Canonical Expansion Research Articles

Uniformizer of the false Tate curve extension of $${\mathbb {Q}}_p$$

Let \(p\ge 3\) be a prime number. In this article, we study the canonical expansion of the primitive \(p^n\)-th root of unity \(\zeta _{p^n}\) in p-adic Mal’cev–Neumann field \({\mathbb {L}}_p\) for \(n\ge 1\). More precisely, we give the explicit formula for the first \(\aleph _0\) terms of the expansion of \(\zeta _{p^n}\), and as an application, we use it to construct a uniformizer of \(K_{2,m}={\mathbb {Q}}_p\left( \zeta _{p^2},p^{1/p^m}\right) \) with \(m\ge 1\).

A series expansion for generalized harmonic functions

We consider a class of generalized harmonic functions in the open unit disc in the complex plane. Our main results concern a canonical series expansion for such functions. Of particular interest is a certain individual generalized harmonic function which suitably normalized plays the role of an associated Poisson kernel.

Software tools for analysis and synthesis of stochastic systems with high availability (XI)

The article proceeds the thematic cycle dedicated to software tools for stochastic systems with high availability (StSHA) functioning at shock disturbances (ShD) and is dedicated to wavelet synthesis according to complex statistical criteria (CsC). Short survey concerning corresponding works for mean square criteria (msc) is given. In Sect.1 basic CsC definitions and approaches are given. Sect. 2 dedicated to CsC wavelet necessary and sufficient conditions of optimality for scalar non-stationary shock StSHA (StSSHA). Methodological support is based on Haar wavelets. Sect. 4 and Sect.5 are devoted to CSK optimization ShStSHA (basic wavelet equations, algorithms, software tools and example). Several advantages of wavelet algorithms and tools are described and stated for complex ShD. Some generalization of CsC algorithms based on wavelet canonical expansion of StP in ShStSHA mentioned.

The possibility to applying simulated series for compile scenario forecasting river runoff

The work shows the possibility of jointed simulation of monthly runoff, precipitation and air temperature with the method of canonical expansion. The method allows to save average runoff values, variability indexes and asymmetry coefficients as well as autocorrelation and intercorrelation coefficients. Simulated values of flow-forming factors can be used to compile scenario forecasts of monthly runoff hydrographs based on meteorological factors.

Визначення оптимальних параметрів машин для виділення насіннєвої маси овоче-баштанних культур

To determine the optimal parameters of machines for isolating the seed mass of vegetable and melon crops, a modeling method is presented, which is based on the nonlinear canonical decomposition of a random vector. A block diagram of the procedure for calculating the parameters of the canonical expansion is presented. The method of mathematical modeling was tested in the study of the technological process of separating melon seeds on an experimental installation. Mathematical models of the third order for purity and loss of seeds are presented, describing the technological process of separation for melon seeds.

Defining Participatory Worlds - Canonical Expansion of Fictional Worlds through Audience Participation

Defining Participatory Worlds - Canonical Expansion of Fictional Worlds through Audience Participation

The Greatest Comic Book of All Time: Symbolic Capital and the Field of American Comic Books, Arresting Development: Comics at the Boundaries of Literature, We Told You So: Comics as Art and Spurgeon with Michael Dean

The Greatest Comic Book of All Time: Symbolic Capital and the Field of American Comic Books, Arresting Development: Comics at the Boundaries of Literature, We Told You So: Comics as Art and Spurgeon with Michael Dean

