A Theorem of Lyapunov states that the range R(μ) of a non–atomic vector measure μ is compact and convex. In this paper we give a condition to detect the dimension of the extremal faces of R(μ) in terms of the Radon–Nikodym derivative of μ with respect to its total variation |μ|: namely R(μ) has an extremal face of dimension less or equal to k if and only if the set (x1, . . . , xk+1) such that ...

the present note is devoted to establish some extremal results for the zeroth-order general randi'{c} index of cacti, characterize the extremal polyomino chains with respect to the aforementioned index, and hence to generalize two already reported results.

The present note is devoted to establish some extremal results for the zeroth-order general Randi'{c} index of cacti, characterize the extremal polyomino chains with respect to the aforementioned index, and hence to generalize two already reported results.

We consider the sub-Riemannian length optimization problem on the group of motions of hyperbolic plane i.e. the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2 dimensional linear control input and energy cost functional. We prove the global controllability of control distribution and use Pontryagin Maximum Principle to obtain the extremal control inpu...

We construct and study the electrically charged, rotating black hole solution in heterotic string theory compactified on a (10 − D) dimensional torus. This black hole is characterized by its mass, angular momentum, and a (36 − 2D) dimensional electric charge vector. One of the novel features of this solution is that for D > 5, its extremal limit saturates the Bogomol’nyi bound. This is in contr...

In this paper we analyse the optimality of broken Pontryagin extremal for an n -dimensional affine control system with a control parameter, taking values in a k dimensional closed ball. We prove the optimality of broken normal extremals when n = 3 and the controllable vector fields form a contact distribution, and when the Lie algebra of the controllable fields is locally orthogonal to the sing...

A bstract Motivated by the connection between pole-skipping phenomena of two point functions and four out-of-time-order correlators, we study for rotating BTZ black holes. In particular, investigate effect rotations on various fields with spin s = 1 / 2 , 3, extending previous research 0 2. We derive an analytic full tower points fermionic ( 2) vector 1) exact holographic Green’s functions. For...

for a bounded linear operator on hilbert space we define a sequence of the so-called weakly extremal vectors‎. ‎we study the properties of weakly extremal vectors and show that the orthogonality equation is valid for weakly extremal vectors‎. ‎also we show that any quasinilpotent operator $t$ has an hypernoncyclic vector‎, ‎and so $t$ has a nontrivial hyperinvariant subspace‎.

We present a consolidated manual of Euclidean gravity approaches for finding the logarithmic corrections to entropy full Kerr-Newman family black holes in both extremal and non-extremal limits. Seeley-DeWitt coefficients quadratic fluctuations concern theory appear be key ingredients this manual. Following manual, we calculate first three generalized Einstein-Maxwell minimally-coupled additiona...

The study of extremal Kähler metric is initiated by the seminar work of Calabi [4], [5]. Let (M, [ω]) be a compact Kähler manifold with fixed Kähler class [ω]. The extremal Kähler metric is the critical point of the Calabi energy C(g) for any Kähler metrics g in the fixed Kähler class [ω], C(g) = M s 2 dµ, where s is the scalar curvature of g. The extremal condition asserts that ¯ ∂∇s = 0. In o...