scholarly journals Low-Degree Approximation of Random Polynomials

2021 ◽  
Author(s):  
Daouda Niang Diatta ◽  
Antonio Lerario
Keyword(s):  
High Probability ◽  
Random Polynomials ◽  
Zero Set ◽  
Small Probability ◽  
Zero Sets ◽  
Low Degree ◽  
Probabilistic Study ◽  
Exponentially Small

AbstractWe prove that with “high probability” a random Kostlan polynomial in $$n+1$$ n + 1 many variables and of degree d can be approximated by a polynomial of “low degree” without changing the topology of its zero set on the sphere $$\mathbb {S}^n$$ S n . The dependence between the “low degree” of the approximation and the “high probability” is quantitative: for example, with overwhelming probability, the zero set of a Kostlan polynomial of degree d is isotopic to the zero set of a polynomial of degree $$O(\sqrt{d \log d})$$ O ( d log d ) . The proof is based on a probabilistic study of the size of $$C^1$$ C 1 -stable neighborhoods of Kostlan polynomials. As a corollary, we prove that certain topological types (e.g., curves with deep nests of ovals or hypersurfaces with rich topology) have exponentially small probability of appearing as zero sets of random Kostlan polynomials.

Set theoretic properties of Loeb measure

1990 ◽  
Vol 55 (3) ◽  
pp. 1022-1036 ◽  
Author(s):  
Arnold W. Miller
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Zero Set ◽  
Zero Sets ◽  
Martin’S Axiom ◽  
Basic Set ◽  
Nonstandard Models ◽  
The Continuum

AbstractIn this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω1. Define cof(H) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that card(⌊log2(H)⌋) ≤ cof (H) ≤ card(2H), where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes M ≼ N and hyperfinite integer H ∈ M such that H is not enlarged by N, 2H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.

scholarly journals Zero sets of Lie algebras of analytic vector fields on real and complex two-dimensional manifolds

2017 ◽  
Vol 39 (4) ◽  
pp. 954-979 ◽  
Author(s):  
MORRIS W. HIRSCH ◽  
F.-J. TURIEL
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Fixed Point Theorems ◽  
Zero Set ◽  
Analytic Vector ◽  
Ground Field ◽  
Zero Sets ◽  
Finite Dimensional ◽  
Differentiable Vector ◽  
Analytic Vector Field

Let$M$be an analytic connected 2-manifold with empty boundary, over the ground field$\mathbb{F}=\mathbb{R}$or$\mathbb{C}$. Let$Y$and$X$denote differentiable vector fields on$M$. We say that$Y$tracks$X$if$[Y,X]=fX$for some continuous function$f:\,M\rightarrow \mathbb{F}$. A subset$K$of the zero set$\mathsf{Z}(X)$is an essential block for$X$if it is non-empty, compact and open in$\mathsf{Z}(X)$, and the Poincaré–Hopf index$\mathsf{i}_{K}(X)$is non-zero. Let${\mathcal{G}}$be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field$X$. Let$K\subset \mathsf{Z}(X)$be an essential block. Assume that if$M$is complex and$\mathsf{i}_{K}(X)$is a positive even integer, no quotient of${\mathcal{G}}$is isomorphic to$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then${\mathcal{G}}$has a zero in$K$(main result). As a consequence, if$X$and$Y$are analytic,$X$is non-trivial, and$Y$tracks$X$, then every essential component of$\mathsf{Z}(X)$meets$\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

scholarly journals Irreversible Physical Process and Information Height

2021 ◽  
Author(s):  
Qin Liu ◽  
Quan Tang ◽  
Yuling Yi ◽  
Yu Feng
Keyword(s):  
Physical Process ◽  
Physical System ◽  
High Probability ◽  
Physical Phenomenon ◽  
Small Probability ◽  
Simplified Models ◽  
Extra Information ◽  
High Information

The time’s arrow of macroscopic physical phenomenon is reflected by irreversible physical process, which essentially occurs from small probability state to high probability state. In this paper, simplified models are proposed to understand the macroscopic physical process. In order to describe the information of a physical system, we defined the full self-information as "information height" to describe the complexity or difficulty of a macrostate of physical system. In this way, the direction of macroscopic physical process is from high information height to low information height. We can judge the direction of physical process by the information height. If we want the macroscopic physical process to evolve from the low information height state to the high information height state, the system need to add extra information and corresponding energy to increase the information height.

scholarly journals Low-Degree Factors of Random Polynomials

2018 ◽  
Vol 32 (2) ◽  
pp. 1076-1104
Author(s):  
Sean O’Rourke ◽  
Philip Matchett Wood
Keyword(s):  
Random Polynomials ◽  
Low Degree

scholarly journals On the zero set of G-equivariant maps

2009 ◽  
Vol 147 (3) ◽  
pp. 735-755
Author(s):  
P-L. BUONO ◽  
M. HELMER ◽  
J. S. W. LAMB
Keyword(s):  
Bifurcation Theory ◽  
Vector Fields ◽  
Isotropy Subgroup ◽  
Zero Set ◽  
Group Representations ◽  
Equivariant Maps ◽  
Zero Sets ◽  
The Difference ◽  
A Finite Group ◽  
Stratified Set

