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

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]

Abstract Strong extrema of functions on some quasi-metric and compact spaces are studied. It is shown that the maximum cardinalities of the set of strong extrema of discontinuous functions can be larger than for continuous ones. We construct quasi-metric spaces with uncountable weight, on which any function has no more than a countable set of strong extrema. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For any real α ∈ [ 0 , 1 ] , Nikiforov (2017) [7] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let u and v be two vertices of a connected graph G . Suppose that u and v are connected by a path w 0 ( = v ) w 1 ⋯ w s − 1 w s ( = u ) where d ( w i ) = 2 for 1 ≤ i ≤ s − 1 . Let G p , s , q ( u , v ) be the graph obtained by attaching the paths P p to u and P q to v . Let s = 0 , 1 . Nikiforov and Rojo (2018) [9] conjectured that ρ α ( G p , s , q ( u , v ) ) < ρ α ( G p − 1 , s , q + 1 ( u , v ) ) if p ≥ q + 2 . In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal A α -spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal A α -spectral radius with fixed order and matching number. These results generalize some known results. [ABSTRACT FROM AUTHOR]

Let C v ( k ; T ) be the number of closed walks of length k starting at vertex v in a tree T . We prove that for any tree T with a given degree sequence π , the vector C ( k ; T ) ≡ ( C v ( k ; T ), v  ∈  V ( T )) is weakly majorized by the vector C ( k ; T π * ) ≡ ( C v ( k ; T π * ) , v ∈ V ( T π * ) ) , where T π * is the greedy tree with the degree sequence π . In addition, for two trees degree sequences π and π ′, if π is majorized by π ′, then C ( k ; T π * ) is weakly majorized by C ( k ; T π ′ * ) . [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of tracking a solution of a reference parabolic equation by a solution of another equation. A stable algorithm based on the extremal shift method is proposed for this problem. The algorithm is designed to work on a sufficiently large time interval where both equations operate. [ABSTRACT FROM AUTHOR]

Results and applications of the method of modules in geometric function theory are presented. The method was originally created by J. A. Jenkins, and further developed in works of the Leningrad-St. Petersburg mathematical school. A retrospective description of the origin of the method is given, and the determining role of Jenkins in the development of the method of the extremal metric is pointed out. [ABSTRACT FROM AUTHOR]

When a singular projective variety Xsing admits a projective crepant resolution Xres and a smoothing Xsm, we say that Xres and Xsm are related by extremal transition. In this article, we study a relationship between the quantum cohomology of Xres and Xsm in some examples. For 3-dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing Xsm is isomorphic to a certain subquotient of the quantum cohomology of a resolution Xres with the quantumvariables of exceptional curves specialized to one. We observe that similar phenomena happen for toric degenerations of Fl(1, 2, 3), Gr(2, 4), and Gr(2, 5) by explicit computations. [ABSTRACT FROM AUTHOR]

We extend the Erdős-Gallai Theorem for Berge paths in r -uniform hypergraphs. We also find the extremal hypergraphs avoiding t -tight paths of a given length and consider this extremal problem for other definitions of paths in hypergraphs. [ABSTRACT FROM AUTHOR]

In 1975, Richard O'Malley proved that every approximately con- tinuous function has approximate extrema, and this result provides an immediate solution to Scottish Book Problem 157. The purpose of this paper is to provide an additional proof of O'Malley's result. [ABSTRACT FROM AUTHOR]

A compact hyperbolic surface of genus g is said to be extremal if it admits an extremal disc, a disc of the largest radius determined only by g. We discuss how many extremal discs are embedded in non-orientable extremal surfaces of genus 6. This is the final genus in our interest because it is already known for g = 3, 4, 5, or g > 6. We show that non-orientable extremal surfaces of genus 6 admit at most two extremal discs. The locus of extremal discs is also obtained for each surface. Consequently non-orientable extremal surfaces of arbitrary genus g ≧. 3 admit at most two extremal discs. Furthermore we determine the groups of automorphisms of non-orientable extremal surfaces of genus 6 with two extremal discs. [ABSTRACT FROM AUTHOR]

The isolation and nondegeneracy of constrained extrema arising in geometric problems and mathematical models of electrostatics are studied. In particular, it is proved that a convex concyclic configuration of polygonal linkages is a nondegenerate maximum of the oriented area. Geometric properties of equilibrium configurations of point charges with Coulomb interaction on convex curves are considered, and methods for constructing them are presented. It is shown that any configuration of an odd number of points on a circle is an equilibrium point for the Coulomb potential of nonzero point charges. The stability of the equilibrium configurations under consideration is discussed. [ABSTRACT FROM AUTHOR]

