scholarly journals Spectral properties of the iterated Laplacian with a potential in a punctured domain

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2897-2900 ◽  
Author(s):  
Gulzat Nalzhupbayeva
Keyword(s):  
Spectral Properties ◽  
Spatial Dimension ◽  
Trace Formulas ◽  
Laplace Operators ◽  
Regularized Trace ◽  
Spatial Dimensions ◽  
Trace Formulae ◽  
Punctured Domain ◽  
Nonharmonic Analysis

In the work we derive regularized trace formulas which were established in papers of Kanguzhin and Tokmagambetov for the Laplace and m-Laplace operators in a punctured domain with the fixed iterating order m 2 N. By using techniques of Sadovnichii and Lyubishkin, the authors in that papers described regularized trace formulae in the spatial dimension d = 2. In this note one claims that the formulas are also true for more general operators in the higher spatial dimensions, namely, 2 ? d ? 2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called ?nonharmonic? analysis.

scholarly journals p-Laplace Operators for Oriented Hypergraphs

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Dong Zhang
Keyword(s):  
Laplace Operator ◽  
Spectral Properties ◽  
General Setting ◽  
Laplace Operators

AbstractThe p-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex p-Laplacian and a hyperedge p-Laplacian are defined for oriented hypergraphs, for all p ≥ 1. Several spectral properties of these operators are investigated.

scholarly journals Partition Function in One, Two, and Three Spatial Dimensions from Effective Lagrangian Field Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Christoph P. Hofmann
Keyword(s):  
Partition Function ◽  
Degrees Of Freedom ◽  
Spatial Dimension ◽  
Lagrangian Method ◽  
Point Of View ◽  
Low Energy ◽  
Effective Theory ◽  
Goldstone Bosons ◽  
Effective Lagrangian ◽  
Spatial Dimensions

The systematic effective Lagrangian method was first formulated in the context of the strong interaction; chiral perturbation theory (CHPT) is the effective theory of quantum chromodynamics (QCD). It was then pointed out that the method can be transferred to the nonrelativistic domain—in particular, to describe the low-energy properties of ferromagnets. Interestingly, whereas for Lorentz-invariant systems the effective Lagrangian method fails in one spatial dimension (ds=1), it perfectly works for nonrelativistic systems in ds=1. In the present brief review, we give an outline of the method and then focus on the partition function for ferromagnetic spin chains, ferromagnetic films, and ferromagnetic crystals up to three loops in the perturbative expansion—an accuracy never achieved by conventional condensed matter methods. We then compare ferromagnets in ds=1, 2, 3 with the behavior of QCD at low temperatures by considering the pressure and the order parameter. The two apparently very different systems (ferromagnets and QCD) are related from a universal point of view based on the spontaneously broken symmetry. In either case, the low-energy dynamics is described by an effective theory containing Goldstone bosons as basic degrees of freedom.

General variational model reduction applied to incompressible viscous flows

2008 ◽  
Vol 617 ◽  
pp. 31-50 ◽  
Author(s):  
THORSTEN BOGNER
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Reynolds Numbers ◽  
Spatial Dimension ◽  
Reduced Model ◽  
Navier Stokes ◽  
Variational Model ◽  
Navier Stokes Equation ◽  
Nonlinear Dynamical ◽  
Density Matrix Renormalization ◽  
Spatial Dimensions

In this paper, a method is introduced that allows calculation of an approximate proper orthogonal decomposition (POD) without the need to perform a simulation of the full dynamical system. Our approach is based on an application of the density matrix renormalization group (DMRG) to nonlinear dynamical systems, but has no explicit restriction on the spatial dimension of the model system. The method is not restricted to fluid dynamics. The applicability is exemplified on the incompressible Navier–Stokes equation in two spatial dimensions. Merging of two equal-signed vortices with periodic boundary conditions is considered for low Reynolds numbers Re≤800 using a spectral method. We compare the accuracy of a reduced model, obtained by our method, with that of a reduced model obtained by standard POD. To this end, error functionals for the reductions are evaluated. It is observed that the proposed method is able to find a reduced system that yields comparable or even superior accuracy with respect to standard POD method results.

scholarly journals Essentially isospectral transformations and their applications

2017 ◽  
Author(s):  
Namig J. Guliyev
Keyword(s):  
Boundary Conditions ◽  
Existence Theorems ◽  
Dirichlet Boundary Conditions ◽  
Trace Formulas ◽  
Asymptotics Of Eigenvalues ◽  
Regularized Trace ◽  
Eigenvalue Parameter ◽  
Nevanlinna Functions ◽  
Sturm Liouville ◽  
Uniqueness And Existence

We define and study the properties of Darboux-type transformations between Sturm–Liouville problems with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

scholarly journals Effective string theory inspired potential and meson masses in higher dimension

2016 ◽  
Vol 94 (12) ◽  
pp. 1282-1288 ◽  
Author(s):  
Sabyasachi Roy ◽  
D.K. Choudhury
Keyword(s):  
Planck Scale ◽  
String Model ◽  
Spatial Dimension ◽  
Wave Functions ◽  
Meson Mass ◽  
Higher Dimension ◽  
Dimensional Dependence ◽  
Spatial Dimensions ◽  
The Masses ◽  
Effective String Theory

Nambu–Goto action in classical bosonic string model for hadrons predicts quark-antiquark potential to be (Lüscher and Weisz. JHEP 07, 049 (2002). doi: 10.1088/1126-6708/2002/07/049 ) V(r) = −(γ/r) + σr + μ0. In this report we present studies of masses of heavy flavour mesons in higher dimension with our recently developed wave functions obtained following string inspired potential. We report the dimensional dependence of the masses of mesons. Our results suggest that as the meson mass increases with the number of extra spatial dimensions, it will attain the Planck scale (∼1019 GeV) asymptotically at an astronomically large spatial dimension (we call it Planck dimension) DPlanck ∼ 1011, which sets the limit of applicability of Schrodinger equation in large dimension.

