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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]

This short paper gives a necessary and sufficient condition for the Daugavet equation ǁI + Tǁ = 1 + ǁTǁ. A new characterization of the solution of the Daugavet equation in terms of invariant affine subspaces is given.We also study the notions of alternatively convex or smooth (acs) and locally uniformly alternatively convex or smooth (luacs). [ABSTRACT FROM AUTHOR]

In the article, we solve an extremal long time span backward heat conduction problem (BHCP) of a 3D nonhomogeneous heat conduction equation with nonhomogeneous boundary conditions in a cuboid. We first derive a time-dependent 3D homogenization function, such that in terms of the new variable by a variable transformation we can find the expansion coefficients in a closed form by using the Fourier sine series method. After a simple regularization technique, a stable analytic series solution of the 3D BHCP is available. We also develop a regularized Fourier sine series solution of temperature in the whole space-time domain. Numerical tests for the BHCPs in a large space-time domain reveal that the present method is very accurate to recover the initial temperature and the whole solution. [ABSTRACT FROM AUTHOR]

In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in Rn: ... for any nonnegative functions ... and ... such that .... We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the ndimensional sphere Sn by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities. [ABSTRACT FROM AUTHOR]

We analyse the local conditions for a wave optical field evolves toward diffusive behaviour, this was obtained by solving the wave-diffusion partial differential equation and identifying regions where the diffusion effects become dominant and non-dominant. Through extremal analysis, the phase function was determined to satisfy the Poisson equation, where the diffusion parameters exhibit electrically charged particle-like features in the neighbourhood of focusing regions. Additionally, the optical field exhibits a self-regulated behaviour driven by diffusion effects These features were corroborated experimentally in regions that act as a source/sinks in the interference fringes. Experimental results are shown. [ABSTRACT FROM AUTHOR]

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

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

Let be an extremal copositive matrix with unit diagonal. Then the minimal zeros of A all have supports of cardinality two if and only if the elements of A are all from the set . Thus the extremal copositive matrices with minimal zero supports of cardinality two are exactly those matrices which can be obtained by diagonal scaling from the extremal unit diagonal matrices characterized by [Hoffman, Pereira, 1973]. [ABSTRACT FROM AUTHOR]

Abstract Addressing spatial variation of extreme precipitation in urban areas is important for urban hydrologic designs. Climatology of urban areas is, in general, different from that of its surroundings and the spatial variation of extreme precipitation within the city exhibits shorter spatial range, especially for short duration events. This work aims at understanding the dependence structure of extreme precipitation within an urban area and its surrounding non-urban areas at various durations. The spatial dependence of precipitation is analysed with three different measures for examining dependence, considering observations from pairs of stations. Further, the spatial precipitation extremes are modelled with the max-stable process to include the dependence structure of spatial extremes. The City of Berlin, Germany, with surrounding non-urban area is considered to demonstrate the methodology. For this case study, the hourly precipitation shows weaker dependence within the city even at small distances, whereas the daily precipitation shows a high degree of dependence. This dependence structure of the daily precipitation gets masked as more and more surrounding non-urban areas are included in the analysis. Further, the extreme precipitation at different durations are modelled considering max-stable process. Different geographical and climatological covariates are considered in the modelling of location and scale parameters of the Generalized Extreme Value distribution. The geographical covariates are seen to be predominant within the city and the climatological covariates prevail when non-urban areas are added. These results suggest the importance of quantification of dependence structure of spatial precipitation at the sub-daily timescales, as well as the need to more precisely model spatial extremes within the urban areas. [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]

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]

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]

For a graph G = ( V , E ) , the eccentric connectivity index of G , denoted by ξ c ( G ), is defined as ξ c ( G ) = ∑ v ∈ V ɛ ( v ) d ( v ) , where ɛ( v ) and d ( v ) are the eccentricity and the degree of v in G , respectively. In this paper, we first establish the sharp lower bound for the eccentric connectivity index in terms of the order and the minimum degree of a connected G , and characterize some extremal graphs, which generalize some known results. Secondly, we characterize the extremal trees having the maximum or minimum eccentric connectivity index for trees of order n with given degree sequence. Finally, we give a sharp lower bound for the eccentric connectivity index in terms of the order and the radius of a unicyclic G , and characterize all extremal graphs. [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]

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

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

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

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

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

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

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

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

We develop an intrinsic geometric approach to the calculus of variations in the Wasserstein space. We show that the flows associated with the Schrödinger bridge with general prior, with optimal mass transport, and with the Madelung fluid can all be characterized as annihilating the first variation of a suitable action. We then discuss the implications of this unified framework for stochastic mechanics: It entails, in particular, a sort of fluid-dynamic reconciliation between Bohm’s and Nelson’s stochastic mechanics. [ABSTRACT FROM AUTHOR]

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

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]

We define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld “coproduct.” This allows us to recover the vector representations recently introduced by Feigin-Jimbo-Miwa-Mukhin [7] and constructed by the author [21] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity. [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]

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

For two graphs G and F , the extremal number of F in G , denoted by ex (G , F) , is the maximum number of edges in a spanning subgraph of G not containing F as a subgraph. Determining ex (K n , F) for a given graph F is a classical extremal problem in graph theory. In 1962, Erdős determined ex (K n , k K 3) , which generalized Mantel's Theorem. On the other hand, in 1974, Bollobás, Erdős, and Straus determined ex (K n 1 , n 2 , ... , n r , K t) , which extended Turán's Theorem to complete multipartite graphs. In this paper, we determine ex (K n 1 , n 2 , ... , n r , k K 3) for r ≥ 4 and 10 k − 4 ≤ n 1 + 4 k ≤ n 2 ≤ n 3 ≤ ⋯ ≤ n r. [ABSTRACT FROM AUTHOR]