Inequalities and separation for Schrödinger type operators in L2(Rn)

1978 ◽  
Vol 79 (3-4) ◽  
pp. 257-265 ◽  
Author(s):  
W. N. Everitt ◽  
M. Giertz
Keyword(s):  
Euclidean Space ◽  
Differential Expression ◽  
Integral Inequalities ◽  
Laplacian Operator ◽  
Generalized Derivatives ◽  
Maximal Domain ◽  
Definition Of ◽  
Schrödinger Type Operators ◽  
Real Euclidean Space

SynopsisThe symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.

scholarly journals Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in ℝn\mathbb{R}^{n}

2020 ◽  
Vol 26 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Euclidean Space ◽  
Convex Functions ◽  
Convex Bodies ◽  
Integral Inequalities ◽  
Divergence Theorem ◽  
Several Variables ◽  
Type Integral ◽  
Bounded Convex

AbstractIn this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space {\mathbb{R}^{n}} for any {n\geq 2}.

On Certain Functional Identities in EN

1971 ◽  
Vol 23 (2) ◽  
pp. 315-324 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Euclidean Space ◽  
Taylor Series ◽  
Dimensional Space ◽  
The Real ◽  
Higher Dimensional ◽  
Isolated Points ◽  
Functional Identities ◽  
Image Position ◽  
Special Case ◽  
Real Euclidean Space

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

scholarly journals Remarks on the Generalized Fractional Laplacian Operator

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 320 ◽  
Author(s):  
Chenkuan Li ◽  
Changpin Li ◽  
Thomas Humphries ◽  
Hunter Plowman
Keyword(s):  
Differential Equations ◽  
Fractional Laplacian ◽  
Laplacian Operator ◽  
Generalized Convolution ◽  
Riesz Fractional Derivative ◽  
Derivative Operator ◽  
Delta Sequence ◽  
Remote Sites ◽  
Definition Of ◽  
Fractional Laplacian Operator

The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( − Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform.

scholarly journals Hermite-Hadamard Type Inequalities Associated with Coordinated((s,m),QC)-Convex Functions

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yu-Mei Bai ◽  
Shan-He Wu ◽  
Ying Wu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Integral Inequalities ◽  
Type Integral ◽  
Definition Of

In this paper, we introduce the definition of coordinated((s,m),QC)-convex function and establish some Hermite-Hadamard type integral inequalities for coordinated((s,m),QC)-convex functions.

On generic hypersurfaces in ℝn

1981 ◽  
Vol 90 (3) ◽  
pp. 389-394 ◽  
Author(s):  
J. W. Bruce
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Dense Subset ◽  
Precise Definition ◽  
Open Dense Subset ◽  
Curves And Surfaces ◽  
Definition Of

In this paper we consider certain questions concerning the differential geometry of generic hypersurfaces in ℝn. Our results prove, for example, that the curve of rib points of a generic surface in ℝ3 has transverse self-intersections.In (4) Porteous discussed (amongst other things) the generic geometry of curves and surfaces in ℝ3. Subsequently Looijenga ((3) and see also (5)) gave a more precise definition of the term generic and showed that an open dense subset of smooth embeddings of manifolds in Euclidean space were indeed generic.

scholarly journals THE SOq(N, R)-SYMMETRIC HARMONIC OSCILLATOR ON THE QUANTUM EUCLIDEAN SPACE ${\bf R}_q^N$ AND ITS HILBERT SPACE STRUCTURE

1993 ◽  
Vol 08 (26) ◽  
pp. 4679-4729 ◽  
Author(s):  
GAETANO FIORE
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Harmonic Oscillator ◽  
Mechanical Model ◽  
Scalar Product ◽  
Space Structure ◽  
Quantum Mechanical ◽  
Quantum Space ◽  
Isotropic Harmonic Oscillator ◽  
Definition Of

We show that the isotropic harmonic oscillator in the ordinary Euclidean space RN (N≥3) admits a natural q-deformation into a new quantum-mechanical model having a q-deformed symmetry (in the sense of quantum groups), SO q(N, R). The q-deformation is the consequence of replacing RN by [Formula: see text] (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over [Formula: see text], which we use for the definition of the scalar product of states.

scholarly journals Certain parameterized inequalities arising from fractional integral operators with exponential kernels

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1707-1724
Author(s):  
Zhengrong Yuan ◽  
Taichun Zhou ◽  
Qiang Zhang ◽  
Tingsong Du
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Cumulative Distribution ◽  
Digamma Function ◽  
Fractional Integral Operators ◽  
Definition Of ◽  
Fractional Type

We utilize the definition of a fractional integral operators, which was presented by Ahmad et al., to investigate a general fractional-type identity with a parameter. We establish certain parameterized fractional integral inequalities based on this identity, and provide two examples to illustrate the obtained results. Also, these results derived in this paper are applied to the estimations of q-digamma function, divergence measures and cumulative distribution function, respectively.

The convergence of a sum of independent random variables

1963 ◽  
Vol 59 (2) ◽  
pp. 411-416
Author(s):  
G. De Barra ◽  
N. B. Slater
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Rate Of Convergence ◽  
Radial Distribution Function ◽  
Random Variables ◽  
Covariance Matrices ◽  
Independent Random Variables ◽  
Second Moments ◽  
Image Position ◽  
Real Euclidean Space

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

scholarly journals On the convergence of a Newton-like method inℝnand the use of Berinde's exit criterion

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
Rabindranath Sen ◽  
Sulekha Mukherjee ◽  
Rajesh Patra
Keyword(s):  
Euclidean Space ◽  
Newton’S Method ◽  
Newton's Method ◽  
Scalar Equation ◽  
Dimensional Euclidean Space ◽  
Definition Of

Berinde has shown that Newton's method for a scalar equationf(x)=0converges under some conditions involving onlyfandf′and notf″when a generalized stopping inequality is valid. Later Sen et al. have extended Berinde's theorem to the case where the condition thatf′(x)≠0need not necessarily be true. In this paper we have extended Berinde's theorem to the class ofn-dimensional equations,F(x)=0, whereF:ℝn→ℝn,ℝndenotes then-dimensional Euclidean space. We have also assumed thatF′(x)has an inverse not necessarily at every point in the domain of definition ofF.

scholarly journals The Observable Representation

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 310
Author(s):  
L. Schulman
Keyword(s):  
Phase Transition ◽  
Euclidean Space ◽  
Complex Systems ◽  
Random Walks ◽  
Stochastic Dynamics ◽  
Dimensional Euclidean Space ◽  
Significant Feature ◽  
Original Space ◽  
Definition Of ◽  
Low Dimensional

The observable representation (OR) is an embedding of the space on which a stochastic dynamics is taking place into a low dimensional Euclidean space. The most significant feature of the OR is that it respects the dynamics. Examples are given in several areas: the definition of a phase transition (including metastable phases), random walks in which the OR recovers the original space, complex systems, systems in which the number of extrema exceed convenient viewing capacity, and systems in which successful features are displayed, but without the support of known theorems.

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