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If $ P(z)=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a n ∏ v = 1 n (z − z v) is a polynomial of degree n having all its zeros in $ |z|\le k, k\ge 1 $ | z | ≤ k , k ≥ 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that \[ \max_{|z|=1}|P'(z)|\ge \frac{2}{1+k^n}\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \] max | z | = 1 | P ′ (z) | ≥ 2 1 + k n ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial $ P(z)=a_0+a_1z+a_2z^2+\cdots +a_nz^n=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n = a n ∏ v = 1 n (z − z v) , that $$\begin{align*} & \max_{|z|=1}|P'(z)| \\ & \ge \left(\frac{2}{1+k^n}+\frac{(|a_n|k^n-|a_0|)(k-1)}{(1+k^n)(|a_n|k^n+k|a_0|)}\right)\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \end{align*} $$ max | z | = 1 | P ′ (z) | ≥ (2 1 + k n + (| a n | k n − | a 0 |) (k − 1) (1 + k n) (| a n | k n + k | a 0 |)) ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)]. [ABSTRACT FROM AUTHOR]

We prove an analogue of the Korneichuk–Stechkin lemma for functions with values in L-spaces. As applications, we obtain sharp Ostrowski type inequalities and solve problems of optimal recovery of identity and convexifying operators, as well as the problem of integral recovery on the classes of L-space valued functions with given majorant of modulus of continuity. The recovery is done based on n mean values of the functions over intervals. Moreover, on the classes of functions with given majorant of modulus of continuity of their Hukuhara type derivative, we solve the problem of optimal recovery of the function and the Hukuhara type derivative. The recovery is done based on n values of the function. We obtain sharp Landau type inequalities and solve an analogue of the Stechkin problem about approximation of unbounded operators by bounded ones and the problem of optimal recovery of an unbounded operator on a class of elements, known with error. Consideration of L-space valued functions gives a unified approach to solution of the mentioned above extremal problems for the classes of multi- and fuzzy-valued functions, and for the classes of functions with values in Banach spaces, in particular random processes, and many other classes of functions. [ABSTRACT FROM AUTHOR]

Brualdi and Hoffman [On the spectral radius of (0, 1)-matrices. Linear Algebra Appl. 1985;65:133–146] proposed the problem of determining the maximal spectral radius of graphs with a given size. In this paper, we consider the Brualdi–Hoffman-type problem for graphs with a given matching number. The maximal Q-spectral radius of graphs with a given size and matching number is obtained, and the corresponding extremal graphs are completely determined. [ABSTRACT FROM AUTHOR]

In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator M −. [ABSTRACT FROM AUTHOR]

Fractional Langevin equation describes the evolution of physical phenomena in fluctuating environments for the complex media systems. It is a sequential fractional differential equation with two fractional orders involving a memory kernel, which leads to non-Markovian dynamics and subdiffusion. Here by establishing a general solution of the linear fractional Langevin equations involving initial conditions with the help of well-known Mittag–Leffler functions and using the special properties of these functions, we construct a new comparison result related to linear fractional Langevin equation. Meanwhile, we investigate the existence of extremal solutions for nonlinear boundary value problems with advanced arguments. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. At last an example is presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

The abstract Cauchy problem for the fractional evolution equation with the Caputo derivative of order and operator , , is considered, where generates a strongly continuous one-parameter semigroup on a Banach space. Subordination formulae for the solution operator are derived, which are integral representations containing a subordination kernel (a scalar probability density function) and a -semigroup of operators. Some properties of the subordination kernel are established and representations in terms of Mainardi function and Lévy extremal stable densities are derived. Applications of the subordination formulae are given with a special focus on the multi-dimensional space-time fractional diffusion equation. [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]

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 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]

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]

A unique type of subcontexts is always present in formal contexts with many concepts: the contranominal scales. We make this precise by giving an upper bound for the number of minimal generators (and thereby for the number of concepts) of contexts without contranominal scales larger than a given size. We give an interpretation of this bound in terms of the Vapnik–Chervonenkis dimension of the concept lattice. Extremal contexts are constructed which meet this bound exactly. They are completely classified. [ABSTRACT FROM AUTHOR]

We give a short argument showing that the set of openness points of a continuous function from a metric space X into a metric space Y is of type Gδ. If X is locally connected and Y := R, the set of nonopenness points (of type Fσ) coincides with the set of points of extrema of f. We discuss which Fσ sets can be equal to the set of points of extrema for a continuous function f from R into R, and we present a short survey of the known results on this topic. [ABSTRACT FROM AUTHOR]

Letdenote the class ofc-cyclic graphs withnvertices, girthandpendant vertices. In this paper, we determine the unique extremal graph with largest signless Laplacian spectral radius and Laplacian spectral radius in the class of connectedc-cyclic graphs withvertices, girthgand at mostpendant vertices, respectively, and the unique extremal graph with largest signless Laplacian spectral radius ofwhenand, and we also identify the unique extremal graph with largest Laplacian spectral radius inin the caseand eitherandgis even orandgis odd. Our results extends the corresponding results of [Sci. Sin. Math. 2010;40:1017–1024, Electron. J. Combin. 2011; 18:p.183, Comput. Math. Appl. 2010;59:376–381, Electron. J. Linear Algebra. 2011;22:378–388 and J. Math. Res. Appl. 2014;34:379–391]. [ABSTRACT FROM AUTHOR]

We propose a new notion called “extremal depth” (ED) for functional data, discuss its properties, and compare its performance with existing concepts. The proposed notion is based on a measure of extreme “outlyingness.” ED has several desirable properties that are not shared by other notions and is especially well suited for obtaining central regions of functional data and function spaces. In particular: (a) the central region achieves the nominal (desired) simultaneous coverage probability; (b) there is a correspondence between ED-based (simultaneous) central regions and appropriate pointwise central regions; and (c) the method is resistant to certain classes of functional outliers. The article examines the performance of ED and compares it with other depth notions. Its usefulness is demonstrated through applications to constructing central regions, functional boxplots, outlier detection, and simultaneous confidence bands in regression problems. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every biderivation of certain triangular rings is the sum of an extremal biderivation and an inner biderivation, using the notion of maximal left ring of quotients. As a consequence, we show that every biderivation of the ring of allupper triangular matrices over a unital ring () is the sum of an extremal biderivation and an inner biderivation, which extends two results of Benkovič and Ghosseiri, respectively. [ABSTRACT FROM PUBLISHER]

Meta-analysis is a valuable tool for combining information from independent studies. However, most common meta-analysis techniques rely on distributional assumptions that are difficult, if not impossible, to verify. For instance, in the commonly used fixed-effects and random-effects models, we take for granted that the underlying study-level parameters are either exactly the same across individual studies or that they are realizations of a random sample from a population, often under a parametric distributional assumption. In this article, we present a new framework for summarizing information obtained from multiple studies and make inference that is not dependent on any distributional assumption for the study-level parameters. Specifically, we assume the study-level parameters are unknown, fixed parameters and draw inferences about, for example, the quantiles of this set of parameters using study-specific summary statistics. This type of problem is known to be quite challenging (see Hall and Miller). We use a novel resampling method via the confidence distributions of the study-level parameters to construct confidence intervals for the above quantiles. We justify the validity of the interval estimation procedure asymptotically and compare the new procedure with the standard bootstrapping method. We also illustrate our proposal with the data from a recent meta-analysis of the treatment effect from an antioxidant on the prevention of contrast-induced nephropathy. [ABSTRACT FROM AUTHOR]

The paper concerns complex-valued functions which are holomorphic in bounded complete n-circular domainsand fulfil some geometric conditions. The familiesof such kind of functions were considered for instance by Bavrin [1,2], Dobrowolska and Liczberski [4], Dziubiński and Sitarski [5], Fukui [6], Higuchi [8], Jakubowski and Kamiński [9], Liczberski and Wrzesień [14], Marchlewska [15,16], Michiwaki [17], and Stankiewicz [22]. The above functions were applied later to research some families of locally biholomorphic mappings in(see for instance Pfaltzgraff and Suffridge [19], Liczberski [12], Hamada, Honda and Kohr [7]). In this paper, we consider an interesting familyof the typewhich separates two Bavrin’s families. These families correspond to the well-known families of convex univalent and close-to-convex univalent functions of one variable, respectively. We defineusing the property of evenness of functions. We obtain forsome embedding theorems relevant to the mentioned separation question. Applying the Minkowski function of, we solve also some extremal problems for functions from. As an application, we give a topologic property of the family. [ABSTRACT FROM PUBLISHER]

