International Journal of Mathematics and Mathematical Sciences

In this paper, we present a continuous mathematical model of alcohol drinking with the influence of private and public addiction treatment centers. We study the dynamical behavior of this model and we discuss the basic properties of the system and determine its basic reproduction number . We also study the sensitivity analysis of model parameters to know the parameters that have a high impact on the reproduction number . The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at drinking-free equilibrium when . When , drinking present equilibrium exists and the system is locally as well as globally asymptotically stable at alcohol present equilibrium .

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In this paper, we give an explicit lower bound for the class number of real quadratic field , where is a square-free integer, using which is the number of odd prime divisors of .

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We consider the complex Grassmannian of k-dimensional subspaces of . There is a natural inclusion . Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion in the space of mappings from to for and in particular to show that the cohomology of contains a truncated algebra , where , for and .

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Frameproof codes were first introduced by Boneh and Shaw in 1998 in the context of digital fingerprinting to protect copyrighted materials. These digital fingerprints are generally denoted as codewords in , where Q is an alphabet of size q and n is a positive integer. A 2-frameproof code is a code C such that any 2 codewords in C cannot form a new codeword under a particular rule. Thus, no pair of users can frame a user who is not a member of the coalition. This paper concentrates on the upper bound for the size of a q-ary 2-frameproof code of length 4. Our new upper bound shows that when q is odd and .

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The aim of this paper is to provide an approximation of the value-at-risk of the multivariate copula associated with financial loss and profit function. A higher dimensional extension of the Taylor–Young formula is used for this estimation in a Euclidean space. Moreover, a time-varying and conditional copula is used for the modeling of the VaR.

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For a locally compact Hausdorff space K and a Banach space X, let be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is , we denote as . If be an extremely regular subspace of and is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of , then the cardinality of K is less than that of S. Moreover, if and is also a subalgebra of , the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal Finally, if and , then K is a continuous image of a subspace of S. Here, is the geometrical parameter introduced by Jarosz in 1989: As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

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