The moments of the number of exits from a simply connected region

1998 ◽  
Vol 30 (1) ◽  
pp. 167-180 ◽  
Author(s):  
Robert Illsley
Keyword(s):  
Integral Formula ◽  
Sufficient Conditions ◽  
Moment Matrix ◽  
Expected Number ◽  
Factorial Moments ◽  
Gaussian Case ◽  
Simply Connected Region ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Stationary Vector

We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossings E(C) of the boundary of a region of ℝp by a stationary vector stochastic process are obtained, along with an explicit formula for E(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝp is given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2 processes are briefly considered.

The moments of the number of exits from a simply connected region

1998 ◽  
Vol 30 (01) ◽  
pp. 167-180 ◽  
Author(s):  
Robert Illsley
Keyword(s):  
Integral Formula ◽  
Sufficient Conditions ◽  
Moment Matrix ◽  
Expected Number ◽  
Factorial Moments ◽  
Gaussian Case ◽  
Simply Connected Region ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Stationary Vector

We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossings E(C) of the boundary of a region of ℝ p by a stationary vector stochastic process are obtained, along with an explicit formula for E(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝ p is given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2 processes are briefly considered.

scholarly journals On dimensions supporting a rational projective plane

2019 ◽  
Vol 11 (03) ◽  
pp. 535-555 ◽  
Author(s):  
Lee Kennard ◽  
Zhixu Su
Keyword(s):  
Projective Plane ◽  
Open Problem ◽  
Sufficient Conditions ◽  
Quadratic Residue ◽  
Bernoulli Numbers ◽  
Existence Results ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Existence Question ◽  
Smooth Closed Manifold

A rational projective plane ([Formula: see text]) is a simply connected, smooth, closed manifold [Formula: see text] such that [Formula: see text]. An open problem is to classify the dimensions at which such a manifold exists. The Barge–Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori–Stong integrality conditions on the Pontryagin numbers. In this paper, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a [Formula: see text]. We then confirm the existence of a [Formula: see text] in two new dimensions and prove several non-existence results using factorization of the numerators of the divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces.

Existence of moments in a stationary stochastic difference equation

1990 ◽  
Vol 22 (1) ◽  
pp. 129-146 ◽  
Author(s):  
Hans Arnfinn Karlsen
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stochastic Difference Equation ◽  
Dependent Variables ◽  
Gaussian Case ◽  
Necessary And Sufficient

The stationary stochastic difference equation Xt = YtXt–1 + Wt is analyzed with emphasis on conditions ensuring that ||Xt||p <∞. Some general results are obtained and then applied to different classes of input processes {(Yt, Wt)}. Especially both necessary and sufficient conditions are given in the Gaussian case. We also obtain results concerning moments of products of dependent variables.

An Inverse Eigenvalue Problem for Spring-Mass Systems with Partial Mass Connected to the Ground

2013 ◽  
Vol 353-356 ◽  
pp. 3308-3311
Author(s):  
Xia Tian ◽  
Chuan Xiao Li
Keyword(s):  
Total Mass ◽  
Sufficient Conditions ◽  
Inverse Eigenvalue Problem ◽  
Spring Stiffness ◽  
Positive Mass ◽  
Mass System ◽  
Inverse Eigenvalue ◽  
Physical Elements ◽  
Necessary And Sufficient ◽  
Simply Connected

Consider the simply connected spring-mass system with partial mass connected to the ground. The inverse mode problem of constructing the physical elements of the system from two eigenpairs, the grounding spring stiffness and total mass of the system is considered. The necessary and sufficient conditions for constructing a physical realizable system with positive mass and stiffness elements are established. If these conditions are satisfied, the grounding spring-mass system may be constructed uniquely. The numerical methods and examples are given finally.

scholarly journals Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

2021 ◽  
Author(s):  
Nikolay Abrosimov ◽  
Bao Vuong
Keyword(s):  
Convex Hull ◽  
Hyperbolic Space ◽  
General Type ◽  
Integral Formula ◽  
Sufficient Conditions ◽  
Gram Matrix ◽  
Dihedral Angles ◽  
Hyperbolic Tetrahedron ◽  
Volume Formula ◽  
Necessary And Sufficient

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

An Inverse Mode Problem for the Rod on Elastic Foundation

2013 ◽  
Vol 353-356 ◽  
pp. 3198-3201
Author(s):  
Xia Tian ◽  
Chuan Xiao Li
Keyword(s):  
Numerical Methods ◽  
Elastic Foundation ◽  
Discrete Model ◽  
Sufficient Conditions ◽  
Spring Stiffness ◽  
Positive Mass ◽  
Mass System ◽  
Physical Elements ◽  
Necessary And Sufficient ◽  
Simply Connected

Consider the rod on elastic foundation. Its discrete model is the simply connected spring-mass system with partial mass connected to the ground. The inverse mode problem of constructing the physical elements of the system from two eigenpairs, the spring stiffness of the system is considered. The necessary and sufficient conditions for constructing a physical realizable system with positive mass and stiffness elements are established. If these conditions are satisfied, the rod on the elastic foundation may be constructed uniquely. The numerical methods and examples are given finally.

scholarly journals On Statistics of Permutations Chosen From the Ewens Distribution

2014 ◽  
Vol 23 (6) ◽  
pp. 889-913
Author(s):  
TATJANA BAKSHAJEVA ◽  
EUGENIJUS MANSTAVIČIUS
Keyword(s):  
Probability Measure ◽  
Sufficient Conditions ◽  
Random Permutation ◽  
Asymptotic Distributions ◽  
Additive Functions ◽  
Factorial Moments ◽  
Permutation Matrices ◽  
Poisson Limit ◽  
Number Of Cycles ◽  
Necessary And Sufficient

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial moments, we establish necessary and sufficient conditions for the weak convergence of distributions to discrete laws. More attention is paid to the Poisson limit distribution. The particular case of the number-of-cycles function is analysed in more detail. The results can be applied to statistics defined on random permutation matrices.

scholarly journals The factorial moments of additive functions with rational argument

2006 ◽  
Vol 81 (3) ◽  
pp. 425-440
Author(s):  
J. Šiaulys ◽  
G. Stepanauskas
Keyword(s):  
Weak Convergence ◽  
Limit Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Additive Functions ◽  
Factorial Moments ◽  
Rational Argument ◽  
Necessary And Sufficient ◽  
The Given

AbstractWe consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that f(p) and f(1/p) ∈ {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and Halász's and Ruzsa's inequalities are used. We present a few examples of application of the given results to some sets of fractions.

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