Network traffic forecasting based on the canonical expansion of a random process

We studied the problem of forecasting network traffic in TCP/IP networks based on statistical observational data. We determined that existing protocols (SNMP, RMON) do not provide long-term forecasting, which is necessary for network upgrades. Regression methods (AR, ARMA, ARIMA, SARIMA), which are the basis of protocols, use only a sequence of values of forecasted series, which makes long-term forecasting impossible. We made a conclusion that there is no universal effective method for forecasting time sequences that describe traffic of a computer network.We developed the model of a forecast of network traffic taking into account features of accumulation of statistical data: presence of a priori trajectories, a posteriori character of forecasting, finiteness of variance. We applied the apparatus of the canonical expansion of a random process, taking into account heterogeneity of traffic. We developed a mathematical apparatus to solve the problem of extrapolation of implementation; we obtained expressions for the estimation of an extrapolation error, and expressions for the reconstruction of a posteriori random process based on modeling. We took into account accuracy of a priori measurements, which makes it possible to use this model in networks with a minimum of diagnostic data. It provides accurate determination of parameters of a random process at control points and the minimum standard approximation error in the intervals between these points.Application of the proposed method based on the canonical decomposition of random processes provides a solution to the problem of long-term forecasting of network traffic. A comparative analysis of forecasting methods indicates that the method of canonical decomposition of a random process comes close to intelligent forecasting methods.

Canonization, Colonization, and the Rise of Neo-Victorianism

Over the last two decades, contemporary culture has witnessed the rapidly increasing popularity of neo-Victorian (literary, filmic, and stage) texts and the development of critical interest in the field. The neo-Victorian genre was initially construed as a subgenre of postmodernism, but more recently has been considered in relation to broader literary and cultural trends. This essay considers the construction of the neo-Victorian canon, and contends that its formation follows the same trajectory as that of the broader literary canon in the early twentieth century, initially excluding or marginalizing particular works (in particular, popular fiction and culture) prior to expanding to include a wider array of texts and genres, including, in recent years, global literatures. I map the development of the neo-Victorian canon to date, including critical debates around the genre, and examine the implications of this in terms of prevailing attitudes towards literary and cultural scholarship, popular culture, and textual colonization, before considering the (sometimes problematic) usage and connotations of the terminology associated with this key genre of contemporary fiction. I consider the potential problems of the genre’s rapid cultural and critical expansion, and in the conclusion to the essay, suggest some possible future directions for neo-Victorian scholarship in terms of the continued canonical expansion and its ethical framework.

The exact analytical solution of the problem on the average number of spikes of the narrowband Gaussian stochastic process

In this article, the problem of the number of spikes (level crossings) of the stationary narrowband Gaussian process has been considered. The process was specified by an exponentially-cosine autocorrelation function. The problem had been solved earlier by Rice in terms of the joint probabilities’ density of the process and its derivative with respect to time, but in our article we obtained the solution using the functional of probabilities’ density (the functional was obtained by Amiantov), as well as an expansion of the canonical stochastic process. In this article, the optimal canonical expansion of a narrowband stochastic process based on the work of Filimonov and Denisov was also considered to solve the problem. The application of all these resources allowed obtaining an exact analytical solution of the problem on spikes of stationary narrowband Gaussian process. The obtained formulae could be used to solve, for example, some problems about the residual resource of some radiotechnical products, about the breaking sea waves and others.

Uncertainty principle and orthogonal condition for the short-time linear canonical transform

The short-time linear canonical transform (STLCT) is a novel time–frequency analysis tool, which maps the time domain signal into the joint time and frequency domain. This short paper investigates its two theoretical problems. The first one relates to the uncertainty principle in one STLCT domain, and the other one is that coefficients of the short-time linear canonical expansion (STLCE) are of incompleteness. To solve these issues, first, we develop the uncertainty principle in one STLCT domain. Second, the orthogonal condition is proposed to guarantee the completeness of the STLCE coefficients.

Infinitesimal affine transformations of a Weil bundle of second order with complete lift connection

We obtain the canonical expansion of an arbitrary infinitesimal affine transformation of a Weil bundle of second order over a differentiable manifold with complete lift connection. We establish necessary and sufficient conditions under which a vector field is an infinitesimal affine transformation.

Nonuniqueness of optical theorem detectors

We demonstrate and discuss the multitude of ways in which the extinct power of a scatterer can be measured. To tie some of the developed results to the classical statement of the optical theorem involving the imaginary part of the forward-scattering amplitude, particular attention is given to plane wave excitation. On the other hand, the general results apply to more general probing fields including near fields carrying evanescent components. Novel optical theorem detectors are derived that are based on the Kirchhoff-Helmholtz and Rayleigh-Sommerfeld-based formulations of diffraction, backpropagation, and boundary-value problems as well as on the canonical multipole expansion. The derived detectors also lead to novel expressions for the extinct power in terms of the incident and scattered fields. Applications of the derived results to scattering power sensing with near-field data are also discussed.

