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Minors play a key role in graph theory, and extremal problems on forbidding minors have attracted appreciable amount of interest in the past decades. In this paper, we focus on spectral extrema of K s , t -minor free graphs, and determine extremal graphs with maximum spectral radius over all K s , t -minor free graphs of sufficiently large order. This generalizes and improves several previous results. For t ≥ s ≥ 2 , our result completely solves Tait's conjecture. For t ≥ s = 1 , our result gives a spectral analogue of a theorem due to Ding, Johnson and Seymour, which determines the maximum number of edges in K 1 , t -minor free connected graphs. Some spectral and structural tools, such as, local edge maximality, local degree sequence majorization and double eigenvectors transformation, are used to characterize structural properties of extremal graphs. [ABSTRACT FROM AUTHOR]

Generalized Turán problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size t in a graph of a fixed order that does not contain any path (or cycle) of length at least a given number. Both of the path-free and cycle-free extremal problems were recently considered and asymptotically solved by Luo. We fully resolve these problems by characterizing all possible extremal graphs. We further extend these results by solving the edge-variant of these problems where the number of edges is fixed instead of the number of vertices. We similarly obtain exact characterization of the extremal graphs for these edge variants. [ABSTRACT FROM AUTHOR]

For positive integers n > d ≥ k , let ϕ (n , d , k) denote the least integer ϕ such that every n -vertex graph with at least ϕ vertices of degree at least d contains a path on k + 1 vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function ϕ (n , d , k) , and conjectured that for any positive integers n > d ≥ k , it holds that ϕ (n , d , k) ≤ ⌊ k − 1 2 ⌋ ⌊ n d + 1 ⌋ + ϵ , where ϵ = 1 if k is odd and ϵ = 2 otherwise. In this paper we determine the values of the function ϕ (n , d , k) exactly. This confirms the above conjecture of Erdős et al. for all positive integers k ≠ 4 and in a corrected form for the case k = 4. Our proof utilizes, among others, a lemma of Erdős et al. [3] , a theorem of Jackson [6] , and a (slight) extension of a very recent theorem of Kostochka, Luo and Zirlin [7] , where the latter two results concern maximum cycles in bipartite graphs. Moreover, we construct examples to provide answers to two closely related questions raised by Erdős et al. [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]

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]

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 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]

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]

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]

We construct an even extremal lattice of rank 64 by means of a generalized quadratic residue code. [ABSTRACT FROM AUTHOR]

Let K s 1 , s 2 , ⋯ , s r ( r ) be the complete r -partite r -uniform hypergraph and e x ( n , K s 1 , s 2 , ⋯ , s r ( r ) ) be the maximum number of edges in any n -vertex K s 1 , s 2 , ⋯ , s r ( r ) -free r -uniform hypergraph. It is well-known in the graph case [19,18] that e x ( n , K s , t ) = Θ ( n 2 − 1 / s ) when t is sufficiently larger than s . In this note, we generalize the above to hypergraphs by showing that if s r is sufficiently larger than s 1 , s 2 , ⋯ , s r − 1 then e x ( n , K s 1 , s 2 , ⋯ , s r ( r ) ) = Θ ( n r − 1 s 1 s 2 ⋯ s r − 1 ) . This follows from a more general Turán type result we establish in hypergraphs, which also improves and generalizes some recent results of Alon and Shikhelman [2] . The lower bounds of our results are obtained by the powerful random algebraic method of Bukh [6] . Another new, perhaps unsurprising insight which we provide here is that one can also use the random algebraic method to construct non-degenerate (hyper-)graphs for various Turán type problems. The asymptotics for e x ( n , K s 1 , s 2 , ⋯ , s r ( r ) ) is also proved by Verstraëte [25] independently with a different approach. [ABSTRACT FROM AUTHOR]

We consider 4-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of 5 vertices spans either 0 or exactly 2 hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically best-possible result using random tournaments. We give a connection between Baber's result and our construction via Paley tournaments and investigate a ‘switching’ operation on tournaments that preserves hypergraphs arising from this construction. [ABSTRACT FROM AUTHOR]

