1/N expansion for scalar fields

Models of scalar field theories with a large number N of isotopic degrees of freedom are considered. A theory of perturbations with respect to a small parameter is developed in the formalism of path integration for a space-time dimensionD =2, 3, 4.The particle spectrum obtained in basic order with respect to N^?1 is compared with the spectrum in the path approach. It is shown that whenD =4 the chiral field model is turned, as a result of renormalization, into a model with four interactions. The limitations of the applicability of the 1/N expansion are discussed.     

One-sided maximal functions and Hp

Let N be the nontangential maximal function of a function u harmonic in the Euclidean half-space Rⁿ × (0, ∞) and let N^? be the nontangential maximal function of its negative part. If u(0, y) = o(y^?n) as y → ∞, then ∥N∥_p ? c_p ∥N^?∥_p, 0 < p < 1, and more. The basic inequality of the paper (Theor. 2.1) can be used not only to derive such global results but also may be used to study the behavior of u near the boundary. Similar results hold for martingales with continuous sample functions. In addition, Theorem 1.3 contains information about the zeros of u. For example, if u belongs to H^p for some 0 < p < 1, then every thick cone in the half-space must contain a zero of u.     

The 1/N Expansion of Colored Tensor Models

In this paper, we perform the 1/N expansion of the colored three-dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with increasingly complicated topologies suppressed by higher and higher powers of N. We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S ³ contribute to the leading order in the large N limit.     

The service Erlangian machine interference model:M/E r /1k/N with balking and reneging

The aim of this paper is to derive the analytical solution of the quene:M/E _r /1/k/N for machine interference system with balking and reneging considering the discipline FIFO. Some measures of effectiveness and some special cases are obtained.     

A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots

A simple and complete solution to determine the distributions of queue lengths at different observation epochs for the model GI^X/Geo/c/N is presented. In the past, various discrete-time queueing models, particularly the multi-server bulk-arrival queues with finite-buffer have been solved using complicated methods that lead to results in a non-explicit form. The purpose of this paper is to present a simple derivation for the model GI^X/Geo/c/N that leads to a complete solution in an explicit form. The same method can also be used to solve the GI^X/Geo/c/N queues with heavy-tailed inter-batch-arrival time distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue lengths at different time epochs. All queue-length distributions are in the form of sums of geometric terms.

     

Analysis of the discrete-time bulk-service queue Geo/G/1/N+B

This paper considers a discrete-time bulk-service queueing system with variable capacity, finite waiting space and independent Bernoulli arrival process: Geo/G^Y/1/N+B. Both the analytic and computational aspects of the distributions of the number of customers in the queue at post-departure, random and pre-arrival epochs are discussed.     

A note on uniqueness problem for meromorphic mappings with 2N + 3 hyperplanes

In this paper,a uniqueness theorem for meromorphic mappings partially sharing 2N+3 hyperplanes is proved.For a meromorphic mapping f and a hyperplane H,set E(H,f) = {z|ν(f,H)(z) 0}.Let f and g be two linearly non-degenerate meromorphic mappings and {Hj}j2=N1+ 3be 2N + 3 hyperplanes in general position such that dim f-1(Hi) ∩ f-1(Hj) n-2 for i = j.Assume that E(Hj,f) E(Hj,g) for each j with 1 j 2N +3 and f = g on j2=N1+ 3f-1(Hj).If liminfr→+∞ 2j=N1+ 3N(1f,Hj)(r) j2=N1+ 3N(1g,Hj)(r) NN+1,then f ≡ g.     

Trigonometric Approximation of SO(N) Loops

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a H?lder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order O(n^-a+e)\mathcal{O}(n^{-\alpha+\epsilon}) for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.     

On real moduli spaces of holomorphic bundles over M-curves

Let F be a genus g curve and σ:F→F a real structure with the maximal possible number of fixed circles. We study the real moduli space N^′=Fix(σ#) where σ#:N→N is the induced real structure on the moduli space N of stable holomorphic bundles of rank 2 over F with fixed non-trivial determinant. In particular, we calculate H^?(N^′,Z) in the case of g=2, generalizing Thaddeus' approach to computing H^?(N,Z).     

Type III1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

We complete the analysis of KMS-states of the Toeplitz algebra T(N?N^×) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1?β?2, the unique KMSβ-state is of type III₁. We prove this by reducing the type classification from T(N?N^×) to that of the symmetric part of the Bost-Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of N?N^× in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of N?N^× on the Nica spectrum, we can also recover all the KMS-states of T(N?N^×) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMSβ-states for β>2 that does not ostensibly come from an action of T on the C^?-algebra.     

A geometric process model for M/M/1 queueing system with a repairable service station

In this paper, we study a geometric process model for M/M/1 queueing system with a repairable service station. By introducing a supplementary variable, some queueing characteristics of the system and reliability indices of the service station are derived. Then a replacement policy N for the service station by which the service station will be replaced following the Nth failure is applied. An optimal replacement policy N¹ for minimizing the long-run average cost per unit time for the service station is then determined.     

An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads

The large sparse linear systems arising from the finite element or finite difference discretization of elliptic PDEs can be solved directly via, e.g., nested dissection or multifrontal methods. Such techniques reorder the nodes in the grid to reduce the asymptotic complexity of Gaussian elimination from O(N ²) to O(N ^1.5) for typical problems in two dimensions. It has recently been demonstrated that the complexity can be further reduced to O(N) by exploiting structure in the dense matrices that arise in such computations (using, e.g., \(\mathcal {H}\) -matrix arithmetic). This paper demonstrates that such accelerated nested dissection techniques become particularly effective for boundary value problems without body loads when the solution is sought for several different sets of boundary data, and the solution is required only near the boundary (as happens, e.g., in the computational modeling of scattering problems, or in engineering design of linearly elastic solids). In this case, a modified version of the accelerated nested dissection scheme can execute any solve beyond the first in O(N _boundary) operations, where N _boundary denotes the number of points on the boundary. Typically, N _boundary ～ N ^0.5. Numerical examples demonstrate the effectiveness of the procedure for a broad range of elliptic PDEs that includes both the Laplace and Helmholtz equations.     

Partial difference equations in m1⩾m2⩾⋯⩾mn⩾0 and their applications to combinatorics

Various discrete functions encountered in Combinatorics are solutions of Partial Difference Equations in the subset of Nⁿ given by m₁?m₂???m_n?0. Given a partial difference equation, it is described how to pass from the standard “easy” solution of an equation in Nⁿ to a solution of the same equation subject to certain “Dirichlet” or “Neumann” boundary conditions in the domain m₁?m₂???m_n?0 and related domains. Applications include a rather quick derivation of MacMahon's generating function for plane partitions, a generalization and q-analog of the Ballot problem, and a joint analog of the Ballot problem and Simon Newcomb's problem.     

A polynomial-time algorithm for computing an optimal admission policy in a GI/M/1/N queue

A polynomial-time algorithm is proposed for computing an optimal admission policy for GI/M/1/N queueing systems. The approach is based upon mathematical programming in which a pointwise minimal value functions is to be found subject to a finite number of DP type constraints. The program is transformed by Fourier-Motzkin elimination into equivalent reduced systems to which a simple, forward-substitution type algorithm can be applied.     

Triviality of Xsplit(N)(Q) for certain congruence classes of N

We give a criterion to check if, given a prime number N, the only rational points of the modular curve X_split(N) are trivial (i.e., cusps or points furnished by complex multiplication). We then prove that this criterion is verified for large enough N satisfying some explicit congruences. To cite this article: P. Parent, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the number of coprime integral solutions of y2 = x3 + k

Let N′(k) denote the number of coprime integral solutions x, y of y² = x³ + k. It is shown that lim sup_k→∞N′(k) ≥ 12.