let $g$ be a connected graph with vertex set $v(g)$‎. ‎the‎ ‎degree resistance distance of $g$ is defined as $d_r(g) = sum_{{u‎,‎v} subseteq v(g)} [d(u)+d(v)] r(u,v)$‎, ‎where $d(u)$ is the degree‎ ‎of vertex $u$‎, ‎and $r(u,v)$ denotes the resistance distance between‎ ‎$u$ and $v$‎. ‎in this paper‎, ‎we characterize $n$-vertex unicyclic‎ ‎graphs having minimum and second minimum degree resista...

We study a natural generalization of the heavy hitters problem in the streaming context. We term this generalization block heavy hitters and define it as follows. We are to stream over a matrix A, and report all rows that are heavy, where a row is heavy if its `1-norm is at least φ fraction of the `1 norm of the entire matrix A. In comparison, in the standard heavy hitters problem, we are requi...

In this paper, a text dependent speaker recognition algorithm based on spectrogram is proposed. The spectrograms have been generated using Discrete Fourier Transform for varying frame sizes with 25% and 50% overlap between speech frames. Feature vector extraction has been done by using the row mean vector of the spectrograms. For feature matching, two distance measures, namely Euclidean distanc...

In the Manhattan Sequence Consensus problem (MSC problem) we are given k integer sequences, each of length l, and we are to find an integer sequence x of length l (called a consensus sequence), such that the maximum Manhattan distance of x from each of the input sequences is minimized. For binary sequences Manhattan distance coincides with Hamming distance, hence in this case the string consens...

High-dimensional vectors are ubiquitous in algorithms and this lecture seeks to introduce some common properties of these vectors. We encounter the so-called curse of dimensionality which refers to the fact that algorithms are simply harder to design in high dimensions and often have a running time exponential in the dimension. We also show that it is possible to reduce the dimension of a datas...

We propose a new definition for the gradient field of a discrete image, defined on a twice finer grid. The differentiation process from the image to its gradient field is viewed as the inverse operation of linear integration, and the proposed mapping is nonlinear. Then, we define the total variation of an image as the `1 norm of its gradient field amplitude. This new definition of the total var...

In this paper we define, for each aspherical orientable 3-manifold M endowed with a torus splitting T , a 2-dimensional fundamental l1-class [M ] whose l1norm has similar properties as the Gromov simplicial volume of M (additivity under torus splittings and isometry under finite covering maps). Next, we use the Gromov simplicial volume of M and the l1-norm of [M ] to give a complete characteriz...

We study an intermittent map which has exactly two ergodic invariant densities. The densities are supported on two subintervals with a common boundary point. Due to certain perturbations, leakage of mass through subsets, called holes, of the initially invariant subintervals occurs and forces the subsystems to merge into one system that has exactly one invariant density. We prove that the invari...

A distribution D on a set S ⊂ Zp -fools polynomials of degree at most d in N variables over Zp if for any such polynomial P , the distribution of P (x) when x is chosen according to D differs from the distribution when x is chosen uniformly by at most in the `1 norm. Distributions of this type generalize the notion of -biased spaces and have been studied in several recent papers. We establish t...

Recently, a distance function was defined on the face-centered cubic and body-centered cubic grids by combining weights and neighbourhood sequences. These distances share many properties with traditional path-based distance functions, such as the city-block distance, but are less rotational dependent. We present conditions for metricity and algorithms to compute the distances.