Completeness theorem for biprobability models

1986 ◽  
Vol 51 (3) ◽  
pp. 586-590 ◽  
Author(s):  
Miodrag D. Rašković
Keyword(s):  
Completeness Theorem ◽  
Probability Measures ◽  
List Type ◽  
Probability Logic ◽  
Admissible Set ◽  
Absolutely Continuous ◽  
Image Position ◽  
Standard Probability ◽  
Natural Way

The aim of the paper is to prove the completeness theorem for biprobability models. This also solves Keisler's Problem 5.4 (see [4]).Let be a countable admissible set and ω ∈ . The logic is similar to the standard probability logic . The only difference is that two types of probability quantifiers and are allowed.A biprobability model is a structure (, μ1, μ2) where is a classical structure without operations and μ1, μ2 are two types of probability measures such that μ1 is absolutely continuous with respect to μ2, i.e. μ1 ≪ μ2.The quantifiers are interpreted in the natural way, i.e.for i = 1, 2. (The measure is the restriction of the completion of to the σ-algebra generated by the measurable rectangles and the diagonal sets Axioms and rules of inference are those of , as listed in [2] with the axiom B4 from [4], with the remark that both P1 and P2 can play the role of P, together with the following axioms:Axioms of continuity.1) .2) .Axiom of absolute continuity:where and Φn = {φ ∈ Φ: φ has n free variables}.

scholarly journals EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

2019 ◽  
Vol 84 (02) ◽  
pp. 452-472 ◽  
Author(s):  
JAROSLAV NEŠETŘIL ◽  
PATRICE OSSONA DE MENDEZ
Keyword(s):  
Relative Size ◽  
Convergent Sequence ◽  
List Type ◽  
Sparse Graphs ◽  
First Order ◽  
Graph Modeling ◽  
Image Position ◽  
Standard Probability ◽  
Convergent Sequences

AbstractA sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:1.If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.2.A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.

A probabilistic interpolation theorem

1985 ◽  
Vol 50 (3) ◽  
pp. 708-713 ◽  
Author(s):  
Douglas N. Hoover
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Probability Logic ◽  
Admissible Set ◽  
First Order ◽  
Probability Spaces ◽  
Image Position ◽  
Dedekind Cuts ◽  
Exact Definition

The probability logic is a logic with a natural interpretation on probability spaces (thus, a logic whose model theory is part of probability theory rather than a system for putting probabilities on formulas of first order logic). Its exact definition and basic development are contained in the paper [3] of H. J. Keisler and the papers [1] and [2] of the author. Building on work in [2], we prove in this paper the following probabilistic interpolation theorem for .Let L be a countable relational language, and let A be a countable admissible set with ω ∈ A (in this paper some probabilistic notation will be used, but ω will always mean the least infinite ordinal). is the admissible fragment of corresponding to A. We will assume that L is a countable set in A, as is usual in practice, though all that is in fact needed for our proof is that L be a set in A which is wellordered in A.Theorem. Let ϕ(x) and ψ(x) be formulas of LAP such thatwhere ε ∈ [0, 1) is a real in A (reals may be defined in the usual way as Dedekind cuts in the rationals). Then for any real d > ε¼, there is a formula θ(x) of (L(ϕ) ∩ L(ψ))AP such thatand

scholarly journals Probabilities in Statistical Mechanics

2017 ◽  
Author(s):  
Wayne Myrvold
Keyword(s):  
Statistical Mechanics ◽  
Equilibrium Distribution ◽  
Probability Measures ◽  
The Status ◽  
History Of ◽  
Statistical Mechanical ◽  
Standard Probability

This chapter reviews selected aspects of the terrain of discussion of the role of probabilities in statistical mechanics. Among the topics addressed are the reasons for introduction of probabilities into statistical mechanics, the status of the standard equilibrium distribution, and the question of interpretation of statistical mechanical probabilities. The chapter starts with a brief history of probabilities in physics and the evolution of statistical mechanics therefrom. The approaches of Boltzmann and Gibbs are presented, and then some approaches to justifying choice of probability measures. The chapter closes with a presentation of possible resolutions to some puzzling aspects of the use of standard probability measures.

