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Differential privacy becomes a standard for evaluating the privacy protection performance for an algorithm these years. However, the definition of differential privacy seems not so easy to understand as the classical k-anonymity and etc. In this paper, we propose a new measure which is more comprehensible. Some properties of such measure are investigated and the relationship between our new definition and differential privacy is studied. [ABSTRACT FROM AUTHOR]

Music plays a prominent role in society and companies have even started studying its aspects for commercial purposes. It is only natural to ask what characteristics make certain songs appealing. While much research has been conducted on the mathematical principles of sound, there has been less focus on analyzing the structure of popular songs from a mathematical perspective. One mathematical tool that researchers have used to study musical structure is seriation, ordering. This paper applies several types of seriation algorithms to conduct a mathematical analysis of the structural qualities of several musical pieces. This paper focuses on 10 popular artists and their musical influences. The artists chosen for this research are linked because of the influences they cite, musical genre, and the popularity of their music. Results show that an artist's songs have a higher quantitatively measured connection with the artists they cite as influences rather than the artists who they never mention as musical influences. [ABSTRACT FROM AUTHOR]

In this paper, we consider the discrete unit square cover (DUSC) problem as follows: given a set 𝒫 of n points and a set 𝒮 of m axis-aligned unit squares in ℝ 2 , the objective is (i) to check whether the union of the squares in 𝒮 covers all the points in 𝒫 , and (ii) if the answer is yes, then select a minimum cardinality subset 𝒮 ∗ ⊆ 𝒮 such that each point in 𝒫 is covered by at least one square in 𝒮 ∗ . For the DUSC problem: (i) we propose a (2 + 4 k − 2) -approximation algorithm, where k (> 2) is an integer parameter that defines a trade-off between the running time and the approximation factor of the algorithm. The running time of our proposed algorithm is O (k m k n + n log n). Our solution of the DUSC problem is based on a simple (1 + 2 k − 2) -approximation algorithm for the subproblem strip square cover (SSC) problem, where all the points in 𝒫 are lying within a horizontal strip of unit height. (ii) we also propose a 2-approximation algorithm, which runs in O (m 4 n) time. The 2-approximation algorithm is based on an algorithm for the subproblem SSC problem. The algorithm for the subproblem is developed using plane sweep and graph search traversal techniques. We also extend this algorithm to get 2-approximation result for the weighted DUSC problem where the squares are assigned weights, and the aim is to choose a subset 𝒮 ∗ ⊆ 𝒮 such that each point in 𝒫 is covered by at least one square in 𝒮 ∗ and the sum of the weights of squares in 𝒮 ∗ is minimized. [ABSTRACT FROM AUTHOR]

In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code (PD-Code). PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan et al. We present the set of PD-codes as a standalone object and discuss its relationship with link diagrams. We give an explicit algorithm for reconstructing a knot diagram on a surface from a PD-code. We also discuss the intrinsic symmetries of PD-codes (i.e. invertibility and chirality). The moves analogous to the Reidemeister moves are also explored, and we show that the given set of PD-codes modulo these combinatorial Reidemeister moves is equivalent to classical link theory. [ABSTRACT FROM AUTHOR]

We present new expressions for the densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix. The series expansions for these expressions involve fewer terms than previously known results. For the trace, we also present a new algorithm that is linear in the size of the matrix and the degree of truncation, which is optimal. [ABSTRACT FROM AUTHOR]

We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups. [ABSTRACT FROM AUTHOR]

Most of the algorithms used for information extraction and for processing the amino acid chains that make up proteins treat them as symbolic chains. Fewer algorithms exploit signal processing techniques that require a numerical representation of amino acid chains. However, these algorithms are very powerful for extracting regularities that cannot be detected when working with a symbolic chain, which may be important for understanding the biological meaning of a sequence or in classification tasks. In this study, a new mathematical representation of amino acid chains is proposed, which is derived using a similarity measure based on the PAM250 amino acid substitution matrix and that generates 20 signals for each protein sequence. Using this representation 20 consensus spectra for a protein family are determined and the relevance of the frequency peaks is established, obtaining a group of significant frequency peaks that manifest common periodicities of the amino acid sequences that belong to a protein family. We also show that the proposed representation in 20 signals can be integrated into Chou's pseudo amino acid composition (PseAAC) and constitute a useful alternative to amino acid physicochemical properties in Chou's PseAAC. [ABSTRACT FROM AUTHOR]

In the smooth case, we prove quasi-flasqueness for the sheaves of all subelliptic multipliers as well as at each of the steps of the Kohn algorithm on a pseudoconvex domain in ℂn. We use techniques by Jean-Claude Tougeron to show that if the domain has a real-analytic defining function, the modified Kohn algorithm involving generating ideals and taking real radicals only in the ring of real-analytic germs yields quasi-coherent sheaves. This sharpens a result obtained by J. J. Kohn in 1979. [ABSTRACT FROM AUTHOR]