Big data is large volume, heterogeneous, distributed data. Big data applications where data collection has grown continuously, it is expensive to manage, capture or extract and process data using existing software tools. For example Weather Forecasting, Electricity Demand Supply, social media and so on. With increasing size of data in data warehouse it is expensive to perform data analysis. Dat...

In this paper, we investigate extremal volumes of non-central slices of the unit cube. The case of central hyperplane sections is known and was studied in [Ha], [Ba1] [He1]. The case of non-central sections, i.e. when we dictate that the hyperplane must be a certain distance t > 0 from the center of the cube, is open in general and the same is true about sections of the unit cube by slabs, with...

Percolation with edge-passage probability p and first-passage percolation are studied for the n-cube Bn = {0, 1} with nearest neighbor edges. For oriented and unoriented percolation, p = e/n and p = 1/n are the respective critical probabilities. For oriented first-passage percolation with i.i.d. edge-passage times having a density of 1 near the origin, the percolation time (time to reach the op...

In 1950 Bang proposed a conjecture which became known as “the plank conjecture”: Suppose that a convex set S contained in the unit cube of <n and touching all its sides is covered by planks. (A plank is a set of the form {(x1, . . . , xn): xj ∈ I } for some j ∈ {1, . . . , n} and a measurable subset I of [0, 1]. Its width is defined as |I |.) Then the sum of the widths of the planks is at least...

Las Vergnas Cube Conjecture states that the cube matroid has exactly one class of orientations. We prove that this conjecture is equivalent to saying that the oriented matroid Aff(C), of the affine dependencies of the n-cube C := {−1, 1} over IR, can be reconstructed from the underlying matroid and one of the following partial lists of signed circuits or cocircuits: 1) the signed circuits of ra...

We prove that n cops can capture (that is, some cop can get less than unit distance from) a robber in a continuous square region with side length less than √ 5n and hence that bn/ √ 5c+ 1 cops can capture a robber in a square with side length n. We extend these results to three dimensions, proving that 0.34869 · · ·n2 +O(n) cops can capture a robber in a n× n× n cube and that a robber can forev...

Composition problem is considered for partition constrained vertex subsets of n dimensional unit cube n E . Generating numerical characteristics of n E subsets partitions is considered by means of the same characteristics in 1 − n dimensional unit cube, and construction of corresponding subsets is given for a special particular case. Using pairs of lower layer characteristic vectors for 1 − n E...

We consider tilings and packings of R by integral translates of cubes [0, 2[, which are 4Z-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension d ≤ 4. For higher dimension, we use random methods for generating some examples. Such a cube packing is called non-extendible if we cannot insert a cube in the complement of the pa...

La diafonia b-adica è una misura quantitativa della irregularità di distribuzione di un insieme di punti nel cubo unità s-dimensionale. In questi appunti mostriamo che la diafonia b-adica (per un numero primo b) di un insieme di N punti nel cubo unità di dimensione s è sempre almeno di ordine (logN)(s−1)/2/N . Questo limite inferiore è il migliore possibile. Abstract The b-adic diaphony is a qu...

This report introduces a novel sparse decomposition model for hyperspectral image reconstruction. The model integrates two well-known sparse structures of hyperspectral images: a small set of signature spectral vectors span all spectral vectors (one at each pixel), and like a standard image, a hyperspectral image is spatially redundant. In our model, a threedimensional hyperspectral cube X is f...