On the local converse theorem and the descent theorem in families

It was proved by Salmon (Geom. of three dimensions (1882), p. 331) that the chords of the curve of intersection of two algebraic surfaces of order m and n. which can be drawn from an arbitrary point,meet the curve upon a surface of order (m — 1) (n — 1); it was proved by Valentiner (Acta Math. 2 (1883), p. 191), and by Noether (Berlin. Abh. (1882), Zur Grundlegung u.s.w., p. 27), that the surface of order (m — 1) (n — 1) is a cone, with vertex at the point from which the chords are drawn; and a converse theorem was given by Halphen (J. de l' école Polyt. 52 (1882), p. 106). But the proofs given by Valentiner and Noether have not the elementary character that seems desirable, Noether's proof in particular depending on the theory of the canonical series upon the curve.

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Exponential Strong Converse for One Helper Source Coding Problem

Entropy ◽

10.3390/e21060567 ◽

2019 ◽

Vol 21 (6) ◽

pp. 567 ◽

Cited By ~ 3

Author(s):

Yasutada Oohama

Keyword(s):

Error Probability ◽

Source Coding ◽

Small Error ◽

Converse Theorem ◽

Correlated Sources ◽

Strong Converse Theorem ◽

Explicit Lower Bound ◽

Source Block ◽

Strong Converse ◽

The One

We consider the one helper source coding problem posed and investigated by Ahlswede, Körner and Wyner. Two correlated sources are separately encoded and are sent to a destination where the decoder wishes to decode one of the two sources with an arbitrary small error probability of decoding. In this system, the error probability of decoding goes to one as the source block length n goes to infinity. This implies that we have a strong converse theorem for the one helper source coding problem. In this paper, we provide the much stronger version of this strong converse theorem for the one helper source coding problem. We prove that the error probability of decoding tends to one exponentially and derive an explicit lower bound of this exponent function.

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Sharp converse theorem ofL p -polynomial approximation

Constructive Approximation ◽

10.1007/bf02075471 ◽

1988 ◽

Vol 4 (1) ◽

pp. 419-433 ◽

Cited By ~ 8

Author(s):

V. Totik

Keyword(s):

Polynomial Approximation ◽

Converse Theorem

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On the converse theorem for Borcherds products

Journal of Algebra ◽

10.1016/j.jalgebra.2013.08.034 ◽

2014 ◽

Vol 397 ◽

pp. 315-342 ◽

Cited By ~ 15

Author(s):

Jan Hendrik Bruinier

Keyword(s):

Converse Theorem

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A CONVERSE THEOREM FOR BORCHERDS PRODUCTS ON

Nagoya Mathematical Journal ◽

10.1017/nmj.2019.3 ◽

2019 ◽

Vol 240 ◽

pp. 237-256 ◽

Cited By ~ 1

Author(s):

JAN HENDRIK BRUINIER ◽

MARKUS SCHWAGENSCHEIDT

Keyword(s):

Modular Form ◽

Converse Theorem ◽

Maass Form ◽

Derivatives Of

We show that every Fricke-invariant meromorphic modular form for $\unicode[STIX]{x1D6E4}_{0}(N)$ whose divisor on $X_{0}(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-functions of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.

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The Saxon–Hutner Theorem, its Converse Theorem and their Applications to Localization and Delocalization Problems in Binary Quasiperiodic Lattices

International Journal of Modern Physics B ◽

10.1142/s0217979297001118 ◽

1997 ◽

Vol 11 (18) ◽

pp. 2157-2182 ◽

Cited By ~ 5

Author(s):

Kazumoto Iguchi

Keyword(s):

Transfer Matrix ◽

Transfer Matrix Method ◽

Matrix Method ◽

Cantor Set ◽

Localized States ◽

Edge States ◽

Converse Theorem ◽

One Dimensional ◽

The One ◽

Fibonacci Lattice

In this paper we discuss the application of the Saxon–Hutner theorem and its converse theorem in one-dimensional binary disordered lattices to the one-dimensional binary quasiperiodic lattices. We first summarize some basic theorems in one-dimensional periodic lattices. We discuss how the bulk and edge states are treated in the transfer matrix method. Second, we review the Saxon–Hutner theorem and prove the converse theorem, using the so-called Fricke identities. Third, we present an alternative approach for a rigorous proof of the existence of a Cantor-set spectrum in the Fibonacci lattice and in the related binary quasiperiodic lattices by means of the theorems together with their trace map with the invariant I. We obtain that if I > 0, then the spectrum is always a Cantor set, which was first proved for the Fibonacci lattice by Sütö and generalized for other quasiperiodic lattices by Bellissard, Iochum, Scopolla, and Testard. Fourth, we rigorously prove the existence of extended states in the spectrum of a class of binary quasiperiodic lattices first studied by Kolář and Ali. Fifth, we discuss the so-called gap labeling theorem emphasized by Bellissard and the classic argument of Kohn and Thouless for localized states in a one-dimensional disordered lattice in terms of the language of the transfer matrix method.

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