Numerical tensor calculus

Acta Numerica ◽  
2014 ◽  
Vol 23 ◽  
pp. 651-742 ◽  
Author(s):  
Wolfgang Hackbusch
Keyword(s):  
Large Scale ◽  
Spatial Dimension ◽  
Grid Function ◽  
Numerical Representation ◽  
Regular Grid ◽  
Tensor Calculus ◽  
Spatial Dimensions ◽  
Tensor Spaces ◽  
Numerical Performance ◽  
Grid Points

The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid withngrid points per direction, a spatial dimensiondyieldsndgrid points. A grid function defined on such a grid is an example of a tensor of orderd. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly ind. In this way, data of sizend= 10001000can also be treated.This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.

Beyond the Assimilation Fixation: Skinner and the Possibility of a Spatial Approach to Twentieth-Century Thai History

2005 ◽  
Vol 1 (2) ◽  
pp. 184-216 ◽  
Author(s):  
Michael Montesano
Keyword(s):  
Twentieth Century ◽  
National Level ◽  
Spatial Dimension ◽  
Chinese Society ◽  
Chinese History ◽  
Spatial Change ◽  
Systems Model ◽  
Spatial Dimensions ◽  
Asian History

AbstractG. William Skinner's early work on the Chinese of Thailand anticipated the spatial concerns that he later brought to the study of Chinese history. The present article revisits Skinner's 1957 classic “Chinese Society in Thailand” to highlight its overlooked spatial dimension and its emphasis on the role of Chinese in patterns of spatial change in Thai history. It then applies the formal approaches pioneered in Skinner's work on spatial dimensions of Chinese history to the Thai case. A two-factor regional-systems model for twentieth-century Thailand is developed in explicit imitation of Skinner's modeling of China's “macroregions.” The model illustrates long-term trends toward the tighter integration of Thailand's Bangkok-centered national-level regional system, the importance of numerous patterns of more local spatial change, the significance of extra-systemic influences on the system, and the role of Chinese as significant participants and agents in each of these processes. Results also suggest the need for further work on spatial dimensions of modern Thai and Southeast Asian history and on the role of Chinese as agents of spatial change in the region.

scholarly journals Resilience Strategy In Contemporary Iraqi City: Baghdad City as a Case Study

2020 ◽  
Vol 55 (1) ◽  
Author(s):  
Enas Dhiyaa Hadi ◽  
Abdul Hussain Alaskary
Keyword(s):  
Formal Theory ◽  
Research Problem ◽  
Spatial Dimension ◽  
Main Research ◽  
Spatial Dimensions ◽  
Semantic Reference ◽  
Key Terms ◽  
Difficult Situations

Over the past decade, the resilience concept has gained great importance in climate change, sustainability, and city disaster researches. To tackle the problem that follows the concept, this dissertation posits a formal theory of resilience. In this sense, resilience provides a semantic reference frame for city risks and disasters. Key terms, which fall under the purview of resilience, are defined. The research problem was crystallized to be formulated as “Our cities of today face many challenges and sudden shocks that are difficult to be predicted,” and “Negligence of the disparities in spatial competence has caused difficult situations to face sudden challenges and shocks to reach the more resilient city.” The aims of this research: 1) to build a conceptual framework for the concept of resilient cities, as well as the determination of spatial competence that has a significant impact on varying levels of resilience in places; 2) to reach a resilient Iraqi city strategy by adopting five more vital areas in Baghdad city. The research hypothesis is as follows: Adopting the presence of spatial competence in the area will facilitate the process of making resilient cities to face risks and sudden shocks. To achieve the research objective, the theoretical framework, built to consist the main research conclusion, was that there is spatial competence and place efficiency in any spatial dimension, which makes it difficult to deal with each place in the same way; every place has its own privacy and accessibility to its best strategies in temporal and spatial dimensions.

scholarly journals Quantum Computing in Four Spatial Dimensions

2019 ◽  
Author(s):  
Arturo Tozzi ◽  
Muhammad Zubair Ahmad ◽  
James F. Peters
Keyword(s):  
Set Theory ◽  
Quantum Hall Effect ◽  
Topological Charge ◽  
Electromagnetic Waves ◽  
Quantum Computer ◽  
Quantum Hall ◽  
Higher Dimensions ◽  
Spatial Dimension ◽  
Optical Computers ◽  
Spatial Dimensions

Relationships among near set theory, shape maps and recent accounts of the Quantum Hall effect pave the way to quantum computations performed in higher dimensions.  We illustrate the operational procedure to build a quantum computer able to detect, assess and quantify a fourth spatial dimension.  We show how, starting from two-dimensional shapes embedded in a 2D topological charge pump, it is feasible to achieve the corresponding four-dimensional shapes, which encompass a larger amount of information.  This novel, relatively straightforward architecture not only permits to increase the amount of available qbits in a fixed volume, but also converges towards a solution to the problem of optical computers, that are not allowed to tackle quantum entanglement through their canonical superposition of electromagnetic waves.

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