The harmonic index of a graph G is defined as the sum of the weights 2/d(u)+d(u) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we present the minimum harmonic indices for unicyclic and bicyclic graphs with n vertices and matching number m (2 ≤ m ≤ [n/2]), respectively. The corresponding extremal graphs are also characterized. [ABSTRACT FROM AUTHOR]

The classical works in function theory (by E. Borel, R. Baire, H. Lebesgue, F. Hausdorff, et al.) laid down the foundation of the classical descriptive theory of functions. Its initial notions are those of a descriptive space and of a measurable function on it. Measurable functions are defined in the (classical) pre-image language. However, a specific range of tasks in the theory of functions, measure theory, and integration theory that emerge on this basis necessitates using an entirely different (postclassical) cover language, which is equivalent to the preimage one in the classical case. By means of the cover language, the general concepts of a prescriptive space and of distributable and uniform functions on it are introduced and their basic properties are studied. [ABSTRACT FROM AUTHOR]

We prove a sharp form of the Trudinger–Moser inequality for the Sobolev space H 1 , n ( R n ) . The sharpness comes from the introduction of an extra factor ‖ u ‖ n n in the classical Trudinger–Moser inequality. Let ℓ ( α ) : = sup { u ∈ H 1 , n ( R n ) : ‖ u ‖ 1 , n = 1 } ⁡ ∫ R n Φ ∘ ν α ( u ) d x , where Φ ( t ) : = e t − ∑ i = 0 n − 1 t i i ! and ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) . The main results read: (1) for 0 ≤ α < 1 we have ℓ ( α ) < ∞ , (2) for α > 1 , ℓ ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α < 1 , an extremal function for ℓ ( α ) exists. [ABSTRACT FROM AUTHOR]

Extremal problems for quadrangles circuminscribed in a circular annulus with the Poncelet porism property are considered. Quadrangles with the maximal and the minimal perimeters are determined. Two conjectures end the paper. [ABSTRACT FROM AUTHOR]

Let Hn be a κ-graph on n vertices. For 0 ≤ ℓ < κ and an ℓ-subset T of V(Hn), define the degree deg(T) of T to be the number of (κ-ℓ)-subsets S such that S ∪ T is an edge in Hn. Let the minimum ℓ-degree of Hn be δ ℓ(Hn) = min{deg(T) : T ⊆ V(Hn) and |T| = ℓ}. Given a family F of κ-graphs, the ℓ-degree Turán number exℓ(n, Ƒ) is the largest δℓ (Hn) over all Ƒ-free κ-graphs Hn on n vertices. Hence, ex0(n, Ƒ) is the Turán number. We define 8467;-degree Turán density to be πκℓ) = limsupn→ ∞ ... In this paper, we show that for κ > ℓ> 1, the set of πκℓ(Ƒ) is dense in the interval [0,1). Hence, there is no "jump" for ℓ-degree Turán density when k > ℓ > 1. We also give a lower bound on πκℓ(Ƒ) in terms of an ordinary Turán density. [ABSTRACT FROM AUTHOR]

The paper of Lavrent'ev [1] was the beginning of geometrical theory of functions of the complex variable. He solved a problem on the product of conformal radiuses of two non-overlapping domains. In many papers (see [2] - [13]) the Lavrent'ev's result are generalized. In this paper are obtained the new results of this direction. [ABSTRACT FROM AUTHOR]

The paper concerns complex-valued functions which are holomorphic in bounded complete n-circular domainsand fulfil some geometric conditions. The familiesof such kind of functions were considered for instance by Bavrin [1,2], Dobrowolska and Liczberski [4], Dziubiński and Sitarski [5], Fukui [6], Higuchi [8], Jakubowski and Kamiński [9], Liczberski and Wrzesień [14], Marchlewska [15,16], Michiwaki [17], and Stankiewicz [22]. The above functions were applied later to research some families of locally biholomorphic mappings in(see for instance Pfaltzgraff and Suffridge [19], Liczberski [12], Hamada, Honda and Kohr [7]). In this paper, we consider an interesting familyof the typewhich separates two Bavrin’s families. These families correspond to the well-known families of convex univalent and close-to-convex univalent functions of one variable, respectively. We defineusing the property of evenness of functions. We obtain forsome embedding theorems relevant to the mentioned separation question. Applying the Minkowski function of, we solve also some extremal problems for functions from. As an application, we give a topologic property of the family. [ABSTRACT FROM PUBLISHER]

Extremal problems for graph homomorphisms have recently become a topic of much research. Let [ABSTRACT FROM AUTHOR]

We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(Ḡ) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p - 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m? [ABSTRACT FROM AUTHOR]

The Grötzsch and Teichmüller well-known extremal problems for the moduli of plane doubly connected domains are extended to the case of domains on Riemann surfaces. [ABSTRACT FROM AUTHOR]

