Convolution Formula Research Articles

Intermittency and a phase transition in a lattice gas model.

On the basis of a lattice gas model and the convolution formula with a cell construction scheme, we demonstrate that intermittency, i.e., the power law behavior of the moments in the rapidity space, is caused by a phase transition between the ordered phase and disordered phase with respect to the particle number distribution. In this model the critical moments are directly connected with an order parameter at the phase transition point. The indices of these moments are simply given by the critical exponent near the critical point. It is pointed out that the critical indices of this phase transition are not constant, but depend on the size of the rapidity interval, if the system is not near enough to the critical point.

Proton spin structure functions and generalized Sullivan processes.

We employ the time-ordered perturbation field theory in the infinite momentum limit, as developed extensively by Drell, Levy, and Yan, to derive the convolution formulas for the proton spin structure functions ${g}_{1}(x,{Q}^{2})$ and ${g}_{2}(x,{Q}^{2})$ suitable for the generalized Sullivan processes, in which the meson in the cloud or the recoiling baryon is struck and smashed by the virtual photon. Following a previous suggestion that the sea quark distributions of a nucleon at low and moderate ${Q}^{2}$ may be attributed primarily to such Sullivan processes, we use the convolution formulas to obtain the first moments ${\ensuremath{\Gamma}}_{p}\ensuremath{\equiv}\ensuremath{\int}{0}^{1}{g}_{1}^{p}(x,{Q}^{2})dx$ and ${\ensuremath{\Gamma}}_{n}\ensuremath{\equiv}\ensuremath{\int}{0}^{1}{g}_{1}^{n}(x,{Q}^{2})dx$, as well as to investigate ${g}_{1}^{p}(x,{Q}^{2})$ and ${g}_{1}^{n}(x,{Q}^{2})$ as a function of $x$. Our results indicate that both the data on ${g}_{1}^{p}(x,{Q}^{2})$ obtained previously by the European Muon Collaboration and the recent data on the deuteron spin structure function ${g}_{1}^{d}(x,{Q}^{2})$ obtained by the Spin Muon Collaboration can be understood and the Bjorken sum rule is satisfied. A comparison between our prediction on ${g}_{1}^{n}(x,{Q}^{2})$ and the recent SLAC data is also given.

Algorithms for the transient simulation of lossy interconnect

In this paper, a new linear-time technique is described for the simulation of lossy lines with frequency-independent R, L, C and G. Exact analytic forms are shown to exist for the frequency-independent lossy line, with application in both the new technique and the conventional convolution method. Numerical convolution formulae that exploit the analytic forms are presented. Experimental results for industrial circuits indicate that the new technique can be 10 and 50 times faster than the convolution and lumped-RLC methods, respectively, for long simulations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improved recursive convolution integral for the analysis of fluorescence decay data: Local approximation of the apparatus response function by a general polynomial

An analytical solution of the recursion formula for convolution of a sum-of-exponentials fluorescence impulse (δ) response with an apparatus response (‘‘lamp’’) function approximated locally by a general polynomial of the nth degree is derived. The five lowest approximations are tested by comparison with properly simulated fluorescence evolution-and-decay data. These were obtained by analytical convolution of selected δ responses with an analytically defined apparatus function chosen closely to simulate typically observable flash-lamp apparatus functions, taking account of the integration within channels inherent in the experimental collection of both ‘‘lamp’’ and fluorescence response curves. A precise comparison of the recovery of various test theoretical mono- and sum-of-exponentials impulse-response parameters for the different approximations was attained by performing all calculations for both simulation and nonlinear least-squares optimization analysis in double precision. The results highlight the advantages of using local higher-order polynomial approximations under various circumstances. The improved time resolution in particular suggests more expedient regimes of data collection than heretofore possible.

Off-shell W-pair production in e+e−-annihilation. Initial state radiation

With a current-splitting technique, we calculate the gauge-invariant initial-state radiation to order O(α) with soft-photon exponentiation for on- and off-shell W-pair production. This result generalizes the convolution formula, which is known from the description of the Z resonance, to the case of the production of two W-bosons. After up to eightfold analytical integrations, a sufficiently smooth integral over three invariant masses remains to be treated numerically. Including the Coulomb singularity, the largest corrections are covered. We discuss the corrections in a large energy range up to √ s = 1 TeV and draw numerical conclusions on their influence on the W-mass determination at LEP 200.

Calculating frequency‐domain data by time‐domain methods

AbstractWe show the derivation of parameters in the frequency domain from time‐domain data. Far‐field characteristics are obtained by a convolution formula with the harmonic field amplitudes, which are obtained via a Fourier transform or by sampling. The electric field of filter ports is expanded into the discrete eigenmodes. By this method, a monochromatic exact open boundary can be formulated and the fields divided into the incident and the reflected part. For wide band operation an a posteriori error correction scheme is presented.

Role of nuclear binding in the European-Muon-Collaboration effect.

We present a new derivation of the convolution formula for the contributions of nuclear binding to the structure functions measured in the deep inelastic scattering of leptons from nuclei. The derivation, which is manifestly covariant, gives a new binding correction. This new correction, which depends on the mass of the recoiling nucleon fragments, gives corrections that are numerically significant, and that improve the agreement between theory and experiment at large x. We conclude that nuclear binding effects may be sufficient to explain the European-Muon-Collaboration effect at large x.

Some aspects of Wiener-Hopf factorization

Wiener-Hopf factorization means many apparently different things, both in theory and in its wide variety of applications. This paper is designed so that almost all of it may be read by non-probabilists, though it makes demands on the reader’s ability to use analogy. It is written in response to requests from people in other fields to give some idea of what probabilists are doing. It gives some reformulations of the probabilistic Wiener-Hopf problem studied by London et al . One reformulation as a problem of simultaneous reduction of quadratic forms is used to motivate another as a Riemann-Hilbert problem. In addition to trying to synthesize various results, it answers affirmatively a question of McGregor as to whether a useful convolution formula which he obtained in a special case holds generally. Section 4 on examples, methods, and their interrelations is the liveliest part of the paper. Though algebra and complex analysis are successful and link perfectly with probability in much of what has so far been achieved, the scope of these methods is very severely limited, and much more challenging problems lie ahead. Motivation for this study derives originally from the practically important fact that integrals of Markov processes often provide better models than Markov processes themselves; but it has obvious pure-mathematical ‘rightness’ too.

European Muon Collaboration effect: Nuclear-binding effect or vivid quark signature?

A relativistic two-level convolution model for deep-inelastic lepton nucleus scattering is presented in which the target nucleus is considered as a composite system of baryon-mesons which are also composite systems of quark-gluons, with each level based on the light-cone quantum field theory, respectively. In this prescription the impulse approximation is justified, the baryon number conservation is naturally guaranteed, and the off-shell ambiguity in the nucleonic approach is fixed. The scaling variable for bound nucleons is derived in a realistic way from the constraint of overall ``energy'' conservation, and a two-level convolution formula for nuclear-structure functions is given. It is shown that the European Muon Collaboration effect seems unable to be explained by the off-shell effect in terms of nucleonic degrees of freedom alone in this prescription.

Risk theory for the compound Poisson process that is perturbed by diffusion

The classical model of collective risk theory is extended in that a diffusion process is added to the compound Poisson process. It is shown that the probabilities of ruin (by oscillation or by a claim) satisfy certain defective renewal equations. The convolution formula for the probability of ruin is derived and interpreted in terms of the record highs of the aggregate loss process. If the distribution of the individual claim amounts are combinations of exponentials, the probabilities of ruin can be calculated in a transparent fashion. Finally, the role of the adjustment coefficient (for example, for the asymptotic formulas) is explained.

On the design and performance of the Czerny-Turner monochromator in ICP-AES

The optical arrangement of the Czerny-Turner (CT) monochromator and the theoretical dependence of dispersion and spectral slit width on wavelength, order and groove density are discussed in detail. Because a given spectral line appears at the grating under higher angles for higher order or higher groove density, the improvement in spectral slit width is always greater than the ratio of the orders or the ratio of the groove densities. Attention is drawn to the magnification of the image of the entrance slit by the CT optics and its influence on spectral slit width and the proper choice of the exit slit width. The improvement of spectral slit width by an apodization mask is theoretically discussed and experimentally verified. An extended convolution formula which includes the magnification factor is proposed and compared with known models. Coma as a source of residual lack of fit and the conditions for coma-free imaging in the CT system are discussed. The theoretically predicted performance of Czerny-Turner monochromators is compared with results obtained with two commercially available 1-m instruments with gratings of 2400 and 3600 linesmm.

Y scaling as a probe of nuclear light-cone dynamics.

The y scaling exhibited in quasielastic electron scattering on nuclei is shown to occur in the same kinematic limit as x scaling in deep-inelastic lepton-nucleon scattering. Using the impulse approximation in a relativistic model, we demonstrate that the scaling function F(y) can be interpreted as the nucleon light-cone momentum distribution and the scaling variable y is related to the light-cone momentum t+ of the nucleon. We also derive the convolution formula for deep-inelastic lepton-nucleus scattering and show that the same F(y) can be extracted from the experimental structure functions of the nucleon and nuclei.

