Growth Series and Random Walks on Some Hyperbolic Graphs

 Consider the tessellation of the hyperbolic plane by m-gons, ℓ per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of proper loops in these graphs. We then apply Grigorchuk’s result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs.     

Analysis of the energetic radiation intensity of a linear antenna array radiating nonsinusoidal signals

We present the energetic radiation intensity (ERI) as the quadratic form of the family of integral operators on a finite interval. The kernel of each operator is the autocorrelation function of the signal, which is radiated in the given direction. Spectral representation of the operators gives a fast-converging series representation of the ERI. For the signals, whose Fourier transforms are rational functions of the frequency, spectral analysis of the operators is reduced to finite-dimensional linear systems. Moreover, for such signals we express the ERI as the linear combination of the monochromatic directivity diagrams, evaluated in the complex poles of the signal’s Fourier transform. For the isotropic array elements and the most important amplitude distributions the ERI is obtained explicitly. We consider in detail a signal given by a truncated decaying exponent. Bibliography: 32 titles. Dedicated to Vasilii Mikhailovich Babich with high respect and gratitude __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 239–267.     

Growth Series and Random Walks on Some Hyperbolic Graphs

 Consider the tessellation of the hyperbolic plane by m-gons, ℓ per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of proper loops in these graphs. We then apply Grigorchuk’s result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs. Received 19 September 2001; in revised form 23 December 2001     

An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions

To construct an effective minimization algorithm for locally Lipschitz functions, we show how to compute a descent direction satisfying Armijo’s condition. We present a finitely terminating algorithm to construct an approximating set for the Goldstein subdifferential leading to the desired descent direction. Using this direction, we propose a minimization algorithm for locally Lipschitz functions and prove its convergence. Finally, we implement our algorithm with matrix laboratory (MATLAB) codes and report our testing results. The comparative numerical results attest to the efficiency of the proposed algorithm.     

The centroid of extended affine and root graded Lie algebras

We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.     

Computing Zeta Functions of Superelliptic Curves in Larger Characteristic

Following Gaudry and Gürel who extended Kedlaya’s algorithm to superelliptic curves, we introduce Harvey’s optimisation for large characteristic p to the superelliptic case. As result, we state the most general algorithm to compute zeta functions that runs soft linear in p ^1/2. We demonstrate its effectiveness using a Magma implementation.     

The Quiver of an Algebra Associated to the Mantaci-Reutenauer Descent Algebra and the Homology of Regular Semigroups

We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola’s results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao’s algebra, which is related to the Mantaci-Reutenauer descent algebra.     

Diversity of firm’s life cycle adapted from the firm’s technology investment decision

The stylized model presented is an optimal control model of technology investment decision of a single product firm. The firm’s technology investment does not have only a long-run positive effect but also a short-run adverse effect on its sales volume. We examine the case of high adverse investment effects where the firm finally leaves the market but we have observed different life cycles till this happens. Depending on the firm’s initial technology stock and sales volume, we compute different firm’s life cycles, which are driven by a trade-off between two strategies: technology versus sales focus strategy. Indifference curves, where managers are indifferent to apply initially technology or sales focus strategies, separate founding conditions of the firm to various classes distinguishable because of the firm’s life cycle.     

The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

We show how ‘test’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger’s constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger’s constant in terms of the inradius of a plane domain. Received: 13 June 2005     

Single-machine scheduling problems with time and position dependent processing times

We consider single-machine scheduling problems with time and position dependent job processing times. In many industrial settings, the processing time of a job changes due to either job deterioration over time or machine/worker’s learning through experiences. In the models we study, each job has its normal processing time. However, a job’s actual processing time depends on when its processing starts and how many jobs have completed before its start. We prove that the classical SPT (Shortest Processing Time) rule remains optimal when we minimize the makespan or the total completion time. For problems of minimizing the total weighted completion time, the maximum lateness, and the discounted total weighted completion time, we present heuristic sequencing rules and analyze the worst-case bounds for performance ratios. We also show that these heuristic rules can be optimal under some agreeable conditions between the normal processing times and job due dates or weights.     

An application of simultaneous diophantine approximation in combinatorial optimization

We present a preprocessing algorithm to make certain polynomial time algorithms strongly polynomial time. The running time of some of the known combinatorial optimization algorithms depends on the size of the objective functionw. Our preprocessing algorithm replacesw by an integral valued-w whose size is polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions asw. As applications we show how existing polynomial time algorithms for finding the maximum weight clique in a perfect graph and for the minimum cost submodular flow problem can be made strongly polynomial. Further we apply the preprocessing technique to make H. W. Lenstra’s and R. Kannan’s Integer Linear Programming algorithms run in polynomial space. This also reduces the number of arithmetic operations used. The method relies on simultaneous Diophantine approximation. This research was done while the authors were visiting the Institute for Operations Research, University of Bonn, West Germany (1984–85), and while the second author was visiting MSRI, Berkeley. Her research was supported in part by NSF Grant 8120790.     

Computing numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions

In this article we compute numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions. A compactly perturbed half-plane Helmholtz problem is used to motivate this calculation, by treating it through integral equation techniques. These require the knowledge of the calculated Green’s function, and lead to a boundary element discretization. The Green’s function is computed using the inverse Fourier operator of its spectral transform, applying an inverse FFT for the regular part, and removing the singularities analytically. Finally, some numerical results for the Green’s function and for a benchmark resonance problem are shown.     

Uniform Shrinking and Expansion under Isotropic Brownian Flows

We study some finite time transport properties of isotropic Brownian flows. Under a certain nondegeneracy condition on the potential spectral measure, we prove that uniform shrinking or expansion of balls under the flow over some bounded time interval can happen with positive probability. We also provide a control theorem for isotropic Brownian flows with drift. Finally, we apply the above results to show that, under the nondegeneracy condition, the length of a rectifiable curve evolving in an isotropic Brownian flow with strictly negative top Lyapunov exponent converges to zero as t→∞ with positive probability. P. Baxendale’s research was supported in part by NSF Grant DMS-05-04853.     

Scattering by magnetic nanocylinders and the method of distorted waves in noncollinear fields

In the perturbation theory framework, we compute the cross section of scattering by a magnetic nanocylinder and a helicoid arbitrarily oriented in an external magnetic field. We are the first to obtain the matrix Green’s function for two media with an interface and noncollinear magnetic fields on the two sides of the interface. We show how to compute scattering by magnetic inclusions in one of the media.     

Optimal Compensation with Hidden Action and Lump-Sum Payment in a Continuous-Time Model

We consider a problem of finding optimal contracts in continuous time, when the agent’s actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal’s utility is infinite, while it becomes finite with hidden action, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization.     

Pivoting in Extended Rings for Computing Approximate Gr?bner Bases

It is well known that in the computation of Gr?bner bases arbitrarily small perturbations in the coefficients of polynomials may lead to a completely different staircase, even if the solutions of the polynomial system change continuously. This phenomenon is called artificial discontinuity in Kondratyev’s Ph.D. thesis. We show how such phenomenon may be detected and even “repaired” by using a new variable to rename the leading term each time we detect a “problem”. We call such strategy the TSV (Term Substitutions with Variables) strategy. For a zero-dimensional polynomial ideal, any monomial basis (containing 1) of the quotient ring can be found with the TSV strategy. Hence we can use TSV strategy to relax term order while keeping the framework of Gr?bner basis method so that we can use existing efficient algorithms (for instance the F ₅ algorithm) to compute an approximate Gr?bner basis. Our main algorithms, named TSVn and TSVh, can be used to repair artificial e{\epsilon}-discontinuities. Experiments show that these algorithms are effective for some nontrivial problems.     

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Image inpainting using complex 2-D dual-tree wavelet transform

The dual-tree complex wavelet transform is a useful tool in signal and image processing. In this paper, we propose a dual-tree complex wavelet transform (CWT) based algorithm for image inpainting problem. Our approach is based on Cai, Chan, Shen and Shen’s framelet-based algorithm. The complex wavelet transform outperforms the standard real wavelet transform in the sense of shift-invariance, directionality and anti-aliasing. Numerical results illustrate the good performance of our algorithm.     

Maximally Equiangular Frames and Gauss Sums

In a finite-dimensional complex Euclidean space, a maximally equiangular frame is a tight frame which has a number of elements equal to the square of the dimension of the space, and in which the inner products of distinct elements are of constant magnitude. Though the general question of their existence remains open, many examples of maximally equiangular frames have been constructed as finite Gabor systems. These constructions involve number theory, specifically Schaar’s identity, which provides a reciprocity formula for quadratic Gauss sums. To be precise, Zauner used Schaar’s identity to compute the spectrum of a chirp-Fourier operator, the eigenvectors of which he conjectured to be well-suited for the construction of maximally equiangular Gabor frames. We provide two new characterizations of such frames, both of which further confirm the relevance of the theory of Gauss sums to this area of frame theory. We also show how the unique time-frequency properties of a particular cyclic chirp function may be exploited to provide a new, short and elementary proof of Schaar’s identity.      

The centroid of the zeroes of a polynomial via certain Laguerre-type operators

In this paper, we present several properties of the centroid of the zeroes of a polynomial. As an illustration, we apply these results to the d-orthogonal polynomials. Finally, we provide the relationship between different centroids of a general monic polynomial and its image under a certain Laguerre–type operator.