The multivariate extremal index function is a measure of the clustering among the extreme values of a multivariate stationary sequence. In this article, we introduce a measure of the degree of clustering of upcrossings in a multivariate stationary sequence, called multivariate upcrossings index, which is a multivariate generalization of the concept of upcrossings index. We derive the main properties of this function, namely the relations with the multivariate extremal index and the clustering of upcrossings. Imposing general local and asymptotic dependence restrictions on the sequence or on its marginals we compute the multivariate upcrossings index from the marginal upcrossings indices and from the joint distribution of a finite number of variables. A couple of illustrative examples are exploited. [ABSTRACT FROM PUBLISHER]

We study the Carathéodory and Kobayashi metrics by way of the method of dual extremal problems in functional analysis. Particularly, incisive results are obtained for convex domains. [ABSTRACT FROM AUTHOR]

LetBφbe a bounded linear functional on the Hardy spaceH1on the bidisc, defined by an essentially bounded function φ. When φ is a continuous function, we study the solution setSφof an extremal problem, that is,Sφ = { f ∈ H1:Bφ(f) = ‖Bφ‖ and ‖ f ‖1 = 1}. [ABSTRACT FROM AUTHOR]

We introduce a certain extremal problem for quasiconformal automorphisms of annuli and give upper and lower estimates for the minimal value of their maximal dilatations. [ABSTRACT FROM AUTHOR]

The article presents solutions to several mathematical problems. Proving n ≥ 0 with n ≠ 3 for the equation p(n) -4p(n-3) + 4p(n-5) -p(n-8) > 0 when p(0) = 1 and p(n) = 0 for n < 0. Proving |a + b + c| can be written as m2 - mn + n2 for some integers m and n when relatively prime integers a, b, c and d satisfy the equation a2 + b2 + c2 + d2 = (a + b + c + d)2 + d).

Miranda and Persson classified all extremal rational elliptic surfaces in characteristic zero. We show that each surface in Miranda and Persson's classification has an integral model with good reduction everywhere (except for those of type X 11(j), which is an exceptional case), and that every extremal rational elliptic surface over an algebraically closed field of characteristic p > 0 can be obtained by reducing one of these integral models mod p. [ABSTRACT FROM PUBLISHER]

The article discusses the problems of the calculus of variations including finding the minimal surfaces of revolution. It provides geometric construction to count the number of smooth extremals that connect the two given endpoints. It gives an explicit formulae for the envelope of the family of smooth extremals which passes through one endpoint.

In this article, some basic characterizations of a non-differentiable η-pseudolinear function are obtained. By means of these basic characterizations, the solution set of the non-differentiable η-pseudolinear extremum problem is characterized. Our study improves naturally and extends some previously known results. [ABSTRACT FROM PUBLISHER]

In this article, we prove an existence result for partial hyperbolic differential equations of fractional order in Banach algebras under Lipschitz and Carathéodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions. [ABSTRACT FROM AUTHOR]

This note provides a new approach to a result of Foregger [T.H. Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear Multilinear Algebra 20 (1987) pp. 377–385] and related earlier results by Keilson [J. Keilson, A theorem on optimum allocation for a class of symmetric multilinear return functions, J. Math. Anal. Appl. 15 (1966), pp. 269–272] and Eberlein [P.J. Eberlein, Remarks on the van der Waerden conjecture, II, Linear Algebra Appl. 2 (1969), pp. 311–320]. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof in [Foregger, 1987] is flawed. [ABSTRACT FROM PUBLISHER]

Some problems of statistics can be reduced to extremal problems of minimizing functionals of smooth functions defined on the cube [0, 1] m , m≥2. In this paper, we study a class of extremal problems that is closely connected to the problem of testing multivariate independence. By solving the extremal problem, we provide a unified approach to establishing weak convergence for a wide class of empirical processes which emerge in connection with testing independence. The use of our result is also illustrated by describing the domain of local asymptotic optimality of some nonparametric tests of independence. [ABSTRACT FROM PUBLISHER]

This article is devoted to the study of the Dirichlet boundary problem for equations with p-Laplacian [image omitted] where n ≥ 1, 1 < p < q < +∞, and a(·) may change the sign in Ω. In the main result, we establish necessary and sufficient conditions when the problem possesses a weak positive solution with λ ∈ [λ1, λ*) where λ1 > 0 is the first eigenvalue of the Dirichlet p-Laplacian and λ* is an extremal point. [ABSTRACT FROM AUTHOR]

