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Let C ℓ be the cycle of order ℓ. The square of C ℓ , denoted by C ℓ 2 , is obtained by joining all pairs of vertices with distance no more than two in C ℓ. Denote by ex (n , F) and spex (n , F) the maximum size and maximum spectral radius over all n -vertex F -free graphs, respectively. The well-known Turán problem asks for ex (n , F) , and Nikiforov in 2010 proposed a spectral counterpart, known as the Brualdi-Solheid-Turán type problem, focusing on determining spex (n , F). In this paper, for any integer ℓ ≥ 6 that is not divisible by 3, we characterize the unique extremal graph with respect to ex (n , C ℓ 2) and spex (n , C ℓ 2) for sufficiently large n , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of a class of casual differential equations on a time scale. An existence theorem for casual differential equations on a time scale is given under mixed Lipschitz and compactness conditions by the fixed point theorem in Banach algebra due to Dhage. Some fundamental differential inequalities on a time scale are also presented which are utilized to investigate the existence of extremal solutions. The comparison principle on casual differential equations on a time scale is established. Our results in this paper extend and improve some well-known results. [ABSTRACT FROM AUTHOR]

For a set L of positive integers, a set system F ⊆ 2 [ n ] is said to be L-close Sperner, if for any pair F, G of distinct sets in F the skew distance s d (F , G) = min { | F \ G | , | G \ F | } belongs to L. We reprove an extremal result of Boros, Gurvich, and Milanič on the maximum size of L-close Sperner set systems for L = { 1 } , generalize it to | L | = 1 , and obtain slightly weaker bounds for arbitrary L. We also consider the problem when L might include 0 and reprove a theorem of Frankl, Füredi, and Pach on the size of largest set systems with all skew distances belonging to L = { 0 , 1 } . [ABSTRACT FROM AUTHOR]

We study aspects of entanglement and extremal surfaces in various families of spacetimes exhibiting cosmological, Big-Crunch, singularities, in particular isotropic AdS Kasner. The classical extremal surface dips into the bulk radial and time directions. Explicitly analysing the extremization equations in the semiclassical region far from the singularity, we find the surface bends in the direction away from the singularity. In the 2-dim cosmologies obtained by dimensional reduction of these and other singularities, we have studied quantum extremal surfaces by extremizing the generalized entropy. The resulting extremization shows the quantum extremal surfaces to always be driven to the semiclassical region far from the singularity. We give some comments and speculations on our analysis. [ABSTRACT FROM AUTHOR]

In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator M −. [ABSTRACT FROM AUTHOR]

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]

Non-singular Kruskal-like coordinates of some black holes space-times in f (R) gravity are presented in this research paper, and are also removed by establishing Kruskal-Szekeres coordinates for Non-extremal case. Carter-like coordinates can also be built for its extreme case. [ABSTRACT FROM AUTHOR]

Consider the semilinear elliptic equation − Δ u = λ f (u) in the unit ball B 1 ⊂ R N , with Dirichlet data u | ∂ B 1 = 0 , where λ ≥ 0 is a real parameter and f is a C 1 positive, nondecreasing and convex function in [ 0 , ∞) such that f (s) / s → ∞ as s → ∞. In this paper we study the behavior of f ′ (u ⁎) near the origin when u ⁎ , the extremal solution of the previous problem associated to λ = λ ⁎ , is singular. This answers to an open problems posed by Brezis and Vázquez [2, Open problem 5]. [ABSTRACT FROM AUTHOR]

In the paper, maximal values x(t) of the solutions x(t) of the linear fifferential equations excited by the Dirac delta function are determined. The analytical solutions of the equations and also the maximal positive values of these solutions are obtained. The analytical formulae enable the design of the system with prescribed properties. The complementary case to the earlier paper is presented. In am earlier paper it was assumed that the roots si are different, and no we consider the case when s1 = s2 = ... =sn. [ABSTRACT FROM AUTHOR]

Fractional Langevin equation describes the evolution of physical phenomena in fluctuating environments for the complex media systems. It is a sequential fractional differential equation with two fractional orders involving a memory kernel, which leads to non-Markovian dynamics and subdiffusion. Here by establishing a general solution of the linear fractional Langevin equations involving initial conditions with the help of well-known Mittag–Leffler functions and using the special properties of these functions, we construct a new comparison result related to linear fractional Langevin equation. Meanwhile, we investigate the existence of extremal solutions for nonlinear boundary value problems with advanced arguments. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. At last an example is presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

We prove convergence of the full extremal process of the two-dimensional scale-inhomogeneous discrete Gaussian free field in the weak correlation regime. The scale-inhomogeneous discrete Gaussian free field is obtained from the 2d discrete Gaussian free field by modifying the variance through a function I:[0,1]→[0,1]. The limiting process is a cluster Cox process. The random intensity of the Cox process depends on the I'(0) through a random measure Y and on the I'(1) through a constant β. We describe the cluster process, which only depends on I'(1), as points of a standard 2d discrete Gaussian free field conditioned to be unusually high. [ABSTRACT FROM AUTHOR]

We consider the nonlinear eigenvalue problem L u = λ f (u) , posed in a smooth bounded domain Ω ⊆ R N with Dirichlet boundary condition, where L is a uniformly elliptic second-order linear differential operator, λ > 0 and f : [ 0 , a f) → R + (0 < a f ⩽ ∞) is a smooth, increasing and convex nonlinearity such that f (0) > 0 and which blows up at a f . First we present some upper and lower bounds for the extremal parameter λ ∗ and the extremal solution u ∗ . Then we apply the results to the operator L A = − Δ + A c (x) with A > 0 and c (x) is a divergence-free flow in Ω. We show that, if ψ A , Ω is the maximum of the solution ψ A (x) of the equation L A u = 1 in Ω with Dirichlet boundary condition, then for any incompressible flow c (x) we have, ψ A , Ω ⟶ 0 as A ⟶ ∞ if and only if c (x) has no non-zero first integrals in H 0 1 (Ω). Also, taking c (x) = − x ρ (| x |) where ρ is a smooth real function on [ 0 , 1 ] then c (x) is never divergence-free in unit ball B ⊂ R N , but our results completely determine the behaviour of the extremal parameter λ A ∗ as A ⟶ ∞. [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]

We study the uniqueness of optimal solutions to extremal graph theory problems. Lov'asz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph. This statement is often referred to as saying that 'every extremal graph theory problem has a finitely forcible optimum'. We present a counterexample to the conjecture. Our techniques also extend to a more general setting involving other types of constraints. [ABSTRACT FROM AUTHOR]

This paper provides an overview of the latest research on the two-sided estimates of classical characteristics of growth of entire functions such as the type and the lower type in terms of the ordinary or average densities of the distribution of zeros. We give also accurate estimates of the type of an entire function, taking into account additionally the step and the lacunarity index of the sequence of zeros. The results under consideration are based on the solution of extremal problems in classes of entire functions with restrictions on the behavior of the zero set. Particular attention is paid to the following important cases of the location of zeros: on a ray, on a straight line, on a number of rays, in the angle, or arbitrarily in the complex plane. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of extremal solutions for periodic boundary value problems for conformable fractional differential equations with deviating arguments. We first build two comparison principles for the corresponding linear equation with deviating arguments. With the help of new comparison principles, some sufficient conditions for the existence of extremal solutions are established by combining the method of lower and upper solutions and the monotone iterative technique. As an application, an example is presented to enrich the main results of this article. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of the maximum and minimum iterative solutions for a tempered fractional turbulent flow model in a porous medium with nonlocal boundary conditions. By introducing a new growth condition and developing an iterative technique, we establish new results on the existence of the maximum and minimum solutions for the considered equation; at the same time, the iterative sequences for approximating the extremal solutions are performed, and the asymptotic estimates of solutions are also derived. [ABSTRACT FROM AUTHOR]

The total number of independent subsets, including the empty set, of a graph, is also termed as the Merrifield–Simmons index (MSI) in mathematical chemistry. The connective eccentricity index (CEI), the multiplicative Wiener index (MWI), and the Wiener polarity index (WPI) are all distance-based topological indices. Some of these topological indices found applications in chemistry. The eccentric complexity of a graph G is defined to be the number of different eccentricities of its vertices. In this paper, we first investigate the relationship between MSI and three other distance-based topological indices (CEI, MWI, WPI). We prove that MSI > WPI for any tree, MSI > CEI for any connected bipartite graph, and MWI > MSI for connected graphs with given restricted condition on size. Then we consider extremal problems of MSI on graphs with small eccentric complexity. Moreover, we prove an inequality relating MSI, independence number and eccentric complexity for some special graph families. Finally, some further problems and conjectures are proposed. [ABSTRACT FROM AUTHOR]

Phenomena treated by non-equilibrium thermodynamics can be very effectively described by thermodynamic variational principles. The remarkable advantage of such an approach consists in possibility to account for an arbitrary number of constraints among state or kinetic variables stemming, e.g., from conservation laws or balance equations. As shown in the current paper, the variational principles can provide original evolution equations for the state variables implicitly respecting the constraints. Moreover, the variational approach allows formulating the problem directly in discrete state variables and deriving their evolution equations without the necessity to solve partial differential equations. The variational approach makes it also possible to use different kinetic variables in formulation of dissipation and dissipation function. [ABSTRACT FROM AUTHOR]

Previously, Erdős, Kierstead and Trotter [5] investigated the dimension of random height 2 partially ordered sets. Their research was motivated primarily by two goals: (1) analyzing the relative tightness of the Füredi–Kahn upper bounds on dimension in terms of maximum degree; and (2) developing machinery for estimating the expected dimension of a random labeled poset on n points. For these reasons, most of their effort was focused on the case 0 < p ≤ 1 / 2 . While bounds were given for the range 1 / 2 ≤ p < 1 , the relative accuracy of the results in the original paper deteriorated as p approaches 1. Motivated by two extremal problems involving conditions that force a poset to contain a large standard example, we were compelled to revisit this subject, but now with primary emphasis on the range 1 / 2 ≤ p < 1 . Our sharpened analysis shows that as p approaches 1, the expected value of dimension increases and then decreases, answering in the negative a question posed in the original paper. Along the way, we apply inequalities of Talagrand and Janson, establish connections with latin rectangles and the Euler product function, and make progress on both extremal problems. [ABSTRACT FROM AUTHOR]

In the paper, maximal values xe( Ï„ ) of the solutions x(t) of the linear differential equations excited by the Dirac delta function are determined. There are obtained the analytical solutions of the equations and also the maximal positive values of these solutions. The obtained sufficient conditions of the positivity of these solutions are defined by the Theorems. There are also formulated the necessary conditions of the positivity of these solutions. The analytical formulae enable the design of the system with prescribed properties [3]. [ABSTRACT FROM AUTHOR]

We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional [ABSTRACT FROM AUTHOR]

A large number of studies have investigated the systems of fuzzy relation equations and inequalities, which have a much wider field of application. In this paper, we study several types of systems of fuzzy relation equations and inequalities consisting of a given family of k-ary fuzzy relations, where natural number k ≥ 2, and three unknown fuzzy relations over complete residuated lattices and meet-continuous lattices. Their solutions are triples of fuzzy relations. For the systems of fuzzy relation inequalities, we give the greatest solutions contained in a given triple of fuzzy relations, the least solutions containing a given triple of fuzzy relations, or give maximal solutions contained in or containing a given triple of fuzzy relations, or belonging to a given interval of triples of fuzzy relations over complete residuated lattices and complete meet-continuous lattices. For the systems of fuzzy relation equations, we present a method of computing maximal solutions contained in a given triple of fuzzy relations and a method of computing minimal solutions containing a given triple of fuzzy relations. Furthermore, we provide some conditions under which there exist the greatest solutions contained in and the least solutions containing a given triple of fuzzy relations. [ABSTRACT FROM AUTHOR]

In this note we prove that McShane and Whitney's Lipschitz extensions are viscosity solutions of Jensen's auxiliary equations which are known to have a key role in Jensen's celebrated proof of uniqueness of infinity harmonic functions, and therefore of absolutely minimizing Lipschitz extensions. To the best of the author's knowledge, this result does not appear to be known in the literature in spite of the vast amount of work on the topic. [ABSTRACT FROM AUTHOR]

A spectral version of extremal graph theory problem is as follows: what is the maximum spectral radius of an H -free graph of order n ? Nikiforov posed a conjecture on spectral extremal value of cycles. In the past decade, much attention has been paid to this conjecture and other questions on spectral extremal graphs. In order to approach to Nikiforov's conjecture, we provide a new conjecture on spectral extremal value involving fans and wheels. Moreover, we show it is true for k = 2. [ABSTRACT FROM AUTHOR]

We give the first correct O(n log n) algorithm for finding a maximum st-flow in a directed planar graph. After a preprocessing step that consists in finding single-source shortest-path distances in the dual, the algorithm consists of repeatedly saturating the leftmost residual s-to-t path. [ABSTRACT FROM AUTHOR]

Regularized learning problems in Banach spaces, which often minimize the sum of a data fidelity term in one Banach norm and a regularization term in another Banach norm, is challenging to solve. We construct a direct sum space based on the Banach spaces for the fidelity term and the regularization term, and recast the objective function as the norm of a quotient space of the direct sum space. We then express the original regularized problem as an optimization problem in the dual space of the direct sum space. It is to find the maximum of a linear function on a convex polytope, which may be solved by linear programming. A solution of the original problem is then obtained by using related extremal properties of norming functionals from a solution of the dual problem. Numerical experiments demonstrate that the proposed duality approach is effective for solving the regularization learning problems. [ABSTRACT FROM AUTHOR]

To extend the applicability of the linearized initial value representation (LIVR) method to lower temperatures and realistic potentials, a generalization to barriers other than the inverted parabola is proposed. The LIVR method calculates rate constants of chemical reactions involving quantum effects by weighting classical trajectories by the Wigner distribution function (WDF) of the Boltzmann-averaged flux operator. These calculations can be performed efficiently if the WDF is available in analytical form, which is the case for harmonic barriers only. The proposed generalization to anharmonic barriers is based on the recognition that above a critical temperature T[sup *] = &hslash;ω/πk[sub B], where ω is the curvature at the top of the barrier and k[sub B] is the Boltzmann constant, the WDF is dominated by an extremal trajectory. The evaluation of WDFs and thus of thermal rate constants is thereby reduced to the search for the extremal path defined by a steepest-descent condition for the WDF. This extremal trajectory is the high-temperature analogue of the instanton (bounce path), which exists for temperatures lower than T[sup *]/2. Explicit formulas are derived for the generation of realistic WDFs and barrier crossing rate constants for symmetric barriers of arbitrary shape. Approximations are introduced that will reduce the extra computational effort required for these anharmonic barriers. They are based on the fact that above the critical temperature the WDF of any anharmonic potential can be represented with good approximation in an analytical form analogous to that of the parabolic barrier by the introduction of effective parameters. Results obtained for two standard model potentials, the quartic potential and the symmetric Eckart barrier, are compared with the well-known parabolic barrier results. The formal and actual temperature limits for calculating tunneling rate constants and the extension of the method to asymmetric barriers are... [ABSTRACT FROM AUTHOR]

A central problem in extremal graph theory is to estimate, for a given graph H , the number of H -free graphs on a given set of n vertices. In the case when H is not bipartite, Erd̋s, Frankl, and Rödl proved that there are 2(1+ o (1))ex(n , H) such graphs. In the bipartite case, however, bounds of the form 2 O (ex(n , H)) have been proven only for relatively few special graphs H. As a 1st attempt at addressing this problem in full generality, we show that such a bound follows merely from a rather natural assumption on the growth rate of n ↦ ex(n , H); an analogous statement remains true when H is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erd̋s and Simonovits. The bounds on the number of H -free hypergraphs are derived from it using the method of hypergraph containers. [ABSTRACT FROM AUTHOR]

Although much research (f. e. [1], [3], [5], [7-15]) has been devoted to the extremal problems of a geometric function theory associated with estimates of functionals defined on systems of non-overlapping domains, however, in the general case the problems remain unsolved. The paper describes the problem of finding the maximum of a functional. This problem is to find a maximum of the product of inner radii of mutually non-overlapping symmetric domains with respect to a unit circle and the inner radius in some positive certain degree of the domain with respect to zero and description of extreme configurations. The topic of the paper is devoted to the study of the problem of the classical direction of the geometric theory of complex variable functions, namely, the extremal problems for non-overlapping domains. [ABSTRACT FROM AUTHOR]

For a graph F, the k-subdivision of F, denoted Fk, is the graph obtained by replacing the edges of F with internally vertex-disjoint paths of length k. In this paper, we prove that ex(n,Kk s,t) = O(n1+ s 1 sk), which is tight for t sufficiently large. This settles a conjecture of Conlon-- Janzer--Lee, and improves on a substantial body of work by Conlon--Janzer--Lee and Jiang--Qiu. [ABSTRACT FROM AUTHOR]

The paper concerns the properties of linear dynamical systems described by linear differential equations, excited by the Dirac delta function. A differential equation of the form anx(n)(t)+· · ·+a1x' (t)+a0x(t) = bmu(m)(t)+· · ·+b1u' (t)+ b0u(t) is considered with ai, bj > 0. In the paper we assume that the polynomials Mn(s) = ansn + · · · + a1s + a0 and Lm(s) = bmsm + · · · + b1s + b0 partly interlace. The solution of the above equation is denoted by x(t,Lm,Mn). It is proved that the function x(t,Lm,Mn) is nonnegative for t ∈ (0,∞), and does not have more than one local extremum in the interval (0,∞) (Theorems 1, 3 and 4). Besides, certain relationships are proved which occur between local extrema of the function x(t,Lm,Mn), depending on the degree of the polynomial Mn(s) or Lm(s) (Theorems 5 and 6). [ABSTRACT FROM AUTHOR]

In this paper, the extremal problems on the phenylene chains with respect to some graph invariants are studied. All the graphs minimizing (resp. maximizing) the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index among all the phenylene chains each of which contains n four-membered rings are identified. [ABSTRACT FROM AUTHOR]

Some extremal problems of the geometric theory of functions of a complex variable related to the estimates of functionals defined on systems of non-overlapping domains are considered. Till now, many such problems have not been solved, though some partial solutions are available. In the paper, the improved method is proposed for solving the problems on extremal decomposition of the complex plane. The main results generalize and strengthen some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane. [ABSTRACT FROM AUTHOR]

This paper studies the existence of extremal solutions for nonlinear fractional differential coupled systems with p-Laplacian operator. The monotone iterative method combined with lower and upper solutions is applied. As an application, an example is presented to illustrate the main result. [ABSTRACT FROM AUTHOR]

The abstract Cauchy problem for the fractional evolution equation with the Caputo derivative of order and operator , , is considered, where generates a strongly continuous one-parameter semigroup on a Banach space. Subordination formulae for the solution operator are derived, which are integral representations containing a subordination kernel (a scalar probability density function) and a -semigroup of operators. Some properties of the subordination kernel are established and representations in terms of Mainardi function and Lévy extremal stable densities are derived. Applications of the subordination formulae are given with a special focus on the multi-dimensional space-time fractional diffusion equation. [ABSTRACT FROM AUTHOR]

Abstract We study directed, weighted graphs G = (V , E) and consider the (not necessarily symmetric) averaging operator (L u) (i) = − ∑ j ∼ i p i j (u (j) − u (i)) , where p i j are normalized edge weights. Given a vertex i ∈ V , we define the diffusion distance to a set B ⊂ V as the smallest number of steps d B (i) ∈ N required for half of all random walks started in i and moving randomly with respect to the weights p i j to visit B within d B (i) steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if u satisfies L u = λ u on V and ε > 0 is so large that B = { i ∈ V : − ε ≤ u (i) ≤ ε } ≠ ∅ , then, for all i ∈ V , d B (i) log ⁡ (1 | 1 − λ |) ≥ log ⁡ (| u (i) | ‖ u ‖ L ∞ ) − log ⁡ (1 2 + ε). d B (i) is a remarkably good approximation of | u | in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture. [ABSTRACT FROM AUTHOR]

Abstract Let Ω be a domain of R n , n ≥ 3. The classical Caffarelli–Kohn–Nirenberg inequality rewrites as the following inequality: for any s ∈ [ 0 , 2 ] and any γ < (n − 2) 2 4 , there exists a constant K (Ω , γ , s) > 0 such that (H S) ∫ Ω | u | 2 ⋆ (s) | x | s d x 2 2 ⋆ (s) ≤ K (Ω , γ , s) ∫ Ω | ∇ u | 2 − γ u 2 | x | 2 d x , for all u ∈ D 1 , 2 (Ω) (the completion of C c ∞ (Ω) for the relevant norm). When 0 ∈ Ω is an interior point, the range (− ∞ , (n − 2) 2 4) for γ cannot be improved: moreover, the optimal constant K (Ω , γ , s) is independent of Ω and there is no extremal for (H S). But when 0 ∈ ∂ Ω , the situation turns out to be drastically different since the geometry of the domain impacts : • the range of γ 's for which (H S) holds. • the value of the optimal constant K (Ω , γ , s) ; • the existence of extremals for (H S). When Ω is smooth, the problem was tackled by Ghoussoub–Robert (2017) where the role of the mean curvature was central. In the present paper, we consider nonsmooth domain with a singularity at 0 modeled on a cone. We show how the local geometry induced by the cone around the singularity influences the value of the Hardy constant on Ω. When γ is small, we introduce a new geometric object at the conical singularity that generalizes the "mean curvature": this allows to get extremals for (H S). The case of larger values for γ will be dealt in the forthcoming paper (Cheikh-Ali, 2018). As an intermediate result, we prove the symmetry of some solutions to singular pdes that has an interest on its own. [ABSTRACT FROM AUTHOR]

Matrices with displacement structure, such as Pick, Vandermonde, and Hankel matrices, appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functions to derive explicit bounds on the singular values of such matrices. For example, we show that the kth singular value of a real n \times n positive definite Hankel matrix, Hn, is bounded by C\rho k/log n\| Hn\| 2 with explicitly given constants C > 0 and \rho > 1, where \| Hn\| 2 is the spectral norm. This means that a real n \times n positive definite Hankel matrix can be approximated, up to an accuracy of \epsilon \| Hn\| 2 with 0 < \epsilon < 1, by a rank \scrO (log n log(1/\epsilon)) matrix. Analogous results are obtained for Pick, Cauchy, real Vandermonde, L\"owner, and certain Krylov matrices. [ABSTRACT FROM AUTHOR]

