Generalized Functions Research Articles

Impact of non-Gaussian beam profiles in the performance of hadron colliders

At the Large Hadron Collider (LHC), the interplay between a series of effects, including intrabeam scattering (IBS), synchrotron radiation, longitudinal beam manipulations, two beam effects (beam-beam, e-cloud) and machine non-linearities, can change the population of the core and tails and lead to non-Gaussian beam distributions, at different periods during the beam cycle. By employing generalised distribution functions, it can be demonstrated that the modified non-Gaussian beam profiles have an impact in the beam emittance evolution by itself and thereby to the collider luminosity. This paper focuses on the estimation of beam distribution modification and the resulting rms beam size due to the combined effect of IBS and synchrotron radiation by employing a Monte-Carlo simulation code which is able to track 3D generalised particle distributions (SIRE). The code is first benchmarked over classical semi-analytical IBS theories and then compared with measurements from the LHC at injection and collision energies, including projections for the High-Luminosity LHC (HL-LHC) high brightness regime. The impact of the distribution shape on the evolution of the bunch characteristics and machine performance is finally addressed.

Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model

In this article we study imaginary Gaussian multiplicative chaos—namely a family of random generalized functions which can formally be written as $e^{iX(x)}$, where $X$ is a log-correlated real-valued Gaussian field on $\mathbf{R}^{d}$, that is, it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions these objects live in, along with their basic stochastic properties, such as moment and tail estimates. After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, that is, the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution. The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.

Interval static analysis of multi-cracked beams with uncertain size and position of cracks

• Interval static analysis on cracked beams with uncertain size/position of cracks. • Uncertainty in size/position of cracks is handled by proper generalized functions. • Monotonicity of response is tested with respect to all uncertain parameters. • Bounds of monotonic response are obtained by sensitivity-based method. • Bounds of non-monotonic response are computed by global optimization technique. This paper deals with beams under static loads, in presence of multiple cracks with uncertain parameters. The crack is modelled as a linearly-elastic rotational spring and, following a non-probabilistic approach, both stiffness and position of the spring are taken as uncertain-but-bounded parameters. A novel approach is proposed to compute the bounds of the response. The key idea is a preliminary monotonicity test, which evaluates sensitivity functions of the beam response with respect to the separate variation of every uncertain parameter within the pertinent interval. Next, two alternative procedures calculate lower and upper bounds of the response. If the response is monotonic with respect to all the uncertain parameters, the bounds are calculated by a straightforward sensitivity-based method making use of the sensitivity functions built in the monotonicity test. In contrast, if the response is not monotonic with respect to even one parameter only, the bounds are evaluated via a global optimization technique. The presented approach applies for every response function and the implementation takes advantage of closed analytical forms for all response variables and related sensitivity functions. Numerical results prove efficiency and robustness of the approach, which provides very accurate bounds even for large uncertainties, avoiding the computational effort required by the vertex method and Monte Carlo simulation.

GREEN TENSOR OF MOTION EQUATIONS OF TWO COMPONENTS BIOT’S MEDIUM BY STATIONARY VIBRATIONS

Here processes of wave propagation in a two-component Biot’s medium are considered which are generated by periodic forces actions. By use Fourier transformation of generalized functions, the Green tensor - a fundamental solutions of oscillation equations of this medium has been constructed. This tensor describes the process of propagation of harmonic waves of a fixed frequency in spaces of dimension N = 1,2,3 under the action of power sources concentrated at the coordinates origin, described by a singular delta -function. Based on it, generalized solutions of these equations are constructed under the action of various sources of periodic perturbations, which are described by both regular and singular generalized functions. For regular acting forces, integral representations of solutions are given that can be used to calculate the stress-strain state of a porous water-saturated medium.

On Fourier - Laplace transform of a class of generalized functions

On Fourier - Laplace transform of a class of generalized functions

Extensions of nonclosable operators and multiplication of distributions

In the paper some new constructions of extensions of nonclosable operators is proposed and several examples of applications are given. One of particular cases of the problem under consideration is the question on multiplication of distributions, a solution to which can be given by introduction of the so-called new generalized functions. It was demonstrated that the main obstacle for multiplication of distributions is nonclosablility of classical multiplication and the construction of new generalized functions is based on the ideas similar to that used under construction of the extension of nonclosable operators.