The role of differential phase contrast in electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 176-177
Author(s):  
E.M. Waddell ◽  
J.N. Chapman ◽  
R.P. Ferrier
Keyword(s):  
Phase Contrast ◽  
Practical Method ◽  
Position Vector ◽  
Differential Phase ◽  
Pure Phase ◽  
Differential Phase Contrast ◽  
The Difference ◽  
Image Position ◽  
First Moment

Dekkers and de Lang (1977) have discussed a practical method of realising differential phase contrast in a STEM. The method involves taking the difference signal from two semi-circular detectors placed symmetrically about the optic axis and subtending the same angle (2α) at the specimen as that of the cone of illumination. Such a system, or an obvious generalisation of it, namely a quadrant detector, has the characteristic of responding to the gradient of the phase of the specimen transmittance. In this paper we shall compare the performance of this type of system with that of a first moment detector (Waddell et al.1977).For a first moment detector the response function R(k) is of the form R(k) = ck where c is a constant, k is a position vector in the detector plane and the vector nature of R(k)indicates that two signals are produced. This type of system would produce an image signal given bywhere the specimen transmittance is given by a (r) exp (iϕ (r), r is a position vector in object space, ro the position of the probe, ⊛ represents a convolution integral and it has been assumed that we have a coherent probe, with a complex disturbance of the form b(r-ro) exp (iζ (r-ro)). Thus the image signal for a pure phase object imaged in a STEM using a first moment detector is b2 ⊛ ▽ø. Note that this puts no restrictions on the magnitude of the variation of the phase function, but does assume an infinite detector.

scholarly journals Early oxygen levels contribute to brain injury in extremely preterm infants

2021 ◽  
Author(s):  
Krista Rantakari ◽  
Olli-Pekka Rinta-Koski ◽  
Marjo Metsäranta ◽  
Jaakko Hollmén ◽  
Simo Särkkä ◽  
...  
Keyword(s):  
Preterm Infants ◽  
Brain Structures ◽  
High Oxygen ◽  
List Type ◽  
Neurodevelopmental Outcomes ◽  
Oxygen Levels ◽  
Arterial Blood ◽  
Extremely Preterm ◽  
Extremely Preterm Infants

Abstract Background Extremely low gestational age newborns (ELGANs) are at risk of neurodevelopmental impairments that may originate in early NICU care. We hypothesized that early oxygen saturations (SpO2), arterial pO2 levels, and supplemental oxygen (FiO2) would associate with later neuroanatomic changes. Methods SpO2, arterial blood gases, and FiO2 from 73 ELGANs (GA 26.4 ± 1.2; BW 867 ± 179 g) during the first 3 postnatal days were correlated with later white matter injury (WM, MRI, n = 69), secondary cortical somatosensory processing in magnetoencephalography (MEG-SII, n = 39), Hempel neurological examination (n = 66), and developmental quotients of Griffiths Mental Developmental Scales (GMDS, n = 58). Results The ELGANs with later WM abnormalities exhibited lower SpO2 and pO2 levels, and higher FiO2 need during the first 3 days than those with normal WM. They also had higher pCO2 values. The infants with abnormal MEG-SII showed opposite findings, i.e., displayed higher SpO2 and pO2 levels and lower FiO2 need, than those with better outcomes. Severe WM changes and abnormal MEG-SII were correlated with adverse neurodevelopment. Conclusions Low oxygen levels and high FiO2 need during the NICU care associate with WM abnormalities, whereas higher oxygen levels correlate with abnormal MEG-SII. The results may indicate certain brain structures being more vulnerable to hypoxia and others to hyperoxia, thus emphasizing the role of strict saturation targets. Impact This study indicates that both abnormally low and high oxygen levels during early NICU care are harmful for later neurodevelopmental outcomes in preterm neonates. Specific brain structures seem to be vulnerable to low and others to high oxygen levels. The findings may have clinical implications as oxygen is one of the most common therapies given in NICUs. The results emphasize the role of strict saturation targets during the early postnatal period in preterm infants.

scholarly journals Universal Qudit Hamiltonians

2021 ◽  
Author(s):  
Stephen Piddock ◽  
Ashley Montanaro
Keyword(s):  
State Energy ◽  
Entangled State ◽  
Quantum Computers ◽  
List Type ◽  
Universal Family ◽  
Finite Dimensional ◽  
Universal Quantum ◽  
Aklt Model ◽  
Almost All ◽  
Natural Way

