Papers by Assaf Goldberger
Linear & Multilinear Algebra, 2002
It is shown that the class of all n × n inverse M-matrices A for which both A and A m 1 are circu... more It is shown that the class of all n × n inverse M-matrices A for which both A and A m 1 are circulant matrices on three symbols is closed under Hadamard products.
Proceedings of the American Mathematical Society, Jan 2, 2009
In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle ... more In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A is an (n+1)×(n+1) nonnegative matrix whose nonzero eigenvalues are: λ 0 ≥ |λ i |, i = 1,. .. , r, r ≤ n, then for all x ≥ λ 0 , (*) r i=0 (x − λ i) ≤ x r+1 − λ r+1 0. To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when 2(r + 1) ≥ (n + 1), while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when n ≤ 4 and when the spectrum of A is real. They also showed that the conjecture is asymptotically true with the dimension. Here we prove a slightly stronger inequality than in (*), from which it follows that the Boyle-Handelman conjecture is true. Actually, we do not start from the assumption that the λ i 's are eigenvalues of a nonnegative matrix, but that λ 1 ,. .. , λ r+1 satisfy λ 0 ≥ |λ i |, i = 1,. .. , r, and the trace conditions: (* *) r i=0 λ k i ≥ 0, for all k ≥ 1. A strong form of the Boyle-Handelman conjecture, conjectured in 2002 by the present authors, says that (*) continues to hold if the trace inequalities in (* *) hold only for k = 1,. .. , r. We further improve here on earlier results of the authors concerning this stronger form of the Boyle-Handelman conjecture.
Linear & Multilinear Algebra, Nov 11, 2021
Every (full) finite Gabor system generated by a unit-norm vector g ∈ C d is a finite unit-norm ti... more Every (full) finite Gabor system generated by a unit-norm vector g ∈ C d is a finite unit-norm tight frame (FUNTF), and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the d 2 corresponding rank one matrices form a basis for the space of d × d matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank one matrices generated by a finite Gabor frame. We also use these results to construct some sets of d 2 unit vectors in C d with a relatively smaller number of distinct inner products.
Czechoslovak Mathematical Journal, Sep 1, 2004
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Journal of Pharmaceutical Sciences, 1997
A general mathematical model was developed to describe drug release from erodible tablets undergo... more A general mathematical model was developed to describe drug release from erodible tablets undergoing surface erosion. The model (which is based on the Hopfenberg equation) takes into account the three dimensions of a tablet dosage form. The model enables the characterization of the release kinetics by one or two erosion rate constants depending on the hydrodynamic conditions of the system. The model can be used to compare the release rates of the drug within different batches produced under different conditions. The model was utilized to evaluate some factors affecting the release rate from erodible tablets consisting of hydroxypropyl methylcellulose and amoxicillin. The release data of amoxicillin measured from the whole tablet followed the equation developed. Amoxicillin release data measured from the planar surface of the tablet appeared to follow zero-order kinetics as predicted by theory for drug release from an erodible device maintaining constant surface area with time.
Linear Algebra and its Applications, Apr 1, 2005
For nonnegative matrices A, the well known Perron-Frobenius theory studies the spectral radius ρ(... more For nonnegative matrices A, the well known Perron-Frobenius theory studies the spectral radius ρ(A). Rump has offered a way to generalize the theory to arbitrary complex matrices. He replaced the usual eigenvalue problem with the equation |Ax| = λ|x| and he replaced ρ(A) by the signed spectral radius, which is the maximum λ that admits a nontrivial solution to that equation. We generalize this notion by replacing the linear transformation A by a map f : C n → R whose coordinates are seminorms, and we use the same definition of Rump for the signed spectral radius. Many of the features of the Perron-Frobenius theory remain true in this setting. At the center of our discussion there is an alternative theorem relating the inequalities f (x) λ|x| and f (x) < λ|x|, which follows from topological principals. This enables us to free the theory from matrix theoretic considerations and discuss it in the generality of seminorms. Some consequences for P-matrices and D-stable matrices are discussed.
