If a and b are positive integers with a ≤ b and a ≡ a mod b, then the set Ma,b = {x ∈ N : x ≡ a mod b or x = 1} is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid M with units M× and any x ∈ M \M× we say that t ∈ N is a factorization length of x if and only if there exist irreducible elements y1, . . . , yt of M and x = y1 · · · yt. Let L(x) = {t1, . ...

The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskĭı, Tsĕıtin, Kreisel, and Lacombe assert the existence of non-empty co-r.e. closed sets devoid of computable points: sets which are even ‘large’ in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary : every non-empty co-r.e. closed real set without ...

Let μ be singular of uncountable cofinality. If μ > 2cf(μ), we prove that in P = ([μ],⊇) as a forcing notion we have a natural complete embedding of Levy(א0, μ+) (so P collapses μ+ to א0) and even Levy(א0,UJbd κ (μ)). The “natural” means that the forcing ({p ∈ [μ] : p closed},⊇) is naturally embedded and is equivalent to the Levy algebra. Moreover we prove more than conjectured: if P fails the ...

A beautiful and influential subject in the study of the question of smoothness of solutions for the Navier – Stokes equations in three dimensions is the theory of partial regularity. A major paper on this topic is Caffarelli, Kohn & Nirenberg [5](1982) which gives an upper bound on the size of the singular set S(u) of a suitable weak solution u. In the present paper we describe a complementary ...

In this paper we study the behavior of the singular set {u = |∇u| = 0}, for solutions u to the free boundary problem ∆u = fχ{u≥0} − gχ{u<0}, with f > 0, f(x) + g(x) < 0, and f, g ∈ Cα. Such problems arises in an eigenvalue optimization for composite membranes. Here we show that if for a singular point z ∈ {u = ∇u = 0}, there are r0 > 0, and c0 > 0 such that the density assumption |{u < 0} ∩Br(z...

A singular hyperbolic set is a partially hyperbolic set with singularities (all hyper-bolic) and volume expanding central direction [MPP1]. We study connected, singular-hyperbolic, attracting sets with dense closed orbits and only one singularity. These sets are shown to be transitive for most C r flows in the Baire's second category sence. In general these sets are shown to be either transitiv...

Easton proved that the behavior of the exponential function 2 at regular cardinals κ is independent of the axioms of set theory except for some simple classical laws. The Singular Cardinals Hypothesis SCH implies that the Generalized Continuum Hypothesis GCH 2 = κ holds at a singular cardinal κ if GCH holds below κ. Gitik and Mitchell have determined the consistency strength of the negation of ...

For an infinite cardinal μ, MAD(μ) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over μ. Erdős and Hechler proved in [7] the consistency of μ ∈ MAD(μ) for a singular cardinal μ and asked if it was ever possible for a singular μ that μ / ∈ MAD(μ), and also whether 2 μ < μ =⇒ μ ∈ MAD(μ) for every singular cardinal μ. We introduce a new method for controlling ...

Abstract: In this note, we introduce the singular value decomposition Geršgorin set, Γ (A), of an N ×N complex matrix A, where N ≤ ∞. For N finite, the set Γ (A) is similar to the standard Geršgorin set, Γ(A), in that it is a union of N closed disks in the complex plane and it contains the spectrum, σ(A), of A. However, Γ (A) is constructed using column sums of singular value decomposition matr...

We study the parabolic obstacle problem ∆u− ut = fχ{u>0}, u ≥ 0, f ∈ L with f(0) = 1 and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition...