scholarly journals HARRINGTON’S PRINCIPLE IN HIGHER ORDER ARITHMETIC

2015 ◽  
Vol 80 (2) ◽  
pp. 477-489 ◽  
Author(s):  
YONG CHENG ◽  
RALF SCHINDLER
Keyword(s):  
Higher Order ◽  
Order Arithmetic ◽  
Image Position

AbstractLet Z2, Z3, and Z4 denote 2nd, 3rd, and 4th order arithmetic, respectively. We let Harrington’s Principle, HP, denote the statement that there is a real x such that every x-admissible ordinal is a cardinal in L. The known proofs of Harrington’s theorem “$Det\left( {{\rm{\Sigma }}_1^1} \right)$ implies 0♯ exists” are done in two steps: first show that $Det\left( {{\rm{\Sigma }}_1^1} \right)$ implies HP, and then show that HP implies 0♯ exists. The first step is provable in Z2. In this paper we show that Z2 + HP is equiconsistent with ZFC and that Z3 + HP is equiconsistent with ZFC + there exists a remarkable cardinal. As a corollary, Z3 + HP does not imply 0♯ exists, whereas Z4 + HP does. We also study strengthenings of Harrington’s Principle over 2nd and 3rd order arithmetic.

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.

scholarly journals A fixed point for the jump operator on structures

2013 ◽  
Vol 78 (2) ◽  
pp. 425-438 ◽  
Author(s):  
Antonio Montalbán
Keyword(s):  
Fixed Point ◽  
Higher Order ◽  
Turing Degrees ◽  
Jump Operator ◽  
Order Arithmetic ◽  
Image Position

AbstractAssuming that 0# exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure such thatwhere is the set of Turing degrees which compute a copy of More interesting than the result itself is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full “nth-order arithmetic for all n, cannot prove the existence of such a structure.

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.

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.

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.

Kernels with only a finite number of characteristic values

1971 ◽  
Vol 70 (2) ◽  
pp. 257-262
Author(s):  
Dale W. Swann
Keyword(s):  
Finite Number ◽  
Higher Order ◽  
Complex Numbers ◽  
Finite Characteristic ◽  
Characteristic Values ◽  
Image Position ◽  
Complex Valued

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we meanand let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfyingFor such kernels, the iterated kernels,are well-defined (1), as are the higher order tracesCarleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinantthe latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).

scholarly journals On Eddington's higher order equations of the gravitational field

1948 ◽  
Vol 8 (2) ◽  
pp. 89-94 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Gravitational Field ◽  
Empty Space ◽  
Higher Order ◽  
Energy Tensor ◽  
Image Position ◽  
Fundamental Equations

Einstein's fundamental equations of the gravitational field arewhere Tμν are the components of the energy tensor and λ is the cosmical constant. In empty space these equations becomewhich may be reduced tosince G = 4λ, by contraction of (2).

scholarly journals Erratum to “Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes”. Math. Proc. Camb. Phil. Soc. (2012), 152, 365–383.

2013 ◽  
Vol 155 (2) ◽  
pp. 375-377
Author(s):  
LUIS J. ALÍAS ◽  
DEBORA IMPERA ◽  
MARCO RIGOLI
Keyword(s):  
Mean Curvature ◽  
Higher Order ◽  
Spacelike Hypersurfaces ◽  
Higher Order Mean Curvature ◽  
Formula Group ◽  
Image Position ◽  
Correct Expression

The proof of Corollary 4⋅3 in our paper [1] is not correct because there is a mistake in the expression given for ∥X* ∧ Y*∥2 on page 374. In fact, the correct expression for this term is \begin{eqnarray*} \norm{X^*\wedge Y^*}^2 & = & \norm{X^*}^2\norm{Y^*}^2-\pair{X^*,Y^*}^2\\ {} & = & 1+\pair{X,T}^2+\pair{Y,T}^2\geq 1, \end{eqnarray*} and then the inequality (4⋅9) is no longer true. Observe that all the previous reasoning before the wrong expression for ∥X* ∧ Y*∥2 is correct.

On the differentials in the Adams spectral sequence

1964 ◽  
Vol 60 (3) ◽  
pp. 409-420 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Spectral Sequence ◽  
Higher Order ◽  
Adams Spectral Sequence ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Cohomology Operations ◽  
Image Position

In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements ofcorresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.

A remark on Zilber's pseudoexponentiation

2006 ◽  
Vol 71 (3) ◽  
pp. 791-798 ◽  
Author(s):  
David Marker
Keyword(s):  
Baire Category ◽  
Positive Answer ◽  
Complex Conjugation ◽  
Closed Set ◽  
Definable Set ◽  
Open Questions ◽  
Novel Approach ◽  
Definable Sets ◽  
Order Arithmetic ◽  
Image Position

When studying the model theory ofthe first observation is that the integers can be defined asSince ∂exp is subject to all of Gödel's phenomena, this is often also the last observation. After Wilkie proved that ℝexp is model complete, one could ask the same question for ∂exp, but the answer is negative.Proposition 1.1. ∂expis not model completeProof. If ∂exp is model complete, then every definable set is a projection of a closed set. Since ∂ is locally compact, every definable set is Fσ. The same is true for the complement, so every definable set is also Gδ. But, since ℤ is definable, ℚ is definable and a standard corollary of the Baire Category Theorem tells us that ℚ is not Gδ.Still, there are several interesting open questions about ∂exp.• Is ℝ definable in ∂exp?• (quasiminimality) Is every definable set countable or co-countable? (Note that this is true in the structure (∂, ℤ, +, ·) where we add a predicate for ℤ).• (Mycielski) Is there an automorphism of ∂exp other than the identity and complex conjugation?1A positive answer to the first question would tell us that ∂exp is essentially second order arithmetic, while a positive answer to the second would say that integers are really the only obstruction to a reasonable theory of definable sets.A fascinating, novel approach to ∂exp is provided by Zilber's [6] pseudoexponentiation. Let L be the language {+, · E, 0, 1}.

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