scholarly journals Inverse of Simple Sets of Polynomials Effective in a Faber Region

1962 ◽  
Vol 65 ◽  
pp. 38-46
Author(s):  
F.R. Barsoum ◽  
M. Nassif
Keyword(s):  
Simple Sets

Entanglement criteria for the three-qubit states

2017 ◽  
Vol 15 (07) ◽  
pp. 1750049 ◽  
Author(s):  
Y. Akbari-Kourbolagh
Keyword(s):  
White Noise ◽  
Tensor Products ◽  
W State ◽  
Entangled States ◽  
Mean Values ◽  
W States ◽  
Pauli Matrices ◽  
The Mean ◽  
Simple Sets ◽  
Qubit States

We present sufficient criteria for the entanglement of three-qubit states. For some special families of states, the criteria are also necessary for the entanglement. They are formulated as simple sets of inequalities for the mean values of certain observables defined as tensor products of Pauli matrices. The criteria are good indicators of the entanglement in the vicinity of three-qubit GHZ and W states and enjoy the capability of detecting the entangled states with positive partial transpositions. Furthermore, they improve the best known result for the case of W state mixed with the white noise. The efficiency of the criteria is illustrated through several examples.

Degrees of classes of RE sets

1976 ◽  
Vol 41 (3) ◽  
pp. 695-696 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Order Theory ◽  
Turing Degree ◽  
Natural Numbers ◽  
First Order ◽  
First Order Theory ◽  
Simple Sets ◽  
Natural Classes

In [3], Martin computed the degrees of certain classes of RE sets. To state the results succinctly, some notation is useful.If A is a set (of natural numbers), dg(A) is the (Turing) degree of A. If A is a class of sets, dg(A) = {dg(A): A ∈ A). Let M be the class of maximal sets, HHS the class of hyperhypersimple sets, and DS the class of dense simple sets. Martin showed that dg(M), dg(HHS), and dg(DS) are all equal to the set H of RE degrees a such that a′ = 0″.Let M* be the class of coinfinite RE sets having no superset in M; and define HHS* and DS* similarly. Martin showed that dg(DS*) = H. In [2], Lachlan showed (among other things) that dg(M*)⊆K, where K is the set of RE degrees a such that a″ > 0″. We will show that K ⊆ dg (HHS*). Since maximal sets are hyperhypersimple, this gives dg(M*) = dg (HHS*) = K.These results suggest a problem. In each case in which dg(A) has been calculated, the set of nonzero degrees in dg(A) is either H or K or the empty set or the set of all nonzero RE degrees. Is this always the case for natural classes A? Natural here might mean that A is invariant under all automorphisms of the lattice of RE sets; or that A is definable in the first-order theory of that lattice; or anything else which seems reasonable.

The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral

2021 ◽  
Vol 15 (01) ◽  
pp. 45-59
Author(s):  
E. M. Bonotto ◽  
M. Federson ◽  
P. Muldowney
Keyword(s):  
Extension Theorem ◽  
Continuous Functions ◽  
Sample Space ◽  
Asset Valuation ◽  
Itô Calculus ◽  
Black Scholes ◽  
Pricing Theory ◽  
Simple Sets ◽  
Random Times ◽  
Riemann Integration

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (3) ◽  
pp. 681-694
Author(s):  
Anne Leggett ◽  
Richard A. Shore
Keyword(s):  
Lattice Theory ◽  
Lattice Structure ◽  
Sufficient Conditions ◽  
Recursion Theory ◽  
Recursively Enumerable ◽  
Countable Ordinal ◽  
Congruence Relations ◽  
General Program ◽  
Simple Sets

One general program of α-recursion theory is to determine as much as possible of the lattice structure of (α), the lattice of α-r.e. sets under inclusion. It is hoped that structure results will shed some light on whether or not the theory of (α) is decidable with respect to a suitable language for lattice theory. Fix such a language ℒ.Many of the basic results about the lattice structure involve various sorts of simple α-r.e. sets (we use definitions which are definable in ℒ over (α)). It is easy to see that simple sets exist for all admissible α. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple α-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal α-r.e. sets. Lerman [6] has shown that maximal α-r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of (α). Here various sorts of simple sets have also proved to be vital tools. The importance of simple α-r.e. sets to the study of the lattice structure of (α) is hence obvious.Lerman [6, Q22] has posed the following problem: Find an admissible α for which all simple α-r.e. sets have the same 1-type with respect to the language ℒ. The structure of (α) for such an α would be much less complicated than that of (ω). Lerman [7] showed that such an α could not be a regular cardinal of L. We show that there is no such admissible α.