AbstractA family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all $$k \geqslant 2$$ k ⩾ 2 and all local dimensions $$d \geqslant 2$$ d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions $$d \geqslant 2$$ d ⩾ 2 (spin $$\geqslant 1/2$$ ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for $$d=3$$ d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.

scholarly journals 332 The diagnostic role of shunt series radiographs (SSR) in children presenting to the children’s emergency department (CED) with suspected ventriculoperitoneal (VP) shunt failure

2020 ◽  
Vol 37 (12) ◽  
pp. 852.3-853
Author(s):  
Angharad Griffiths ◽  
Ikechukwu Okafor ◽  
Thomas Beattie
Keyword(s):  
Clinical Outcome ◽  
Sensitivity And Specificity ◽  
Clinical Information ◽  
Flow Diagram ◽  
List Type ◽  
Shunt Failure ◽  
Vp Shunt ◽  
Wide Range ◽  
Diagnostic Role

Aims/Objectives/BackgroundVP shunts are used to drain CSF from the cranial vault because of a wide range of pathologies and, like any piece of hardware, can fail. Traditionally investigations include SSR and CT. This project examines the role of SSR in evaluating children with suspected VP shunt failure.Primary outcome: Sensitivity and specificity of SSR in children presenting to the CED with concern for shunt failure.Methods/DesignConducted in a single centre, tertiary CED of the national Irish Neurosurgical(NS) referral centre (ED attendance:>50,000 patients/year). 100 sequential SSR requested by the CED were reviewed. Clinical information was extracted from electronic requests. Shunt failure was defined by the need for NS intervention(Revision).Abstract 332 Figure 1Abstract 332 Figure 2Results/ConclusionsSensitivity and specificity is presented in figure 1 (two by two table).100 radiographs performed in 84 children.22% shunts revised (see flow diagram).7 SSR’s were abnormal.85% (n=6) shunts revised. [5 following abnormal CT].Of the normal SSR’s; 16 had abnormal CT and revised.85/100 received CT.64 of 85 CT’s (75%) were normal.□6 of the 64 had focal shunt concern.SSR’s shouldn’t be used in isolation. NPV&PPV, Sensitivity&Specificity is low.SSR’s are beneficial where there’s concern over focal shunt problems (injury/pain/swelling) or following abnormal CT.VP shunt failure is not well investigated with SSR alone.SSR’s could be omitted where there is no focal shunt concern/after normal CT (without impacting clinical outcome) reducing radiation exposure and reduce impact on CED’s.59 SSR’s could have been avoided without adverse clinical outcome.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Oxidation of Terpenoids to Achieve High-Value Flavor and Fragrances—Questioning Microalgae Oxidative Capabilities in the Biotransformation of the Sesquiterpene Valencene and of Selected Natural Apocarotenoids

Chemistry ◽  
2021 ◽  
Vol 3 (3) ◽  
pp. 821-830
Author(s):  
Davide De Simeis ◽  
Stefano Serra ◽  
Alessandro Di Fonzo ◽  
Francesco Secundo
Keyword(s):  
Experimental Data ◽  
Natural Products ◽  
Aqueous Medium ◽  
Photochemical Reaction ◽  
Substrate Oxidation ◽  
Market Size ◽  
Chlorella Sp ◽  
General Opinion ◽  
Natural Way

Natural flavor and fragrance market size is expected to grow steadily due to the rising consumer demand of natural ingredients. This market request is guided by the general opinion that the production of natural compounds leads to a reduction of pollution, with inherent advantages for the environment and people’s health. The biotransformation reactions have gained high relevance in the production of natural products. In this context, few pieces of research have described the role of microalgae in the oxidation of terpenoids. In this present study, we questioned the role of microalgal based oxidation in the synthesis of high-value flavors and fragrances. This study investigated the role of three different microalgae strains, Chlorella sp. (211.8b and 211.8p) and Chlorococcum sp. (JB3), in the oxidation of different terpenoid substrates: α-ionone, β-ionone, theaspirane and valencene. Unfortunately, the experimental data showed that the microalgal strains used are not responsible for the substrate oxidation. In fact, our experiments demonstrate that the transformation of the four starting compounds is a photochemical reaction that involves the oxygen as oxidant. Even though these findings cast a shadow on the use of these microorganisms for an industrial purpose, they open a new possible strategy to easily obtain nootkatone in a natural way by just using an aqueous medium, oxygen and light.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

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