We compute an asymptotic estimate of a lower bound of the number of k-convex polyominoes of semiperimeter p. This approximation can be written as μ(k)p4p where μ(k) is a rational fraction of k which up to μ(k) is the asymptotics of convex polyominoes. A polyomino is a connected set of unit square cells drawn in the plane Z × Z [7]. The size of a polyomino is the number of its cells. A central p...

We can generalize some of the constructions in the previous section to higher order curves. One problem with using parabolas as the basic curve element for modelling is that a parabola is a convex curve, convexity being defined as the curve lying on one one side of its tangent lines. This means that in order to model a curve that is not convex like S-curve one must use piecewise parabolic curve...

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities P(n−1/2 ∑n i=1 Xi ∈ A) where X1, . . . , Xn are independent random vectors in R and A is a hyperrectangle, or, more generally, a sparsely conve...

k-means clustering is a popular approach to clustering. It is easy to implement and intuitive but has the disadvantage of being sensitive to initialization due to an underlying non-convex optimization problem. In this paper, we derive an equivalent formulation of k-means clustering. The formulation takes the form of a `0-regularized least squares problem. We then propose a novel convex, relaxed...

In this paper, we prove some fixed points properties and demiclosedness principle for a Banach operator in uniformly convex hyperbolic spaces. We further propose an iterative scheme for approximating a fixed point of a Banach operator and establish some strong and $Delta$-convergence theorems for such operator in the frame work of uniformly convex hyperbolic spaces. The results obtained in this...

Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $mathscr{B}$ (resp. $varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with  $mathscr{B}$ (resp. $varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbase...

 The main purpose of this paper is to study the existence of afixed point in locally convex topology generated by fuzzy n-normed spaces.We prove our main results, a fixed point theorem for a self mapping and acommon xed point theorem for a pair of weakly compatible mappings inlocally convex topology generated by fuzzy n-normed spaces. Also we givesome remarks in locally convex topology generate...

We show that the problem of designing a quantum information processing error correcting procedure can be cast as a bi-convex optimization problem, iterating between encoding and recovery, each being a semidefinite program. For a given encoding operator the problem is convex in the recovery operator. For a given method of recovery, the problem is convex in the encoding scheme. This allows us to ...

In this note we introduce a new family of convex polyhedra which we call the family of generalized deltahedra or in short, the family of g-deltahedra. Here a g-deltahedron is a convex polyhedron in Euclidean 3-space whose each face is an edge-to-edge union of some triangles each being congruent to a given regular triangle. Steiner’s famous icosahedral conjecture (1841) says that among all conve...

By a result of Helton and McCullough [HM12], open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D◦ L of a linear matrix inequality (LMI) L(X) 0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called “Perfect” Positivstellensatz. For example, given a generic convex fre...