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Given k$$ k $$ graphs G1,...,Gk$$ {G}_1,\dots, {G}_k $$ over a common vertex set of size n$$ n $$, what is the maximum value of ∑i∈[k]e(Gi)$$ {\sum}_{i\in \left[k\right]}e\left({G}_i\right) $$ having no "colorful" copy of H$$ H $$, that is, a copy of H$$ H $$ containing at most one edge from each Gi$$ {G}_i $$? Keevash, Saks, Sudakov, and Verstraëte denoted this number as exk(n,H)$$ {\mathrm{ex}}_k\left(n,H\right) $$ and completely determined exk(n,Kr)$$ {\mathrm{ex}}_k\left(n,{K}_r\right) $$ for large n$$ n $$. In fact, they showed that, depending on the value of k$$ k $$, one of the two natural constructions is always the extremal construction. Moreover, they conjectured that the same holds for every color‐critical graphs, and proved it for 3‐color‐critical graphs. They also asked to classify the graphs H$$ H $$ that have only these two extremal constructions. We prove their conjecture for 4‐color‐critical graphs and for almost all r$$ r $$‐color‐critical graphs when r>4$$ r>4 $$. Moreover, we show that for every non‐color‐critical non‐bipartite graphs, none of the two natural constructions is extremal for certain values of k$$ k $$. [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]

We study the following question: How few edges can we delete from any H $H$‐free graph on n $n$ vertices to make the resulting graph k $k$‐colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For H $H$ any fixed odd cycle, we determine the answer up to a constant factor when n $n$ is sufficiently large. We also prove an upper bound when H $H$ is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges. [ABSTRACT FROM AUTHOR]

A graph H$$ H $$ is common if the number of monochromatic copies of H$$ H $$ in a 2‐edge‐coloring of the complete graph is asymptotically minimized by the random coloring. The classification of common graphs is one of the most intriguing problems in extremal graph theory. We study the notion of weakly locally common graphs considered by Csóka, Hubai, and Lovász [arXiv:1912.02926], where the graph is required to be the minimizer with respect to perturbations of the random 2‐edge‐coloring. We give a complete analysis of the 12 initial terms in the Taylor series determining the number of monochromatic copies of H$$ H $$ in such perturbations and classify graphs H$$ H $$ based on this analysis into three categories:Graphs of Class I are weakly locally common.Graphs of Class II are not weakly locally common.Graphs of Class III cannot be determined to be weakly locally common or not based on the initial 12 terms. As a corollary, we obtain new necessary conditions on a graph to be common and new sufficient conditions on a graph to be not common. [ABSTRACT FROM AUTHOR]

Finding the maximum number of induced cycles of length k in a graph on n vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidický and Pfender answered the question in the case k=5. In this paper we determine precisely, for all sufficiently large n, the maximum number of induced 5‐cycles that an n‐vertex planar graph can contain. [ABSTRACT FROM AUTHOR]

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

This paper 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]

Abstract: We improve the upper bound on the Ramsey number R(5, 5) from R ( 5 , 5 ) ≤ 49 to R ( 5 , 5 ) ≤ 48. We also complete the catalog of extremal graphs for R(4, 5). [ABSTRACT FROM AUTHOR]

We show that a tree of order n has at most [ABSTRACT FROM AUTHOR]

We prove that for every fixed k, the number of occurrences of the transitive tournament Tr k of order k in a tournament [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

We study the topic of 'extremal' planar graphs, defining [ABSTRACT FROM AUTHOR]

The extremogram is a useful tool for measuring extremal dependence and checking model adequacy in a time series. We define the extremogram in the spatial domain when the data is observed on a lattice or at locations distributed as a Poisson point process in d-dimensional space. We establish a central limit theorem for the empirical spatial extremogram. We show these conditions are applicable for max-moving average processes and Brown-Resnick processes and illustrate the empirical extremogram's performance via simulation. We also demonstrate its practical use with a data set related to rainfall in a region in Florida and ground-level ozone in the eastern United States. [ABSTRACT FROM AUTHOR]

A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of K n is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large p, every maximum triangle-free subgraph of G( n, p) is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for [ABSTRACT FROM AUTHOR]

