Abstract: The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function’s structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model. We derive a notion of the coherence of a signal with respect to a dictionary from our characterization of the approximation errors of a pursuit. The dictionary elements slected during the initial iterations of a pursuit correspond to a function’s coherent structures. The tail of the expansion, on the other hand, corresponds to a noise which is characterized by the invariant measure of the pursuit map. When using a suitable dictionary, the expansion of a function into its coherent structures yields a compact approximation. We demonstrate a denoising algorithm based on coherent function expansions.

Abstract: We describe efficient constructions for various cryptographic primitives (both in private-key and in public-key cryptography). We show these constructions to be at least as secure as the decisional version of the Diffie-Hellman assumption or as the assumption that factoring is hard. Our major result is a new construction of pseudo-random functions such that computing their value at any given point involves two multiple products. This is much more efficient than previous proposals. Furthermore, these functions have the advantage of being in TC/sup 0/ (the class of functions computable by constant depth circuits consisting of a polynomial number of threshold gates) which has several interesting applications. The simple algebraic structure of the functions implies additional features. In particular, we show a zero-knowledge proof for statements of the form "y=f/sub s/(x)" and "y/spl ne/f(x)" given a commitment to a key s of a pseudo-random function f/sub s/.

Abstract: A complex mathematical model that produces output values from input values is now commonly called a computer model. This thesis considers the problem of finding the global optimum of the response with few function evaluations. A small number of function evaluations is desirable since the computer model is often expensive (time consuming) to evaluate. The function to be optimized is modeled as a stochastic process from initial function evaluations. Points are sampled sequentially according to a criterion that combines promising prediction values with prediction uncertainty. Some graphical tools are given that allow early assessment about whether the modeling strategy will work well. The approach is generalized by introducing a parameter that controls how global versus local the search strategy is. Strategies to conduct the optimization in stages and for optimization subject to constraints on additional response variables are presented. Special consideration is given to the stopping criterion of the global optimization algorithm. The problem of achieving a tolerance on the global minimum can be represented by determining whether the first order statistic of N dependent variables is greater than a certain value. An algorithm is developed that quickly determines bounds on the probability of this event. A strategy to explore high-dimensional data informally through effect plots is presented. The interpretation of the plots is guided by pointwise standard errors of the effects which are developed. When used in the context of global optimization, the graphical analysis sheds light on the number and location of local optima.

Abstract: Coping with uncertainty in dynamical systems has recently received some attention in artificial intelligence (AI), particularly in the fields of qualitative and model-based reasoning. In this paper, we propose an approach to modelling and simulation of uncertain dynamics which is based on the following ideas: We consider (linguistic) descriptions of uncertain functional relationships characterizing the behavior of some dynamical system. Based on a certain interpretation of such rule-based models, we derive a fuzzy function $\tilde{F}$. It will be shown that all (reasonable) fuzzy functions can be approximated to any degree of accuracy in this way. The function $\tilde{F}$ is then used as the "fuzzy" right hand side of a set of differential equations, which leads us to consider fuzzy initial value problems. We are going to propose an interpretation of such problems. Moreover, several aspects of simulation methods for characterizing the set of all system behaviors compatible with this interpretation will be...

Abstract: When studying some process or development in different subjects or units--be it biological, chemical or physical--we usually see a typical pattern, common to all curves. Yet there is variation both in amplitude and dynamics between curves. Following some ideas of structural analysis introduced by Kneip and Gasser, we study a method--dynamic time warping with a proper cost function--for estimating the shift or warping function from one curve to another to align the two functions. For some models this method can identify the true shift functions if the data are noise free. Noisy data are smoothed by a nonparametric function estimate such as a kernel estimate. It is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity. Some simulation results are presented to illustrate the performance of this method.

Abstract: The second-order statistical properties of complex signals are usually characterized by the covariance function. However, this is not sufficient for a complete second-order description, and it is necessary to introduce another moment called the relation function. Its properties, and especially the conditions that it must satisfy, are analyzed both for stationary and nonstationary signals. This leads to a new perspective concerning the concept of complex white noise as well as the modeling of any signal as the output of a linear system driven by a white noise. Finally, this is applied to complex autoregressive signals, and it is shown that the classical prediction problem must be reformulated when the relation function is taken into consideration.

