Let S(n, d) and C(n, d) denote, respectively, the stacked and the cyclic d-dimensional polytopes on n vertices. Furthermore, fi(P ) denotes the number of i-dimensional faces of a polytope P . Theorem 1. Let P be a d-dimensional simplicial polytope. Suppose that fr(S(n1, d)) ≤ fr(P ) ≤ fr(C(n2, d)) for some integers n1, n2 and r ≤ d − 2. Then, fs(S(n1, d)) ≤ fs(P ) ≤ fs(C(n2, d)) for all s such ...

In this supplementary document, we prove two theorems that support the algorithms described in [1] and we present experimental results from systematic study of the performance of Crane when the value of the tuneable parameters are varied. First, in Theorem 1, we provide bound on the score of a state function that can be obtained by extending that state function, in terms of simple statistics of...

We improve the theorem of Beck giving a lower bound on number $k$-flats spanned by set points in real space, and Elekes T\'oth incidences between space.

Recently there has been some interest in the combinatorics of the geometry of the Hamming space, e.g., [10], and in particular, in tilings of this space [8]. Here, we investigate partitions of the Hamming space into spheres with possibly different radii. Such a partition is sometimes called a generalized perfect code, see e.g. [1, 3, 6, 13, 15]. Generalized spherepacking bounds can be found in ...

We propose to interpret Levinson’s theorem as an index theorem. This exhibits its topological nature. It furthermore leads to a more coherent explanation of the corrections due to resonances at thresholds.

The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then ( ∫ fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound...

In this paper, we give a constant $C$ in \cite[Theorem 1.2]{sha2014bounding} by using an explicit Baker's inequality, hence have bound of the integral points on modular curves.

In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand’s lower bound on the correlatio...

Abstract. A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i > 0 interior lattice points. We give a new proof for this theorem and classify polygons with maximal volume. Then we show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimension. From a theorem of Victor Batyrev the finiteness of ...