the wiener polarity index wp(g) of a molecular graph g of order n is the number ofunordered pairs of vertices u, v of g such that the distance d(u,v) between u and v is 3. in anearlier paper, some extremal properties of this graph invariant in the class of catacondensedhexagonal systems and fullerene graphs were investigated. in this paper, some new bounds forthis graph invariant are presented....

‎We consider the oscillator group equipped with‎ ‎a biinvariant Lorentzian metric‎. ‎Some geometrical properties of this space and the harmonicity properties of left-invariant vector fields on this space are determined‎. ‎In some cases‎, ‎all these vector fields are critical points for the energy functional‎ ‎restricted to vector fields‎. ‎Left-invariant vector fields defining harmonic maps are...

for a simple connected graph $g$ with $n$-vertices having laplacian eigenvalues‎ ‎$mu_1$‎, ‎$mu_2$‎, ‎$dots$‎, ‎$mu_{n-1}$‎, ‎$mu_n=0$‎, ‎and signless laplacian eigenvalues $q_1‎, ‎q_2,dots‎, ‎q_n$‎, ‎the laplacian-energy-like invariant($lel$) and the incidence energy ($ie$) of a graph $g$ are respectively defined as $lel(g)=sum_{i=1}^{n-1}sqrt{mu_i}$ and $ie(g)=sum_{i=1}^{n}sqrt{q_i}$‎. ‎in th...

We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...

This article presents a fairly complete account on various topological and metrical aspects of feedback stabilization for single-input single-output (SISO) continuous discrete time linear time-invariant (LTI) systems. In particular, we prove that the set stabilizing output gains SISO system with n states have at most [n/2] connected components. Furthermore, our analysis yields an algorithm dete...

We introduce a new symmetry, light-cone reflection (LCR), which interchanges timelike and spacelike intervals. Our motivation is to provide reason, based on why tachyons might exist, with emphasis application neutrinos. show that LCR, combined translations, leads much larger symmetry. construct an LCR-invariant Lagrangian discuss some of its properties. In simple example, we find complete symme...

We study the statistical properties of soft gamma repeater SGR 1935+2154. find that cumulative distributions duration, waiting time, fluence and flux can be well fitted by bent power law. In addition, probability density functions fluctuations follow Tsallis $q$-Gaussian distribution. The $q$ values keep steady for different temporal scale intervals, indicating a scale-invariant structure burst...

We equip a knot K with set of colored bonds, that is, intervals properly embedded into . Such construction can be viewed as structure topologically models closed protein chain including any type bridges connecting the backbone residues. introduce an invariant such bonded knots respects HOMFLYPT relation, namely, skein module knots. show rigid version is freely generated by ?-curves and handcuff...

The study of the partition function in M-theory involves the use of index theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed as a boundary, this is given by secondary index invariants such as the Atiyah–Patodi–Singer eta-invariant, the Chern–Simons invariant, or the Adams e-invariant. If the eleven-dimensional manifold itself has a boundary, the resulting ten-dimensi...