We investigate stable solutions of elliptic equations of the type [image omitted] where n ≥ 2, s ∈ (0, 1), λ ≥0 and f is any smooth positive superlinear function. The operator (- Δ)s stands for the fractional Laplacian, a pseudo-differential operator of order 2s. According to the value of λ, we study the existence and regularity of weak solutions u. [ABSTRACT FROM AUTHOR]

In this article we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them, here called rated extremal principles. These developments are in the core geometric theory of variational analysis. We present their applications to calculus and optimality conditions for problems with infinitely many constraints. [ABSTRACT FROM AUTHOR]

We develop necessary optimality conditions using techniques from variational analysis for a bilevel optimization problem without assuming the lower-level problem satisfying neither the Mangasarian Fromovitz constraint qualification nor the partial calmness condition [D.L. Ye and D.L. Zhu, Optimality conditions for bilevel programming problems, Optimization 33 (1995), pp. 9–27; J.J. Ye and J.J. Zhu, A note on optimality conditions for bilevel programming problems, Optimization 39 (1997), pp. 361–366]. Reducing the problem into a one-level nonlinear and nonsmooth programme, we use the approximate extremal principle [B.S. Mordukhovich, Variational Analysis and Generalized Differentiation: I. Basic Theory, Grundlehren Series, Fundamental Principles of Mathematical Sciences, Vol. 330, Springer, Berlin, 2006; B.S. Mordukhovich, Variational Analysis and Generalized Differentiation: II. Applications, Grundlehren Series, Fundamental Principles of Mathematical Sciences, Vol. 331, Springer, Berlin, 2006] to get fuzzy and exact optimality conditions. [ABSTRACT FROM AUTHOR]

We consider some extremal problems for the solution of the generalized heat equation similar to the well-known Hadamard three-circle theorem. These problems are connected with optimal recovery problems. We construct a family of optimal recovery methods for the solution of the generalized heat equation using the information about inaccurate data. [ABSTRACT FROM AUTHOR]

Let G be a graph and let Q(G) be its signless Laplacian matrix. When G is a unicyclic (respectively, bipartite) graph we obtain sharp upper and lower bounds for the permanent of Q(G) in terms of the order of G. Improved bounds are obtained in terms of the given girth of G. In each of these cases, we characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with a sequence of two set valued optimization problems in which the feasible region of the first one (the upper-level problem) is determined implicitly by the solution set of the second (the lower-level problem). Since bilevel programming problems are in general nonconvex problems even if the problem data are convex, we use the exact extremal principle and the approximate extremal principle introduced by Mordukhovich [14, 15] in order to get optimality conditions for this bilevel problem. [ABSTRACT FROM AUTHOR]

The article offers the authors' insights on the mathematical presentation of numerical and geometrical pattern-type problems in middle school mathematics curricula. The authors mention that the presentation involves portrayal of patterns including nature, purpose, and mathematical soundness of several problems. The authors also discuss related topics such as the presentation of algebra in view of the context of functions and patterns and the concept of absolute value in mathematics.

Dynamic optimisation provides a unified framework for improving process operations while taking operational constraints into account. In the presence of uncertainty, measurements can be incorporated into the optimisation framework for tracking the optimum. For non-singular control problems, neighbouring-extremal (NE) control can be used to force the first-order variation of the necessary conditions of optimality (NCO) to zero along interior arcs. An extension of NE control to singular control problems has been proposed in the companion paper for single-input problems. In this article, a generalisation to multiple-input systems is presented. In order for these controllers to be tractable from a real-time optimisation perspective, an approximate NE feedback law is proposed, whose application guarantees, under mild assumptions, that the first-order variation of the NCO converges to zero exponentially. The performance of multi-input NE control is illustrated by the case study of a steered car. [ABSTRACT FROM AUTHOR]

Stock returns exhibit heavy tails and volatility clustering. These features, motivating the use of GARCH models, make it difficult to predict times and sizes of losses that might occur. Estimation of losses, like the Value-at-Risk, often assume that returns, normalized by the level of volatility, are Gaussian. Often under ARMA-GARCH modeling, such scaled returns are heavy tailed and show extremal dependence, whose strength reduces when increasing extreme levels. We model heavy tails of scaled returns with generalized Pareto distributions, while extremal dependence can be reduced by declustering data. [ABSTRACT FROM AUTHOR]

This note presents geometric and physical interpretations of the sufficient condition for a critical point to be a strict relative extremum: [image omitted]. The role of the double derivative fxy in this inequality will be highlighted in these interpretations. [ABSTRACT FROM AUTHOR]

