Zeros of derivatives of sum of Dirichlet L-functions

A linear combination L(s) of two Dirichlet L-functions has infinitely many complex zeros in Res<0. In this note we prove an infinity of complex zeros of L(k)(s) in the same region.     

Numerical analysis of 3D vortical cavity flow

The vortical flows of an incompressible fluid in a rectangular three-dimensional container with a large spanwise aspect ratio driven by a moving solid lid are studied using a combined compact finite difference (CCD) scheme with high accuracy and high resolution. The study focuses on the change of the steady flow structures in the cavity with Reynolds numbers ranging from 100 to 850. The results of the flow in the cavity with a spanwise aspect ratio 6.55 show that several stable closed streamlines localized near the symmetric plane are found at Re ≥500, while a closed stable streamline is found near the side wall at Re ≤300. The change of the flow pattern present in this system affects the diffusion properties in the flow but seems to have no qualitative effect on the global flow properties which include energy dissipation in the cavity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of landslide model with time delay and periodic parameter perturbations

In present paper, we analyze the dynamics of a single-block model on an inclined slope with Dieterich–Ruina friction law under the variation of two new introduced parameters: time delay T_d and initial shear stress μ. It is assumed that this phenomenological model qualitatively simulates the motion along the infinite creeping slope. The introduction of time delay is proposed to mimic the memory effect of the sliding surface and it is generally considered as a function of history of sliding. On the other hand, periodic perturbation of initial shear stress emulates external triggering effect of long-distant earthquakes or some non-natural vibration source. The effects of variation of a single observed parameter, T_d or μ, as well as their co-action, are estimated for three different sliding regimes: β < 1, β = 1 and β > 1, where β stands for the ratio of long-term to short-term stress changes. The results of standard local bifurcation analysis indicate the onset of complex dynamics for very low values of time delay. On the other side, numerical approach confirms an additional complexity that was not observed by local analysis, due to the possible effect of global bifurcations. The most complex dynamics is detected for β < 1, with a complete Ruelle–Takens–Newhouse route to chaos under the variation of T_d, or the co-action of both parameters T_d and μ. These results correspond well with the previous experimental observations on clay and siltstone with low clay fraction. In the same regime, the perturbation of only a single parameter, μ, renders the oscillatory motion of the block. Within the velocity-independent regime, β = 1, the inclusion and variation of T_d generates a transition to equilibrium state, whereas the small oscillations of μ induce oscillatory motion with decreasing amplitude. The co-action of both parameters, in the same regime, causes the decrease of block’s velocity. As for β > 1, highly-frequent, limit-amplitude oscillations of initial stress give rise to oscillatory motion. Also for β > 1, in case of perturbing only the initial shear stress, with smaller amplitude, velocity of the block changes exponentially fast. If the time delay is introduced, besides the stress perturbation, within the same regime, the co-action of T_d (T_d < 0.1) and small oscillations of μ induce the onset of deterministic chaos.     

Symmetric spaces with maximal projection constants

In this paper we introduce a special class of finite-dimensional symmetric subspaces of L₁, so-called regular symmetric subspaces. Using this notion, we show that for any k?2, there exist k-dimensional symmetric subspaces of L₁ which have maximal projection constant among all k-dimensional symmetric spaces. Moreover, L₁ is a maximal overspace for these spaces (see Theorems 4.4 and 4.5.) Also a new asymptotic lower bound for projection constants of symmetric spaces is obtained (see Theorem 5.3). This result answers the question posed in [12, p. 36] (see also [15, p. 38]) by H. Koenig and co-authors. The above results are presented both in real and complex cases.     

An asymmetric Orlicz centroid inequality for probability measures

Using M-addition,an asymmetric Orlicz centroid inequality for absolutely continuous probability measures is established corresponding to Paouris and Pivovarov’s recent result on the symmetric case.As an application,we extend Haberl and Schuster’s asymmetric Lp centroid inequality from star bodies to compact sets.     

Predicting vortex-induced vibration from driven oscillation results

Two-dimensional simulations of flow past both an elastically-mounted cylinder and an externally-driven oscillating cylinder were performed at a Reynolds number of Re = 200. The results were compared to determine if the oscillations of the driven-oscillation model were consistent with the oscillations observed in the elastically-mounted system. It was found that while this is the case, there is considerable sensitivity to input forcing. This sensitivity could explain observed discrepancies between experimental results for the two systems.     

Sur la cohomologie à support des fibrés en droites sur les variétés symétriques complètes

Let X be a complete symmetric variety, i.e., the wonderful compactification of a symmetric G-homogeneous space (where G is a simply connected semi-simple linear algebraic group). If L is a line bundle over X and if C is a Bialynicki-Birula cell of codimension c in X, then the Lie algebra $ \mathfrak{g} $ of G operates naturally on the cohomology group with support H _C ^c (L). A necessary condition on C for the existence of a finite-dimensional simple subquotient of this $ \mathfrak{g} $ -module is given. As applications one calculates the Euler–Poincaré characteristic of L over X, estimates the higher cohomology group H ^d (X, L), d ≥ 0, and obtains the exact formulas in some cases including that of the complete conic variety.     

Algebraic shifting of cyclic polytopes and stacked polytopes

Gil Kalai introduced the shifting-theoretic upper bound relation as a method to generalize the g-theorem for simplicial spheres by using algebraic shifting. We will study the connection between the shifting-theoretic upper bound relation and combinatorial shifting. Also, we will compute the exterior algebraic shifted complex of the boundary complex of the cyclic d-polytope as well as of a stacked d-polytope. It will turn out that, in both cases, the exterior algebraic shifted complex coincides with the symmetric algebraic shifted complex.     

Non-split sums of coefficients of GL(2)-automorphic forms

Given a cuspidal automorphic form π on GL₂, we study smoothed sums of the form $\sum\nolimits_n {{a_\pi }({n^2} + d)V({n \over x})} $ . The error term we get is sharp in that it is uniform in both d and Y and depends directly on bounds towards Ramanujan for forms of half-integral weight and Selberg eigenvalue conjecture. Moreover, we identify (at least in the case where the level is square-free) the main term as a simple factor times the residue as s = 1 of the symmetric square L-function L(s, sym² π). In particular there is no main term unless d > 0 and π is a dihedral form.     

Long time behavior of heat kernels of operators with unbounded drift terms

Given a second-order elliptic operator on Rd, with bounded diffusion coefficients and unbounded drift, which is the generator of a strongly continuous semigroup on L²(Rd) represented by an integral, we study the time behavior of the integral kernel and prove estimates on its decay at infinity. If the diffusion coefficients are symmetric, a local lower estimate is also proved.     

Differential recursion relations for Laguerre functions on symmetric cones

Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R⁺. This family forms an orthogonal basis for the subspace of L-invariant functions in L²(Ω,dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L²(Ω,dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T(Ω)=Ω+iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R⁺.     

An assessment of turbulence models applied to the simulation of a two-dimensional submerged jet

This paper is concerned with the investigation of the performance of different turbulence models in the numerical prediction of transient flow caused by a confined submerged jet. Several widely used models, i.e., the standard k–ε, RNG k–ε, low Reynolds number k–ε models and the differential Reynolds stress model, as included in CFD codes, were compared with each other for a two-dimensional, incompressible, turbulent jet flow and with reported experimental data. A flapping oscillation was predicted regardless of the model used. A chosen Strouhal (St) number definition brought the fundamental frequencies from both the experiments and computations into close proximity. However, different turbulence models have exhibited quite different behaviours in terms of the frequency and regularity of the oscillation and in terms of the scale and duration of the vortices generated. All versions of the k–ε model yielded regular oscillations, which agree with experimental observations. On the other hand, the Reynolds stress (RS) model produced a complex pattern but a slower dissipation of vortices. In addition, some aspects of gridding and inflow representation are also discussed.     

Cutoff for the Ising model on the lattice

Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in L ¹ on a system of size n is O(logn). Whether in this regime there is cutoff, i.e. a sharp transition in the L ¹-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at (c+o(1))logn for some fixed c>0, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For ?² this carries all the way to the critical temperature. Specifically, for fixed d≥1, the continuous-time Glauber dynamics for the Ising model on (?/n?) ^d with periodic boundary conditions has cutoff at (d/2λ _∞)logn, where λ _∞ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating L ¹-mixing to L ²-mixing of projections of the chain, which enables the application of logarithmic-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems, e.g. gas hard-core, Potts, anti-ferromagentic Ising, arbitrary boundary conditions, etc.     

