36 research outputs found
Evaluating Matrix Functions by Resummations on Graphs: the Method of Path-Sums
We introduce the method of path-sums which is a tool for exactly evaluating a
function of a discrete matrix with possibly non-commuting entries, based on the
closed-form resummation of infinite families of terms in the corresponding
Taylor series. If the matrix is finite, our approach yields the exact result in
a finite number of steps. We achieve this by combining a mapping between matrix
powers and walks on a weighted directed graph with a universal graph-theoretic
result on the structure of such walks. We present path-sum expressions for a
matrix raised to a complex power, the matrix exponential, matrix inverse, and
matrix logarithm. We show that the quasideterminants of a matrix can be
naturally formulated in terms of a path-sum, and present examples of the
application of the path-sum method. We show that obtaining the inversion height
of a matrix inverse and of quasideterminants is an NP-complete problem.Comment: 23 pages, light version submitted to SIAM Journal on Matrix Analysis
and Applications (SIMAX). A separate paper with the graph theoretic results
is available at: arXiv:1202.5523v1. Results for matrices over division rings
will be published separately as wel
Discrete Quantum Walks on Graphs and Digraphs
This thesis studies various models of discrete quantum walks on graphs and digraphs via a spectral approach.
A discrete quantum walk on a digraph is determined by a unitary matrix , which acts on complex functions of the arcs of . Generally speaking, is a product of two sparse unitary matrices, based on two direct-sum decompositions of the state space. Our goal is to relate properties of the walk to properties of , given some of these decompositions.
We start by exploring two models that involve coin operators, one due to Kendon, and the other due to Aharonov, Ambainis, Kempe, and Vazirani. While is not defined as a function in the adjacency matrix of the graph , we find exact spectral correspondence between and . This leads to characterization of rare phenomena, such as perfect state transfer and uniform average vertex mixing, in terms of the eigenvalues and eigenvectors of . We also construct infinite families of graphs and digraphs that admit the aforementioned phenomena.
The second part of this thesis analyzes abstract quantum walks, with no extra assumption on . We show that knowing the spectral decomposition of leads to better understanding of the time-averaged limit of the probability distribution. In particular, we derive three upper bounds on the mixing time, and characterize different forms of uniform limiting distribution, using the spectral information of .
Finally, we construct a new model of discrete quantum walks from orientable embeddings of graphs. We show that the behavior of this walk largely depends on the vertex-face incidence structure. Circular embeddings of regular graphs for which has few eigenvalues are characterized. For instance, if has exactly three eigenvalues, then the vertex-face incidence structure is a symmetric -design, and is the exponential of a scalar multiple of the skew-symmetric adjacency matrix of an oriented graph. We prove that, for every regular embedding of a complete graph, is the transition matrix of a continuous quantum walk on an oriented graph
Grassmann Integral Representation for Spanning Hyperforests
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set,
and show that a class of Grassmann integrals permits an expansion in terms of
spanning hyperforests. Special cases provide the generating functions for
rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All
these results are generalizations of Kirchhoff's matrix-tree theorem.
Furthermore, we show that the class of integrals describing unrooted spanning
(hyper)forests is induced by a theory with an underlying OSP(1|2)
supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J.
Phys.
Sublinear Computation Paradigm
This open access book gives an overview of cutting-edge work on a new paradigm called the “sublinear computation paradigm,” which was proposed in the large multiyear academic research project “Foundations of Innovative Algorithms for Big Data.” That project ran from October 2014 to March 2020, in Japan. To handle the unprecedented explosion of big data sets in research, industry, and other areas of society, there is an urgent need to develop novel methods and approaches for big data analysis. To meet this need, innovative changes in algorithm theory for big data are being pursued. For example, polynomial-time algorithms have thus far been regarded as “fast,” but if a quadratic-time algorithm is applied to a petabyte-scale or larger big data set, problems are encountered in terms of computational resources or running time. To deal with this critical computational and algorithmic bottleneck, linear, sublinear, and constant time algorithms are required. The sublinear computation paradigm is proposed here in order to support innovation in the big data era. A foundation of innovative algorithms has been created by developing computational procedures, data structures, and modelling techniques for big data. The project is organized into three teams that focus on sublinear algorithms, sublinear data structures, and sublinear modelling. The work has provided high-level academic research results of strong computational and algorithmic interest, which are presented in this book. The book consists of five parts: Part I, which consists of a single chapter on the concept of the sublinear computation paradigm; Parts II, III, and IV review results on sublinear algorithms, sublinear data structures, and sublinear modelling, respectively; Part V presents application results. The information presented here will inspire the researchers who work in the field of modern algorithms
Skew Howe duality and limit shapes of Young diagrams
We consider the skew Howe duality for the action of certain dual pairs of Lie
groups on the exterior algebra as a probability measure on Young diagrams by the
decomposition into the sum of irreducible representations. We prove a
combinatorial version of this skew Howe for the pairs , ,
, and using crystal bases, which allows us to interpret the skew
Howe duality as a natural consequence of lattice paths on lozenge tilings of
certain partial hexagonal domains. The -representation multiplicity is
given as a determinant formula using the Lindstr\"om-Gessel-Viennot lemma and
as a product formula using Dodgson condensation. These admit natural
-analogs that we show equals the -dimension of a -representation (up
to an overall factor of ), giving a refined version of the combinatorial
skew Howe duality. Using these product formulas (at ), we take the
infinite rank limit and prove the diagrams converge uniformly to the limit
shape.Comment: 54 pages, 12 figures, 2 tables; v2 fixed typos in Theorem 4.10, 4.14,
shorter proof of Theorem 4.6 (thanks to C. Krattenthaler), proved of
Conjecture 4.17 in v
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum