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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

This paper is concerned with some stochastic properties of a novel rock–paper–scissors model. Firstly, the global existence of an unique positive solution of the stochastic model is obtained. Then we demonstrate the positive solution of the model is stochastically bounded. Besides, some sufficient conditions for population to be stochastically permanent and extinct are derived with the use of some appropriate Lyapunov functions. At last, some numerical simulations are carried out to illustrate our theoretical analysis results.

In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

In this paper, we consider precoder designs for multiuser multiple-input-single-output (MISO) broadcasting channels. Instead of using a traditional linear zero-forcing (ZF) precoder, we propose a generalized ZF (GZF) precoder in conjunction with successive dirty-paper coding (DPC) for data-transmissions, namely, the GZF-DP precoder, where the suffix \lq{}DP\rq{} stands for \lq{}dirty-paper\rq{}. The GZF-DP precoder is designed to generate a band-shaped and lower-triangular effective channel $\vec{F}$ such that only the entries along the main diagonal and the $\nu$ first lower-diagonals can take non-zero values. Utilizing the successive DPC, the known non-causal inter-user interferences from the other (up to) $\nu$ users are canceled through successive encoding. We analyze optimal GZF-DP precoder designs both for sum-rate and minimum user-rate maximizations. Utilizing Lagrange multipliers, the optimal precoders for both cases are solved in closed-forms in relation to optimal power allocations. For the sum-rate maximization, the optimal power allocation can be found through water-filling, but with modified water-levels depending on the parameter $\nu$. While for the minimum user-rate maximization that measures the quality of the service (QoS), the optimal power allocation is directly solved in closed-form which also depends on $\nu$. Moreover, we propose two low-complexity user-ordering algorithms for the GZF-DP precoder designs for both maximizations, respectively. We show through numerical results that, the proposed GZF-DP precoder with a small $\nu$ ($\leq\!3$) renders significant rate increments compared to the previous precoder designs such as the linear ZF and user-grouping based DPC (UG-DP) precoders., Comment: 31 pages, 13 figures, submitted to IEEE Transactions on Wireless Communications in Aug. 2016

Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.

In this study, we investigate a discrete-time version of logit dynamics, as applied to the rock–paper–scissors (RPS) game. First, we show that around the Nash equilibrium point, an attracting closed invariant curve appears due to the Neimark–Sacker bifurcation. Next, near the resonance point, we find a period-three attracting cycle, which can be thought of as a counterpart to the cyclically stable set in the RPS game with best response dynamics. Moreover, we show that the cycle can coexist with an attracting closed invariant curve, a period-three saddle cycle, and the attracting or repelling Nash equilibrium point. Finally, we use the codimension-two bifurcation theory to specify the set of heteroclinic bifurcations that destroy the coexistence of the attractors.

Based on symbolic and numeric manipulations, a model simplification technique is proposed in this paper for the linear fractional representation (LFR) and for the differential algebraic representation introduced by Trofino and Dezuo (2013). This representation is needed for computational Lyapunov stability analysis of uncertain rational nonlinear systems. The structure of the parameterized rational Lyapunov function is generated from the linear fractional representation (LFR) of the system model. The developed method is briefly compared to the n-D order reduction technique known from the literature. The proposed model transformations does not affect the structure of Lyapunov function candidate, preserves the well-posedness of the LFR and guarantees that the resulting uncertainty block is at most the same dimensional as the initial one. The applicability of the proposed method is illustrated on two examples.

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models., Comment: 36 pages, 8 figures

Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.

In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as $$t\rightarrow T$$ (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]).

A criteria due to Toivo Nieminen concerning selfadjoint linear operators in complex Hilbert spaces is extended to the case of linear relations.

This letter presents closed-form capacity bounds for an exponential version of the dirty paper channel (EDPC). The EDPC can be used to model the non-coherent Gaussian dirty paper channel (GDPC). First, a superposition modulation-like technique is used to obtain a capacity upper bound for the EDPC. Then, a closed-form capacity lower bound for the EDPC is obtained by using a Costa-like strategy. Using this lower bound and exploiting the asymptotic properties of high signal to noise ratios (SNRs), we obtain the high-SNR capacity of the EDPC which is intriguingly analogous to the complex-valued GDPC. Finally, a closed-form capacity lower bound for the EDPC is given which is useful in the weak interference regime.

We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

We investigate the dynamics of three-strategy (rock–paper–scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval $$T$$ . Taking $$T$$ as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an analysis of the resulting limit cycles using Lindstedt’s method.

The authors of [1] have become aware that several equations in [1] were presented with errors. As confirmed in this brief note, neither do the errors invalidate the game theoretic approach proposed in [1] , nor violate the equilibrium of the approach. However, the parameter selection, which is required to preserve the validity and unique equilibrium of the approach, is affected by the errors. We correct the equations with detailed mathematical derivations in the note.

In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{\zeta(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{\eta(s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form., Comment: 17 pages, no figures

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker–Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker–Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker–Planck equation from a Chapman–Kolmogorov equation, but no proof was offered that a Chapman–Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker–Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker–Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker–Planck pdes and Chapman–Kolmogorov equations for stochastic processes with finite memory.

In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock-scissors-paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May-Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka-Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.

International audience; The huge potential of ultra wide band impulse radio (UWB IR) with multiple access was initially evaluated using the standard Gaussian approximation which assumed that multi user interference could be approximated as classical Gaussian noise. Later the standard Gaussian approximation has been proved to be invalid in several cases. Thus the performance of the classical UWB IR correlation receiver dramatically reduces. We show in this paper how the non Gaussianity of the multi user interference can be exploited, turning the non Gaussianity into an advantage, increasing the performance with respect to the Gaussian interference case which was supposed to be too optimistic until now.

In this short note an addition to the paper [A.G. Bratsos, M. Ehrhardt, I.Th. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrodinger equations, Appl. Math. Comput. 197(1) (2008) 190-205] using the modulus of the terms evaluated from the Adomian decomposition method on p. 194 and their relation to the convergence of the resulting series is presented. Conclusions for the accuracy of the approximated solution are derived.

This paper considers certain neutral differential equation with the distributed delayddt[x(t)+c(t)x([email protected])]+p(t)x(t)+q(t)x([email protected])+r(t)@!"t"-"@d^tx(s)ds=0.Without imposing a nonnegative restriction on coefficients of the equation, we establish a delay-dependent asymptotic stability condition for the equation by introducing a new Lyapunov function. The condition presented here is different from that of Park [J.H. Park, LMI optimization approach to asymptotic stability of certain neutral delay differential equation with time-varying coefficients, Appl. Math. Comput. 160 (2005) 355-361] in the sense that it is dependent on all delays of the equation. Numerical examples illustrate effectiveness and sharpness of our theoretical result.

In this paper we analyze the population dynamics of bacteria competing by anti-bacterial toxins (bacteriocins). Three types of bacteria involved in these dynamics can be distinguished: toxin producers, resistant bacteria and sensitive bacteria. Their interplay can be regarded as a R ock- S cissors- P aper - game (RSP). Here, this is modeled by a reasonable three-dimensional Lotka- Volterra ($L$V) type differential equation system. In contrast to earlier approaches to modeling the RSP game such as replicator equations, all interaction terms have negative signs because the interaction between the three different types of bacteria is purely competitive, either by toxification or by competition for nutrients. The model allows one to choose asymmetric parameter values. Depending on parameter values, our model gives rise to a stable steady state, a stable limit cycle or a heteroclinic orbit with three fixed points, each fixed point corresponding to the existence of only one bacteria type. An alternative model, the May - Leonard model, leads to coexistence only under very restricted conditions. We carry out a comprehensive analysis of the generic stability conditions of our model, using, among other tools, the Volterra-Lyapunov method. We also give biological interpretations of our theoretical results, in particular, of the predicted dynamics and of the ranges for parameter values where different dynamic behavior occurs. For example, one result is that the intrinsic growth rate of the producer is lower than that of the resistant while its growth yield is higher. This is in agreement with experimental results for the bacterium Listeria monocytogenes.

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral ∫ab ∫-11 f(x, y)/x-y dxdy, where (a,b) ≡ (-1,1) or (a,b) ≡ (a,-1) or (a,b) ≡ (1, b) and f(x,y) is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

A l-Bayesian erasure test (BET) based on order statistics with diversity reception for noncoherent M-ary frequency shift keying (FSK) signaling is examined. In the l-BET, the M-ary FSK symbol is erased whenever the a posteriori probability of correct decision, which is a function of the l-largest sums of energy detector outputs, is less than a threshold. For binary FSK signaling in Rayleigh flat fading channels, the 2-BET is found to be the same as the difference threshold test (DTT), which declares an erasure whenever the difference between the largest and the second largest sums of energy detector outputs is less than a threshold. The output threshold test (OTT), which declares an erasure whenever the largest sum of energy detector outputs is less than a threshold, is proved equivalent to the 1-BET for the Rayleigh flat fading channel. Closed-form expressions for the probabilities of correct decision and symbol error for the DTT, Viterbi's ratio threshold test (RTT), and the OTT are derived when square-law combining (SLC) diversity is used. Several erasure tests including the Z-BET, the RTT, the OTT, and the DTT are compared in terms of probability of codeword error. The 2-BET provides almost the same performance as the BET that requires all the sums of energy detector outputs. The 2-BET provides power gains of 0.9 dB and 0.7 dB over the RTT for (7, 3) and (15, 7) Reed-Solomon (RS) codes, respectively, when the probability of codeword error is 10-3 and the diversity order is one.

