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On the Correspondence between Approximations and Similarity [chapter]

Patrick Doherty, Andrzej Szałas
2004 Lecture Notes in Computer Science  
Similarity spaces are used to define neighborhoods around individuals and these in turn are used to define approximate sets and relations.  ...  This paper focuses on the use and interpretation of approximate databases where both rough sets and indiscernibility partitions are generalized and replaced by approximate relations and similarity spaces  ...  The meaning of inclusion A S + ⊆ A S ⊕ Consider the very basic requirement in rough set theory that the lower approximation of a set should be contained in its upper approximation, i.e., for any set A  ... 
doi:10.1007/978-3-540-25929-9_16 fatcat:wbjtcq6uefbepdu4ksq6izj4u4

Rough-Neuro Computing [chapter]

Lech Polkowski, Andrzej Skowron
2001 Lecture Notes in Computer Science  
Our approach is based on rough sets, rough mereology and information granule calculus.  ...  (o(ag, t), in) is a parameterized approximation space [10] corresponding to the i − th argument of o(ag, t) and AS(o(ag, t), out) is a parameterized approximation space [10] for the output of o(ag,  ...  In our model we assume that for any agent ag and its operation o(ag) of arity n there are approximation spaces AS 1 (o(ag), in), ..., AS n (o(ag), in) which will filter (approximately) the granules received  ... 
doi:10.1007/3-540-45554-x_6 fatcat:f347dtmfjre25iezj2abigh6ry

Conditional Expectation and the Approximation of Labelled Markov Processes [chapter]

Vincent Danos, Josée Desharnais, Prakash Panangaden
2003 Lecture Notes in Computer Science  
The key idea is to approximate a system by a coarse-graining of the state space and using averages of the transition probabilities.  ...  The use of averages and expectations may well also make the approximations more robust.  ...  Any formulas of L 0 that are satisfied by any approximant of P are satisfied by the process P itself and any formula satisfied by P is satisfied by some approximant.  ... 
doi:10.1007/978-3-540-45187-7_31 fatcat:fc7tvnji2nbgpghxynxl65laq4

Convergence of numerical differentiation

R.A Lorentz
1980 Journal of Approximation Theory  
Let n, be the set of polynomials of degree not exceeding n and let M, be the polynomial interpolation to fat a given set of points x1, x2 ,..., x" We approximate DOf by D&J This is called a numerical differentiation  ...  We study the pointwise convergence of DJvf" to D, for two choices of the set of points: for equispaced points and for the extrema of the Chebycheff polynomials.  ...  By numerical differentiation we mean the replacement of the derivative evaluated at the point zero by the approximate expression where --I -.:< x1 < x2 < ... < x, .< 1 and the 3(i are real.  ... 
doi:10.1016/0021-9045(80)90026-x fatcat:lubygcvkeveilcpagebklgj6be

Model Theory and the mean ergodic theorem for abelian unitary actions [article]

Eduardo Dueñez, José Iovino
2014 arXiv   pre-print
We explore connections between von Neumann's mean ergodic theorem and concepts of model theory.  ...  As an application we present a proof Wiener's generalization of von Neumann's result in which the group acting on the Hilbert space H is any abelian group of unitary transformations of H.  ...  This means that approximate truth of L[S]-formulas "transfers" between S and S.  ... 
arXiv:1409.6208v1 fatcat:wmkjdelki5hkdb2hdyu5qd7tca

Monte Carlo Cubature Construction [article]

Satoshi Hayakawa
2020 arXiv   pre-print
In this study, we show that cubature formulas can be constructed for probability measures provided that we have an i.i.d. sampler from the measure and the mean values of given test functions.  ...  However, the construction of cubature formulas has not generally been known, and existing examples only represent the particular domains of integrands, such as hypercubes and spheres.  ...  In each entry, "A (B)" means (estimated N ) = A and ds,m = B.  ... 
arXiv:2001.00843v2 fatcat:dcxw7rihungvxp2o55mmhznegm

Page 3131 of Mathematical Reviews Vol. , Issue 2001E [page]

2001 Mathematical Reviews  
Summary: “We estimate moduli of continuity of functions satisfy- ing the Lipschitz condition with random exponents in the sense of both convergence in probability and convergence in mean.  ...  3131 41 APPROXIMATIONS AND EXPANSIONS following general problem: Let F be a Banach space on a prob- ability space and let C, C F be a sequence of nonempty closed convex sets converging to a set C.  ... 

