Pseudo-Hermitian (so called PT symmetric) Hamiltonians are featured within a re-formulated Witten's supersymmetric quantum mechanics. An unusual form of the supersymmetric partnership between the spiked harmonic oscillators is described. In the review [1] of the Witten's (or so called supersymmetric) quantum mechanics one finds the specific Fock vacuum q|0 = exp(−q 2 /2)/ √ π 0 , q ∈ (−∞, ∞). (...

max(u,v)∈E |f(u)− f(v)| if p =∞. If G is connected, then the only functions f satisfying ||f ||E,p = 0 are constant functions, so || · ||E,p is a norm on each linear space of functions on VG which does not contain constants. Usually we shall consider the subspace in the space of all functions on VG given by ∑ v∈V f(v)dv = 0. The obtained normed space will be called a Sobolev space on G and will...

Throughout this paper the normal n denotes the interior normal direction; i.e., n = ( f (x1); 1)= p 1 + jf (x1)j when x approaches from above (denoted by ) and n = (f (x1); 1)= p 1 + jf (x1)j from below (denoted by ). In a recent paper [2] the present authors studied the asymptotic behavior of any solution of (1.2){(1.4) in a neighborhood of the tip X(t). Taking for simplicity t = 0 and assumin...

• Positivity: N(v) ≥ 0 with equality if and only if v = 0. • Positive Homogeneity: N(αv) = |α|N(v). • Triangle Inequality: N(x1 + x2) ≤ N(x1) +N(x2). If N is a norm for V then we call ρ N (x1, x2) := N(x1−x2) the associated distance function (or metric) for V . A vector space V together with some a choice of norm is called a normed space, and the norm is usually denoted by ‖ ‖. If V is complete...

The bulk structure of the ?-Al5Fe2 intermetallic compound consists in an ordered framework pentagonal antiprims with composition Al2Fe inside which nearly continuous chains fractionally occupied Al sites exist. At low temperature, ordering these channel atoms can occur, leading to various superstructures. Although Al5Fe2 phase is technological importance being present protective coating for ste...

In this paper, vectors are column vectors. Unless otherwise specified, the norm of a matrix ‖M‖ is the Frobenius norm, defined by ‖M‖ = √∑ i,jM 2 ij . The all ones vector is denoted 1 and the all zeros vector is denoted 0. The all ones matrix is denoted J and the all zeros matrix is denoted O. The n× n identity matrix is denoted In, or simply I if its size is understood from context. The vector...

We produce infinite families of knots $\{K^i\}_{i\geq 1}$ for which the set cables $\{K^i_{p,1}\}_{i,p\geq is linearly independent in knot concordance group. arrange that these examples lie arbitrarily deep solvable and bipolar filtrations group, denoted by $\{F_n\}$ $\{B_n\}$ respectively. As a consequence, this result cannot be reached any combination algebraic invariants, Casson-Gordon Heega...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

Let G be a simple graph with order n and size m. The quantity $$M_1(G)=\sum _{i=1}^{n}d^2_{v_i}$$ is called the first Zagreb index of G, where $$d_{v_i}$$ degree vertex $$v_i$$ , for all $$i=1,2,\dots ,n$$ . signless Laplacian matrix $$Q(G)=D(G)+A(G)$$ A(G) D(G) denote, respectively, adjacency diagonal degrees G. $$q_1\ge q_2\ge \dots \ge q_n\ge 0$$ eigenvalues largest eigenvalue $$q_1$$ spectr...