2,799 research outputs found
Jump numbers, hyperrectangles and Carlitz compositions
Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 1998.A thesis submitted to the Faculty of Science, University of the Witwatersrand,
Johannesburg, in fulfilment of the requirements for the degree of
Doctor of Philosophy.
Johannesburg 1998Let A = (aij) be an m x n matrix. There is a natural way to associate a
poset PA with A. A jump in a linear extension of PA is a pair of consecutive
elements which are incomparable in Pa. The jump number of A is the minimum
number of jumps in any linear extension of PA. The maximum jump
number over a class of n x n matrices of zeros and ones with constant row and
column sum k, M (n, k), has been investigated in Chapter 2 and 3. Chapter
2 deals with extremization problems concerning M (n ,k). In Chapter 3, we
obtain the exact values for M (11,k). M(n,Q), M (n,n-3) and M(n,n-4).
The concept of frequency hyperrectangle generalizes the concept of latin
square. In Chapter 4 we derive a bound for the maximum number of mutually
orthogonal frequency hyperrectangles. Chapter 5 gives two algorithms to
construct mutually orthogonal frequency hyperrectangles.
Chapter 6 is devoted to some enumerative results about Carlitz compositions
(compositions with different adjacent parts)
Wavelet filters and infinite-dimensional unitary groups
In this paper, we study wavelet filters and their dependence on two numbers,
the scale N and the genus g. We show that the wavelet filters, in the
quadrature mirror case, have a harmonic analysis which is based on
representations of the C^*-algebra O_N. A main tool in our analysis is the
infinite-dimensional group of all maps T -> U(N) (where U(N) is the group of
all unitary N-by-N matrices), and we study the extension problem from low-pass
filter to multiresolution filter using this group.Comment: AMS-LaTeX; 30 pages, 2 tables, 1 picture. Invited lecture by
Jorgensen at International Conference on Wavelet Analysis and Its
Applications, Zhongshan University, Guangzhou, China, in November 1999.
Changes: Some references have been added and some technical points in several
proofs have been clarified in this new revised versio
Genuinely multipartite entangled states and orthogonal arrays
A pure quantum state of N subsystems with d levels each is called
k-multipartite maximally entangled state, written k-uniform, if all its
reductions to k qudits are maximally mixed. These states form a natural
generalization of N-qudits GHZ states which belong to the class 1-uniform
states. We establish a link between the combinatorial notion of orthogonal
arrays and k-uniform states and prove the existence of several new classes of
such states for N-qudit systems. In particular, known Hadamard matrices allow
us to explicitly construct 2-uniform states for an arbitrary number of N>5
qubits. We show that finding a different class of 2-uniform states would imply
the Hadamard conjecture, so the full classification of 2-uniform states seems
to be currently out of reach. Additionally, single vectors of another class of
2-uniform states are one-to-one related to maximal sets of mutually unbiased
bases. Furthermore, we establish links between existence of k-uniform states,
classical and quantum error correction codes and provide a novel graph
representation for such states.Comment: 24 pages, 7 figures. Comments are very welcome
Row-column factorial designs and mutually orthogonal frequency rectangles
A (full) qįµ factorial design with replication Ī» is the multi-set containing all possible q-ary sequences of length k, each occurring exactly Ī» times. An m Ć n row-column factorial design is any arrangement of Ī» replicates of the qįµ factorial design in an m Ć n array. We say that the design has strength t if each row and column is an orthogonal array of strength t. We denote such a design by Iā (m,n,q,t).
A frequency rectangle of type FR(m,n;q) is an m Ć n array based on a symbol set S of size q, such that each element of S appears exactly n/q times in each row and m/q times in each column. Two frequency rectangles of the same type are said to be orthogonal if each possible pair of symbols appears the same number of times when the two arrays are superimposed. By kāMOFR(m,n;q) we mean a set of k frequency rectangles of type FR(m,n;q) in which every pair is orthogonal.
