Academic literature on the topic 'Kato square-root problems'

This paper presents a review of results on nonlocal problems, functional-differential equations, and their applications, obtained during several last years. The following research areas are covered: the Kato square root problem for functional-differential operators, Vlasov equations and their applications to the modelling of high-temperature plasma, specific properties of differential-difference equations with incommensurable translations, degenerate functional-differential equations and their applications, functional-differential equations with contractions and extensions of independent variables, and operational methods for parabolic and elliptic functional-differential equations.

AbstractThe aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let {L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)}, where {w\in A_{2}} and A is an {n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the {L^{2}(w)}-Kato square root problem obtaining that {\sqrt{L_{w}}} is equivalent to the gradient on {L^{2}(w)}. The same authors in collaboration with the second named author of this paper studied the {L^{p}(w)}-boundedness of operators that are naturally associated with {L_{w}}, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in {L^{p}(v\,dw)} for {v\in A_{\infty}(w)}), and in particular a class of “degeneracy” weights w was found in such a way that the classical {L^{2}}-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on {L^{p}(w)} and on {L^{p}(v\,dw)}, with {v\in A_{\infty}(w)}, of the conical square functions that one can construct using the heat or Poisson semigroup associated with {L_{w}}. As a consequence of our methods, we find a class of degeneracy weights w for which {L^{2}}-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with {L_{w}}.

AbstrakStatus gizi adalah kondisi fisiologis tubuh terkait konsumsi makanan dan penggunaan zat gizi oleh tubuh. Hingga sekarang, masalah gizi masih umum terjadi terutama di negara berkembang. Status gizi pada siswa sekolah dasar penting karena dapat mempengaruhi kognitif dan capaian pembelajaran siswa. Salah satu akar masalah gizi adalah kemiskinan yang terkait dengan sosiodemografi yang meliputi status sosial dan ekonomi. Tujuan penelitian ini adalah untuk menganalisis hubungan karakteristik sosiodemografi dan status gizi siswa sekolah dasar di Desa Wokam dan Desa Karangguli, Kabupaten Kepualaun Aru, Maluku. Penelitian ini menggunakan rancangan cross-sectional dan jenis penelitian ini adalah analitik observasional. Penilaian status gizi menggunakan grafik berat badan terhadap tinggi badan dengan kriteria Waterlow. Data mengenai sosiodemografi dikumpulkan dengan wawancara. Hubungan antara karakteristik sosiodemografi dan status gizi dinilai dengan uji statistik chi-square. Dari 106 sampel, 73 siswa (68,9%) memiliki status gizi normal dan 33 siswa (31,1%) memiliki status gizi kurang. Uji statistik tidak menunjukkan hubungan yang signifikan antara karakteristik sosiodemografi dan status gizi di Desa Wokam dan Karangguli, Kabupaten Kepulauan Aru. Kata kunci : hubungan, siswa sekolah dasar, sosiodemografi, status gizi AbstractNutritional status is a physiological condition of the body related to food consumption and requirements of the body. Until now, malnutrition is a common problem in developing countries. Nutritional status in school age children is important because it can affect cognitive ability and student achievement. One of the root problems of malnutrition is poverty which is related to sociodemographic including social and economic status. The purpose of this study is to analyze the correlation between sociodemographic characteristics and nutritional status in elementary school children in Wokam and Karangguli Village, Aru Islands Regency, Maluku. This study used cross-sectional study design and the type of this study is analytical observational. Nutritional status was assessed using weight to stature growth chart with Waterlow criteria. Data about sociodemographic were collected by interview. Correlation between sociodemographic characteristics and nutritional status were analyzed using chi-square test. From 106 samples, 73 students (68,9%) have normal nutritional status and 33 students (31,1%) were wasted. No significant correlation was found between sociodemographic characteristics and nutritional status in Wokam and Karangguli Village, Aru Islands Regency. Keywords: correlation, elementary school students, nutritional status, sociodemographic

This paper presents some results of rainfall forecast in the monsoon onset period in South Vietnam, with the use of ensemble Kalman filter to assimilate observation data into the initial field of the model. The study of rainfall forecasts are experimented at the time of Southern monsoon outbreaks for 3 years (2005, 2008 and 2009), corresponding to 18 cases. In each case, there are five trials, including satellite wind data assimilation, upper-air sounding data assimilation, mixed data (satellite wind+upper-air sounding data) assimilation and two controlled trials (one single predictive test and one multi-physical ensemble prediction), which is equivalent to 85 forecasts for one trial. Based on the statistical evaluation of 36 samples (18 meteorological stations and 18 trials), the results show that Kalman filter assimilates satellite wind data to forecast well rainfall at 48 hours and 72 hours ranges. With 24 hour forecasting period, upper-air sounding data assimilation and mixed data assimilation experiments predicted better rainfall than non-assimilation tests. The results of the assessment based on the phase prediction indicators also show that the ensemble Kalman filter assimilating satellite wind data and mixed data sets improve the rain forecasting capability of the model at 48 hours and 72 hour ranges, while the upper-air sounding data assimilation test produces satisfactory results at the 72 hour forecast range, and the multi-physical ensemble test predicted good rainfall at 24 hour and 48 hour forecasts. The results of this research initially lead to a new research approach, Kalman Filter Application that assimilates the existing observation data into input data of the model that can improve the quality of rainfall forecast in Southern Vietnam and overall country in general.References Bui Minh Tuan, Nguyen Minh Truong, 2013. 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Local ensemble transform Kalman filtering with an AGCM at a T159/L48 resolution. Mon. Wea. Rev., 135, 3841-3861.Nguyen Khanh Van, Tong Phuc Tuan, Vuong Van Vu, Nguyen Manh Ha, 2013. The heavy rain differences based on topo-geographical analyse at Coastal Central Region, from Thanh Hoa to Khanh Hoa. J. Sciences of the Earth, 35, 301-309.Nguyen Minh Truong, Bui Minh Tuan, 2013. A case study on summer monsoon onset prediction for southern Vietnam in 2012 using the RAMS model. VNU Journal of Science, 29(1S), 179-186.Phillips N.A., 1960b. Numerical weather prediction. Adv. Computers, 1, 43-91, Kalnay 2004.Phillips N., 1960a. On the problem of the initial data for the primitive equations, Tellus, 12, 121126.Phuong Nguyen Duc, 2013. Experiment on combinatorial Kalman filtering method for WRF model to forecast heavy rain in central region in Vietnam. 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Tellus A., 60, 113-130.Tanaka M., 1992. Intraseasonal oscillation and the onset and retreat dates of the summer monsoon east, southeast Asia and the western Pacific region using GMS high cloud amount data. J. Meteorol. Soc. Japan, 70, 613-628.Tan Tien Tran, Nguyen Thi Thanh, 2011. The MODIS satellite data assimilation in the WRF model to forecast rainfall in the central region. VNU Journal of Science, Natural Sciences and Technology, 27(3S), 90-95.Tao S., Chen L., 1987. A review of recent research on East summer monsoon in China, Monsoon Meteorology. C. P. Changand T. N. Krishramurti, Eds, Oxford University Press, Oxford, 60-92.Tippett M.K., Anderson J.L., Bishop C.H., Hamill T.M., Whitaker J.S., 2003. Ensemble square root filters. Mon. Wea. Rev., 131, 1485.Thuy Kieu Thi, Giam Nguyen Minh, Dung Dang Van, 2013. Using WRF model to forecast heavy rainfall events on September 2012 in Dong Nai River Basin. 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<p class="Judul2">Abstrak</p><p class="Abstrak">Perkembangan pengambilan keputusan telah semakin cepat dan metode yang digunakan beragam. Sehingga perlu adanya pengambilan keputusan yang tidak hanya cepat tapi juga tepat. Salah satunya yaitu penerapan sistem pendukung keputusan pada Computer Maintenance Management System (CMMS). Penelitian ini melakukan uji data dengan menggunakan Association Rule, Fisher Exact One Side P-Value dan double one side p-value dalam satu permasalahan yang sama pada Computer Maintenance Management System (CMMS). Dengan tujuan mencari pola hubungan antara gejala dan akar permasalahan, untuk membuktikan adanya hubungan kedua variabel. Pada penelitian yang dilakukan sebelumnya (Sukmana, Bulaili) tentang Association Rule dan Pearson Chi-Square pada CMMS menunjukkan adanya beberapa rule yang tereliminasi karena adanya beberapa rule yang tidak masuk dalam kriteria untuk penerapan pada Chi-Square. Sehingga dalam penelitian ini bertujuan menggunakan metode Chi-Square lain dengan maksud membuktikan dan memperkuat adanya hubungan atau relasi antara gejala dan akar permasalahan dalam CMMS. Dan diharapkan pembuktian ini akan menunjukkan hasil lebih baik dalam hal hubungan antara gejala dan akar permasalahannya memiliki korelasi.</p><p><strong>Kata kunci</strong>: <em>Association Rule, Fisher Exact Test</em></p><p><em><br /></em></p><p class="Judul2">Abstract</p><p><em>Decision-making has been growth rapidly and many methods can used. Thus, how to apply that methode not only fast but also right. One of implementation decision making is decision support system in Computer Maintenance Management System (CMMS). This research using data test with Association Rule and Fisher Exact One Side P-Value from same problems in Computer Maintenance Management System (CMMS). Object from this research to get pattern of association between symptom and root cause, to prove relation those variable. Previous research (Sukmana, Bulaili) about Association Rule and Pearson Chi-Square in CMMS show some rule was elimanated from calculation, becaise that is rule out of way from criteria in Chi-Square. So this research has objective to use another methode to observe and get strength evidence about correlation between symptom and root cause in CMMS. And hope with this result of test can make strength hypothesis about relation between symptom and root cause. </em></p><p><strong>Keywords</strong>: <em>Association Rule, Fisher Exact Test</em><em></em></p>

