sets and singletons

1999 ◽  
Vol 64 (2) ◽  
pp. 590-616
Author(s):  
Kai Hauser ◽  
W. Hugh Woodin
Keyword(s):  
Core Model ◽  
Real Numbers ◽  
The Third ◽  
Sacks Forcing ◽  
Projective Hierarchy ◽  
Definable Elements

AbstractWe extend work of H. Friedman, L. Harrington and P. Welch to the third level of the projective hierarchy. Our main theorems say that (under appropriate background assumptions) the possibility to select definable elements of non-empty sets of reals at the third level of the projective hierarchy is equivalent to the disjunction of determinacy of games at the second level of the projective hierarchy and the existence of a core model (corresponding to this fragment of determinacy) which must then contain all real numbers. The proofs use Sacks forcing with perfect trees and core model techniques.

A minimal counterexample to universal baireness

1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Canonical Model ◽  
Core Model ◽  
Real Numbers ◽  
The Real ◽  
Minimal Counterexample ◽  
General Method ◽  
The Way

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

COMPUTING STRENGTH OF STRUCTURES RELATED TO THE FIELD OF REAL NUMBERS

2017 ◽  
Vol 82 (1) ◽  
pp. 137-150 ◽  
Author(s):  
GREGORY IGUSA ◽  
JULIA F. KNIGHT ◽  
NOAH DAVID SCHWEBER
Keyword(s):  
Analytic Function ◽  
Real Numbers ◽  
Computing Power ◽  
The Real ◽  
The Third ◽  
Ordered Field ◽  
Image Position

AbstractIn [8], the third author defined a reducibility $\le _w^{\rm{*}}$ that lets us compare the computing power of structures of any cardinality. In [6], the first two authors showed that the ordered field of reals ${\cal R}$ lies strictly above certain related structures. In the present paper, we show that $\left( {{\cal R},exp} \right) \equiv _w^{\rm{*}}{\cal R}$. More generally, for the weak-looking structure ${\cal R}$ℚ consisting of the real numbers with just the ordering and constants naming the rationals, all o-minimal expansions of ${\cal R}$ℚ are equivalent to ${\cal R}$. Using this, we show that for any analytic function f, $\left( {{\cal R},f} \right) \equiv _w^{\rm{*}}{\cal R}$. (This is so even if $\left( {{\cal R},f} \right)$ is not o-minimal.)

scholarly journals The SIR model towards the data

2021 ◽  
Vol 136 (8) ◽  
Author(s):  
Ignazio Lazzizzera
Keyword(s):  
Generation Time ◽  
Reproduction Number ◽  
Removal Rate ◽  
Sir Model ◽  
Infectious Period ◽  
Third Wave ◽  
Real Numbers ◽  
The Third ◽  
Study Case ◽  
Good Agreement

AbstractIn this work, the SIR epidemiological model is reformulated so to highlight the important effective reproduction number, as well as to account for the generation time, the inverse of the incidence rate, and the infectious period (or removal period), the inverse of the removal rate. The aim is to check whether the relationships the model poses among the various observables are actually found in the data. The study case of the second through the third wave of the Covid-19 pandemic in Italy is taken. Given its scale invariance, initially the model is tested with reference to the curve of swab-confirmed infectious individuals only. It is found to match the data, if the curve of the removed (that is healed or deceased) individuals is assumed underestimated by a factor of about 3 together with other related curves. Contextually, the generation time and the removal period, as well as the effective reproduction number, are obtained fitting the SIR equations to the data; the outcomes prove to be in good agreement with those of other works. Then, using knowledge of the proportion of Covid-19 transmissions likely occurring from individuals who didn’t develop symptoms, thus mainly undetected, an estimate of the real numbers of the epidemic is obtained, looking also in good agreement with results from other, completely different works. The line of this work is new, and the procedures, computationally really inexpensive, can be applied to any other national or regional case besides Italy’s study case here.