Abstract: A set of rules is given for dealing with WKB expansions in the one-dimensional analytic case, whereby such expansions are not considered as approximations but as exact encodings of wave functions, thus allowing for analytic continuation with respect to whichever parameters the potential function depends on, with an exact control of small exponential effects. These rules, which include also the case when there are double turning points, are illustrated on various examples, and applied to the study of bound state or resonance spectra. In the case of simple oscillators, it is thus shown that the Rayleigh–Schrodinger series is Borel resummable, yielding the exact energy levels. In the case of the symmetrical anharmonic oscillator, one gets a simple and rigorous justification of the Zinn-Justin quantization condition, and of its solution in terms of “multi-instanton expansions.”

Abstract: We study a class of methods for solving convex programs, which are based on nonquadratic augmented Lagrangians for which the penalty parameters are functions of the multipliers. This gives rise to Lagrangians which are nonlinear in the multipliers. Each augmented Lagrangian is specified by a choice of a penalty function $\varphi$ and a penalty-updating function $\pi$. The requirements on $\varphi$ are mild and allow for the inclusion of most of the previously suggested augmented Lagrangians. More importantly, a new type of penalty/barrier function (having a logarithmic branch glued to a quadratic branch) is introduced and used to construct an efficient algorithm. Convergence of the algorithms is proved for the case of $\pi$ being a sublinear function of the dual multipliers. The algorithms are tested on large-scale quadratically constrained problems arising in structural optimization.

Abstract: A dynamical model of capital exchange is introduced in which a specified amount of capital is exchanged between two individuals when they meet. The resulting time dependent wealth distributions are determined for a variety of exchange rules. For ``greedy'' exchange, an interaction between a rich and a poor individual results in the rich taking a specified amount of capital from the poor. When this amount is independent of the capitals of the two traders, a mean-field analysis yields a Fermi-like scaled wealth distribution in the long-time limit. This same distribution also arises in greedier exchange processes, where the interaction rate is an increasing function of the capital difference of the two traders. The wealth distribution in multiplicative processes, where the amount of capital exchanged is a finite fraction of the capital of one of the traders, are also discussed. For random multiplicative exchange, a steady state wealth distribution is reached, while in greedy multiplicative exchange a non-steady power law wealth distribution arises, in which the support of the distribution continuously increases. Finally, extensions of our results to arbitrary spatial dimension and to growth processes, where capital is created in an interaction, are presented.

Abstract: We have attempted to find the global minima of clusters containing between 20 and 80 atoms bound by the Morse potential as a function of the range of the interatomic force. The effect of decreasing the range is to destabilize strained structures, and hence the global minimum changes from icosahedral to decahedral to face-centred-cubic as the range is decreased. For N>45 the global minima associated with a long-ranged potential have polytetrahedral structures involving defects called disclination lines. For the larger clusters the network of disclination lines is disordered and the global minimum has an amorphous structure resembling a liquid. The size evolution of polytetrahedral packings enables us to study the development of bulk liquid structure in finite systems. As many experiments on the structure of clusters only provide indirect structural information, these results will be very useful in aiding the interpretation of experiment. They also provide candidate structures for theoretical studies using more specific and computationally expensive descriptions of the interatomic interactions. Furthermore, Morse clusters provide a rigorous testing ground for global optimization methods.

Abstract: An image is a function, f(x, y) , of the independent space variables x and y . The global Fourier spectrum of the image is a complex function F(k x , k y ) of the wave numbers k x and k y . The global spectrum may be viewed as a construct of the spectra of an arbitrary number of segments of f(x, y) , leading to the concept of a local spectrum at every point of f(x, y) . The two-dimensional S transform is introduced here as a method of computation of the local spectrum at every point of an image. In addition to the variables x and y , the 2-D S transform retains the variables k x and k y , being a complex function of four variables. Visualisation of a function of four variables is difficult. We skirt around this by removing one degree of freedom, through examination of ‘slices’. Each slice of the 2-D S transform would then be a complex function of three variables, with separate amplitude and phase components. By ranging through judiciously chosen slice locations the entire S transform can be examined. Images with strictly periodic patterns are best analysed with a global Fourier spectrum. On the other hand, the 2-D S transform would be more useful in spectral characterisation of aperiodic or random patterns.

Abstract: Consequences of species-area relationships (SARs) of the form S = cA are derived. One consequence is an endemics-area relationship (EAR); it is of the same power-law form as the SAR but with an exponent z' that is a function only of z and that always exceeds unity. An explicit formula is derived for the dependence of species turnover in space on census plot size, interplot distance, and the SAR exponent z; this formula can be used to determine z over spatial scales that are too large to permit direct estimate of z by censusing of nested patches. The areal dependence of link-species patterns observed in food webs is also examined; SARs are shown to imply the approximate, but not exact, area-independence of link-species relationships of the form L = kSy, where L is the number of trophic links and y < 2. A relationship between the average range of species in a habitat patch and the exponent z is also derived, leading to the result that average range is a decreasing function of both patch area and z.

Abstract: We present a new approach to clustering, based on the physical properties of an inhomogeneous ferromagnet. No assumption is made regarding the underlying distribution of the data. We assign a Potts spin to each data point and introduce an interaction between neighboring points, whose strength is a decreasing function of the distance between the neighbors. This magnetic system exhibits three phases. At very low temperatures it is completely ordered; all spins are aligned. At very high temperatures the system does not exhibit any ordering and in an intermediate regime clusters of relatively strongly coupled spins become ordered, whereas different clusters remain uncorrelated. This intermediate phase is identified by a jump in the order parameters. The spin-spin correlation function is used to partition the spins and the corresponding data points into clusters. We demonstrate on three synthetic and three real data sets how the method works. Detailed comparison to the performance of other techniques clearly indicates the relative success of our method.

Abstract: We generalize some results of Ford and Roman constraining the possible behaviors of the renormalized expected stress-energy tensor of a free massless scalar field in two-dimensional Minkowski spacetime. Ford and Roman showed that the energy density measured by an inertial observer, when averaged with respect to the observers proper time by integrating against some weighting function, is bounded below by a negative lower bound proportional to the reciprocal of the square of the averaging time scale. However, the proof required a particular choice for the weighting function. We extend the Ford-Roman result in two ways. (i) We calculate the optimum (maximum possible) lower bound and characterize the state which achieves this lower bound; the optimum lower bound differs by a factor of six from the bound derived by Ford and Roman for their choice of smearing function. (ii) We calculate the lower bound for arbitrary, smooth positive weighting functions. We also derive similar lower bounds on the spatial average of energy density at a fixed moment of time.

Abstract: We show that various possible versions of the Brjuno function, based on different kinds of continued fraction developments, are all equivalent and we study their regularity (L p, BMO and Holder) properties, through a systematic analysis of the functional equation which they fulfill.

Abstract: The author (1997) introduced a large family of one-unit contrast functions to be used in independent component analysis (ICA). In this paper, the family is analyzed mathematically in the case of a finite sample. Two aspects of the estimators obtained using such contrast functions are considered: asymptotic variance, and robustness against outliers. An expression for the contrast function that minimizes the asymptotic variance is obtained as a function of the probability densities of the independent components. Combined with robustness considerations, these results provide strong arguments in favor of the use of contrast functions based on slowly growing functions, and against the use of kurtosis, which is the classical contrast function.

Abstract: The Support Vector (SV) method is a new general method of function estimation which does not depend explicitly on the dimensionality of input space It was applied for pattern recognition, regression estimation, and density estimation problems as well as for problems of solving linear operator equations In this article we describe the general idea of the SV method and present theorems demonstrating that the generalization ability of the SV method is based on factors which classical statistics do not take into account We also describe the SV method for density estimation in a set of functions defined by a mixture of an infinite number of Gaussians

Abstract: Recently, Peng considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained differentiable optimization reformulation of VIP. In this paper, we generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the D-gap function. We give conditions under which any stationary point of the D-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the D-gap function.

