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.

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 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.

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 A Study of Nonlinear Boundary Value Problem

2021 ◽  
Author(s):  
Noureddine Bouteraa ◽  
Habib Djourdem
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Higher Order ◽  
Nonlinear Boundary Value Problem ◽  
Point Boundary ◽  
Error Estimations ◽  
Fractional Differential

In this chapter, firstly we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-order fractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions and the error estimations are also given. Secondly, we cover the multi-valued case of our problem. We investigate it for nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler. Two illustrative examples are presented at the end to illustrate the validity of our results.

scholarly journals Equivalence of two fixed-point semantics for definitional higher-order logic programs

2017 ◽  
Vol 668 ◽  
pp. 27-42 ◽  
Author(s):  
Angelos Charalambidis ◽  
Panos Rondogiannis ◽  
Ioanna Symeonidou
Keyword(s):  
Fixed Point ◽  
Higher Order ◽  
Order Logic ◽  
Logic Programs ◽  
Higher Order Logic ◽  
Fixed Point Semantics

Eventually infinite time Turing machine degrees: infinite time decidable reals

2000 ◽  
Vol 65 (3) ◽  
pp. 1193-1203 ◽  
Author(s):  
P.D. Welch
Keyword(s):  
Turing Machine ◽  
Initial Segment ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Turing Machines ◽  
Jump Operator ◽  
Infinite Time

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

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 An EKF-Based Fixed-Point Iterative Filter for Nonlinear Systems

Sensors ◽  
2019 ◽  
Vol 19 (8) ◽  
pp. 1893
Author(s):  
Feng ◽  
Feng ◽  
Wen
Keyword(s):  
Fixed Point ◽  
Kalman Filter ◽  
Nonlinear Systems ◽  
Iterative Method ◽  
Taylor Series ◽  
Nonlinear Filtering ◽  
Higher Order ◽  
Point Function ◽  
State Estimate ◽  
Linearized System

In this paper, a fixed-point iterative filter developed from the classical extended Kalman filter (EKF) was proposed for general nonlinear systems. As a nonlinear filter developed from EKF, the state estimate was obtained by applying the Kalman filter to the linearized system by discarding the higher-order Taylor series items of the original nonlinear system. In order to reduce the influence of the discarded higher-order Taylor series items and improve the filtering accuracy of the obtained state estimate of the steady-state EKF, a fixed-point function was solved though a nested iterative method, which resulted in a fixed-point iterative filter. The convergence of the fixed-point function is also discussed, which provided the existing conditions of the fixed-point iterative filter. Then, Steffensen’s iterative method is presented to accelerate the solution of the fixed-point function. The final simulation is provided to illustrate the feasibility and the effectiveness of the proposed nonlinear filtering method.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

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