Nonlinear differential equations satisfied by certain classical modular forms

A unified treatment is given of low-weight modular forms on ?? ₀(N), N = 2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under ?? ₀(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard?CFuchs equations of triangle subgroups of PSL(2, R), which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with ??(1) is treated.     

Weighted matching with pair restrictions

The weighted matroid parity problems for the matching matroid and gammoids are among the very few cases for which the weighted matroid parity problem is polynomial time solvable. In this work we extend these problems to a general revenue function for each pair, and show that the resulting problem is still solvable in polynomial time via a standard weighted matching algorithm. We show that in many other directions, extending our results further is impossible (unless P = NP). One consequence of the new polynomial time algorithm is that it demonstrates, for the first time, that a prize-collecting assignment problem with “pair restriction” is solved in polynomial time. The prize collecting assignment problem is a relaxation of the prize-collecting traveling salesman problem which requires, for any prescribed pair of nodes, either both nodes of the pair are matched or none of them are. It is shown that the prize collecting assignment problem is equivalent to the prize collecting cycle cover problem which is hence solvable in polynomial time as well.     

Δ-Monotone arrangements of real numbers

Given a set of M × N real numbers, can these always be labeled as x_i,j; i = 1,…, M; j = 1,…, N; such that x_i+1,j+1 ? x_i+1,j ? x_i,j+1 + x_ij ≥ 0, for every (i, j) where 1 ≤ i ≤ M ? 1, 1 ≤ j ≤ N ? 1? For M = N = 3, or smaller values of M, N it is shown that there is a “uniform” rule. However, for max(M, N) > 3 and min(M, N) ≥ 3, it is proved that no uniform rule can be given. For M = 3, N = 4 a way of labeling is demonstrated. For general M, N the problem is still open although, for a special case where all the numbers are 0's and 1's, a solution is given.     

Stability of the planar equilibrium solutions of a restricted 1 + N body problem

We started our studies with a planar Eulerian restricted four-body problem (ERFBP) where three masses move in circular orbits such that their configuration is always collinear. The fourth mass is small and does not influence the motion of the three primaries. In our model we assume that one of the primaries has mass 1 and is located at the origin and two masses of size µ rotate around it uniformly. The problem was studied in [3], where it was shown that there exist noncollinear equilibria, which are Lyapunov stable for small values of µ. KAM theory is used to establish the stability of the equilibria. Our computations do not agree with those given in [3], although our conclusions are similar. The ERFBP is a special case of the 1 + N restricted body problem with N = 2. We are able to do the computations for any N and find that the stability results are very similar to those for N = 2. Since the 1 + N body configuration can be stable when N > 6, these results could be of more significance than for the case N = 2.     

Steady-state probability of the randomized server control system with second optional service, server breakdowns and startup

This paper deals with the 〈N,p〉-policy M/G/1 queue with server breakdowns and general startup times, where customers arrive to demand the first essential service and some of them further demand a second optional service. Service times of the first essential service channel are assumed to follow a general distribution and that of the second optional service channel are another general distribution. The server breaks down according to a Poisson process and his repair times obey a general distribution in the first essential service channel and second optional service channel, respectively. The server operation starts only when N (N≥1) customers have accumulated, he requires a startup time before each busy period. When the system becomes empty, turn the server off with probability p (p∈[0,1]) and leave it on with probability (1?p). The method of maximum entropy principle is used to develop the approximate steady-state probability distribution of the queue length in the M/G(G, G)/1 queueing system. A study of the derived approximate results, compared to the established exact results for three different 〈N,p〉-policy queues, suggests that the maximum entropy principle provides a useful method for solving complex queueing systems.     

Multiperfect numbers with identical digits

Let g?2. A natural number N is called a repdigit in base g if all of the digits in its base g expansion are equal, i.e., if for some m?1 and some D∈{1,2,…,g−1}. We call N perfect if σ(N)=2N, where σ denotes the usual sum-of-divisors function. More generally, we call N multiperfect if σ(N) is a proper multiple of N. The second author recently showed that for each fixed g?2, there are finitely many repdigit perfect numbers in base g, and that when g=10, the only example is N=6. We prove the same results for repdigit multiperfect numbers.     

