In this note, we establish $L^p$-bounds for the semigroup $e^{-tq^w(x,D)}$, $t \ge 0$, generated by a quadratic differential operator $q^w(x,D)$ on $\mathbb{R}^n$ that is Weyl quantization of complex-valued form $q$ defined phase space $\mathbb{R}^{2n}$ with non-negative real part $\operatorname{Re} q 0$ and trivial singular space. Specifically, show $e^{-tq^w(x,D)}$ bounded from $L^p(\mathbb{R...

From an abstract algebraic perspective, an explanation for this can be given as follows: since √ D is irrational, the polynomial t −D is irreducible over Q. Since the ring Q[t] is a PID, the irreducible element t − D generates a maximal ideal (t−D), so that the quotient Q[t]/(t−D) is a field. Moreover, the map Q[ √ D]→ Q[t]/(t −D) which is the identity on Q and sends √ D 7→ t is an isomorphism ...

Motivated from studies on anomalous diffusion, we show that the memory function $M(t)$ of complex materials, their creep compliance follows a power law, $J(t)\sim t^q$ with $q\in \mathbb{R}^+$, is fractional derivative Dirac delta function, $\frac{\mathrm{d}^q\delta(t-0)}{\mathrm{d}t^q}$ \mathbb{R}^+$. This leads to finding inverse Laplace transform $s^q$ for any \mathbb{R}^+$ $\frac{\mathrm{d}...

In this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation$$int_{0}^{1}b(q)D^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$using a fixed point theorem in the Banach algebras. This proof is given in two cases of the continuous and discontinuous function $g$, under the generalized Lipschitz and Caratheodory conditions.

Enumeration of derangements in the symmetric group $\mathfrak{S}_n$ is classical. Extensions enumerative results to hyperoctahedral $B_n$ are combinatorially sound. That even-signed permutation $D_n$ remains largely unexplored. Let $d_n^D(q) = \sum_{\sigma \in \mathcal{D}_n^D} q^{\operatorname{maj}(\sigma)}$ be generating function by their major indices. We study this work properties $d_n^D(q)$...