The complexity of computing, via threshold circuits, the and of fixed-dimension $k\times k$ matrices with integer or rational entries is studied. We call these two problems ${\mathrm{𝖨𝖬𝖯}}_{𝗄}$ and ${\mathrm{𝖬𝖯𝖮𝖶}}_{𝗄}$, respectively, for short. We prove that: (i) For $k\ge 2$, ${\mathrm{𝖨𝖬𝖯}}_{𝗄}$ does not belong to ${\mathrm{TC}}^0$, unless ${\mathrm{TC}}^0={\mathrm{NC}}^1$.newline (ii) For : ${\mathrm{𝖨𝖬𝖯}}_{\mathsf{2}}$ belongs to ${\mathrm{TC}}^0$ while, for $k\ge 3$, ${\mathrm{𝖨𝖬𝖯}}_{𝗄}$ does not belong to ${\mathrm{TC}}^0$, unless ${\mathrm{TC}}^0={\mathrm{NC}}^1$. (iii) For any , ${\mathrm{𝖬𝖯𝖮𝖶}}_{𝗄}$ belongs to ${\mathrm{TC}}^0$.

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\mathrm{e}}^{i\widehat{s}·\overrightarrow{v}}$ in terms of spherical harmonics ${\left\{Y_{\ell ,m}\left(\widehat{s}\right)\right\}}_{\left|m\right|\le \ell \le \infty }$. We consider the truncated series where the summation is performed over the $(\ell ,m)$'s satisfying $\left|m\right|\le \ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than as soon as satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K{ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where is the Lambert function and $C,K,\delta ,\gamma$ are pure positive constants. Numerical experiments show that this asymptotic...

We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, $\frac{{\mathrm{e}}^{i|\overrightarrow{u}-\overrightarrow{v}|}}{4\pi i|\overrightarrow{u}-\overrightarrow{v}|}$, which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices $\ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than as soon as satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K\left(\alpha \right){ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where  is the Lambert function, $K\left(\alpha \right)$ depends...

We consider the random vector $u(t,\underline{x})=(u(t,x_1),\cdots ,u(t,x_d))$, where are distinct points of ${ℝ}^2$ and denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for $u(t,\underline{x})$. We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize...