Strictly layered feedforward networks with binary neurons are viewed as maps from the vertex set of an n cube to the vertex set of an l cube With only one output neuron in principle they can realize any Boolean function on n inputs We address the problem of determining the necessary and su cient numbers of hidden units for this task by using separability properties of a ne oriented hyperplane a...

A new algebraic cubature formula of degree 2n+1 for the product Chebyshev measure in the d-cube with ≈ n/2 nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree n in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpo...

We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with $p$-faces $n$-cube (for $n>p$) and stabilizer constraints faces dimension $(p\pm1)$. The code identifying antipodal resulting complex encodes one logical qubit into $N = 2^{n-p-1} \tbinom{n}{p}$ physical displays local testability soundness $\Omega(1/\log(N))$ beating current state-of-the-art $1/\log...

We show that half-integral polytopes obtained as the convex hull of a random set of half-integral points of the 0/1 cube have rank as high as Ω(log n/ log log n) with positive probability — even if the size of the set relative to the total number of half-integral points of the cube tends to 0. The high rank is due to certain obstructions. We determine the exact threshold number, when these obst...

Define a cubical complex to be a collection of integer-aligned unit cubes in d dimensions. Lebowitz and Mazel (1998) proved that there are between (C1d) and (C2d) complexes containing a fixed cube with connected boundary of (d − 1)volume n. In this paper we narrow these bounds to between (C3d) and (C4d). We also show that there are nn/(2d(d−1))+o(1) connected complexes containing a fixed cube w...

We determine the maximal hyperplane sections of regular n-simplex, if distance to centroid is fairly large, i.e. larger than midpoint edges. Similar results for n-cube and l1n-ball were obtained by Moody, Stone, Zach Zvavitch Liu Tkocz. The hyperplanes in these three cases are perpendicular vectors from vertices. For smaller distances -in a well-defined range- we show that at least locally maxi...

The Möbius cube MQn proposed by Cull et al. is an alternative to the popular hypercube network. Recently, MQn was shown to be pancyclic, i.e., cycles of any lengths at least four can be embedded into it. Due to the importance of the fault tolerance in the parallel processing area, in this paper, we study an injured MQn with mixed node and link faults. We show that it is (n − 2)-fault-tolerant p...

This paper deals with bases of tropical Plücker functions defined on a Boolean (hyper)cube and is devoted to a combinatorial description of a wide class of such bases via a relationship to certain classes of wiring and rhombus tiling diagrams. For a positive integer n, let [n] denote the ordered set of elements 1, 2, . . . , n. Consider a real-valued function f on the subsets of [n], or on the ...

The crossed cube CQn introduced by Efe has many properties similar to those of the popular hypercube. However, the diameter of CQn is about one half of that of the hypercube. Failures of links and nodes in an interconnection network are inevitable. Hence, in this paper, we consider the hybrid fault-tolerant capability of the crossed cube. Letting fe and fv be the numbers of faulty edges and ver...

In this note we present a construction which improves the best known bound on minimal dispersion of large volume boxes in unit cube. Let d>1. The T⊂[0,1]d is defined as supremum taken over all axis parallel cube do not intersect T. n points infimum T such that |T|=n. Define “large volume” regime class volumes 14<r≤12. inverse denoted N(r,d). When large, upper N(r,d) order (r−14)−1. presented yi...