Semi-explicit methods for isospectral flows

In this paper we propose semi-explicit schemes based on Taylor methods for the solution of the isospectral equation L=[B,L] for d×d real matrices L, while reproducing the isospectrality of the exact equation. Although the theoretical solution may be symmetric, the proposed schemes usually do not retain symmetry of the underlying flow. We present techniques that allow us to decrease the breakdown in symmetry.     

An immersed boundary-lattice Boltzmann method combined with a robust lattice spring model for solving flow–structure interaction problems

An immersed boundary (IB)-lattice Boltzmann method (LBM) combined with a robust lattice spring model (LSM) was developed for modeling fluid–elastic body interactions. To include the effects of viscous flow forces on the deformation of a flexible body, rotational invariant springs were connected regularly inside the deformable body with square lattices. Fluid–solid interactions were due to an additional force density in the lattice Boltzmann equation enhanced by the split-forcing approach. To check the validity and accuracy of the numerical method, the flow over a rigid plate and the deformation of a cantilever beam were investigated. To demonstrate the capability of the new method, different test cases were examined. The deformation of a two-dimensional flexible vertical plate in a laminar cross-flow stream at different conditions was analyzed. The simulations were performed for different boundary conditions imposed on the elastic plate, namely, fixed-end corners and fixed middle point. Different flow conditions such as “steady flow regime”, “vortex shedding flow regime”, and the limit of “rigid body motion” were examined using the new IB-LBM-LSM approach. A general formulation for evaluating the deformation of the elastic body was also introduced, in which the position of the LSM nodes (inside the body) was updated implicitly at each time step. Two dimensionless groups, namely capillary number (Ca) and Reynolds number (Re), were used for parametric study of the behavior of the flow around the deformable plate. It was found that for low Reynolds numbers (Re < 50) and when the middle of the plate was fixed, decreasing the capillary number led to a decrease in the drag coefficient. The fluctuation of the plate during the vortex shedding flow regime was also explored. It was found that when the middle of the plate was fixed, the critical Reynolds number for the initiation of vortex shedding increased. For Re > 100, the Strouhal number was observed to increase with the decrease in capillary number.     

Pointwise bornological spaces

With each metric space (X,d) we can associate a bornological space (X,Bd) where Bd is the set of all subsets of X with finite diameter. Equivalently, Bd is the set of all subsets of X that are contained in a ball with finite radius. If the metric d can attain the value infinite, then the set of all subsets with finite diameter is no longer a bornology. Moreover, if d is no longer symmetric, then the set of subsets with finite diameter does not coincide with the set of subsets that are contained in a ball with finite radius. In this text we will introduce two structures that capture the concept of boundedness in both symmetric and non-symmetric extended metric spaces.     

Semiconductor ring laser subject to delayed optical feedback: Bifurcations and stability

We study semiconductor ring lasers subject to delayed optical feedback from one or two short external cavities. In case of two cavities we consider the feedback strengths and phases to be either symmetric or asymmetric feedback. When feedback is symmetric, the laser operates in a bi-directional continuous wave or periodic regime for most parameter values. Only for some small parameter regions complex dynamics, such as quasi-periodicity and chaos are obtained. When the feedback is asymmetric complex dynamical regimes, including chaos, are obtained in large parameter regions. We explain complex dynamical regimes obtained in both symmetric and asymmetric feedback cases by linear stability of the stationary solutions calculated using DDE-BIFTOOL.     

Asymptotic relations betweenL- andM-estimators in the linear model

We obtain Bahadur-type representations for one-stepL-estimators,M- and one-stepM-estimators in the linear model. The order of the remainder terms in these representations depends on the smooth-ness of the weight function forL-estimators and on the smoothness of the ψ-function forM- and one-stepM-estimators. We use the representations to investigate the asymptotic relations between these estimators. In particular, we show that asymptotically equivalentL- andM-estimators of the slope parameter exist even when the underlying distribution is asymmetric. It is important to consider the asymmetric case for both practical and robustness reasons: first, there is no compelling argument which precludes asymmetric distributions from arising in practice, and, secondly, even if a symmetric model can be posited, it is important to allow for the possibility of mild (and therefore difficult to detect) departures from the symmetric model.