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In this paper we show that every non-cycle finite transitive directed graph has a Cuntz–Krieger family whose WOT-closed algebra is
. This is accomplished through a new construction that reduces this problem to in-degree 2-regular graphs, which is then treated by applying the periodic Road Colouring Theorem of Béal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.
We show how geometric conjugacy can be viewed in terms of the dual group. We explain the centre of a semi-simple group in terms of the affine root system. We explain Lusztig’s Jordan decomposition of characters. We express the characters of the general linear and unitary groups in term of Deligne–Lusztig characters and give the character table of GL2.
After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using cohomological methods. We refer to the books by Carter and Digne-Michel for proofs of some fundamental properties, like orthogonality relations and degree formulae. Based on these results, we develop in some detail the basic formalism of Lusztig's book, which leads to a classification of the irreducible characters of finite groups of Lie type in terms of a fundamental Jordan decomposition. Using the general theory about regular embeddings in Chapter 1, we state and discuss that Jordan decomposition in complete generality, that is, without any assumption on the center of the underlying algebraic group. The final two sections give an introduction to the problems of computing Green functions and characteristic functions of character sheaves.
After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig‘s result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of the parametrisation and the properties of unipotent characters of finite reductive groups and related data like Fourier matrices and eigenvalues of Frobenius. We then describe the decomposition of Lusztig induction and collect the most recent results on its commutation with Jordan decomposition. We end the chapter with a survey of the character theory of finite disconnected reductive groups.
In this paper, an algorithm for extracting and localizing a radar pulse in a noisy environment is described. The algorithm combines two powerful tools: wavelet denoising and the short-time Fourier transform (STFT) analysis with statistical-based threshold. We aim to detect radar pulses transmitted by any radar in blind mode regardless of the intra-pulse modulation and parametric features. The use of the proposed technique makes the detection and localization of radar pulses possible under very low signal-to-noise ratio conditions (−18 dB), which leads to a reduction of the required signal power or alternatively extends the detection range of radar systems. Radar classes pattern-based analysis is used in blind mode to decrease the probability of false alarm.
This paper provides a toolbox for the credibility analysis of frequency risks, with allowance for the seniority of claims and of risk exposure. We use Poisson models with dynamic and second-order stationary random effects that ensure nonnegative credibilities per period. We specify classes of autocovariance functions that are compatible with positive random effects and that entail nonnegative credibilities regardless of the risk exposure. Random effects with nonnegative generalized partial autocorrelations are shown to imply nonnegative credibilities. This holds for ARFIMA(0, d, 0) models. The AR(p) time series that ensure nonnegative credibilities are specified from their precision matrices. The compatibility of these semiparametric models with log-Gaussian random effects is verified. Gaussian sequences with ARFIMA(0, d, 0) specifications, which are then exponentiated entrywise, provide positive random effects that also imply nonnegative credibilities. Dynamic random effects applied to Poisson distributions are retained as products of two uncorrelated and positive components: the first is time-invariant, whereas the autocovariance function of the second vanishes at infinity and ensures nonnegative credibilities. The limit credibility is related to the three levels for the length of the memory in the random effects. The limit credibility is less than one in the short memory case, and a formula is provided.
This study contributes to a growing literature body of studies aimed at explaining socio-economic-related health inequality in non-communicable diseases (NCDs), with a focus on older people who are commonly affected by socio-economic gradient in later life. It identifies factors associated with self-reported NCDs and examines socio-economic-related health inequality in self-reported NCDs between rural and urban Vietnamese older people. This cross-sectional study utilised data from the Viet Nam Ageing Survey. A sample of 2,682 older people aged 60 and over (urban = 703, rural = 1,979) was analysed. Concentration indices were computed to measure socio-economic inequalities in self-reported NCDs. Concentration index decomposition analysis was performed to determine the relative contributions of the determinants to explaining those inequalities. Significant socio-economic inequalities in self-reported NCDs favouring the rich were found, in which the degree of inequality was more pronounced in urban areas than in their rural counterparts. Household wealth and social health insurance were the main drivers contributing to increased socio-economic inequalities in self-reported NCDs in urban and rural areas, respectively. Among disadvantaged groups, older people living alone, with lowest wealth and with social health insurance had highest probability of reporting at least one NCD for both areas. Public policies aimed at narrowing wealth gaps and expanding and improving principle roles of social health insurance should prioritise the most disadvantaged groups in order to achieve health equality.
