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We consider collision of two particles 1 and 2 near the horizon of the extremal Reissner–Nordström (RN) black hole that produces two other particles 3 and 4. There exists such a scenario that both new particles fall into a black hole. One of them emerges from the white hole horizon in the asymptotically flat region, and the other one oscillates between turning points. However, the unbounded energies E at infinity (super-Penrose process (SPP)) turn out to be impossible for any finite angular momenta L 3 , 4 . In this sense, the situation for such a white hole scenarios is opposite to the black hole ones, where the SPP is found earlier to be possible for the RN metric even for all L i = 0. However, if L 3 , 4 themselves are unbounded, the SPP does exist for white holes. [ABSTRACT FROM AUTHOR]

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

A Roman dominating function (RDF) on a graph is a function satisfying the condition that every vertex with is adjacent to at least one vertex of for which . The weight of a RDF is the sum , and the minimum weight of a RDF is the Roman domination number . A subset of is a -independent set of if every vertex of has at most one neighbor in The maximum cardinality of a -independent set of is the -independence number Both parameters are incomparable in general, however, we show that if is a tree, then . Moreover, all extremal trees attaining equality are characterized. [ABSTRACT FROM AUTHOR]

We discuss critical and subcritical general maximizing problems associated with a Trudinger-Moser type inequality for a class of weighted Sobolev spaces. The key of our approach relies on a concentration-compactness type result and sharp estimates for the concentration level of a Moser-type functional. In particular, we give an explicit concentrating sequence that is maximizing for the maximal concentration level. As an application, we prove the existence of a nontrivial solution for a family of quasilinear elliptic equations with critical growth. [ABSTRACT FROM AUTHOR]

Let be a smooth compact Riemannian manifold of dimension and let to be a closed submanifold of dimension . In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on within the framework of Brezis-Marcus-Shafrir [Extremal functions for Hardy's inequality with weight, J. Funct. Anal. 171 (2000) 177-191]. In particular, we provide necessary and sufficient conditions for existence of minimizers. [ABSTRACT FROM AUTHOR]

In this paper, we give a result on the properness of the -energy, which answers a question of Song-Weinkove [On the convergence and singularities of the -flow with applications to the Mabuchi energy, Comm. Pure Appl. Math. 61(2) (2008) 210-229] in any dimensions. Moreover, we extend our previous result on the properness of -energy in [H. Li, Y. Shi and Y. Yao, A criterion for the properness of the -energy in a general Kähler class, Math. Ann. 361(1-2) (2015) 135-156] to the case of extremal -energy associated to extremal Kähler metrics. [ABSTRACT FROM AUTHOR]

In this paper, we establish a weighted Hardy-Littlewood-Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev-Hardy trace inequality with -biharmonic operator. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Trudinger-Moser inequality in the Sobolev space . Setting we prove: for for , and there exist extremal functions for if . Blow-up analysis, elliptic estimates and a version of compactness result due to Lions are used to prove (1) and (3). The proof of (2) is based on computations of testing functions which are a combination of eigenfunctions with the Moser sequence. [ABSTRACT FROM AUTHOR]

Let K be a field and let S=K[x1,...,xn] be a polynomial ring over K. Let be a finitely generated graded free S-module with basis {e1,...,er} in degrees f1,...,fr such that f1 ≤ f2 ≤ ⋯ ≤ fr. We examine some classes of squarefree monomial submodules of F. Hence, we focalize our attention on the Betti table of such classes in order to analyze the behavior of their extremal Betti numbers. [ABSTRACT FROM AUTHOR]

For a cuspidal newform of weight and a prime of , the deformation problem for its associated mod Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the -adic deformation problem for is unobstructed, then is not congruent mod to a newform of lower level. [ABSTRACT FROM AUTHOR]

Consider the Liouville-Gelfand type problems with nonlinear Neumann boundary conditions where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem. [ABSTRACT FROM AUTHOR]

A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in Hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints. [ABSTRACT FROM AUTHOR]

We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+n, characterized by the fact that they admit a C1 parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class Ck, then the surface is regularly immersed away from a closed singular set of Euclidean Hausdorff dimension at most 1 + 1/k, and that this bound is sharp. We also show that, generically with respect to a natural topology, the singular set of a timelike weakly extremal cylinder in ℝ1+n is one-dimensional if n = 2, and it is empty if n ≥ 4. For n = 3, timelike weakly extremal surfaces exhibit an intermediate behavior. [ABSTRACT FROM AUTHOR]

In this paper, we give the expression of the least square solution of the linear quaternion matrix equation AXB = C subject to a consistent system of quaternion matrix equations D1X = F1, X E2 = F2, and derive the maximal and minimal ranks and the least- norm of the above mentioned solution. The finding of this paper extends some known results in the literature. [ABSTRACT FROM AUTHOR]

Employing higher-order perturbation theory, we find a new class of perturbative extremal rotating black hole solutions with Born-Infeld electric charge in odd D dimensional spacetime. The seed solution is an odd-dimensional extremal Myers-Perry black hole with equal angular momenta to which a perturbative, nonlinear, electric Born-Infeld field charge q is added maintaining the extremality condition. The perturbations are performed up to third-order. We also study some physical properties of these black holes. In particular, it is shown that the values of the gyromagnetic ratio of the black holes are modified by the perturbative parameter q and the Born-Infeld parameter β. [ABSTRACT FROM AUTHOR]

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are determined using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with φ(m) > 24. In particular the lattice P48n is the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field. [ABSTRACT FROM AUTHOR]

We prove that any neighborhood N of a positive twist knot , m ≥ 3 and odd with extremal Euler number can be thickened to a standard neighborhood of a maximal Thurston-Bennequin number Legendrian representative of . [ABSTRACT FROM AUTHOR]

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,...,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ {2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ {2Δ(G), 19} if G is a graph such that mad(G) ≤ 5. [ABSTRACT FROM AUTHOR]

We single out every longest path of n-1 order that solves each of the Longest Euclidean Hamiltonian Path Problems on the even nth root of the unity, by means of a geometric and arithmetic procedure. This identification is done regardless of planar rotations and orientation. In addition, the uniqueness of the Euclidean Hamiltonian cycle that resolves the Maximum Traveling Salesman Problem is shown. [ABSTRACT FROM AUTHOR]

Let Ω be a smooth bounded domain in ℝN with N ≥ 1. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of Ω. In particular we provide sufficient and necessary conditions on the existence of minimizers. [ABSTRACT FROM AUTHOR]

We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entangled state can be detected by one of these witnesses, and this witness detects a unique set of entangled states among those. Therefore, they provide a minimal set of witnesses to detect any kind of entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entangled states with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one-parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps in three-dimensional matrix algebra. [ABSTRACT FROM AUTHOR]

We consider Kerr-Newman-AdS-dS black holes near extremality and work out the near-horizon geometry of these near-extremal black holes. We identify the exact U(1)L×U(1)R isometries of the near-horizon geometry and provide boundary conditions enhancing them to a pair of commuting Virasoro algebras. The conserved charges of the corresponding asymptotic symmetries are found to be well-defined and nonvanishing and to yield central charges cL≠0 and cR = 0. The Cardy formula subsequently reproduces the Bekenstein-Hawking entropy of the black hole. This suggests that the near-extremal Kerr-Newman-AdS-dS black hole is holographically dual to a non-chiral two-dimensional conformal field theory. [ABSTRACT FROM AUTHOR]

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kähler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the Hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-Kähler metrics saturating LeBrun's estimates. [ABSTRACT FROM AUTHOR]

We will study the dependence of λ(a, b), half-eigenvalues of the one-dimensional p-Laplacian, on potentials $(a, b)\in ({\mathcal L}^\gamma, {\mathcal L}^\gamma)$, 1 ≤ γ ≤ ∞, where ${\mathcal L}^\gamma = L^\gamma([0, 1], {\mathbb R})$. Two results are obtained. One is the continuity of half-eigenvalues in $(a, b)\in({\mathcal L}^\gamma, w_\gamma)^2$, where wγ is the weak topology in ${\mathcal L}^\gamma$ space. The other is the continuous differentiability of half-eigenvalues in $(a, b)\in ({\mathcal L}^\gamma, \|\cdot\|_\gamma)^2$, where | ⋅ |γ is the Lγ norm of ${\mathcal L}^\gamma$. These results will be used to study extremal problems of half-eigenvalues in future work. [ABSTRACT FROM AUTHOR]

