scholarly journals ON IDEMPOTENT ULTRAFILTERS IN HIGHER-ORDER REVERSE MATHEMATICS

2015 ◽  
Vol 80 (1) ◽  
pp. 179-193 ◽  
Author(s):  
ALEXANDER P. KREUZER
Keyword(s):  
Reverse Mathematics ◽  
Higher Order ◽  
Hindman’S Theorem ◽  
Image Position ◽  
Idempotent Ultrafilter ◽  
Order Extension

AbstractWe analyze the strength of the existence of idempotent ultrafilters in higher-order reverse mathematics.Let $\left( {{{\cal U}_{{\rm{idem}}}}} \right)$ be the statement that an idempotent ultrafilter on ℕ exists. We show that over $ACA_0^\omega$, the higher-order extension of ACA0, the statement $\left( {{{\cal U}_{{\rm{idem}}}}} \right)$ implies the iterated Hindman’s theorem (IHT) and we show that $ACA_0^\omega + \left( {{{\cal U}_{{\rm{idem}}}}} \right)$ is ${\rm{\Pi }}_2^1$-conservative over $ACA_0^\omega + IHT$ and thus over $ACA_0^ +$.

scholarly journals TRANSFINITE RECURSION IN HIGHER REVERSE MATHEMATICS

2015 ◽  
Vol 80 (3) ◽  
pp. 940-969 ◽  
Author(s):  
NOAH SCHWEBER
Keyword(s):  
Reverse Mathematics ◽  
Higher Order ◽  
Steel Type ◽  
Base Theory ◽  
Higher Types ◽  
Flexible Framework ◽  
Image Position ◽  
Transfinite Recursion

AbstractIn this paper we investigate the reverse mathematics of higher-order analogues of the theory $$ATR_0$$ within the framework of higher order reverse mathematics developed by Kohlenbach [11]. We define a theory $$RCA_0^3$$, a close higher-type analogue of the classical base theory $$RCA_0$$ which is essentially a conservative subtheory of Kohlenbach’s base theory $$RCA_{\rm{0}}^\omega$$. Working over $$RCA_0^3$$, we study higher-type analogues of statements classically equivalent to $$ATR_0$$, including open and clopen determinacy, and examine the extent to which $$ATR_0$$ remains robust at higher types. Our main result is the separation of open and clopen determinacy for reals, using a variant of Steel’s tagged tree forcing; in the presentation of this result, we develop a new, more flexible framework for Steel-type forcing.

scholarly journals NON-PRINCIPAL ULTRAFILTERS, PROGRAM EXTRACTION AND HIGHER-ORDER REVERSE MATHEMATICS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250002 ◽  
Author(s):  
ALEXANDER P. KREUZER
Keyword(s):  
Extraction Method ◽  
Reverse Mathematics ◽  
Higher Order ◽  
Program Extraction ◽  
Order Arithmetic ◽  
Order Extension

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher-order arithmetic. Let [Formula: see text] be the statement that a non-principal ultrafilter on ℕ exists and let [Formula: see text] be the higher-order extension of ACA0. We show that [Formula: see text] is [Formula: see text]-conservative over [Formula: see text] and thus that [Formula: see text] is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly [Formula: see text] statement ∀ f ∃ g A qf(f, g) in [Formula: see text] a realizing term in Gödel's system T can be extracted. This means that one can extract a term t ∈ T, such that ∀ f A qf(f, t(f)).

Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

Irreducibility of Bernoulli Polynomials of Higher Order

1962 ◽  
Vol 14 ◽  
pp. 565-567 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Positive Integer ◽  
Bernoulli Number ◽  
Bernoulli Polynomials ◽  
Constant Term ◽  
Rational Numbers ◽  
Higher Order ◽  
Image Position

The Bernoulli polynomials of order k, where k is a positive integer, are defined byBm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

scholarly journals The reverse mathematics of Hindman’s Theorem for sums of exactly two elements

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 253-263 ◽  
Author(s):  
Barbara F. Csima ◽  
Damir D. Dzhafarov ◽  
Denis R. Hirschfeldt ◽  
Carl G. Jockusch, Jr. ◽  
Reed Solomon ◽  
...  
Keyword(s):  
Reverse Mathematics ◽  
Hindman’S Theorem

Extension of Lifschitz' realizability to higher order arithmetic, and a solution to a problem of F. Richman

1991 ◽  
Vol 56 (3) ◽  
pp. 964-973 ◽  
Author(s):  
Jaap van Oosten
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Church’S Thesis ◽  
Second Order Arithmetic ◽  
Church's Thesis ◽  
Order Arithmetic ◽  
Image Position

AbstractF. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic HAH:and if not, whether assuming Church's Thesis CT and Markov's Principle MP would help. Blass and Scedrov gave models of HAH in which this principle, which we call RP, is not valid, but their models do not satisfy either CT or MP.In this paper a realizability topos Lif is constructed in which CT and MP hold, but RP is false. (It is shown, however, that RP is derivable in HAH + CT + MP + ECT0, so RP holds in the effective topos.) Lif is a generalization of a realizability notion invented by V. Lifschitz. Furthermore, Lif is a subtopos of the effective topos.

Free discontinuity problems generated by singular perturbation: the n−dimensional case

2000 ◽  
Vol 130 (3) ◽  
pp. 449-469 ◽  
Author(s):  
R. Alicandro ◽  
M. S. Gelli
Keyword(s):  
Singular Perturbation ◽  
Hessian Matrix ◽  
Higher Order ◽  
Dimensional Case ◽  
Free Discontinuity Problems ◽  
Limiting Behaviour ◽  
Image Position

We provide an approximation of some free discontinuity problems by local functionals with a singular perturbation of higher order. More precisely, we study the limiting behaviour of energies of the form where Hu denotes the Hessian matrix of u.

scholarly journals THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

2018 ◽  
Vol 83 (2) ◽  
pp. 817-828 ◽  
Author(s):  
ERIC P. ASTOR
Keyword(s):  
Reverse Mathematics ◽  
Previous Article ◽  
Asymptotic Density ◽  
Turing Degrees ◽  
Image Position ◽  
Immune Set

AbstractIn a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

Hindman's theorem, ultrafilters, and reverse mathematics

2004 ◽  
Vol 69 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Jeffry L. Hirst
Keyword(s):  
Reverse Mathematics ◽  
Boolean Algebras ◽  
Natural Numbers ◽  
Hindman’S Theorem ◽  
Power Set

AbstractAssuming CH. Hindman [2] showed that the existence of certain ultrafilters on the power set of the natural numbers is equivalent to Hindman's Theorem. Adapting this work to a countable setting formalized in RCA0, this article proves the equivalence of the existence of certain ultrafilters on countable Boolean algebras and an iterated form of Hindman's Theorem, which is closely related to Milliken's Theorem. A computable restriction of Hindman's Theorem follows as a corollary.

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