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The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. Several structural properties of the trees with minimal ABC index were proved recently, however the complete characterization of the minimal-ABC trees is still an open problem. It is known that minimal-ABC trees can have at most one pendent path of length 3. It is also known that the minimal-ABC trees that have a pendent path of length 3 do not contain so-called B k -branches, with k ≥ 4, and do not contain more than two B 2 -branches. Here, we improve the latter result by showing that minimal-ABC trees of order larger than 168 and with a pendent path of length 3 do not contain B 2 -branches. Moreover, we show that trees with minimal ABC index with a pendent path of length 3 do not contain B 1 -branches. [ABSTRACT FROM AUTHOR]

Nonlinear programming is based on a collection of definitions, theorems, and principles that must be clearly understood if the available nonlinear pro-gramming methods are to be used effectively. This article begins with the definition of the gradient vector, the Hessian matrix, and the various types of extrema (maxima and minima). [ABSTRACT FROM AUTHOR]

In this paper, we deal with a coupled system of nonlinear fractional differential equations, which involve the Riemann-Liouville derivatives of different fractional orders. By using the monotone iterative technique combined with the method of upper and lower solutions, we not only obtain the existence of extremal system of solutions, but also establish iterative sequences for approximating the solutions. As an application, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

Bang-singular controls may appear in optimal control problems where the control enters the system linearly. We analyze a discretization of the first-order system of necessary optimality conditions written in terms of a variational inequality (or: inclusion) under appropriate assumptions including second-order optimality conditions. For the so-called semilinear case, it is proved that the discrete control has the same principal bang-singular-bang structure as the reference control and, in $$L_{1}$$ topology, the convergence is of order one w.r.t. the stepsize. [ABSTRACT FROM AUTHOR]

The Krein–Milman theorem states that every compact convex subset in a locally compact convex space is the closure of the convex hull of its extreme points. Inspired by this result, we investigate the existence of extreme points in compact convex subsets of asymmetric normed spaces. We focus our attention in the finite dimensional case, giving a geometric description of all compact convex subsets of a finite dimensional asymmetric normed space. [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]

We identify the exposed points of the unit ball of the dual space of the ball algebra. As a corollary, we show that the predual of H∝(Bd) has no extreme points in its unit ball. [ABSTRACT FROM AUTHOR]

On the space of weighted radial Sobolev space, the following generalization of Moser–Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If β ∈ [ 0 , 1 ) and w 0 ( x ) = | log | x | | β then sup ∫ B ∣ ∇ u ∣ 2 w 0 ≤ 1 , u ∈ H 0 , r a d 1 ( w 0 , B ) ∫ B e α u 2 1 − β d x < ∞ , if and only if α ≤ α β = 2 [ 2 π ( 1 − β ) ] 1 1 − β . We prove the existence of an extremal function for the above inequality for the critical case when α = α β thereby generalizing the result of Carleson–Chang who proved the case when β = 0 . [ABSTRACT FROM AUTHOR]

In this work, we propose a modified four circulant construction for self-dual codes and a bordered version of the construction using the properties of λ -circulant and λ -reverse circulant matrices. By using the constructions on F 2 , we obtain new binary codes of lengths 64 and 68. We also apply the constructions to the ring R 2 and considering the F 2 and R 1 -extensions, we obtain new singly-even extremal binary self-dual codes of lengths 66 and 68. More precisely, we find 3 new codes of length 64, 13 new codes of length 66 and 21 new codes of length 68. These codes all have weight enumerators with parameters that were not known to exist in the literature. [ABSTRACT FROM AUTHOR]

In the article solution of the problem of extremal value of x(τ) is presented, for the n-th order linear systems. The extremum of x(τ) is considered as a function of the roots s1, s2, ... sn of the characteristic equation. The obtained results give a possibility of decomposition of the whole n-th order system into a set of 2-nd order systems. [ABSTRACT FROM AUTHOR]

For a cuspidal newform of weight and a prime of , the deformation problem for its associated mod Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the -adic deformation problem for is unobstructed, then is not congruent mod to a newform of lower level. [ABSTRACT FROM AUTHOR]

A graph G = ( V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3 n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true for unbalanced ones. We prove that the maximal possible size of bipartite 1-planar graphs whose one partite set is much smaller than the other one tends towards 2 n rather than 3 n. In particular, we prove that if the size of the smaller partite set is sublinear in n, then | E| = (2 + o(1)) n, while the same is not true otherwise. [ABSTRACT FROM AUTHOR]

In this paper, a new method to interpolate a sequence of ordered points with conic splines is presented. The degree of continuity at the joints of the resulting splines can reach G 3 ; and the splines are faired by decreasing curvature extrema. The construction is not based on parametrization, but based on basic geometric elements, such as the directions of tangents and chord lengths. The weight of Rational Quadratic Bézier Spline is translated into an equivalent form called chord–tangent ratio. The main idea of the new method is converting the geometric construction problem into a conditional extrema problem through representing the curvature and the curvature changing rate at joints with tangent arguments and chord–tangent ratios, and then deriving the unknowns by solving the conditional extremum problem. Experiments show that splines constructed by the new method perform well not only in terms of continuity, but also in smoothness. [ABSTRACT FROM AUTHOR]

In 1978, Gutman proposed the concept of graph energy, defined as the sum of the absolute values of eigenvalues of the adjacency matrix of a molecular graph, which is related to the energy of π -electrons in conjugated hydrocarbons. Recently, Gutman and Wagner proposed the concept of matching energy and pointed out that the chemical applications of matching energy go back to the 1970s. In this paper, we study the extremal values of the matching energy and characterize the graphs with minimal matching energy among all tricyclic graphs with a given diameter. Our methods can help to find more extremal values for other classes of molecular networks and the results suggest the structures with extremal energies. [ABSTRACT FROM AUTHOR]

In this paper, we study the Wiener index (i.e., the total distance or the transmission number) of not necessarily strongly connected digraphs. In order to do so, if there is no directed path from u to v , we follow the convention d ( u , v ) = 0 , which was independently introduced in several studies of directed networks. Under this assumption we naturally generalize the Wiener theorem, as well as a relation between the Wiener index and betweenness centrality to directed graphs. We formulate and study conjectures about orientations of undirected graphs which achieve the extremal values of Wiener index. [ABSTRACT FROM AUTHOR]

A game starts with the empty graph on n vertices, and two players alternate adding edges to the graph. Only moves which do not create a triangle are valid. The game ends when a maximal triangle-free graph is reached. The goal of one player is to end the game as soon as possible, while the other player is trying to prolong the game. With optimal play, the length of the game (number of edges played) is called the K 3 game saturation number. In this paper we prove an upper bound for this number. [ABSTRACT FROM AUTHOR]

Measurements on quantum channels are described by so-called process positive operator valued measures, or process POVMs. We study implementing schemes of extremal process POVMs. As it turns out, the corresponding measurement must satisfy certain extremality property, which is stronger than the usual extremality given by the convex structure. This property motivates the introduction and investigation of the A-convex structure of POVMs, which generalizes both the usual convex and C*-convex structures. We show that extremal points and faces of the set of process POVMs are closely related to A-extremal points and A-faces of POVMs, for a certain subalgebra A. We also give a characterization of A-extremal and A-pure POVMs. [ABSTRACT FROM AUTHOR]

For an integer-valued compound Poisson process with geometrically distributed jumps of a certain sign [these processes are called almost upper (lower) semicontinuous] defined on a finite regular Markov chain, we establish relations (without projections) for the moment-generating functions of extrema and their complements. Unlike the relations obtained earlier in terms of projections, the proposed new relations for the moment-generating functions are determined by the inversion of the perturbed matrix cumulant function. These matrix relations are expressed via the moment-generating functions for the distributions of the corresponding jumps. [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]

The recently proved ‘no short hair’ theorem asserts that, if a spherically-symmetric static black hole has hair, then this hair (the external fields) must extend beyond the null circular geodesic (the “photonsphere”) of the corresponding black-hole spacetime: r field > r null . In this paper we provide compelling evidence that the bound can be violated by non -spherically symmetric hairy black-hole configurations. To that end, we analytically explore the physical properties of cloudy Kerr–Newman black-hole spacetimes – charged rotating black holes which support linearized stationary charged scalar configurations in their exterior regions. In particular, for given parameters { M , Q , J } of the central black hole, we find the dimensionless ratio q / μ of the field parameters which minimizes the effective lengths (radii) of the exterior stationary charged scalar configurations (here { M , Q , J } are respectively the mass, charge, and angular momentum of the black hole, and { μ , q } are respectively the mass and charge coupling constant of the linearized scalar field). This allows us to prove explicitly that (non-spherically symmetric non-static) composed Kerr–Newman-charged-scalar-field configurations can violate the no-short-hair lower bound. In particular, it is shown that extremely compact stationary charged scalar ‘clouds’, made of linearized charged massive scalar fields with the property r field → r H , can be supported in the exterior spacetime regions of extremal Kerr–Newman black holes (here r field is the peak location of the stationary scalar configuration and r H is the black-hole horizon radius). Furthermore, we prove that these remarkably compact stationary field configurations exist in the entire range s ≡ J / M 2 ∈ ( 0 , 1 ) of the dimensionless black-hole angular momentum. In particular, in the large-mass limit they are characterized by the simple dimensionless ratio q / μ = ( 1 − 2 s 2 ) / ( 1 − s 2 ) . [ABSTRACT FROM AUTHOR]

Freeman’s centralization (Freeman, 1978) for a given centrality index is a measure of how central a vertex is regarding to how central all the other vertices are with respect to the given index. The transmission of a vertex v in a graph G is equal to the sum of distances between v and all other vertices of G . In this paper we study the centralization of transmission, in particular, we determine the graphs on n vertices which attain the maximum or minimum value. Roughly, the maximizing graphs are comprised of a path which has one end glued to a clique of similar order. The minimizing family of extremal graphs consists of three paths of almost the same length, glued together in one end-vertex. We conclude the paper with some problems for possible further work. [ABSTRACT FROM AUTHOR]

For a normalized analytic function $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:z<1\}$, the estimate of the integral means $$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{f(re^{i{\it\theta}})^{2}}\end{eqnarray}$$ is an important quantity for certain problems in fluid dynamics, especially when the functions $f(z)$ are nonvanishing in the punctured unit disk $\mathbb{D}\setminus \{0\}$. Let ${\rm\Delta}(r,f)$ denote the area of the image of the subdisk $\mathbb{D}_{r}:=\{z\in \mathbb{C}:z

Maria Ivette Gomes was born July 21, 1948 in Almada, Portugal. After obtaining a BSc in Pure Mathematics in 1970 from the University of Lisbon, Ivette moved in to the University of Sheffield where she completed her PhD in Statistics in 1978, under the supervision of Clive Anderson. Research on statistics of extremes in Portugal has now a long tradition, with Ivette being one of the founders of the movement, along with Tiago de Oliveira (1928-1992). Ivette was co-founder of the Portuguese Statistical Society, and acted as its President from 1990 until 1994; since 1975 she played a key role in the Center of Statistics and Applications (CEAUL), being its scientific coordinator from 1999 until 2006. She was Professor at the Department of Statistics and Operations Research, University of Lisbon, from 1986 until her retirement in 2011. She has been an influential and distinguished scientist in the field of statistics of extremes, through her research, teaching and supervision activities, as well an ambassador of the 'school of extremes' in Portugal. [ABSTRACT FROM AUTHOR]

We obtain a number of new general properties, related to the closedness of the class of long-tailed distributions under convolutions, that are of interest themselves and may be applied in many models that deal with 'plus' and/or 'max' operations on heavy-tailed random variables. We analyse the closedness property under convolution roots for these distributions. Namely, we introduce two classes of heavy-tailed distributions that are not long-tailed and study their properties. These examples help to provide further insights and, in particular, to show that the properties to be both long-tailed and so-called 'generalised subexponential' are not preserved under the convolution roots. This leads to a negative answer to a conjecture of Embrechts and Goldie (J. Austral. Math. Soc. (Ser. A) 29, 243-256 , Stoch. Process. Appl. 13, 263-278 ) for the class of long-tailed and generalised subexponential distributions. In particular, our examples show that the following is possible: an infinitely divisible distribution belongs to both classes, while its Lévy measure is neither long-tailed nor generalised subexponential. [ABSTRACT FROM AUTHOR]

