Symbolic Covariance Principal Component Analysis and Visualization for Interval-Valued Data

This article proposes a new approach to principal component analysis (PCA) for interval-valued data. Unlike classical observations, which are represented by single points in p-dimensional space ?^p, interval-valued observations are represented by hyper-rectangles in ?^p, and as such, have an internal structure that does not exist in classical observations. As a consequence, statistical methods for classical data must be modified to account for the structure of the hyper-rectangles before they can be applied to interval-valued data. This article extends the classical PCA method to interval-valued data by using the so-called symbolic covariance to determine the principal component (PC) space to reflect the total variation of interval-valued data. The article also provides a new approach to constructing the observations in a PC space for better visualization. This new representation of the observations reflects their true structure in the PC space. Supplementary materials for this article are available online.     

Lanczos,Householder transformations,and implicit deflation for fast and reliable dominant singular subspace computation

Many applications, such as subspace‐based models in information retrieval and signal processing, require the computation of singular subspaces associated with the k dominant, or largest, singular values of an m×n data matrix A, where k?min(m,n). Frequently, A is sparse or structured, which usually means matrix–vector multiplications involving A and its transpose can be done with much less than ??(mn) flops, and A and its transpose can be stored with much less than ??(mn) storage locations. Many Lanczos‐based algorithms have been proposed through the years because the underlying Lanczos method only accesses A and its transpose through matrix–vector multiplications. We implement a new algorithm, called KSVD, in the Matlab environment for computing approximations to the singular subspaces associated with the k dominant singular values of a real or complex matrix A. KSVD is based upon the Lanczos tridiagonalization method, the WY representation for storing products of Householder transformations, implicit deflation, and the QR factorization. Our Matlab simulations suggest it is a fast and reliable strategy for handling troublesome singular‐value spectra. Copyright © 2001 John Wiley & Sons, Ltd.     

On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces

This paper is devoted to the study of the Cauchy problem of incompressible magneto‐hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n?3, we establish the global well‐posedness of the Cauchy problem of an incompressible magneto‐hydrodynamics system for small data and the local one for large data in the Besov space ? (?ⁿ), 1?p<∞ and 1?r?∞. Meanwhile, we also prove the weak–strong uniqueness of solutions with data in ? (?ⁿ)∩L²(?ⁿ) for n/2p+2/r>1. In the case of n=2, we establish the global well‐posedness of solutions for large initial data in homogeneous Besov space ? (?²) for 2<p<∞ and 1?r<∞. Copyright © 2008 John Wiley & Sons, Ltd.     

Bayesian Analysis of Geostatistical Models With an Auxiliary Lattice

The Gaussian geostatistical model has been widely used for modeling spatial data. However, this model suffers from a severe difficulty in computation: it requires users to invert a large covariance matrix. This is infeasible when the number of observations is large. In this article, we propose an auxiliary lattice-based approach for tackling this difficulty. By introducing an auxiliary lattice to the space of observations and defining a Gaussian Markov random field on the auxiliary lattice, our model completely avoids the requirement of matrix inversion. It is remarkable that the computational complexity of our method is only O(n), where n is the number of observations. Hence, our method can be applied to very large datasets with reasonable computational (CPU) times. The numerical results indicate that our model can approximate Gaussian random fields very well in terms of predictions, even for those with long correlation lengths. For real data examples, our model can generally outperform conventional Gaussian random field models in both prediction errors and CPU times. Supplemental materials for the article are available online.     

Restricted partitions and q-Pell numbers

In this paper, we provide new combinatorial interpretations for the Pell numbers p _n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p _n . By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p _n . Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.     

On the Transient Behavior of the M/M/1 and M/M/1/M Queues

This paper gives a transient analysis of the classic M/M/1 and M/M/1/K queues. Our results are asymptotic as time and queue length become simultaneously large for the infinite capacity queue, and as the system’s storage capacity K becomes large for the finite capacity queue. We give asymptotic expansions for p_n(t), which is the probability that the system contains n customers at time t. We treat several cases of initial conditions and different traffic intensities. The results are based on (i) asymptotic expansion of an exact integral representation for p_n(t) and (ii) applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of p_n(t).     

The coexponent of a regular p -group

A sharp bound is derived for the nilpotency class of a regular p-group in terms of its coexponent, and is used to show that the number of groups of order pⁿ with a given fixed coexponent, is independent of n, for p and n sufficiently large. Explicit formulae are calculated in the case of coexponent 3.     

Lanczos and the Riemannian SVD in information retrieval applications

Variations of the latent semantic indexing (LSI) method in information retrieval (IR) require the computation of singular subspaces associated with the k dominant singular values of a large m × n sparse matrix A, where k?min(m,n). The Riemannian SVD was recently generalized to low‐rank matrices arising in IR and shown to be an effective approach for formulating an enhanced semantic model that captures the latent term‐document structure of the data. However, in terms of storage and computation requirements, its implementation can be much improved for large‐scale applications. We discuss an efficient and reliable algorithm, called SPK‐RSVD‐LSI, as an alternative approach for deriving the enhanced semantic model. The algorithm combines the generalized Riemannian SVD and the Lanczos method with full reorthogonalization and explicit restart strategies. We demonstrate that our approach performs as well as the original low‐rank Riemannian SVD method by comparing their retrieval performance on a well‐known benchmark document collection. Copyright 2004 John Wiley & Sons, Ltd.     