A geometric method is described to characterize the different kinds of extremals in optimal control theory. This comes from the use of a presymplectic constraint algorithm starting from the necessary conditions given by Pontryagin's Maximum Principle. The algorithm must be run twice so as to obtain suitable sets that once projected must be compared. Apart from the design of this general algorithm useful for any optimal control problem, it is shown how to classify the set of extremals and, in particular, how to characterize the strict abnormality. An example of strict abnormal extremal for a particular control-affine system is also given. [ABSTRACT FROM AUTHOR]

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included. [ABSTRACT FROM AUTHOR]

This review is an elementary introduction to the concepts of the recently developed asymptotic methods and new developments. Particular attention is paid throughout the paper to giving an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, variational iteration method, parameter-expansion method, exp-function method, homotopy perturbation method, and ancient Chinese mathematics as well. Subsequently, nanomechanics in textile engineering and E-infinity theory in high energy physics, Kleiber's 3/4 law in biology, possible mechanism in spider-spinning process and fractal approach to carbon nanotube are briefly introduced. Bubble-electrospinning for mass production of nanofibers is illustrated. There are in total more than 280 references. [ABSTRACT FROM AUTHOR]

We consider the problem of regularity of the extremal positive solution of the Dirichlet problem for superlinear elliptic equations with a strong non-linearity and a positive parameter. We prove the smoothness of the extremal solution for some classes of nonlinearities. These classes are generalizations of the model examples of the exponential and of the power functions. We pay a special attention to cases where the regularity of the extremal solution holds for the same dimensions as for the model cases. [ABSTRACT FROM AUTHOR]

In this paper we describe a genetic algorithm (GA) for the uncapacitated multiple allocation p-hub center problem (UMApHCP). Binary coding is used and genetic operators adapted to the problem are constructed and implemented in our GA. Computational results are presented for the standard hub instances from the literature. It can be seen that proposed GA approach reaches all solutions that are proved to be optimal so far. The solutions are obtained in a reasonable amount of computational time, even for problem instances of higher dimensions. [ABSTRACT FROM AUTHOR]

We classify the cones of curves of Fano varieties of dimension greater or equal than five and (pseudo)index dim X - 3, describing the number and type of their extremal rays. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded smooth domain in ℝ2, and λ1(Ω) the first eigenvalue of the Laplacian with Dirichlet boundary condition in Ω. Then Adimurthi and Druet show that for any 0 ≤ α < λ1(Ω) \[ \sup_{u\in H_0^1(\Omega),\,\|\nabla u\|_2=1}\int_\Omega e^{4\pi(1+\alpha\|u\|_2^2)u^2}dx<+\infty. \] We prove in this paper that there exist extremal functions for the above inequality. In other words, we show that \[ \sup_{u\in H_0^1(\Omega),\,\|\nabla u\|_2=1}\int_\Omega e^{4\pi(1+\alpha\|u\|_2^2)u^2}dx \] is attained for any 0 ≤ α < λ1(Ω). [ABSTRACT FROM AUTHOR]

Let (M,g) be a 2-dimensional compact Riemannian manifold with boundary. In this paper, we use the method of blowing up analysis to prove the existence of the extremal functions for some Moser–Trudinger inequalities on (M,g). [ABSTRACT FROM AUTHOR]

We show that a certain natural class of tangles 'generate the collection of all tangles with respect to composition'. This result is motivated by, and describes the reasoning behind, the 'uniqueness assertion' in Jones' theorem on the equivalence between extremal subfactors of finite index and what we call 'subfactor planar algebras' here. This result is also used to identify the manner in which the planar algebras corresponding to M⊂M1 and Nop⊂Mop are obtained from that of N⊂M. Our results also show that 'duality' in the category of extremal subfactors of finite index extends naturally to the category of 'general' planar algebras (not necessarily finite-dimensional or spherical or connected or C*, in the terminology of Jones). [ABSTRACT FROM AUTHOR]

For the L[sup p] (p > 1) version of the Caffarelli-Kohn-Nirenberg inequalities we prove a symmetry breaking result for the extremal functions. [ABSTRACT FROM AUTHOR]

Deals with a singular extremal rays with a G[sub m] X SL action. House models; Evidence of an arbitrary large second Betti number in the minimal projective 3-fold house models; Criterion for projectivity.