Sivaram K. Narayan

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We consider frames in a finite-dimensional Hilbert space H n where frames are exactly the spanning sets of the vector space. The diagram vector of a vector in R 2 was previously defined using polar coordinates and was used to characterize tight frames in R 2 in a geometric fashion. Reformulating the definition of a diagram vector in R 2 we provide a natural(More)
The minimum vector rank (mvr) of a graph over a field F is the smallest d for which a faithful vector representation of G exists in Fd . For simple graphs, minimum semidefinite rank (msr) and minimum vector rank differ by exactly the number of isolated vertices. We explore the relationship between msr and mvr for multigraphs and show that a result linking(More)
Let G = (V, E) be a multigraph with no loops on the vertex set V = {1, 2, . . . , n}. Define S+(G) as the set of symmetric positive semidefinite matrices A = [aij ] with aij 6= 0, i 6= j, if ij ∈ E(G) is a single edge and aij = 0, i 6= j, if ij / ∈ E(G). Let M+(G) denote the maximum multiplicity of zero as an eigenvalue of A ∈ S+(G) and mr+(G) = |G|−M+(G)(More)
Let G = (V, E) be a multigraph with no loops on the vertex set V = {1, 2,. .. , n}. Define S + (G) as the set of symmetric positive semidefinite matrices A = [a ij ] with a ij = 0, i = j, if ij ∈ E(G) is a single edge and a ij = 0, i = j, if ij / ∈ E(G). Let M + (G) denote the maximum multiplicity of zero as an eigenvalue of A ∈ S + (G) and mr + (G) = |G| −(More)
A magic square M is an n-by-n array of numbers whose rows, columns, and the two diagonals sum to µ called the magic sum. If all the diagonals including broken diagonals sum to µ then the magic square is said to be pandiagonal. A regular magic square satisfies the condition that the entries symmetrically placed with respect to the center sum to 2µ n. If the(More)
A tight frame in R<sup>n</sup> is a redundant system which has a reconstruction formula similar to that of an orthonormal basis. For a unit-norm frame F = {f<sub>i</sub>}<sup>k</sup><sub>i=1</sub>, a scaling is a vector c = (c(l),..., c(k)) &#x03B5; R<sup>k</sup>&#x2265;0 such that {c(i)f<sub>i</sub>}<sup>k</sup><sub>i=1</sub> is a tight frame in(More)
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