Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, $L=F(\sqrt{d})$ a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, $M_f$, $M_g$ their matrices. We say that the pair $(f,g)$ is a k-pair if there exist $S\in G{L}_n(L)$ such that all the entries of the $k\times k$ upper-left corner of the matrices $S{M}_f{S}^t$ and $S{M}_g{S}^t$ are in F. We give certain criteria to determine whether a given pair $(f,g)$ is a k-pair. We consider the transfer ${\mathrm{cor}}_{L(t)/F(t)}$ determined by the $F(t)$-linear map $s:L(t)\to F(t)$ with $s(1)=0$, $s(\sqrt{d})=1$, and prove that if $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a $[\frac{k+1}{2}]$-pair. If, additionally, the form $f+tg$ does not have a totally isotropic subspace of dimension $p+1$ over $L(t)$, we show that $(f,g)$ is a $(k?2p)$-pair. In particular, if the form $f+tg$ is anisotropic, and $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a k-pair.     

Sharp bounds for the spectral radius of digraphs

Let $G=(V\text{,}E)$ be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius $\rho (G)$ of G is the largest eigenvalue of its adjacency matrix. In this paper, the following sharp bounds on $\rho (G)$ have been obtained.$\min \{\sqrt{t_i^{+}{t}_j^{+}}:(v_i\text{,}{v}_j)\in E\}?\rho (G)?\max \{\sqrt{t_i^{+}{t}_j^{+}}:(v_i\text{,}{v}_j)\in E\}$where G is strongly connected and $t_i^{+}$ is the average 2-outdegree of vertex $v_i$. Moreover, each equality holds if and only if G is average 2-outdegree regular or average 2-outdegree semiregular.     

New identities for 7-cores with prescribed BG-rank

Let $\pi$ be a partition. BG-rank$(\pi )$ is defined as an alternating sum of parities of parts of $\pi$ [A. Berkovich, F.G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc. 358 (2006) 703–726. [1]]. Berkovich and Garvan [The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear in $〈$http://arxiv.org/abs/math/0602362$〉$] found theta series representations for the t-core generating functions ${\sum }_{n?0}{a}_{t,j}(n)q^n$, where $a_{t,j}(n)$ denotes the number of t-cores of n with $\text{BG-rank}=j$. In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme values of BG-rank. In this paper we discuss representations of this type for all 7-cores with prescribed BG-rank. We make an essential use of the Ramanujan modular equations of degree seven [B.C. Berndt, Ramanujan's Notebooks, Part III, Springer, New York, 1991] to prove a variety of new formulas for the 7-core generating function$\prod _{j?1}\frac{(1-q^{7j}{)}^7}{(1-q^j)}\text{.}$These formulas enable us to establish a number of striking inequalities for $a_{7,j}(n)$ with $j=-1,0,1,2$ and $a_7(n)$, such as$a_7(2n+2)?2a_7(n),a_7(4n+6)?10a_7(n)\text{.}$Here $a_7(n)$ denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only.     

The generic gradient-like structure of certain asymptotically autonomous semilinear parabolic equations

We consider asymptotically autonomous semilinear parabolic equations

$u_t+Au=f(t,u).$

Suppose that $f(t,.)\to f^{\pm }$ as $t\to \pm \infty$, where the semiflows induced by

(*)$u_t+Au=f^{\pm }(u)$

are gradient-like. Under certain assumptions, it is shown that generically with respect to a perturbation g with $g(t)\to 0$ as $\left|t\right|\to \infty$, every solution of

$u_t+Au=f(t,u)+g(t)$

is a connection between equilibria $e^{\pm }$ of (*) with $m(e^{?})\ge m(e^{+})$. Moreover, if the Morse indices satisfy $m(e^{?})=m(e^{+})$, then u is isolated by linearization.     

On a discrete bilinear singular operator

We prove that for a large class of functions P and Q, the discrete bilinear operator $T_{P,Q}(f,g)(n)={\sum }_{m\in \mathbb{Z}?\{0\}}f(n?P(m))g(n?Q(m))\frac{1}{m}$ is bounded from $l^2\times l^2$ into $l^{1+?,\infty }$ for any $?\in (0,1]$.