State complexity of power

The number of states in a deterministic finite automaton (DFA) recognizing the language $L^k$, where $L$ is regular language recognized by an $n$-state DFA, and $k?2$ is a constant, is shown to be at most $n{2}^{(k?1)n}$ and at least $(n?k)2^{(k?1)(n?k)}$ in the worst case, for every $n>k$ and for every alphabet of at least six letters. Thus, the state complexity of $L^k$ is $\Theta \left(n{2}^{(k?1)n}\right)$. In the case $k=3$ the corresponding state complexity function for $L^3$ is determined as $\frac{6n?3}{8}{4}^n?(n?1)2^n?n$ with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of $L^k$ is demonstrated to be $nk$. This bound is shown to be tight over a two-letter alphabet.     

Continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system on RN

In this article, we use the so-called difference estimate method to investigate the continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system with a general nonlinearity. Firstly, under weak assumptions on the noise coefficient, we prove the existence of a pullback attractor in $L^2\left({\mathbb{R}}^N\right)\times L^2\left({\mathbb{R}}^N\right)$ by using the tail estimate method and a certain compact embedding on bounded domains. Secondly, although the difference of the first component of solutions possesses at most $p$-times integrability where $p$ is the growth exponent of the nonlinearity, we overcome the absence of higher-order integrability and establish the continuity of solutions in $(L^p\left({\mathbb{R}}^N\right)\cap H^1\left({\mathbb{R}}^N\right))\times L^2\left({\mathbb{R}}^N\right)$ with respect to the initial values belonging to $L^2\left({\mathbb{R}}^N\right)\times L^2\left({\mathbb{R}}^N\right)$. As an application of the result on the continuity, the existence of a pullback attractor in $(L^p\left({\mathbb{R}}^N\right)\cap H^1\left({\mathbb{R}}^N\right))\times L^2\left({\mathbb{R}}^N\right)$ is proved for arbitrary $N\ge 1$ and $p>2$.     

Existence of ground states for a Kirchhoff type problem without 4-superlinear condition

We consider the existence of ground state solutions for the Kirchhoff type problem

where $a,b>0$, $N=1,2,3$ and $2<p<2^1$. Here we are interested in the case that $2<p\le 4$ since the existence of ground state for $4<p\le 2^1$ is easily obtained by a standard variational argument. Our method is based on a Poho$\stackrel{?}{z}$aev type identity.     

Innovation based on Gaussian elimination to compute generalized inverse AT,S(2)

In this paper, we execute elementary row and column operations on the partitioned matrix $\left(\begin{array}{cc}\hfill GAG\hfill & \hfill G\hfill \\ \hfill G\hfill & \hfill 0\hfill \end{array}\right)$ into $\left(\begin{array}{cc}\hfill \left(\begin{array}{cc}\hfill I_s\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ?A_{T,S}^{\left(2\right)}\hfill \end{array}\right)$to compute generalized inverse $A_{T,S}^{\left(2\right)}$ of a given complex matrix $A$, where $G$ is a matrix such that $R\left(G\right)=T$ and $N\left(G\right)=S$. The total number of multiplications and divisions operations is $T(m,n,s)=2m{n}^2+\frac{4m?s?1}{2}ns+(m?s)ns+mns$ and the upper bound of $T(m,n,s)$ is less than $6m{n}^2?\frac{3}{2}{n}^3?\frac{1}{2}{n}^2$ when $n\le m$. A numerical example is shown to illustrate that this method is correct.     

On the applicability of Postʼs lattice

For decision problems $\Pi (B)$ defined over Boolean circuits using gates from a restricted set B only, we have $\Pi (B){?}_m^{{\mathrm{AC}}^0}\Pi (B^{\prime })$ for all finite sets B and $B^{\prime }$ of gates such that all gates from B can be computed by circuits over gates from $B^{\prime }$. In this note, we show that a weaker version of this statement holds for decision problems defined over Boolean formulae, namely that $\Pi (B){?}_m^{{\mathrm{NC}}^2}\Pi (B^{\prime }\cup \{\wedge ,\vee \})$ and $\Pi (B){?}_m^{{\mathrm{NC}}^2}\Pi (B^{\prime }\cup \{0,1\})$ for all finite sets B and $B^{\prime }$ of Boolean functions such that all $f\in B$ can be defined in $B^{\prime }$.     

Eigenvalues of a real supersymmetric tensor

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 $m\mathrm{th}$-order $n$-dimensional supersymmetric tensor where $m$ is even has exactly $n{(m-1)}^{n-1}$ eigenvalues, and the number of its E-eigenvalues is strictly less than $n{(m-1)}^{n-1}$ when $m\ge 4$. 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 ${(m-1)}^{n-1}$. The $n{(m-1)}^{n-1}$ eigenvalues are distributed in $n$ disks in $C$. The centers and radii of these $n$ 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.