AbstractLet G be a finite group acting on vector spaces V and W and consider a smooth G-equivariant mapping f: V → W. This paper addresses the question of the zero set of f near a zero x with isotropy subgroup G. It is known from results of Bierstone and Field on G-transversality theory that the zero set in a neighbourhood of x is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near x using only information from the representations V and W. We define an index s(Σ) for isotropy subgroups Σ of G which is the difference of the dimension of the fixed point subspace of Σ in V and W. Our main result states that if V contains a subspace G-isomorphic to W, then for every maximal isotropy subgroup Σ satisfying s(Σ) > s(G), the zero set of f near x contains a smooth manifold of zeros with isotropy subgroup Σ of dimension s(Σ). We also present partial results in the case of group representations V and W which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of G-reversible equivariant vector fields.

scholarly journals Perceiving Prospects Properly

2016 ◽  
Vol 106 (7) ◽  
pp. 1601-1631 ◽  
Author(s):  
Jakub Steiner ◽  
Colin Stewart
Keyword(s):  
Information Processing ◽  
Prospect Theory ◽  
High Probability ◽  
Winner's Curse ◽  
Small Probability ◽  
Winner’S Curse

When an agent chooses between prospects, noise in information processing generates an effect akin to the winner's curse. Statistically unbiased perception systematically overvalues the chosen action because it fails to account for the possibility that noise is responsible for making the preferred action appear to be optimal. The optimal perception pattern exhibits a key feature of prospect theory, namely, overweighting of small probability events (and corresponding underweighting of high probability events). This bias arises to correct for the winner's curse effect. (JEL D11, D81, D82, D83)

scholarly journals Strong measure zero sets without Cohen reals

1993 ◽  
Vol 58 (4) ◽  
pp. 1323-1341 ◽  
Author(s):  
Martin Goldstern ◽  
Haim Judah ◽  
Saharon Shelah
Keyword(s):  
Measure Zero ◽  
Zero Set ◽  
Zero Sets ◽  
Cohen Real ◽  
Strong Measure ◽  
Strong Measure Zero

AbstractIf ZFC is consistent, then each of the following is consistent with :(1) X ⊆ ℝ is of strong measure zero iff ∣X∣ ≤ ℵ1 + there is a generalized Sierpinski set.(2) The union of ℵ many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size ℵ2 + there is no Cohen real over L.

Decomposition of Reflexive Differential-Difference Polynomial Systems

2013 ◽  
Vol 380-384 ◽  
pp. 1645-1648 ◽  
Author(s):  
Zhi Wang Gan ◽  
Meng Zhou
Keyword(s):  
Polynomial Systems ◽  
Differential Polynomial ◽  
Decomposition Algorithm ◽  
Difference Operators ◽  
Zero Set ◽  
Finitely Generated ◽  
Zero Sets ◽  
Term Order ◽  
Difference Polynomial ◽  
Generalized Term Order

This paper presents a zero decomposition algorithm for reflexive difference and differential polynomial systems. The problem of negative exponent of difference operators in DD-polynomials is solved by means of generalized term order. Our algorithm would be used to decompose the zero set of a finitely generated reflexive DD-polynomial set into the union of zero sets of coherent chains.

Remark on Zero Sets of Holomorphic Functions in Convex Domains of Finite Type

2010 ◽  
Vol 53 (2) ◽  
pp. 311-320 ◽  
Author(s):  
Michał Jasiczak
Keyword(s):  
Holomorphic Function ◽  
Finite Type ◽  
Holomorphic Functions ◽  
Bounded Mean Oscillation ◽  
Carleson Measures ◽  
Zero Set ◽  
Convex Domains ◽  
Zero Sets ◽  
Analytic Subvariety ◽  
Mean Oscillation

AbstractWe prove that if the (1, 1)-current of integration on an analytic subvariety V ⊂ D satisfies the uniform Blaschke condition, then V is the zero set of a holomorphic function ƒ such that log |ƒ| is a function of bounded mean oscillation in bD. The domain D is assumed to be smoothly bounded and of finite d’Angelo type. The proof amounts to non-isotropic estimates for a solution to the -equation for Carleson measures.

Quick Consult: Quick Consult: Acromioplasty

2002 ◽  
Vol 7 (3) ◽  
pp. 4-5
Keyword(s):  
Glasgow Coma Scale ◽  
Outcome Assessment ◽  
High Probability ◽  
Vegetative State ◽  
Behavioral Impairment ◽  
Whole Person ◽  
Ordinal Scales ◽  
Probability Of Death ◽  
The Individual ◽  
Assessment Scales

Abstract Different jurisdictions use the AMA Guides to the Evaluation of Permanent Impairment (AMA Guides) for different purposes, and this article reviews a specific jurisdictional definition in the Province of Ontario of catastrophic impairment that incorporates the AMA Guides. In Ontario, a whole person impairment (WPI) exceeding 54% or a mental or behavioral impairment of Class 4 or 5 qualifies the individual for catastrophic benefits, and individuals who do not meet the test receive a lesser benefit. By inference, this establishes a parity threshold among dissimilar injuries and dissimilar outcome assessment scales for benefits. In Ontario, the Glasgow Coma Scale (GCS) identifies patients who have a high probability of death or of severely disabled survival. The GCS recognizes gradations of vegetative state and disability, but translating the gradations for rating individual impairment on ordinal scales into a method of assessing percentage impairments cannot be done reliably, as explained in the AMA Guides, Fifth Edition. The AMA Guides also notes that mental and behavioral impairment in Class 4 (marked impairment) or 5 (extreme impairment) indicates “catastrophic impairment” by significantly impeding useful functioning (Class 4) or significantly impeding useful functioning and implying complete dependency on another person for care (Class 5). Translating the AMA Guides guidelines into ordinal scales cannot be done reliably.

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