A convex geometric hypergraph (abbreviated cgh) consists of a collection of subsets of a strictly convex set of points in the plane. Extremal problems for cgh's have been extensively studied in the literature, and in this paper we consider their corresponding saturation problems. We asymptotically determine the saturation number of two geometrically disjoint r -tuples. Further, amongst the eight nonisomorphic 3-uniform cgh's on two edges, we determine the saturation number for seven of these up to order of magnitude and the eighth up to a log factor. [ABSTRACT FROM AUTHOR]

For graphs G and H , an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H . H -colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H -colorings. In this paper, we prove several results in this area. First, we find a class of graphs H with the property that for each H ∈ H , the n -vertex tree that minimizes the number of H -colorings is the path P n . We then present a new proof of a theorem of Sidorenko, valid for large n , that for every H the star K 1 , n − 1 is the n -vertex tree that maximizes the number of H -colorings. Our proof uses a stability technique which we also use to show that for any non-regular H (and certain regular H ) the complete bipartite graph K 2 , n − 2 maximizes the number of H -colorings of n -vertex 2-connected graphs. Finally, we show that the cycle C n has the most proper q -colorings among all n -vertex 2-connected graphs. [ABSTRACT FROM AUTHOR]

We investigate the existence of the least and greatest solutions to measure differential equations, as well as the relation between the extremal solutions and lower or upper solutions. Along the way, we prove a fairly general local existence theorem and an analogue of Peano's uniqueness theorem for measure differential equations. Finally, we show that the general theory is applicable in the study of differential equations with impulses or dynamic equations on time scales. [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]

We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are quite short. [ABSTRACT FROM AUTHOR]

Let λ ⁎ > 0 denote the supremum possible value of λ such that { Δ 2 u = λ f ( u ) in B 1 , u = ∂ u ∂ n = 0 on ∂ B 1 } has a classical solution, where B 1 is the unit ball in R N , n is the exterior unit normal vector, and f ∈ C 1 ( R ) is nondecreasing and satisfies f ( 0 ) > 0 and f ( t ) / t → + ∞ as t → + ∞ . For λ = λ ⁎ this problem possesses a weak solution u ⁎ , the so-called extremal solution. We establish the regularity of this extremal solution for N ≤ 10 . For N ≥ 11 we establish that lim r → 0 ⁡ r N − 8 2 ( u ⁎ ) ′ ( r ) = lim r → 0 ⁡ r N − 10 2 u ⁎ ( r ) = 0 for N ≤ 19 and lim r → 0 ⁡ r N − 9 2 ( u ⁎ ) ′ ( r ) = lim r → 0 ⁡ r N − 11 2 u ⁎ ( r ) = 0 for N ≥ 20 . Our regularity results do not depend on the specific nonlinearity f . [ABSTRACT FROM AUTHOR]

Max-stable processes are widely used to model spatial extremes. These processes exhibit asymptotic dependence meaning that the large values of the process can occur simultaneously over space. Recently, inverted max-stable processes have been proposed as an important new class for spatial extremes which are in the domain of attraction of a spatially independent max-stable process but instead they cover the broad class of asymptotic independence. To study the extreme values of such processes we use the conditioned approach to multivariate extremes that characterises the limiting distribution of appropriately normalised random vectors given that at least one of their components is large. The current statistical methods for the conditioned approach are based on a canonical parametric family of location and scale norming functions. We study broad classes of inverted max-stable processes containing processes linked to the widely studied max-stable models of Brown–Resnick and extremal- t , and identify conditions for the normalisations to either belong to the canonical family or not. Despite such differences at an asymptotic level, we show that at practical levels, the canonical model can approximate well the true conditional distributions. [ABSTRACT FROM AUTHOR]

Let B n + be the set of all connected bipartite bicyclic graphs with n vertices. The Estrada index of a graph G is defined as EE ( G ) = ∑ i = 1 n e λ i , where λ 1 , λ 2 , … , λ n are the eigenvalues of the adjacency matrix of G , and the Kirchhoff index of a graph G is defined as Kf ( G ) = ∑ i < j r i j , where r i j is the resistance distance between vertices v i and v j in G . The complement of G is denoted by G ‾ . In this paper, sharp upper bound on EE ( G ) (resp. Kf ( G ‾ ) ) of graph G in B n + is established. The corresponding extremal graphs are determined, respectively. Furthermore, by means of some newly created inequalities, the graph G in B n + with the second maximal EE ( G ) (resp. Kf ( G ‾ ) ) is identified as well. It is interesting to see that the first two bicyclic graphs in B n + according to these two orderings are mainly coincident. [ABSTRACT FROM AUTHOR]

