scholarly journals Approximation Set of the Interval Set in Pawlak's Space

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qinghua Zhang ◽  
Jin Wang ◽  
Guoyin Wang ◽  
Feng Hu
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Granular Computing ◽  
Research Work ◽  
Optimal Approximation ◽  
Binary Relations ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Uncertain Set ◽  
Approximation Set

The interval set is a special set, which describes uncertainty of an uncertain concept or setZwith its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two approximations (i.e., upper approximation setR¯(Z) and lower approximation setR_(Z)) are presented, respectively. The disadvantages of using upper-approximation setR¯(Z) or lower-approximation setR_(Z) as approximation sets of the uncertain set (uncertain concept)Zare analyzed, and a new method for looking for a better approximation set of the interval setZis proposed. The conclusion that the approximation setR0.5(Z) is an optimal approximation set of interval setZis drawn and proved successfully. The change rules ofR0.5(Z) with different binary relations are analyzed in detail. Finally, a kind of crisp approximation set of the interval setZis constructed. We hope this research work will promote the development of both the interval set model and granular computing theory.

Topological structures of a type of granule based covering rough sets

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3129-3141
Author(s):  
Yan-Lan Zhang ◽  
Chang-Qing Li
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Equivalence Relations ◽  
Topological Spaces ◽  
Topological Properties ◽  
Binary Relations ◽  
Upper Approximation ◽  
Approximation Operators

Rough set theory is one of important models of granular computing. Lower and upper approximation operators are two important basic concepts in rough set theory. The classical Pawlak approximation operators are based on partition and have been extended to covering approximation operators. Covering is one of the fundamental concepts in the topological theory, then topological methods are useful for studying the properties of covering approximation operators. This paper presents topological properties of a type of granular based covering approximation operators, which contains seven pairs of approximation operators. Then, topologies are induced naturally by the seven pairs of covering approximation operators, and the topologies are just the families of all definable subsets about the covering approximation operators. Binary relations are defined from the covering to present topological properties of the topological spaces, which are proved to be equivalence relations. Moreover, connectedness, countability, separation property and Lindel?f property of the topological spaces are discussed. The results are not only beneficial to obtain more properties of the pairs of covering approximation operators, but also have theoretical and actual significance to general topology.

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

scholarly journals Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽  
2021 ◽  
Author(s):  
Seungbum Jo ◽  
Rahul Lingala ◽  
Srinivasa Rao Satti
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Total Order ◽  
Two Dimensional ◽  
Information Theoretic ◽  
Cartesian Tree ◽  
Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

On the Modal μ-Calculus Over Finite Symmetric Graphs

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Giovanna D’Agostino ◽  
Giacomo Lenzi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Satisfiability Problem ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Symmetric Graphs ◽  
Trivial Consequence

AbstractIn this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

scholarly journals Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

2018 ◽  
Vol 28 (3) ◽  
pp. 365-387
Author(s):  
S. CANNON ◽  
D. A. LEVIN ◽  
A. STAUFFER
Keyword(s):  
Markov Chain ◽  
Relaxation Time ◽  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Mixing Time ◽  
Perpendicular Bisector ◽  
Unit Square

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

scholarly journals On Some Types of Covering-Based ℐ , T -Fuzzy Rough Sets and Their Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Mohammed Atef ◽  
José Carlos R. Alcantud ◽  
Hussain AlSalman ◽  
Abdu Gumaei
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Practical Application ◽  
Fuzzy Rough Sets ◽  
Fuzzy Rough Set ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Numerical Example ◽  
Variable Precision

The notions of the fuzzy β -minimal and maximal descriptions were established by Yang et al. (Yang and Hu, 2016 and 2019). Recently, Zhang et al. (Zhang et al. 2019) presented the fuzzy covering via ℐ , T -fuzzy rough set model ( FC ℐ T FRS ), and Jiang et al. (Jiang et al., in 2019) introduced the covering through variable precision ℐ , T -fuzzy rough sets ( CVP ℐ T FRS ). To generalize these models in (Jiang et al., 2019 and Zhang et al. 2019), that is, to improve the lower approximation and reduce the upper approximation, the present paper constructs eight novel models of an FC ℐ T FRS based on fuzzy β -minimal (maximal) descriptions. Characterizations of these models are discussed. Further, eight types of CVP ℐ T FRS are introduced, and we investigate the related properties. Relationships among these models are also proposed. Finally, we illustrate the above study with a numerical example that also describes its practical application.

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