In this article, we shall consider the growth at infinity of a sequence (Pn) of entire functions of bounded orders. Our results extend a result of J. Muller and A. Yavrian for the growth near infinity of a polynomial sequence. Given a sequence of entire functions of bounded orders Pn(z), we found a nearly optimal condition, given in terms of zeros of Pn, for which (kn) we have [image omitted] for all compact sets K ⊂  (Theorem 3). Exploring the growth of a sequence of entire functions of bounded orders lead naturally to an extremal function which is similar to the Siciak's extremal function (Section 6). [ABSTRACT FROM AUTHOR]

In this paper we propose residual-based tests for the null hypothesis of cointegration with a structural break against the alternative of no cointegration. The Lagrange Multiplier (LM) test is proposed and its limiting distribution is obtained for the case in which the timing of a structural break is known. Then the test statistic is extended to deal with a structural break of unknown timing. The test statistic, a plug-in version of the test statistic for known timing, replaces the true break point by the estimated one. We show the limiting properties of the test statistic under the null as well as the alternative. Critical values are calculated for the tests by simulation methods. Finite-sample simulations show that the empirical size of the test is close to the nominal one unless the regression error is very persistent and that the test rejects the null when no cointegrating relationship with a structural break is present. We provide empirical examples based on the present-value model, the term structure model, and the money-output relationship model. [ABSTRACT FROM AUTHOR]

During the great period of Greek geometry, an intriguing construction challenge was posed by Apollonius. Some eighteen centuries later, there was revived interest in the problem, and it continues today. François Viète, Isaac Newton, and Joseph-Diaz Gergonne were among the many mathematicians who solved the problem. They lived in different eras and had very different approaches. Viète stayed close to geometric fundamentals, Newton used his grasp of conics, and Gergonne employed a new understanding of inversion geometry properties. [ABSTRACT FROM AUTHOR]

In this article, the authors prove an existence theorem for periodic boundary value problems for first order functional differential equations (FDE) in Banach algebras under mixed generalized Lipschitz and Carathéodory conditions. The existence of extremal positive solutions is also proved under certain monotonicity conditions. An example illustrating the results is included. [ABSTRACT FROM AUTHOR]

We study symmetry, existence, and uniqueness properties of extremal functions for the weighted Sobolev inequality where x &Element; &reals; m , y &Element; &reals; k with m , k ≥ 1 and n = m + k , α > 0, and Q = m + k (α + 1). [ABSTRACT FROM AUTHOR]

This note presents a brief and partial review of the work of Broom, Cannings and Vickers 1. It also presents some simple examples of an extension of the their formalism to non-symmetric matrices. [ABSTRACT FROM AUTHOR]

Let { X n } be a stationary sequence subject to failures, meaning that some of the variables will be either withdrawn or somehow replaced. We discuss some extremal properties of three different failure models assuming that initially { X n } has an extremal index and satisfies appropriate dependence conditions. Special care is taken whenever X n is integer valued. [ABSTRACT FROM AUTHOR]

We study some aspects of the relationship between algebras associated with graphs and automorphism groups. We study an algebra generated by the adjacent matrix of a graph and the all ones matrix, and derive a lower bound for the rank of the automorphism group of a graph. If a graph attains the equality in the above bound, it is called extremal . We also describe some properties and examples of extremal graphs. [ABSTRACT FROM AUTHOR]

Let X be a nondegenerate subvariety of degree d and codimension e in the projective space P n . If X is smooth, any multisecant line to X cuts X along a 0-dimensional scheme of length at most d - e + 1. Moreover, smooth varieties X having a ( d - e + 1)-secant line (an extremal secant line) have been completely classified, extending del Pezzo and Bertini classification of varieties of minimal degree. In this article, we almost completely classify possibly singular varieties having an extremal secant line, without any assumptions on the singularities of X . First, we show that, if e ? 2, a multisecant line to X meets X along a 0-dimensional scheme of length at most d - e + 1. Then, we completely classify singular varieties having a ( d - e + 1)-secant line for e ? 3. A partial result is provided in case e = 3. [ABSTRACT FROM AUTHOR]

For projective codimension two surfaces and threefolds whose singular locus is one dimensional, we get the sharp Castelnuovo-Mumford regularity bound in terms of degrees of defining equations and give the classification of nearly extremal cases. This is a generalization of the result of Bertram et al. [ABSTRACT FROM AUTHOR]