The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If G = ( V G , E G ) is a simple connected graph, then the total reciprocal edge-eccentricity (REE) of G is defined as ξ e e ( G ) = ∑ u v ∈ E G ( 1 / ε G ( u ) + 1 / ε G ( v ) ) , where ε G ( v ) is the eccentricity of the vertex v . In this paper we first introduced four edge-grafting transformations to study the mathematical properties of the reciprocal edge-eccentricity of G . Using these elegant mathematical properties, we characterize the extremal graphs among n -vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. Some sharp bounds on the reciprocal edge-eccentricity of trees are determined. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a class of elliptic equations of the form: − Δ u − λ | x | 2 u = h ( x ) u − q + μ W ( x ) u p , x ∈ Ω \ { 0 } with Dirichlet boundary conditions, where 0 ∈ Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary ∂Ω, μ > 0 is a parameter, 0 < λ < Λ = ( N − 2 ) 2 4 , 0 < q < 1 < p < 2 ⁎ − 1 = N + 2 N − 2 , h ( x ) > 0 and W ( x ) is a given function with the set { x ∈ Ω : W ( x ) > 0 } of positive measure. Using variational methods, we establish some existence and multiplicity of positive solutions and provide uniform estimates of extremal values for the problem. [ABSTRACT FROM AUTHOR]

In this work, quadratic double and quadratic bordered double circulant constructions are applied to F 4 + u F 4 as well as to F 4 , and as a result, extremal binary self-dual codes of length 56 and 64 are obtained. The binary extension theorems as well as the ring extension version are used to obtain seven extremal self-dual binary codes of length 58, twenty-four extremal self-dual binary codes of length 66 and twenty-nine extremal self-dual binary codes of length 68, all with new weight enumerators, updating the list of all the known extremal self-dual codes in the literature. [ABSTRACT FROM AUTHOR]

Given a finite poset P , we consider the largest size La ( n , P ) of a family F of subsets of [ n ] : = { 1 , … , n } that contains no subposet P . This continues the study of the asymptotic growth of La ( n , P ) ; it has been conjectured that for all P , π ( P ) : = lim n → ∞ ⁡ La ( n , P ) / ( n ⌊ n 2 ⌋ ) exists and equals a certain integer, e ( P ) . This is known to be true for paths, and for several more general families of posets, while for the simple diamond poset D 2 , even the existence of π frustratingly remains open. Here we develop theory to show that π ( P ) exists and equals the conjectured value e ( P ) for many new posets P . We introduce a hierarchy of properties for posets, each of which implies π = e , and some implying more precise information about La ( n , P ) . The properties relate to the Lubell function of a family F of subsets, which is the average number of times a random full chain meets F . We present an array of examples and constructions that possess the properties. [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]

We consider the extremal problem of maximizing a point value ∣ f ( z ) ∣ at a given point z ∈ G by some positive definite and continuous function f on a locally compact Abelian group (LCA group) G , where for a given symmetric open set Ω ∋ z , f vanishes outside Ω and is normalized by f ( 0 ) = 1 . This extremal problem was investigated in R and R d and for Ω a 0-symmetric convex body in a paper of Boas and Kac in 1945. Arestov, Berdysheva and Berens extended the investigation to T d , where T : = R / Z . Kolountzakis and Révész gave a more general setting, considering arbitrary open sets, in all the classical groups above. Also they observed, that such extremal problems occurred in certain special cases and in a different, but equivalent formulation already a century ago in the work of Carathéodory and Fejér. Moreover, following observations of Boas and Kac, Kolountzakis and Révész showed how the general problem can be reduced to equivalent discrete problems of “Carathéodory–Fejér type” on Z or Z m : = Z / m Z . We extend their results to arbitrary LCA groups. [ABSTRACT FROM AUTHOR]