Considering extreme quantiles is a popular way to understand the tail of a distribution. While they have been extensively studied for univariate distributions, much less has been done for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile or a multivariate distribution tail should be. In this paper, we focus on extreme geometric quantiles. In Girard and Stupfler () Intriguing properties of extreme geometric quantiles, their asymptotics are established, both in direction and magnitude, under suitable integrability conditions, when the norm of the associated index vector tends to one. In this paper, we study extreme geometric quantiles when the integrability conditions are not fulfilled, in a framework of regular variation. [ABSTRACT FROM AUTHOR]

Slepian models are derived for a stochastic process observed at level crossings of a moving average driven by a gaussian or Laplace noise. In particular, a Slepian model for the noise - the Slepian noise - is developed. For Laplace moving average process a method of sampling from the Slepian noise is also obtained by a Gibbs sampler. This facilitates comparison of behavior at crossing of a level between a gaussian process and a non-gaussian one and allows to study a random processes sampled at crossings of a non-gaussian moving average process. In a numerical study based on the method it is observed that the behavior of a non-gaussian moving average process at high level crossings is fundamentally different from that for the gaussian case, which is in line with some recent theoretical results on the subject. [ABSTRACT FROM AUTHOR]

The notion of a phantom distribution function (phdf) was introduced by O'Brien (Ann. Probab. 15, 281-292 ()). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any α-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationary sequences exhibiting weak dependence, what allows the use of attractive models beyond mixing. The case of discontinuous marginals is also discussed for α-mixing. Special attention is paid to examples of processes which admit a continuous phantom distribution function while their extremal index is zero. We show that Asmussen (Ann. Appl. Probab. 8, 354-374 ) and Roberts et al. (Extremes. 9, 213-229 ) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type. [ABSTRACT FROM AUTHOR]

The extremal index θ, a measure of the degree of local dependence in the extremes of a stationary process, plays an important role in extreme value analyses. We estimate θ semiparametrically, using the relationship between the distribution of block maxima and the marginal distribution of a process to define a semiparametric model. We show that these semiparametric estimators are simpler and substantially more efficient than their parametric counterparts. We seek to improve efficiency further using maxima over sliding blocks. A simulation study shows that the semiparametric estimators are competitive with the leading estimators. An application to sea-surge heights combines inferences about θ with a standard extreme value analysis of block maxima to estimate marginal quantiles. [ABSTRACT FROM AUTHOR]

Fractional Brownian motion is a non-Markovian Gaussian process Xt, indexed by the Hurst exponent H. It generalizes standard Brownian motion (corresponding to H=1/2). We study the probability distribution of the maximum m of the process and the time tmax at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting H=1/2+ε. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of H test these analytical predictions and show excellent agreement, even for large ε. [ABSTRACT FROM AUTHOR]

The symmetry/asymmetry of functions providing sharp constants in the embedding theorems[InlineMediaObject not available: see fulltext.] for various r and k is studied. The sharp constants for all r > k in the cases k = 4 and k = 6 are calculated explicitly as well. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

We consider the trajectory tracking problem for a dynamical system described by a parabolic differential equation with memory. We present an algorithm for solving this problem based on the Krasovskii extremal shift method. [ABSTRACT FROM AUTHOR]

The isolation and nondegeneracy of constrained extrema arising in geometric problems and mathematical models of electrostatics are studied. In particular, it is proved that a convex concyclic configuration of polygonal linkages is a nondegenerate maximum of the oriented area. Geometric properties of equilibrium configurations of point charges with Coulomb interaction on convex curves are considered, and methods for constructing them are presented. It is shown that any configuration of an odd number of points on a circle is an equilibrium point for the Coulomb potential of nonzero point charges. The stability of the equilibrium configurations under consideration is discussed. [ABSTRACT FROM AUTHOR]

