scholarly journals Forcing with Δ perfect trees and minimal Δ-degrees

1981 ◽  
Vol 46 (4) ◽  
pp. 803-816 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Upper Bounds ◽  
Natural Numbers ◽  
Cantor Space ◽  
Standard Notation ◽  
General Method

This paper is a sequel to [3] and it contains, among other things, proofs of the results announced in the last section of that paper.In §1, we use the general method of [3] together with reflection arguments to study the properties of forcing with Δ perfect trees, for certain Spector pointclasses Γ, obtaining as a main result the existence of a continuum of minimal Δ-degrees for such Γ's, under determinacy hypotheses. In particular, using PD, we prove the existence of continuum many minimal Δ½n+1-degrees, for all n.Following an idea of Leo Harrington, we extend these results in §2 to show the existence of minimal strict upper bounds for sequences of Δ-degrees which are not too far apart. As a corollary, it is computed that the length of the natural hierarchy of Δ½n+1-degrees is equal to ω when n ≥ 1. (By results of Sacks and Richter the length of the natural hierarchy of -degrees is known to be equal to the first recursively inaccessible ordinal.)We will follow in this paper standard notation and terminology, as in Moschovakis' book [7]. Letters e, i, j, k, l, m, n vary over the set of natural numbers ω, a, b, c over the Cantor space 2ω and α, β, γ, δ, … over the set of reals ωω. Finally ξ, η, κ, λ always denote ordinals.

scholarly journals Hat Guessing Numbers of Degenerate Graphs

10.37236/9449 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Xiaoyu He ◽  
Ray Li
Keyword(s):  
Upper Bounds ◽  
General Method

Recently, Farnik asked whether the hat guessing number $\mathrm{HG}(G)$ of a graph $G$ could be bounded as a function of its degeneracy $d$, and Bosek, Dudek, Farnik, Grytczuk and Mazur showed that $\mathrm{HG}(G)\ge 2^d$ is possible. We show that for all $d\ge 1$ there exists a $d$-degenerate graph $G$ for which $\mathrm{HG}(G) \ge 2^{2^{d-1}}$. We also give a new general method for obtaining upper bounds on $\mathrm{HG}(G)$. The question of whether $\mathrm{HG}(G)$ is bounded as a function of $d$ remains open.

scholarly journals Computability and the algebra of fields: Some affine constructions

1980 ◽  
Vol 45 (1) ◽  
pp. 103-120 ◽  
Author(s):  
J. V. Tucker
Keyword(s):  
Algebraic Structure ◽  
Algebraic System ◽  
Finiteness Condition ◽  
Coordinate Systems ◽  
Natural Numbers ◽  
Algebraic Foundation ◽  
Image Position ◽  
Technical Idea ◽  
Natural Way ◽  
General Method

A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure A = (A; σ1,…, σk) is computable if it possesses a recursive coordinate system in the following precise sense: associated to A there is a pair (α, Ω) consisting of a recursive set of natural numbers Ω and a surjection α: Ω → A so that (i) the relation defined on Ω by n ≡α m iff α(n) = α(m) in A is recursive, and (ii) each of the operations of A may be effectively followed in Ω, that is, for each (say) r-ary operation σ on A there is an r argument recursive function on Ω which commutes the diagramwherein αr is r-fold α × … × α.This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra—an isomorphism invariant possessed of all finite algebraic systems—and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.

scholarly journals Some inequalities arising from analytic summability of functions

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3223-3230
Author(s):  
Sh. Saadat ◽  
M.H. Hooshmand
Keyword(s):  
Functional Equation ◽  
Holomorphic Function ◽  
Upper Bound ◽  
Bernoulli Polynomials ◽  
Upper Bounds ◽  
Bernoulli Numbers ◽  
Trigonometric Functions ◽  
Natural Numbers ◽  
Sums Of Powers ◽  
The Difference

