137,090 research outputs found
Quadratic maps between modules
We introduce a notion of -quadratic maps between modules over a
commutative ring which generalizes several classical notions arising in
linear algebra and group theory. On a given module such maps are
represented by -linear maps on a certain module . The structure of
this module is described in term of the symmetric tensor square ,
the degree 2 component of the divided power algebra over ,
and the ideal of generated by the elements , . The
latter is shown to represent quadratic derivations on which arise in the
theory of modules over square rings. This allows to extend the classical notion
of nilpotent -group of class 2 with coefficients in a 2-binomial ring to
any ring . We provide a functorial presentation of and several exact
sequences embedding the modules and .Comment: 22 page
Dobinski-type relations and the Log-normal distribution
We consider sequences of generalized Bell numbers B(n), n=0,1,... for which
there exist Dobinski-type summation formulas; that is, where B(n) is
represented as an infinite sum over k of terms P(k)^n/D(k). These include the
standard Bell numbers and their generalizations appearing in the normal
ordering of powers of boson monomials, as well as variants of the "ordered"
Bell numbers. For any such B we demonstrate that every positive integral power
of B(m(n)), where m(n) is a quadratic function of n with positive integral
coefficients, is the n-th moment of a positive function on the positive real
axis, given by a weighted infinite sum of log-normal distributions.Comment: 7 pages, 2 Figure
Local rigidity of infinite-dimensional Teichmüller spaces
This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space , for such a Riemann surface , is isomorphic to the Banach space of summable sequence, . This implies that whenever and are Riemann surfaces that are not analytically finite, and in particular are not necessarily homeomorphic, then is isomorphic to . It is known from V. Markovic that if there is a linear isometry between and , for two Riemann surfaces and of non-exceptional type, then this isometry is induced by a conformal mapping between and . As a corollary to this rigidity theorem presented here, taking the Banach duals of and shows that the space of holomorphic quadratic differentials on , is isomorphic to the Banach space of bounded sequences, . As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent
Fixed-rank Rayleigh Quotient Maximization by an PSK Sequence
Certain optimization problems in communication systems, such as
limited-feedback constant-envelope beamforming or noncoherent -ary
phase-shift keying (PSK) sequence detection, result in the maximization of a
fixed-rank positive semidefinite quadratic form over the PSK alphabet. This
form is a special case of the Rayleigh quotient of a matrix and, in general,
its maximization by an PSK sequence is -hard. However, if the
rank of the matrix is not a function of its size, then the optimal solution can
be computed with polynomial complexity in the matrix size. In this work, we
develop a new technique to efficiently solve this problem by utilizing
auxiliary continuous-valued angles and partitioning the resulting continuous
space of solutions into a polynomial-size set of regions, each of which
corresponds to a distinct PSK sequence. The sequence that maximizes the
Rayleigh quotient is shown to belong to this polynomial-size set of sequences,
thus efficiently reducing the size of the feasible set from exponential to
polynomial. Based on this analysis, we also develop an algorithm that
constructs this set in polynomial time and show that it is fully
parallelizable, memory efficient, and rank scalable. The proposed algorithm
compares favorably with other solvers for this problem that have appeared
recently in the literature.Comment: 15 pages, 12 figures, To appear in IEEE Transactions on
Communication
Construction of -ary Sequence Families of Period and Cross-Correlation of -ary m-Sequences and Their Decimated Sequences
학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2015. 2. 노종선.This dissertation includes three main contributions: a construction of a new family of -ary sequences of period with low correlation, a derivation of the cross-correlation values of decimated -ary m-sequences and their decimations, and an upper bound on the cross-correlation values of ternary m-sequences and their decimations.
First, for an odd prime and an odd integer , a new family of -ary sequences of period with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and . The upper bound on the maximum value of the magnitude of the correlation of the family is shown to be by using the generalized Kloosterman sums. The family size is four times the period of sequences, .
Second, based on the work by Helleseth \cite{Helleseth1}, the cross-correlation values between two decimated m-sequences by 2 and are derived, where is an odd prime and is an integer. The cross-correlation is at most 4-valued and their values are . As a result, for , a new sequence family with the maximum correlation value and the family size is obtained, where is the period of sequences in the family.
Lastly, the upper bound on the cross-correlation values of ternary m-sequences and their decimations by is investigated, where is an integer and the period of m-sequences is . The magnitude of the cross-correlation is upper bounded by . To show this, the quadratic form technique and Bluher's results \cite{Bluher} are employed. While many previous results using quadratic form technique consider two quadratic forms, four quadratic forms are involved in this case. It is proved that quadratic forms have only even ranks and at most one of four quadratic forms has the lowest rank .Abstract i
Contents iii
List of Tables vi
List of Figures vii
1. Introduction 1
1.1. Background 1
1.2. Overview of Dissertation 9
2. Sequences with Low Correlation 11
2.1. Trace Functions and Sequences 11
2.2. Sequences with Low Autocorrelation 13
2.3. Sequence Families with Low Correlation 17
3. A New Family of p-ary Sequences of Period (p^n−1)/2 with Low Correlation 21
3.1. Introduction 22
3.2. Characters 24
3.3. Gaussian Sums and Kloosterman Sums 26
3.4. Notations 28
3.5. Definition of Sequence Family 29
3.6. Correlation Bound 30
3.7. Size of Sequence Family 35
3.8. An Example 38
3.9. Related Work 40
3.10. Conclusion 41
4. On the Cross-Correlation between Two Decimated p-ary
m-Sequences by 2 and 4p^{n/2}−2 44
4.1. Introduction 44
4.2. Decimated Sequences of Period (p^n−1)/2 49
4.3. Correlation Bound 53
4.4. Examples 59
4.5. A New Sequence Family of Period (p^n−1)/2 60
4.6. Discussions 61
4.7. Conclusion 67
5. On the Cross-Correlation of Ternary m-Sequences of Period 3^{4k+2} − 1 with Decimation (3^{4k+2}−3^{2k+1}+2)/4 + 3^{2k+1} 69
5.1. Introduction 69
5.2. Quadratic Forms and Linearized Polynomials 71
5.3. Number of Solutions of x^{p^s+1} − cx + c 78
5.4. Notations 79
5.5. Quadratic Form Expression of the Cross-Correlation Function 80
5.6. Ranks of Quadratic Forms 83
5.7. Upper Bound on the Cross-Correlation Function 89
5.8. Examples 93
5.9. Related Works 94
5.10. Conclusion 94
6. Conclusions 96
Bibliography 98
초록 109Docto
On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is
investigated, and characterized up to a measure zero set, by means of the
Bombieri-Taylor conjecture, for Bragg peaks, and of another conjecture that we
call Aubry-Godr\`eche-Luck conjecture, for the singular continuous component.
The decomposition of the Fourier transform of the weighted Dirac comb is
obtained in terms of tempered distributions. We show that the asymptotic
arithmetics of the -rarefied sums of the Thue-Morse sequence (Dumont;
Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the
fractality of sum-of-digits functions, play a fundamental role in the
description of the singular continous part of the spectrum, combined with some
classical results on Riesz products of Peyri\`ere and M. Queff\'elec. The
dominant scaling of the sequences of approximant measures on a part of the
singular component is controlled by certain inequalities in which are involved
the class number and the regulator of real quadratic fields.Comment: 35 pages In honor of the 60-th birthday of Henri Cohe
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