scholarly journals UNCOUNTABLE TREES AND COHEN -REALS

2019 ◽  
Vol 84 (3) ◽  
pp. 877-894 ◽  
Author(s):  
GIORGIO LAGUZZI
Keyword(s):  
Strong Relationship ◽  
Standard Case ◽  
Regularity Properties ◽  
Uncountable Cardinals ◽  
Image Position

AbstractWe investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show ${\bf{\Sigma }}_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slight contrast with the standard case.

Regularity results for equilibria in a variational model for fracture

1997 ◽  
Vol 127 (5) ◽  
pp. 889-902 ◽  
Author(s):  
Emilio Acerbi ◽  
Irene Fonseca ◽  
Nicola Fusco
Keyword(s):  
Broad Class ◽  
Variational Model ◽  
Classical Solutions ◽  
Scalar Case ◽  
Regularity Properties ◽  
Bulk Energy ◽  
Relaxed Energy ◽  
Image Position ◽  
Regularity Results ◽  
Vector Valued

SynopsisIn recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms, of the formwhere Ω is an open, bounded subset of ℝN, q ≧1, g ∈ L∞ (Ω ℝN), λ, β > 0, the bulk energy density F is quasiconvex, K⊂ℝN is closed, and the admissible deformation u:Ω→ ℝN is C1 in Ω\K One of the main issues has to do with regularity properties of the ‘crack site’ K for a minimising pair (K, u). In the scalar case, i.e. when uΩ→ ℝ, similar models were adopted to image segmentation problems, and the regularity of the ‘edge’ set K has been successfully resolved for a quite broad class of convex functions F with growth p > 1 at infinity. In turn, this regularity entails the existence of classical solutions. The methods thus used cannot be carried out to the vectorial case, except for a very restrictive class of integrands. In this paper we deal with a vector-valued case on the plane, obtaining regularity for minimisers of corresponding to polyconvex bulk energy densities of the formwhere the convex function h grows linearly at infinity.

scholarly journals UNIVERSAL CLASSES NEAR ${\aleph _1}$

2018 ◽  
Vol 83 (04) ◽  
pp. 1633-1643 ◽  
Author(s):  
MARCOS MAZARI-ARMIDA ◽  
SEBASTIEN VASEY
Keyword(s):  
Sufficient Conditions ◽  
List Type ◽  
Type Number ◽  
Abstract Elementary Classes ◽  
Uncountable Cardinal ◽  
Elementary Classes ◽  
Universal Classes ◽  
Uncountable Cardinals ◽  
Image Position

AbstractShelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:Theorem. Assume ${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.(1)If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) < 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.(2)If ψ is categorical in ${\aleph _1}$, then ψ is categorical in all uncountable cardinals.The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

scholarly journals On a conjecture of Dobrinen and Simpson concerning almost everywhere domination

2006 ◽  
Vol 71 (1) ◽  
pp. 119-136 ◽  
Author(s):  
Stephen Binns ◽  
Bjørn Kjos-Hanssen ◽  
Manuel Lerman ◽  
Reed Solomon
Keyword(s):  
Set Theory ◽  
Computable Function ◽  
Turing Degrees ◽  
Fair Coin ◽  
Almost Everywhere ◽  
Regularity Properties ◽  
Turing Reducibility ◽  
Image Position ◽  
Almost All ◽  
Generic Extensions

Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the Lebesgue measure on 2ω in reverse mathematics. In this article, we examine one of their conjectures concerning these notions.Throughout this article, ≤T denotes Turing reducibility and μ denotes the Lebesgue (or “fair coin”) probability measure on 2ω given byA property holds almost everywhere or for almost all X ∈ 2ω if it holds on a set of measure 1. For f, g ∈ ωω, f dominatesg if ∃m∀n < m(f(n) > g(n)).(Dobrinen, Simpson). A set A ∈ 2ωis almost everywhere (a.e.) dominating if for almost all X ∈ 2ω and all functions g ≤TX, there is a function f ≤TA such that f dominates g. A is uniformly almost everywhere (u.a.e.) dominating if there is a function f ≤TA such that for almost all X ∈ 2ω and all functions g ≤TX, f dominates g.There are several trivial but useful observations to make about these definitions. First, although these properties are stated for sets, they are also properties of Turing degrees. That is, a set is (u.)a.e. dominating if and only if every other set of the same degree is (u.)a.e. dominating. Second, both properties are closed upwards in the Turing degrees. Third, u.a.e. domination implies a.e. domination. Finally, if A is u.a.e. dominating, then there is a function f ≤TA which dominates every computable function.

