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We study the problem of fairly allocating a set of indivisible goods among n agents with additive valuations. Envy freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of an EFX allocation has not been settled and is one of the most important problems in fair division. Toward resolving this question, many impressive results show the existence of its relaxations. In particular, it is known that 0.618-EFX allocations exist and that EFX allocation exists if we do not allocate at most (n-1) goods. Reducing the number of unallocated goods has emerged as a systematic way to tackle the main question. For example, follow-up works on three- and four-agents cases, respectively, allocated two more unallocated goods through an involved procedure. In this paper, we study the general case and achieve sublinear numbers of unallocated goods. Through a new approach, we show that for every ε∈(0,1/2] , there always exists a (1−ε) -EFX allocation with sublinear number of unallocated goods and high Nash welfare. For this, we reduce the EFX problem to a novel problem in extremal graph theory. We define the notion of rainbow cycle number R(·) in directed graphs. For all d∈N, R(d) is the largest k such that there exists a k-partite graph G=(∪i∈[k]Vi,E) , in which each part has at most d vertices (i.e., |Vi|≤d for all i∈[k]); for any two parts Vi and Vj, each vertex in Vi has an incoming edge from some vertex in Vj and vice versa; and there exists no cycle in G that contains at most one vertex from each part. We show that any upper bound on R(d) directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on R(d) , yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation. Funding: J. Garg was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1942321]. R. Mehta was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1750436]. [ABSTRACT FROM AUTHOR]

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 generalized Turán number of 2S'" by Yanjiao Liu and Jianhua Yin explores the concept of the generalized Turán number and its application to graphs. The authors focus on determining the value of ex(n; Ks; 2S') for different scenarios. The article presents the proof of Theorem 1.4, which establishes the lower and upper bounds for ex(n; Ks; 2S `) based on the extremal graph Kn. The text provides several claims and proofs to support the main theorem. [Extracted from the article]

In this paper we continue with our investigation of principal and antiprincipal solutions at infinity solutions of a dynamic symplectic system. The paper is a continuation of part I appeared in Differential Equations and Applications in 2022, where we have presenteded a theory of genera of conjoined bases for symplectic dynamic systems on time scales and its connections with principal solutions at infinity and antiprincipal solutions at infinity for these systems together with some basic properties of this new concept on time scales. Here we provide a characterization of all principal solutions of dynamic symplectic system at infinity in the given genus in terms of the initial conditions and a fixed principal solution at infinity from this genus. Further, we provide a characterization of all antiprincipal solutions of dynamic symplectic system at infinity in the given genus in terms of the initial conditions and a fixed principal solution at infinity from this genus. We also establish mutual limit properties of principal and antiprincipal solutions at infinity. [ABSTRACT FROM AUTHOR]

The paper proves a theorem about the structure of the distribution function on which the extremum of the fractionally linear functional is reached in the presence of an uncountable number of linear constraints. The problem of finding an extremal distribution function arises when determining the optimal control strategy in a class of Markov homogeneous randomized control strategies. The structure of extremal functions is described by a finite number of parameters; hence, the problem is greatly simplified since it is reduced to the search for an extremum of some function. [ABSTRACT FROM AUTHOR]

The conditional matching preclusion number of a graph G , denoted by m p 1 G , is the minimum number of edges whose deletion results in the graph with no isolated vertices that has neither perfect matching nor almost-perfect matching. In this paper, we first give some sharp upper and lower bounds of conditional matching preclusion number. Next, the graphs with large and small conditional matching preclusion numbers are characterized, respectively. In the end, we investigate some extremal problems on conditional matching preclusion number. [ABSTRACT FROM AUTHOR]

Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, the (edge-)Mostar index, and the (vertex-)PI index. For these indices, the corresponding polynomials were also defined, i.e., the (edge-)Szeged polynomial, the Mostar polynomial, the PI polynomial, etc. It is well known that, by evaluating the first derivative of such a polynomial at  x = 1 , we obtain the related topological index. The aim of this paper is to introduce and investigate a new graph polynomial of two variables, which is called the SMP polynomial, such that all three vertex versions of the above-mentioned indices can be easily calculated using this polynomial. Moreover, we also define the edge-SMP polynomial, which is the edge version of the SMP polynomial. Various properties of the new polynomials are studied on some basic families of graphs, extremal problems are considered, and several open problems are stated. Then, we focus on the Cartesian product, and we show how the (edge-)SMP polynomial of the Cartesian product of n graphs can be calculated using the (weighted) SMP polynomials of its factors. [ABSTRACT FROM AUTHOR]

Extremal problems for multiple time delay hyperbolic systems are presented. The optimal boundary control problems for hyperbolic systems in which multiple time delays appear both in the state equations and in the Neumann boundary conditions are solved. The time horizon is fixed. Making use of Dubovicki-Milutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived. [ABSTRACT FROM AUTHOR]

This paper introduces a new concept of Caputo type interval-valued fractional conformable calculus. Based on this, some theorems and properties related to fractional conformable calculus are presented. As an application, this paper attempts to investigate an initial value problem of functional integro-differential equations with Caputo type interval-valued fractional conformable derivatives. We not only prove the existence of at least one solution, the extremal solutions and the unique solution of the above problem, but also establish some monotone iterative sequences converging to the extremal solutions. More meaningful and important, some comparison principles and generalized Gronwall inequality related to the proposed problem are also proved. Two examples are presented to illustrate the main results. We firmly believe that the concept of interval-valued fractional conformable calculus, new comparison principles and generalized Gronwall inequality introduced here can be conveniently applied to qualitative analysis of a variety of new interval-valued fractional conformable biological models, economic models, etc., which will better describe and explain various complex phenomena derived from the real world. [ABSTRACT FROM AUTHOR]

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A block code is called a locally recoverable code (LRC for short) with $(r,\delta)$ -locality, if subject to any $\delta -1$ erasure failures, every symbol in the encoding can still be recovered by accessing at most $r$ other symbols. Recently, it was discovered by several authors that a $q$ -ary optimal $(r,\delta)$ -LRC, i.e., an LRC achieving the generalized Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$ -ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special cases). In this paper, we further investigate constructions of optimal $(r,\delta)$ -LRCs along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and the work in Jin (2019), we equip parity-check matrices with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure that produces an optimal LRC must obey certain disjoint property for subsets of $\mathbb {F}_{q}$. We manage to show the existence of these disjoint subsets. Thus, this yields an optimal $(r,\delta)$ -LRC with code length $\Omega \left({q^{\frac {\delta }{2}\left({1+\frac {1}{\lfloor \frac {d-1}{\delta }\rfloor -1}}\right)}}\right)\vphantom {\bigg)_{j}}$ , where $d$ is the minimum distance. In particular, to our surprise, for $\delta =2$ this disjoint condition is equivalent to a well-studied problem in extremal graph theory. With the help of extremal graph theory, we succeed to improve all of the best known results in Guruswami et al. (2018) for $d\geq 7$. In addition, for $d=6$ , we are able to remove the constraint required in Jin (2019) that $q$ is even. [ABSTRACT FROM AUTHOR]

The proposed distribution is validated by COVID-19 data from the UK and is evaluated for goodness-of-fit using the empirical survival Jensen-Shannon divergence and the Kolmogorov-Smirnov two-sample test statistic. 35741558 23 Jaenada M., Miranda P., Pardo L. Robust Test Statistics Based on Restricted Minimum Rényi's Pseudodistance Estimators. In [[26]], the authors focus on a general family of measures of divergence and purpose a restricted minimum divergence estimator under constraints and a new double-index (dual) divergence test statistic which is thoroughly examined. The present Special Issue of I Entropy i , entitled Information and Divergence Measures, covers various aspects and applications in the general area of Information and Divergence Measures. [Extracted from the article]

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]

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]

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]

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]

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]

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]

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]

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 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 The Adimurthi–Druet [1] inequality is an improvement of the standard Moser–Trudinger inequality by adding a L 2 -type perturbation, quantified by α ∈ [ 0 , λ 1) , where λ 1 is the first Dirichlet eigenvalue of Δ on a smooth bounded domain. It is known [3,10,14,19] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter α approaches λ 1. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as α → λ 1. [ABSTRACT FROM AUTHOR]

Abstract We prove two results: 1. A graph G on at least seven vertices with a vertex v such that G − v is planar and t triangles satisfies | E (G) | ≤ 3 | V (G) | − 9 + t ∕ 3. 2. For p = 2 , 3 , ... , 9 , a triangle-free graph G on at least 2 p − 5 vertices with no K p -minor satisfies | E (G) | ≤ (p − 2) | V (G) | − (p − 2) 2. [ABSTRACT FROM AUTHOR]

Abstract It is known that a positive Boolean function f depending on n variables has at least n + 1 extremal points, i.e. minimal ones and maximal zeros. We show that f has exactly n + 1 extremal points if and only if it is linear read-once. The class of linear read-once functions is known to be the intersection of the classes of read-once and threshold functions. Generalizing this result we show that the class of linear read-once functions is the intersection of read-once and Chow functions. We also find the set of minimal read-once functions which are not linear read-once and the set of minimal threshold functions which are not linear read-once. In other words, we characterize the class of linear read-once functions by means of minimal forbidden subfunctions within the universe of read-once and the universe of threshold functions. Within the universe of threshold functions the importance of linear read-once functions is due to the fact that they attain the minimum value of the specification number, which is n + 1 for functions depending on n variables. In 1995 Anthony et al. conjectured that for all other threshold functions the specification number is strictly greater than n + 1. We disprove this conjecture by exhibiting a threshold non-linear read-once function depending on n variables whose specification number is n + 1. [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]

An extremal Kähler metric with finite singularities on a compact Riemann surface is often called an HCMU metric. It can be regarded as a generalization of CSC (constant scalar curvature) metric. In this paper we will prove that any HCMU metric is a pullback of an HCMU metric on S 2 with only two singularities (we call it football) by a multi-valued holomorphic function. Then we study the monodromy group of the multi-valued function. [ABSTRACT FROM AUTHOR]

A connected graph is said to be a cactus if each of its blocks is either a cycle or an edge. Let C n be the set of all n -vertex cacti with circumference at least 4, and let C n , k be the set of all n -vertex cacti containing exactly k ⩾ 1 cycles where n ⩾ 3 k + 1 . In this paper, lower bounds on the difference between the (revised) Szeged index and Wiener index of graphs in C n (resp. C n , k ) are proved. The minimum and the second minimum values on the difference between the Szeged index and Wiener index of graphs among C n are determined. The bound on the minimum value is strengthened in the bipartite case. A lower bound on the difference between the revised Szeged index and Wiener index of graphs among C n , k is also established. Along the way the corresponding extremal graphs are identified. [ABSTRACT FROM AUTHOR]

In this paper, we determine the extremal graphs with respect to the F-index among the classes of connected prism like graphs. [ABSTRACT FROM AUTHOR]

This paper considers the degree-diameter problem for undirected circulant graphs. For degrees 10 and 11, newly discovered families of circulant graphs of arbitrary diameter are presented which are largest known and are conjectured to be extremal. They are also the largest-known Abelian Cayley graphs of these degrees. For each such family, the order of every graph in the family is defined by a quintic polynomial function of the diameter which is specific to the family. The elements of the generating set for each graph are similarly defined by a set of polynomials in the diameter. The existence of the graphs in the degree 10 families has been proved for all diameters. These graphs are consistent with a conjecture on the order of extremal Abelian Cayley and circulant graphs of any degree and diameter. [ABSTRACT FROM AUTHOR]

In this paper, we construct new barriers for the Pucci extremal operators with unbounded RHS. The geometry of these barriers is given by a Harnack inequality up to the boundary type estimate. Under the possession of these barriers, we prove a new quantitative version of the Hopf–Oleĭnik Lemma for quasilinear elliptic equations with g -Laplace type growth. Finally, we prove (sharp) regularity for ω -semiconvex supersolutions for some nonlinear PDEs. These results are new even for second order linear elliptic equations in nondivergence form. Moreover, these estimates extend and improve a classical a priori estimate proven by L. Caffarelli, J.J. Kohn, J. Spruck and L. Nirenberg in [13] in 1985 as well as a more recent result on the C 1 , 1 regularity for convex supersolutions obtained by C. Imbert in [33] in 2006. [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]

Motivation: Although there is a rich literature on methods for assessing the impact of functional predictors, the focus has been on approaches for dimension reduction that do not suit certain applications. Examples of standard approaches include functional linear models, functional principal components regression and cluster-based approaches, such as latent trajectory analysis. This article is motivated by applications in which the dynamics in a predictor, across times when the value is relatively extreme, are particularly informative about the response. For example, physicians are interested in relating the dynamics of blood pressure changes during surgery to postsurgery adverse outcomes, and it is thought that the dynamics are more important when blood pressure is significantly elevated or lowered. Results: We propose a novel class of extrema-weighted feature (XWF) extraction models. Key components in defining XWFs include the marginal density of the predictor, a function up-weighting values at extreme quantiles of this marginal, and functionals characterizing local dynamics. Algorithms are proposed for fitting of XWF-based regression and classification models, and are compared with current methods for functional predictors in simulations and a blood pressure during surgery application. XWFs find features of intraoperative blood pressure trajectories that are predictive of postoperative mortality. By their nature, most of these features cannot be found by previousmethods. [ABSTRACT FROM AUTHOR]

