Minimal Translation Hypersurfaces in Euclidean Space

The graph of a function f defined in some open set of the Euclidean space of dimension (p + q) is said to be a translation graph if f may be expressed as the sum of two independent functions ? and ψ defined in open sets of the Euclidean spaces of dimension p and q, respectively. We obtain a useful expression for the mean curvature of the graph of f in terms of the Laplacian, the gradient of ? and ψ as well as of the mean curvatures of their graphs. We study translation graphs having zero mean curvature, that is, minimal translation graphs, by imposing natural conditions on ? and ψ, like harmonicity, minimality and eikonality (constant norm of the gradient), giving several examples as well as characterization results.     

A new proof of a theorem of Petersen

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature n- 1. By developing some new techniques, Colding(1996) proved that the following three conditions are equivalent: 1)dGH(M, S~n) → 0; 2) the volume of M Vol(M) → Vol(S~n); 3) the radius of M rad(M) →π. By developing a different technique, Petersen(1999) gave the 4th equivalent condition, namely he proved that the n + 1-th eigenvalue of M, λ_(n+1)(M) → n, is also equivalent to the radius of M, rad(M) →π, and hence the other two.In this paper, we use Colding's techniques to give a new proof of Petersen's theorem. We expect our estimates will have further applications.     

Optimal Control of an <Emphasis Type="Italic">M/E</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>/1 Queueing System with a Removable Service Station

This paper studies the optimal operation of an M/E _k /1 queueing system with a removable service station under steady-state conditions. Analytic closed-form solutions of the controllable M/E _k /1 queueing system are derived. This is a generalization of the controllable M/M/1, the ordinary M/E _k /1, and the ordinary M/M/1 queueing systems in the literature. We prove that the probability that the service station is busy in the steady-state is equal to the traffic intensity. Following the construction of the expected cost function per unit time, we determine the optimal operating policy at minimum cost.     

Some Numerical Data for Single-Server Queues Involving Deterministic Input Arrangements

The steady-state parameters of the bulk input queue D _c /M/1 and the Erlang service queue D/E _c /1 have been tabulated for C = 1(1)6(2)12(4)20 and 25, 50 and 100 and for ρ = 0·1(0·1)0·9. The tabulation includes the mean waiting time, idle time and queue size. In addition the queue D/E _c /1 has been compared with the queue M/E _c /1 to indicate the gains to be achieved by regularizing the arrival mechanism for a given E _c service facility.     

The Transient Behaviour of a Cyclic Queue

This paper studies a cyclic queueing problem arising in civil engineering earthmoving projects. There are m excavators and n trucks which queue to be loaded by the excavators. The problem is therefore equivalent to a machine interference problem with m operators and n machines. The size of the haul fleet must be optimized, so as to minimize the cost per unit volume of earth moved. Graphs which show regions of optimal n in the parameter space are presented for the steady-state D/D/1, M/D/1 and E _k /M/1 models. The effect of operating the M/M/m system in short work shifts, so that the initial conditions and transient behaviour are important, is then analysed and quantified as a correction to the optimal number of trucks in the haul fleet from the steady-state solution.     

Sharp Bounds for Exponential Approximations of NWUE Distributions

Let F be an NWUE distribution with mean 1 and G be the stationary renewal distribution of F. We would expect G to converge in distribution to the unit exponential distribution as its mean goes to 1. In this paper, we derive sharp bounds for the Kolmogorov distance between G and the unit exponential distribution, as well as between G and an exponential distribution with the same mean as G. We apply the bounds to geometric convolutions and to first passage times.     

The 〈G<Emphasis Type="Italic">/M/r/</Emphasis>FIFO〉 Machine-Interference Model with State-Dependent Speeds

This article treats a version of the multiple machine-interference problem with r operatives under FIFO repair discipline. The running times of machine i are supposed to be identically and arbitrarily distributed random variables with density function f _i (x), i = 1,…, n. The repair times of all machines are assumed to be identically and exponentially distributed random variables with mean 1/μ. The paper provides the main steady-state operational characteristics of the system when the running and repair speeds are dependent on the number of machines in working order.     

An analog of Wald’s identity for random walks with infinite mean

We deduce an analog of the classical Wald’s identity ES _τ = EτEξ in the case of the infinite mean of summands. We find the conditions on τ under which Emin(S _τ , x) ～ EτE min(ξ, x) as x→∞.     

On some mean values for the divisor function and the Riemann zeta-function

Let Δ(x) and E(x) denote respectively the error terms in the summatory formula for the divisor function and in the mean square formula for ζ(s) on the critical line. We consider some general mean values for Δ(x) and E(x) and discover interesting differences between these two functions. In particular, this yields evidence that E(x) is more negative than Δ(x).     

On the Relative Waiting Times in the GI/M/s and the GI/M/1 Queueing Systems

The expected steady-state waiting time, Wq(s), in a GI/M/s system with interarrival-time distribution H(·) is compared with the mean waiting time, Wq, in an "equivalent" system comprised of s separate GI/M/1 queues each fed by an interarrival-time distribution G(·) with mean arrival rate equal to 1/s times that of H(·). For H(·) assumed to be Exponential, Gamma or Deterministic three possible relationships between H(·) and G(·) are considered: G(·) can be of the "same type" as H(·); G(·) can be derived from H(·) by assigning new arrivals to the individual channels in a cyclic order; and G(·) may be obtained from H(·) by assigning customers probabilistically to the different queues. The limiting behaviour of the ratio R = Wq/Wq(s) is studied for the extreme values (1 and 0) of the common traffic intensity, ρ. Closed form results, which depend on the forms of H(·) and G(·) and on the relationships between them, are derived. It is shown that Wq is greater than Wq(s) by a factor of at least (s + 1)/2 when ρ approaches one, and that R is at least s(s!) when ρ tends to zero. In the latter case, however, R goes to infinity (!) in most cases treated. The results may be used to evaluate the effect on the waiting times when, for certain (non-queueing) reasons, it is needed to partition a group of s servers into several small groups.     

Tong-type identity and the mean square of the error term for an extended Selberg class

In 1956, Tong established an asymptotic formula for the mean square of the error term of the summatory function of the Piltz divisor function d3(n). The aim of this paper is to generalize Tong's method to a class of Dirichlet series L(s) which satisfies a functional equation. Let a(n) be an arithmetical function related to a Dirichlet series L(s), and let E(x) be the error term of ′n xa(n). In this paper, after introducing a class of Diriclet series with a general functional equation(which contains the well-known Selberg class), we establish a Tong-type identity and a Tong-type truncated formula for the error term of the Riesz mean of the coefficients of this Dirichlet series L(s). This kind of Tong-type truncated formula could be used to study the mean square of E(x) under a certain assumption. In other words, we reduce the mean square of E(x) to the problem of finding a suitable constant σ*which is related to the mean square estimate of L(s). We shall represent some results of functions in the Selberg class of degrees 2–4.     

Boundary Crossing Probabilities for General Exponential Families

We consider parametric exponential families of dimension K on the real line. We study a variant of boundary crossing probabilities coming from the multi-armed bandit literature, in the case when the real-valued distributions form an exponential family of dimension K. Formally, our result is a concentration inequality that bounds the probability that B ^ψ (θ? _n , θ*) ≥ f(t/n)/n, where θ* is the parameter of an unknown target distribution, θ? _n is the empirical parameter estimate built from n observations, ψ is the log-partition function of the exponential family and B ^ψ is the corresponding Bregman divergence. From the perspective of stochastic multi-armed bandits, we pay special attention to the case when the boundary function f is logarithmic, as it is enables to analyze the regret of the state-of-the-art KL-ucb and KL-ucb+ strategies, whose analysis was left open in such generality. Indeed, previous results only hold for the case when K = 1, while we provide results for arbitrary finite dimension K, thus considerably extending the existing results. Perhaps surprisingly, we highlight that the proof techniques to achieve these strong results already existed three decades ago in the work of T. L. Lai, and were apparently forgotten in the bandit community. We provide a modern rewriting of these beautiful techniques that we believe are useful beyond the application to stochastic multi-armed bandits.     