Structural questions with GERT-networks

LetN be a GERT network with sinks, representing the termination of the project. There are two natural questions to ask: Cans be activated at all? And if so, is the activation ofs sure? After having shown the first question to be NP-complete and the second to be NP-hard, we state some structural results for the bipartite case which can be reduced to a flow problem. From this we derive a theoretical algorithm for the general case, which is — not surprisingly — exponential. Moreover, a connection to reliability analysis is revealed.     

Relationship between strictly collapsible and perfect contingency tables

Whittemore (1978) conjectured that an N-dimensional contingency table p is strictly collapsible over each factor with respect to the set of remaining factors if and only if p has a certain factorization. I prove this conjecture for N = 3 and show by counterexamples that it is false for N > 3.     

On the enumeration of chains in regular chain-groups

Let N be a regular chain-group on E (see W. T. Tutte, Canad. J. Math.8 (1956), 13–28); for instance, N may be the group of integer flows or tensions of a directed graph with edge-set E). It is known that the number of proper Z_λ-chains of N (λ ∈ Z, λ ≥ 2) is given by a polynomial in λ, P(N, λ) (when N is the chain-group of integer tensions of the connected graph G, λP(N, λ) is the usual chromatic polynomial of G). We prove the formula: P(N, λ) = Σ_{[E′]∈O(N)⁺/～}Q(R_[E′](N), λ), where O(N)⁺ is the set of orientations of N with a proper positive chain, ～ is a simple equivalence relation on O(N)⁺ (sequence of reversals of positive primitive chains), and Q(R_[E′](N), λ) is the number of chains with values in [1, λ ? 1] in any reorientation of N associated to an element of [E′]. Moreover, each term Q(R_[E′](N), λ) is a polynomial in λ. As applications we obtain: P(N, 0) = (?1)^r(N)∥O(N)⁺/～∥; P(N, ?1) = (?1)^r(N)∥O(N)⁺∥ (a result first proved by Brylawski and Lucas); P(N, λ + 1) ≥ P(N, λ) for λ ≥ 2, λ ∈ Z. Our result can also be considered as a refinement of the following known fact: A regular chain-group N has a proper Z_λ-chain iff it has a proper chain in [?λ + 1, λ ? 1].     

Estimates for the Derivatives of the Poisson Kernel on Nilpotent Meta-Abelian Groups

Let S be a semi direct product \(S=N\rtimes A\) where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic with ? ^k , k?>?1. We consider a class of second order left-invariant differential operators on S of the form \(\mathcal{L}_\alpha=L^a+\Delta_\alpha,\) where α?∈?? ^k , and for each a?∈?? ^k , L ^a is left-invariant second order differential operator on N and \(\Delta_\alpha=\Delta-\langle\alpha,\nabla\rangle,\) where Δ is the usual Laplacian on ? ^k . Using some probabilistic techniques (skew-product formulas for diffusions on S and N respectively, the concept of the derivative of a measure, etc.) we obtain an upper bound for the derivatives of the Poisson kernel for \(\mathcal{L}_\alpha.\) During the course of the proof we also get an upper estimate for the derivatives of the transition probabilities of the evolution on N generated by L ^σ(t), where σ is a continuous function from [0,?∞?) to ? ^k .     

Optimal control of a removable server in an M/G/1 queue with finite capacity

We consider an M/G/1 system with finite capacity L in which the removable server applies a (ν, N) policy; a classical cost structure is imposed and the total expected cost per unit time in the steady state is considered. For the M/M/1 situation, Hersh and Brosh [3] analysed the policies with 0⩾ν<N⩽L and established that the best of them is characterized either by ν = 0 or by N = L. By a different and quite easy way and for a general service time distribution, we prove that an optimal policy has the form (ν = 0, 0 ⩽ N ⩽ L + 1), where the (0, 0) and (0, L + 1) policies consist in never closing or never opening the station respectively. Moreover, we describe a precise technique to analyse the policy space and to determine easily the optimal policy.     

General study of the distribution of N-tuples of letters or words based on the distributions of the single letters or words

This paper establishes the general relation between the distribution of N-tuples of letters (e.g., N-truncations, N-grams) or words (e.g., N-word phrases) and the distributions of the single letters or words. Here the very general case is treated: the case where there is dependence on the place i in the N-tuple (i = 1,…, N) in the sense that, for each i = 1,…, N, a different distribution of the letters or words is supposed.Concrete calculations are performed in the important case of Zipfian distributions (i.e., power laws) for the single letters or words. In this case, we prove that the distribution of the N-tuples (N-fixed) is the sum of power laws.     