Hyperhypersimple supersets in admissible recursion theory

1983 ◽  
Vol 48 (1) ◽  
pp. 185-192
Author(s):  
C. T. Chong
Keyword(s):  
Recursion Theory ◽  
Order Type ◽  
Complete Characterization ◽  
Point Of View ◽  
Extensive Literature ◽  
Careful Study ◽  
Finite Sets ◽  
Natural Class ◽  
Recursively Enumerable Set ◽  
Simple Sets

Let α be an admissible ordinal. An α-recursively enumerable set H is hyper-hypersimple (hh-simple) if its lattice of α-r.e. supersets forms a Boolean algebra. In [3], Chong and Lerman characterized the class ℋ() of hh-simple -r.e. sets as precisely those -r.e. sets whose complements are unbounded and of order type less than . Perhaps a nice example of such a set is {σ∣σ is not for any n < ω}. It follows that all hh-simple sets in are nonhyperregular and therefore of degree 0′. That ℋ() is a natural class to study can be seen from the role played by its ω-counterpart in the study of decision problems and automorphisms of ℰ*(ω), the lattice of ω-r.e. sets modulo finite sets (Soare [13] gives an extensive literature on these topics). In α-recursion theory the existence of hh-simple sets is not an all pervasive phenomenon, and there is as yet no complete characterization of the admissible ordinal α for which ℋ(α) is nonempty. While this situation is admittedly unsatisfactory, we feel that the lattice ℰ*(α) of α-r.e. sets modulo α*-finite sets for which ℋ(α) ≠ ∅ deserves a careful study. Indeed armed with some understanding over the last few years of the general theory of admissible ordinals, it is tempting to focus one's attention on some specific ordinals whose characteristics admit a more detailed analysis of the fine structure of sets and degrees. From this point of view, and ℋ() are natural objects of study since the former is a typical example of a non-Σ2-projectible, Σ2-inadmissible ordinal, while the latter is important for the investigations of automorphisms over ℰ*().

scholarly journals Data-driven analysis of the number of Lennard-Jones types needed in a force field

2020 ◽  
Author(s):  
Michael Schauperl ◽  
Sophie Kantonen ◽  
Lee-Ping Wang ◽  
Michael Gilson
Keyword(s):  
Force Field ◽  
Predictive Accuracy ◽  
Force Fields ◽  
Global Optimum ◽  
Data Driven ◽  
Organic Liquids ◽  
Field Development ◽  
Force Field Development ◽  
Lennard Jones ◽  
Simple Sets

<p>We optimized force fields with smaller and larger sets of chemically motivated Lennard-Jones types against the experimental properties of organic liquids. Surprisingly, we obtained results as good as or better than those from much more complex typing schemes from exceedingly simple sets of LJ types; e.g. a model with only two types of hydrogen and only one type apiece for carbon, nitrogen and oxygen.</p><p>The results justify sharply limiting the number of parameters to be optimized in future force field development work, thus reducing the risks of overfitting and the difficulties of reaching a global optimum in the multidimensional parameter space. They thus increase our chances of arriving at well-optimized force fields that will improve predictive accuracy, with applications in biomolecular modeling and computer-aided drug design. The results also prove the feasibility and value of a rigorous, data-driven approach to advancing the science of force field development.</p>

scholarly journals Computational Behavioral Models in Public Goods Games with Migration Between Groups

2021 ◽  
Author(s):  
Marco Tomassini ◽  
Alberto Antonioni
Keyword(s):  
Numerical Simulation ◽  
Public Goods ◽  
Laboratory Experiments ◽  
Replicator Dynamics ◽  
Behavioral Models ◽  
Public Goods Games ◽  
Heterogeneous Behavior ◽  
Simulation Results ◽  
Human Participants ◽  
Simple Sets

Abstract In this study we have simulated numerically two models of linear Public Goods Games where players are equally distributed among a given number of groups. Agents play in their group by using two simple sets of rules that are inspired by the observed behavior of human participants in laboratory experiments. In addition, unsatisfied agents have the option of leaving their group and migrating to a new random one through probabilistic choices. Stochasticity, and the introduction of two types of players in the population, help simulate the heterogeneous behavior that is often observed in experimental work. The numerical simulation results of the corresponding dynamical systems show that being able to leave a group when unsatisfied favors contribution and avoids free-riding to a good extent in a range of the enhancement factor where defection would prevail without migration. Our numerical simulation results are qualitatively in line with known experimental data when human agents are given the same kind of information about themselves and the other players in the group. This is usually not the case with customary mathematical models based on replicator dynamics or stochastic approaches. As a consequence, models like the ones described here may be useful for understanding experimental results and also for designing new experiments by first running cheap simulations instead of doing costly preliminary laboratory work. The downside is that models and their simulation tend to be less general than standard mathematical approaches.

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