Let be one solution to $${\partial }_{t}y(t,x)-\sum _{i,j=1}^n{\partial }_j\left(a_{ij}\left(x\right){\partial }_{i}y(t,x)\right)=h(t,x),0<t<T,x\in \Omega$$ with a non-homogeneous term , and ${y|}_{(0,T)\times \partial \Omega }=0$, where $\Omega \subset {ℝ}^n$ is a bounded domain. We discuss an inverse problem of determining unknown functions by $\{{\partial }_{\nu }y\left(h_{\ell }\right){|}_{(0,T)\times {\Gamma }_0}$, $y\left(h_{\ell }\right)(\theta ,·){\}}_{1\le \ell \le {\ell }_0}$ after selecting input sources $h_1,...,h_{{\ell }_0}$ suitably, where ${\Gamma }_0$ is an arbitrary subboundary, ${\partial }_{\nu }$ denotes the normal derivative, $0<\theta <T$ and ${\ell }_0\in \mathbb{N}$. In the case of ${\ell }_0={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1,...,h_{{\ell }_0})$ from a set $\mathscr{H}\subset {\left\{C_0^{\infty }((0,T)\times \omega )\right\}}^{{\ell }_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take ${\ell }_0=(n+3)n/2$ by making special choices...

We consider a class of semilinear elliptic equations of the form 15.7cm -${\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$ where $\epsilon >0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show variational methods that if is sufficiently small and is not constant, then ([see full textsee full text]) admits infinitely many of such solutions,...

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 ${\mathrm{TC}}^0$.

In this paper, we consider a Borel measurable function on the space of $m\times n$ matrices $f:M^{m\times n}\to \overline{ℝ}$ taking the value $+\infty$, such that its rank-one-convex envelope $Rf$ is finite and satisfies for some fixed $p>1$: $$-c_0\le Rf\left(F\right)\le {c(1+\parallel F\parallel }^p)\text{for}\text{all}F\in M^{m\times n},$$ where $c,c_0>0$. Let $Ø$ be a given regular bounded open domain of ${ℝ}^n$. We define on $W^{1,p}(Ø;{ℝ}^m)$ the functional $$I\left(u\right)={\int }_{Ø}f\left(\nabla u\left(x\right)\right)dx.$$ Then, under some technical restrictions on $f$, we show that the relaxed functional $\overline{I}$ for the weak topology of $W^{1,p}(Ø;{ℝ}^m)$ has the integral representation: $$\overline{I}\left(u\right)={\int }_{Ø}Q\left[Rf\right]\left(\nabla u\left(x\right)\right)dx,$$ where for a given function $g$, $Qg$ denotes...