Abstract Let C ℓ be a cycle of length ℓ and S n , k = K k ∨ K n − k ‾ , the join graph of a complete graph of order k and an empty graph on n − k vertices, and S n , k + be the graph obtained from S n , k by adding an edge in the independent set of S n , k. Nikiforov conjectured that for a given integer k ≥ 2 , any graph G of sufficiently large order n with spectral radius μ (G) ≥ μ (S n , k) (or μ (G) ≥ μ (S n , k +)) contains C 2 k + 1 or C 2 k + 2 (or C 2 k + 2), unless G = S n , k (or G = S n , k +). In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k ≥ 2 , any graph G of sufficiently large order n with spectral radius μ (G) ≥ μ (S n , k) (or μ (G) ≥ μ (S n , k +)) contains a cycle C ℓ with ℓ ≥ 2 k + 1 (or C ℓ with ℓ ≥ 2 k + 2), unless G = S n , k (or G = S n , k +). These results also imply a result given by Nikiforov in (2010) [3, Theorem 2]. [ABSTRACT FROM AUTHOR]

Abstract We consider an initial–boundary value problem for a quasilinear parabolic system of hemivariational inequalities which is not necessarily coercive. The system exhibits full dependence on the gradient of the solution and is doubly coupled on both the source and multivalued terms. Based on sub-supersolutions, truncation functions, and nonsmooth analysis we establish an existence and enclosure result for weak solutions within a trapping region. The existence of extremal solutions is proved without imposing any monotonicity conditions on lower order terms. Moreover, we present a sufficient condition to construct effectively a trapping region and show the existence of positive solutions. [ABSTRACT FROM AUTHOR]

We consider the inverse problem in magnetostatics for recovering the moment of a planar magnetization from measurements of the normal component of the magnetic field at a distance from the support. Such issues arise in studies of magnetic material in general and in paleomagnetism in particular. Assuming the magnetization is a measure with L2-density, we construct linear forms to be applied on the data in order to estimate the moment. These forms are obtained as solutions to certain extremal problems in Sobolev classes of functions, and their computation reduces to solving an elliptic differential-integral equation, for which synthetic numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Relationships between steepest descent lines and gradient extremals on potential surfaces are elucidated. It is shown that gradient extremals are the curves which connect those points where the steepest descent lines have zero curvature. This condition gives rise to a direct method for the global determination of gradient extremals which is illustrated on the Müller–Brown surface. Furthermore, explicit expressions are obtained for the derivatives of the steepest-descent-line curvatures and, from them, for the gradient extremal tangents. With the help of these formulas, a new gradient extremal following algorithm is formulated. [ABSTRACT FROM AUTHOR]

Let F denote a set of graphs. A graph G is said to be F -free if it does not contain any element of F as a subgraph. The Turán number is the maximum possible number of edges in an F -free graph with n vertices. It is well known that classical Turán type extremal problem aims to study the Turán number of fixed graphs. In 2010, Nikiforov [17] proposed analogously a spectral Turán type problem which asks to determine the maximum spectral radius of an F -free graph with n vertices. It attracts much attention and many such problems remained elusive open even after serious attempts, and so they are considered as one of the most intriguing problems in spectral extremal graph theory. It is interesting to consider another spectral Turán type problem which asks to determine the maximum spectral radius of an F -free graph with m edges. Denote by G (m , F) the set of F -free graphs with m edges having no isolated vertices. Each of the graphs among G (m , F) having the largest spectral radius is called a maximal graph. Let θ p , q , r be a theta graph formed by connecting two distinct vertices with three independent paths of length p , q and r , respectively (length refers to the number of edges). In this paper, we firstly determine the unique maximal graph among G (m , θ 1 , 2 , 3) and G (m , θ 1 , 2 , 4) , respectively. Then we determine all the maximal graphs among G (m , C 5) (resp. G (m , C 6)) excluding the book graph. These results extend some earlier results. [ABSTRACT FROM AUTHOR]