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Articles 1 - 7 of 7
Full-Text Articles in Mathematics
Siegel’S Lemma Outside Of A Union Of Varieties, Lenny Fukshansky
Siegel’S Lemma Outside Of A Union Of Varieties, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the AMS Special Session on Number Theory, October 2008.
On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
CMC Faculty Publications and Research
A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of Z², as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum …
Cosamp: Iterative Signal Recovery From Incomplete And Inaccurate Samples, Deanna Needell, J. A. Tropp
Cosamp: Iterative Signal Recovery From Incomplete And Inaccurate Samples, Deanna Needell, J. A. Tropp
CMC Faculty Publications and Research
Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm calledCoSaMP that delivers the same guarantees as the best optimization-based approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(Nlog2N), where N is the length …
On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at Institut de Mathématiques in Bordeaux, France, June 2008.
Uniform Uncertainty Principle And Signal Recovery Via Regularized Orthogonal Matching Pursuit, Deanna Needell, Roman Vershynin
Uniform Uncertainty Principle And Signal Recovery Via Regularized Orthogonal Matching Pursuit, Deanna Needell, Roman Vershynin
CMC Faculty Publications and Research
This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.
Frobenius Number, Covering Radius, And Well-Rounded Lattices, Lenny Fukshansky, Sinai Robins
Frobenius Number, Covering Radius, And Well-Rounded Lattices, Lenny Fukshansky, Sinai Robins
CMC Faculty Publications and Research
Lecture given at the Joint Mathematics Meeting in San Diego, January 2008.
Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky
Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky
CMC Faculty Publications and Research
Let N >= 2 be an integer, F a quadratic form in N variables over (Q) over bar, and Z subset of (Q) over bar (N) an L-dimensional subspace, 1 <= L <= N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F). This provides an analogue over (Q) over bar of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over (Q) over bar. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over (Q) over bar. This extends previous results of the author over number fields. All bounds on height are explicit.