Abstract: Given angles , we discuss various extremal problems over the class of polynomials endowed with the norm [Copyright &y& Elsevier]

Abstract: For a fixed bipartite graph and given , we determine the threshold which guarantees that any -vertex graph with at least edges contains vertex-disjoint copies of . In the proof, we use a variant of a technique developed by Komlós [J. Komlós, Tiling Turán theorems, Combinatorica 20 (2) (2000) 203–218]. [Copyright &y& Elsevier]

Abstract: The family of subsets of is 2-union free if it does not contain three different sets A, B, C such that , and is union free if it does not contain different sets , , …, such that . Let , be the maximum size of a union free, union free family, respectively. Simple construction shows that , which slightly improves the lower bound obtained by Kleitman if n is odd. Another more complicated construction further improves this lower bound. Union free family of m subsets of naturally corresponds to an matrix with independent rows. The result is related to maximum number m of independent rows in some matrix with n columns. [Copyright &y& Elsevier]

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turán-type results) or on finding spanning subgraphs (Dirac-type results). In this paper, we are interested in finding intermediate-sized subgraphs. We investigate minimum degree conditions under which a graph G contains squared paths and squared cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends results of Fan and Kierstead [J. Combin. Theory Ser. B 63 (1995) 55–64] and of Komlós, Sarközy and Szemerédi [Random Structures Algorithms 9 (1996) 193–211] concerning the containment of a spanning squared path and a spanning squared cycle, respectively. Our results show that such minimum degree conditions constitute not merely an interpolation between the corresponding Turán-type and Dirac-type results, but exhibit other interesting phenomena. [ABSTRACT FROM PUBLISHER]

Generalized quantum instruments correspond to measurements where the input and output are either states or more generally quantum circuits. These measurements describe any quantum protocol including games, communications, and algorithms. The set of generalized quantum instruments with a given input and output structure is a convex set. Here, we investigate the extremal points of this set for the case of finite dimensional quantum systems and generalized instruments with finitely many outcomes. We derive algebraic necessary and sufficient conditions for extremality. [ABSTRACT FROM AUTHOR]

The aim of this article is to study the homological properties of deficiently extremal Gorenstein algebras. We prove that if R/ I is an odd deficiently extremal Gorenstein algebra with pure minimal free resolution, then the codimension of R/ I must be odd. As an application, the structure of pure minimal free resolution of a nearly extremal Gorenstein algebra is obtained. [ABSTRACT FROM AUTHOR]

We propose a new approach to the solution of certain extremal problems in the geometric theory of functions of a complex variable. [ABSTRACT FROM AUTHOR]

We give solutions to some extremal problems involving distance function in mixed normed spaces of harmonic functions on the unit ball in Rn. [ABSTRACT FROM AUTHOR]

Inspired by recent work on the atom-bond connectivity ( ABC) index we propose here a new topological index, augmented Zagreb index ( AZI). The tight upper and lower bounds for chemical trees are obtained. Moreover, it has been shown that among all trees the star has the minimum AZI value. Characterizing trees with maximal augmented Zagreb index remains an open problem for future research. [ABSTRACT FROM AUTHOR]

The notion of a boundary class is a useful notion in the investigation of the complexity of extremal problems on graphs. One boundary class is known for the independent set problem and three boundary classes are known for the dominating set problem. In this paper it is proved that the set of boundary classes for the 3-colouring problem is infinite. [ABSTRACT FROM AUTHOR]

Necessary conditions of extremum (from the times of Fermat and Lagrange till our times) for extremal problems where smoothness is interlaced with convexity, and some type of regularity takes place, correspond to a unique general principle, which is due to Lagrange. This report is devoted to the Lagrange principle in the theory of optimization. [ABSTRACT FROM AUTHOR]

The energetic SCF(selfconsistent field) functionals (the SCF energy(I), the rotational increment of the SCF energy(II) and the rotational gradient of of SCF Lagrange multipliers(III)) are fitted by the catastrophe structures(CSs), allowing applications of the of the catastrophe theory(CT) in the study of the SCF theory and method. Through (I)-(II), in the SCF theory, rise qualitatively distinct problems by CT. They reduce themselves to a unique general problem with the stages: (a) the finding of the polynomial form (I)-(III) possibly to be approximated by the catastrophe potentials(CPs) (or the unfoldings of the catastrophe); (b) the identification of the parameter a of the behaviour(or state) for (a); (c) the inducing of the k-jets and the corresponding CPs; (d) the research of the extremum (minimum) of the critical points of CSs that fit the hypersurfaces (I)-(III); (e) the computation and analysis of the integral quadratic error as a measure of the degree of the fitting of (I)-(III) by the CSs; (f) the interpretation of the general results (a)-(e) by the peculiarity of the SCF problem risen by (I)-(III). [ABSTRACT FROM AUTHOR]