is a polynomial with at least one irrational coefficient on non-constant terms,
is any real number and, for
$a\in [0,\infty )$
is the fractional part of
. With the help of a method recently introduced by Wu, we show that the closure of
must have full Hausdorff dimension.
The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.
Block decomposition of
spaces with weighted Hausdorff content is established for
and the Fefferman–Stein type inequalities are shown for fractional integral operators and some variants of maximal operators.
This paper generalizes the Kunita–Watanabe decomposition of an
space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in
. This result is also the solution of an optimization problem in
. First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.
It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.
In this paper, we introduce a new large family of Lévy-driven point processes with (and without) contagion, by generalising the classical self-exciting Hawkes process and doubly stochastic Poisson processes with non-Gaussian Lévy-driven Ornstein–Uhlenbeck-type intensities. The resulting framework may possess many desirable features such as skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the ‘contagion’ or feedback effects, which could be very useful for modelling event arrivals in finance, economics, insurance, and many other fields. We characterise the distributional properties of this new class of point processes and develop an efficient sampling method for generating sample paths exactly. Our simulation scheme is mainly based on the distributional decomposition of the point process and its intensity process. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of our scheme. Moreover, we use portfolio risk management as an example to show the applicability and flexibility of our algorithms.
This chapter generalizes and extends the development of operator-adapted wavelets (gamblets) and their resulting multiresolution decompositions from Sobolev spaces to Banach spaces equipped with a quadratic norm and a nonstandard dual pairing. The fundamental importance of the Schur complement is elucidated and the geometric nature of gamblets is presented from two views: one regarding basis transformations derived from the nesting, and the other the linear transformations associated with these basis transformations. A table of gamblet identities is presented.
At the cost of some redundancy, to facilitate accessibility, the multiresolution decomposition and inversion of symmetric positive definite (SPD) matrices on finite-dimensional Euclidean space are developed in the Gamblet Transform and Decomposition framework.
Wavelets adapted to a given self-adjoint elliptic operator are characterized by the requirement that they block-diagonalize the operator into uniformly well-conditioned and sparse blocks. These operator-adapted wavelets (gamblets) are constructed as orthogonalized hierarchies of nested optimal recovery splines obtained from classical/simple prewavelets (e.g., ~Haar) used as hierarchies of measurement functions. The resulting gamblet decomposition of an element in a Sobolev space is described and analyzed.
This chapter establishes the exponential decay of gamblets under an appropriate notion of distance derived from subspace decomposition in a way that generalizes domain decomposition in the computation of PDEs. The first steps present sufficient conditions for localization based on a generalization of the Schwarz subspace decomposition and iterative correction method introduced by Kornhuber and Yserentant and the LOD method of Malqvist and Peterseim. However, when equipped with nonconforming measurement functions, one cannot directly work in the primal space, but instead one has to find ways to work in the dual space. Therefore, the next steps present necessary and sufficient conditions expressed as frame inequalities in dual spaces that, in applications to linear operators on Sobolev spaces, are expressed as Poincaré, inverse Poincaré, and frame inequalities.
The computation of gamblets is accelerated by localizing their computation in a hierarchical manner (using a hierarchy of distances), and the approximation errors caused by these localization steps are bounded based on three properties: nesting, the well-conditioned nature of the linear systems solved in the Gamblet Transform, and the exponential decay of the gamblets. These efficiently computed, accurate, and localized gamblets are shown to produce a Fast Gamblet Transform of near-linear complexity. Application to the three primary classes of measurement functions in Sobolev spaces are developed.