Characterization of the passive mechanical properties of spine muscles across species

Passive mechanical properties differ between muscle groups within a species. Altered functional demands can also shift the passive force-length relationship. The extent that passive mechanical properties differ within a muscle group (e.g. spine extensors) or between homologous muscles of different species is unknown. It was hypothesized that multifidus, believed to specialize in spine stabilization, would generate greater passive tensile stresses under isometric conditions than erector spinae, which have more generalized functions of moving and stabilizing the spine; observing greater multifidus moduli in different species would strengthen this hypothesis. Permeabilized fibre bundles (n = 337) from the multifidus and erector spinae of mice, rats, and rabbits were mechanically tested. A novel logistic function was fit to the experimental data to fully characterize passive stress and modulus. Species had the greatest effect on passive muscle parameters with mice having the largest moduli at all lengths. Rats generated less passive stress than rabbits due to a shift of the passive force-length relationship towards longer muscle lengths. Rat multifidus generated slightly greater stresses than erector spinae, but no differences were observed between mouse muscles. The secondary objective was to determine the parameters required to simulate the passive force-length relationship. Experimental data were compared to the passive muscle model in OpenSim. The default OpenSim model, optimized for hindlimb muscles, did not fit any of the spine muscles tested; however, the model could accurately simulate experimental data after adjusting the input parameters. The optimal parameters for modelling the passive force-length relationships of spine muscles in OpenSim are presented.

Three Categories of Set-Valued Generalizations From Fuzzy Sets to Interval-Valued and Atanassov Intuitionistic Fuzzy Sets

Many different notions included in the fuzzy set literature can be expressed in terms of functionals defined over collections of tuples of fuzzy sets. During the past decades, different authors have independently generalized those definitions to more general contexts, like interval-valued fuzzy sets and Atanassov intuitionistic fuzzy sets. These generalized versions can be introduced either through a list of axioms or in a constructive manner. We can divide them into two further categories: set-valued and point-valued generalized functions. Here, we deal with constructive set-valued generalizations. We review a long list of functions, sometimes defined in quite different contexts, and we show that we can group all of them into three main different categories, each of them satisfying a specific formulation. We respectively call them the set-valued extension, the max–min extension, and the max–min varied extension. We conclude that the set-valued extension admits a disjunctive interpretation, whereas the max–min extension can be interpreted under an ontic perspective. Finally, the max–min varied extension provides a kind of compromise between both approaches.

On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis

Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of successful approaches to building of such a theory consists in introduction of spaces of the above-mentioned functions in such a way that the dual pairing between test and generalized functions is generated by integration with respect to some probability measure. First it was the Gaussian measure, then it were realized numerous generalizations. In particular, important results can be obtained if one uses the Levy white noise measure, the corresponding theory is called the Levy white noise analysis.
 In the Gaussian case one can construct spaces of test and generalized functions and introduce some important operators (e.g., stochastic integrals and derivatives) on these spaces by means of a so-called chaotic representation property (CRP): roughly speaking, any square integrable random variable can be decomposed in a series of repeated Itos stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP, but there are different generalizations of this property.
 In this paper we deal with one of the most useful and challenging generalizations of the CRP in the Levy analysis, which is proposed by E.W. Lytvynov, and with corresponding spaces of nonregular generalized functions. The goal of the paper is to introduce a natural product (a Wick product) on these spaces, and to study some related topics. Main results are theorems about properties of the Wick product and of Wick versions of holomorphic functions. In particular, we prove that an operator of stochastic differentiation satisfies the Leibniz rule with respect to the Wick multiplication. In addition we show that the Wick products and the Wick versions of holomorphic functions, defined on the spaces of regular and nonregular generalized functions, constructed by means of Lytvynov's generalization of the CRP, coincide on intersections of these spaces.
 Our research is a contribution in a further development of the Levy white noise analysis.