Let $r \geq s \geq 0$ be integers and $G$ be an $r$-graph. The higher inclusion matrix $M_s^r(G)$ is a ${\{ }0,1{\} }$-matrix with rows indexed by the edges of $G$ and columns indexed by the subsets of $V(G)$ of size $s$: the entry corresponding to an edge $e$ and a subset $S$ is $1$ if $S \subseteq e$ and $0$ otherwise. Following a question of Frankl and Tokushige and a result of Keevash, we define the rank-extremal function ${\rm {rex}} (n, t, r, s)$ as the maximum number of edges of an $r$-graph $G$ having ${\rm {rk}}\, M_s^r(G) \leq \binom {n}{s} - t$. For $t$ at most linear in $n,$ we determine this function as well as the extremal $r$-graphs. The special case $t=1$ answers a question of Keevash. [ABSTRACT FROM PUBLISHER]

This study deals with the investigation of the magnitude-frequency responses of all-pole fractional order systems in the viewpoint of extrema existence in these responses. In this investigation, a sufficient algebraic condition, two necessary algebraic conditions, and a necessary and sufficient algebraic condition are obtained to guarantee the non-existence of extrema in the magnitude-frequency response of all-pole fractional order systems. Some examples are presented to show the effectiveness of the results of the paper. [ABSTRACT FROM AUTHOR]

A graph is K2, 3 -saturated if it has no subgraph isomorphic to K2, 3, but does contain a K2, 3 after the addition of any new edge. We prove that the minimum number of edges in a K2, 3-saturated graph on [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

A proper edge coloring of a graph G without isolated edges is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The neighbor-distinguishing index of G is the minimum number ndi( G) of colors in a neighbor-distinguishing edge coloring of G. Zhang, Liu, and Wang in 2002 conjectured that [ABSTRACT FROM AUTHOR]

It is shown that any graph with maximum degree Δ in which the average degree of the induced subgraph on the set of all neighbors of each vertex exceeds [ABSTRACT FROM AUTHOR]

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

Let [ABSTRACT FROM AUTHOR]

We study the degree-diameter problem for claw-free graphs and 2-regular hypergraphs. Let [ABSTRACT FROM AUTHOR]

New extremal odd unimodular lattices in dimension 36 are constructed. Some new odd unimodular lattices in dimension 36 with long shadows are also constructed. [ABSTRACT FROM AUTHOR]

We firstly establish some new theorems on time scales, and then, by employing them together with a new comparison result and the monotone iterative technique, we show the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem: u▿(t) = ƒ(t, u, ʃt0 g(t,s)▿s), t ε [0, a]T, u(0) = u(ρ(a)), where T is a time scale. [ABSTRACT FROM AUTHOR]

The Wiener polarity index of a connected graph G is defined as the number of its pairs of vertices that are at distance three. By introducing some graph transformations, in different way with that of Huang et al., 2013, we determine the minimum Wiener polarity index of unicyclic graphs with any given maximum degree and girth, and characterize extremal graphs. These observations lead to the determination of the minimum Wiener polarity index of unicyclic graphs and the characterization of the extremal graphs. [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

We introduce the class of n-extremal holomorphic maps, a class that generalizes both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence ν, where ν≥0, of necessary conditions for solvability, prove that they are of strictly increasing strength and show that n−3 is insufficient for the solvability of an n-point problem for n≥3. We propose the conjecture that condition n−2 is necessary and sufficient for the solvability of an n-point interpolation problem for Γ and we explore the implications of this conjecture.We introduce a classification of rational Γ-inner functions, that is, analytic functions from the disc into Γ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of Γ. The classes are related to n-extremality and the conditions ν; we prove numerous strict inclusions between the classes. [ABSTRACT FROM PUBLISHER]

In this paper, we present the existence of extremal solutions of set-valued differential equations with feedback control on semilinear Hausdorff space under Hukuhara derivative which is developed under the form DHX(t) = F(t,X(t),H(t,X(t))), X(0) = X0, for allt € [0,T] with the monotone iterative technique and we will verify that monotone sequence of approximate solutions converging uniformly to the solution of the problem, that is useful for optimization problems. [ABSTRACT FROM AUTHOR]

A (m, n)-bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell D(n, a, b) consists of the path Pn-a-b together with a independent vertices adjacent to one pendent vertex of Pn−a−b and b independent vertices adjacent to the other pendent vertex of Pn−a−b. In this paper, firstly, we show that, among (m, n)-bipartite graphs (m ≤ n), the complete bipartite graph Km,n has minimal Kirchhoff index and the tree dumbbell D(m + n, [n − (m + 1)/2], [n − (m + 1)/2]) has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order l, the complete bipartite graph K[l/2],l−[l/2] has minimal Kirchhoff index and the path Pl has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of (m, n)-bipartite graphs and bipartite graphs of order l are obtained by computing the Kirchhoff index of these extremal graphs. [ABSTRACT FROM AUTHOR]

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

A cusp form f(z) of weight k for SL2(ℤ) is determined uniquely by its first ℓ := dim Sk Fourier coefficients. We derive an explicit bound on the nth coefficient of f in terms of its first ℓ coefficients. We use this result to study the non-negativity of the coefficients of the unique modular form of weight k with Fourier expansion In particular, we show that k=81632 is the largest weight for which all the coefficients of Fk, 0(z) are non-negative. This result has applications to the theory of extremal lattices. [ABSTRACT FROM PUBLISHER]

The article presents a study on the development of half-space finite element and a transmitting boundary for water-saturated layered strata with the use of paraxial boundary condition. It states that the development of half-space finite element and a transmitting boundary was verified by comparing the dynamic stiffness of impermeable and permeable rigid strip foundations. It mentions that the study derived the exact dynamic stiffness of a half-space in plane strain and formulated the second-order paraxial approximation of stiffness. It adds that the numerical method that was described in the study maintains the strengths and weaknesses of the finite element method and can be applied to the problems of soil-structure interaction in a water-saturated layered half-space.

Recently, Balacheff [‘A local optimal diastolic inequality on the two-sphere’, J. Topol. Anal. 2 (2010) 109–121] proved that the Calabi–Croke sphere made of two flat 1-unit-side equilateral triangles glued along their boundaries is a local extremum for the length of the shortest closed geodesic among the Riemannian spheres with conical singularities of fixed area. We give an alternative proof of this theorem, which does not make use of the uniformization theorem and carries over to the Lipschitz distance topology. Furthermore, we extend the result to Finsler metrics. [ABSTRACT FROM AUTHOR]

We study spectral properties of a family of quasiperiodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp (2005). [ABSTRACT FROM AUTHOR]

The article offers suggestions on how the change point methods may be applied to detect changes in annual maximal and minimal temperatures. The data sets that have been examined were obtained from a CD-ROM that was a part of the book edited by Camuffo and Jones. Several illustrations are presented to show the periods of measurement together for the annual minima and maxima and to show the behavior of the series under study. It also presents the results for temperature series under study. It also shows that the approximate critical values obtained by the permutation principle vary only little.

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

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

Let G( k, n) be the set of simple graphs (i.e. without multiple edges or loops) that have n vertices and the minimum degree of vertices is k. The Randić index of a graph G is: , where δ u is the degree of vertex u and the summation extends over all edges ( uv) of G. Using linear programming, we find the extremal graphs or give good bounds for this index when the number n k of vertices of degree kis n− k+ t, for 0⩽ t⩽ k and k⩽ n/2. We also prove that for n k⩾ n− k, ( k⩽ n/2) the minimum value of the Randić index is attained for the graph . [ABSTRACT FROM AUTHOR]

We study the following question: given an open set Ω, symmetric about 0, and a continuous, integrable, positive definite function f, supported in Ω and with f(0) = 1, how large can ∫ f be? This problem has been studied so far mostly for convex domains Ω in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and ℤ d. We exhibit upper bounds for ∫ f assuming geometric properties of Ω of two types: (a) packing properties of Ω and (b) spectral properties of Ω. Several examples and applications of the main theorems are shown. In particular, we recover and extend several known results concerning convex domains in Euclidean space. Also, we investigate the question of estimating ∫Ω f over possibly dispersed sets solely in dependence of the given measure m :=|Ω| of Ω. In this respect we show that in ℝ and ℤ the integral is maximal for intervals. [ABSTRACT FROM AUTHOR]

A “perfect matching” of a graph G with n vertices is a set of ⌊n/2⌋ independent edges of G. In the present study, we succeeded in determining the trees whose complements have the extremal number of “perfect matchings” for two different group of trees. Some further problems are also posed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [ABSTRACT FROM AUTHOR]

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective $n$-dimensional varieties, with $n \geq 4$, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed. The examples considered come from particular special homaloidal and subhomaloidal linear systems, which are usually degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron. The first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of $\mathbf{P}^{m}$, with $3 \leq m \leq 5$. We get another series of examples by considering special cubic hypersurfaces through some surfaces in $\mathbf{P}^5$, or some 3-folds in $\mathbf{P}^7$ having one apparent double point. The last examples come from an intriguing birational elementary extremal contraction in dimension 6, studied by Semple and Tyrrell and fully described in the last section of the paper. [ABSTRACT FROM AUTHOR]

