Abstract Using the $H^{\infty }$ -functional calculus for quaternionic operators, we show how to generate fractional powers of some densely defined differential operators order $m\geq 1$ , acting on right linear Hilbert space $L^{2}(\Omega,\mathbb {C}\otimes \mathbb {H})$ . The that consider are type \begin{align*} T=i^{m-1}\left(a_1(x) e_1\partial_{x_1}^{m}+a_2(x) e_2\partial_{x_2}^{m}+a_3(x) ...

We study the problem of constructing explicit families matrices which cannot be expressed as a product few sparse matrices. In addition to being natural mathematical question on its own, this appears in various incarnations computer science; most significant context lower bounds for algebraic circuits compute linear transformations, matrix rigidity and data structure bounds. first show, every c...

We consider a spin-balanced degenerate gas of spin-1/2 fermions whose dynamics is governed by low-energy $P$-wave interactions, characterized the scattering volume $a_1$ and effective momentum $r_1$. The energy per particle $\bar{\cal{E}}$ in many-body system calculated resumming ladder diagrams comprising both particle-particle hole-hole intermediate states, following novel advances recently d...

Exponential stability of the second order linear delay differential equation in $x$ and $u$-control $$ \ddot{x}(t)+a_1(t)\dot{x}(h_1(t))+a_2(t)x(h_2(t))+a_3(t)u(h_3(t))=0 is studied, where indirect feedback control $\dot{u}(t)+b_1(t)u(g_1(t))+b_2(t)x(g_2(t))=0$ connects $u$ with solution. Explicit sufficient conditions guarantee that both decay exponentially.

The aim of this paper is to analyze the asymptotic behavior eigenvalues and eigenvectors particular sequences products involving two square real matrices $A$ $B$, namely form $B^kA$, as $k\rightarrow \infty$. This analysis represents a detailed deepening case within general theory on finite families $\mathcal{F} = \{ A_1, \ldots, A_m \}$ already available in literature. Bachmann-Landau symbols ...

In the following: x denotes any real number greater than 1 ; y, c, t denote real positive numbers; ¿ = log (x); d, n, m, v, A denote integers satisfying w^O, d>0, m>0, 0<v^k, (v,k) = l; Aim, v, k) denotes the set of integers miv + nk) for varying n; fim, v, A, x, c) denotes the number of integers in A im, v, k) less than or equal to x and prime to primes greater than x°; y¡/(x, k) denotes the n...