The Shape of Orthogonal Cycles in Three Dimensions

Let σ be a directed cycle whose edges have each been assigned a desired direction in 3D (East, West, North, South, Up, or Down) but no length. We say that σ is a shape cycle. We consider the following problem. Does there exist an orthogonal representation Γ of σ in 3D space such that no two edges of Γ intersect except at common endpoints and such that each edge of Γ has the direction specified in σ? If the answer is positive, we say that σ is simple. This problem arises in the context of extending orthogonal graph drawing techniques from 2D to 3D. We give a combinatorial characterization of simple shape cycles that yields linear time recognition and drawing algorithms.     

<Emphasis Type="Italic">IFPBM</Emphasis>s and their application to multiple attribute group decision making

The Bonferroni mean (BM) had been generalized for its capacity to capture the interrelationship between input arguments. In order to obtain much more information in the process of group decision making, especially in the cases that the relationships between the fused data are considered, this paper combines the power average operator with the intuitionistic fuzzy Bonferroni mean (IFBM) and develops the intuitionistic fuzzy power Bonferroni mean (IFPBM) and the weighted intuitionistic fuzzy power Bonferroni mean (WIFPBM). We investigate the desirable properties of these new extensions of BM and discuss their special cases. We give a comparison of the new extensions of BM with the corresponding existing IFBMs. Furthermore, the detailed steps of multiple attribute group decision making with the presented IFPBM or WIFPBM are given and numerical examples are illustrated to show the validity and feasibility of the new approaches.     

An approximation scheme for a problem of search for a vector subset

We consider the following clustering problem: Given a vector set, find a subset of cardinality k and minimum square deviation from its mean. The distance between the vectors is defined by the Euclideanmetric. We present an approximation scheme (PTAS) that allows us to solve this problem with an arbitrary relative error ? in time O(n ^2/?+1(9/?)^3/? d), where n is the number of vectors of the input set and d denotes the dimension of the space.     

Holomorphic Minorants of Plurisubharmonic Functions

Let φ be a plurisubharmonic function on a pseudoconvex domain D in an n-dimensional complex space. We show that there exists a nonzero holomorphic function f on D such that some local mean value of φ with logarithmic additional terms majorizes log|f|. A similar problem is discussed for a locally integrable function on D in terms of balayage of positive measures.     

On Schrödinger Propagator for the Special Hermite Operator

We establish a Strichartz type estimate for the Schrödinger propagator e ^it? for the special Hermite operator ? on ? ⁿ . Our method relies on a regularization technique. We show that no admissibility condition is required on (q,p) when 1≤q≤2.     

Path <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript> algebras of positively graded quivers

Let A be a path A_∞-algebra over a positively graded quiver Q: We prove that the derived category of A is triangulated equivalent to the derived category of kQ; which is viewed as a DG algebra with trivial differential. The main technique used in the proof is Koszul duality for DG algebras.     

Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

In (0,T)×Ω, Ω open subset of ? ⁿ , n≥2, we consider a parabolic operator P=? _t ?? _x δ(t,x)? _x , where the (scalar) coefficient δ(t,x) is piecewise smooth in space yet discontinuous across a smooth interface S. We prove a global in time, local in space Carleman estimate for P in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient δ at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface. In the high-frequency regime we use Calderón projectors. In the low-frequency regime we follow a more classical approach. Because of the parabolic nature of the problem we need to introduce Weyl-Hörmander anisotropic metrics, symbol classes and pseudo-differential operators. Each frequency regime and the associated technique require a different calculus. A global in time and space Carleman estimate on (0,T)×M, M a manifold, is also derived from the local result.     

Uniform approximations for the M/G/1 queue with subexponential processing times

This paper studies the asymptotic behavior of the steady-state waiting time, W _∞, of the M/G/1 queue with Subexponential processing times for different combinations of traffic intensities and overflow levels. In particular, we provide insights into the regions of large deviations where the so-called heavy-traffic approximation and heavy-tail asymptotic hold. For queues whose service time distribution decays slower than \(e^{-\sqrt{t}}\) we identify a third region of asymptotics where neither the heavy-traffic nor the heavy-tail approximations are valid. These results are obtained by deriving approximations for P(W _∞>x) that are either uniform in the traffic intensity as the tail value goes to infinity or uniform on the positive axis as the traffic intensity converges to one. Our approach makes clear the connection between the asymptotic behavior of the steady-state waiting time distribution and that of an associated random walk.