Advances in Applied Mathematics, Aug 1, 2006
Let A, B and C = U 1 AU * 1 + U 2 BU * 2 be hermitian (or real symmetric) matrices, where U 1 and... more Let A, B and C = U 1 AU * 1 + U 2 BU * 2 be hermitian (or real symmetric) matrices, where U 1 and U 2 are unitary (respectively orthogonal). Our goal is to develop as concrete as possible expression to the probability distribution of the spectrum of C where U i are drawn from the unitary (respectively orthogonal) group with respect to the Haar measure. While we do this for the unitary group, using representation theory of U n , we can only accomplish the same for the real symmetric case where B has rank 1, by performing explicit calculations. Here is what we do in the unitary case. For a given n by n matrix A over the field of complex numbers we study the operator E N A = uAu * ⊗N du, where the integration is taken over the unitarian group with respect to the Haar measure. When A has nonnegative spectrum, we show that E N A , as N tends to infinity, is concentrated around some simple S N × U n submodule of (C n) ⊗N determined only by the spectra of A. Along the proof we reprove a generalization of Heckman to the convexity theorem of Horn. The Schur-Weyl duality and the technique of Bernstein polynomials approximations are our tools. Using this we compute the distribution, induced by a sum of two hermitian orbits on the set of hermitian orbits in terms of asymptotic Littlewood-Richardson coefficients times an asymptotic version of the Weyl dimensional formula. More precisely, translation of the lattice permutation and semistandard Young tableaux rules to a set of linear inequalities, enable us to express the density of the distribution above as multiplication of volumes of two concrete polytopes.
arXiv (Cornell University), Jun 2, 2021
Every (full) finite Gabor system generated by a unit-norm vector g ∈ C d is a finite unit-norm ti... more Every (full) finite Gabor system generated by a unit-norm vector g ∈ C d is a finite unit-norm tight frame (FUNTF), and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the d 2 corresponding rank one matrices form a basis for the space of d × d matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank one matrices generated by a finite Gabor frame. We also use these results to construct some sets of d 2 unit vectors in C d with a relatively smaller number of distinct inner products.
arXiv (Cornell University), Mar 1, 2019
The aim of this work is to construct families of weighing matrices via their automorphism group a... more The aim of this work is to construct families of weighing matrices via their automorphism group action. This action is determined from the 0, 1, 2-cohomology groups of the underlying abstract group. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley Conference, the Projective-Space, the Grassmannian, and the Flag-Variety weighing matrices. We develop a general theory relying on low dimensional group-cohomology for constructing automorphism group actions, and in turn obtain structured matrices that we call Cohomology-Developed matrices. This "Cohomology-Development" generalizes the Cocyclic and Group Developments. The Algebraic structure of modules of Cohomology-Developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of Quasiproducts, which is a generalization of the Kronecker-product.
arXiv (Cornell University), Mar 1, 2019
The aim of this work is to construct families of weighing matrices via their automorphism group a... more The aim of this work is to construct families of weighing matrices via their automorphism group action. The matrices can be reconstructed from the 0, 1, 2-cohomology groups of the underlying automorphism group. We use this mechanism to (re)construct the matrices out of abstract group datum. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley Conference, the Projective-Space, the Grassmannian, and the Flag-Variety weighing matrices. We develop a general theory relying on low dimensional group-cohomology for constructing automorphism group actions, and in turn obtain structured matrices that we call Cohomology-Developed matrices. This "Cohomology-Development" generalizes the Cocyclic and Group Developments. The Algebraic structure of modules of Cohomology-Developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of Quasiproducts, which is a generalization of the Kronecker-product.
