From Differential Equations to the Construction of New Wavelet-Like Bases

In this paper, an approach is introduced based on differential operators to construct wavelet-like basis functions. Given a differential operator L with rational transfer function, elementary building blocks are obtained that are shifted replicates of the Green's function of L. It is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential B-spline representation. The corresponding B-splines are entirely specified by their poles and zeros; they are compactly supported, have an explicit analytical form, and generate multiresolution Riesz bases. Moreover, they satisfy generalized refinement equations with a scale-dependent filter and lead to a representation that is dense in $L _{ 2 }$ . This allows us to specify a corresponding family of semi-orthogonal exponential spline wavelets, which provides a major extension of earlier polynomial spline constructions. These wavelets are completely characterized, and it is proven that they satisfy the following remarkable properties: 1) they are orthogonal across scales and generate Riesz bases at each resolution level; 2) they yield unconditional bases of $L _{ 2 }$ -either compactly supported (B-spline-type) or with exponential decay (orthogonal or dual-type); 3) they have N vanishing exponential moments, where N is the order of the differential operator; 4) they behave like multiresolution versions of the operator L from which they are derived; and 5) their order of approximation is (N − M), where N and M give the number of poles and zeros, respectively. Last but not least, the new wavelet-like decompositions are as computationally efficient as the classical ones. They are computed using an adapted version of Mallat's filterbank algorithm, where the filters depend on the decomposition level

Operator-Like Wavelet Bases of L2 L _{ 2 } (Rd (\mathbb{R} ^{ d } )

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator

BSLIM: Spectral Localization by Imaging with Explicit B0 B _{ 0 } Field Inhomogeneity Compensation

Magnetic resonance spectroscopy imaging (MRSI) is an attractive tool for medical imaging. However, its practical use is often limited by the intrinsic low spatial resolution and long acquisition time. Spectral localization by imaging (SLIM) has been proposed as a non-Fourier reconstruction algorithm that incorporates spatial a priori information about spectroscopically uniform compartments. Unfortunately, the influence of the magnetic field inhomogeneity—in particular, the susceptibility effects at tissues' boundaries—undermines the validity of the compartmental model. Therefore, we propose BSLIM as an extension of SLIM with field inhomogeneity compensation. A $B _{ 0 }$ -field inhomogeneity map, which can be acquired rapidly and at high resolution, is used by the new algorithm as additional a priori information. We show that the proposed method is distinct from the generalized SLIM (GSLIM) framework. Experimental results of a two-compartment phantom demonstrate the feasibility of the method and the importance of inhomogeneity compensation

Factors that Predetermined Activation of Plague Epizooties in the Pre-Caspian Sandy Natural Focus in 2014

In 2014, in the central coastal part of the Pre-Caspian sandy natural plague focus, activation of epizootic process in the complex rodent settlements occurred. Objective of the study was to identify the factors that predetermined the development of intensive and extensive plague epizooties. Utilized were operational data from the Astrakhan, Elista, and Dagestan plague control stations, and personal investigation records gathered in the focus. Consequently to the high baseline numbers of rodents and fleas, mild winter weather in 2013-2014 population density of the vectors and carriers of the infection significantly exceeded long-term average annual indexes in the following spring. Total abundance rates reached up to 31.1 specimens per a hectare, being lowered up to 17.8 by the fall. Stock of the main vector of plague, N. laeviceps flea, amounted to 350 and to 120 per a hectare in the spring and autumn, respectively. All in all, in the course of the three rolling seasons isolated were 56 Y. pestis cultures: 17 strains - from the rodents, and 39 - from the fleas. The overall epizootic area totaled to 2300 km2. Plague epizooties were registered in the zone of sustained natural focality within the bonds of the Ilmen-Pridelta, Primorsky, and Chernozemelsky landscape-epizootic regions in the administrative territory of the Limansky District in the Astrakhan Region, Lagansky and Chernozemelsky Regions of the Republic of Kalmykia, and Tarumovsky Region of the Republic of Dagestan. Results of monitoring over the epizootic manifestations and plague carrier and vector abundance dynamics provided the basis for scheduling and performance of anti-epidemic activities, as well as for forecasting of the epidemic situation development in the focus

Эколингвистические проблемы Северного Кавказа в контексте языковой политики

The paper is devoted to the study of language ecology of the North Caucasus in the context of language policy. The following problems have been identified among the main ecolinguistic issues: many languages of the peoples of Russia are facing the threat of extinction or are endangered; the actual ethnic composition of the population and the number of languages in the country have not been identified yet; in the conditions of irresistible globalization there is little chance of small ethnic groups’ languages survival, the groups having neither written language nor the language of official status; ethnicity is doomed to extinction as their language is falling out of use, yet it is the main bearer of the cultural code of the people. The paper concludes that the threat to the Russian language and culture from other peoples and their cultures seems exaggerated. In relation to the urgency of preserving the Russian language as the state language and the language of interethnic communication in the country as a whole as well as in the republics, it is vital to consider the importance of preservation of all the languages spoken by the peoples of Russia and create the conditions for their further developmentСтатья посвящена проблеме изучения экологии языка народов Северного Кавказа в контексте языковой политики. Среди основных эколингвистических проблем выделены следующие: многие языки народов России подошли или подходят к грани исчезновения; до сих пор не определены реальный этнический состав населения страны и число языков в стране; в условиях непреодолимой глобализации невероятно мало шансов на выживание языков малочисленных этносов, к тому же бесписьменных и не имеющих статуса государственных; этнос обречен на угасание и вымирание по мере угасания своего языка, являющегося главным носителем культурного кода народа. В работе делается вывод, что угроза русскому языку и русской культуре со стороны других народов и их культур представляется преувеличенной. При важности сохранения отношения к русскому языку как общегосударственному, как к языку межнационального общения в стране в целом и внутри республик необходимо осознавать значимость сохранения всех языков, на которых говорят народы России, и создавать условия для их дальнейшего развити

Exponential-Spline Wavelet Bases

We build a multiresolution analysis based on shift-invariant exponential B-spline spaces. We construct the basis functions for these spaces and for their orthogonal complements. This yields a new family of wavelet-like basis functions of $L _{ 2 }$ , with some remarkable properties. The wavelets, which are characterized by a set of poles and zeros, have an explicit analytical form (exponential spline). They are nonstationary is the sense that they are scale-dependent and that they are not necessarily the dilates of one another. They behave like multi-scale versions of some underlying differential operator L; in particular, they are orthogonal to the exponentials that are in the null space of L. The corresponding wavelet transforms are implemented efficiently using an adaptation of Mallat's filterbank algorithm

Эколингвистические проблемы Северного Кавказа в контексте языковой политики

The paper is devoted to the study of language ecology of the North Caucasus in the context of language policy. The following problems have been identified among the main ecolinguistic issues: many languages of the peoples of Russia are facing the threat of extinction or are endangered; the actual ethnic composition of the population and the number of languages in the country have not been identified yet; in the conditions of irresistible globalization there is little chance of small ethnic groups’ languages survival, the groups having neither written language nor the language of official status; ethnicity is doomed to extinction as their language is falling out of use, yet it is the main bearer of the cultural code of the people. The paper concludes that the threat to the Russian language and culture from other peoples and their cultures seems exaggerated. In relation to the urgency of preserving the Russian language as the state language and the language of interethnic communication in the country as a whole as well as in the republics, it is vital to consider the importance of preservation of all the languages spoken by the peoples of Russia and create the conditions for their further developmentСтатья посвящена проблеме изучения экологии языка народов Северного Кавказа в контексте языковой политики. Среди основных эколингвистических проблем выделены следующие: многие языки народов России подошли или подходят к грани исчезновения; до сих пор не определены реальный этнический состав населения страны и число языков в стране; в условиях непреодолимой глобализации невероятно мало шансов на выживание языков малочисленных этносов, к тому же бесписьменных и не имеющих статуса государственных; этнос обречен на угасание и вымирание по мере угасания своего языка, являющегося главным носителем культурного кода народа. В работе делается вывод, что угроза русскому языку и русской культуре со стороны других народов и их культур представляется преувеличенной. При важности сохранения отношения к русскому языку как общегосударственному, как к языку межнационального общения в стране в целом и внутри республик необходимо осознавать значимость сохранения всех языков, на которых говорят народы России, и создавать условия для их дальнейшего развити

Generalized L-Spline Wavelet Bases

We build wavelet-like functions based on a parametrized family of pseudo-differential operators $L _{ v }$ that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of $L _{ v }$ , generate a family of L-spline spaces. These spaces have the approximation order equal to the order of the underlying operator. A sequence of embedded spaces is obtained by choosing a dyadic scale progression a = $2 ^{ i }$ . The consecutive inclusion of the spaces yields the refinement equation, where the scaling filter depends on scale. The generalized L-wavelets are then constructed as basis functions for the orthogonal complements of spline spaces. The vanishing moment property of conventional wavelets is generalized to the vanishing null space element property. In spite of the scale dependence of the filters, the wavelet decomposition can be performed using an adapted version of Mallat's filterbank algorithm