Finite Unions Of Sets Research Articles

ON THE CANONICAL RAMSEY THEOREM OF ERDŐS AND RADO AND RAMSEY ULTRAFILTERS

We give a characterizations of Ramsey ultrafilters on ω in terms of functions \(f:{{\omega }^{n}} \to \omega \) and their ultrafilter extensions. To do this, we prove that for any partition \(\mathcal{P}\) of \({{[\omega ]}^{n}}\) there is a finite partition \(\mathcal{Q}\) of \({{[\omega ]}^{{2n}}}\) such that any set \(X \subseteq \omega \) that is homogeneous for \(\mathcal{Q}\) is a finite union of sets that are canonical for \(\mathcal{P}\).

The Fourier Dimension is Not Finitely Stable

The Fourier dimension is not, in general, stable under finite unions of sets. Moreover, the stability of the Fourier dimension on particular pairs of sets is independent from the stability of the compact Fourier dimension.

Outer Minkowski content for some classes of closed sets

We find conditions ensuring the existence of the outer Minkowski content for d-dimensional closed sets in \({\mathbb{R}^d}\) , in connection with regularity properties of their boundaries. Moreover, we provide a class of sets (including all sufficiently regular sets) stable under finite unions for which the outer Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positive reach belong to this class.

On Set Covariance and Three-Point Test Sets

The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of \(\mathbb{R}^d \)is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic is computed.

Curvatures and Currents for Unions of Sets with Positive Reach, II

A class of subsets of ℝ d which can berepresented as locally finite unions of sets with positive reach isconsidered. It plays a role in PDE's on manifolds with singularities.For such a set, the unit normal cycle (determining the d − 1curvature measures) is introduced as a (d − 1)-currentsupported by the unit normal bundle and its properties are established.It is shown that, under mild additional assumptions, the unit normalcycle (and, hence, also the curvature measures) of such a set can beapproximated by that of a close parallel body or, alternatively, by themirror image of that of the closure of the complement of the parallelbody (which has positive reach). Finally, the mixed curvature measuresof two sets of this class are introduced and a translative integralgeometric formula for curvature measures is proved.

Block Coordinate Relaxation Methods for Nonparametric Wavelet Denoising

An important class of nonparametric signal processing methods entails forming a set of predictors from an overcomplete set of basis functions associated with a fast transform (e.g., wavelet packets). In these methods, the number of basis functions can far exceed the number of sample values in the signal, leading to an ill-posed prediction problem. The “basis pursuit” denoising method of Chen, Donoho, and Saunders regularizes the prediction problem by adding an l 1 penalty term on the coefficients for the basis functions. Use of an l 1 penalty instead of l 2 has significant benefits, including higher resolution of signals close in time/frequency and a more parsimonious representation. The l 1 penalty, however, poses a challenging optimization problem that was solved by Chen, Donoho and Saunders using a novel application of interior-point algorithms (IP). This article investigates an alternative optimization approach based on block coordinate relaxation (BCR) for sets of basis functions that are the finite union of sets of orthonormal basis functions (e.g., wavelet packets). We show that the BCR algorithm is globally convergent, and empirically, the BCR algorithm is faster than the IP algorithm for a variety of signal denoising problems.

Solution set properties for static errors-in-variables problems

This paper examines and refutes a conjecture to the effect that the solution set for a general (real) static errors-in-variables problem is a finite union of sets that are described by a finite number of linear inequalities. The conjecture is disproved by detailed examination of particular errors-in-variables problems with four variables. The solution set in this case is described by five surfaces, all intersecting in straight lines, but in general one of these surfaces is not flat.

ASH'S TYPE II THEOREM, PROFINITE TOPOLOGY AND MALCEV PRODUCTS: PART I

This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After reviewing the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture — also verified by Ash — it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH1H2…Hn, where g ∈ G and each Hi is a finitely generated subgroup of G. This significantly extends classical results of M. Hall. Finally, we return to the roots of this problem and give connections with the complexity theory of finite semigroups. We show that the largest local complexity function in the sense of Rhodes and Tilson is computable.

