scholarly journals Weighted integrability of polyharmonic functions in the higher-dimensional case

2021 ◽  
Vol 14 (7) ◽  
pp. 2047-2068
Author(s):  
Congwen Liu ◽  
Antti Perälä ◽  
Jiajia Si
Keyword(s):  
Dimensional Case ◽  
Polyharmonic Functions ◽  
Higher Dimensional ◽  
Weighted Integrability

A Graphical Exposition of the Ising Problem

1971 ◽  
Vol 12 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

scholarly journals AN ALTERNATIVE DIMENSIONAL REDUCTION PRESCRIPTION

1996 ◽  
Vol 11 (13) ◽  
pp. 1037-1045 ◽  
Author(s):  
J.D. EDELSTEIN ◽  
C. NÚÑEZ ◽  
F.A. SCHAPOSNIK ◽  
J.J. GIAMBIAGI
Keyword(s):  
Dimensional Reduction ◽  
Dimensional Space ◽  
Dimensional Case ◽  
Green Functions ◽  
Two Dimensional ◽  
Spatial Coordinate ◽  
Higher Dimensional

We propose an alternative dimensional reduction prescription which in respect with Green functions corresponds to dropping the extra spatial coordinate. From this, we construct the dimensionally reduced Lagrangians both for scalars and fermions, discussing bosonization and supersymmetry in the particular two-dimensional case. We argue that our proposal is in some situations more physical in the sense that it maintains the form of the interactions between particles thus preserving the dynamics corresponding to the higher-dimensional space.

TIME LOWER BOUNDS FOR PERMUTATION ROUTING ON MULTI-DIMENSIONAL BUSED MESHES

1993 ◽  
Vol 03 (02) ◽  
pp. 129-138
Author(s):  
STEVEN CHEUNG ◽  
FRANCIS C.M. LAU
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Routing Problem ◽  
Higher Dimensional ◽  
Low Dimensional ◽  
Permutation Routing

We present time lower bounds for the permutation routing problem on three- and higher-dimensional n x…x n meshes with buses. We prove an (r–1)n/r lower bound for the general case of an r-dimensional bused mesh, r≥2, which is not as strong for low-dimensional as for higher-dimensional cases. We then use a different approach to construct a 0.705n lower bound for the three-dimensional case.

The structure of rank-one convex quadratic forms on linear elastic strains

2003 ◽  
Vol 133 (1) ◽  
pp. 213-224 ◽  
Author(s):  
Kewei Zhang
Keyword(s):  
Quadratic Forms ◽  
Morse Index ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Linear Elastic ◽  
Direct Sum ◽  
Higher Dimensional ◽  
Rank One ◽  
Finite Set ◽  
Elastic Strains

We classify the Morse indices for rank-convex quadratic forms defined on the space of linear elastic strains in two- and three-dimensional linear elasticity. For the higher-dimensional case n > 3, we give a universal lower bound of the largest possible Morse index and various upper bound of this index. We show in the three-dimensional case that the Morse index is at most 1, and in this case the nullity cannot exceed 2. Examples are given that show that the estimates can be reached. We apply the results to study the critical points for smooth rank-one convex functions defined on the space of linear strains. We also examine an example and construct a quasiconvex function that vanishes in a finite set in the direct sum of the null subspace and the negative subspace of the rank-one quadratic form.

Higher Dimensional Asymptotic Cycles

2003 ◽  
Vol 55 (3) ◽  
pp. 636-648 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Invariant Measure ◽  
Compact Manifold ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Stationary Points ◽  
Dimensional Case ◽  
One Dimensional ◽  
Smooth Flow ◽  
Higher Dimensional ◽  
The One

AbstractGiven a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure τ, Sullivan has defined an element of Hp(Mn; R). This generalized the notion of a μ-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure μ. In this one-dimensional case there was a natural 1—1 correspondence between transversal invariant measures τ and invariant measures μ when one had a smooth flow without stationary points.For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

scholarly journals Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1880
Author(s):  
Artur Kobus ◽  
Jan L. Cieśliński
Keyword(s):  
Building Block ◽  
Differential Calculus ◽  
Dimensional Space ◽  
Dimensional Case ◽  
Main Building ◽  
Higher Dimensional ◽  
Algebraic Operations ◽  
Low Dimensional

The scator space, introduced by Fernández-Guasti and Zaldívar, is endowed with a product related to the Lorentz rule of addition of velocities. The scator structure abounds with definitions calculationally inconvenient for algebraic operations, like lack of the distributivity. It occurs that situation may be partially rectified introducing an embedding of the scator space into a higher-dimensonal space, that behaves in a much more tractable way. We use this opportunity to comment on the geometry of automorphisms of this higher dimensional space in generic setting. In parallel, we develop commutative-hypercomplex analogue of differential calculus in a certain, specific low-dimensional case, as also leaned upon the notion of fundamental embedding, therefore treating the map as the main building block in completing the theory of scators.

scholarly journals On the Explicite Defining Relations of Abelian Schemes of Level Three

1966 ◽  
Vol 27 (1) ◽  
pp. 143-157 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Abelian Varieties ◽  
Dimensional Case ◽  
Plane Curves ◽  
Complex Numbers ◽  
Defining Relations ◽  
Higher Dimensional ◽  
Dimension One ◽  
Abelian Schemes

It is known classically that abelian varieties of dimension one over the field of complex numbers may be expressed by non-singular Hesse’s canonical cubic plane curves, The purpose of the present paper is to generalize this idea to higher dimensional case.

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