Exact frequency response of two-node coupled bending-torsional beam element with attachments

• Coupled bending-torsional dynamics of beams with dampers/masses is addressed. • Exact analytical frequency response is built for any harmonic point/polynomial loads. • Exact two-node 6 × 6 dynamic stiffness matrix and 6 × 1 load vector are built in closed form. • Dynamic stiffness matrix and load vector hold same size for any number of dampers/masses and loads. • Modal frequency responses and modal impulse responses are derived by complex modal analysis. This paper addresses the frequency response of coupled bending-torsional beams carrying an arbitrary number of in-span viscoelastic dampers and attached masses. Using the elementary coupled bending-torsion theory, along with appropriate generalized functions to treat the discontinuities of the response variables at the application points of dampers/masses, exact analytical expressions are derived for the frequency response of the beam under harmonically-varying, arbitrarily-placed point/polynomial loads. On this basis, the exact 6 × 6 dynamic stiffness matrix and 6 × 1 load vector of a two-node coupled bending-torsional beam finite element, with any number of in-span dampers/masses and harmonic loads, are obtained in a closed analytical form. Finally, the modal frequency response functions of the beam are built by a complex modal analysis approach, upon deriving pertinent orthogonality conditions for the modes. In this context, the modal impulse response functions are also obtained for time-domain analysis under arbitrary loads.

Sufficiency for nondifferentiable multiobjective programming problem

This paper introduces a class of new generalized functions of the concept of invex of σ(B, ϕ) − V − type I for nondifferentiable locally Lipschitz functions by using the tools of Clarke subdifferential. These functions are used to derive the sufficient optimality conditions for a class of nondifferentiable multiobjective nonlinear programming problems with inequality constraints where the objective and constraint functions are locally Lipschitz.

Invariant generalized functions supported on an orbit

We study the space of invariant generalized functions supported on an orbit of the action of a real algebraic group on a real algebraic manifold. This space is equipped with the Bruhat filtration. We study the generating function of the dimensions of the filtras, and give some methods to compute it. To illustrate our methods we compute those generating functions for the adjoint action of GL3(C). Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently in [Sak16].

Uniqueness of solutions to some quasilinear elliptic equations whose Hamiltonian has natural growth in the gradient

The paper discusses uniqueness of solutions to stationary elliptic problems of the type $$\begin{aligned} A(u)+H(u)=f\in {\mathcal {D}}'(\Omega ), \end{aligned}$$ where $$\Omega \ \in R^{N},\ $$ $$u\in W^{1,p}(\Omega )\ (1\le p\le +\infty ),\ A(u)\ $$ is an elliptic operator, $$H(u)\ $$ is an Hamiltonian that grows with $$\left| {\nabla u}\right| ^{p}$$ and f is given. Methods introduced in Artola (Boll UMI 6(5-B):51–71, 1986), (Proceedings of the International Conference on Generalized Functions, (ICGF 2000). Cambridge Scientific Publishers, Cambridge, 51–92, 2004), (Ricerche di Matematica XLIV, fasc. 2:400–420, 1995) for quasilinear parabolic or elliptic equations, together with properties for some continuity moduli, are used to improve some results from Barles and Murat (Arch Ration Mech Anal 133(1):77–101, 1995) for bounded solutions and from Barles and Porretta (Ann Scuola Norm Sup Pisa Cl Sci 5(1):107–136, 2006), Lions (J Anal Math 45: 234–254, 1985) for unbounded solutions, when 1 $$\le p\le 2.$$ Unilateral problems are considered and the case where f depends on the solution u is also discussed.

Bogoliubov-Born-Green-Kirkwood-Yvon chain and kinetic equations for the level dynamics in an externally perturbed quantum system

Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing an approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in the case of an adiabatically slow external perturbation, though it is not restricted to adiabatic systems. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of particles with cubic repulsion. Equilibrium statistical mechanics of this Pechukas gas allows us to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the Pechukas gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e., approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and interlevel transition amplitudes, they allow us to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as $1/N$, where $N$ is the number of the ``Pechukas gas particles'' (i.e., energy levels in the system).

