scholarly journals Exponential Dichotomies of a Non-Classical Equation in Spaces of Differential Forms on a Two-Dimensional Torus with "Noises"

2019 ◽  
Vol 6 (3) ◽  
pp. 26-38
Author(s):  
O.G. Kitaeva ◽  
Keyword(s):  
Differential Forms ◽  
Classical Equation ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Exponential Dichotomies

scholarly journals Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Pavlo Gavrylenko ◽  
Alessandro Tanzini
Keyword(s):  
Integrable System ◽  
Gauge Theories ◽  
The Self ◽  
Free Fermion ◽  
Two Dimensional ◽  
Isomonodromic Deformation ◽  
Specific Time ◽  
Dimensional Torus ◽  
Isomonodromic Deformations ◽  
Deformation Problem

AbstractWe study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

scholarly journals Stable Arcs Connecting Polar Cascades on a Torus

2021 ◽  
Vol 17 (1) ◽  
pp. 23-37
Author(s):  
O. V. Pochinka ◽  
◽  
E. V. Nozdrinova ◽  
Keyword(s):  
Two Dimensional ◽  
Dimensional Torus ◽  
Positive Orientation ◽  
Stable Isotopic ◽  
Orientation Type

In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.

scholarly journals Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

2020 ◽  
pp. 2150006
Author(s):  
Denis Bonheure ◽  
Jean Dolbeault ◽  
Maria J. Esteban ◽  
Ari Laptev ◽  
Michael Loss
Keyword(s):  
Magnetic Field ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Euclidean Spaces ◽  
Hardy Inequalities ◽  
Interpolation Inequalities ◽  
Nonlinear Interpolation ◽  
Magnetic Potentials ◽  
Aharonov Bohm

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

scholarly journals Thermodynamics of the Katok map

2017 ◽  
Vol 39 (3) ◽  
pp. 764-794 ◽  
Author(s):  
Y. PESIN ◽  
S. SENTI ◽  
K. ZHANG
Keyword(s):  
Limit Theorem ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Existence Of Equilibrium ◽  
Equilibrium Measures ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Continuous Potential ◽  
Uniqueness Of Equilibrium ◽  
Unstable Direction

We effect the thermodynamical formalism for the non-uniformly hyperbolic $C^{\infty }$ map of the two-dimensional torus known as the Katok map [Katok. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2)110(3) 1979, 529–547]. It is a slow-down of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $\unicode[STIX]{x1D711}_{t}=-t\log \mid df|_{E^{u}(x)}|$ for any $t\in (t_{0},\infty )$, $t\neq 1$, where $E^{u}(x)$ denotes the unstable direction. We show that $t_{0}$ tends to $-\infty$ as the domain of the perturbation shrinks to zero. Finally, we establish exponential decay of correlations as well as the central limit theorem for the equilibrium measures associated to $\unicode[STIX]{x1D711}_{t}$ for all values of $t\in (t_{0},1)$.

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