In this paper, we prove the light-likeness of boundaries of smooth extremal surfaces of mixed type in general physical space–time R 1 + n ( n > 1 ) , in particular we improve Gu's theorem on the light-likeness of boundaries of extremal surfaces in R 1 + 2 and prove the light-likeness of boundaries of smooth extremal surfaces of mixed type in general physical space–times. As a consequence, we show that a curve moving in a physical space–time keeps its like-property and the boundary only exists when its world sheet at the initial time has light-like points. This implies that any extremal surface of mixed type is generated by an “initial curve of mixed type”. [ABSTRACT FROM AUTHOR]

We examine the two elliptic systems given by ( G ) λ , γ − Δ u = λ f ′ ( u ) g ( v ) , − Δ v = γ f ( u ) g ′ ( v ) in Ω , and ( H ) λ , γ − Δ u = λ f ( u ) g ′ ( v ) , − Δ v = γ f ′ ( u ) g ( v ) in Ω , with zero Dirichlet boundary conditions and where λ , γ are positive parameters. We show that for general nonlinearities f and g the extremal solutions associated with ( G ) λ , γ are bounded, provided Ω is a convex domain in R N where N ≤ 3 . In the case of a radial domain, we show the extremal solutions are bounded provided N < 10 . The extremal solutions associated with ( H ) λ , γ are bounded in the case where f is a general nonlinearity, g ( v ) = ( v + 1 ) q for 1 < q < ∞ and when Ω is a bounded convex domain in R N for N ≤ 3 . Certain regularity results are also obtained in higher dimensions for ( G ) λ , γ and ( H ) λ , γ for the case of explicit nonlinearities of the form f ( u ) = ( u + 1 ) p and g ( v ) = ( v + 1 ) q . [ABSTRACT FROM AUTHOR]

For positive integers w and k , two vectors A and B from Z w are called k-crossing if there are two coordinates i and j such that A [ i ] − B [ i ] ≥ k and B [ j ] − A [ j ] ≥ k . What is the maximum size of a family of pairwise 1-crossing and pairwise non- k -crossing vectors in Z w ? We state a conjecture that the answer is k w − 1 . We prove the conjecture for w ≤ 3 and provide weaker upper bounds for w ≥ 4 . Also, for all k and w , we construct several quite different examples of families of desired size k w − 1 . This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove that for toric varieties the uniform K-stability is a necessary condition for the existence of extremal metrics. [Copyright &y& Elsevier]

Abstract: In Luo et al. (2012), an extremal graph theory problem is proposed for group connectivity: for an abelian group with and an integer , find , where is the maximum number so that every -vertex simple graph with at most edges is not -connected. In this paper, we determine the values for where is an odd integer and for , each of which solves some open problem proposed in Luo et al. (2012). Furthermore, the values also imply a characterization of -connected graphic sequences. [Copyright &y& Elsevier]

Abstract: In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on . These extremal functions minimize the -distance to the original function, where is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function , where , in the general framework of de Branges spaces of entire functions. We then specialize the construction to a particular family of homogeneous de Branges spaces to approach the multidimensional Euclidean case. Finally, we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter against suitable measures. Applications of the results presented here include multidimensional versions of Hilbert-type inequalities, extremal one-sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one-sided approximations by polynomials for a class of functions on the sphere with an axis of symmetry. [Copyright &y& Elsevier]

Abstract: We consider the properties of weak cancellation, -surjectivity, good index theory, and -injectivity, for the class of extremally rich -algebras, and for the smaller class of isometrically rich -algebras. We establish all four properties for isometrically rich -algebras and for extremally rich -algebras that are either purely infinite or of real rank zero, -injectivity in the real rank zero case following from a prior result of H. Lin. We also show that weak cancellation implies the other properties for extremally rich -algebras and that the class of extremally rich -algebras with weak cancellation is closed under extensions. Moreover, we consider analogous properties which replace the group with the extremal K-set as well as two versions of -surjectivity. [Copyright &y& Elsevier]

In this paper, we consider a compact Kähler manifold with extremal Kähler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kähler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kähler metrics on the projectivization of the dual vector bundle. [ABSTRACT FROM AUTHOR]

Highlights: [•] Only the second harmonic of a lightcurve due to shape can exceed a peak-to-peak amplitude of ∼0.3 magnitude. [•] Lightcurves of lower amplitude are formally ambiguous. [•] Large amplitude lightcurves with more than two pairs of extrema may be due to complex (non-principal axis) rotation. [•] An example of each is presented. [Copyright &y& Elsevier]