arXiv (Cornell University), May 24, 2015
The set of diagonals of real symmetric matrices of given non negative spectrum is endowed with a ... more The set of diagonals of real symmetric matrices of given non negative spectrum is endowed with a measure which is obtained by the push forward of the Haar measure of the real orthogonal group. We prove that the Radon Nicodym derivation of this measure with respect to the relative Euclidean measure is approximated by the coefficients of a sequence of zonal sphere polynomials corresponding with the given spectrum. There is a striking similarity between the role of the zonal sphere polynomials in the orthogonal case, and that of the Schur function in the Hermitian case. Following this we obtain a combinatorial approximation for the probability of real symmetric matrix of a given spectrum to appear as the sum of two real symmetric matrices, each of a given spectrum. In addition we obtain a real orthogonal analogue to the Zuber Itzykson Harish Chandra integration formula.
PharmacoEconomics, 2007
Uncertainty in the decision-making process for reimbursement of health technologies could be redu... more Uncertainty in the decision-making process for reimbursement of health technologies could be reduced if additional information were available. Although methods to evaluate the monetary value of the uncertainty have been previously described, an economic evaluation of alternative methods to acquire additional information has not yet been thoroughly explored. Should resources be allocated to a retrospective study design or to a
Fingerprinting is a widely-used technique for efficiently verifying that two files are identical.... more Fingerprinting is a widely-used technique for efficiently verifying that two files are identical. More generally, linear sketching is a form of lossy compression (based on random projections) that also enables the "dissimilarity" of nonidentical files to be estimated. Many sketches have been proposed for dissimilarity measures that decompose coordinatewise such as the Hamming distance between alphanumeric strings, or the Euclidean distance between vectors. However, virtually nothing is known on sketches that would accommodate alignment errors. With such errors, Hamming or Euclidean distances are rendered useless: a small misalignment may result in a file that looks very dissimilar to the original file according such measures. In this paper, we present the first linear sketch that is robust to a small number of alignment errors. Specifically, the sketch can be used to determine whether two files are within a small Hamming distance of being a cyclic shift of each other. Furthermore, the sketch is homomorphic with respect to rotations: it is possible to construct the sketch of a cyclic shift of a file given only the sketch of the original file. The relevant dissimilarity measure, known as the shift distance, arises in the context of embedding edit distance and our result addressed an open problem [26, Question 13] with a rather surprising outcome. Our sketch projects a length n file into D(n) • polylog n dimensions where D(n) n is the number of divisors of n. The striking fact is that this is near-optimal, i.e., the D(n) dependence is inherent to a problem that is ostensibly about lossy compression. In contrast, we then show that any sketch for estimating the edit distance between two files, even when small, requires sketches whose size is nearly linear in n. This lower bound addresses a long-standing open problem on the low distor-* Supported by NSF CAREER Award CCF-0953754. † This work was supported by a Google award.
arXiv (Cornell University), Jun 1, 2020
A Formal Orthogonal Pair is a pair (A, B) of symbolic rectangular matrices such that AB T = 0. It... more A Formal Orthogonal Pair is a pair (A, B) of symbolic rectangular matrices such that AB T = 0. It can be applied for the construction of Hadamard and Weighing matrices. In this paper we introduce a systematic way for constructing such pairs. Our method involves Representation Theory and Group Cohomology. The orthogonality property is a consequence of non-vanishing maps between certain cohomology groups. This construction has strong connections to the theory of Association Schemes and (weighted) Coherent Configurations. Our techniques are also capable for producing (anti-) amicable pairs. A handful of examples are given.
