Uncertainty Distribution of Some Composite Uncertain Variables

2017 ◽  
Vol 25 (04) ◽  
pp. 545-555
Author(s):  
Chongshuang Chen ◽  
Jiayin Tang ◽  
Jianbo Xiao ◽  
Lei Huang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Measurable Function ◽  
Uncertain Variable ◽  
Uncertainty Distribution ◽  
Uncertain Variables ◽  
Convex Upward

In this paper, we named the composition by a real-valued measurable function and an uncertain variable as a composite uncertain variable. We focused on the uncertainty distribution for two kinds of composite uncertain variables. The conclusions show: (1) it exists a lower bound when the composed function is continuous and strictly monotonically decreasing at first and then strictly monotonically increasing (e.g. convex downward functions); (2) it exists an upper bound when the composed function is continuous and strictly monotonically increasing at first and then strictly monotonically decreasing (e.g. convex upward functions).

A new measure of indeterminacy for uncertain variables with application to portfolio selection

2021 ◽  
pp. 1-5
Author(s):  
Jin Liu ◽  
Jinsheng Xie ◽  
Hamed Ahmadzade ◽  
Mehran Farahikia
Keyword(s):  
Probability Theory ◽  
Portfolio Selection ◽  
Uncertainty Theory ◽  
Random Variable ◽  
Uncertain Variable ◽  
Uncertainty Distribution ◽  
Uncertain Variables ◽  
Selection Problems ◽  
Exponential Entropy

Entropy is a measure for characterizing indeterminacy of a random variable or an uncertain variable with respect to probability theory and uncertainty theory, respectively. In order to characterize indeterminacy of uncertain variables, the concept of exponential entropy for uncertain variables is proposed. For computing the exponential entropy for uncertain variables, a formula is derived via inverse uncertainty distribution. As an application of exponential entropy, portfolio selection problems for uncertain returns are optimized via exponential entropy-mean models. For better understanding, several examples are provided.

scholarly journals A New Formula for Calculating Uncertainty Distribution of Function of Uncertain Variables

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2429
Author(s):  
Yuxing Jia ◽  
Yuer Lv ◽  
Zhigang Wang
Keyword(s):  
Uncertainty Theory ◽  
Monotone Function ◽  
Uncertain Variable ◽  
Mathematical Tool ◽  
Human Beings ◽  
Uncertainty Distribution ◽  
Uncertain Variables ◽  
New Ideas ◽  
New Formula ◽  
Theory Uncertainty

As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key approach in the characterization of an uncertain variable. This paper shows a new formula to calculate the uncertainty distribution of strictly monotone function of uncertain variables, which breaks the habitual thinking that only the former formula can be used. In particular, the new formula is symmetrical to the former formula, which shows that when it is too intricate to deal with a problem using the former formula, the problem can be observed from another perspective by using the new formula. New ideas may be obtained from the combination of uncertainty theory and symmetry.

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.

An entropy-based global sensitivity analysis for the structures with both fuzzy variables and random variables

2012 ◽  
Vol 227 (2) ◽  
pp. 195-212 ◽  
Author(s):  
Tang Zhangchun ◽  
Lu Zhenzhou ◽  
Pan Wang ◽  
Zhang Feng
Keyword(s):  
Failure Probability ◽  
Uncertainty Propagation ◽  
Random Variables ◽  
Uncertain Variable ◽  
Importance Measure ◽  
Similar Measure ◽  
Practical Applications ◽  
Fuzzy Variables ◽  
Global Sensitivity ◽  
Uncertain Variables

Based on the entropy of the uncertain variable, a novel importance measure is proposed to identify the effect of the uncertain variables on the system, which is subjected to the combination of random variables and fuzzy variables. For the system with the mixture of random variables and fuzzy variables, the membership function of the failure probability can be obtained by the uncertainty propagation theory first. And then the effect of each input variable on the output response of the system can be evaluated by measuring the shift between entropies of two membership functions of the failure probability, obtained before and after the uncertainty elimination of the input variable. The intersecting effect of the multiple input variables can be calculated by the similar measure. The mathematical properties of the proposed global sensitivity indicators are investigated and proved in detail. A simple example is first employed to demonstrate the procedure of solving the proposed global sensitivity indicators and then the influential variables of four practical applications are identified by the proposed global sensitivity indicators.

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