Gaussian‐based kernels

AbstractWe derive a class of higher‐order kernels for estimation of densities and their derivatives, which can be viewed as an extension of the second‐order Gaussian kernel. These kernels have some attractive properties such as smoothness, manageable convolution formulae, and Fourier transforms. One important application is the higher‐order extension of exact calculations of the mean integrated squared error. The proposed kernels also have the advantage of simplifying computations of common window‐width selection algorithms such as least‐squares cross‐validation. Efficiency calculations indicate that the Gaussian‐based kernels perform almost as well as the optimal polynomial kernels when die order of the derivative being estimated is low.

Structure functions of nuclei in the "instant" form of dynamics.

Convolution formulas for the inelastic structure functions of a nucleus are derived using the ``instant'' form of relativistic particle dynamics, and are contrasted with other convolution approaches. Using only physical nucleon constituents and harmonic oscillator shell model wave functions, the magnitude of the dip in the European Muon Collaboration ratio in the intermediate range of x is reproduced. The calculations of this ratio fail to come back up towards unity as x decreases below 0.4, and possible reasons for this are discussed.

Two-level convolution formula for nuclear structure function

A two-level convolution formula for the nuclear structure function is derived in considering the nucleus as a composite system of baryon-mesons which are also composite systems of quark-gluons again. The results show that the European Muon Colaboration effect can not be explained by the nuclear effects as nucleon Fermi motion and nuclear binding contributions.

Identification of the acoustic ocean transfer function in the Tyrrhenian Sea. II: Physical considerations

In the process of statistical analysis of signature/response pairs collected in the Tyrrhenian Sea during the Napoli 85 Trial, a novel deconvolution method has been developed for identifying the ocean transfer function (OTF) when its bandwidth is larger than the bandwidth of the emitted pulse. The analysis itself is described in the preceding paper [L. Ju. Fradkin, J. Acoust. Soc. Am. 87, XXXX (1990)]. Here, the underlying convolution formula relating the received signal to the signature as well as other important physical constraints are derived under the assumptions (reasonable in the analyzed case) that the emitted pulse contains no low frequencies, and that it travels along the paths none of which reverses the phase (i.e., no reflections or caustics are involved). The results of processing a representative signature/response pair with the proposed method are discussed briefly. It is shown that for the given source/receiver configuration at least two deep sea (and hence non-phase-reversing) paths exist. It is also shown that the ‘‘internal waves’’ model, rejected by other investigators in the course of their analysis of the environmental data, would not allow for existence of the second macropath. It is concluded that in certain circumstances, the proposed deconvolution method could be used to assess models of medium fluctuations.

Convolution theory of a phase transition between hadronic and quark matter and the characteristic multiplicity distributions.

A convolution form of the free energy density of the thermodynamic system of hadrons is given on the basis of the convolution formula which yields a general form of the statistical bootstrap equation. It is shown that in the thermodynamic limit this free energy density has a typical singular form, which presents a phase transition between hadronic and quark matter, in the same way as magnetic transitions and classical liquid-gas transitions. The characteristic multiplicity distribution is predicted as a signal of the disordered state.

Nuclear binding effect in deep-inelastic lepton scattering

We show that the convolution formula for the structure function F 2 A of the nucleus used in our previous work follows from a fully relativistic consideration of the problem. We demonstrate that the contribution of nuclear binding to F 2 A does not depend on the flux factor. We therefore conclude that our previous results give the correct magnitude of the effect of nuclear binding F 2 A .

Multifold analogues of Gould-Hsu inversions and their applications

By means of difference calculus, a multifold analog of Gould-Hsu inversion is established. As concrete examples, some multifold inverse relations are presented. By developing a standard proof technique for combinatorial identities, we demonstrate Mohanty-Hanna's convolution formulas almost mechanically. In addition, an application of reciprocal formulas to interpolation process is also sketched.

Factorial functions and stirling numbers of fractional orders

The aim of this paper is to study the binomial coefficients ( xn ), the factorial polynomials [x]n and [x]n, the Stirling numbers of first and second kind, namely s(n,k) and S(n,k), in the case that n ∈ ℕ is replaced by real α ∈ ℝ. In the course of the paper, the Vandermonde convolution formula is presented in an infinite series frame, the binomial coefficient function ( xa ), α ∈ ℝ, is sampled in terms of the binomial coefficients ( xk ) for k ∈ ℕo, Bell numbers of fractional orders are introduced. Emphasis is placed on the fractional order Stirling numbers s(α,k) and S(α,k), first studied here. Some applications of the S(α,k) are given.