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The number of states in a deterministic finite automaton (DFA) recognizing the language , where is regular language recognized by an -state DFA, and is a constant, is shown to be at most and at least in the worst case, for every and for every alphabet of at least six letters. Thus, the state complexity of is . In the case the corresponding state complexity function for is determined as with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of is demonstrated to be . This bound is shown to be tight over a two-letter alphabet. 相似文献
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Mingqing Zhai Guanglong Yu Jinlong Shu 《Computers & Mathematics with Applications》2010,59(1):376-381
Let be the class of bicyclic graphs on vertices with girth . Let be the subclass of consisting of all bicyclic graphs with two edge-disjoint cycles and . This paper determines the unique graph with the maximal Laplacian spectral radius among all graphs in and , respectively. Furthermore, the upper bound of the Laplacian spectral radius and the extremal graph for are also obtained. 相似文献
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In this paper, we execute elementary row and column operations on the partitioned matrix into to compute generalized inverse of a given complex matrix , where is a matrix such that and . The total number of multiplications and divisions operations is and the upper bound of is less than when . A numerical example is shown to illustrate that this method is correct. 相似文献
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Eigenvalues of a real supersymmetric tensor 总被引:3,自引:0,他引:3
In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An -order -dimensional supersymmetric tensor where is even has exactly eigenvalues, and the number of its E-eigenvalues is strictly less than when . We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by . The eigenvalues are distributed in disks in . The centers and radii of these disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations. 相似文献
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Aleksandar Ilić 《Computers & Mathematics with Applications》2010,59(8):2776-2783
Let be a simple undirected graph with the characteristic polynomial of its Laplacian matrix , . It is well known that for trees the Laplacian coefficient is equal to the Wiener index of , while is equal to the modified hyper-Wiener index of the graph. In this paper, we characterize -vertex trees with given matching number which simultaneously minimize all Laplacian coefficients. The extremal tree is a spur, obtained from the star graph with vertices by attaching a pendant edge to each of certain non-central vertices of . In particular, minimizes the Wiener index, the modified hyper-Wiener index and the recently introduced Incidence energy of trees, defined as , where are the eigenvalues of signless Laplacian matrix . We introduced a general transformation which decreases all Laplacian coefficients simultaneously. In conclusion, we illustrate on examples of Wiener index and Incidence energy that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution. 相似文献
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Fenglong Sun Lishan Liu Yonghong Wu 《Computers & Mathematics with Applications》2018,75(10):3685-3701
In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations: where , with being the principal eigenvalue for on and . By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) ; (ii) , where is a nonnegative constant; (iii) , where involves the -norm or -norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level. 相似文献
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In this research paper using the Chebyshev expansion, we explicitly determine the best uniform polynomial approximation out of (the space of polynomials of degree at most ) to a class of rational functions of the form on , where is the first kind of Chebyshev polynomial of degree and . In this way we give some new theorems about the best approximation of this class of rational functions. Furthermore we obtain the alternating set of this class of functions. 相似文献