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Let E ⊂ R n be a compact set, and f : E → R. How can we tell if there exists a convex extension F ∈ C 1 , 1 (R n) of f , i.e. satisfying F | E = f | E ? Assuming such an extension exists, how small can one take the Lipschitz constant Lip (∇ F) : = sup x , y ∈ R n , x ≠ y ⁡ | ∇ F (x) − ∇ F (y) | | x − y | ? We provide an answer to these questions for the class of strongly convex functions by proving that there exist constants k # ∈ R and C > 0 depending only on the dimension n , such that if for every subset S ⊂ E , # S ≤ k # , there exists an η -strongly convex function F S ∈ C 1 , 1 (R n) satisfying F S | S = f | S and Lip (∇ F S) ≤ M , then there exists an η C -strongly convex function F ∈ C c 1 , 1 (R n) satisfying F | E = f | E , and Lip (∇ F) ≤ C M 2 / η. Further, we prove a Finiteness Principle for the space of convex functions in C 1 , 1 (R) and that the sharp finiteness constant for this space is k # = 5. [ABSTRACT FROM AUTHOR]

• The shape of the organ is important for myofibroblasts formation. • The remodeling requires the system additional energy. • The system seeks the lowest energy state for equilibrium. • The convex surface is preferable than the concave one for myofibroblasts formation. Myofibroblasts preferentially accumulate on the convex and not on the concave surfaces of the murine cardiac lobe during lung remodeling after pneumonectomy. This clear difference in function due to the organ shape is most likely mediated by the various mechanical forces generated on the lung's surface. For breathing, the lobe cyclically change its configuration. The cyclic deformation requires energy, depending on the local configuration of the lobe (e.g., convex vs. concave). Considering mechanical contributions to the internal energy of the system and according to the second law of thermodynamics, the system seeks the lowest energy state for equilibrium. Although additional energy for remodeling is required, the system chooses such remodeling sites that minimize the total energy of the new equilibrium state. To test this idea, an idealized, concave-convex configuration of the lobe is assumed. The lobe is made of two homogeneous and isotropic materials of different mechanical properties, the bulk parenchyma and the pleura, a thin, mesothelial cell layer surrounding it. While the whole system cyclically changes shape during breathing, we calculated the amount of mechanical energy per unit volume at the parenchyma-pleural interface where, we believe, myofibroblasts preferentially accumulate. Comparison between convex and concave surfaces indicates that convex surfaces store a lower amount of mechanical energy than the concave ones. We also show that any additional energy for remodeling is preferably done at the convex surface where the lowest new energy equilibrium state is achieved. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

Much work has been done to identify which binary codes can be represented by collections of open convex or closed convex sets. While not all binary codes can be realized by such sets, here we prove that every binary code can be realized by convex sets when there is no restriction on whether the sets are all open or closed. We achieve this by constructing a convex realization for an arbitrary code with k nonempty codewords in R k − 1 . This result justifies the usual restriction of the definition of convex neural codes to include only those that can be realized by receptive fields that are all either open convex or closed convex. We also show that the dimension of our construction cannot in general be lowered. [ABSTRACT FROM AUTHOR]

We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane triangulation G with n vertices can be embedded in R 2 in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a 4 n 3 × 8 n 5 × ζ ( n ) integer grid, where ζ ( n ) ≤ ( 500 n 8 ) τ ( G ) and τ ( G ) denotes the shedding diameter of G , a quantity defined in the paper. [ABSTRACT FROM AUTHOR]

How does the brain encode spatial structure? One way is through hippocampal neurons called place cells, which become associated to convex regions of space known as their receptive fields: each place cell fires at a high rate precisely when the animal is in the receptive field. The firing patterns of multiple place cells form what is known as a convex neural code. How can we tell when a neural code is convex? To address this question, Giusti and Itskov identified a local obstruction, defined via the topology of a code's simplicial complex, and proved that convex neural codes have no local obstructions. Curto et al. proved the converse for all neural codes on at most four neurons. Via a counterexample on five neurons, we show that this converse is false in general. Additionally, we classify all codes on five neurons with no local obstructions. This classification is enabled by our enumeration of connected simplicial complexes on 5 vertices up to isomorphism. Finally, we examine how local obstructions are related to maximal codewords (maximal sets of neurons that co-fire). Curto et al. proved that a code has no local obstructions if and only if it contains certain “mandatory” intersections of maximal codewords. We give a new criterion for an intersection of maximal codewords to be non-mandatory, and prove that it classifies all such non-mandatory codewords for codes on up to five neurons. [ABSTRACT FROM AUTHOR]

In this paper we study a class of convex sets which are called closed pseudo-cones and study a new duality of this class. It turns out that the duality characterizes closed pseudo-cones and is essentially the only possible abstract duality of them. The characterization of the duality is corresponding to the classification of endomorphisms closed pseudo-cones. [ABSTRACT FROM AUTHOR]

For a separable unital C ⁎ -algebra A and a separable McDuff II 1 -factor M , we show that the space H om w ( A , M ) of weak approximate unitary equivalence classes of unital ⁎-homomorphisms A → M may be considered as a closed, bounded, convex subset of a separable Banach space – a variation on N. Brown's convex structure H om ( N , R U ) . Many separable unital C ⁎ -algebras, including all (separable unital) nuclear C ⁎ -algebras, have the property that for any McDuff II 1 -factor M , H om w ( A , M ) is affinely homeomorphic to the trace space of A . In general H om w ( A , M ) and the trace space of A do not share the same data (several examples are provided). We characterize extreme points of H om w ( A , M ) in many cases, and we give two different conditions – one necessary and the other sufficient – for extremality in general. The universality of C ⁎ ( F ∞ ) is reflected in the fact that for any unital separable A , H om w ( A , M ) may be embedded as a face in H om w ( C ⁎ ( F ∞ ) , M ) . We also extend Brown's construction to apply more generally to H om ( A , M U ) . [ABSTRACT FROM AUTHOR]

In 1987 the author gave an example of a non convex Chebyshev set S in the incomplete inner product space E consisting of the vectors in l 2 which have at most a finite number of non zero terms. In this paper, we show that the closure of S in the Hilbert space completion l 2 of E is not Chebyshev in l 2 . [ABSTRACT FROM AUTHOR]

We prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorithmic and combinatorial applications. Our main application is the first poly-time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor. This new algorithm is a particular instance of new polynomial time deterministic algorithms related to the multiple partial differentiation of polynomials given by evaluation oracles. [ABSTRACT FROM AUTHOR]

• Existing methods are described in consistent notations, with focus on error control. • A preliminary validation case enables to rule out methods not usable in practice. • A comparison on randomly-generated arrays of spheroids enables to determine which method is the most efficient in average. A review of the existing methods to compute the minimal distance between two ellipsoids has been conducted in order to retain the most adequate one within the context of Particle-Resolved Direct Numerical Simulations for particle-laden flows. First, all methods have been implemented and the corresponding algorithms are reported. Furthermore, a procedure has been systematically suggested to control the error associated with each method, when such control was not explicitly available. In a second phase, a two-ellipsoid configuration where an analytical solution is known has been used to perform an error study. This allows to assess the accuracy and consistency of each method, regarding required criteria defined in this paper. The methods that do not verify these criteria have been ruled out. Finally, the remaining methods have been studied on a benchmark of randomly-generated arrays of mono-dispersed spheroids, with aspect ratios ranging from 1 6 to 6 and volume fractions ranging from 0.05 to 0.25. For each method, the spheroidal packings have been sized to measure a statistically significant computing time. Such procedure enabled to study with generality the computing-time dependency of one method on the aspect ratio, the volume fraction, and the desired accuracy. The most efficient method for a given value of these parameters has then been identified. [ABSTRACT FROM AUTHOR]

Abstract: One of the classical results from Szegö shows that if is analytic and univalent in the unit disk , then the section of is univalent in . The exact (largest) radius of the univalence of remains an open problem. On the other hand, not much is known in the case of harmonic univalent functions. It is then natural to consider the class of normalized harmonic mappings in the unit disk satisfying the condition for , where . Functions in are known to be univalent and close-to-convex in . In this paper, we first show that each is convex in the disk , and then determine the value of so that the partial sums of are convex in . [Copyright &y& Elsevier]

Abstract: We present several inequalities for where is the incomplete gamma function. One of our theorems states that the inequalities hold for all nonnegative real numbers if and only if and . Here, denotes the power sum of order t. This extends and complements a result published by Ismail and Laforgia in 2006. [Copyright &y& Elsevier]

Abstract: We consider a nonlinear (possibly) degenerate elliptic operator where the field a and the function b are (unnecessarily strictly) monotonic and a satisfies a very mild ellipticity assumption. For a given boundary datum ϕ we prove the existence of the maximum and the minimum of the solutions and formulate a Haar–Radò type result, namely a continuity property for these solutions that may follow from the continuity of ϕ. In the homogeneous case we formulate some generalizations of the Bounded Slope Condition and use them to obtain the Lipschitz or local Lipschitz regularity of solutions to . We prove the global Hölder regularity of the solutions in the case where ϕ is Lipschitz. [Copyright &y& Elsevier]

Abstract: J.J. Sylvester''s four-point problem asks for the probability that four points chosen uniformly at random in the plane have a triangle as their convex hull. Using a combinatorial classification of points in the plane due to Goodman and Pollack, we generalize Sylvester''s problem to one involving reduced expressions for the long word in . We conjecture an answer of 1/4 for this new version of the problem. [Copyright &y& Elsevier]

Abstract: For a construction is given for convex bodies K and L in such that the orthogonal projection onto the subspace contains a translate of for every direction u, while the volumes of K and L satisfy . A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection onto a k-dimensional subspace ξ contains a translate of , while the mth intrinsic volumes of K and L satisfy for all . For each , we then define the collection to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms). It is subsequently shown that, if , and if the orthogonal projection contains a translate of for every k-dimensional subspace ξ of , then . The families , called k-cylinder bodies of , form a strictly increasing chain where is precisely the collection of centrally symmetric compact convex sets in , while is the collection of all compact convex sets in . Members of each family are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of are shown to satisfy certain geometric inequalities. Related open questions are also posed. [Copyright &y& Elsevier]

Abstract: In this paper we consider sufficient conditions in order to stochastically compare random vectors of multivariate mixture models. In particular we consider stochastic and convex orders, the likelihood ratio order, and the hazard rate and mean residual life dynamic orders. Applications to proportional hazard models and mixture models in risk theory are also given. [Copyright &y& Elsevier]

Abstract: For each we precisely evaluate the main Bellman functions associated with the local estimates of the dyadic maximal operator on . Actually we do that in the more general setting of tree-like maximal operators and with respect to general convex and increasing growth functions. We prove that these Bellman functions equal to analogous extremal problems for the Hardy operator which can be viewed as a symmetrization principle for such operators. Under certain mild conditions on the growth functions we show that for the latter extremals exist (although for the original Bellman functions do not) and analyzing them we give a determination of the corresponding Bellman function. [Copyright &y& Elsevier]

Abstract: We consider the classes of analytic functions introduced recently by K.I. Noor which are defined by conditions joining ideas of close-to-convex and of bounded boundary rotation functions. We investigate coefficients estimates and radii of convexity. [Copyright &y& Elsevier]

Abstract: Using Ruscheweyh derivative and convolution operator, we introduce a new subclass of analytic functions defined in the unit disc. Some inclusion results, a radius problem and some other interesting properties of this class are investigated. [Copyright &y& Elsevier]

Abstract: Briot–Bouquet differential subordinations play a prominent role in the theory of differential subordinations. In this article we consider the dual problem of Briot–Bouquet differential superordinations. Let β and γ be complex numbers, and let Ω be any set in the complex plane C. The function p analytic in the unit disk U is said to be a solution of the Briot–Bouquet differential superordination if The authors determine properties of functions p satisfying this differential superordination and also some generalized versions of it. In addition, for sets and in the complex plane the authors determine properties of functions p satisfying a Briot–Bouquet sandwich of the form Generalizations of this result are also considered. [Copyright &y& Elsevier]

Abstract: This paper investigates the following two-point singular boundary value problems (BVP): where and are fixed given numbers; is a parameter. The results obtained are the global structure of solutions and exact number of solutions when or and q is sufficiently small. [Copyright &y& Elsevier]

Abstract: The main object of the present paper is to show some interesting relations for certain class of analytic functions by using subordination theorem. [Copyright &y& Elsevier]

Abstract: Carlson and Shaffer [SIAM J. Math. Anal. 15 (1984) 737–745] defined a convolution operator on the class A of analytic functions involving an incomplete beta function as . We use this operator to introduce certain classes of analytic functions in the unit disk and study their properties including some inclusion results, coefficient and radius problems. It is shown that these classes are closed under convolution with convex functions. [Copyright &y& Elsevier]

Abstract: For analytic functions and which satisfy the subordination , J.E. Littlewood [Proc. London Math. Soc. 23 (1925) 481–519] has shown some interesting results for integral means of and . The object of the present paper is to derive some applications of integral means by J.E. Littlewood. We also show interesting examples for our theorems. [Copyright &y& Elsevier]

Let F={P(m,F); m∈MF} be a multidimensional steep natural exponential family parameterized by its domain of the means MF and let VF(m) be its variance function. This paper studies the boundary behaviour of VF. Necessary and sufficient conditions on a point m of ∂MF are given so that VF admits a continuous extension VF(m) to the point m . It is also shown that the existence of VF(m) implies the existence of a limit distribution P(m,F) concentrated on an exposed face of MF containing m. The relation between VF(m) and P(m,F) is established and some illustrating examples are given. [Copyright &y& Elsevier]

The traditional solution to the Minkowski problem for polytopes involves two steps. First, the existence of a polytope satisfying given boundary data is demonstrated. In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of Minkowski''s inequality, a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context. In this article we adapt the classical argument to prove both the existence theorem of Minkowski and his mixed volume inequality simultaneously, thereby providing a new proof of Minkowski''s inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry. [Copyright &y& Elsevier]

Conditions are determined for the starlikeness of the Libera transform of functions of bounded turning. In addition, several other differential subordinations and differential inequalities are considered. [Copyright &y& Elsevier]

We prove, that for each r ∈ N , n ∈ N and s ∈ N there are a collection { y i } i = 1 2 s of points y 2 s < y 2 s − 1 < ⋯ < y 1 < y 2 s + 2 π ≕ y 0 and a 2 π - periodic function f ∈ C (∞) (R) , such that (1) f ′ ′ (t) ∏ i = 1 2 s (t − y i) ≥ 0 , t ∈ [ y 2 s , y 0 ] , and for each trigonometric polynomial T n of degree ≤ n (of order ≤ 2 n + 1), satisfying (2) T n ′ ′ (t) ∏ i = 1 2 s (t − y i) ≥ 0 , t ∈ [ y 2 s , y 0 ] , the inequality n r − 1 ‖ f − T n ‖ C (R) ≥ c r ‖ f (r) ‖ C (R) holds, where c r > 0 is a constant, depending only on r. Moreover, we prove, that for each r = 0 , 1 , 2 and any such collection { y i } i = 1 2 s there is a 2 π - periodic function f ∈ C (r) (R) , such that (− 1) i − 1 f is convex on [ y i , y i − 1 ] , 1 ≤ i ≤ 2 s , and, for each sequence { T n } n = 0 ∞ of trigonometric polynomials T n , satisfying (2) , we have lim sup n → ∞ n r ‖ f − T n ‖ C (R) ω 4 (f (r) , 1 ∕ n) = + ∞ , where ω 4 is the fourth modulus of continuity. [ABSTRACT FROM AUTHOR]

We say that a class F consisting of analytic functions f (z) = ∑ n = 0 ∞ a n z n in the unit disk D : = { z ∈ C : | z | < 1 } satisfies a Bohr phenomenon if there exists r f ∈ (0 , 1) such that ∑ n = 1 ∞ | a n z n | ≤ d (f (0) , ∂ f (D)) for every function f ∈ F and | z | = r ≤ r f , where d is the Euclidean distance. The largest radius r f is the Bohr radius for the class F. In this paper, we establish the Bohr phenomenon for the classes consisting of Ma-Minda type starlike functions and Ma-Minda type convex functions as well as for the class of starlike functions with respect to a boundary point. [ABSTRACT FROM AUTHOR]

For 0 < p < + ∞ and an analytic function f (z) in the disk | z | < R , let M p , φ (f , r) be the area integral means of f with respect to the weighted area measure φ ′ (| z | 2) d A (z). We show that (M p , φ (f , r)) 1 p is a convex function of r if f and φ satisfy certain conditions. [ABSTRACT FROM AUTHOR]

Neural codes, represented as collections of binary strings, encode neural activity and show relationships among stimuli. Certain neurons, called place cells, have been shown experimentally to fire in convex regions in space. A natural question to ask is: Which neural codes can arise as intersection patterns of convex sets? While past research has established several criteria, complete conditions for convexity are not yet known for codes with more than four neurons. We classify all neural codes with five neurons as convex/non-convex codes. Furthermore, we investigate which of these codes can be represented by open versus closed convex sets. Interestingly, we find a code which is an open but not closed convex code and demonstrate a minimal example for this phenomenon. [ABSTRACT FROM AUTHOR]