Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

Stravinsky's Chords (II)

Tempo ◽  
1966 ◽  
pp. 2-9
Author(s):  
G. W. Hopkins
Keyword(s):  
Harmonic Analysis ◽  
Harmonic Functions ◽  
Extended Study ◽  
Middle Period ◽  
Soldier's Tale

In The Soldier's Tale there were instances of musical parody which could not possibly be elucidated by simple harmonic analysis within the work's own stylistic terms of reference; for example, it was necessary to allude to Bach's style in order to discuss the harmonies of Stravinsky's chorales. While it is not always correct to view neo-classical works as parodies, generically they present a ‘gloss’ on earlier styles—so that it becomes impossible to analyse such works without reference back, sometimes even to a particular work of a particular forerunner. Clearly an extended study of the use of classical harmonic functions in the works of Stravinsky's middle period would be an undertaking worthy of some months of research and several chapters of exegesis. Here I shall content myself with some instances of his use of cadential formulae.

Randall Dougherty and Alexander S. Kechris. The complexity of antidifferentiation. Advances in mathematics, vol. 88 (1991), pp. 145–169. - Ferenc Beleznay and Matthew Foreman. The collection of distal flows is not Borel. American journal of mathematics, vol. 117 (1995), pp. 203–239. - Ferenc Beleznay and Matthew Foreman. The complexity of the collection of measure-distal transformations. Ergodic theory and dynamical systems, vol. 16 (1996), pp. 929–962. - Howard Becker. Pointwise limits of subsequences and sets. Fundamenta mathematicae, vol. 128 (1987), pp. 159–170. - Howard Becker, Sylvain Kahane, and Alain Louveau. Some complete sets in harmonic analysis. Transactions of the American Mathematical Society, vol. 339 (1993), pp. 323–336. - Robert Kaufman. PCA sets and convexity Fundamenta mathematicae, vol. 163 (2000), pp. 267–275). - Howard Becker. Descriptive set theoretic phenomena in analysis and topology. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematical Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 1–25.

2001 ◽  
Vol 7 (3) ◽  
pp. 385-388
Author(s):  
Gabriel Debs
Keyword(s):  
New York ◽  
Dynamical Systems ◽  
Harmonic Analysis ◽  
Ergodic Theory ◽  
Set Theory ◽  
American Mathematical Society ◽  
And Topology ◽  
Mathematical Sciences ◽  
Descriptive Set ◽  
The Continuum

scholarly journals MIXED NORM INEQUALITIES FOR SOME DIRECTIONAL MAXIMAL OPERATORS

2012 ◽  
Vol 86 (3) ◽  
pp. 448-455
Author(s):  
DAH-CHIN LUOR
Keyword(s):  
Harmonic Analysis ◽  
Maximal Functions ◽  
Maximal Operators ◽  
Mixed Norm ◽  
Weighted Inequalities ◽  
Norm Inequalities ◽  
One Dimensional

AbstractMixed norm inequalities for directional operators are closely related to the boundedness problems of several important operators in harmonic analysis. In this paper we prove weighted inequalities for some one-dimensional one-sided maximal functions. Then by applying these results, we establish mixed norm inequalities for directional maximal operators which are defined from these one-dimensional maximal functions. We also estimate the constants in these inequalities.

scholarly journals Harmonic Functions on Multiplicative Graphs and Interpolation Polynomials

10.37236/1506 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Alexei Borodin ◽  
Grigori Olshanski
Keyword(s):  
Symmetric Group ◽  
Harmonic Analysis ◽  
Harmonic Functions ◽  
Symmetric Functions ◽  
Multivariate Interpolation ◽  
Infinite Symmetric Group ◽  
Interpolation Polynomials

We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type integrals.

scholarly journals A harmonic analysis of exponential decay

2020 ◽  
Author(s):  
David Lawunmi ◽  
Soodamani Ramalingam
Keyword(s):  
Harmonic Analysis ◽  
Exponential Function ◽  
Exponential Decay ◽  
Harmonic Functions ◽  
Exponential Functions ◽  
Single Exponential Function ◽  
Single Exponential

We analyse the decay of a single exponential function and develop an algorithm to determine the exponent and the constant, C, (C exp(-kt)) associated with this function . In essence this approach involves `transforming' exponential functions into harmonic functions. This manoeuvre allows techniques that are used to analyse harmonic functions to be used to characterise decaying exponential functions.

The Axiom of Choice and the Partition Principle from Dialectica Categories

2020 ◽  
Author(s):  
Samuel G Da Silva
Keyword(s):  
Set Theory ◽  
Open Problem ◽  
Axiom Of Choice ◽  
Full Generality ◽  
Cardinal Invariants ◽  
Partition Principle ◽  
The Axiom Of Choice ◽  
The Continuum

Abstract The method of morphisms is a well-known application of Dialectica categories to set theory (more precisely, to the theory of cardinal invariants of the continuum). In a previous work, Valeria de Paiva and the author have asked how much of the Axiom of Choice is needed in order to carry out the referred applications of such method. In this paper, we show that, when considered in their full generality, those applications of Dialectica categories give rise to equivalents (within $\textbf{ZF}$) of either the Axiom of Choice ($\textbf{AC}$) or Partition Principle ($\textbf{PP}$)—which is a consequence of $\textbf{AC}$ whose precise status of its relationship with$\textbf{AC}$ itself is an open problem for more than a hundred years.

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