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On a uniform grid on the real axis, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space , , of two-way real sequences with the least value of the norm of a linear formally self-adjoint differential operator of order with constant real coefficients. In case of even , the value of the least norm in the space , , of the extremal interpolant is calculated exactly if the grid step and the averaging step are related by the inequality . [ABSTRACT FROM AUTHOR]

The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved problems are also discussed. [ABSTRACT FROM AUTHOR]

The systematic study of Turán-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. (Turán problems for Edge-ordered graphs, 2021). They conjectured that the extremal functions of edge-ordered forests of order chromatic number 2 are n 1 + o (1) . Here we resolve this conjecture proving the stronger upper bound of n 2 O (log n) . This represents a gap in the family of possible extremal functions as other forbidden edge-ordered graphs have extremal functions Ω (n c) for some c > 1 . However, our result is probably not the last word: here we conjecture that the even stronger upper bound of n log O (1) n also holds for the same set of extremal functions. [ABSTRACT FROM AUTHOR]

Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let P 2 , 2 → be the orientation of C 4 which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of P 2 , 2 → , and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let P k → and C k → denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of C k → -free digraphs of order n for all n , k ∈ N ∗ , as well as the extremal digraphs attaining this maximum size. Similar result is obtained for P k → where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of C 4 except P 2 , 2 → . In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. [ABSTRACT FROM AUTHOR]

The Sombor index is a novel topological index, which was introduced by Gutman and defined for a graph G as S O (G) = ∑ u v ∈ E (G) d u 2 + d v 2 , where d u = d G (u) denotes the degree of vertex u in graph G. Extremal problems on the Sombor index for trees with a given diameter has been considered by Chen et al. (MATCH Commun Math Comput Chem 87:23–49, 2022) and Li et al. (Appl Math Comput 416:126731, 2022). As an extension of the results introduced above, we determine the maximum Sombor indices for unicyclic graphs with a fixed order and given diameter. [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]

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]

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]

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]

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

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. The aim of this paper is to prove several important properties of weighted Sobolev spaces: separability, reflexivity, uniform convexity, duality and Markov-type inequalities. [ABSTRACT FROM AUTHOR]

The scalar finite-dimensional case has been discussed in the first part of this work series, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. Based on this preliminary result, the present paper dedicates itself to further study the regularity conditions for vector constrained extremum problems in a Euclidean space. The case of infinite-dimensional image will be the subject of a subsequent paper. [ABSTRACT FROM AUTHOR]

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts. [ABSTRACT FROM AUTHOR]

In this paper, by virtue of the image space analysis, the general scalar robust optimization problems under the strictly robust counterpart are considered, among which, the uncertainties are included in the objective as well as the constraints. Besides, on the strength of a corrected image in a new type, an equivalent relation between the uncertain optimization problem and its image problem is also established, which provides an idea to tackle with minimax problems. Furthermore, theorems of the robust weak alternative as well as sufficient characterizations of robust optimality conditions are achieved on the frame of the linear and nonlinear (regular) weak separation functions. Moreover, several necessary and sufficient optimality conditions, especially saddle point sufficient optimality conditions for scalar robust optimization problems, are obtained. Finally, a simple example for finding a shortest path is included to show the effectiveness of the results derived in this paper. [ABSTRACT FROM AUTHOR]

This note aims at stimulating research on set-valued optimization through image space analysis. [ABSTRACT FROM AUTHOR]

In Part I, sufficient and necessary optimality conditions and the image regularity conditions of constrained scalar and vector extremum problems are reviewed for Image Space Analysis. Part II presents the main feature of the duality and penalization of constrained scalar and vector extremum problems by virtue of Image Space Analysis. In the light, as said in Part I and Part II, to describe the state of Image Space Analysis for constrained optimization, and to stress that it allows us to unify and generalize the several topics of Optimization, in this Part III, we continue to give an exhaustive literature review on separation functions, gap functions and error bounds for generalized systems. Part III also throws light on some research gaps and concludes with the scope of future research in this area. [ABSTRACT FROM AUTHOR]

By means of the image space analysis, a necessary and sufficient condition is established for the existence of vector extrema. In the scalar case, the sufficient part is shown to shrink to a known one. [ABSTRACT FROM AUTHOR]

Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part. [ABSTRACT FROM AUTHOR]

Based on recent results on image space analysis, the paper aims at describing a fixed point approach to vector optimization problems. Possible extensions to the bi-level vector optimization are discussed. [ABSTRACT FROM AUTHOR]

The problems of extremal decomposition with free poles on a circle are well known in the geometric theory of functions of a complex variable. One of such problems is the problem of maximum of the functionalInγ=rγB00∏k=1nrBk,ak,where γ ∈ (0, n], B0, B1, B2,...,Bn, n ≥ 2, are pairwise disjoint domains in C¯,a0=0,ak=1,k=1,n¯ are different points of the circle, r(B, a) is the inner radius of the domain B ⊂ C¯ relative to the point a ∈ B. We consider a more general problem, in which the restriction ak=1,k=1,n¯, is replaced by a more general condition. [ABSTRACT FROM AUTHOR]

For a set of matroids M, let exM(n) be the maximum size of a simple rank-n matroid in M. We prove that, for any finite field F, if M is a minor-closed class of F-representable matroids of bounded branch-width, then limn→ ∞exM(n)/n exists and is a rational number, ∆. We also show that exM(n) - ∆n is periodic when n is sufficiently large and that exM is achieved by a subclass of M of bounded path-width. [ABSTRACT FROM AUTHOR]

Extremal problems on the number of j-independent sets in uniform simple hypergraphs are studied. Nearly optimal results on the maximum number of independent sets for the class of simple regular hypergraphs and on the minimum number of independent sets for the class of simple hypergraphs with given average degree of vertices are obtained. [ABSTRACT FROM AUTHOR]

This article concerns the recovery of the operators by noisy information in the case that their norms are defined by integrals over infinite intervals. We study the conditions under which the dual extremal problem (often nonconvex) can be solved using the Lagrange function minimality condition. [ABSTRACT FROM AUTHOR]

Sharp estimates of a product of the inner radii of pairwise disjoint domains are obtained. In particular, the extremal problem in the case of any finite number of free poles on a radial system of points is solved. [ABSTRACT FROM AUTHOR]

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdös, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs (in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H( n, δ ) be the set of graphs G with n vertices and minimum degree δ , and J( n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a( n, δ ) = min{ δ( G) | G ∈ H( n, δ )}, b( n, δ ) = max{ δ( G) | G ∈ H( n, δ )}, α( n, Δ) = min{ δ( G) | G ∈ J( n, Δ)} and β( n, Δ) = max{ δ( G) | G ∈ J( n, Δ)}. In particular, we obtain bounds for b( n, δ ) and we compute the precise value of a( n, δ ), α( n, Δ) and β( n, Δ) for all values of n, δ and Δ, respectively. [ABSTRACT FROM AUTHOR]

Applications of the method of modules of curve families to extremal problems of geometric function theory are presented. For these problems, the associated quadratic differentials have a large number of free poles. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

Heatwaves are defined as a set of hot days and nights that cause a marked short-term increase in mortality. Obtaining accurate estimates of the probability of an event lasting many days is important. Previous studies of temporal dependence of extremes have assumed either a first-order Markov model or a particularly strong form of extremal dependence, known as asymptotic dependence. Neither of these assumptions is appropriate for the heatwaves that we observe for our data. A first-order Markov assumption does not capture whether the previous temperature values have been increasing or decreasing and asymptotic dependence does not allow for asymptotic independence, a broad class of extremal dependence exhibited by many processes including all non-trivial Gaussian processes. This paper provides a kth-order Markov model framework that can encompass both asymptotic dependence and asymptotic independence structures. It uses a conditional approach developed for multivariate extremes coupled with copula methods for time series. We provide novel methods for the selection of the order of the Markov process that are based upon only the structure of the extreme events. Under this new framework, the observed daily maximum temperatures at Orleans, in central France, are found to be well modelled by an asymptotically independent third-order extremal Markov model. We estimate extremal quantities, such as the probability of a heatwave event lasting as long as the devastating European 2003 heatwave event. Critically our method enables the first reliable assessment of the sensitivity of such estimates to the choice of the order of the Markov process. [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]

