Nontrivial Polydispersity Exponents in Aggregation Models

We consider the scaling solutions of Smoluchowski's equation of irreversible aggregation, for a non gelling collision kernel. The scaling mass distribution f(s) diverges as s^{-tau} when s->0. tau is non trivial and could, until now, only be computed by numerical simulations. We develop here new general methods to obtain exact bounds and good approximations of $\tau$. For the specific kernel KdD(x,y)=(x^{1/D}+y^{1/D})^d, describing a mean-field model of particles moving in d dimensions and aggregating with conservation of ``mass'' s=R^D (R is the particle radius), perturbative and nonperturbative expansions are derived. For a general kernel, we find exact inequalities for tau and develop a variational approximation which is used to carry out the first systematic study of tau(d,D) for KdD. The agreement is excellent both with the expansions we derived and with existing numerical values. Finally, we discuss a possible application to 2d decaying turbulence.Comment: 16 pages (multicol.sty), 6 eps figures (uses epsfig), Minor corrections. Notations improved, as published in Phys. Rev. E 55, 546

Measuring similarity between business process models. In:

Abstract. Quality aspects become increasingly important when business process modeling is used in a large-scale enterprise setting. In order to facilitate a storage without redundancy and an efficient retrieval of relevant process models in model databases it is required to develop a theoretical understanding of how a degree of behavioral similarity can be defined. In this paper we address this challenge in a novel way. We use causal footprints as an abstract representation of the behavior captured by a process model, since they allow us to compare models defined in both formal modeling languages like Petri nets and informal ones like EPCs. Based on the causal footprint derived from two models we calculate their similarity based on the established vector space model from information retrieval. We validate this concept with an experiment using the SAP Reference Model and an implementation in the ProM framework

Asymptotics of self-similar solutions to coagulation equations with product kernel

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(\xi,\eta)= (\xi \eta)^{\lambda}$ with $\lambda \in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2\lambda} g(x)$ is bounded above and below as $x \to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_{\lambda} x^{-1+2\lambda} g(x)$ in the limit $\lambda \to 0$. It turns out that $h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x)$ as $x \to 0$. As $x$ becomes larger $h$ develops peaks of height $1/\lambda$ that are separated by large regions where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x \to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE

DeepAlign: Alignment-based Process Anomaly Correction using Recurrent Neural Networks

In this paper, we propose DeepAlign, a novel approach to multi-perspective process anomaly correction, based on recurrent neural networks and bidirectional beam search. At the core of the DeepAlign algorithm are two recurrent neural networks trained to predict the next event. One is reading sequences of process executions from left to right, while the other is reading the sequences from right to left. By combining the predictive capabilities of both neural networks, we show that it is possible to calculate sequence alignments, which are used to detect and correct anomalies. DeepAlign utilizes the case-level and event-level attributes to closely model the decisions within a process. We evaluate the performance of our approach on an elaborate data corpus of 252 realistic synthetic event logs and compare it to three state-of-the-art conformance checking methods. DeepAlign produces better corrections than the rest of the field reaching an overall $F_1$ score of $0.9572$ across all datasets, whereas the best comparable state-of-the-art method reaches $0.6411$

Know What You Stream: Generating Event Streams from CPN Models in ProM 6

Abstract. The field of process mining is concerned with supporting the analysis, improvement and understanding of business processes. A range of promising techniques have been proposed for process mining tasks such as process discovery and conformance checking. However there are challenges, originally stemming from the area of data mining, that have not been investigated extensively in context of process mining. In particular the incorporation of data stream mining techniques w.r.t. process mining has received little attention. In this paper, we present new developments that build on top of previous work related to the integration of data streams within the process mining framework ProM. We have developed means to use Coloured Petri Net (CPN) models as a basis for eventstream generation. The newly introduced functionality greatly enhances the use of event-streams in context of process mining as it allows us to be actively aware of the originating model of the event-stream under analysis

