Contributions in Mathematical and Computational Sciences Volume 1
... the 3-sphere. Modern topology has also obtained information on high-dimensional knots, that is, embeddings of an n-sphere in an (n + 2)-sphere with n larger than one. In algebra, representations of quantum groups lead to a multitude of knot invariants. Based on ideas of B. Mazur in number theory, on ...
... the 3-sphere. Modern topology has also obtained information on high-dimensional knots, that is, embeddings of an n-sphere in an (n + 2)-sphere with n larger than one. In algebra, representations of quantum groups lead to a multitude of knot invariants. Based on ideas of B. Mazur in number theory, on ...
On the limiting spectral distribution for a large class of symmetric
... The limiting spectral distribution for symmetric matrices with correlated entries received a lot of attention in the last two decades. The starting point is deep results for symmetric matrices with correlated Gaussian entries by Khorunzhy and Pastur [13], Boutet de Monvel et al [6], Boutet de Monvel ...
... The limiting spectral distribution for symmetric matrices with correlated entries received a lot of attention in the last two decades. The starting point is deep results for symmetric matrices with correlated Gaussian entries by Khorunzhy and Pastur [13], Boutet de Monvel et al [6], Boutet de Monvel ...
Introduction to Flocking {Stochastic Matrices}
... Suppose G 2 G is neighbor shared. Then any pair of vertices (i, j) must be reachable from a {common neighbor} vertex k. Suppose for some integer p 2 {2, 3, ..., n -1}, each subset of p vertices is reachable from a single vertex. Let {v1, v2, ..., vp} be any any such set and let v be a vertex from w ...
... Suppose G 2 G is neighbor shared. Then any pair of vertices (i, j) must be reachable from a {common neighbor} vertex k. Suppose for some integer p 2 {2, 3, ..., n -1}, each subset of p vertices is reachable from a single vertex. Let {v1, v2, ..., vp} be any any such set and let v be a vertex from w ...
Parallel numerical linear algebra
... A very simple model for the time it takes to move n data items from one location to another is + n, where 0 ; . One way to describe is the start up time of the operation; another term for this is latency. The incremental time per data item moved is ; its reciprocal is called bandwidth. ...
... A very simple model for the time it takes to move n data items from one location to another is + n, where 0 ; . One way to describe is the start up time of the operation; another term for this is latency. The incremental time per data item moved is ; its reciprocal is called bandwidth. ...
Tranquilli, G.B.; (1965)On the normality of independent random variables implied by intrinsic graph independence without residues."
... Therefore by the Lemma proved above (7), the cumulants A (j) are finite r ...
... Therefore by the Lemma proved above (7), the cumulants A (j) are finite r ...
Homework 1. Solutions 1 a) Let x 2 + y2 = R2 be a circle in E2. Write
... b) det G = A(u, v)D(u, v) − B(u, v)C(u, v) = AD − B 2 6= 0 since it is non-degenerate (see the solution of exercise 1) c) Consider quadratic form G(x, x) = gik xi xk = Ax2 +2Bxy+Dy 2 . (We already know that B = C) Positive -definiteness means that G(x, x) > 0 for all x 6= 0. In particular if we put ...
... b) det G = A(u, v)D(u, v) − B(u, v)C(u, v) = AD − B 2 6= 0 since it is non-degenerate (see the solution of exercise 1) c) Consider quadratic form G(x, x) = gik xi xk = Ax2 +2Bxy+Dy 2 . (We already know that B = C) Positive -definiteness means that G(x, x) > 0 for all x 6= 0. In particular if we put ...
Multiple fundamental frequency estimation based on sparse
... The purpose of sparse coding is to approximate the solution of the following ( P 0 ) problem: ...
... The purpose of sparse coding is to approximate the solution of the following ( P 0 ) problem: ...
On Positive Integer Powers of Toeplitz Matrices
... Greville, 1983). In 1987, Tamir Shalom derived an equivalent necessary and sufficient condition for a nonsingular Toeplitz matrix to have a Toeplitz inverse, and proved that the statement “All positive powers of a nonsingular Toeplitz matrix are Toeplitz matrices” is equivalent to the statement “The i ...
... Greville, 1983). In 1987, Tamir Shalom derived an equivalent necessary and sufficient condition for a nonsingular Toeplitz matrix to have a Toeplitz inverse, and proved that the statement “All positive powers of a nonsingular Toeplitz matrix are Toeplitz matrices” is equivalent to the statement “The i ...
Package `GeneralizedUmatrix`
... res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) # Stress = KruskalStress(InputDistances, as.matrix(dist(ProjectedPoints))) #resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) #plotTopographicMap(resUmatrix$Umatrix,resUmatrix$Bestma ...
... res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) # Stress = KruskalStress(InputDistances, as.matrix(dist(ProjectedPoints))) #resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) #plotTopographicMap(resUmatrix$Umatrix,resUmatrix$Bestma ...
Bipartie Matchings
... There exists a perfect matching in the graph G if and only if the adjacency matrix contains a set of n 1’s, no two of which are in the same column or row. In other words, if all other entries were 0 a permutation matrix would result. Consider the function called the permanent of A, defined as follow ...
... There exists a perfect matching in the graph G if and only if the adjacency matrix contains a set of n 1’s, no two of which are in the same column or row. In other words, if all other entries were 0 a permutation matrix would result. Consider the function called the permanent of A, defined as follow ...
Non-negative matrix factorization
NMF redirects here. For the bridge convention, see new minor forcing.Non-negative matrix factorization (NMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.NMF finds applications in such fields as computer vision, document clustering, chemometrics, audio signal processing and recommender systems.