The class of generalized reflectionless Schrodinger potentials was introduced by Marchenko-Lundina and was analyzed by Kotani. We state and prove various results concerning those stationary ergodic processes of Schrodinger potentials which are contained in this class.

*Correspondence: wadade@se.kanazawa-u.ac.jp 3Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan Full list of author information is available at the end of the article The purpose of this note is to clarify the novelty of the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’ which was published in the J. Inequal. Appl. :,  []. After this paper was published, the authors were informed of the references [–], and [], the results of which partly overlap with those of []. In the paper [], the authors established the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces H n p p,q(R); see Section  in [] for the precise definition of H n p p,q(R). The main theorem in [] is stated as follows. Theorem A [, Theorem .] Let n ∈ N,  < p

A counterexample is provided in this short note to show that some of local convergence theorems established in [D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton’s method with third order convergence, Appl. Math. Comput. 183 (2006) 659–684] are not always true. Some mistakes in the proofs of these theorems are pointed out.

V. G. Maz’ya and J. Nagel found for certain classes of weighted Sobolev norms (defined using the Fourier transform) equivalent Slobodeckij-type difference representations. We extend these considerations to a wider class of anisotropic norms which arise in the theory of Markov processes. In particular we show that these Sobolev norms are equivalent to Dirichlet norms.

Liseo and Loperfido [A note on reference priors for the scalar skew-normal distribution. J Statist Plann Inference. 2006;136(2):373–389] studied some peculiar features of default Bayes analysis of the scalar skew-normal model. In particular, they showed that, by considering the simplest model with a single unknown parameter λ of skewness, the reference – or Jeffreys’ – prior for this parameter is proper. They proved that tails of Jeffreys’ prior are of order O(λ−3/2). But they made a mistake in their proof. In this note, we will modify their proof.

for their paper “Preconditioning Indefinite Systems in Interior Point Methods for Optimization”, published in Volume 28, pages 149–171. This paper describes a class of indefinite preconditioners for reduced KKT systems arising in quadratic and nonlinear optimization with interior point methods. Spectral analysis of preconditioners is given and the improvements resulting from the use of primal-dual regularization are demonstrated. Computational results are reported for the application of the preconditioner to a variety of medium-size convex quadratic programming problems. Work on the paper began during the Summer of 2001, about a year after the authors had first met when Zilli invited Gondzio to give a series of lectures on interior point methods in Padova in September, 2000. The authors had complementary research backgrounds: Bergamaschi and Zilli worked on the theory and implementation of iterative methods for linear and nonlinear systems of equations [2], while Gondzio worked on the design and implementation of interior point methods for large-scale optimization [3]. This was essential since together, the three authors could tackle in depth a problem which required expertice in several areas. At that time Bergamaschi and Zilli were impressed with the recent development of Lukysan and Vlycek [6], who applied conjugate gradients to indefinite systems, and with the later analysis of Keller, Gould and Wathen [4] of indefinite constraint preconditioners. They convinced Gondzio to look into the issue. In the meantime, Rozlozn´ok and Simoncini [7] came up with an improved understanding of issues connected with indefinite preconditioning (in the context of saddle point problems arising in partial differential equation); these developments led the authors to take a closer look at the problem. Starting in July, 2001, the authors began to incorporate into Gondzio’s interior point code HOPDM (higher order primal dual methods) four iterative techniques: conjugate gradients, BiCGstab, GMRES and QMR. A description of these schemes can be found in Kelley’s book [5] and the references therein. The authors realized that the key difficulty in the solution of augmented systems of Newton equations in interior point methods for quadratic and nonlinear programming (otherwise known as reduced KKT systems) consisted in the loss of sparsity caused by the presence of the Hessian matrix Q in the system � (Q + � 1

In this paper, it is shown that von Neumann continuous geometry may be regarded as the first attempt towards formulating a general quantum spacetime geometry akin to that of Cantorian spacetime E(∞) and noncommutative geometry.

The authors would like to correct the proof of Theorem 3(a) in the original paper “On the asymptotic behaviour of the 2D Navier-Stokes equations with Navier friction conditions towards Euler equations.” [ZAMM 89(10), 810-822, 2009].

We point out errors and oversights in a paper by Giordano, Laforgia, and Pecaric [3] on inequalities involving Bessel functions.

