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A geometric lemma for complex polynomial curves with applications in Fourier restriction theory
[article]

2020
*
arXiv
*
pre-print

The aim of this paper is to prove

arXiv:2003.14140v1
fatcat:wazwgt7ptnafbcxr7532e4jyva
*a*uniform Fourier*restriction*estimate for certain 2-dimensional surfaces in R^2n. ... These surfaces are the image of complex*polynomial*curves γ(z) = (p_1(z), ..., p_n(z)), equipped*with*the complex equivalent to the affine arclength measure. ...*A*fact that will be extremely relevant is that all the monomials of Schur*polynomials*are positive. Our*application*of this fact is the following*lemma*:*Lemma*3. ...##
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A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions
[article]

2010
*
arXiv
*
pre-print

We give

arXiv:0909.4727v2
fatcat:xdezqawno5d4jmq24wuritcvsu
*a*"regularity*lemma*" for degree-d*polynomial*threshold functions (PTFs) over the Boolean cube -1,1^n. ... As an*application*of this regularity*lemma*, we prove that for any constants d ≥ 1, ≥ 0, every degree-d PTF over n variables has can be approximated to accuracy eps by*a*constant-degree PTF that has integer ... This*lemma*says that*with*probability at least 1/(2C d ) over*a*random*restriction*ρ fixing variables [ℓ], the*polynomial*p ρ is τ ′ -regular, so the third statement of*Lemma*11 holds. ...##
###
Constant depth circuits, Fourier transform, and learnability

1993
*
Journal of the ACM
*

This approximation allows the algorithm to predict,

doi:10.1145/174130.174138
fatcat:iwwlqkfxazgijjobyhhevsk4su
*with*high probability, the value of the function on other randomly chosen inputs.*A*preliminary version of this paper was published in ... An important ingredient of the proof is Hastad's switching*lemma*[8]. ... We view the*restriction*p as obtained by first having*a*random*restriction**with*Pr[ * ] = 1/10, and then d -1 consecutive*restrictions*each*with*Pr[ * ] = 1/(10 s). ...##
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Every super-polynomial proof in purely implicational minimal logic has a polynomially sized proof in classical implicational propositional logic
[article]

2015
*
arXiv
*
pre-print

In this article we show how any formula

arXiv:1505.06506v3
fatcat:qqogaxiknrghhonggq66ltzyny
*A**with**a*proof in minimal implicational logic that is super-*polynomially*sized has*a**polynomially*-sized proof in classical implicational propositional logic . ... Suppose that*a*first*application*of*lemmas*5, 6 and 7 does not provide us*with**a*Π → having such*polynomially*bounded sub-derivation Π k+1 , then Π k+1 itself is super-*polynomially*sized on || δ ||. ... premisses of ¬-*Restricted*-Elim rule*applications*. ...##
###
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

2019
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

Our pseudorandom multiswitching

doi:10.4230/lipics.approx-random.2019.45
dblp:conf/approx/ServedioT19
fatcat:gtiom7wivve6rhihqxwkzb3b4u
*lemma*-*a*randomness-efficient algorithm for sampling*restrictions*that simultaneously simplify all circuits in*a*family -achieves the parameters obtained by the (full randomness ... The key enabling ingredient in our approach is*a*new pseudorandom multi-switching*lemma*. ... when they are "hit*with**a*random*restriction*." ...##
###
Separable endomorphisms and higher-order commutators

1971
*
Linear Algebra and its Applications
*

If -cp' is separable

doi:10.1016/0024-3795(71)90022-x
fatcat:gvzf2ofg2neatlcw7orvlwuqri
*with*semisimple part u, then R(*A*,) = K(*A*.).Proof.Let o and p be the semisimple and nilpotent parts of z"'.Since o is*a**polynomial*in r@, it follows from*Lemma*2.2 that R(*A*,) E K(*A*. ... Thus we complete the proof of the theorem by showing, under the given*restrictions*on the characteristic of F, that (iv) implies (i).*a**polynomial*in cp*with*coefficients in F. ...##
###
A Regularity Lemma, and Low-Weight Approximators, for Low-Degree Polynomial Threshold Functions

2010
*
2010 IEEE 25th Annual Conference on Computational Complexity
*

We give

doi:10.1109/ccc.2010.28
dblp:conf/coco/DiakonikolasSTW10
fatcat:2bbdzaxqkzcl3e2fqsczurscay
*a*"regularity*lemma*" for degree-d*polynomial*threshold functions (PTFs) over the Boolean cube {−1, 1} n . ... As an*application*of this regularity*lemma*, we prove that for any constants d ≥ 1, > 0, every degree-d PTF over n variables can be approximated to accuracy by*a*constantdegree PTF that has integer weights ... In the first stage, the initial*application*of*Lemma*12 results in*a*tree T 1 . ...##
###
Kummer congruences for expressions involving generalized Bernoulli polynomials

2002
*
Journal de Théorie des Nombres de Bordeaux
*

As

doi:10.5802/jtnb.353
fatcat:l5vdeyr7djgpralvwxuwf7i334
*a*weaker form of this*lemma*, we have the congruence*with*the same*restrictions*on*a*and t as in the*lemma*, but for all positive m E Z. ... As*a*weaker form of this*lemma*, we have*with*the same*restrictions*on*a*and t, but for all positive m E Z. ...##
###
Markov Inequalities for Polynomials with Restricted Coefficients

2009
*
Journal of Inequalities and Applications
*

Essentially sharp Markov-type inequalities are known for various classes of

doi:10.1155/2009/808720
fatcat:kmpnk4nenvhkzpnk6zkgqkafva
*polynomials**with*constraints including constraints of the coefficients of the*polynomials*. ... For N and δ > 0 we introduce the class F n,δ as the collection of all*polynomials*of the form P x n k h*a*k x k ,*a*k ∈ Z, |*a*k | ≤ n δ , |*a*h | max h≤k≤n |*a*k |. ... . < ∞, 0 < p < ∞, f*a*,b f L ∞*a*,b ess sup x∈*a*,b f x . set of all algebraic*polynomials*of degree at most n*with*real coefficients. ...##
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Polynomials of best approximation on an infinite interval

1930
*
Transactions of the American Mathematical Society
*

By

doi:10.1090/s0002-9947-1930-1501569-6
fatcat:5nxexhfzrbf4dgn4dkw5en24j4
*Lemma*3, 7r0' úc(n2Imn)llm so that the inequality (8) results by successive*applications*of*Lemma*1*with*(*a*, b) replaced by (*a*+i¡e,*a*+-¿e) and t by fe. This establishes*Lemma*4. ... ,*a*), (*a*, «•) and (-°°, -*a*) respectively satisfies the conditions of*Lemma*2*with*M = M(*a*), so that there exists*a**polynomial*^"(x) such that the first integral on the right in the inequality /»CO /»CO ... Then by*Lemma*2 of §6, (p)(i) satisfies*a*Lipschitz condition of order u*with*coefficient Z,(-B+/2)(p+u)/2 on -oo <í<oo so that by Theorem J, there exists*a**polynomial*pn (t) of degree n such that on ...##
###
Polynomials of Best Approximation on an Infinite Interval

1930
*
Transactions of the American Mathematical Society
*

By

doi:10.2307/1989357
fatcat:iof5e2kjfzagvdjntccpasonfa
*Lemma*3, 7r0' úc(n2Imn)llm so that the inequality (8) results by successive*applications*of*Lemma*1*with*(*a*, b) replaced by (*a*+i¡e,*a*+-¿e) and t by fe. This establishes*Lemma*4. ... ,*a*), (*a*, «•) and (-°°, -*a*) respectively satisfies the conditions of*Lemma*2*with*M = M(*a*), so that there exists*a**polynomial*^"(x) such that the first integral on the right in the inequality /»CO /»CO ... Then by*Lemma*2 of §6, (p)(i) satisfies*a*Lipschitz condition of order u*with*coefficient Z,(-B+/2)(p+u)/2 on -oo <í<oo so that by Theorem J, there exists*a**polynomial*pn (t) of degree n such that on ...##
###
A New Class of Generating Functions for Hypergeometric Polynomials

1970
*
Proceedings of the American Mathematical Society
*

*Applications*are given for the classical

*polynomials*containing parameters, such as Jacobi, ultraspherical, and Laguerre

*polynomials*, as well as for Hermite and Bessel

*polynomials*. ...

*A*new class of generating functions for generalized hypergeometric

*polynomials*is obtained. ...

*A*generating function for the generalized hypergeometric

*polynomials*, R%M(x), defined by (1.5),

*with*

*a*-c as

*a*positive integer, is given by[1 + g(t)]~° iZ --r-Rn' (x)t" = zZ An(

*a*, b, c, I) •"+i+p77+,+3 ...

##
###
A new class of generating functions for hypergeometric polynomials

1970
*
Proceedings of the American Mathematical Society
*

*Applications*are given for the classical

*polynomials*containing parameters, such as Jacobi, ultraspherical, and Laguerre

*polynomials*, as well as for Hermite and Bessel

*polynomials*. ...

*A*new class of generating functions for generalized hypergeometric

*polynomials*is obtained. ...

*A*generating function for the generalized hypergeometric

*polynomials*, R%M(x), defined by (1.5),

*with*

*a*-c as

*a*positive integer, is given by[1 + g(t)]~° iZ --r-Rn' (x)t" = zZ An(

*a*, b, c, I) •"+i+p77+,+3 ...

##
###
Improved pseudorandom generators from pseudorandom multi-switching lemmas
[article]

2018
*
arXiv
*
pre-print

Our pseudorandom multi-switching

arXiv:1801.03590v1
fatcat:yodu5mvlvnfwxnlip3d3f5oppm
*lemma*---*a*randomness-efficient algorithm for sampling*restrictions*that simultaneously simplify all circuits in*a*family---achieves the parameters obtained by the (full ... an ε-PRG for the class of S-sparse F_2*polynomials**with*seed length 2^O(√( S))·(1/ε). ...*with*low-degree F 2*polynomials*at its leaves under*a*suitable pseudorandom*restriction*. ...##
###
Further results on prime entire functions

1974
*
Transactions of the American Mathematical Society
*

Let H denote the set of all the entire functions f(z) of the form: f(z) = h(z)eM + k(z) where p(z) is

doi:10.1090/s0002-9947-1974-0349972-3
fatcat:qxs5uggnpnhyjah4xgtfyrroxq
*a*nonconstant*polynomial*of degree m, and*A*(# 0), k (# constant) are two entire functions of order ... In this paper,*a*necessary and sufficient condition for*a*function in H to be*a*prime is established. Several generalizations of known results follow. ... Charles Osgood for his suggestions and fruitful discussions*with*regard to the proof of Theorem 1. ...
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