Actions, finite state sets and applications to trees

Let M be a monoid acting on a set X; for any x∊X and A⊆X we put x −1 A = {m∊M/xm∊A}. Call A⊆X finite state if card {x −1 A/x∊X} <∞. The finite state subsets of T Σvia the action T Σ × P Σ→T Σare the recognizable forests (P Σis the monoid of all Σ-trees with just one leaf labeled by a variable x). Next we prove that the recognizability of forests is equivalent to the finiteness of a certain “syntactic” monoid A Mezei's-like theorem for trees is established: the finite state subsets of T Σ × Tг are exactly the finite unions of sets of sets of the form B × C B∊Rec(T Σ) and C∊Rec(T г) Another characterization of such relations is given using bimorphisms.

On the structure of semialgebraic sets over p-adic fields

In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]:Let V ⊂ Rm be a real algebraic set, and let U ⊂ Rm be an open set defined by finitely many polynomial inequalities:Lemma 3.1. If U ∩ V contains points arbitrarily close to the origin (that is if 0 ∈ Closure (U ∩ V)) then there exists a real analytic curvewith p(0) = 0 and with p(t) ∈ U ∩ V for t &gt; 0.Of course, this result will also apply to semialgebraic sets (finite unions of sets U ∩ V), and by Tarski's theorem such sets are exactly the sets obtained from real varieties by means of the finite Boolean operations and the projection maps Rn+1 → Rn. If, in this tiny extension of Milnor's result, we replace ‘R’ everywhere by ‘Qp’, we obtain a p-adic curve selection lemma, a version of which we will prove in this essay. Semialgebraic sets, in the p-adic context, may be defined just as they are over the reals: namely, as those sets obtained from p-adic varieties by means of the finite Boolean operations and the projection maps . Analytic maps are maps whose coordinate functions are given locally by convergent power series.

On the structure of semialgebraic sets over p-adic fields

In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]:Let V ⊂ Rm be a real algebraic set, and let U ⊂ Rm be an open set defined by finitely many polynomial inequalities:Lemma 3.1. If U ∩ V contains points arbitrarily close to the origin (that is if 0 ∈ Closure (U ∩ V)) then there exists a real analytic curvewith p(0) = 0 and with p(t) ∈ U ∩ V for t &gt; 0.Of course, this result will also apply to semialgebraic sets (finite unions of sets U ∩ V), and by Tarski's theorem such sets are exactly the sets obtained from real varieties by means of the finite Boolean operations and the projection maps Rn+1 → Rn. If, in this tiny extension of Milnor's result, we replace ‘R’ everywhere by ‘Qp’, we obtain a p-adic curve selection lemma, a version of which we will prove in this essay. Semialgebraic sets, in the p-adic context, may be defined just as they are over the reals: namely, as those sets obtained from p-adic varieties by means of the finite Boolean operations and the projection maps . Analytic maps are maps whose coordinate functions are given locally by convergent power series.

Inclusions for Classes of Lacunary Sets

A sequence, a1 &lt; a2 &lt; a3 &lt; …, of positive integers is called lacunary if the difference sequence dn = an+l — an tends to infinity as n → ∞.In several recent papers we have made use of these sequences in analysis and combinatorics. In [6] we show that the class of all sets which are either finite or the range of a lacunary sequence is “full” in the sense that if (tk) is a real sequence and for each then (tk) is an l1 sequence, that is,In [3] the class of all finite unions of sets of is shown to consist of exactly those sets of integers, A, whose characteristic sequence, χA, is in the well known summability space bs + c0. More recently, in [1], we study lacunary sequences in connection with the conjecture of P.

Piecewise invertible dynamical systems

The aim of the paper is the investigation of piecewise monotonic maps T of an interval X. The main tool is an isomorphism of (X, T) with a topological Markov chain with countable state space which is described by a 0–1-transition matrix M. The behavior of the orbits of points in X under T is very similar to the behavior of the paths of the Markov chain. Every irreducible submatrix of M gives rise to a T-invariant subset L of X such that L is the set ω(x) of all limit points of the orbit of an x∈X. The topological entropy of L is the logarithm of the spectral radius of the irreducible submatrix, which is a l 1-operator. Besides these sets L there are two T-invariant sets Y and P, such that for all x∈X the set ω(x) is either contained in one of the sets L or in Y or in P. The set P is a union of periodic orbits and Y is contained in a finite union of sets ω(y) with y∈X and has topological entropy zero. This isomorphism of (X, T) with a topological Markov chain is also an important tool for the investigation of T-invariant measures on X. Results in this direction, which are published elsewhere, are described at the end of the paper. Furthermore, a part of the proofs in the paper is purely topological without using the order relation of the interval X, so that some results hold for more general dynamical systems (X, T).

