g) is locally isometric to S 3 × T n − 3 \mathbb {S}^{3}\times \mathbb {T}^{n-3} . Similar results are established for non-compact cases as ( S 3 × R n − 3 , g S 3 + g R n − 3 ) (\mathbb {S}^{3}\times \mathbb {R}^{n-3},g_{\mathbb {S}^{3}}+g_{\mathbb {R}^{n-3}}) being model spaces. We observe that the results differ significantly when n = 4 n=4 compared to n ≥ 5 n\geq 5 . Our results imply that the ϵ \epsilon -gap length extremality of the standard S 3 \mathbb {S}^3 is stable under the Riemannian product with R m \mathbb {R}^m , 1 ≤ m ≤ 4 1\leq m\leq 4 (see D 3 D_{3} . Question in Gromov’s paper [Foundations of mathematics and physics one century after Hilbert, Spring, Cham, 2018, p. 153]).

A Stancu type extension of the Cheney-Sharma Chlodovsky operators

In this paper we introduce a Stancu type extension of the Cheney-Sharma Chlodovsky operators based on the ideas presented by Cătinaș and Buda, Bostanci and Bașcanbaz-Tunca, respectively Söylemez and Tașdelen. For this new operators we study some approximation and convexity properties and the preservation of the Lipschitz constant and order. Finally, we study approximation properties of the new operators with the help of Korovkin type theorems.

A Birkhoff–Kellogg Type Theorem for Discontinuous Operators with Applications

By means of fixed point index theory for multivalued maps, we provide an analogue of the classical Birkhoff–Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general and can be applied, for example, to eigenvalues and parameter problems for ordinary differential equations with discontinuities. We illustrate in detail this fact for a class of second-order boundary value problem with deviated arguments and discontinuous terms. In a specific example, we explicitly compute the terms that occur in our theory.

Parseval-Goldstein type theorems for integral transforms in a general setting

Parseval-Goldstein type theorems for integral transforms in a general setting

Strong completeness of a class of 𝐿²(𝑇)-type Riesz spaces

The L p , 1 ≤ p ≤ ∞ L^{p}, 1\le p\le \infty , spaces have been generalized to the setting of Riesz spaces as L p ( T ) {L}^{p}(T) spaces, on which there are R ( T ) R(T) -valued norms. The strong sequential completeness of the space L 1 ( T ) {L}^{1}(T) and the strong completeness of L ∞ ( T ) {L}^{\infty }(T) with resepct to their respective R ( T ) R(T) -valued norms were established by Kuo, Rodda, and Watson. In the current work, the T T -strong completeness of L 2 ( T ) {L}^{2}(T) is established via the Riesz–Fischer type theorem given by Kalauch, Kuo, and Watson. It is also shown that the conditional expectation operator T T is a weak order unit for the T T -strong dual.

Generalized Szász-Mirakian Type Operators

In this paper, we are proposing certain modifications of Szász-Mirakian type operators and study their approximation properties. We also give a Voronovskaya type theorem for these operators. Keywords: Linear positive operators, Schurer type operator, Szász-Mirakian operators, Voronovskaya type theorem.

The Kontorovich - Lebedev Transformation on Sobolev Type Spaces

The Kontorovich-Lebedev transformation $$(KLf)(x)=\int_0^\inftyK_{i\tau}(x)f(\tau)d\tau, \;\, x \in {\mathbf R}_+$$ is consideredas an operator, which maps the weighted space $L_p(\mathbf R_+;$$\omega(\tau)d\tau), \;\, 2 \le p \le \infty$ into the Sobolevtype space $S_p^{N, \alpha}({\mathbf R}_+)$ with the finite norm$$||u||_{S_p^{N,\alpha}({\mathbf R}_+)}= \biggl( \sum_{k= 0}^N\int_0^\infty |A_x^k u|^p x^{\alpha_k p -1} dx\biggr)^{1/p} <\infty,$$where $\alpha= (\alpha_0, \alpha_1, \dots, \alpha_N), \alpha_k \in{\mathbf R}, k=0, \dots, N$, and $ A_x$ is the differentialoperator of the form$$A_x u= x^2u(x) - x\frac{d}{dx}\biggl[x\frac{du}{dx }\biggr], $$and $A_x^k$ means $k$-th iterate of $A_x, \ A_x^0u= u$. Elementary properties for the space $S_p^{N, \alpha} ({\mathbf R}_+)$ are derived. Boundedness and inversion properties for the Kontorovich-Lebedev transform are studied. In the Hilbert case ($p=2$) the isomorphism between these spaces is established for the special type of weights and Plancherel's type theorem is proved. 2000 Mathematics Subject Classification. 44A15, 46E35, 26D10

On a Sobolev Type Theorem for the Generalized Riesz Potential Generated by the Generalized Shift Operator on Morrey Space

On a Sobolev Type Theorem for the Generalized Riesz Potential Generated by the Generalized Shift Operator on Morrey Space

Weighted p-basic harmonic forms and its applications

In this paper, we study the weighted p-basic harmonic forms on a weighted Riemannian foliation (M,g,F,e−fν) for some basic function f. At the same time, we prove that there is no non-trivial Lfp-weighted p-basic harmonic form under some assumptions about the generalized weighted curvature. Finally, we consider the Liouville type theorem for (F,F′)(p,f)-harmonic map between (M,g,F,e−fν) and (M′,g′,F′) as applications.

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations