Periods of generalized Fermat curves

Let $k,n\ge 2$ be integers. A generalized Fermat curve of type $(k,n)$ is a compact Riemann surface S that admits a subgroup of conformal automorphisms $H\le \text{Aut}(S)$ isomorphic to ${\mathbb{Z}}_k^n$, such that the quotient surface $S/H$ is biholomorphic to the Riemann sphere $\stackrel{?}{\mathbb{C}}$ and has $n+1$ branch points, each one of order k. There exists a good algebraic model for these objects, which makes them easier to study. Using tools from algebraic topology and integration theory on Riemann surfaces, we find a set of generators for the first homology group of a generalized Fermat curve. Finally, with this information, we find a set of generators for the period lattice of the associated Jacobian variety.     

The effective geometry Monte Carlo algorithm: Applications to molecular communication

In this work, we address the systematic biases and random errors stemming from finite step sizes encountered in diffusion simulations. We introduce the Effective Geometry Monte Carlo (EG-MC) simulation algorithm which modifies the geometry of the receiver. We motivate our approach in a 1D toy model and then apply our findings to a spherical absorbing receiver in a 3D unbounded environment. We show that with minimal computational cost the impulse response of this receiver can be precisely simulated using EG-MC. Afterwards, we demonstrate the accuracy of our simulations and give tight constraints on the single free parameter in EG-MC. Finally, we comment on the range of applicability of our results. While we present the EG-MC algorithm for the specific case of molecular diffusion, we believe that analogous methods with effective geometry manipulations can be utilized to approach a variety of problems in other branches of physics such as condensed matter physics and cosmological large scale structure simulations.     

Geometry dependence of the elongation behavior of a polypropylene melt through a micro die

A series of rheological experiments was performed for a polypropylene (PP) melt to explore its elongation behavior through a capillary die. Using an advanced twin-bore capillary rheometer with dies measuring 1.0, 0.5, and 0.25 mm in diameter, the experiments were performed at 210, 220, and 230 °C. The results indicated that the temperature of the PP melt had a significant effect on its extensional viscosity. The different decreases in the extensional viscosity values in the tested dies revealed the geometry dependence of the extensional viscosity. In the case of PP in the 0.25 mm die at 210 °C, the extensional viscosity values under different extensional strain rates were much higher than those in the other dies. Only in the 1.0 mm die did the relationship between the extensional viscosity of PP and its temperature obey the Arrhenius equation due to the larger die size which related to a slight size effect on its elongation behavior. The calculated deviations of the extensional viscosity in the tested dies demonstrated that the increasing pressure applied to the PP melt in the micro channel was related to the geometry dependence of the elongation behavior of the PP melt. The change in the extensional viscosity eventually relied on the interaction of the die geometry, the temperature, and the extensional stress of the PP melt.     

Symmetry and the golden ratio in the analysis of a regular pentagon

The regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems.     

Noether's Theorem on Surfaces

In this article we prove a version of Noether's Theorem (of Calculus of Variations) which is valid for a general regular (compact) surface. As a special feature, the Lie group of transformations is allowed to act on the Cartesian product of the surface and the functional space. Additionally, we apply the Theorem to a problem in Classical Differential Geometry of surfaces. The given application is actually an example showing how Noether's Theorem can be used to construct invariant properties of the solutions to variational problems defined on surfaces, or equivalently, of the solutions to the associated Euler-Lagrange equations resulting from them.     

Geometric structure of b 2,2-orbihedra and interpolation of operators

We compute the normals to faces of b _2,2-orbihedra and apply the results to Operator Interpolation Theory.     

Homoclinic bifurcation for a general state-dependent Kolmogorov type predator–prey model with harvesting

In this paper, a general Kolmogorov type predator–prey model is considered. Together with a constant-yield predator harvesting, the state dependent feedback control strategies which take into account the impulsive harvesting on predators as well as the impulsive stocking on the prey are incorporated in the process of population interactions. We firstly study the existence of an order-1 homoclinic cycle for the system. It is shown that an order-1 positive periodic solution bifurcates from the order-1 homoclinic cycle through a homoclinic bifurcation as the impulsive predator harvesting rate crosses some critical value. The uniqueness and stability of the order-1 positive periodic solution are derived by applying the geometry theory of differential equations and the method of successor function. Finally, some numerical examples are provided to illustrate the main results. These results indicate that careful management of resources and harvesting policies is required in the applied conservation and renewable resource contexts.     

Infinitely many singular interactions on noncompact manifolds

We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004).     

A finite element method based comparative fracture assessment of carbon black and silica filled elastomers: Reinforcing efficacy of carbonaceous fillers in flexible composites

This study is focused on numerical investigation on fracture behaviors of carbon black (CB) and silica filled elastomeric composites. Finite element analysis (FEA) in compliance with multi-specimen method is used to calculate J-integral and geometry factor of the rubber composites up to a displacement of 20 mm for single edge notch in tension (SENT) and double edge notch in tension (DENT) specimens. An empirical relationship between crack tip opening displacement (CTOD) and crack advancement is established depending on notch to width ratio (NWR). The stress contours across the notches for SENT and DENT specimens is discussed briefly. It is found that fracture propagation resistance of CB filled elastomer is 125% more than that of silica filled elastomer. Although, Silica filled elastomer have good tensile strength and crosslink density but it fails to replace carbon black in terms of fracture properties. The critical J-integral for CB filled elastomer is 18.7% and 32.2% more than silica filled elastomer for SENT and DENT specimens respectively. The effect of specimen type on various fracture properties is also explored. The factor of safety is found to be significantly more in case of CB filled elastomers making them less vulnerable to crack propagation and catastrophic failure.