scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

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 A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

2002 ◽  
Vol 65 (2) ◽  
pp. 253-258 ◽  
Author(s):  
Hitoshi Tanaka
Keyword(s):  
Sobolev Space ◽  
Maximal Function ◽  
60Th Birthday ◽  
One Dimensional ◽  
Partial Derivatives ◽  
First Order ◽  
Littlewood Maximal Function ◽  
The One ◽  
Derivatives Of

Dedicated to Professor Kôzô Yabuta on the occasion of his 60th birthdayJ. Kinnunen proved that of P > 1, d ≤ 1 and f is a function in the Sobolev space W1,P(Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function ℳf belong to LP(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.

scholarly journals Extended Conformal Symmetry of the One-Dimensional Bose Gas

1997 ◽  
Vol 12 (29) ◽  
pp. 2153-2159 ◽  
Author(s):  
Milena Maule ◽  
Stefano Sciuto
Keyword(s):  
Cartan Subalgebra ◽  
Conformal Symmetry ◽  
Hard Core ◽  
Bose Gas ◽  
Effective Hamiltonian ◽  
Coupling Constant ◽  
Free Fermions ◽  
One Dimensional ◽  
First Order ◽  
The One

We show that the low-lying excitations of the one-dimensional Bose gas are described, at all orders in a 1/N expansion and at the first order in the inverse of the coupling constant, by an effective Hamiltonian written in terms of an extended conformal algebra, namely the Cartan subalgebra of the [Formula: see text] algebra. This enables us to construct the first interaction term which corrects the Hamiltonian of the free fermions equivalent to a hard-core boson system.

scholarly journals Non-unitary dynamics of Sachdev-Ye-Kitaev chain

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Chunxiao Liu ◽  
Pengfei Zhang ◽  
Xiao Chen
Keyword(s):  
Conformal Symmetry ◽  
Entangled State ◽  
One Dimensional ◽  
First Order ◽  
Large N ◽  
First Order Phase Transition ◽  
Free Fermion Model ◽  
Kitaev Model ◽  
The One ◽  
First Order Phase

We construct a series of one-dimensional non-unitary dynamics consisting of both unitary and imaginary evolutions based on the Sachdev-Ye-Kitaev model. Starting from a short-range entangled state, we analyze the entanglement dynamics using the path integral formalism in the large N limit. Among all the results that we obtain, two of them are particularly interesting: (1) By varying the strength of the imaginary evolution, the interacting model exhibits a first order phase transition from the highly entangled volume law phase to an area law phase; (2) The one-dimensional free fermion model displays an extensive critical regime with emergent two-dimensional conformal symmetry.

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 THE EFFECT OF OFF-RAMP ON THE ONE-DIMENSIONAL CELLULAR AUTOMATON TRAFFIC FLOW WITH OPEN BOUNDARIES

2004 ◽  
Vol 18 (16) ◽  
pp. 2347-2360 ◽  
Author(s):  
HAMID EZ-ZAHRAOUY ◽  
ZOUBIR BENRIHANE ◽  
ABDELILAH BENYOUSSEF
Keyword(s):  
Cellular Automaton ◽  
Traffic Flow ◽  
Average Density ◽  
Constant Current ◽  
Current Phase ◽  
One Dimensional ◽  
First Order ◽  
Open Boundaries ◽  
Flow Phase ◽  
The One

The effect of the position of the off-ramp (way out), on the traffic flow phase transition is investigated using numerical simulations in the one-dimensional cellular automaton traffic flow model with open boundaries using parallel dynamics. When the off-ramp is located between two critical positions ic1 and ic2 the current increases with the extracting rate β0, for β0<β0c1, and exhibits a plateau (constant current) for β0c1<β0<β0c2 and decreases with β0 for β0>β0c2. However, the density undergoes two successive first order transitions: from high density to plateau current phase at β0=β0c1; and from average density to the low one at β0=β0c2. In the case of two off-ramps located respectively at i1 and i2, these transitions occur only when i2-i1 is smaller than a critical value. Phase diagrams in the (α,β0), (β,β0) and (i1,β0) planes are established. It is found that the transitions between free traffic (FT), congested traffic (CT) and plateau current (PC) phases are of first order. The first order line transition in (i1,β0)-phase diagram terminates by an end point above which the transition disappears.

scholarly journals Approximate Symmetries of the Harry Dym Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Mehdi Nadjafikhah ◽  
Parastoo Kabi-Nejad
Keyword(s):  
Lie Algebra ◽  
Optimal System ◽  
Differential Invariants ◽  
Approximate Symmetries ◽  
One Dimensional ◽  
First Order ◽  
Optimal System Of Subalgebras ◽  
Dym Equation ◽  
The One ◽  
Harry Dym Equation

We derive the first-order approximate symmetries for the Harry Dym equation by the method of approximate transformation groups proposed by Baikov et al. (1989, 1996). Moreover, we investigate the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. We compute the one-dimensional optimal system of subalgebras as well as point out some approximately differential invariants with respect to the generators of Lie algebra and optimal system.

Restricted orbit equivalence for ergodic ${\Bbb Z}^{d}$ actions I

1997 ◽  
Vol 17 (5) ◽  
pp. 1083-1129 ◽  
Author(s):  
JANET WHALEN KAMMEYER ◽  
DANIEL J. RUDOLPH
Keyword(s):  
Equivalence Relation ◽  
Dimensional Case ◽  
Orbit Equivalence ◽  
One Dimensional ◽  
Ergodic Transformations ◽  
Higher Dimensional ◽  
The One ◽  
Equivalence Invariant ◽  
Classical Entropy

In [R1] a notion of restricted orbit equivalence for ergodic transformations was developed. Here we modify that structure in order to generalize it to actions of higher-dimensional groups, in particular ${\Bbb Z}^d$-actions. The concept of a ‘size’ is developed first from an axiomatized notion of the size of a permutation of a finite block in ${\Bbb Z}^d$. This is extended to orbit equivalences which are cohomologous to the identity and, via the natural completion, to a notion of restricted orbit equivalence. This is shown to be an equivalence relation. Associated to each size is an entropy which is an equivalence invariant. As in the one-dimensional case this entropy is either the classical entropy or is zero. Several examples are discussed.

One-dimensional Hardy-type inequalities in many dimensions

1998 ◽  
Vol 128 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Gord Sinnamon
Keyword(s):  
Integral Operators ◽  
Weighted Norm ◽  
Weighted Inequalities ◽  
Weighted Norm Inequalities ◽  
Averaging Operators ◽  
One Dimensional ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Higher Dimensional ◽  
The One

Weighted inequalities for certain Hardy-type averaging operators in are shown to be equivalent to weighted inequalities for one-dimensional operators. Known results for the one-dimensional operators are applied to give weight characterisations, with best constants in some cases, in the higher-dimensional setting. Operators considered include averages over all dilations of very general starshaped regions as well as averages over all balls touching the origin. As a consequence, simple weight conditions are given which imply weighted norm inequalities for a class of integral operators with monotone kernels.

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