Journal of Algebraic Combinatorics, Nov 9, 2021
A (classical) partial Hadamard Matrix is an $$m\times n$$ m × n matrix H with values in $$\{-1,1\... more A (classical) partial Hadamard Matrix is an $$m\times n$$ m × n matrix H with values in $$\{-1,1\}$$ { - 1 , 1 } such that $$HH^T=nI_m$$ H H T = n I m . If $$m=n$$ m = n , we say that H is a (full) Hadamard Matrix. In this paper, we address the following Completion Problem: Can a given partial Hadamard matrix be completed to a full Hadamard matrix? And is this completion unique? As soon as $$m\ge n/2$$ m ≥ n / 2 , and under reasonable assumptions, we give a practical algorithm for finding the completion if it exists, or deciding that there is no completion with a great level of certainty. The algorithm works well in practice, at least for $$n<150$$ n < 150 , and depends on an unproven conjecture on the performance of the LLL algorithm. We apply a version of this algorithm to solve the Gram square-root problem: Solving $$XX^T=G$$ X X T = G for X over the integers. As an application, we show how to reconstruct a weighing matrix W (23, 16) from a $$13\times 13$$ 13 × 13 submatrix.
arXiv (Cornell University), Apr 19, 2023
In this paper we describe an algorithm for generating all the possible P IW (m, n, k)-integer m ×... more In this paper we describe an algorithm for generating all the possible P IW (m, n, k)-integer m × n Weighing matrices of weight k up to Hadamard equivalence. Our method is efficient on a personal computer for small size matrices, up to m ≤ n = 12, and k ≤ 50. As a by product we also improved the nsoks [Rie06] algorithm to find all possible representations of an integer k as a sum of n integer squares. We have implemented our algorithm in Sagemath and as an example we provide a complete classification for n = m = 7 and k = 25. Our list of IW (7, 25) can serve as a step towards finding the open classical weighing matrix W (35, 25).
arXiv (Cornell University), Dec 8, 2022
Given N points X = {x k } N k=1 on the unit circle in R 2 and a number 0 ≤ p ≤ ∞ we investigate t... more Given N points X = {x k } N k=1 on the unit circle in R 2 and a number 0 ≤ p ≤ ∞ we investigate the minimizers of the functional N k,ℓ=1 | x k , x ℓ | p. While it is known that each of these minimizers is a spanning set for R 2 , less is known about their number as a function of p and N especially for relatively small p. In this paper we show that there is unique minimum for this functional for all p ≤ log 3/ log 2 and all odd N ≥ 3. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for N ≥ 3 odd, there exists a sequence of number of points log 3/ log 2 = p 1 < p 2 <. .. < p N ≤ 2 so that a unique (up to some isometries) minimizer exists on each sub-intervals (p k , p k+1).
Cornell University - arXiv, Jun 1, 2020
A Formal Orthogonal Pair is a pair (A, B) of symbolic rectangular matrices such that AB T = 0. It... more A Formal Orthogonal Pair is a pair (A, B) of symbolic rectangular matrices such that AB T = 0. It can be applied for the construction of Hadamard and Weighing matrices. In this paper we introduce a systematic way for constructing such pairs. Our method involves Representation Theory and Group Cohomology. The orthogonality property is a consequence of non-vanishing maps between certain cohomology groups. This construction has strong connections to the theory of Association Schemes and (weighted) Coherent Configurations. Our techniques are also capable for producing (anti-) amicable pairs. A handful of examples are given.
arXiv: Functional Analysis, Jun 2, 2021
Every (full) finite Gabor system generated by a unit-norm vector g ∈ C d is a finite unit-norm ti... more Every (full) finite Gabor system generated by a unit-norm vector g ∈ C d is a finite unit-norm tight frame (FUNTF), and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the d 2 corresponding rank one matrices form a basis for the space of d × d matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank one matrices generated by a finite Gabor frame. We also use these results to construct some sets of d 2 unit vectors in C d with a relatively smaller number of distinct inner products.
arXiv: Group Theory, 2019
The aim of this work is to construct families of weighing matrices via automorphisms and cohomolo... more The aim of this work is to construct families of weighing matrices via automorphisms and cohomology. We study some well known families such as Payley's conference and Hadamard matrices and Projective Space weighing matrices, and put them in the context of a general theory. As a consequence, we get the new family of Grassmannian weighing matrices. Our theory generalizes the theory of Cocyclic matrices.
Uploads
Papers by Assaf Goldberger