A multiscale constitutive model for intergranular stress corrosion cracking in type 304 austenitic stainless steel

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A log-free zero-density estimate and small gaps in coefficients of LL-functions

Let $L(s, \pi\times\pi^\prime)$ be the Rankin--Selberg $L$-function attached to automorphic representations $\pi$ and $\pi^\prime$. Let $\tilde{\pi}$ and $\tilde{\pi}^\prime$ denote the contragredient representations associated to $\pi$ and $\pi^\prime$. Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of $L(s, \pi\times\tilde{\pi})$ and $L(s, \pi^\prime\times\tilde{\pi}^\prime)$, we prove a log-free zero-density estimate for $L(s, \pi\times\pi^\prime)$ which generalises a result due to Fogels in the context of Dirichlet $L$-functions. We then employ this log-free estimate in studying the distribution of the Fourier coefficients of an automorphic representation $\pi$. As an application we examine the non-lacunarity of the Fourier coefficients $b_f(p)$ of a modular newform $f(z)=\sum_{n=1}^{\infty} b_f(n) e^{{2\pi i n z}}$ of weight $k$, level $N$, and character $\chi$. More precisely for $f(z)$ and a prime $p$, set $j_f(p):=\max_{x;~x> p} J_{f} (p, x)$, where $J_{f} (p, x):=\#\{{\rm prime}~q;~a_{\pi}(q)=0~{\rm for~all~}p<q\leq x\}.$ We prove that $j_f(p)\ll_{f, \theta} p^\theta$ for some $0<\theta<1$

An analog feedback associative memory

A method for the storage of analog vectors, i.e., vectors whose components are real-valued, is developed for the Hopfield continuous-time network. An important requirement is that each memory vector has to be an asymptotically stable (i.e. attractive) equilibrium of the network. Some of the limitations imposed by the continuous Hopfield model on the set of vectors that can be stored are pointed out. These limitations can be relieved by choosing a network containing visible as well as hidden units. An architecture consisting of several hidden layers and a visible layer, connected in a circular fashion, is considered. It is proved that the two-layer case is guaranteed to store any number of given analog vectors provided their number does not exceed 1 + the number of neurons in the hidden layer. A learning algorithm that correctly adjusts the locations of the equilibria and guarantees their asymptotic stability is developed. Simulation results confirm the effectiveness of the approach

On the Exceptional Gauged WZW Theories

We consider two different versions of gauged WZW theories with the exceptional groups and gauged with any of theirs null subgroups. By constructing suitable automorphism, we establish the equivalence of these two theories. On the other hand our automorphism, relates the two dual irreducible Riemannian globally symmetric spaces with different characters based on the corresponding exceptional Lie groups.Comment: 5 pages, LaTeX fil