We discuss metrics with holonomy G2 by presenting a few crucial examples and review a series of G2 manifolds constructed via solvable Lie groups, obtained in [15]. These carry two related distinguished metrics, one negative Einstein and the other in the conformal class of a Ricci-flat metric, plus other features considered definitely worth investigating.

Two toy Hamiltonians of anharmonic oscillators are considered. Both of them are found exactly solvable not only in the limit of infinite spatial dimension D but also in the next D ≫ 1 approximation. This makes all their bound states tractable via perturbation series in 1/ √ D ≪ 1.

We show how, given a non-Hermitian Hamiltonian H, we can generate new operators sequentially, producing virtually infinite chain of Hamiltonians which are isospectral to H and H† whose eigenvectors easily deduce in an almost automatic way; no ingredients necessary other than its eigensystem. To set off the keep it running, use, for first time our knowledge, series maps all connected different m...

‎let $s$ be a subset of a finite group $g$‎. ‎the bi-cayley graph ${rm bcay}(g,s)$ of $g$ with respect to $s$ is an undirected graph with vertex set $gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid xin g‎, ‎ sin s}$‎. ‎a bi-cayley graph ${rm bcay}(g,s)$ is called a bci-graph if for any bi-cayley graph ${rm bcay}(g,t)$‎, ‎whenever ${rm bcay}(g,s)cong {rm bcay}(g,t)$ we have $t=gs^alpha$ for some $...

‎let $g$ be a finite group‎. ‎we say that the derived covering number of $g$ is finite if and only if there exists a positive integer $n$ such that $c^n=g'$ for all non-central conjugacy classes $c$ of $g$‎. ‎in this paper we characterize solvable groups $g$ in which the derived covering number is finite‎.‎

The study of the convergence of power series expansions of energy eigenvalues for anharmonic oscillators in quantum mechanics differs from general understanding, in the case of quasi-exactly solvable potentials. They provide examples of expansions with finite radius and suggest techniques useful to analyze more generic potentials.

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence latt...

Time series motifs are repeated similar subseries in one or multiple time series data. Time series anomalies are unusual subseries in one or multiple time series data. Finding motifs and anomalies in time series data are closely related problems and are useful in many domains, including medicine, motion capture, meteorology, and finance. This work presents a novel approach for both the motif di...