In Chapter 4, we give the necessary and sufficient conditions when a row-column factorial design of strength 1 exists. We show that an array of type Iā (m,n,q,1) exists if and only if (a) q|m, q|n and qįµ|mn; (b) (k,q,m,n) ā (2,6,6,6) and (c) if (k,q,m) = (2,2,2) then 4 divides n. In Chapter 5, we discuss designs of strength 2 and above. We solve the case completely when t = 2 and q is a prime power: we show that there exists an array of type Iā(qį“¹,qį“ŗ,q,2) if and only if k ā¤ M + N, k ā¤ (qį“¹ - 1)/(q - 1) and (k,M,q) ā (3,2,2). We also show that Iā+Ī±(2Ī±b,2įµ,2,2) exists whenever Ī± ā„ 2 and 2Ī± + Ī± + 1 ā¤ k < 2Ī±b - Ī±, assuming there exists a Hadamard matrix of order 4b. For strength 3 we restrict ourselves to the binary case, solving it completely when q is a power of 2.
In Chapter 6, our focus is on mutually orthogonal frequency rectangles (MOFR). We use orthogonal arrays and Hadamard matrices to construct sets of MOFR. We also describe a new form of orthogonality for a set of frequency rectangles. We say that a kāMOFR(m,n;q) is tāorthogonal if each subset of size t, when superimposed, forms a qįµ factorial design with replication mn/qįµ. A set of vectors over a finite field is said to be t-independent if each subset of size t is linearly independent. We describe a relationship between a set of tāorthogonal MOFR and a set of t-independent vectors. We use known results from coding theory and related literature to formulate a table for the size of a set of t-independent vectors of length N ā¤ 16, over Fā. We also describe a method to construct a set of (p - 1)āMOFR(2p,2p;2) where p is an odd prime, improving known lower bounds for all p ā„ 19
Qudit surface codes and gauge theory with finite cyclic groups
Surface codes describe quantum memory stored as a global property of
interacting spins on a surface. The state space is fixed by a complete set of
quasi-local stabilizer operators and the code dimension depends on the first
homology group of the surface complex. These code states can be actively
stabilized by measurements or, alternatively, can be prepared by cooling to the
ground subspace of a quasi-local spin Hamiltonian. In the case of spin-1/2
(qubit) lattices, such ground states have been proposed as topologically
protected memory for qubits. We extend these constructions to lattices or more
generally cell complexes with qudits, either of prime level or of level
for prime and , and therefore under tensor
decomposition, to arbitrary finite levels. The Hamiltonian describes an exact
gauge theory whose excitations
correspond to abelian anyons. We provide protocols for qudit storage and
retrieval and propose an interferometric verification of topological order by
measuring quasi-particle statistics.Comment: 26 pages, 5 figure
Algebraic geometry in experimental design and related fields
The thesis is essentially concerned with two subjects corresponding to the two grants under which the author was research assistant in the last three years. The one presented first, which cronologically comes second, addresses the issues of iden- tifiability for polynomial models via algebraic geometry and leads to a deeper understanding of the classical theory. For example the very recent introduction of the idea of the fan of an experimental design gives a maximal class of models identifiable with a given design. The second area develops a theory of optimum orthogonal fractions for Fourier regression models based on integer lattice designs. These provide alternatives to product designs. For particular classes of Fourier models with a given number of interactions the focus is on the study of orthogonal designs with attention given to complexity issues as the dimension of the model increases. Thus multivariate identifiability is the field of concern of the thesis. A major link between these two parts is given by Part III where the algebraic approach to identifiability is extended to Fourier models and lattice designs. The approach is algorithmic and algorithms to deal with the various issues are to be found throughout the thesis.
Both the application of algebraic geometry and computer algebra in statistics and the analysis of orthogonal fractions for Fourier models are new and rapidly growing fields. See for example the work by Koval and Schwabe (1997) [42] on qualitative Fourier models, Shi and Fang (1995) [67] on Āæ/-designs for Fourier regression and Dette and Haller (1997) [25] on one-dimensional incomplete Fourier models. For algebraic geometry in experimental design see Fontana, Pistone and Rogantin (1997) [31] on two-level orthogonal fractions, Caboara and Robbiano (1997) [15] on the inversion problem and Robbiano and Rogantin (1997) [61] on distracted fractions. The only previous extensive application of algebraic geometry in statistics is the work of Diaconis and Sturmfels (1993) [27] on sampling from conditional distributions
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