6,689 research outputs found
Dense sets of integers with prescribed representation functions
Let A be a set of integers and let h \geq 2. For every integer n, let r_{A,
h}(n) denote the number of representations of n in the form n=a_1+...+a_h,
where a_1,...,a_h belong to the set A, and a_1\leq ... \leq a_h. The function
r_{A,h} from the integers Z to the nonnegative integers N_0 U {\infty} is
called the representation function of order h for the set A. We prove that
every function f from Z to N_0 U {\infty} satisfying liminf_{|n|->\infty} f
(n)\geq g is the representation function of order h for some sequence A of
integers, and that A can be constructed so that it increases "almost" as slowly
as any given B_h[g] sequence. In particular, for every epsilon >0 and g \geq
g(h,epsilon), we can construct a sequence A satisfying r_{A,h}=f and A(x)\gg
x^{(1/h)-epsilon}.Comment: 10 page
Flexible Multi-layer Sparse Approximations of Matrices and Applications
The computational cost of many signal processing and machine learning
techniques is often dominated by the cost of applying certain linear operators
to high-dimensional vectors. This paper introduces an algorithm aimed at
reducing the complexity of applying linear operators in high dimension by
approximately factorizing the corresponding matrix into few sparse factors. The
approach relies on recent advances in non-convex optimization. It is first
explained and analyzed in details and then demonstrated experimentally on
various problems including dictionary learning for image denoising, and the
approximation of large matrices arising in inverse problems
-adic quotient sets
For , the question of when is dense in the positive real numbers has been examined by
many authors over the years. In contrast, the -adic setting is largely
unexplored. We investigate conditions under which is dense in the
-adic numbers. Techniques from elementary, algebraic, and analytic number
theory are employed in this endeavor. We also pose many open questions that
should be of general interest.Comment: 24 page
Semicontinuity for representations of one-dimensional Cohen-Macaulay rings
We show that the number of parameters for CM-modules of prescribed rank is
semi-continuous in families of CM rings of Krull dimension 1. This transfers a
result of Knoerrer from the commutative to the not necessarily commutative
case. For this purpose we introduce the notion of ``dense subrings'' which
seems rather technical but, nevertheless, useful. It enables the construction
of ``almost versal'' families of modules for a given algebra and the definition
of the ``number of parameters''. The semi--continuity implies, in particular,
that the set of so-called ``wild algebras'' in any family is a countable union
of closed subsets. A very exciting problem is whether it is actually closed,
hence whether the set of tame algebras is open. However, together with the
results of a former paper of the authors the semi-continuity implies that tame
is indeed an open property for curve singularities (commutative CM rings). An
analogous procedure leads to the semicontinuity of the number of parameters in
other cases, like representations of finite dimensional algebras or finite
dimensional bimodules.Comment: LaTeX2
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