scholarly journals A dichotomy of sets via typical differentiability

2020 ◽  
Vol 8 ◽  
Author(s):  
Michael Dymond ◽  
Olga Maleva
Keyword(s):  
Euclidean Space ◽  
Lipschitz Function ◽  
Analytic Subset ◽  
Typical Behaviour ◽  
Opposite Extreme

Abstract We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function: namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has a zero-length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: in any such coverable set, a typical Lipschitz function is everywhere severely non-differentiable.

scholarly journals Lipschitz Functions on Expanders are Typically Flat

2013 ◽  
Vol 22 (4) ◽  
pp. 566-591 ◽  
Author(s):  
RON PELED ◽  
WOJCIECH SAMOTIJ ◽  
AMIR YEHUDAYOFF
Keyword(s):  
Exponential Decay ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Expander Graphs ◽  
Double Exponential ◽  
Random Integer ◽  
Typical Behaviour ◽  
Good Expansion

This work studies the typical behaviour of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations. For instance, we show that a uniformly chosen M-Lipschitz function takes only M+1 values on most of the graph, with a double exponential decay for the probability of taking other values.

The French Directory: Mirage of the Moderates

2017 ◽  
Author(s):  
R. R. Palmer
Keyword(s):  
The Political ◽  
Middle Classes ◽  
Constitutional Regime ◽  
Opposite Extreme

This chapter details events following the end of the Terror and the political and emotional crisis of the Year II. The question that a great many Frenchmen put to themselves both in France and in the emigration, and a question to which observers throughout Europe and America awaited the answer, was whether some kind of moderate or constitutional regime would be durably established. The next four years showed that constitutional quietude was still far away. The difficulty was that not everyone agreed on what either moderation or justice should consist in. Justice, for some, required the punishment of all revolutionaries and their sympathizers. For others, it meant a continuing battle against kings, priests, aristocrats, and the comfortable middle classes. Both groups saw in “moderation” a mere tactic of the opposition, and moderates as the dupes of the opposite extreme. Compromise for them meant the surrender of principle. It meant truckling with an enemy that could never be trusted, and had no real intention of compromise.

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

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