A graph $$G$$ is $$F$$ -saturated if it has no $$F$$ as a subgraph, but does contain $$F$$ after the addition of any new edge. The saturation number, $$sat(n,F)$$ , is the minimum number of edges of a graph in the set of all $$F$$ -saturated graphs with order $$n$$ . In this paper, we determine the saturation number $$sat(n,P_5\cup tP_2)$$ for $$n\ge 3t+8$$ and characterize the extremal graphs for $$n>(18t+76)/5$$ . [ABSTRACT FROM AUTHOR]

The article discusses the solutions for elliptic problem involving sublinear term and peaking phenomenon. It mentions that all solutions have compact support such as the least energy solution which radially symmetric and decreases with respect to problem. It notes that the least energy solution has a unique maximum point which converges to a harmonic center of omega. .

In 1993 Hong asked what are the best bounds on the k 'th largest eigenvalue λ k ( G ) of a graph G of order n . This challenging question has never been tackled for any 2 < k < n . In the present paper tight bounds are obtained for all k > 2 , and even tighter bounds are obtained for the k 'th largest singular value λ k ⁎ ( G ) . Some of these bounds are based on Taylor's strongly regular graphs, and others on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus–Gaddum problems of the kind: How large can λ k ( G ) + λ k ( G ¯ ) be? These constructions are successful also in another open question: How large can the Ky Fan norm λ 1 ⁎ ( G ) + ⋯ + λ k ⁎ ( G ) be? Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs. In the final section, several results and problems are restated for ( − 1 , 1 ) -matrices, which seem to provide a more natural ground for such research than graphs. Many of the results in the paper are paired with open questions and problems for further study. [ABSTRACT FROM AUTHOR]

The Moser-Trudinger embedding has been generalized by Adimurthi and Sandeep to the following weighted version: if $$\Omega \subset \mathbb {R}^2$$ is bounded, $$\alpha >0$$ and $$\beta \in [0,2)$$ are such that then We prove that the supremum is attained, generalizing a well-known result by Flucher, who has proved the case $$\beta =0$$ . [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]

We obtain supremum of the $$k$$ -th normalized Steklov eigenvalues of all rotationally symmetric conformal metrics on $$[0,T]\times \mathbb {S}^1$$ , $$k>1$$ . This generalizes the corresponding result of Fraser and Schoen for the case $$k=1$$ . We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We obtain some results on the comparison of the normalized Steklov eigenvalues of rotationally symmetric metrics and general conformal metrics on $$[0,T]\times \mathbb {S}^1$$ . We also construct an example of a conformal metric on $$[0,T]\times \mathbb {S}^1$$ whose first normalized Steklov eigenvalue is larger than that of the corresponding rotationally symmetric conformal metric. [ABSTRACT FROM AUTHOR]

The harmonic index of a graph G is defined as the sum of the weights 2/d(u)+d(u) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we present the minimum harmonic indices for unicyclic and bicyclic graphs with n vertices and matching number m (2 ≤ m ≤ [n/2]), respectively. The corresponding extremal graphs are also characterized. [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]

The classical works in function theory (by E. Borel, R. Baire, H. Lebesgue, F. Hausdorff, et al.) laid down the foundation of the classical descriptive theory of functions. Its initial notions are those of a descriptive space and of a measurable function on it. Measurable functions are defined in the (classical) pre-image language. However, a specific range of tasks in the theory of functions, measure theory, and integration theory that emerge on this basis necessitates using an entirely different (postclassical) cover language, which is equivalent to the preimage one in the classical case. By means of the cover language, the general concepts of a prescriptive space and of distributable and uniform functions on it are introduced and their basic properties are studied. [ABSTRACT FROM AUTHOR]

We provide a new proof of a result of X. X. Chen and G. Tian [5]: for a polarized extremal Kähler manifold, the minimum of the modified K-energy is attained at an extremal metric. The proof uses an idea of C. Li [16] adapted to the extremal metrics using some weighted balanced metrics. [ABSTRACT FROM AUTHOR]

