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We suggest a purely combinatorial approach to a general problem in system reliability. We show how to determine if a given vector can be the signature of a system, and in the affirmative case exhibit such a system in terms on its structure function. The method employs results from the theory of simplicial sets, and provides a full characterization of signature vectors.
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The $F$-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focussing on the number of non-free direct summands. In this paper, we provide several methods to compute the (generalized) $F$-signature of a Hibi ring which is a special class of toric rings. In particular, we show that it can be computed by counting the elements in the symmetric group satisfying certain conditions. As an application, we also give the formula of the (generalized) $F$-signature for some Segre products of polynomial rings.
In this paper our aim is to characterize the set of extreme points of the set of all n-dimensional copulas (n > 1). We have shown that a copula must induce a singular measure with respect to Lebesgue measure in order to be an extreme point in the set of n-dimensional copulas. We also have discovered some sufficient conditions for a copula to be an extreme copula. We have presented a construction of a small subset of n-dimensional extreme copulas such that any n-dimensional copula is a limit point of that subset with respect to weak convergence. The applications of such a theory are widespread, finding use in many facets of current mathematical research, such as distribution theory, survival analysis, reliability theory and optimization purposes. To illustrate the point further, examples of how such extremal representations can help in optimization have also been included.
Given a domain G, a reflection vector field d(.) on the boundary of G, and drift and dispersion coefficients b(.) and sigma(.), let L be the usual second-order elliptic operator associated with b(.) and sigma(.). Under suitable assumptions that, in particular, ensure that the associated submartingale problem is well posed, it is shown that a probability measure $pi$ on bar{G} is a stationary distribution for the corresponding reflected diffusion if and only if $pi (partial G) = 0$ and $int_{bar{G}} L f (x) pi (dx) leq 0$ for every f in a certain class of test functions. Moreover, the assumptions are shown to be satisfied by a large class of reflected diffusions in piecewise smooth multi-dimensional domains with possibly oblique reflection.
For a field $mathbb{F}$, the notion of $mathbb{F}$-tightness of simplicial complexes was introduced by Kuhnel. Kuhnel and Lutz conjectured that any $mathbb{F}$-tight triangulation of a closed manifold is the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only $mathbb{F}$-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is $mathbb{F}$-tight if and only if it is $mathbb{F}$-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is $mathbb{F}$-tight if and only if it is $mathbb{F}$-orientable, neighbourly and stacked. In consequence, the Kuhnel-Lutz conjecture is valid in dimension $leq 3$.
We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).
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