We present a parametrised family of groups of order $p^8$, one for each prime $p>3$, and show that the number of descendants of order $p^9$ is not PORC.

In this paper, we consider the relations between Bernoulli polynomials, Legendre polynomials and combinatoric convolution sums of divisor functions. In addition, we give examples of approximate normal distribution derived from combinatoric convolution sums of divisor functions.

This research clarifies the characteristics of radiation fields associated with biophotons on the basis of Hamiltonian dynamics which describes phenomenological aspects of biophotons signals.

In this paper, we give q-analogies of classical Kummer, Lucas and ASH (Anton, Stickelberger, Hensel)'s results on binomial coefficients modulo primes. Our results generalize the previous result by T. Cai (2001)

The celebrated Cahn–Hilliard (CH) equation was proposed to model the process of phase separation in binary alloys by Cahn and Hilliard. Since then the equation has been extended to a variety of chemical, physical, biological, and other engineering fields such as spinodal decomposition, diblock copolymer, image inpainting, multiphase fluid flows, microstructures with elastic inhomogeneity, tumor growth simulation, and topology optimization. Therefore, it is important to understand the basic mechanism of the CH equation in each modeling type. In this paper, we review the applications of the CH equation and describe the basic mechanism of each modeling type with helpful references and computational simulation results.

Cryptographic primitives are fundamental building blocks for designing security protocols to achieve confidentiality,authentication, integrity and non-repudiation. It is not toomuch to say that the selection and integration of appropriate cryptographic primitives into the security protocols determines the largest part of the efficiency and energy consumption of the wireless sensor network (WSN). There are a number of surveys on security issues on WSNs, which, however, did not focus on public-key cryptographic primitives in WSNs. In this survey, we provide a deeper understanding of public-key cryptographic primitives in WSNs including identity-based cryptography and discuss their main directions and some open research issues that can be further pursued. We investigate state-of-the-art software implementation results We investigate state-of-the-art software implementation results of public-key cryptographic primitives in terms of execution time, energy consumption and resource occupation on constrained wireless devices choosing popular IEEE 802.15.4-compliant WSN hardware platforms, used in real-life deployments. This survey provides invaluable insights on public-key cryptographic primitives on WSN platforms, and solutions to find tradeoffs between ost, performance and security for designing security protocols in WSNs.

Our algorithm precomputes a primitive rs-th root of unity   where s is the largest positive integer satisfying rsjq   1, and is applicable for the cases when s is small.
The proposed algorithm requires one exponentiation for the r-th root computation and is favorably compared to the existing algorithms..
Our algorithm precomputes a primitive rs-th root of unity   where s is the largest positive integer satisfying rsjq   1, and is applicable for the cases when s is small.
The proposed algorithm requires one exponentiation for the r-th root computation and is favorably compared to the existing algorithms.

Using self-consistent three-dimensional (3D) N-body simulations, we investigate the physical properties of nonaxisymmetric features in a disk galaxy created by a tidal interaction with its companion. The primary galaxy consists of a stellar disk, a bulge, and a live halo, corresponding to Milky-Way-type galaxies, while the companion is represented by a halo alone. We vary the companion mass and the pericenter distance to explore situations with differing tidal strength parameterized by either the relative tidal force P or the relative imparted momentum S. We find that the formation of a tidal tail in the outer parts requires or . A stronger interaction results in a stronger, less wound tail that forms earlier. Similarly, a stronger tidal forcing produces stronger, more loosely wound spiral arms in the inner parts. The arms are approximately logarithmic in shape, with both amplitude and pitch angle decaying with time. The derived pattern speed decreases with radius and is close to the curve at late time, with Ω and κ denoting the angular and epicycle frequencies, respectively. This suggests that the tidally induced spiral arms are most likely kinematic density waves weakly modified by self-gravity. Compared to the razor-thin counterparts, arms in the 3D models are weaker, have a smaller pitch angle, and wind and decay more rapidly. The 3D density structure of the arms is well described by the concentrated and sinusoidal models when the arms are in the nonlinear and linear regimes, respectively. We demonstrate that dynamical friction between interacting galaxies transfers the orbital angular momentum of one galaxy to the spin angular momentum of the companion halo.

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We describe directed searches for continuous gravitational waves (GWs) in data from the sixth Laser Interferometer Gravitational-wave Observatory (LIGO) science data run. The targets were nine young supernova remnants not associated with pulsars; eight of the remnants are associated with non-pulsing suspected neutron stars. One target's parameters are uncertain enough to warrant two searches, for a total of 10. Each search covered a broad band of frequencies and first and second frequency derivatives for a fixed sky direction. The searches coherently integrated data from the two LIGO interferometers over time spans from 5.3–25.3 days using the matched-filtering -statistic. We found no evidence of GW signals. We set 95% confidence upper limits as strong (low) as 4 × 10⁻²⁵ on intrinsic strain, 2 × 10⁻⁷ on fiducial ellipticity, and 4 × 10⁻⁵ on r-mode amplitude. These beat the indirect limits from energy conservation and are within the range of theoretical predictions for neutron-star ellipticities and r-mode amplitudes.