ASSIGNING AN ISOMORPHISM TYPE TO A HYPERDEGREE

2019 ◽  
Vol 85 (1) ◽  
pp. 325-337
Author(s):  
HOWARD BECKER
Keyword(s):  
Isomorphism Type ◽  
Formula One ◽  
Image Position ◽  
Scott Rank

AbstractLet L be a computable vocabulary, let XL be the space of L-structures with universe ω and let $f:{2^\omega } \to {X_L}$ be a hyperarithmetic function such that for all $x,y \in {2^\omega }$, if $x{ \equiv _h}y$ then $f\left( x \right) \cong f\left( y \right)$. One of the following two properties must hold. (1) The Scott rank of f (0) is $\omega _1^{CK} + 1$. (2) For all $x \in {2^\omega },f\left( x \right) \cong f\left( 0 \right)$.

scholarly journals COMPUTABLE STRUCTURES IN GENERIC EXTENSIONS

2016 ◽  
Vol 81 (3) ◽  
pp. 814-832 ◽  
Author(s):  
JULIA KNIGHT ◽  
ANTONIO MONTALBÁN ◽  
NOAH SCHWEBER
Keyword(s):  
Positive Result ◽  
Generic Extension ◽  
Ground Model ◽  
Rigid Structure ◽  
Basic Properties ◽  
Image Position ◽  
Scott Rank ◽  
The Universe ◽  
Generic Extensions ◽  
Countable Structure

AbstractIn this paper, we investigate connections between structures present in every generic extension of the universe V and computability theory. We introduce the notion of generic Muchnik reducibility that can be used to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of generic presentability, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making ω2 countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentable by a forcing notion that does not make ω2 countable has a copy in the ground model. We also show that any countable structure ${\cal A}$ that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V, as does any structure ${\cal B}$ generically Muchnik reducible to a structure ${\cal A}$ of cardinality ℵ1. The former positive result yields a new proof of Harrington’s result that counterexamples to Vaught’s conjecture have models of power ℵ1 with Scott rank arbitrarily high below ω2. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.

A Mixed Parseval's Equation And a Generalized Hankel Transformation of Distributions

1989 ◽  
Vol 41 (2) ◽  
pp. 274-284 ◽  
Author(s):  
J. J. Betancor
Keyword(s):  
Dual Space ◽  
Integral Transform ◽  
Generalized Functions ◽  
Generalized Function ◽  
Hankel Transformation ◽  
Formula One ◽  
Image Position ◽  
Complex Valued

Let an integral transform T﹛f﹜ of a complex valued function f(x) defined over the interval (0, ∞) be defined as One of the most usual procedures to extend the classical transform (l.a) to generalized functions consists in constructing a space A of testing functions over (0, ∞) which is closed with respect to the classical transform (l.a) and then the corresponding transform of the generalized function/ of the dual space of A is defined through This approach has been followed by L. Schwartz [13] and A. H. Zemanian [20], amongst others.

A HYPERSTABILITY RESULT FOR THE CAUCHY EQUATION

2013 ◽  
Vol 89 (1) ◽  
pp. 33-40 ◽  
Author(s):  
JANUSZ BRZDĘK
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Normed Space ◽  
Functional Equations ◽  
Cauchy Equation ◽  
Cauchy Functional Equation ◽  
Real Numbers ◽  
Formula One ◽  
Image Position

AbstractWe prove a hyperstability result for the Cauchy functional equation$f(x+ y)= f(x)+ f(y)$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function$f$, mapping a normed space${E}_{1} $into a normed space${E}_{2} $, and for all real numbers$r, s$with$r+ s\gt 0$one of the following two conditions must be valid:$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = \infty , &&\displaystyle\end{eqnarray*}$$$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = 0. &&\displaystyle\end{eqnarray*}$$In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem.

scholarly journals Twisted group algebras and their representations

1964 ◽  
Vol 4 (2) ◽  
pp. 152-173 ◽  
Author(s):  
S. B. Conlon
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Group Algebras ◽  
Linear Combinations ◽  
Twisted Group Algebra ◽  
Formula One ◽  
Twisted Group ◽  
Image Position ◽  
A Finite Group ◽  
Group Product

Let be a finite group, a field. A twisted group algebra A() on over is an associative algebra whose elements are the formal linear combinations and in which the product (A)(B) is a non-zero multiple of (AB), where AB is the group product of A, B ∈: . One gets the ordinary group algebra () by taking each fA, B ≠ 1.

