Explicit Bounds for Hermite Polynomials in the Oscillatory Region

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157000000310 ◽

2000 ◽

Vol 3 ◽

pp. 307-314 ◽

Cited By ~ 6

Author(s):

William H. Foster ◽

Ilia Krasikov

Keyword(s):

Quadratic Forms ◽

Hermite Polynomials ◽

Polynomial Inequalities ◽

Explicit Bounds

AbstractWe apply a method of positive quadratic forms based on polynomial inequalities to establish sharp explicit bounds on the envelope of Hermite polynomials in the oscillatory region |x| < (2k – 3/2)1/2.

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27.—Generating Sets for Fuchsian Groups

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0080454100009687 ◽

1975 ◽

Vol 72 (4) ◽

pp. 317-326 ◽

Cited By ~ 3

Author(s):

J. H. H. Chalk ◽

B. G. A. Kelly

Keyword(s):

Quadratic Forms ◽

Fundamental Region ◽

Fuchsian Groups ◽

Quaternion Algebras ◽

Generating Sets ◽

Complete Set ◽

Explicit Bounds

SynopsisFor a class of Fuchsian groups, which includes integral automorphs of quadratic forms and unit groups of indefinite quaternion algebras, it is shown that the geometry of a suitably chosen fundamental region leads to explicit bounds for a complete set of generators.

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Markov type polynomial inequality for some generalized Hermite weight

Tatra Mountains Mathematical Publications ◽

10.2478/v10127-011-0030-4 ◽

2011 ◽

Vol 49 (1) ◽

pp. 111-118

Author(s):

Branislav Ftorek ◽

Mariana Marˇcokov´A

Keyword(s):

Weight Function ◽

Hermite Polynomials ◽

Polynomial Inequality ◽

Markov Type ◽

Polynomial Inequalities ◽

Weighted Polynomial Inequalities ◽

Special Case

ABSTRACT In this paper we study some weighted polynomial inequalities of Markov type in L2-norm. We use the properties of the system of generalized Hermite polynomials . The polynomials H(α)n (x) are orthogonal in ℝ = (−∞,∞) with respect to the weight function . The classical Hermite polynomials Hn(x) present the special case for α = 0.

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Quadratic Forms with Applications to Algebraic Geometry and Topology

10.1017/cbo9780511526077 ◽

1995 ◽

Cited By ~ 64

Author(s):

Albrecht Pfister

Keyword(s):

Algebraic Geometry ◽

Quadratic Forms ◽

And Topology

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Key Recovery Attacks on Multivariate Public Key Cryptosystems Derived from Quadratic Forms over an Extension Field

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e100.a.18 ◽

2017 ◽

Vol E100.A (1) ◽

pp. 18-25 ◽

Cited By ~ 3

Author(s):

Yasufumi HASHIMOTO

Keyword(s):

Quadratic Forms ◽

Public Key ◽

Extension Field ◽

Key Recovery ◽

Public Key Cryptosystems ◽

Multivariate Public Key ◽

Key Recovery Attacks

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Existence

10.23943/princeton/9780691166902.003.0016 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Structure Theorem ◽

Quadratic Extension ◽

Quadratic Space ◽

Division Algebras ◽

Separable Extension ◽

Valued Fields ◽

Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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Totally Wild Quadratic Forms of Type E7

10.23943/princeton/9780691166902.003.0015 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Quadratic Space ◽

Trace Map ◽

Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

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Linked Tori, II

10.23943/princeton/9780691166902.003.0006 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Integral Domain ◽

Quadratic Space ◽

Small Observation ◽

Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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