Monte Carlo cubature construction

Satoshi Hayakawa
2020 Japan journal of industrial and applied mathematics  
In this study, we show that cubature formulas can be constructed for probability measures provided that we have an i.i.d. sampler from the measure and the mean values of given test functions.  ...  However, the construction of cubature formulas has not generally been known, and existing examples only represent the particular domains of integrands, such as hypercubes and spheres.  ...  To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.  ... 
doi:10.1007/s13160-020-00451-x fatcat:gr7prx7vknex5g43xg6wxl3req

Page 5230 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
It is shown that best approximations from a sun in a uniformly convex Banach space locally satisfy this strong uniqueness condition.  ...  The basic analytical tool they employ is a set of operators they term “averaging projectors”, which are defined on appropriate function spaces and seem to be of some interest in themselves.  ... 

A kernel function for Signal Temporal Logic formulae [article]

Luca Bortolussi, Giuseppe Maria Gallo, Laura Nenzi
2020 arXiv   pre-print
Such a kernel allows us to embed the space of formulae into a Hilbert space, and opens up the use of kernel-based machine learning algorithms in the context of STL.  ...  We show an application of this idea to a regression problem in formula space for probabilistic models.  ...  As we will see in the experiments, this choice allows the effective use of this kernel to perform regression on the formula space for approximating the satisfaction probability and the expected robustness  ... 
arXiv:2009.05484v1 fatcat:3icmzumaa5a2pf3euki337mfke

Researches on Rough Truth of Rough Axioms Based on Granular Computing

Lin Yan, Shuo Yan
2014 Journal of Software  
The concept, rough truth, is first presented in rough logic. It is a logical value, and lies between truth and falsity.  ...  The conclusions show that each rough axiom is roughly true at every state of the structure, which means that each rough axiom is roughly valid. This is the desired result.  ...  Let |(a, v)|={u | u∈D, and u satisfies (a, v)} that is subset of D, and consists of the elements satisfying (a, v). In rough logic, the set |(a, v)| is referred to as the meaning of (a, v).  ... 
doi:10.4304/jsw.9.2.265-273 fatcat:4pmol4ivkvadlaodo3ie45plpq

Some notes on an expansion theorem of Paley and Wiener

R. J. Duffin, J. J. Eachus
1942 Bulletin of the American Mathematical Society  
In the first part of this paper we show that, conversely, if a set {g n } approximately satisfies Parseval's formula then there exists at least one orthonormal set which it is "near."  ...  The interest of this criterion is that it guarantees the set {g n } to have expansion properties similar to an orthonormal set. 2 In particular, they show that the set \g n ) approximately satisfies Parseval's  ...  In the first part of this paper we show that, conversely, if a set {g n } approximately satisfies Parseval's formula then there exists at least one orthonormal set which it is "near."  ... 
doi:10.1090/s0002-9904-1942-07797-4 fatcat:cf4b2u4jebhjbkgfdiv3gn3hwu

Logics for approximate and strong entailments

Francesc Esteva, Lluís Godo, Ricardo O. Rodríguez, Thomas Vetterlein
2012 Fuzzy sets and systems (Print)  
We consider two kinds of similarity-based reasoning and formalise them in a logical setting.  ...  We develop two logical calculi based on the notions of approximate and of strong entailment, respectively. * This paper is a fully revised and extended version of the conference paper [4] , presented at  ...  Esteva and Godo also acknowledge partial support of the Spanish project FFI2008-03126-E/FILO and Rodríguez acknowledges the projects CyT-UBA X484 and the re-  ... 
doi:10.1016/j.fss.2011.09.005 fatcat:pkoi5g74yfhzfiiigbzhhl6bx4

On pp elimination and stability in a continuous setting [article]

Nicolas Chavarria Gomez, Anand Pillay
2021 arXiv   pre-print
We conclude that the continuous logic theory of such a structure is stable.  ...  By an approximation to φ we mean a CL-formula obtained from φ by replacing every appearance of any closed set D in φ by an approximation to D.  ...  The CL-structure M is said to be κ-saturated, if whenever Γ(x) is a collection of CL-formulas over a subset A of M − of cardinality < κ and Γ is finitely approximately satisfiable in M, then Γ is satisfiable  ... 
arXiv:2107.14329v2 fatcat:yhiz2h76i5a4dbzdja3zf3esba

Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations

V. Sakbaev, A. Yaakbarieh
2012 American Journal of Computational Mathematics  
We establish that the Laplas operator with perturbation by symmetrised linear hall of displacement argument operators is the generator of unitary group in the Hilbert space of square integrable functions  ...  The representation of semigroup of Cauchy problem solutions for considered functional differential equation is given by the Feynman formulas.  ...  Chernoff's Approximation of Solutions of the Cauchy Problem which is satisfied for arbitrary function from the dense in space 0 u H linear manifold   R  0 D C  .  ... 
doi:10.4236/ajcm.2012.24040 fatcat:57zrj6hblvdyjc5iuif5mb4raq
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