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We study covering sumsets of a prime field based on its multiplicative structure. By developing various sufficient analytic and algebraic criteria for their existence, it is shown that covering sumsets arise in two main families, namely in the form of complementary sumsets and in the form of double sumsets. In each case, the abundance of covering sumsets is supported by providing asymptotically growing lower bounds on their number which in turn point out a rich array of fruitful connections to seemingly unrelated topics such as the Titchmarsh divisor problem, Mersenne primes, Fermat quotients, partitions into cycles, quadratic reciprocity, Gauss and Jacobi sums, and density results in class field theory resulting from Chebotarev's theorem. Moreover, representations of an element taken from a prime field, in terms of the sums in a covering sumset, furnish us with new formulas for the class numbers of quadratic fields, Bernoulli numbers and Bernoulli polynomials. In this way, curious tendencies among the number of representations are discovered over half intervals. Lastly, our findings show in different circumstances that the summands of a covering sumset can seldom form an arithmetic progression, thereby indicating a tension between additive and multiplicative structures in a prime field. [ABSTRACT FROM AUTHOR]

We study tails of prime counting functions. Our approach leads to representations with a main term and an error term for the asymptotic size of each tail. It is further shown that the main term is of a specific shape and can be written discretely as a sum involving probabilities of certain events belonging to a perturbed binomial distribution. The limitations of the error term in our representation give us equivalent conditions for various forms of the Riemann hypothesis, for classical type zero-free regions in the case of the Riemann zeta function and the size of semigroups of integers in the sense of Beurling. Inspired by the works of Panaitopol, asymptotic companions pertaining to the magnitude of specific prime counting functions are obtained in terms of harmonic numbers, hyperharmonic numbers and the number of indecomposable permutations. By introducing the notion of asymptotic convexity and fusing it with a nice generalization of an inequality of Ramanujan due to Hassani, we arrive at a curious asymptotic inequality for the classical prime counting function at any convex combination of its arguments and further show that quotients arising from prime counting functions of progressions furnish examples of asymptotically convex, but not convex functions. [ABSTRACT FROM AUTHOR]

Answering a question of Erdös, Komlós proved in 1968 that almost all n×n Bernoulli matrices are nonsingular as n→∞. In this paper, we offer a new perspective on the question of Erdös by studying n×n matrices with prime number entries in an almost all sense. Precisely, it is shown that, as x→∞, the probability of randomly choosing a nonsingular n×n matrix among all n×n matrices with prime number entries that are ≤x is 1. If A is a unitary matrix, then it is well known that |detA|=1. However, the converse is far from being true. As a remedy of this defect, we search for necessary and sufficient conditions for being a unitary matrix by teaming up determinant with trace. In this way, we are led to simple characterizations of unitary matrices in the set of normal matrices. The question of which nonsingular commuting complex matrices with real eigenvalues have the same characteristic polynomial is formulated via determinant and trace conditions. Finally, through a study of eigenvectors, we obtain new characterizations of Hermitian and normal matrices. Our approach to proving these results benefits from a modular interpretation of nonsingularity and the spectral theorem for normal operators together with equality cases of classical inequalities such as the arithmetic-geometric mean inequality and the Cauchy-Schwarz inequality. [ABSTRACT FROM AUTHOR]

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan's constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan's constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers. [ABSTRACT FROM AUTHOR]

Purpose: Context-aware systems for the operating room (OR) provide the possibility to significantly improve surgical workflow through various applications such as efficient OR scheduling, context-sensitive user interfaces, and automatic transcription of medical procedures. Being an essential element of such a system, surgical action recognition is thus an important research area. In this paper, we tackle the problem of classifying surgical actions from video clips that capture the activities taking place in the OR. Methods: We acquire recordings using a multi-view RGBD camera system mounted on the ceiling of a hybrid OR dedicated to X-ray-based procedures and annotate clips of the recordings with the corresponding actions. To recognize the surgical actions from the video clips, we use a classification pipeline based on the bag-of-words (BoW) approach. We propose a novel feature encoding method that extends the classical BoW approach. Instead of using the typical rigid grid layout to divide the space of the feature locations, we propose to learn the layout from the actual 4D spatio-temporal locations of the visual features. This results in a data-driven and non-rigid layout which retains more spatio-temporal information compared to the rigid counterpart. Results: We classify multi-view video clips from a new dataset generated from 11-day recordings of real operations. This dataset is composed of 1734 video clips of 15 actions. These include generic actions (e.g., moving patient to the OR bed) and actions specific to the vertebroplasty procedure (e.g., hammering). The experiments show that the proposed non-rigid feature encoding method performs better than the rigid encoding one. The classifier's accuracy is increased by over 4 %, from 81.08 to 85.53 %. Conclusion: The combination of both intensity and depth information from the RGBD data provides more discriminative power in carrying out the surgical action recognition task as compared to using either one of them alone. Furthermore, the proposed non-rigid spatio-temporal feature encoding scheme provides more discriminative histogram representations than the rigid counterpart. To the best of our knowledge, this is also the first work that presents action recognition results on multi-view RGBD data recorded in the OR. [ABSTRACT FROM AUTHOR]

Using a specific evaluation method for L-functions, it is shown that character values and weighted averages of Gauss sums can be well approximated by linear combinations of the algebraic parts of special values of L-functions under correct parity conditions. This is a surprising universal property of these special values since the coefficients of the resulting combinations turn out to be independent of the character but its parity. [ABSTRACT FROM AUTHOR]

Motivated by the classical work of Ramanujan and recent work of Berndt and Zaharescu, we establish certain infinite families of identities relating values of Dirichlet L-functions at positive integers to Gauss sums and trigonometric sums twisted with characters. [ABSTRACT FROM AUTHOR]

We prove a Lipschitz type summation formula with periodic coefficients. Using this formula, representations of the values at positive integers of Dirichlet L-functions with periodic coefficients are obtained in terms of Bernoulli numbers and certain sums involving essentially the discrete Fourier transform of the periodic function forming the coefficients. The non-vanishing of these L-functions at s = 1 are then investigated. There are additional applications to the Fourier expansions of Eisenstein series over congruence subgroups of $${SL_2(\mathbb{Z})}$$ and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given. [ABSTRACT FROM AUTHOR]

We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the corresponding L-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain Dirichlet L-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series. [ABSTRACT FROM AUTHOR]