scholarly journals Universality probability of a prefix-free machine

2012 ◽  
Vol 370 (1971) ◽  
pp. 3488-3511 ◽  
Author(s):  
George Barmpalias ◽  
David L. Dowe
Keyword(s):  
Turing Degree ◽  
Real Numbers ◽  
Arithmetical Hierarchy ◽  
Halting Problem ◽  
The Real ◽  
The Third ◽  
Free Machine

We study the notion of universality probability of a universal prefix-free machine, as introduced by C. S. Wallace. We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in the arithmetical hierarchy of complexity. Furthermore, we give a computational characterization of the real numbers that are universality probabilities of universal prefix-free machines.

scholarly journals Affine functions and series with co-inductive real numbers

2007 ◽  
Vol 17 (1) ◽  
pp. 37-63 ◽  
Author(s):  
YVES BERTOT
Keyword(s):  
Search Engine ◽  
Proof System ◽  
Real Numbers ◽  
Mechanical Verification ◽  
The Third ◽  
Proof Search ◽  
Inductive Type ◽  
Affine Functions ◽  
Inductive Types ◽  
Definition Of

We extend the work of A. Ciaffaglione and P. di Gianantonio on the mechanical verification of algorithms for exact computation on real numbers, using infinite streams of digits implemented as a co-inductive type. Four aspects are studied. The first concerns the proof that digit streams correspond to axiomatised real numbers when they are already present in the proof system. The second re-visits the definition of an addition function, looking at techniques to let the proof search engine perform the effective construction of an algorithm that is correct by construction. The third concerns the definition of a function to compute affine formulas with positive rational coefficients. This is an example where we need to combine co-recursion and recursion. Finally, the fourth aspect concerns the definition of a function to compute series, with an application on the series that is used to compute Euler's number e. All these experiments should be reproducible in any proof system that supports co-inductive types, co-recursion and general forms of terminating recursion; we used the COQ system (Dowek et al. 1993; Bertot and Castéran 2004; Giménez 1994).

scholarly journals Gaging the Neural Path of the Universal Grammar by the Logos Heuristics

2019 ◽  
Vol 7 (6) ◽  
pp. 85
Author(s):  
Antonio Cassella
Keyword(s):  
Common Sense ◽  
Quantum Physics ◽  
Universal Grammar ◽  
Charles Sanders Peirce ◽  
Focused Attention ◽  
Real Numbers ◽  
The Third ◽  
The Earth ◽  
The Common ◽  
Classical Computing

The logos heuristics (“Λ”) derived from the fix of autistics—e.g., the strain to tap infinity—can help us gage the Universal Grammar that Charles Sanders Peirce implied in his “Semeiotics” since 1868. Knowledge of that Grammar equates the return of “Quetzal-coatl” in Mesoamerican legends. If that “bird-serpent/sky-land” returned with the Common Sense in Peirce’s “Pragmaticism,” social science would befit Dharma. In the Third Attention of Dharma, the 2nd restores the 1st attention; infinity, finiteness; quantum, classical computing; doubt, certainty; hyperspace, spacetime; flail, crook; yin, yang; mother, father; water, stones; moon, sun; urim, thummim; nagual, tonal; Shiva, Vishnu; novelty, sameness; mindfullness, focused attention; learning, memory; fantasy, reality; unknown, known; simultaneity, sequence; lying, candor; patience, anger; fluidity, bluntness; less-than-perfection, perfection; witch, muggle; imaginary, real numbers; wine, bread; jury, law; flexibility, rigidity; hope, faith; arch, column; bow, arrow; quantum physics, gravity; and pragmatics in cerebellar microcomplexes, grammar in the neocortex. If the return of Common Sense distanced us from the desire to overwin and replace other species, our young will regain the Earth that hosted saber-toothed cats.

scholarly journals How to solve third degree equations without moving to complex numbers

2020 ◽  
Vol 12 ◽  
pp. 123-131
Author(s):  
Antoni Leon Dawidowicz
Keyword(s):  
High School ◽  
16Th Century ◽  
Complex Numbers ◽  
Algebraic Equations ◽  
Current Version ◽  
Elementary Functions ◽  
Real Numbers ◽  
Fourth Degree ◽  
The Third ◽  
Algebraic Operations

