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

Jaume de Dios Pont
2020 arXiv   pre-print
The aim of this paper is to prove 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.  ...

### A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions [article]

Ilias Diakonikolas, Rocco A. Servedio, Li-Yang Tan, Andrew Wan
2010 arXiv   pre-print
We give 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

Nathan Linial, Yishay Mansour, Noam Nisan
1993 Journal of the ACM
This approximation allows the algorithm to predict, 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 .  ...  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).  ...

### Every super-polynomial proof in purely implicational minimal logic has a polynomially sized proof in classical implicational propositional logic [article]

Edward Hermann Haeusler
2015 arXiv   pre-print
In this article we show how any formula 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

Rocco A. Servedio, Li-Yang Tan, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization
Our pseudorandom multiswitching 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

Donald W. Robinson
1971 Linear Algebra and its Applications
If -cp' is separable 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

Ilias Diakonikolas, Rocco A. Servedio, Li-Yang Tan, Andrew Wan
2010 2010 IEEE 25th Annual Conference on Computational Complexity
We give 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

Glenn J. Fox
2002 Journal de Théorie des Nombres de Bordeaux
As 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

Feilong Cao, Shaobo Lin
2009 Journal of Inequalities and Applications
Essentially sharp Markov-type inequalities are known for various classes of 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.  ...

### Polynomials of best approximation on an infinite interval

James M. Earl
1930 Transactions of the American Mathematical Society
By 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

James M. Earl
1930 Transactions of the American Mathematical Society
By 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

David Zeitlin
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

David Zeitlin
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]

Rocco A. Servedio, Li-Yang Tan
2018 arXiv   pre-print
Our pseudorandom multi-switching 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

Fred Gross, Chung Chun Yang
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 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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