Lyrics in Congregational Song: A Biblical and Historical Survey

Is there a standard as to what constitutes biblical, God-honoring congregational song lyrics? In this paper, I will seek to address this question through a review of relevant biblical principles and positions of prominent leaders throughout church history. The Bible provides a language of worship in the Psalms and includes some information regarding singing in the New Testament. Biblical principles have been applied throughout the centuries by various church fathers. The synthesis of biblical principles and historical contributions will produce a set of guidelines by which congregational song lyrics may be evaluated for the purpose of contextual application in modern services

On the Existence and Frequency Distribution of the Shell Primes

This research presents the results of a study on the existence and frequency distribution of the shell primes defined herein as prime numbers that result from the calculation of the "half-shell" of an p-dimensional entity of the form $n^p-(n-1)^p$ where power $p$ is prime and base $n$ is the realm of the positive integers. Following the introduction of the shell primes, we will look at the results of a non-sieving application of the Euler zeta function to the prime shell function as well as to any integer-valued polynomial function in general which has the ability to produce prime numbers when power $p$ is prime. One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no known way to sieve composite numbers out of the product term in this famous equation. Such would be case when an infinite series of numbers to be analyzed are calculated by a polynomial expression that yields successively increasing positive integer values and which has as its input domain the positive integers themselves. In such cases there may not be an intuitive way to eliminate the composite terms from the product term in the Euler zeta function equation by either scaling a previous prime number calculation or by employing predictable values of the domain of the function which would render outputs of the polynomial prime. So the best one may be able to hope for in these cases is to calculate some value to be added or subtracted from unity in the numerator above the product term in the Euler Zeta function to make both sides of that equation equal with the expectation that that value could be used to predict the number of prime numbers that exist as outputs of the polynomial function for some limit less than or equal to x of the input domain.Comment: Version 5: Spelling corrections only ("Riemann" on page 5

A Non-Sieving Application of the Euler Zeta Function

One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no known way to sieve composite numbers out of the product term in this famous equation. Such would be case when an infinite series of numbers to be analyzed are calculated by a polynomial expression that yields successively increasing positive integer values and which has as its input domain the positive integers themselves. In such cases there may not be an intuitive way to eliminate the composite terms from the product term on the right-hand side of the Euler zeta function equation by either scaling a previous prime number calculation or by employing predictable values of the domain of the function which would make future outputs of the polynomial prime. So the best one may be able to hope for in this case is to determine some value to be added or subtracted from unity in the numerator above the product term to make both sides of the equation equal in the hope that that value can be used to predict the number of prime numbers that exist as outputs of the polynomial function for some limit less than or equal to x of the input domain.Comment: V3 added a modified version of the big Zeta equation that adapts to integer-valued functions for which the first term generated is not unity. This version also generalizes Theorem 1 to include all integer-valued polynomials, even those which do not generate prime numbers, thus opening up the potential for the M-series to be used to predict other properties of integer-valued polynomial functions. arXiv admin note: substantial text overlap with arXiv:1510.0102

Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods

We present a robust and efficient target-based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint-based mesh adaptation over uniform and residual-based mesh refinement, and secondly to investigate the efficiency of the global error estimate

Singlet-doublet fermion and triplet scalar dark matter with radiative neutrino masses

We present a detailed study of a combined singlet-doublet fermion and triplet scalar model for dark matter. These models have only been studied separately in the past. Together, they form a simple extension of the Standard Model that can account for dark matter and explain the existence of neutrino masses, which are generated radiatively. This holds even if singlet-doublet fermions and triplet scalars never contribute simultaneously to the dark matter abundance. However, this also implies the existence of lepton flavour violating processes. In addition, this particular model allows for gauge coupling unification. The new fields are odd under a new $\mathbb{Z}_2$ symmetry to stabilise the dark matter candidate. We analyse the dark matter, neutrino mass and lepton flavour violation aspects both separately and in conjunction, exploring the viable parameter space of the model. This is done using a numerical random scan imposing successively the neutrino mass and mixing, relic density, Higgs mass, direct detection, collider and lepton flavour violation constraints. We find that dark matter in this model is fermionic for masses below about 1 TeV and scalar above. The narrow mass regions found previously for the two separate models are enlarged by their coupling. While coannihilations of the weak isospin partners are sizeable, this is not the case for fermions and scalars despite their often similar masses due to the relatively small coupling of the two sectors, imposed by the small neutrino masses. We observe a high degree of complementarity between direct detection and lepton flavour violation experiments, which should soon allow to fully probe the fermionic dark matter sector and at least partially the scalar dark matter sector.Comment: 24 pages, 12 figures; version accepted by and published in JHE

Singlet-doublet/triplet dark matter and neutrino masses

In these proceedings, we present a study of a combined singlet--doublet fermion and triplet scalar model for dark matter (DM). Together, these models form a simple extension of the Standard Model (SM) that can account for DM and explain the existence of neutrino masses, which are generated radiatively. However, this also implies the existence of lepton flavour violating (LFV) processes. In addition, this particular model allows for gauge coupling unification. The new fields are odd under a new $\mathbb{Z}_2$ symmetry to stabilise the DM candidate. We analyse the DM, neutrino mass and LFV aspects, exploring the viable parameter space of the model. This is done using a numerical random scan imposing successively the neutrino mass and mixing, relic density, Higgs mass, direct detection, collider and LFV constraints. We find that DM in this model is fermionic for masses below about 1 TeV and scalar above. We observe a high degree of complementarity between direct detection and LFV experiments, which should soon allow to fully probe the fermionic DM sector and at least partially the scalar DM sector.Comment: 4 pages, 1 figure; contribution to the 2019 EW session of the 54th Rencontres de Moriond (summary of arXiv:1812.11133

Homology of E_n Ring Spectra and Iterated THH

We describe an iterable construction of THH for an E_n ring spectrum. The reduced version is an iterable bar construction and its n-th iterate gives a model for the shifted cotangent complex at the augmentation, representing reduced topological Quillen homology of an augmented E_n algebra.Comment: Some additional exposition added. Minor correction