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Register a new userUpper bounds for Z$_1$-eigenvalues of generalized Hilbert tensors
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In this paper, we introduce the concept of Z$_1$-eigenvalue to infinite dimensional generalized Hilbert tensors (hypermatrix) $mathcal{H}_lambda^{infty}=(mathcal{H}_{i_{1}i_{2}cdots i_{m}})$, $$ mathcal{H}_{i_{1}i_{2}cdots i_{m}}=frac{1}{i_{1}+i_{2}+cdots i_{m}+lambda}, lambdain mathbb{R}setminusmathbb{Z}^-; i_{1},i_{2},cdots,i_{m}=0,1,2,cdots,n,cdots, $$ and proved that its $Z_1$-spectral radius is not larger than $pi$ for $lambda>frac{1}{2}$, and is at most $frac{pi}{sin{lambdapi}}$ for $frac{1}{2}geq lambda>0$. Besides, the upper bound of $Z_1$-spectral radius of an $m$th-order $n$-dimensional generalized Hilbert tensor $mathcal{H}_lambda^n$ is obtained also, and such a bound only depends on $n$ and $lambda$.
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In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $mathcal{H}_{n}=(mathcal{H}_{i_{1}i_{2}cdots i_{m}})$, $$ mathcal{H}_{i_{1}i_{2}cdots i_{m}}=frac{1}{i_{1}+i_{2}+cdots i_{m}-m+a}, ain mathbb{R}setminusmathbb{Z}^-; i_{1},i_{2},cdots,i_{m}=1,2,cdots,n, $$ and show that its $H$-spectral radius and its $Z$-spectral radius are smaller than or equal to $M(a)n^{m-1}$ and $M(a)n^{frac{m}{2}}$, respectively, here $M(a)$ is a constant only dependent on $a$. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for $ageq1$. For an $m$-order infinite dimensional generalized Hilbert tensor $mathcal{H}_{infty}$ with $a>0$, we prove that $mathcal{H}_{infty}$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^{1}$ into $l^{p} (1<p<infty)$. The upper bounds of norm of corresponding positively homogeneous operators are obtained.
In this paper, we study the spectrums of faithful dimension pairs on a closed Finsler manifold and obtain a Gromov type and a Buser type lower bounds for eigenvalues. Furthermore, for the Lusternik-Schnirelmann spectrum, we not only obtain a better lower bound, but also estimate the multiplicity of each eigenvalue.
A uniform hypergraph $mathcal{H}$ is corresponding to an adjacency tensor $mathcal{A}_mathcal{H}$. We define an Estrada index of $mathcal{H}$ by using all the eigenvalues $lambda_1,dots,lambda_k$ of $mathcal{A}_mathcal{H}$ as $sum_{i=1}^k e^{lambda_i}$. The bounds for the Estrada indices of uniform hypergraphs are given. And we characterize the Estrada indices of $m$-uniform hypergraphs whose spectra of the adjacency tensors are $m$-symmetric. Specially, we characterize the Estrada indices of uniform hyperstars.
We consider integer programs (IP) defined by equations and box constraints, where it is known that determinants in the constraint matrix are one measure of complexity. For example, Artmann et al. showed that an IP can be solved in strongly polynomial time if the constraint matrix is bimodular, that is, the determinants are bounded in absolute value by two. Determinants are also used to bound the $ell_1$-distance between IP solutions and solutions of its linear relaxation. One of the first works to quantify the complexity of IPs with bounded determinants was that of Heller, who identified the maximum number of differing columns in a totally unimodular constraint matrix. So far, each extension of Hellers bound to general determinants has been exponential in the determinants or the number of equations. We provide the first column bound that is polynomial in both values. As a corollary, we give the first $ell_1$-distance bound that is polynomial in the determinants and the number of equations. We also show a tight bound on the number of differing columns in a bimodular constraint matrix; this is the first tight bound since Hellers result. Our analysis reveals combinatorial properties of bimodular IPs that may be of independent interest, in particular in recognition algorithms for IPs with bounded determinants.
Verification theorems are key results to successfully employ the dynamic programming approach to optimal control problems. In this paper we introduce a new method to prove verification theorems for infinite dimensional stochastic optimal control problems. The method applies in the case of additively controlled Ornstein-Uhlenbeck processes, when the associated Hamilton-Jacobi-Bellman (HJB) equation admits a mild solution. The main methodological novelty of our result relies on the fact that it is not needed to prove, as in previous literature, that the mild solution is a strong solution, i.e. a suitable limit of classical solutions of the HJB equation. To achieve our goal we prove a new type of Dynkin formula, which is the key tool for the proof of our main result.
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