Vector Hyper-Delta Distribution and Recommendations on How to Apply It

2021 ◽  
Vol 55 (S1) ◽  
pp. 1-7
Author(s):  
V. A. Smagin ◽  
V. P. Bubnov
Keyword(s):  
Delta Distribution

scholarly journals Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hansol Park
Keyword(s):  
Influence Function ◽  
Equilibrium Solution ◽  
Phase Locking ◽  
Small Diameter ◽  
High Dimensional ◽  
Dimensional Sphere ◽  
Sphere Model ◽  
Dirac Delta ◽  
Delta Distribution ◽  
Winfree Model

<p style='text-indent:20px;'>We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death(COD) state from the exponential <inline-formula><tex-math id="M1">\begin{document}$ \ell^1 $\end{document}</tex-math></inline-formula>-stability and the existence of the equilibrium solution.</p>

scholarly journals On the Convolution Equation Related to the Diamond Klein-Gordon Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Amphon Liangprom ◽  
Kamsing Nonlaopon
Keyword(s):  
Convolution Equation ◽  
Dirac Delta ◽  
Delta Distribution ◽  
Singular Distribution ◽  
Klein Gordon ◽  
The Relationship

We study the distributioneαx(♢+m2)kδform≥0, where(♢+m2)kis the diamond Klein-Gordon operator iteratedktimes,δis the Dirac delta distribution,x=(x1,x2,…,xn)is a variable inℝn, andα=(α1,α2,…,αn)is a constant. In particular, we study the application ofeαx(♢+m2)kδfor solving the solution of some convolution equation. We find that the types of solution of such convolution equation, such as the ordinary function and the singular distribution, depend on the relationship betweenkandM.

Superstatistics of the Klein–Gordon equation in deformed formalism for modified Dirac delta distribution

2018 ◽  
Vol 33 (10n11) ◽  
pp. 1850060 ◽  
Author(s):  
S. Sargolzaeipor ◽  
H. Hassanabadi ◽  
W. S. Chung
Keyword(s):  
Deformation Parameter ◽  
Gordon Equation ◽  
Wave Functions ◽  
Boltzmann Factor ◽  
Dirac Delta ◽  
Cornell Potential ◽  
Delta Distribution ◽  
Klein Gordon ◽  
Aharonov Bohm ◽  
Klein Gordon Equation

The Klein–Gordon equation is extended in the presence of an Aharonov–Bohm magnetic field for the Cornell potential and the corresponding wave functions as well as the spectra are obtained. After introducing the superstatistics in the statistical mechanics, we first derived the effective Boltzmann factor in the deformed formalism with modified Dirac delta distribution. We then use the concepts of the superstatistics to calculate the thermodynamics properties of the system. The well-known results are recovered by the vanishing of deformation parameter and some graphs are plotted for the clarity of our results.

PERTURBED SCHRÖDINGER EQUATION WITH SINGULAR POTENTIAL AND INITIAL DATA

2006 ◽  
Vol 08 (04) ◽  
pp. 433-452 ◽  
Author(s):  
MIRJANA STOJANOVIĆ
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Singular Potential ◽  
Singular Integral Equations ◽  
Uniqueness Result ◽  
Stability Estimates ◽  
Delta Distribution ◽  
The Green Function ◽  
Regularized Derivatives

We consider linear Schrödinger equation perturbed by delta distribution with singular potential and the initial data. Due to the singularities appearing in the equation, we introduce two kinds of approximations: the parameter's approximation for potential and the initial data given by mollifiers of different growth and the approximation for the Green function for Schrödinger equation with regularized derivatives. These approximations reduce the perturbed Schrödinger equation to the family of singular integral equations. We prove the existence-uniqueness theorems in Colombeau space [Formula: see text], 1 ≤ p,q ≤ ∞, employing novel stability estimates (w.r.) to singular perturbations for ε → 0, which imply the statements in the framework of Colombeau generalized functions. In particular, we prove the existence-uniqueness result in [Formula: see text] and [Formula: see text] algebra of Colombeau.

scholarly journals On regularizations of the Dirac delta distribution

2016 ◽  
Vol 305 ◽  
pp. 423-447 ◽  
Author(s):  
Bamdad Hosseini ◽  
Nilima Nigam ◽  
John M. Stockie
Keyword(s):  
Dirac Delta ◽  
Delta Distribution

scholarly journals Completeness and Nonclassicality of Coherent States for Generalized Oscillator Algebras

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Kevin Zelaya ◽  
Oscar Rosas-Ortiz ◽  
Zurika Blanco-Garcia ◽  
Sara Cruz y Cruz
Keyword(s):  
Coherent States ◽  
Beam Splitter ◽  
Delta Function ◽  
Oscillator Algebra ◽  
Closure Relation ◽  
Classical Analog ◽  
Delta Distribution ◽  
Nonclassical Properties ◽  
Generalized Oscillator ◽  
Oscillator Algebras

The purposes of this work are (1) to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form and (2) to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are P-represented by a delta function. In (1) we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer G-function. This property automatically defines the delta distribution as the P-representation of such states. Then, in principle, there must be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the su(1,1) Lie algebra that is realized as a deformation of the oscillator algebra. In (2), we use a beam splitter to show that the nonlinear coherent states exhibit properties like antibunching that prohibit a classical description for them. We also show that these states lack second-order coherence. That is, although the P-representation of the nonlinear coherent states is a delta function, they are not full coherent. Therefore, the systems associated with the generalized oscillator algebras cannot be considered “classical” in the context of the quantum theory of optical coherence.

scholarly journals Evaluation of inverse integral transforms for undergraduate physics students

2012 ◽  
Vol 90 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Aaron Farrell ◽  
Brandon P. van Zyl ◽  
Zachary MacDonald
Keyword(s):  
Nuclear Physics ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Simple Approach ◽  
General Representation ◽  
Central Idea ◽  
Physics Students ◽  
Dirac Delta ◽  
Delta Distribution ◽  
Inverse Transform

We provide a simple approach to the analytical evaluation of inverse integral transforms that does not require any knowledge of complex analysis. The central idea behind our method is to reduce the inverse transform to the solution of an ordinary differential equation. We illustrate the utility of our approach by providing examples of the evaluation of transforms without the use of tables. We also demonstrate how the method may be used to obtain a general representation of a function in the form of a series involving the Dirac delta distribution and its derivatives, which has applications in quantum mechanics, semiclassical, and nuclear physics.

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