Analytic summability of functions was introduced by the second author in 2016. He utilized Bernoulli numbers and polynomials for a holomorphic function to construct analytic summability. The analytic summand function f? (if exists) satisfies the difference functional equation f?(z) = f (z) + f?(z-1). Moreover, some upper bounds for f? and several inequalities between f and f? were presented by him. In this paper, by using Alzer?s improved upper bound for Bernoulli numbers, we improve those upper bounds and obtain some inequalities and new upper bounds. As some applications of the topic, we obtain several upper bounds for Bernoulli polynomials, sums of powers of natural numbers, (e.g., 1p+2p+3p+...+rp ? 2p! ?p+1 (e?r-1)) and several inequalities for exponential, hyperbolic and trigonometric functions.

Upper bounds for analytic summand functions and related inequalities

2021 ◽  
Vol 71 (5) ◽  
pp. 1103-1112
Author(s):  
Soodeh Mehboodi ◽  
M. H. Hooshmand
Keyword(s):  
Special Functions ◽  
Bernoulli Polynomials ◽  
Upper Bounds ◽  
Bernoulli Numbers ◽  
Natural Numbers ◽  
Bernoulli Polynomials And Numbers

Abstract The topic of analytic summability of functions was introduced and studied in 2016 by Hooshmand. He presented some inequalities and upper bounds for analytic summand functions by applying Bernoulli polynomials and numbers. In this work we apply upper bounds, represented by Hua-feng, for Bernoulli numbers to improve the inequalities and related results. Then, we observe that the inequalities are sharp and leave a conjecture about them. Also, as some applications, we use them for some special functions and obtain many particular inequalities. Moreover, we arrived at the inequality 1 p + 2 p + 3 p + ⋯ + r p ≤ 1 2 r p + 1 3 r p + 1 ( p + 1 ) + 2 3 p ! π p + 1 sinh ⁡ ( π r ) $1^p + 2^p + 3^p + \dots + r^p \leq \frac{1}{2}r^p + \frac{1}{3}\frac{r^{p+1}}{(p+1)} + \frac{2}{3}\frac{p!}{\pi^{p+1}}\sinh(\pi r)$ , for r sums of power of natural numbers, if p ∈ ℕ e and analogously for the odd case.

A General Method to Determine Asymptotic Capacity Upper Bounds for Wireless Networks

2019 ◽  
Vol 6 (1) ◽  
pp. 2-15
Author(s):  
Canming Jiang ◽  
Yi Shi ◽  
Y. Thomas Hou ◽  
Wenjing Lou ◽  
Sastry Kompella ◽  
...  
Keyword(s):  
Wireless Networks ◽  
Upper Bounds ◽  
General Method

scholarly journals Distribuion of Leading Digits of Numbers II

2019 ◽  
Vol 14 (1) ◽  
pp. 19-42
Author(s):  
Yukio Ohkubo ◽  
Oto Strauch
Keyword(s):  
Distribution Function ◽  
Real Number ◽  
Prime Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Natural Numbers ◽  
Increasing Functions

AbstractIn this paper, we study the sequence (f (pn))n≥1,where pn is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=logb xr and f (x)=logb(x log x)r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (Ni)i≥1 of the natural numbers. Especially for f (x)= logb xr, we obtain the effective results for an upper bound of D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).

scholarly journals General Upper Bounds on the Runtime of Parallel Evolutionary Algorithms*

2014 ◽  
Vol 22 (3) ◽  
pp. 405-437 ◽  
Author(s):  
Jörg Lässig ◽  
Dirk Sudholt
Keyword(s):  
Evolutionary Algorithms ◽  
Upper Bounds ◽  
Constant Factor ◽  
Structured Populations ◽  
Fitness Level ◽  
Performance Guarantees ◽  
Parallel Evolutionary Algorithms ◽  
Spatially Structured Populations ◽  
Boolean Optimization ◽  
General Method

