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The *Main* *Result*

"... � � � Given r>2, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coefficients mapping the “cube ” with real coordinates from [−r, r] into[−t, t]. This directly translates to a nice statement in l ..."

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coefficients mapping the “cube” with real variables from [−r, r] into[−t, t]. Robin Pemantle has established a rough upper bound. Here, utilizing Chebyshev polynomials, we establish a reasonably good upper bound. Namely, in this paper we prove our

*main**result*and some related ones, applications of which###
OF THE *MAIN* *RESULTS*

"... ABSTRACT. In this paper we study the limit cycles of polynomial vector fields in R3 which bifurcates from three different kinds of two dimensional centers (non-degenerate and degenerate). The study is down using the averaging theory. AMS (MOS) Subject Classification. 37G15, 37D45 ..."

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ABSTRACT. In this paper we study the limit cycles of polynomial vector fields in R3 which bifurcates from three different kinds of two dimensional centers (non-degenerate and degenerate). The study is down using the averaging theory. AMS (MOS) Subject Classification. 37G15, 37D45

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*Main* *Results* Applications Notation

, 2013

"... I A: an associative algebra over a field k with a finite complete set of primitive orthogonal idempotents E = {ei}ni=1; that is, 1A = ∑n i=1 ei. I G: a finite group. I A group homomorphism ρ: G → Aut(A). I The skew group algebra AG = A⊗k kG as vector spaces, with multiplication determined by: (a ⊗ g ..."

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I A: an associative algebra over a field k with a finite complete set of primitive orthogonal idempotents E = {ei}ni=1; that is, 1A = ∑n i=1 ei. I G: a finite group. I A group homomorphism ρ: G → Aut(A). I The skew group algebra AG = A⊗k kG as vector spaces, with multiplication determined by: (a ⊗ g) · (b ⊗ h) = ag(b) ⊗ gh, where g(b): = ρ(g)(b). I Examples: regular group algebras, algebra of matrices, etc. Liping Li Representations of modular skew group algebras

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Introduction and *main* *result*

, 1990

"... Laboratory experiments on gravitation are usually performed with objects of constant density, so that the analysis of the forces concerns only the geometry of their shape. In ..."

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Laboratory experiments on gravitation are usually performed with objects of constant density, so that the analysis of the forces concerns only the geometry of their shape. In

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*Main* *Result* Outline of the Proof

"... ◮ Relay channel was introduced in 1971 by van der Meulen, but the capacity is still open in general. ..."

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◮ Relay channel was introduced in 1971 by van der Meulen, but the capacity is still open in general.

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1. The *main* *results*

, 2000

"... There is a gradient map associated to any reduced homogeneous polynomial h ∈ C[x0,..., xn] of degree d> 0, namely grad(h) : D(h) → P n, (x0:...: xn) ↦ → (h0(x) :...: hn(x)) where D(h) = {x ∈ P n; h(x) ̸ = 0} is the principal open set associated to h and hi = ∂h ∂xi. Our first result is the foll ..."

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There is a gradient map associated to any reduced homogeneous polynomial h ∈ C[x0,..., xn] of degree d> 0, namely grad(h) : D(h) → P n, (x0:...: xn) ↦ → (h0(x) :...: hn(x)) where D(h) = {x ∈ P n; h(x) ̸ = 0} is the principal open set associated to h and hi = ∂h ∂xi. Our first

*result*###
3 *Main* *Results*

, 2009

"... • Large audio/video collections exist on the Internet • Many containing music and/or speech • How can search engines effectively index this content? 2 ..."

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• Large audio/video collections exist on the Internet • Many containing music and/or speech • How can search engines effectively index this content? 2

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1. FORMULATION OF THE *MAIN* *RESULT*

"... In dimension d ≥ 5, we consider the differential operator ..."

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§1. *Main* *results*

, 1996

"... Abstract: Let ε> 0. We prove that there exists an operator Tε: ℓ2 → ℓ2, such that for any polynomial P we have ‖P(T) ‖ ≤ (1 + ε)‖P ‖∞, but which is not similar to a contraction, i.e. there does not exist an invertible operator S: ℓ2 → ℓ2 such that ‖S−1TεS ‖ ≤ 1. This answers negatively a questi ..."

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Abstract: Let ε> 0. We prove that there exists an operator Tε: ℓ2 → ℓ2, such that for any polynomial P we have ‖P(T) ‖ ≤ (1 + ε)‖P ‖∞, but which is not similar to a contraction, i.e. there does not exist an invertible operator S: ℓ2 → ℓ2 such that ‖S−1TεS ‖ ≤ 1. This answers negatively a question attributed to Halmos after his well