The medical image retrieval plays an important role in medical diagnosis where a physician can retrieve most similar images from template images against a query image of a particular patient. In this letter, a new and efficient image features descriptor based on the local diagonal extrema pattern (LDEP) is proposed for CT image retrieval. The proposed approach finds the values and indexes of the local diagonal extremas to exploit the relationship among the diagonal neighbors of any center pixel of the image using first-order local diagonal derivatives. The intensity values of the local diagonal extremas are compared with the intensity value of the center pixel to utilize the relationship of central pixel with its neighbors. Finally, the descriptor is formed on the basis of the indexes and comparison of center pixel and local diagonal extremas. The consideration of only diagonal neighbors greatly reduces the dimension of the feature vector which speeds up the image retrieval task and solves the “Curse of dimensionality” problem also. The LDEP is tested for CT image retrieval over Emphysema-CT and NEMA-CT databases and compared with the existing approaches. The superiority in terms of performance and efficiency in terms of speedup of the proposed method are confirmed by the experiments. [ABSTRACT FROM AUTHOR]

For every real number γ ≥ 1 we denote by K( γ) the least possible value of the constant term of an even nonnegative trigonometric polynomial with monotone coefficients such that all its coefficients, save for the constant term, are not lesser than 1 and the sum of these coefficients equals γ. In this paper, the asymptotic estimate of K( γ) is found and some extremal problems on the minimum of the constant term of an even nonnegative trigonometric polynomial are studied. [ABSTRACT FROM AUTHOR]

We give a combinatorial description of extremal generators of the supereigenvector cone { x : A ⊗ x ≥ x } in max algebra. [ABSTRACT FROM AUTHOR]

Erdős, Gallai, and Tuza posed the following problem: given an n -vertex graph G , let τ 1 ( G ) denote the smallest size of a set of edges whose deletion makes G triangle-free, and let α 1 ( G ) denote the largest size of a set of edges containing at most one edge from each triangle of G . Is it always the case that α 1 ( G ) + τ 1 ( G ) ≤ n 2 / 4 ? We also consider a variant on this conjecture: if τ B ( G ) is the smallest size of an edge set whose deletion makes G bipartite, does the stronger inequality α 1 ( G ) + τ B ( G ) ≤ n 2 / 4 always hold? By considering the structure of a minimal counterexample to each version of the conjecture, we obtain two main results. Our first result states that any minimum counterexample to the original Erdős–Gallai–Tuza Conjecture has “dense edge cuts”, and in particular has minimum degree greater than n / 2 . This implies that the conjecture holds for all graphs if and only if it holds for all triangular graphs (graphs where every edge lies in a triangle). Our second result states that α 1 ( G ) + τ B ( G ) ≤ n 2 / 4 whenever G has no induced subgraph isomorphic to K 4 − , the graph obtained from the complete graph K 4 by deleting an edge. Thus, the original conjecture also holds for such graphs. [ABSTRACT FROM AUTHOR]

This paper is devoted to constructing binary quantum stabilizer codes based on the binary extremal self-dual code of parameters $$[48, 24, 12]$$ by Steane's construction. First, we provide an explicit generator matrix for the unique self-dual $$[48,24,12]$$ code to see it as a one-generator quasi-cyclic one and obtain six optimal self-orthogonal codes of parameters $$[48 - t, 24 - t, 12]$$ for $$1 \le t \le 6$$ with dual distances from 11 to 7 by puncturing the $$[48,24,12]$$ code. Second, a special type of subcode structures for self-orthogonal codes is investigated, and then ten derived dual chains are designed. Third, twelve binary quantum codes are constructed from these derived dual pairs within dual chains using Steane's construction. Ten of them, $$[[42,10,8]], [[44,11,8]], [[45,10,8]], [[45,8,9]], [[46,12,8]]$$ , $$[[46,9,9]], [[46,5,10]], [[48,13,8]], [[48,9,9]]$$ , and $$[[48,4,11]]$$ , achieve as good parameters as the best known ones with comparable lengths and dimensions. Two other codes of parameters $$[[47,4,11]]$$ and $$[[48,3,12]]$$ are record breaking in the sense that they improve on the best known ones with the same lengths and dimensions in terms of distance. [ABSTRACT FROM AUTHOR]