We prove that if the function determining a linear functional over the Hardy space is analytic on the disk of radius greater than 1 then the extremal function of this functional is analytic on the same disk. [ABSTRACT FROM AUTHOR]

This paper deals with an extremal problem for harmonic functions in the unit ball of $$\mathbf {R}^n$$ . We are concerned with the pointwise sharp estimates for the gradient of real-valued bounded harmonic functions. Our main result may be formulated as follows. The sharp constants in the estimates for the absolute value of the radial derivative and the modulus of the gradient of a bounded harmonic function coincide near the boundary of the unit ball. This result partially confirms a conjecture posed by D. Khavinson. [ABSTRACT FROM AUTHOR]

We consider the nilpotent sub-Riemannian problem with growth vector (2, 3, 5, 8). We describe and study abnormal extremals orthogonal to the cube of the distribution. We analyze the geometric properties of a two-dimensional surface in the state space on which the corresponding abnormal trajectories define optimal synthesis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of separating value and tantiles of continuous random variables and examine their extremal properties and extremal properties of quantiles. We obtain estimates for the differences between expected values, medians, and separating values of continuous random variables. [ABSTRACT FROM AUTHOR]

We consider the extremal problem of maximizing functions u in the class of real-valued biconvex functions satisfying a boundary condition ψ on a product of the unit ball with itself, with the $\ell^{p}$ -norm. In 1986, Burkholder explicitly found the maximal function for $p=2$ . In this paper, we find some characterizations of such extremal functions. We establish that sufficiently smooth solutions to the convex extremal problems with given boundary values are affine on line segments and the domain D is foliated by such segments. [ABSTRACT FROM AUTHOR]

Erdős' well-known problem on the maximum absolute value of the derivative of a polynomial on a connected lemniscate is extended to the case of a rational function. Moreover, under the assumption that certain lemniscates are connected, a sharp upper bound for the absolute value of the derivative of a rational function at any point in the plane different from the poles is found. The role of the extremal function is played by an appropriate Zolotarev fraction. [ABSTRACT FROM AUTHOR]

We give the solution of the Delsarte extremal problem for even entire functions of exponential type that are Jacobi transforms and prove the uniqueness of the extremal function. The quadrature Markov formula on the half-line with zeros of the modified Jacobi function are used. [ABSTRACT FROM AUTHOR]

The author analyzes the conditions of obtaining exact solutions to quadratic extremum problems of general type by using SDP-relaxations. Based on the known results for exactness of dual bounds (Lagrangian relaxations) and their relationship with SDP-relaxations, several conditions are formulated, under which the values of optimal objective function of the quadratic extremal problem and its SDP-relaxation are equal. [ABSTRACT FROM AUTHOR]

An optimal control problem with state constraints is considered. Some properties of extremals to the Pontryagin maximum principle are studied. It is shown that, from the conditions of the maximum principle, it follows that the extended Hamiltonian is a Lipschitz function along the extremal and its total time derivative coincides with its partial derivative with respect to time. [ABSTRACT FROM AUTHOR]

