In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on GL_2) in new cases with joint ramifications. The construction involves minimal vectors, rather than newforms and their variants. This paper gives a uniform treatment for the matrix algebra and division algebra cases under mild assumptions, and establishes an explicit relation between the size of the local integral and the finite conductor C(π×π_{χ^{−1}}). As an application, we combine the test vector results with the relative trace formula, and prove a hybrid type subconvexity bound which can be as strong as the Weyl bound in proper range.

In this paper, we discuss high order finite difference weighted essentially non-oscillatory (WENO) schemes, coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration, for solving the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintanance of third order spatial / temporal accuracy when the limiters are applied to a third order finite difference scheme and third order TVD-RK time discretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth order accuracy.

In 1975, P. Erd\H os proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on $n$ vertices in which any two cycles are of different lengths. Let $M_n$ be the set of simple graphs on $n$ vertices in which any two cycles are of different lengths and with the edges of $f^{\ast}(n)$. Let $mc(n)$ be the maximum cycle length for all $G \in M_n$. In this paper, it is proved that for $n$ sufficiently large, $mc(n)\leq \frac{15}{16}n.$ \par We make the following conjecture: \par \bigskip \noindent{\bf Conjecture.} $$\lim_{n \rightarrow \infty} {mc(n)\over n}= 0.$$

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Given a fixed$p$≠2, we prove a simple and effective characterization of all radial multipliers of $ \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) $ , provided that the dimension$d$is sufficiently large. The method also yields new$L$^{$q$}space-time regularity results for solutions of the wave equation in high dimensions.