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

We rebut Kowalenko's claims in 2010 that he proved the irrationality of Euler's constant, and that his rational series for it is new., Comment: 3 pages including a referee's report; added Goldbach's theorem and a reference for it

The paper is mainly a survey of the topic how to approximate the probability density of the parameter estimator in a nonlinear regression model. A short presentation of the geometry of the model and a heuristic discussion of the model and a heuristic discussion of the “irregularities” of the estimates are given. In the model with Gaussian errors we present the asymptotic normal approximation, the approximationby the second order Edgeworth expansion, a conditional density of BARNDORFF-NIELSEN, and mainly the approximation called “flat” or “saddlepoint” approximation, which will be shown to have several interesting properties. Further, we present the possibility of improving the approximation in some models, the extension of the approximation to some cases of nongaussian errors, and besides the maximum likelihood estimator we consider also the weighted least-squares estimator, with the weights not depending on the error concariance matrix.

van Lenthe, Broer, and Rashid made comments on our 2009 paper [Song et al., J. Comput. Chem. 2009, 30, 399] by criticizing that we did not properly reference the work by Broer and Nieuwpoort in 1988 [Broer and Nieuwpoort, Theor. Chim. Acta. 1988, 73, 405], and we favorably compared our valence bond self-consistent field (VBSCF) algorithm with theirs. However, both criticisms are unjustified insignificant. The Broer–Nieuwpoort algorithm, properly cited in our paper, is for the evaluations of matrix elements between determinants of nonorthogonal orbitals. Stating that this algorithm “can be used for an orbital optimization” afterwards [van Lenthe et al., submitted] is not a plausible way to require more credits or even criticize others. While we stand by our statement that our algorithms scales at O(m4) and van Lenthe et al.'s approximate Newton Raphson algorithm scales at O(mN5) (here m and N are the numbers of basis functions and electrons), as we discussed in our original paper, it becomes obvious that any strict comparison among different algorithms is difficult, unproductive, and counteractive. © 2012 Wiley Periodicals, Inc.

I enjoyed reading Professor Desmond’s paper. It makes clear a number of impor- tant issues which are often obscured in the literature. I note in passing that the pseudolikelihood (5.5) has better small-sample behavior if it is combined with REML, see Carroll and Ruppert (1988, pp. 75-76) and Breslow and Clayton (1993). My main comments are tangentially concerned with the material in Section 7 on case-control studies. There are two ways to estimate the covariance matrix (3.3): using the structure of the model (e.g., homoscedastic linear, logistic, Poisson, etc.) or nonparametrically via the method of moments (there are analogs to the parametric

The development of computational fluid dynamics (CFD) can be traced back as far as the early 1900’s. The pioneering efforts by Richardson [1], Courant, Friedrichs, and Lewy [2], Southwell [3], Von Neumann [4], Lax [5], as well as Godunov [6] address the fundamental issues in numerical analyses for CFD. It is immediately clear that a major portion of these efforts was focused on one of the most difficult problems in resolving the discontinuous fluid phenomena in a discretized space — the Riemann problem [7]. As it will be seen later, it remains the most studied problem in CFD. However, if one is interested in the viscous flow simulation, Thom [8] probably obtained the first-ever numerical solution by solving the partial differential equation for a low speed flow past a circular cylinder. For a scholarly description of the CFD historical perspective, the books by Roach [9] and Tannehill, Anderson, and Pletcher [10] are highly recommended.Copyright © 2003 by ASME

A hypergeometric transformation formula is developed that simultaneously simplifies and generalizes arguments and identities in a previous paper of Rao et al. [An entry of Ramanujan on hypergeometric series in his notebooks, J. Comput. Appl. Math. 173(2) (2005) 239–246].

The two papers in this issue deal with differential equations, one with the numerical solution of partial differential equations, and the other one with analytic solutions for ordinary differential equations. In his paper "From Functional Analysis to Iterative Methods", Robert Kirby is concerned with linear systems arising from discretizations of partial differential equations (PDEs). Specifically, the PDEs are elliptic and describe boundary value problems; the discretizations are done via finite elements, and at issue is the convergence rate of iterative methods for solving the linear systems. The author's approach is to go back to the underlying variational problem in a Hilbert space, and to make ample use of the Riesz representation theorem. This point of view results in short and elegant proofs, as well as the construction of efficient preconditioners. The general theory is illustrated with two concrete model problems of PDEs for convection diffusion and planar elasticity. This paper will appeal to anybody who has an interest in the numerical solution of PDEs. In 1963 the mathematician/meteorologist Edward Lorenz formulated a system of three coupled nonlinear ordinary differential equations, whose long-term behavior is described by an attractor with fractal structure. You can see a beautiful rendition of the thus named Lorenz attractor on the cover of this issue. Although it is "easy" to plot solutions of the Lorenz system, it is much harder to determine them mathematically. This is what motivated the paper "Complex Singularities and the Lorenz Attractor" by Divakar Viswanath and Sonmez Sahutoglu. Their idea is to allow the time variable to be complex, rather than real; to focus on singular solutions; and to express these singular solutions in terms of so-called psi series. After all is said and done, the authors end up with a two-parameter family of complex solutions to the Lorenz system. This a highly readable and very enjoyable paper, with concrete steps for future research, and connections to thunderstorms and analytic function theory.