We study balls of homogeneous cubics on $${\mathbb {R}}^n$$ , $$n = 2,3$$ which are bounded by unity the unit sphere. For 2$$ we completely describe facial structure this norm ball, while for 3$$ classify all extremal points and some families faces.

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. A version of Perelman’s collapsing theorem states: “Let {M3 i } be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and diam(M3 i ) ≥ c0 > 0. Suppose that all unit metric balls in M3 i have very small volume at mos...

Bidimensionality theory was introduced by Demaine et al. [JACM 2005 ] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been sucessfully applied to yield subexponential time parameterized algorithms, EPTASs and linear kernels for many problems on families of graphs excluding a fixed graph H as a minor. In this paper we use several of th...

The contact number of a packing finitely many balls in Euclidean d-space is the touching pairs packing. A prominent subfamily sphere packings formed by so-called totally separable packings: here, called if any two can be separated hyperplane such that it disjoint from interior each ball Bezdek et al. (Discrete Math. 339(2) (2016), 668–676) upper bounded numbers n unit terms and d. In this paper...

j This is one of the 11 Masters of Medicine" series, the introductory volume of which we reviewed in March. Everything as been done to make the record of the life of this truly great ^>an as complete as possible ; and although the material at the author's disposal was somewhat scanty, yet we have a consecutive and very readable biography supplemented by wellhosen extracts from Harvey's works. T...

We establish a new homological lower bound for the Thurston norm on 1-cohomology of 3-manifolds. This generalizes previous results of C. McMullen, S. Harvey, and the author. We also establish an analogous lower bound for 1-cohomology of 2-dimensional CW-complexes. AMS Classification 57M27, 57M20, 57M05

This book constitutes an impassioned plea for providing substitute family life for deprived young children. It has the backing of Miss Balls' experience in the Northumberland Children's Department, where she was able to promote a policy of closing nurseries and enlisting the help of foster parents. The question arises whether her championship of the deprived child, illustrated by selective mate...

We asymptotically estimate the variance of number lattice points in a thin, randomly rotated annulus lying on surface sphere. This partially resolves conjecture Bourgain, Rudnick, and Sarnak. also obtain estimates that are valid for all balls annuli not too small. Our results have several consequences: Linnik sums two squares "microsquare", Bourgain Rudnick small sphere, covering radius distrib...

We prove two lower bounds for the volumes of balls in a Riemannian manifold. If (Mn, g) is a complete Riemannian manifold with filling radius at least R, then it contains a ball of radius R and volume at least δ(n)Rn . If (Mn, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most δ(n)V ol(M,hyp), then the universal cover of (M, g) contains a unit ball with vol...

The study shows that the volumes and surfaces of n-balls, n-simplices, n-orthoplices are holomorphic functions n, which makes those objects omnidimensional, is well defined in any complex dimension. Applications these formulas to omnidimensional polytopes inscribed circumscribed about n-balls reveal previously unknown properties geometric objects. In particular, for 0<n<1, larger than cir...