We present a general method for analyzing the runtime of parallel evolutionary algorithms with spatially structured populations. Based on the fitness-level method, it yields upper bounds on the expected parallel runtime. This allows for a rigorous estimate of the speedup gained by parallelization. Tailored results are given for common migration topologies: ring graphs, torus graphs, hypercubes, and the complete graph. Example applications for pseudo-Boolean optimization show that our method is easy to apply and that it gives powerful results. In our examples the performance guarantees improve with the density of the topology. Surprisingly, even sparse topologies such as ring graphs lead to a significant speedup for many functions while not increasing the total number of function evaluations by more than a constant factor. We also identify which number of processors lead to the best guaranteed speedups, thus giving hints on how to parameterize parallel evolutionary algorithms.

scholarly journals Constructive decidability of classical continuity

2014 ◽  
Vol 25 (7) ◽  
pp. 1578-1589
Author(s):  
MARTÍN ESCARDÓ
Keyword(s):  
Uniform Continuity ◽  
Constructive Mathematics ◽  
Natural Numbers ◽  
Cantor Space ◽  
Universal Quantifiers ◽  
The One ◽  
One Point Compactification ◽  
Excluded Middle

We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof does not require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weakening of the notion of (dis)continuity, where the existential quantifiers are replaced by negated universal quantifiers. We also show that the classical continuity of all functions is equivalent to the negation of the weak limited principle of omniscience. We use this to relate uniform continuity and searchability of the Cantor space.

Minimal complementation below uniform upper bounds for the arithmetical degrees

1996 ◽  
Vol 61 (4) ◽  
pp. 1158-1192
Author(s):  
Masahiro Kumabe
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Arithmetical Function ◽  
Natural Numbers

This paper was inspired by Lerman [15] in which he proved various properties of upper bounds for the arithmetical degrees. We discuss the complementation property of upper bounds for the arithmetical degrees. In Lerman [15], it is proved that uniform upper bounds for the arithmetical degrees are jumps of upper bounds for the arithmetical degrees. So any uniform upper bound for the arithmetical degrees is not a minimal upper bound for the arithmetical degrees. Given a uniform upper bound a for the arithmetical degrees, we prove a minimal complementation theorem for the upper bounds for the arithmetical degrees below a. Namely, given such a and b < a which is an upper bound for the arithmetical degrees, there is a minimal upper bound for the arithmetical degrees c such that b ∪ c = a. This answers a question in Lerman [15]. We prove this theorem by different methods depending on whether a has a function which is not dominated by any arithmetical function. We prove two propositions (see §1), of which the theorem is an immediate consequence.Our notation is almost standard. Let A ⊕ B = {2n∣n ∈ A} ∪ {2n + 1∣n + 1∣n ∈ B} for any sets A and B. Let ω be the set of nonnegative natural numbers.

A General Method for Spectrometer Design in the Scoff Approximation

1976 ◽  
Vol 34 ◽  
pp. 538-539
Author(s):  
J. R. Fields
Keyword(s):  
Electron Microscope ◽  
Transmission Electron Microscope ◽  
Energy Analysis ◽  
Incident Beam ◽  
Scanning Transmission Electron Microscope ◽  
Chemical Identification ◽  
Scanning Transmission Electron ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
General Method

The energy analysis of electrons scattered by a specimen in a scanning transmission electron microscope can improve contrast as well as aid in chemical identification. In so far as energy analysis is useful, one would like to be able to design a spectrometer which is tailored to his particular needs. In our own case, we require a spectrometer which will accept a parallel incident beam and which will focus the electrons in both the median and perpendicular planes. In addition, since we intend to follow the spectrometer by a detector array rather than a single energy selecting slit, we need as great a dispersion as possible. Therefore, we would like to follow our spectrometer by a magnifying lens. Consequently, the line along which electrons of varying energy are dispersed must be normal to the direction of the central ray at the spectrometer exit.

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