ISOMORPHISM ON HYP

2016 ◽  
Vol 81 (2) ◽  
pp. 395-399
Author(s):  
SY-DAVID FRIEDMAN
Keyword(s):  
Equivalence Relation ◽  
Elementary Equivalence ◽  
Limit Ordinal ◽  
Image Position ◽  
Scott Rank

AbstractWe show that isomorphism is not a complete ${\rm{\Sigma }}_1^1$ equivalence relation even when restricted to the hyperarithmetic reals: If E1 denotes the ${\rm{\Sigma }}_1^1$ (even ${\rm{\Delta }}_1^1$) equivalence relation of [4] then for no Hyp function f do we have xEy iff f(x) is isomorphic to f(y) for all Hyp reals x,y. As a corollary to the proof we provide for each computable limit ordinal α a hyperarithmetic reduction of ${ \equiv _\alpha }$ (elementary-equivalence for sentences of quantifier-rank less than α) on arbitrary countable structures to isomorphism on countable structures of Scott rank at most α.

scholarly journals Subexponentially increasing sums of partial quotients in continued fraction expansions

2015 ◽  
Vol 160 (3) ◽  
pp. 401-412 ◽  
Author(s):  
LINGMIN LIAO ◽  
MICHAŁ RAMS
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Point Of View ◽  
Partial Quotient ◽  
Formula One ◽  
Formula Group ◽  
Partial Quotients ◽  
Image Position ◽  
Continued Fraction Expansions ◽  
Increasing Function

AbstractWe investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S_{n}(x)=\sum_{j=1}^n a_{j}(x)$, where x = [a1(x), a2(x), . . .] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function ϕ : $\mathbb{N}$ → $\mathbb{N}$, one is interested in the Hausdorff dimension of the set E_\varphi = \left\{x\in (0,1): \lim_{n\to\infty} \frac {S_n(x)} {\varphi(n)} =1\right\}. Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case exp(nγ), γ ∈ [1/2, 1). We show that when γ ∈ [1/2, 1), Eϕ has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at ϕ(n) = exp(n1/2). In a similar way, the distribution of the largest partial quotient is also studied.

scholarly journals Sum of Many Dilates

2015 ◽  
Vol 25 (3) ◽  
pp. 460-469 ◽  
Author(s):  
GEORGE SHAKAN
Keyword(s):  
Formula One ◽  
Coprime Integers ◽  
Image Position

We show that for any coprime integers λ1, . . ., λkand any finiteA⊂$\mathbb{Z}$, one has|\lambda_1 \cdot A + \cdots + \lambda_k \cdot A| \geq (|\lambda_1| + \cdots + |\lambda_k|)|A|- C,whereConly depends on λ1, . . ., λk.

A critical virus production rate for efficiency of oncolytic virotherapy

2020 ◽  
pp. 1-16 ◽  
Author(s):  
YOUSHAN TAO ◽  
MICHAEL WINKLER
Keyword(s):  
Initial Data ◽  
Virus Production ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Threshold Value ◽  
Oncolytic Virotherapy ◽  
Formula One ◽  
Image Position ◽  
Initial Boundary Value

In a planar smoothly bounded domain $\Omega$ , we consider the model for oncolytic virotherapy given by $$\left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array} \right.$$ with positive parameters $ D_w $ , $ D_z $ and $\beta$ . It is firstly shown that whenever $\beta \lt 1$ , for any choice of $M \gt 0$ , one can find initial data such that the solution of an associated no-flux initial-boundary value problem, well known to exist globally actually for any choice of $\beta \gt 0$ , satisfies $$u\ge M \qquad \mbox{in } \Omega\times (0,\infty).$$ If $\beta \gt 1$ , however, then for arbitrary initial data the corresponding is seen to have the property that $$\liminf_{t\to\infty} \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$ This may be interpreted as indicating that $\beta$ plays the role of a critical virus replication rate with regard to efficiency of the considered virotherapy, with corresponding threshold value given by $\beta = 1$ .

A second paper “On the interpolation theorem for the logic of constant domains”

1983 ◽  
Vol 48 (3) ◽  
pp. 595-599 ◽  
Author(s):  
E.G.K. López-Escobar
Keyword(s):  
Interpolation Theorem ◽  
Cut Elimination ◽  
Property A ◽  
Formula One ◽  
Correct Rule ◽  
Tableau System ◽  
Image Position ◽  
The Right ◽  
Constant Domains ◽  
Subformula Property

It was brought to our attention by M. Fitting that Beth's semantic tableau system using the intuitionistic propositional rules and the classical quantifier rules produces a correct but incomplete axiomatization of the logic CD of constant domains. The formulawhere T is a truth constant, being an instance of a formula which is valid in all Kripke models with constant domains but which is not provable without cut.From the Fitting formula one can immediately obtain that the sequentalthough provable in the system GD outlined in [3], does not have a cut-free proof (in the system GD).If the only problem with GD were the sequent S0, then we could extend GD to the system GD+ by adding the following (correct) rule:Since the new rule still satisfies the subformula property a cut elimination theorem for GD+ would be a step in the right direction for a syntactical proof for the interpolation theorem for the logic of constant domains (cf. Gabbay [2]; see also §4). Unfortunately, one can show that the sequentwhere P is a propositional parameter (or formula without x free) has a derivation in GD+, but does not have a cut-free derivation (in GD+). Of course, we could extend GD+ to GD++ by adding the following correct (and with the subformula property) rule:But then we can find a sequent S2 which, although provable with cut in GD++, does not have a cut-free derivation in GD++.

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

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