Asymptotic conditional probabilities: The non-unary case

1996 ◽  
Vol 61 (1) ◽  
pp. 250-276 ◽  
Author(s):  
Adam J. Grove ◽  
Joseph Y. Halpern ◽  
Daphne Koller
Keyword(s):  
Lower Bounds ◽  
Predicate Symbol ◽  
Upper And Lower Bounds ◽  
Degrees Of Belief ◽  
Conditional Probabilities ◽  
Unary Predicate ◽  
First Order

AbstractMotivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1, …, N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kiĭ [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.

scholarly journals An exact expression of positive periodic solution for a first-order singular equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yun Xin ◽  
Xiaoxiao Cui ◽  
Jie Liu
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Lower Bounds ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Positive Periodic Solution ◽  
Exact Expression ◽  
Upper And Lower Bounds ◽  
Singular Equation ◽  
First Order

Abstract The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

scholarly journals A New Method to Study the Periodic Solutions of the Ordinary Differential Equations Using Functional Analysis

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 677 ◽  
Author(s):  
Kadry ◽  
Alferov ◽  
Ivanov ◽  
Korolev ◽  
Selitskaya
Keyword(s):  
Functional Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Lower Bounds ◽  
New Method ◽  
Upper And Lower Bounds ◽  
First Order ◽  
Periodic Problems ◽  
Existence And Stability

In this paper, a new theorems of the derived numbers method to estimate the number of periodic solutions of first-order ordinary differential equations are formulated and proved. Approaches to estimate the number of periodic solutions of ordinary differential equations are considered. Conditions that allow us to determine both upper and lower bounds for these solutions are found. The existence and stability of periodic problems are considered.

An axiomatization for a class of two-cardinal models

1977 ◽  
Vol 42 (2) ◽  
pp. 174-178 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Predicate Symbol ◽  
Unary Predicate ◽  
First Order ◽  
Order Language

In this note we give a simple recursive axiomatization for the class of structures of type (ℶω ℵ0). This solves a problem of Vaught which is Problem 13 in the book [1] of Chang and Keisler. The same technique is used to get a recursive axiomatization for the class of κ-like structures where κ is strongly ω-inaccessible.Let us fix throughout some recursive first-order language L, and until further notice let us suppose that included in L is a distinguished unary predicate symbol U. For cardinals κ and λ with κ ≥ λ ≥ ℵ0, we say the structure has type (κ, λ) if card(A)= κ and card . Let K(κ, λ) be the class of all structures of type (κ, λ). For each ordinal α define 2ακby 20κ = κ, and 2ακ= ⋃ {2λ: λ = 2βκ for some β < α} when α > 0. Let Vaught proved the following theorem in [7].Theorem (Vaught). Suppose a is a sentence such that for each n < ω there are κ, λ with κ > 2λn and a model of σ of type (κ, λ). Then whenever κ ≥ λ ≥ ℵ0, the sentence σ has a model of type (κ, λ).

The two-cardinal problem for languages of arbitrary cardinality

2010 ◽  
Vol 75 (3) ◽  
pp. 785-801
Author(s):  
Luis Miguel ◽  
Villegas Silva
Keyword(s):  
Predicate Symbol ◽  
Unary Predicate ◽  
Transfer Theorem ◽  
First Order ◽  
Order Language

AbstractLet ℒ be a first-order language of cardinality κ++ with a distinguished unary predicate symbol U. In this paper we prove, working on L, the two cardinal transfer theorem (κ+,κ) ⇒ (κ++, κ+) for this language. This problem was posed by Chang and Keisler more than twenty years ago.

scholarly journals Bounds on Heavy-to-Heavy Weak Decay Form Factors

2001 ◽  
Vol 16 (supp01a) ◽  
pp. 416-418
Author(s):  
Cheng-Wei Chiang
Keyword(s):  
Heavy Quark ◽  
Lower Bounds ◽  
Sum Rules ◽  
Form Factors ◽  
Weak Decay ◽  
Upper And Lower Bounds ◽  
Effective Theory ◽  
Heavy Quark Effective Theory ◽  
First Order ◽  
Decay Form

We provide upper and lower bounds on the semileptonic weak decay form factors for B → D(*) and Λb → Λc decays by utilizing inclusive heavy quark effective theory sum rules. These bounds are calculated to second order in ΛQCD/mQ and first order in αs. The [Formula: see text] corrections to the bounds at zero recoil are also presented.

Equivalent Composite Laminated Beams With Various Elastic Constitutive Models

1999 ◽  
Author(s):  
Izhak Sheinman ◽  
Yeoshua Frostig
Keyword(s):  
Lower Bounds ◽  
Plane Stress ◽  
Constitutive Models ◽  
Upper And Lower Bounds ◽  
Two Dimensional ◽  
Laminated Beams ◽  
Cylindrical Bending ◽  
One Dimensional ◽  
First Order ◽  
Shear Deformable

Abstract Equivalent one-dimensional constitutive models of composite laminated beams with shear deformation are derived from the classical laminate two-dimensional using first-order shear deformable theory. The present cylindrical bending constitutive models can be used — with much greater accuracy than their well known plane-strain and plane-stress counterparts — as upper and lower bounds, to one of which the behavior tends depending on the width-to-length ratio; this aspect was investigated and results are presented.

TWO PROCESSES FOR TWO PRICES

2014 ◽  
Vol 17 (01) ◽  
pp. 1450005 ◽  
Author(s):  
DILIP B. MADAN ◽  
WIM SCHOUTENS
Keyword(s):  
Lower Bounds ◽  
Markov Processes ◽  
Stochastic Dominance ◽  
Upper And Lower Bounds ◽  
One Dimensional ◽  
First Order ◽  
Order Stochastic Dominance ◽  
Bid Price ◽  
Increasing Process ◽  
Exchange Traded Fund

Postulating additivity of bid and ask prices for claims comonotone with a long or short stock position, two pricing processes are identified from data on bid and ask prices for options. It is observed that there are two separate put call parity relations in place, with the ask price for call less bid prices for put delivering an ask price for the forward-stock. Likewise the ask for puts less the bid for calls identifies the bid for the forward-stock. Two processes are introduced to determine bid and ask prices for claims comonotone with a long or short position in the stock. For a claim comonotone with a long position one uses the so-called increasing process for the ask price and the so-called decreasing process for the bid price and vice versa for a claim comonotone with a short position. As candidates for the two processes one may employ any of the traditional one-dimensional Markov processes. We illustrate the theory by using a Sato process, a model known to produce a smile conforming fit over strike and maturity. The two processes are observed to have marginals related by first order stochastic dominance. The increasing process dominates the decreasing process in this sense. These two processes are also used to construct upper and lower bounds for bid and ask prices for claims not comonotone with a long or short stock position. The two processes and their properties are illustrated with data on bid and ask prices for options on the exchange traded fund, SPY, that is the Standard and Poors' Depository Receipt tracking the S&P 500 index.

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