During the Renaissance, the theory of algebraic equations developed in Europe. It is about finding a solution to the equation of the formanxn + . . . + a1x + a0 = 0,represented by coefficients subject to algebraic operations and roots of any degree. In the 16th century, algorithms for the third and fourth-degree equations appeared. Only in the nineteenth century, a similar algorithm for thehigher degree was proved impossible. In (Cardano, 1545) described an algorithm for solving third-degree equations. In the current version of this algorithm, one has to take roots of complex numbers that even Cardano didnot know.This work proposes an algorithm for solving third-degree algebraic equations using only algebraic operations on real numbers and elementary functions taught at High School.

New oscillation criteria for third order nonlinear neutral difference equations

2011 ◽  
Vol 61 (4) ◽  
Author(s):  
S. Saker
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Nonnegative Integer ◽  
Real Numbers ◽  
Third Order ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Oscillation Criteria ◽  
The Third ◽  
Positive Integers

AbstractIn this paper, we are concerned with oscillation of the third-order nonlinear neutral difference equation $\Delta (c_n [\Delta (d_n \Delta (x_n + p_n x_{n - \tau } ))]^\gamma ) + q_n f(x_{g(n)} ) = 0,n \geqslant n_0 ,$ where γ > 0 is the quotient of odd positive integers, c n, d n, p n and q n are positive sequences of real numbers, τ is a nonnegative integer, g(n) is a sequence of nonnegative integers and f ∈ C(ℝ,ℝ) such that uf(u) > 0 for u ≠ 0. Our results extend and improve some previously obtained ones. Some examples are considered to illustrate the main results.

scholarly journals About Quotient Orders and Ordering Sequences

2017 ◽  
Vol 25 (2) ◽  
pp. 121-139
Author(s):  
Sebastian Koch
Keyword(s):  
Finite Subset ◽  
Finite Sequence ◽  
Relational Structure ◽  
Real Numbers ◽  
Natural Restriction ◽  
Relational Structures ◽  
The Third ◽  
Finite Sequences ◽  
Directed Walks ◽  
Ordered Pairs

Summary In preparation for the formalization in Mizar [4] of lotteries as given in [14], this article closes some gaps in the Mizar Mathematical Library (MML) regarding relational structures. The quotient order is introduced by the equivalence relation identifying two elements x, y of a preorder as equivalent if x ⩽ y and y ⩽ x. This concept is known (see e.g. chapter 5 of [19]) and was first introduced into the MML in [13] and that work is incorporated here. Furthermore given a set A, partition D of A and a finite-support function f : A → ℝ, a function Σf : D → ℝ, Σf (X)= ∑x∈X f(x) can be defined as some kind of natural “restriction” from f to D. The first main result of this article can then be formulated as: $$\sum\limits_{x \in A} {f(x)} = \sum\limits_{X \in D} {\Sigma _f (X)\left( { = \sum\limits_{X \in D} {\sum\limits_{x \in X} {f(x)} } } \right)} $$ After that (weakly) ascending/descending finite sequences (based on [3]) are introduced, in analogous notation to their infinite counterparts introduced in [18] and [13]. The second main result is that any finite subset of any transitive connected relational structure can be sorted as a ascending or descending finite sequence, thus generalizing the results from [16], where finite sequence of real numbers were sorted. The third main result of the article is that any weakly ascending/weakly descending finite sequence on elements of a preorder induces a weakly ascending/weakly descending finite sequence on the projection of these elements into the quotient order. Furthermore, weakly ascending finite sequences can be interpreted as directed walks in a directed graph, when the set of edges is described by ordered pairs of vertices, which is quite common (see e.g. [10]). Additionally, some auxiliary theorems are provided, e.g. two schemes to find the smallest or the largest element in a finite subset of a connected transitive relational structure with a given property and a lemma I found rather useful: Given two finite one-to-one sequences s, t on a set X, such that rng t ⊆ rng s, and a function f : X → ℝ such that f is zero for every x ∈ rng s \ rng t, we have ∑ f o s = ∑ f o t.

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