Complete solving of explicit evaluation of Gauss sums in the index 2 case

Let p be a prime number,N be a positive integer such that gcd(N,p) = 1,q = pf where f is the multiplicative order of p modulo N.Let χ be a primitive multiplicative character of order N over finite field Fq.This paper studies the problem of explicit evaluation of Gauss sums G(χ) in the "index 2 case"(i.e.[(Z/NZ):p] = 2).Firstly,the classification of the Gauss sums in the index 2 case is presented.Then,the explicit evaluation of Gauss sums G(χλ)(1 λ N-1) in the index 2 case with order N being general even integer(i.e.N = 2r·N0,where r,N0 are positive integers and N0 3 is odd) is obtained.Thus,combining with the researches before,the problem of explicit evaluation of Gauss sums in the index 2 case is completely solved.     

Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then $2^r\le N$. Here we state a stronger conjecture about varieties of square-zero upper triangular $N\times N$ matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N<8$ or $r<3$ without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N>4$ and $r=2$.     

Nearrings whose set of N-subgroups is linearly ordered

We consider zerosymmetric right nearrings whose lattice of N-subgroups is linearly ordered by inclusion and in which for every n ∈ N, there is x ∈ N such that x * n = n. All such nearrings with finitely many N-subgroups are constructed.     

A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p∈(1,N), Ω⊂RN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(N−p,N). We show in the first part that for every p∈[2N/(N+2),N)∩(1,N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L²(Ω), and hence, the associated first order Cauchy problem is well posed on Lq(Ω) for every q∈[1,∞). In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.     

Construction of D-optimal designs for N ≡ 2 mod 4 using block-circulant matrices

Circulant matrices of order t with elements circulant matrices of order s are used for the construction of D-optimal saturated designs of order N = 2st. A number of new designs are so constructed. The optimal design for N = 66 is constructed for the first time.     

Solving Equations of Free Vibration for a Cylindrical Shell Rotating on Rollers by the Fourier Method

The small free vibrations of an infinite circular cylindrical shell rotating about its axis at a constant angular velocity are considered. The shell is supported on n absolutely rigid cylindrical rollers equispaced on its circle. The roller-supported shell is a model of an ore benefication centrifugal concentrator with a floating bed. The set of linear differential equations of vibrations is sought in the form of a truncated Fourier series containing N terms along the circumferential coordinate. A system of 2N–n linear homogeneous algebraic equations with 2N–n unknowns is derived for the approximate estimation of vibration frequencies and mode shapes. The frequencies ω _k , k = 1, 2, …, 2N–n, are positive roots of the (2N–n)th-order algebraic equation D(ω²) = 0, where D is the determinant of this set. It is shown that the system of 2N–n equations is equivalent to several independent systems with a smaller number of unknowns. As a consequence, the (2N–n)th-order determinant D can be written as a product of lower-order determinants. In particular, the frequencies at N = n are the roots of algebraic equations of an order is lower than 2 and can be found in an explicit form. Some frequency estimation algorithms have been developed for the case of N > n. When N increases, the number of found frequencies also grows, and the frequencies determined at N = n are refined. However, in most cases, the vibration frequencies can not be found for N > n in an explicit form.     

Bilinear approach to <Emphasis Type="Italic">N</Emphasis> = 2 supersymmetric KdV equations

The N = 2 supersymmetric KdV equations are studied within the framework of Hirota bilinear method. For two such equations, namely N = 2, a = 4 and N = 2, a = 1 supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear Bäcklund transformation is given for the N = 2, a = 1 supersymmetric KdV equation.     

The extrinsic holonomy Lie algebra of a parallel submanifold

We investigate parallel submanifolds of a Riemannian symmetric space N. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full, complete, parallel submanifolds of N, usually called “1-fullness” of M. Furthermore, for every parallel submanifold \({M\subset N}\) we consider the pullback bundle T N| _M with the linear connection induced by \({\nabla^N}\) . Then there exists a distinguished parallel subbundle \({\mathcal {O}M}\) , usually called the “second osculating bundle” of M. Given a parallel isometric immersion from a symmetric space M into N, we can describe the “extrinsic” holonomy Lie algebra of \({\mathcal {O} M}\) by means of the second fundamental form and the curvature tensor of N at some fixed point. If moreover N is simply connected and M is even a full symmetric submanifold of N, then we will calculate the “extrinsic” holonomy Lie algebra of T N| _M in an explicit form.