3,874 research outputs found
Inverting graphs of rectangular matrices
AbstractThis paper addresses the question of determining the class of rectangular matrices having a given graph as a row or column graph. We also determine equivalent conditions on a given pair of graphs in order for them to be the row and column graphs of some rectangular matrix. In connection with these graph inversion problems we discuss the concept of minimal inverses. This concept turns out to have two different forms in the case of one-graph inversion. For the two-graph case we present a method of determining when an inverse is minimal. Finally we apply the two-graph theorem to a class of energy related matrices
Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)
We consider a product of an arbitrary number of independent rectangular
Gaussian random matrices. We derive the mean densities of its eigenvalues and
singular values in the thermodynamic limit, eventually verified numerically.
These densities are encoded in the form of the so called M-transforms, for
which polynomial equations are found. We exploit the methods of planar
diagrammatics, enhanced to the non-Hermitian case, and free random variables,
respectively; both are described in the appendices. As particular results of
these two main equations, we find the singular behavior of the spectral
densities near zero. Moreover, we propose a finite-size form of the spectral
density of the product close to the border of its eigenvalues' domain. Also,
led by the striking similarity between the two main equations, we put forward a
conjecture about a simple relationship between the eigenvalues and singular
values of any non-Hermitian random matrix whose spectrum exhibits rotational
symmetry around zero.Comment: 50 pages, 8 figures, to appear in the Proceedings of the 23rd Marian
Smoluchowski Symposium on Statistical Physics: "Random Matrices, Statistical
Physics and Information Theory," September 26-30, 2010, Krakow, Polan
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Constructing a Distance Sensitivity Oracle in O(n^2.5794 M) Time
M}, in O(n^2.5286 M) time. This algorithm is crucial in the preprocessing algorithm of our DSO. Our solution improves the O(n^2.6865 M) time bound in [Ren, ESA 2020], and matches the current best time bound for computing all-pairs shortest paths
Connected Substitutes and Invertibility of Demand
We consider the invertibility of a nonparametric nonseparable demand system. Invertibility of demand is important in several contexts, including identification of demand, estimation of demand, testing of revealed preference, and economic theory requiring uniqueness of market clearing prices. We introduce the notion of "connected substitutes" and show that this structure is sufficient for invertibility. The connected substitutes conditions require weak substitution between all goods and sufficient strict substitution to necessitate treating them in a single demand system. These conditions are satisfied in many standard models, have transparent economic interpretation, and allow us to show invertibility without functional form restrictions, smoothness assumptions, or strong domain restrictions.Demand, Invertibility, Connected substitutes
Quantum transport on two-dimensional regular graphs
We study the quantum-mechanical transport on two-dimensional graphs by means
of continuous-time quantum walks and analyse the effect of different boundary
conditions (BCs). For periodic BCs in both directions, i.e., for tori, the
problem can be treated in a large measure analytically. Some of these results
carry over to graphs which obey open boundary conditions (OBCs), such as
cylinders or rectangles. Under OBCs the long time transition probabilities
(LPs) also display asymmetries for certain graphs, as a function of their
particular sizes. Interestingly, these effects do not show up in the marginal
distributions, obtained by summing the LPs along one direction.Comment: 22 pages, 11 figure, acceted for publication in J.Phys.
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