Une notion de multizêtas finis associée au Frobenius du groupe fondamental de [formula omitted

We show that multiple harmonic sums of the form∑apk<n1<…<nd<(a+1)pk1n1s1…ndsd,for a∈Z,k∈N⁎,sd,…,s1∈N⁎, admit a simple canonical expansion in terms of p-adic multiple zeta values. More generally, we interpret geometrically the multiplication by p of the upper bound of a multiple harmonic sum. This is equivalent to the inversion of the sums of series that express p-adic multiple zeta values. The result leads to the definition of a notion of finite multiple zeta values that is of geometric origin; it gives a framework to study the algebraic properties of those multiple harmonic sums whose upper bound is a power of a prime number. The result also has applications to a conjecture of Kaneko and Zagier.

Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle

Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and \(q=1-1/\langle n \rangle + \Delta n^2/\langle n \rangle^2\). For the binomial distribution of \(n\) one obtains \(q=1-1/k\), for the negative binomial \(q=1+1/(k+1)\). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion \(\omega \ll E\). For general systems the average phase-space volume ratio \(\langle e^{S(E-\omega)}/e^{S(E)}\rangle\) to second order delivers \(q=1-1/C+\Delta \beta^2/\langle \beta \rangle^2\) with \(\beta=S^{\prime}(E)\) and \(C=dE/dT\) heat capacity. However, \(q \ne 1\) leads to non-additivity of the Boltzmann–Gibbs entropy, \(S\). We demonstrate that a deformed entropy, \(K(S)\), can be constructed and used for demanding additivity, i.e., \(q_K=1\). This requirement leads to a second order differential equation for \(K(S)\). Finally, the generalized \(q\)-entropy formula, \(K(S)=\sum p_i K(-\ln p_i)\), contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, \(\Delta\beta^2\) we obtain a novel entropy formula.

The Canonical Debate Today: Crossing Disciplinary and National Boundaries

The Canonical Debate Today: Crossing Disciplinary and National Boundaries

Continuous channel modeling in shape of some space transformation operators

Continuous channel phenomenological model is suggested. Linear continuous channel models in shape of metric space transformation operators with specified basis and also special case of coordinate function basis of Pugachev integral canonical expansion are designed.

Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions

This paper is devoted to the problem of ergodicity of p-adic dynamical systems. We solved the problem of characterization of ergodicity and measure preserving for (discrete) p-adic dynamical systems for arbitrary prime p for iterations based on 1-Lipschitz functions. This problem was open since long time and only the case p=2 was investigated in details. We formulated the criteria of ergodicity and measure preserving in terms of coordinate functions corresponding to digits in the canonical expansion of p-adic numbers. (The coordinate representation can be useful, e.g., for applications to cryptography.) Moreover, by using this representation we can consider non-smooth p-adic transformations. The basic technical tools are van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theorems by presenting concrete classes of ergodic functions. As is well known, p-adic spaces have the fractal (although very special) structure. Hence, our study covers a large class of dynamical systems on fractals. Dynamical systems under investigation combine simplicity of the algebraic dynamical structure with very high complexity of behavior.

A canonical expansion of the product of two Stanley symmetric functions

We study the problem of expanding the product of two Stanley symmetric functions F w ?F u into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_{w}=\lim_{n\to\infty}\mathfrak{S}_{1^{n}\times w}$ , and study the behavior of the expansion of $\mathfrak {S}_{1^{n}\times w}\cdot \mathfrak {S}_{1^{n}\times u}$ into Schubert polynomials as n increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. We then study some other related stability properties, providing a second proof of the main result.

Group modeling of monthly runoff hydrographs

The possibility of group simulation of monthly runoff hydrograph by canonical expansion method is demonstrated. The simulation procedure takes into account the coefficients of autocorrelation and mutual correlation and enables one to reproduce the ratios between parameters Cν and Cs for monthly time intervals. The model preserves the similarity between within-year distributions of the measured and simulated series.