Given a set V of points in $${\mathbb {R}}^d$$ , two points p, q from V form a double-normal pair, if the set V lies between two parallel hyperplanes that pass through p and q, respectively, and that are orthogonal to the segment pq. In this paper we study the maximum number $$N_d(n)$$ of double-normal pairs in a set of n points in $${\mathbb {R}}^d$$ . It is not difficult to get from the famous Erdős-Stone theorem that $$N_d(n) = \frac{1}{2}(1-1/k)n^2+o(n^2)$$ for a suitable integer $$k = k(d)$$ and it was shown in a paper by Pach and Swanepoel that $$\lceil d/2\rceil \le k(d)\le d-1$$ and that asymptotically $$k(d)\gtrsim d-O(\log d)$$ . In this paper we sharpen the upper bound on k( d), which, in particular, gives $$k(4)=2$$ and $$k(5)=3$$ in addition to the equality $$k(3)=2$$ established by Pach and Swanepoel. Asymptotically we get $$k(d)\le d- \log _2k(d) = d - (1+ o(1)) \log _2k(d)$$ and show that this problem is connected with the problem of determining the maximum number of points in $${\mathbb {R}}^d$$ that form only acute (or non-obtuse) angles. [ABSTRACT FROM AUTHOR]

Methods for solving the multidimensional multiextremal optimization problems using the nested optimization scheme are considered. A novel approach for solving the multidimensional multiextremal problems based on the adaptive nested optimization has been proposed. This approach enables to develop methods of the global optimum search which are more efficient in comparison with the ones on the base of the traditional nested optimization scheme. The new approach provides advantages due to better usage of the information on the problem in the course of optimization. A general scheme of a adaptive nested optimization is described. A theoretical substantiation of the method convergence is given for the case when for solving the univariate subproblems within the nested scheme an information algorithm of global search is used. Results of numerical experiments on the well-known classes of the test multiextremal functions confirming the efficiency of the proposed scheme are presented. [ABSTRACT FROM AUTHOR]

This work considers a parallel algorithm for solving multidimensional multiextremal optimization problems. This algorithm uses Peano-type space filling curves for dimension reduction. Conditions of non-redundant parallelization of the algorithm are considered. Efficiency of the algorithm on modern computing systems with the use of graphics processing units (GPUs) is investigated. Speedup of the algorithm using GPU as compared with the same algorithm implemented on CPU only is demonstrated experimentally. Computational experiments are carried out on a series of several hundred multidimensional multiextremal problems. [ABSTRACT FROM AUTHOR]

We give several general theorems concerning positive definite solutions of Riemann-Hilbert problems on the real line. Furthermore, as an example, we apply our theory to the characteristic function of a class of Lévy processes and we find the distribution of their extrema at a given stopping time. [ABSTRACT FROM AUTHOR]

By establishing a comparison theorem and applying the monotone iterative technique combined with the method of lower and upper solutions, we investigate the existence of extremal solutions of the initial value problem for fractional q-difference equation involving Caputo derivative. An example is presented to illustrate the main result. [ABSTRACT FROM AUTHOR]

The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs. Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques. This connection provided a new proof of Turán classical result on the Turán density of complete graphs. Since then, Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs. Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. They showed that if G is a 3-uniform graph with m edges containing a clique of order t − 1, then λ( G) = λ([ t − 1]) provided $$\left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) \leqslant m \leqslant \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)$$. They also conjectured: If G is an r-uniform graph with m edges not containing a clique of order t − 1, then λ( G) < λ([ t − 1]) provided $$\left( {\begin{array}{*{20}{c}} {t - 1} \\ r \end{array}} \right) \leqslant m \leqslant \left( {\begin{array}{*{20}{c}} {t - 1} \\ r \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ {r - 1} \end{array}} \right)$$. It has been shown that to verify this conjecture for 3-uniform graphs, it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with $$m = \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)$$. Regarding this conjecture, we show: If G is a left-compressed 3-uniform graph on the vertex set [ t] with m edges and |[ t − 1] E( G)| = p, then λ( G) < λ([ t− 1]) provided $$m = \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)$$ and t ≥ 17 p/2 + 11. [ABSTRACT FROM AUTHOR]

The article explores the estimation of the remainder in the asymptotic solution of an extremal problem which involves nonnegative trigonometric polynomials. Topics mentioned include the formula to denote the lower bound for real numbers, asymptotic relations which prove the existence of a positive absolute constant C, and the relationship between functions provided in the article.