Errata: On a Θ-Weyl sum (Nagoya Math. J. Vol. 52 (1973), 163–172

We study the long-time behavior of bound quantum systems whose classical dynamics is chaotic and put forward two conjectures. Conjecture A states that the autocorrelation function C(t)=<Ψ(0)|Ψ(t)> of a delocalized initial state |Ψ(0)> shows characteristic fluctuations, which we identify with a universal signature of temporal quantum chaos. For example, for the (appropriately normalized) value distribution of S~|C(t)| we predict the distribution P(S)=(π/2)Se-πS2/4. Conjecture B gives the best possible upper bound for a generalized Weyl sum and is related to the extremely large recurrence times in temporal quantum chaos. Numerical tests carried out for numerous chaotic systems confirm nicely the two conjectures and thus provide strong evidence for temporal quantum chaos.

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Correction to “Isospectral surfaces of small genus,” (Nagoya Math. J. Vol. 107 (1987), 13-24)

Nagoya Mathematical Journal ◽

10.1017/s0027763000001860 ◽

1990 ◽

Vol 117 ◽

pp. 227-227 ◽

Cited By ~ 1

Author(s):

Robert Brooks ◽

Richard Tse

Keyword(s):

Fundamental Domain ◽

Nagoya Math ◽

Small Genus

It was brought to our attention by Zoran Luicic and Milica Stojanovic, via Peter Gilkey, that some of the diagrams in our paper are not correct.The particular problems are the gluing diagrams for the pair of isospectral surfaces of genus 4, which occur on page 20. It is easy to check that the gluing diagrams given there give rise to a surface of the wrong genus. The problem arose because of carelessness in some of the identifications of some of the edges of the fundamental domain.

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Bounds for the cubic Weyl sum

Journal of Mathematical Sciences ◽

10.1007/s10958-010-0184-7 ◽

2010 ◽

Vol 171 (6) ◽

pp. 813-823

Author(s):

D. R. Heath-Brown

Keyword(s):

Weyl Sum

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Correction to “Gromov’s convergence theorem and its application” (Nagoya Math. J. Vol. 100 (1985), 11–48)

Nagoya Mathematical Journal ◽

10.1017/s002776300000146x ◽

1989 ◽

Vol 114 ◽

pp. 173-174 ◽

Cited By ~ 1

Author(s):

Atsushi Katsuda

Keyword(s):

Convergence Theorem ◽

Nagoya Math

In the proof of Lemma 12.2, the following inequality (p. 43, line 11) is incorrect.This should be corrected as follows.

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Errata for “L2-boundedness of the Cauchy transform on smooth non-Lipschitz curves”, Nagoya Math J. 130 (1993), 123–147

Nagoya Mathematical Journal ◽

10.1017/s0027763000005122 ◽

1995 ◽

Vol 137 ◽

pp. 195-195

Author(s):

Hyeonbae Kang ◽

Jinkeun Seo

Keyword(s):

Symmetry Condition ◽

Cauchy Transform ◽

Extra Condition ◽

Standard Argument ◽

Nagoya Math

Note that an extra condition of symmetry of f, namely, the boundedness of f(x) = f(− x) is added to the hypothesis of the Lemma. Lemma 2.1 was used in three places in the paper to prove that the functions F, G*, and f given in pages 140, 141, and 144, respectively, belong to BMO. It can be checked by a standard argument that these functions satisfy the symmetry condition.

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MINOR ARC MOMENTS OF WEYL SUMS

Glasgow Mathematical Journal ◽

10.1017/s0017089512000365 ◽

2012 ◽

Vol 55 (1) ◽

pp. 97-113 ◽

Cited By ~ 2

Author(s):

M. P. HARVEY

Keyword(s):

Diophantine Equations ◽

Diagonal Form ◽

Norm Form ◽

Goldbach Problem ◽

Weyl Sum ◽

Diophantine Problems

AbstractWe obtain an improved bound for the 2k-th moment of a degree k Weyl sum, restricted to a set of minor arcs, when k is small. We then present some applications of this bound to some Diophantine problems, including a case of the Waring–Goldbach problem, and a particular family of Diophantine equations defined as the sum of a norm form and a diagonal form.

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Explicit logarithmic formulas of special values of hypergeometric functions 3F2

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500408 ◽

2019 ◽

Vol 22 (05) ◽

pp. 1950040

Author(s):

Masanori Asakura ◽

Toshifumi Yabu

Keyword(s):

Linear Combination ◽

Hypergeometric Function ◽

Hypergeometric Functions ◽

Rational Numbers ◽

Explicit Description ◽

Generalized Hypergeometric Function ◽

Algebraic Numbers ◽

Nagoya Math ◽

Special Values ◽

Geometric Study

In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions [Formula: see text], to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of [Formula: see text] of the generalized hypergeometric function is a [Formula: see text]-linear combination of log of algebraic numbers if rational numbers [Formula: see text] satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it.

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