On Parallel Thinning Algorithms: Minimal Non-simple Sets, P-simple Points and Critical Kernels

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.

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Degrees of classes of RE sets

Journal of Symbolic Logic ◽

10.1017/s0022481200051252 ◽

1976 ◽

Vol 41 (3) ◽

pp. 695-696 ◽

Cited By ~ 7

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.

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Automatic Correction of Ma and Sonka's Thinning Algorithm Using P-Simple Points

IEEE Transactions on Pattern Analysis and Machine Intelligence ◽

10.1109/tpami.2010.27 ◽

2010 ◽

Vol 32 (6) ◽

pp. 1148-1152 ◽

Cited By ~ 10

Author(s):

Christophe Lohou ◽

Julien Dehos

Keyword(s):

Automatic Correction ◽

Thinning Algorithm ◽

Simple Points

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

Proceedings of the Singapore National Academy of Science ◽

10.1142/s2591722621400068 ◽

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.

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