scholarly journals Subelliptic operators on Lie groups: regularity

1994 ◽  
Vol 57 (2) ◽  
pp. 179-229 ◽  
Author(s):  
A. F. M. Ter Elst ◽  
Derek W. Robinson
Keyword(s):  
Lie Algebra ◽  
Group Representation ◽  
Positive Definite Matrix ◽  
Continuous Measure ◽  
Bounded Operators ◽  
Optimal Regularity ◽  
Elliptic Regularity ◽  
Subelliptic Operators ◽  
Regularity Properties ◽  
Image Position

AbstractLet (ℋ,G, U) be a continuous representation of the Lie groupGby bounded operatorsg↦U(g)on the Banach space ℋ and let (ℋ,g, dU) denote the representation of the Lie algebragobtained by differentiation. Ifa1,…, ad′is a Lie algebra basis ofgandAi= dU(ai)then we examine elliptic regularity properties of the subelliptic operatorswhere (cij) is a real-valued strictly positive-definite matrix andc0, c1,…, cd′∈ C. We first introduce a family of Lipschitz subspaces ℋγ, γ > 0, of ℋ which interpolate between theCn-subspaces of the representation and for which the parameter γ is a continuous measure of differentiability. Secondly, we give a variety of characterizations of the spaces in terms of the semigroup generated by the closureofHand the group representation. Thirdly, for sufficiently large values of Rec0the fractional powers of the closure ofHare defined, and we prove that D()γ⊆γ′, for γ′ < 2γ/rwhereris the rank of the basis. Further we establish that 2γ/ris the optimal regularity value and it is attained for unitary representations or for the representations obtained by restrictingUto ℋγ. Many other regularity properties are obtained.

scholarly journals Modulus of nearly uniform smoothness and Lindenstrauss formulae

1995 ◽  
Vol 37 (2) ◽  
pp. 143-153 ◽  
Author(s):  
Tomás Domínguez Benavides
Keyword(s):  
Banach Space ◽  
Strong Relationship ◽  
Uniform Convexity ◽  
Measures Of Noncompactness ◽  
Conjugate Space ◽  
Uniform Smoothness ◽  
Image Position ◽  
Smooth Space

AbstractThe Lindenstrauss formulawhich states a strong relationship between the (Clarkson) modulus of uniform convexity δx of a Banach space X and the modulus of uniform smoothness px* of the conjugate space X*, is well known. Following the idea of the definitions of nearly uniform smooth space by S. Prus and modulus of uniform smoothness we define a modulus of nearly uniform smoothness and prove some Lindenstrauss type formulae concerning this modulus and the modulus of nearly uniform convexity for some measures of noncompactness.

Partitioning pairs of countable sets of ordinals

1990 ◽  
Vol 55 (3) ◽  
pp. 1019-1021 ◽  
Author(s):  
Dan Velleman
Keyword(s):  
Partition Theorem ◽  
Partition Relations ◽  
Uncountable Cardinals ◽  
Image Position

In [3], Todorčević showed that ω1 ⇸ [ω1]ω12. In this paper we use similar methods to prove an analogous partition theorem for Pω1(λ), for certain uncountable cardinals λ.Recall that ω1 → [ω1]ω12, means that for every function f: [ω1]2 → ω1 there is a set A ∈ [ω1]ω1 such that f“[A]2 ≠ ω1, and of course ω1 ⇸ [ω1]ω12, is the negation of this statement. For partition relations on Pω1(→) it is customary to partition only those pairs of sets in which the first set is a subset of the second. Thus for A ⊆ Pω1(λ) we defineWe will write Pω1(λ) → [unbdd]λ2 to mean that for every function f: [Pω1(λ)]⊂2 → λ there is an unbounded set A ⊆ Pω1(λ) such that f“[A]⊂2 ≠ λ, and again Pω1(λ) ⇸ [unbdd]λ2 is the negation of this statement.

Regularity for subquadratic parabolic systems: higher integrability and dimension estimates

2010 ◽  
Vol 140 (6) ◽  
pp. 1269-1308 ◽  
Author(s):  
Christoph Scheven
Keyword(s):  
Structure Function ◽  
Polynomial Growth ◽  
Time Derivative ◽  
Parabolic Systems ◽  
Spatial Gradient ◽  
Higher Integrability ◽  
Regularity Properties ◽  
Growth Properties ◽  
Spatial Derivatives ◽  
Image Position

We consider weak solutions of parabolic systems of the typewhere the structure function b is differentiable with respect to x and satisfies standard ellipticity and growth properties with polynomial growth rate p ∊ (2n/(n + 2), 2). We investigate regularity properties of the solution, including the existence of second-order spatial derivatives, the existence of the time derivative and the higher integrability of the spatial gradient. As an application, we derive dimension estimates for the singular set of solutions of homogeneous parabolic systems. More precisely, we establish the boundprovided the structure function depends Höolder continuously on the space variable with Höolder exponent β ∊(0, 1].

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

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