On the structure of Sidon sets

LetG be any connected compact group with dual objectĜ. We give in this paper a new proof that the union of any two Sidon sets inĜ is again a Sidon set. We also show that any Sidon subset ofĜ is the union of a set whose elements have bounded degree with a finite union of sets which satisfy a quasi-independence condition.

On Certain Polynomial-Time Truth-Table Reducibilities of Complete Sets to Sparse Sets

Let $\Sigma $ be a finite alphabet. A set $S \subset \Sigma ^ * $ is called sparse if the number of members of S having length at most n is bounded above by a polynomial in n. Let $ \leqq _m^P $ denote polynomial-time many-one reducibility, and let $ \leqq _{bptt}^P $ denote the more general polynomial-time bounded positive truth-table reducibility. We prove: (1) $ \leqq _{bptt}^P $ reducibility of a coNP-hard set to a sparse set implies ${\text{NP}} = {\text{P}}$, and (2) $ \leqq _{bptt}^P $ reducibility of an NP-hard set to a sparse set which is itself in NP implies ${\text{NP}} = {\text{P}}$. (1) generalizes Fortune’s result [F]. He proved it for the case of $ \leqq _m^P $ reducibility. (2), for the case of $ \leqq _m^P $ reducibility, was proved by Mahaney [M], even without assuming that the sparse set itself is in NP. Our results imply that if a coNP-hard set is a finite union of sets which are $ \leqq _m^P $ reducible to sparse sets, then ${\text{NP}} = {\text{P}}$. We then investigate a certain nonpositive polynomial-time truth-table reducibility of NP-hard sets to sparse sets, and obtain new results regarding the structure of sets hard for NP or for ETIME. Finally, we investigate the possibility of existence of sets in NP–coNP which are $ \leqq _m^P $ reducible to sparse sets. Some of our techniques involve generalizations of and variations on the techniques of Berman [B], Fortune [F] and Mahaney [M].

Value distribution for holomorphic maps inC n

The study of value distribution for holomorphic or meromorphic functions of one complex variable culminated in the very beautiful and deep Nevanlinna theory, which improved upon practically all previously known results. In recent years, value distribution for holomorphic and meromorphic functions in C" and generalizations have been extensively developed, based essentially on a similar theory for one variable (cf. Griffiths [3]), and the results follow at least qualitatively and in general quantitatively the classical one variable theory. As soon as one begins to consider holomorphic maps, however, certain anomalies begin to appear. Perhaps the most perplexing of these is the famous Fatou-Bieberbach map, which is a holomorphic homeomorphism of C 2 into C 2 with Jacobian identically one such that the image omits an open set (cf. Bochner and Martin [2]). In order to develop a coherent theory of value distribution for holomorphic maps, one would like to find sets such that the image of C" under a holomorphic map always intersects these sets. The Fatou-Bieberbach map shows that the image can omit any bounded set, but there are unbounded sets which cannot be omitted. For instance, the theory of holomorphic curves shows that if F : C " ~ C n is a non-degenerate holomorphic map, then the image cannot omit (n + 2) hyperplanes in general position (cf. Vitter [7] for proofs and references and Griffiths [3] for generalizations). These sets are of real dimension ( 2 n 2) in C". Our purpose here is to show that there exist other families of unbounded sets which are in some sense much smaller such that the image of C" under any nondegenerate holomorphic map must intersect these sets. These sets will be finite unions of sets of g~ dimension n in Ir" : {w~C":~eg~(w)= . . .=~eg , , (w)=O, holomorphic in C"}. Since the detailed description of these sets is rather complicated, we will not attempt it here in the introduction. Similar results are obtained for bounded domains D in C ~ which have cg2 boundaries provided that IIF[I grows sufficiently fast near the boundary of D. This leads us to introduce the notion of "essential sets" for families of nondegenerate analytic maps, which we define in the following way: if ~ is a family of non-degenerate holomorphic maps mapping a domain D C C" into C", then a set ECC" will be said to be essential for ~" if for every F ~ and every compact