Initial layers of Zp-extensions and Greenberg's conjecture

Let p be a prime number, K a finite abelian extension of Q containing p-th roots of unity and K _n the n-th layer of the cyclotomic Z _p -extension of K. Under some conditions we construct an element of K _n from an ideal class of the maximal real subfield of K _n . We determine whether its p-th root is contained by some Z _p -extension of K _n or not for each n, using the zero of p-adic L-function and the order of the ideal class group of the maximal real subfield of K _m for sufficiently large m. Received: 13 February 1998 / Revised version: 30 September 1998     

Enumeration of Full Graphs: Onset of the Asymptotic Region

A full graph on n vertices, as defined by Fulkerson, is a representation of the intersection and containment relations among a system of n sets. It has an undirected edge between vertices representing intersecting sets and a directed edge from a to b if the corresponding set A contains B;. Kleitman, Lasaga, and Cowen gave a unified argument to show that asymptotically, almost all full graphs can be obtained by taking an arbitrary undirected graph on the n vertices, distinguishing a clique in this graph, which need not be maximal, and then adding directed edges going out from each vertex in the clique to all vertices to which there is not already an existing undirected edge. Call graphs of this type members of the dominant class. This article obtains the first upper and lower bounds on how large n has to be, so that the asymptotic behavior is indeed observed. It is shown that when n > 170, the dominant class predominates, while when n < 17, the full graphs in the dominant class compose less than half of the total number of realizable full graphs on n vertices.     

Fast detection of common sequence structure patterns in RNAs

We developed a dynamic programming approach for computing common exact sequential and structural patterns between two RNAs, given their sequences and their secondary structures. An RNA consists of a sequence of nucleotides and a secondary structure defined via bonds linking together complementary nucleotides. It is known that secondary structures are more preserved than sequences in the evolution of RNAs.We are able to compute all patterns between two RNAs in time O(nm) and space O(nm), where n and m are the lengths of the RNAs. Our method is useful for describing and detecting local motifs. It is especially suitable for finding similar regions of large RNAs that do not share global similarities. An implementation is available in C++ and can be obtained by contacting one of the authors.     

The falsity of the reconstruction conjecture for tournaments

The conjecture that for all sufficiently large p any tournament of order p is uniquely reconstructable from its point-deleted subtournaments is shown to be false. Counterexamples are presented for all orders of the form 2ⁿ + 1 and 2ⁿ + 2. The largest previously known counterexamples were of order 8.     

A parallel wilf algorithm for complex zeros of a polynomial

A global recursive bisection algorithm is described for computing the complex zeros of a polynomial. It has complexityO(n ³ p) wheren is the degree of the polynomial andp the bit precision requirement. Ifn processors are available, it can be realized in parallel with complexityO(n ² p); also it can be implemented using exact arithmetic. A combined Wilf-Hansen algorithm is suggested for reduction in complexity.     

Large sets of resolvable MTS and DTS of orders pn + 2

An LRMTS(v) [resp., LRDTS(v)] is a large set consisting of v − 2 [resp., 3(v − 2)] disjoint resolvable Mendelsohn (resp., directed) triple systems of order v. In this article, we give a method to construct LRMTS(pⁿ + 2) and LRDTS(pⁿ + 2), where pⁿ is a prime power and pⁿ ≡ 1 (mod 6). Using the method and a recursive construction v → 3v, some unknown LRMTS(v) and LRDTS(v) are obtained such as for v = 69, 123, 141, 159, and 3^km, where k ≥ 1, m ϵ {7, 13, 37, 55, 57, 61, 65, 67}. © 1996 John Wiley & Sons, Inc.     

Randomly coloring random graphs

We consider the problem of generating a coloring of the random graph ??_n,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are βΔ colors available, where Δ is the maximum degree of the graph, and we wish to determine the least β = β(p) such that the distribution is close to uniform in O(n log n) steps of the chain. This problem has been previously studied for ??_n,p in cases where np is relatively small. Here we consider the “dense” cases, where np ε [ω ln n, n] and ω = ω(n) → ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on β(p) than can be achieved using more general techniques. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

The fitting length of solvableH pn-groups

For a (finite) groupG and some prime powerp ⁿ, theH _p ⁿ -subgroupH _pn (G) is defined byH _p ⁿ (G)=〈xεG|x ^pn≠1〉. A groupH≠1 is called aH _p ⁿ -group, if there is a finite groupG such thatH is isomorphic toH _p ⁿ (G) andH _p ⁿ (G)≠G. It is known that the Fitting length of a solvableH _p ⁿ -group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH _p ⁿ -groups having Fitting lengthn, this result is “almost” best possible.     

On the threshold for k-regular subgraphs of random graphs

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Rényi random graph model G(n,p) is at most the threshold for the appearance of a nonempty (k+2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly 2/n for large n and moderately large k. The result is proved by using Tutte’s necessary and sufficient condition for a graph to have a k-factor.     

Optimal location with equitable loads

The problem considered in this paper is to find p locations for p facilities such that the weights attracted to each facility will be as close as possible to one another. We model this problem as minimizing the maximum among all the total weights attracted to the various facilities. We propose solution procedures for the problem on a network, and for the special cases of the problem on a tree or on a path. The complexity of the problem is analyzed, O(n) algorithms and an O(pn ³) dynamic programming algorithm are proposed for the problem on a path respectively for p=2 and p>2 facilities. Heuristic algorithms (two types of a steepest descent approach and tabu search) are proposed for its solution. Extensive computational results are presented.     

n-Summable Valuated p n -Socles and Primary Abelian Groups

Generalizing the classical concept of a valuated vector space, we introduce the notion of a valuated p ⁿ -socle. A valuated p ⁿ -socle is said to be n-summable if it is isometric to the valuated direct sum of countable valuated groups. Many properties of these objects are established, and in particular, they are shown to be completely classifiable using Ulm invariants, providing a strong connection with the theory of direct sums of countable abelian p-groups. The resulting theory is then applied to the category of primary abelian groups.