A Survey of Numerical Solutions to the Coagulation Equation

We present the results of a systematic survey of numerical solutions to the coagulation equation for a rate coefficient of the form A_ij \propto (i^mu j^nu + i^nu j^mu) and monodisperse initial conditions. The results confirm that there are three classes of rate coefficients with qualitatively different solutions. For nu \leq 1 and lambda = mu + nu \leq 1, the numerical solution evolves in an orderly fashion and tends toward a self-similar solution at large time t. The properties of the numerical solution in the scaling limit agree with the analytic predictions of van Dongen and Ernst. In particular, for the subset with mu > 0 and lambda < 1, we disagree with Krivitsky and find that the scaling function approaches the analytically predicted power-law behavior at small mass, but in a damped oscillatory fashion that was not known previously. For nu \leq 1 and lambda > 1, the numerical solution tends toward a self-similar solution as t approaches a finite time t_0. The mass spectrum n_k develops at t_0 a power-law tail n_k \propto k^{-tau} at large mass that violates mass conservation, and runaway growth/gelation is expected to start at t_crit = t_0 in the limit the initial number of particles n_0 -> \infty. The exponent tau is in general less than the analytic prediction (lambda + 3)/2, and t_0 = K/[(lambda - 1) n_0 A_11] with K = 1--2 if lambda > 1.1. For nu > 1, the behaviors of the numerical solution are similar to those found in a previous paper by us. They strongly suggest that there are no self-consistent solutions at any time and that runaway growth is instantaneous in the limit n_0 -> \infty. They also indicate that the time t_crit for the onset of runaway growth decreases slowly toward zero with increasing n_0.Comment: 41 pages, including 14 figures; accepted for publication in J. Phys.

Insulator-to-metal crossover induced by local spin fluctuations in strongly correlated systems

We study the simplified Hubbard (SH) model in the presence of a transverse field in the infinite dimension limit. The relevant one-particle Green's functions of the model are obtained by means a perturbative treatment of the hopping and of the transverse field around the atomic limit. We consider an analytical solution for the impurity problem. It is shown that this solution is very accurate in describing the spectral properties of the heavy-particles of the SH for intermediate and strong values of the on-site Coulomb interaction $U$. We find that for large values of $U$ an insulator-metal transition takes place as a function of the transverse field. We analyze the metallic phase through the behavior of the density of states and of the optical conductivity and static resistivity. Our results for the latter quantity agree with what is observed in experiments on $Bi_2Sr_2CuO_y$.Comment: 6 pages, 5 figures, to appear in Journal of Physics: Condensed Matte

An evolutionary technique to approximate multiple optimal alignments

The alignment of observed and modeled behavior is an essential aid for organizations, since it opens the door for root-cause analysis and enhancement of processes. The state-of-the-art technique for computing alignments has exponential time and space complexity, hindering its applicability for medium and large instances. Moreover, the fact that there may be multiple optimal alignments is perceived as a negative situation, while in reality it may provide a more comprehensive picture of the model’s explanation of observed behavior, from which other techniques may benefit. This paper presents a novel evolutionary technique for approximating multiple optimal alignments. Remarkably, the memory footprint of the proposed technique is bounded, representing an unprecedented guarantee with respect to the state-of-the-art methods for the same task. The technique is implemented into a tool, and experiments on several benchmarks are provided.Peer ReviewedPostprint (author's final draft

Charge-transfer metal-insulator transitions in the spin-one-half Falicov-Kimball model

The spin-one-half Falicov-Kimball model is solved exactly on an infinite-coordination-number Bethe lattice in the thermodynamic limit. This model is a paradigm for a charge-transfer metal-insulator transition where the occupancy of localized and delocalized electronic orbitals rapidly changes at the metal-insulator transition (rather than the character of the electronic states changing from insulating to metallic as in a Mott-Hubbard transition). The exact solution displays both continuous and discontinuous (first-order) transitions.Comment: 22 pages including 4 figures(eps), RevTe

Approximate computation of alignments of business processes through relaxation labelling

A fundamental problem in conformance checking is aligning event data with process models. Unfortunately, existing techniques for this task are either complex, or can only be applicable to restricted classes of models. This in practice means that for large inputs, current techniques often fail to produce a result. In this paper we propose a method to approximate alignments for unconstrained process models, which relies on the use of relaxation labelling techniques on top of a partial order representation of the process model. The implementation on the proposed technique achieves a speed-up of several orders of magnitude with respect to the approaches in the literature (either optimal or approximate), often with a reasonable trade-off on the cost of the obtained alignment.Peer ReviewedPostprint (author's final draft