A new generalization of the weighted majorization theorem for n-convex functions is given, by using a generalization of Taylor's formula. Bounds for the remainders in new majorization identities are given by using...
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A new generalization of the weighted majorization theorem for n-convex functions is given, by using a generalization of Taylor's formula. Bounds for the remainders in new majorization identities are given by using the. Cebysev type inequalities. Mean value theorems and n-exponential convexity are discussed for functionals related to the new majorization identities.
In the article, we present several majorization theorems for strongly convex functions and give their applications in inequality theory. The given results are the improvement and generalization of the earlier results.
In the article, we present several majorization theorems for strongly convex functions and give their applications in inequality theory. The given results are the improvement and generalization of the earlier results.
This paper presents a simple method to order trees, unicyclic graphs and bicyclic graphs according to their spectral radii, which is based on the application of the "majorization theorem" of tree, unicyclic ...
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This paper presents a simple method to order trees, unicyclic graphs and bicyclic graphs according to their spectral radii, which is based on the application of the "majorization theorem" of tree, unicyclic and bicyclic graphs. respectively. Moreover, we obtain some new results on the order of spectral radii of unicyclic and bicyclic graphs by employing this new method. (C) 2011 Elsevier Ltd. All rights reserved.
Let pi = (d(1), d(2), . . . , d(n)) and pi' = (d(1)', d(2)', . . . , d(n)') be two different non-increasing degree sequences. We write pi (sic) pi', if and only if Sigma(n)(i=1) d(i) = Sigma(n)(i=1...
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Let pi = (d(1), d(2), . . . , d(n)) and pi' = (d(1)', d(2)', . . . , d(n)') be two different non-increasing degree sequences. We write pi (sic) pi', if and only if Sigma(n)(i=1) d(i) = Sigma(n)(i=1) d'(i,) and Sigma(j)(i=1) d(i) <= Sigma(j)(i=1) d'(i) for all j = 1, 2, . . . , n. Let Gamma (pi) be the class of connected graphs with degree sequence pi. The second Zagreb index of a graph G is denoted by M-2(G) = Sigma uv is an element of E(G) d(u)d(v). In this paper, we characterize an extremal unicyclic graph that achieves the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence, and we also prove that if pi (sic) pi', pi and pi' are unicyclic degree sequences and U* and U** have the maximum second Zagreb indices in Gamma (pi) and Gamma (pi'), respectively, then M-2 (U*) < M-2 (U**). Furthermore, we determine the first to ninth largest second Zagreb indices together with the corresponding extremal unicyclic graphs in the class of unicyclic graphs on n >= 17 vertices. (C) 2013 Elsevier B.V. All rights reserved.
Recently, various approximate design problems for low-degree trigonometric regression models on a partial circle have been solved. In this paper we consider approximate and exact optimal design problems for first-orde...
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Recently, various approximate design problems for low-degree trigonometric regression models on a partial circle have been solved. In this paper we consider approximate and exact optimal design problems for first-order trigonometric regression models without intercept on a partial circle. We investigate the intricate geometry of the non-convex exact trigonometric moment set and provide characterizations of its boundary. Building on these results we obtain a solution of the exact -optimal design problem. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of observations.
Let G = (V, E) be a connected graph, and d(u) the degree of vertex u is an element of V. We define the general Z-type index of G as Z(alpha,beta)(G) = Sigma (uv is an element of E) [d (u) + d (v ) - beta](alpha), wher...
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Let G = (V, E) be a connected graph, and d(u) the degree of vertex u is an element of V. We define the general Z-type index of G as Z(alpha,beta)(G) = Sigma (uv is an element of E) [d (u) + d (v ) - beta](alpha), where alpha and beta are two real numbers. This generalizes several famous topological indices, such as the first and second Zagreb indices, the general sum-connectivity index, the reformulated first Zagreb index, and the general Platt index, which have successful applications in QSPR/QSAR research. Hence, we are able to study these indices in a unified approach. Let C(pi) the set of connected graphs with degree sequence pi. In the present paper, under different conditions of alpha and beta, we show that: (1) There exists a so-called BFS-graph having extremal Z(alpha,beta) index in C(pi);(2) If pi. is the degree sequence of a tree, a unicyclic graph, or a bicyclic graph, with minimum degree 1, then there exists a special BFS-graph with extremal Z(alpha,beta) index in C(pi.);(3) The so-called majorization theorem of Z(alpha,beta) holds for trees, unicyclic graphs, and bicyclic graphs. As applications of the above results, we determine the extremal graphs with maximum Z(alpha,beta) index for alpha > 1 and beta <= 2 in the set of trees, unicyclic graphs, and bicyclic graphs with given number of pendent vertices, maximum degree, independence number, matching number, and domination number, respectively. These extend the main results of some published papers.
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