The usual tight-binding (TB) approximation, employed in electronic structure calculations, is extended to include the more-distant neighbor sites than the first. In doing so, a simple power law is adopted to describe the bond and overlap contributions, which enables the energy dispersion relation to be obtained in a closed analytic form. The resulting energy band structure has markedly different features from its TB counterpart to which it reduces in the appropriate limits. In particular, local extrema always occur at the TB stationary values, although these may not both be band edges. Another intermediate critical point may arise, which is one of the band edges, the other being located at one of the TB extrema. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 341–346, 2003 [ABSTRACT FROM AUTHOR]

Let X[sub k] be a stationary time series and y [sub k] =X[sub kM] be the sub-sampled series corresponding to a fixed systematic sampling interval M > 1. In this paper, we use a point process approach to study the effect of the sub sampling on the extremal properties of Y[sub k] when X[sub k] is a linear process with heavy-tailed innovations. We prove complete point process convergence theorems which enable us to give in detail the weak limiting behaviour of maxima of the sub-sampled process and to compare it with that of the original process. The results both exemplify the findings of a study by Robinson and Tawn (2000) and offer more precise details for the class of linear models. Motivation comes from the comparison of schemes for monitoring financial and environmental processes. [ABSTRACT FROM AUTHOR]

We characterize the extremal functions for Heisenberg type inequalities of the form ‖f‖1γ‖f‖2δ≤C‖|x|2f‖1α‖|ξ|f^‖2β. The method also yields an alternative and elementary derivation of the sharp constant in Nash's inequality, originally found by Carlen and Loss. [ABSTRACT FROM PUBLISHER]

The class of bilinear time series models is an obvious generalization of linear ARMA models and has found many applications in time series modeling. It is known that the sample paths of even the simplest bilinear process may have sudden bursts of large negative and positive values that vary in form and amplitude depending on the model parameters. Yet, little is known about the extremal properties of this class. In this paper, we look at the extremal properties of bilinear processes and explain how model parameters affect the extremal behavior. [ABSTRACT FROM AUTHOR]

In general thickness varies along the cut edge of a lens. There is some interest in being able to predict the locations of the thickest and thinnest points on the edge. One purpose of this paper was to formulate the general problem mathematically of finding the locations of thickness extrema on the edge of an arbitrary lens cut into an arbitrary shape. A second purpose was to illustrate how the problem can be solved. In particular, the problem is solved completely and explicitly for what is probably the simplest case, the straight cut edge. A component-free matrix expression for the position of the extremum is derived by employing Lagrange multipliers and the concept of the generalized inverse of a matrix. The equation applies to spheres, cylinders and sphero-cylinders and allows for the presence of prism as well. Along some edges the thickness is constant. Along other edges the thickness varies linearly; there are no extrema except for the practical extrema at the two ends of the edge: thickest at the one and thinnest at the other. The mathematical conditions for these two cases are presented. Equivalent to the matrix equation for the position vector of the thickness extremum is a pair of scalar equations expressed in terms of components of the matrices. The pair is useful when locating extrema using manual calculations. On the other hand, the component-free matrix equation is useful in other circumstances such as for mathematical manipulations and when using computer software that handles matrices. The former is used in a number of numerical examples; the latter was used to check the answers by computer. The mathematical techniques described here are likely to find application in other areas of ophthalmic and visual optics [ABSTRACT FROM AUTHOR]

The article presents a study which examines the sharpness of theorem 0.2. to prove the Infinite Ramsey theorem. It mentions that the analysis of unary's relations will result to the generation of counterexamples in expressibility theory. It presents the formula which explores the situation on all countable and closed under besideness and Boolean operations.

Necessary and sufficient conditions are obtained for the weak convergence ( W →) of the sample extremal quotient of independent and identically distributed random variables as the number of observations tends to infinity. The paper fully characterizes the class of possible non-degenerate limit distribution functions of the extremal quotient, under general conditions. [ABSTRACT FROM AUTHOR]

The article studies dynamics under uncertainty. This is an attempt to develop a general approach to the continuous time stochastic processes that arise in dynamic economics from the maximizing behavior of agents. A class of discounted infinite horizong maximum problems is considered. The analysis builds on the result of a study concerning the characterization of the extrema of stochastic variational problems over a finite horizon and on the author's own investigations of the stability properties of the equations of dynamic economics.