Szemerédi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erdős–Stone–Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work. [ABSTRACT FROM AUTHOR]

Abstract: In 1965 Erdős conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative for and n large enough. [Copyright &y& Elsevier]

Abstract: The average sensitivity of a Boolean function is the expectation, given a uniformly random input, of the number of input bits which when flipped change the output of the function. A -CNF is a CNF in which every clause contains at most literals. It has recently been shown by the author Amano (2011) [1] that the average sensitivity of a -CNF is at most . This bound is tight since the parity function on variables has the average sensitivity . In this paper, we consider the problem to determine the extremal formulas achieving this bound. We give a class of such formulas that contains a double exponential (in ) number of non-isomorphic ones. This class captures all formulas, with only one exception, that we have obtained so far. We also give the complete list for and as well as several structural properties of such extremal formulas. [Copyright &y& Elsevier]

Abstract: Binary optimal codes often contain optimal or near-optimal subcodes. In this paper we show that this is true for the family of self-dual codes. One approach is to compute the optimum distance profiles (ODPs) of linear codes, which was introduced by Luo et al. (2010). One of our main results is the development of general algorithms, called the Chain Algorithms, for finding ODPs of linear codes. Then we determine the ODPs for the Type II codes of lengths up to 24 and the extremal Type II codes of length 32, give a partial result of the ODP of the extended quadratic residue code of length 48. We also show that there does not exist a subcode of for , and we find a first example of a doubly-even self-complementary code. [Copyright &y& Elsevier]

Abstract: Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a d-regular graph on n vertices is at most by the Kahn–Zhao theorem [7,13]. Relaxing the regularity constraint to a minimum degree condition, Galvin [5] conjectured that, for , the number of independent sets in a graph with is at most that in . In this paper, we give a lower bound on the number of independent sets in a d-regular graph mirroring the upper bound in the Kahn–Zhao theorem. The main result of this paper is a proof of a strengthened form of Galvinʼs conjecture, covering the case as well. We find it convenient to address this problem from the perspective of . From this perspective, we show that the number of complete subgraphs of a graph G on n vertices with , where with , is bounded above by the number of complete subgraphs in . [Copyright &y& Elsevier]

Abstract: In this paper we establish the boundedness of the extremal solution in dimension of the semilinear elliptic equation , in a general smooth bounded domain , with Dirichlet data , where is a positive, nondecreasing and convex function in such that as . In addition, we prove that, for , the extremal solution . This gives , if and , if . [Copyright &y& Elsevier]

Abstract: It is well known that solutions of ordinary differential equations are continuously dependent on finite-dimensional parameters in equations. In this paper we study the dependence of solutions and eigenvalues of second-order linear measure differential equations on measures as an infinitely dimensional parameter. We will provide two fundamental results, which are the continuity and continuous Fréchet differentiability in measures when the weak⁎ topology and the norm topology of total variations for measures are considered respectively. In some sense the continuity result obtained in this paper is the strongest one. As an application, we will give a natural, simple explanation to extremal problems of eigenvalues of Sturm–Liouville operators with integrable potentials. [Copyright &y& Elsevier]

Abstract: We introduce a modification of Voiculescu’s free entropy which coincides with the variant of Voiculescu’s free entropy on extremal states, but is a concave upper semi-continuous function on the trace state space. We also extend the orbital free entropy of Hiai et al. (2009) [8] to non-hyperfinite multivariables and prove freeness in the case of additivity of Voiculescu’s entropy (or vanishing of our extended orbital entropy). [Copyright &y& Elsevier]

Abstract: The extremal dependence of a random vector describes the tail behaviors of joint probabilities of the random vector with respect to that of its margins, and has been often studied by using the tail dependence function of its copula. A tail density approach is introduced in this paper to analyze extremal dependence of the copulas that are specified only by densities. The relation between the copula tail densities and regularly varying densities are established, and the tail densities of Archimedean and copulas are derived explicitly. The tail density approach becomes especially effective for extremal dependence analysis on a vine copula, for which the tail density can be written recursively in the product form of tail densities of bivariate baseline copulas and densities of bivariate linking copulas. [Copyright &y& Elsevier]

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