In this paper we worked with the relative divergence of type
s
,
s
∈ ℝ, which include Kullback-Leibler divergence and the Hellinger and χ
2
distances as particular cases. We give here a study of the ...sym- metrized divergences in additive and multiplicative forms. Some ba-sic properties as symmetry, monotonicity and log-convexity are estab-lished. An important result from the Convexity Theory is also proved.
Let p(n) denote the partition function and let Δ be the difference operator with respect to n. In this paper, we obtain a lower bound for Δ 2 log p ( n - 1 ) / ( n - 1 ) n - 1 , leading to a proof of ...a conjecture of Sun on the log-convexity of { p ( n ) / n n } n ≥ 60 . Using the same argument, it can be shown that for any real number α , there exists an integer n ( α ) such that the sequence { p ( n ) / n α n } n ≥ n ( α ) is log-convex. Moreover, we show that lim n → + ∞ n 5 2 Δ 2 log p ( n ) n = 3 π / 24 . Finally, by finding an upper bound for Δ 2 log p ( n - 1 ) n - 1 , we establish an inequality on the ratio p ( n - 1 ) n - 1 p ( n ) n .
In the present note, we have given a new integral identity via Conformable fractional integrals and some further properties. We have proved some integral inequalities for different kinds of convexity ...via Conformable fractional integrals. We have also showed that special cases of our findings gave some new inequalities involving Riemann-Liouville fractional integrals.
The main purpose of this paper is to introduce various convexity concepts in terms of a positive Chebyshev system ω and give a systematic investigation of the relations among them. We generalize a ...celebrated theorem of Bernstein–Doetsch to the setting of ω-Jensen convexity. We also give sufficient conditions for the existence of discontinuous ω-Jensen affine functions. The concept of Wright convexity is extended to the setting of Chebyshev systems, as well, and it turns out to be an intermediate convexity property between ω-convexity and ω-Jensen convexity. For certain Chebyshev systems, we generalize the decomposition theorems of Wright convex and higher-order Wright convex functions obtained by C. T. Ng in 1987 and by Maksa and Páles in 2009, respectively.
A set of agents has to make a decision about the provision of a public good and its financing. Agents have heterogeneous values for the public good and each agent's value is private information. An ...agenda-setter has the right to make a proposal about a public-good level and a vector of contributions. For the proposal to be approved, only the favourable votes of a subset of agents are needed. If the proposal is not approved, a type-dependent outside option is implemented. I characterize the optimal public-good provision and the coalition-formation for any outside option in dominant strategies. Optimal public-good provision might be a non-monotonic function of the outside option public-good level. Moreover, the optimal coalition might be a non-convex set of types.
Given a graph G and a set S⊆V(G), the Δ-interval of S, SΔ, is the set formed by the vertices of S and every w∈V(G) forming a triangle with two vertices of S. If SΔ=S, then S is Δ-convex of G; if ...SΔ=V(G), then S is a Δ-interval set of G. The Δ-interval number of G is the minimum cardinality of a Δ-interval set and the Δ-convexity number of G is the maximum cardinality of a proper Δ-convex subset of V(G). In this work, we show that the problem of computing the Δ-convexity number is W1-hard and NP-hard to approximate within a factor O(n1−ɛ) for any constant ɛ>0 even for graphs with diameter 2 and that the problem of computing the Δ-interval number is NP-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the Δ-convexity number of chordal graphs and for computing the Δ-interval number of block graphs. We also present results on the Δ-hull, Δ-interval and Δ-convexity numbers concerning the three standard graph products, namely, the Cartesian, the strong and the lexicographic products, in function of these and well-studied parameters of the operands.
B-convexity is defined as a suitable Peano-Kuratowski limit of linear convexities. An alternative idempotent convex structure called inverse B-convexity was recently proposed in the literature. This ...paper continues and extends some investigation started in these papers. In particular we focus on the Ky-Fan inequality and prove the existence of a Nash equilibrium for inverse B-convex games. This we do by considering a suitable “harmonic” topological structure which allows to establish a KKM theorem as well as some important related properties. Among other things a coincidence theorem is established. The paper also establishes fixed point results and Nash equilibriums properties in the case where two different convex topological structures are merged. It follows that one can consider a large class of games where the players may optimize their payoff subject to different forms of convexity. Among other things an inverse B-convex version of the Debreu-Gale-Nikaido theorem is proposed.
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