On form factors and correlation functions in twistor space

In this paper, we continue our study of form factors and correlation functions of gauge-invariant local composite operators in the twistor-space formulation of mathcal{N}=4 super Yang-Mills theory. Using the vertices for these operators obtained in our recent papers [1, 2], we show how to calculate the twistor-space diagrams for general NkMHV form factors via the inverse soft limit, in analogy to the amplitude case. For general operators without α indices, we then reexpress the NMHV form factors from the position-twistor calculation in terms of momentum twistors, deriving and expanding on a relation between the two twistor formalisms previously observed in the case of amplitudes. Furthermore, we discuss the calculation of generalized form factors and correlation functions as well as the extension to loop level, in particular providing an argument promised in [3].

Observables, Evolution Equation, and Stationary States Equation in the Joint Probability Representation of Quantum Mechanics

Symplectic and optical joint probability representations of quantum mechanics are considered, in which the functions describing the states are the probability distributions with all random arguments (except the argument of time ). The general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations is given. Taking the Gaussian functions as the distributions of the tomographic parameters the correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators in the form of singular and regular generalized functions are derived. Evolution equations and stationary states equations for symplectic and optical joint probability distributions are obtained.

The chain rule applied to generalized functions for use in teaching electric circuits

Generalized functions, in particular the Heaviside unit step H( t) and the Dirac delta impulse δ( t), are valuable teaching tools in many contexts of electric circuits. However, engineering undergraduate courses and textbooks normally cover only an introduction to these functions along with their basic properties, without tackling the rigorous mathematical framework of the theory of distributions. In this work, the Heaviside function is used to represent a square wave signal f( t), and the Dirac function appears when the derivative is calculated. The steps for obtaining f ′( t) compare a graphical method with analytical procedures that employ the chain rule. The topic is further extended with the presentation of a theorem and a proposed corollary related to the study of the chain rule applied to general functions involving H( t) and δ( t). Thus, the work provides an appropriate mathematical support for the intuitive graphical method of analysis, which is very familiar in engineering practice.

ON WEIGHTED GENERALIZED FUNCTIONS ASSOCIATED WITH QUADRATIC FORMS

In this article we consider certain types of weighted generalized functions associated with nondegenerate quadratic forms. Such functions and their derivatives are used for constructing fundamental solutions of iterated ultra-hyperbolic equations with Bessel operator and for constructing negative real powers of ultra-hyperbolic operators with Bessel operator.

Generalization of a stability domain estimation method for nonlinear discrete systems

In this paper, we present a generalization approach of an algebraic existing method to estimate stability regions for discrete nonlinear polynomial systems of degree 3. The existing method is based on the enlargement of a guaranteed stability region by applying various steps of a proposed algorithm. Its main limitation is that the initial result has only been subsequently developed in a particular case of a single iteration. The stability domain obtained is consequently not the widest one. Our main contribution in this paper is to develop generalized functions that allow the enlargement of the guaranteed stability region after k iterations, for any value of k. A required fundamental tool is developed and consists in a general formula allowing to give the result of the Kronecker power calculation of two matrices sum. The advantages of this generalization are to reach a larger region of asymptotic stability and to improve the existing methods results. Two application examples illustrate the proposed method.

The Internet and Generalized Functions of the Public Sphere: Transformative Potentials From a Comparative Perspective

Almost since the advent of the Internet, there has been great interest in analyzing and understanding online communication from the perspective of public sphere theory. The question of whether the properties of the Internet and, specifically, social media actually contribute to the public sphere is the matter of ongoing and somewhat heated scientific debate. The aim of the article is twofold. First, we propose a hierarchical model of generalized functions of public sphere. On a theoretical level, we interweave different strands of thought on the public sphere, and the resulting model is more inclusive and less rigid than each of those strands on their own. We identify four generalized functions: identity building, agenda-setting, control and criticism, and deliberation. The Internet does not contribute equally to these functions and we evaluate the impact of the Internet on each of these functions as a diminishing marginal utility. Second, we empirically explore the plausibility of our model in a global comparative analysis with focus on the Internet. With the help of macro-level variables which indicate the structural preconditions for a public sphere, we identify the highest possible function of the public sphere for each country to which the Internet can potentially contribute. Based on this approach, future research can be contextualized: case-study-based research can plausibly articulate expectations regarding the impact of the Internet on the public sphere.