Sterling stirling play

In this paper we analyze a recently proposed impartial combinatorial ruleset that is played on a permutation of the set \(\left[ n\right] \). We call this ruleset Stirling Shave. A procedure utilizing the ordinal sum operation is given to determine the nim value of a given normal play position. Additionally, we enumerate the number of permutations of \(\left[ n\right] \) which are \(\mathcal {P}\)-positions. The formula given involves the Stirling numbers of the first-kind. We also give a complete analysis of the Misère version of Stirling Shave using Conway’s genus theory. An interesting by-product of this analysis is insight into how the ordinal sum operation behaves in Misère Play.     

Exponential stability for laminated beams with a frictional damping

We study a laminated beam which consists of two identical layers of uniform thickness, taking into account that an adhesive of small thickness is bonding the two surfaces thereby producing an interfacial slip. We show that when the frictional damping acts on the effective rotation angle there is no need for any other kind of internal or boundary control to produce exponential stability for the system.     

Lattice points in bodies of revolution II

In [3] it was shown that when a three-dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomplish this, however, it was necessary to assume a non-vanishing condition on the third derivative of the generatrix. In this article we drop this condition, showing that the aforementioned bound holds for a wider family of revolution bodies, which includes those with analytic boundary. A novelty in our approach is that, besides the usual analytic methods, it requires studying some Diophantine properties of the Taylor coefficients of the phase on the Fourier transform side.     

Linear combinations of prime powers in X-coordinates of Pell equations

The Ramanujan Journal - We give q-analogues of three Ramanujan-type series for $$1/\pi $$ from q-analogues of ordinary WZ pairs. The first one is a new q-analogue of the following Ramanujan’s...     

A simple comparison between Skorokhod & Russo-Vallois integration for insider trading

We consider a simplified version of the problem of insider trading in a financial market. We approach it by means of anticipating stochastic calculus and compare the use of the Skorokhod and the Russo-Vallois forward integrals within this context. We conclude that, while the forward integral yields results with a clear financial meaning, the Skorokhod integral does not provide a suitable formulation for this problem.     

Properties of interpolatory quadrature with equidistant nodes on the unit circle

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error.     

Optimization model for a production,inventory, distribution and routing problem in small furniture companies

Production and distribution are two key decisions in supply chain planning. In order to achieve an effective operational performance, it is important for these two decisions to be integrated, especially in supply chains with low inventory levels. In this paper, we propose a mixed integer programming model to integrate production, inventory, distribution and routing decisions in a single framework. The model was inspired by small Brazilian furniture companies and focuses on production and distribution decisions at an operational level. In particular, we consider a scenario in which only one production line and one vehicle, which makes multiple trips over the planning horizon, are available to produce items and deliver final products, respectively. We also take into account some features rarely considered in the literature, but commonly found in real-world applications, such as producing and stocking multiple items, distribution routes extending over one or more periods, multiple time windows and customers’ due dates. Computational tests on a set of randomly generated instances were carried out using a well-known optimization software and six relax-and-fix heuristics, which explore different criteria for partitioning and fixing variables. We also implemented two hybrid heuristics in which an initial solution is first constructed and then fed into the optimization software to improve it. The results showed that one relax-and-fix and the two hybrid heuristics performed better than the solver on the largest instances.     

Equivariant Classification of <Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>-symplectic Surfaces

Inspired by Arnold’s classification of local Poisson structures [1] in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of b^m-Poisson structures which can be also visualized using differential forms with singularities as b^m-symplectic structures. In this paper we extend the classification scheme in [24] for b^m-symplectic surfaces to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct b^m-symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.     

Choreographies in the <Emphasis Type="Italic">n</Emphasis>-vortex Problem

We consider the equations of motion of n vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, corresponds to choreographies of n vortices. We include numerical results for all cases, for various values of n, and we provide key details on the computational approach.     

Frames of exponentials and sub-multitiles in LCA groups

In this note, we investigate the existence of frames of exponentials for $L^2(\mathrm{\Omega })$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $\mathrm{\Omega }?\stackrel{?}{G}$ with respect to a uniform lattice Γ of $\stackrel{?}{G}$ guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of Γ. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.