In this paper, by introducing a homogenous kernel of -4-degree, we establish a new Hilbert-type integral inequality with multi-parameter and a best constant factor. As applications, the equivalent form, the reverse forms and some particular results are given correspondingly. [ABSTRACT FROM AUTHOR]

For an arbitrary fixed segment [α, β] ⊂ R and given r ∈ N, A r, A0, and p > 0, we solve the extremal problem on the set of all functions x∈ L r ∞ such that ∥ x( r)∥∞ ≤ A r and L( x) p ≤ A0, where In the case where p = ∞ and k ≥ 1, this problem was solved earlier by Bojanov and Naidenov. [ABSTRACT FROM AUTHOR]

Some problems on extremal decomposition in families of nonoverlapping domains containing systems of biangles with free vertices on a circle are considered. Simultaneously, some progress in solving the classical problem on the maximum of a well-known conformal invariant is achieved. This exhibits the role of symmetric configurations in extremal problems under consideration. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The capacity approach and symmetrization method arc adapted to some extremal decomposition problems on the unit disk or an annulus. The problems on the maximum product of the interior radii of pairwise nonoverlapping domains and the maximum product of the Robin radii of such domains are considered. New invariants with respect to the Möbius transformations of the Riemann sphere are introduced. In particular, for these invariants problems on extremal decomposition with free poles on the unit circle are investigated. Bibliography: 19 titles. [ABSTRACT FROM AUTHOR]

The p-harmonic integrals for homeomorphisms between punctured balls are considered. Surprisingly, in spite of the rotational symmetry of the domains and the rotational invariance of the p-harmonic integral the infimum of the energy is not always attained within radially symmetric maps. [ABSTRACT FROM AUTHOR]

This article aims to test and compare the performances of several estimators of the extremal coefficient using simulations of various spatial model. Section 2 of the article summarizes the multivariate extreme values definitions and some properties of the coefficient. Then Section 3 introduces the four estimators. They are compared in Section 4 through a large simulation study. Two applications with real data in the climate field are investigated in Section 5. The first one is related to maxima temperatures and the second one is related to maxima precipitations. Finally, Section 6 provides a conclusion and some perspectives.

We prove that if A and B* are subnormal operators and X is a bounded linear operator such that AX -XB is a Hilbert-Schmidt operator, then f(A)X -Xf(B) is also a Hilbert-Schmidt operator and ∥f(A)X - Xƒ(B)∥2 ≤ L∥AX - XB∥2 for ƒ belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ϵ ℒ(ℋ) is such that SX - XT belongs to a norm ideal (J, ∥·∥J), and we prove that ƒ(S)X - Xƒ(T) ϵ J and ∥f(S)X - Xƒ(T)∥J ≤ C∥SX - XT∥J for ƒ being in a certain class of functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider extremal problems for numerical positive series. The terms of these series are pairwise products of the elements of two sequences, one of which is fixed and the other varies within a given set of sequences. We obtain exact solutions for a number of such problems. As one of the possible applications of the results obtained, we find solutions of some extremal problems related to best n-term approximations of periodic functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an operator H μ ( z) on L ∞(Δ) and obtain some of its properties. Some applications of this operator to the extremal problem of quasiconformal mappings are given. In particular, a sufficient condition for a point τ in the universal Teichmüller space T(Δ) to be a Strebel point is obtained. [ABSTRACT FROM AUTHOR]

We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets. [ABSTRACT FROM AUTHOR]

The paper presents a simple extremal metric approach to problems on extremal decomposition in families of systems of nonoverlapping domains of different types. The approach is based on using general properties of trajectories of the associated quadratic differentials for simpler extremal decomposition problems. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

New definitions of normal (in the sense of Grötzsch) annular and circular domains with radial slits are considered. Their geometric and functional properties are established. Arguments based on the method of extremal metric in the sense of Fuglede allow one to extend these properties to the n-dimensional case, n ≥ 2. The properties established provide a means for simplifying the proofs of the related results by Strebel and Reich and for bringing them to a complete form. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

The paper considers the problem on extremal decomposition of a multiply connected domain G ⊂ ℂ in the case where the associated quadratic differential is a perfect square. It is shown that in the case considered, the value of the functional for this extremal decomposition is the least one in a certain class of decompositions. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

Abstract: We prove two cases of a strong version of Dejean''s conjecture involving extremal letter frequencies. The results are that there exist an infinite -free word over a 5 letter alphabet with letter frequency and an infinite -free word over a 6 letter alphabet with letter frequency . [Copyright &y& Elsevier]