The generalized A B C index of a graph G , denoted by A B C α ( G ) , is defined as the sum of weights ( d i + d j − 2 d i d j ) α over all edges v i v j of G , where α is an arbitrary non-zero real number, and d i is the degree of vertex v i of G . In this paper, we first prove that the generalized A B C index of a connected graph will increase with addition of edge(s) if α < 0 or 0 < α ≤ 1 ∕ 2 , which provides a useful tool for the study of extremal properties of the generalized A B C index. By means of this result, we then characterize the graphs having the maximal A B C α value for α < 0 among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number. Our work extends some previously known results. [ABSTRACT FROM AUTHOR]

Let N ≥ 2 , W 1 , N ( R N ) be the standard Sobolev space. For any real numbers τ > 0 and p ≥ N , we denote λ N , p , τ = inf u ∈ W 1 , N ( R N ) , u ≢ 0 ⁡ ∫ R N ( | ∇ u | N + τ | u | N ) d x ( ∫ R N | u | p d x ) N / p and define a norm in W 1 , N ( R N ) by ‖ u ‖ N , p , α , τ = ( ∫ R N ( | ∇ u | N + τ | u | N ) d x − α ( ∫ R N | u | p d x ) N / p ) 1 / N for any fixed α , 0 ≤ α < λ N , p , τ . If τ > 0 , 0 < β < 1 , 0 < α < λ N , p , τ , p ≥ N ≥ 2 , and 0 ≤ γ ≤ α N , then using blow-up analysis, we prove that the supremum sup u ∈ W 1 , N ( R N ) , ‖ u ‖ N , p , α , τ ≤ 1 ⁡ ∫ R N 1 | x | N β ( e γ ( 1 − β ) | u | N N − 1 − ∑ k = 0 N − 2 γ k ( 1 − β ) k | u | k N N − 1 k ! ) d x can be attained by some function u ∈ W 1 , N ( R N ) with ‖ u ‖ N , p , α , τ ≤ 1 , where α N = N ω N − 1 1 / ( N − 1 ) and ω N − 1 is the area of the unit sphere in R N . This improves an inequality of Adimurthi and Yang and the result I obtained recently with Yang. [ABSTRACT FROM AUTHOR]

We determine the extremal curves of curvature energy functionals which generalize a variational problem studied by Blaschke in R 3 . The generalization is made by extending both, the lagrangian energy itself, and the ambient space (to riemannian and lorentzian 3-space forms). Then, we show that constant mean curvature (CMC) invariant surfaces in 3-space forms can be constructed, locally, as the evolution of the above extremals under their Binormal flow with appropriate velocity. Moreover, we see that these surfaces are intrinsically described by a warping function satisfying an Ermakov–Milne–Pinney equation. Finally, we use the previous findings to extend known results, on isometric deformations of CMC surfaces and the Lawson's correspondence of CMC cousins, to our background spaces. [ABSTRACT FROM AUTHOR]

An extremal element x in a Lie algebra g is an element for which the space [ x , [ x , g ] ] is contained in the linear span of x . Long root elements in classical Lie algebras are examples of extremal elements. Lie algebras generated by extremal elements lead to geometries on points and lines which can be characterized as root shadow spaces of buildings. In this paper we show that, in case the rank of this building is at least 3, the Lie algebra is (up to isomorphism) uniquely defined by this geometry. This provides us with a geometric characterization of (most of the) classical Lie algebras. [ABSTRACT FROM AUTHOR]

The typical extremal problem asks how large a structure can be without containing a forbidden substructure. The Erdős–Rothschild problem, introduced in 1974 by Erdős and Rothschild in the context of extremal graph theory, is a coloured extension, asking for the maximum number of colourings a structure can have that avoid monochromatic copies of the forbidden substructure. The celebrated Erdős–Ko–Rado theorem is a fundamental result in extremal set theory, bounding the size of set families without a pair of disjoint sets, and has since been extended to several other discrete settings. The Erdős–Rothschild extensions of these theorems have also been studied in recent years, most notably by Hoppen, Koyakayawa and Lefmann for set families, and Hoppen, Lefmann and Odermann for vector spaces. In this paper we present a unified approach to the Erdős–Rothschild problem for intersecting structures, which allows us to extend the previous results, often with sharp bounds on the size of the ground set in terms of the other parameters. In many cases we also characterise which families of vector spaces asymptotically maximise the number of Erdős–Rothschild colourings, thus addressing a conjecture of Hoppen, Lefmann and Odermann. [ABSTRACT FROM AUTHOR]

We say a family of subsets of { 1 , 2 , … , n } is antipodal if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of { 1 , 2 , … , n } (of any size). Our inequality implies that for any k ∈ N , among all such families of size 2 k , a family consisting of the union of two antipodal ( k − 1 ) -dimensional subcubes has the smallest possible edge boundary. [ABSTRACT FROM AUTHOR]