Abstract: A method for estimating a real function that describes a phenomenon occurring in a space of any dimensionality is disclosed. The function is estimated by taking a series of measurements of the phenomenon being described and using those measurements to construct an expansion that has a manageable number of terms. A reduction in the number of terms is achieved by using an approximation that is defined as an expansion on kernel functions, the kernel functions forming an inner product in Hilbert space. By finding the support vectors for the measurements one specifies the expansion functions. The number of terms in an estimation according to the present invention is generally much less than the number of observations of the real world phenomenon that is being estimated. In one embodiment, the function estimation method may be used to reconstruct a radiation density image using Positron Emission Tomography (PET) scan measurements.

Abstract: Flux estimates for faint sources or transients are systematically biased high because there are far more truly faint sources than bright. Corrections which account for this effect are presented as a function of signal-to-noise ratio and the (true) slope of the faint-source number-flux relation. The corrections depend on the source being originally identified in the image in which it is being photometered. If a source has been identified in other data, the corrections are different; a prescription for calculating the corrections is presented. Implications of these corrections for analyses of surveys are discussed; the most important is that sources identified at signal-to-noise ratios of four or less are practically useless.

Abstract: We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x ) (log log x ) is not asymptotic as x →+∞ to a composition of semialgebraic functions, log and exp.

Abstract: A global assessment of carbon flux in the world ocean is one of the major undertakings of the Joint Global Ocean Flux Study (JGOFS). This has to be undertaken using historical in situ data of primary productivity. As required by the temporal and spatial scales involved in a global study, it can be conveniently done by combining, through appropriate models, remotely sensed information (chlorophyll a, temperature) with basic information about the parameters related to the carbon uptake by phytoplanktonic algae. This requires a better understanding as well as a more extended knowledge of these parameters which govern the radiative energy absorption and utilization by algae in photosynthesis. The measurement of the photosynthetic response of algae (the photosynthesis (P) versus in-adiance (£) curves), besides being less shiptime consuming than in situ primary production experiments, allows the needed parameters to be derived and systematically studied as a function of the physical, chemical and ecological conditions. The aim of the present paper is to review the sig- nificance of these parameters, especially in view of their introduction into models, to analyze the causes of their variations in the light of physiological considerations, and finally to provide methodo- logical recommendations for meaningful determinations, and interpretation, of the data resulting from /"versus £ determinations. Of main concern are the available and usable irradiance, the chloro- phyll a-specific absorption capabilities of the algae, the maximum light utilization coefficient (a), the maximum quantum yield (4>m), the maximum photosynthetic rate (Pm) and the light saturation index (£k). The potential of other, non-intrusive, approaches, such as the stimulated variable fluorescence, or the sun-induced natural fluorescence techniques is also examined.

Abstract: Role, often designated by a given title, e.g., manager, has been one of the most common means of defining occupational health nursing practice. A function based model provides an opportunity to reframe the occupational health nurse as a member of the management team. This descriptive study characterized the functions of a random sample (40%) of members of the American Association of Occupational Health Nurses from eight Midwestern states (463) in 1994. With a 78% response rate, the most frequently performed function for all respondents was “evaluate status of employees returning to work after absence” (68%). The relative frequencies for functions performed by the associates degree nurses were very similar to those for diploma nurses (r=.889 based on a perfect relationship value of 1). Subjects with baccalaureates in nursing performed more educational programming than subjects with non-nursing baccalaureates who performed more frequently in a policy area. Type of masters preparation represented different functional activities. The department to which the respondent reported affected functions. Reviewing function by salary level revealed a linear relationship with certain functions by frequency. To facilitate the investigation of the role construct based on functions, the researchers conducted a principle components analysis of the data. Four principle components were found representing groups of functions that tended to be performed by the same sets of respondents. The functions in each component tended to cluster around common skills as defined by Hersey (1988).