17,470 research outputs found
The Morse-Sard theorem revisited
Let be positive integers with . We establish an abstract
Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous
result of De Pascale's for Sobolev functions with and, on the other hand, also the following
new result: if satisfies
for every
(that is, is a Stepanov function), then the set
of critical values of is Lebesgue-null in . In the case that
we also show that this limiting condition holding for every
, where is a set of zero
-dimensional Hausdorff measure for some , is
sufficient to guarantee the same conclusion.Comment: We corrected some misprints and made some changes in the introductio
The structure theory of set addition revisited
In this article we survey some of the recent developments in the structure
theory of set addition.Comment: 38p
How many zeros of a random polynomial are real?
We provide an elementary geometric derivation of the Kac integral formula for
the expected number of real zeros of a random polynomial with independent
standard normally distributed coefficients. We show that the expected number of
real zeros is simply the length of the moment curve
projected onto the surface of the unit sphere, divided by . The
probability density of the real zeros is proportional to how fast this curve is
traced out. We then relax Kac's assumptions by considering a variety of random
sums, series, and distributions, and we also illustrate such ideas as integral
geometry and the Fubini-Study metric.Comment: 37 page
Improving Strategies via SMT Solving
We consider the problem of computing numerical invariants of programs by
abstract interpretation. Our method eschews two traditional sources of
imprecision: (i) the use of widening operators for enforcing convergence within
a finite number of iterations (ii) the use of merge operations (often, convex
hulls) at the merge points of the control flow graph. It instead computes the
least inductive invariant expressible in the domain at a restricted set of
program points, and analyzes the rest of the code en bloc. We emphasize that we
compute this inductive invariant precisely. For that we extend the strategy
improvement algorithm of [Gawlitza and Seidl, 2007]. If we applied their method
directly, we would have to solve an exponentially sized system of abstract
semantic equations, resulting in memory exhaustion. Instead, we keep the system
implicit and discover strategy improvements using SAT modulo real linear
arithmetic (SMT). For evaluating strategies we use linear programming. Our
algorithm has low polynomial space complexity and performs for contrived
examples in the worst case exponentially many strategy improvement steps; this
is unsurprising, since we show that the associated abstract reachability
problem is Pi-p-2-complete
Singular Oscillatory Integrals on R^n
Let Pd,n denote the space of all real polynomials of degree at most d on R^n.
We prove a new estimate for the logarithmic measure of the sublevel set of a
polynomial P in Pd,1. Using this estimate, we prove a sharp estimate for a
singular oscillatory integral on R^n.Comment: final version, 10 pages, small typos corrected, one reference added.
To appear in Math.
Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited
Polynomial interpretations are a useful technique for proving termination of
term rewrite systems. They come in various flavors: polynomial interpretations
with real, rational and integer coefficients. As to their relationship with
respect to termination proving power, Lucas managed to prove in 2006 that there
are rewrite systems that can be shown polynomially terminating by polynomial
interpretations with real (algebraic) coefficients, but cannot be shown
polynomially terminating using polynomials with rational coefficients only. He
also proved the corresponding statement regarding the use of rational
coefficients versus integer coefficients. In this article we extend these
results, thereby giving the full picture of the relationship between the
aforementioned variants of polynomial interpretations. In particular, we show
that polynomial interpretations with real or rational coefficients do not
subsume polynomial interpretations with integer coefficients. Our results hold
also for incremental termination proofs with polynomial interpretations.Comment: 28 pages; special issue of RTA 201
Dimension of zero weight space: An algebro-geometric approach
Let G be a connected, adjoint, simple algebraic group over the complex
numbers with a maximal torus T and a Borel subgroup B containing T. The study
of zero weight spaces in irreducible representations of G has been a topic of
considerable interest; there are many works which study the zero weight space
as a representation space for the Weyl group. In this paper, we study the
variation on the dimension of the zero weight space as the irreducible
representation varies over the set of dominant integral weights for T which are
lattice points in a certain polyhedral cone. The theorem proved here asserts
that the zero weight spaces have dimensions which are piecewise polynomial
functions on the polyhedral cone of dominant integral weights.Comment: 19 page
Calder\'{o}n commutators and the Cauchy integral on Lipschitz curves revisited III. Polydisc extensions
This article is the last in a series of three papers, whose scope is to give
new proofs to the well known theorems of Calder\'{o}n, Coifman, McIntosh and
Meyer. Here we extend the results of the previous two papers to the polydisc
setting. In particular, we solve completely an open question of Coifman from
the early eighties.Comment: 25 page
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