The primary focus of this thesis is to consider Kato square root problems for various divergence-form operators on manifolds. This is the study of perturbations of second-order differential operators by bounded, complex, measurable coefficients. In general, such operators are not self-adjoint but uniformly elliptic. The Kato square root problem is then to understand when the square root of such an operator, which exists due to uniform ellipticity, is comparable to its unperturbed counterpart. A remarkably adaptable operator-theoretic framework due to Axelsson, Keith and McIntosh sits in the background of this work. This framework allows us to take a powerful first-order perspective of the problems which we consider in a geometric setting. Through a well established procedure, we reduce these problems to the study of quadratic estimates. Under a set of natural conditions, we prove quadratic estimates for a class of operators on vector bundles over complete measure metric spaces. The first kind of estimates we prove are global, and we establish them on trivial vector bundles when the underlying measure grows at most polynomially. The second kind are local, and there, we allow the vector bundle to be non-trivial but bounded in an appropriate sense. Here, the measure is allowed to grow exponentially. An important consequence of obtaining quadratic estimates on measure metric spaces is that it allows us to consider subelliptic operators on Lie groups. The first-order perspective allows us to reduce the subelliptic problem to a fully elliptic one on a sub-bundle. As a consequence, we are able to solve a homogeneous Kato square root problem for perturbations of subelliptic operators on nilpotent Lie groups. For general Lie groups we solve a similar inhomogeneous problem. In the situation of complete Riemannian manifolds, we consider uniformly elliptic divergence-form operators arising from connections on vector bundles. Under a set of assumptions, we show that the Kato square root problem can be solved for such operators. As a consequence, we solve this problem on functions under the condition that the Ricci curvature and injectivity radius are bounded. Assuming an additional lower bound for the curvature endomorphism on forms, we solve a similar problem for perturbations of inhomogeneous Hodge-Dirac operators. A theorem for tensors is obtained by additionally assuming boundedness of a second-order Riesz transform. Motivated by the study of these Kato problems, where for technical reasons it is useful to know the density of compactly supported functions in the domains of operators, we study connections and their divergence on a vector bundle. Through a first-order formulation, we show that this density property holds for the domains of these operators if the metric and connection are compatible and the underlying manifold is complete. We also show that compactly supported functions are dense in the second-order Sobolev space on complete manifolds under the sole assumption that the Ricci curvature is bounded below, improving a result that previously required an additional lower bound on the injectivity radius.

The general theme of this thesis is the harmonic analysis of Schr¨odinger operators and its applications. We will focus on two distinct but related open problems in this field. The first problem is the construction of potential dependent averaging operators and will be primarily considered in the first part of this thesis. Here, a Hardy-Littlewood type maximal operator adapted to the Schrodinger operator L := −∆ + |x| 2 and acting on L 2 (R n ) is constructed. This is achieved through the use of the Gaussian grid ∆γ 0 , constructed in [42] with the Ornstein-Uhlenbeck operator in mind. At the scale of this grid, the maximal operator will resemble the classical Hardy-Littlewood operator. At a larger scale, the constituent averaging operators of the maximal function are decomposed over the cubes from ∆γ 0 and weighted appropriately. Through this maximal function, a new class of weights is defined, A+ p , with the property that for any w ∈ A+ p the heat maximal operator associated with L is bounded from L p (w) to itself. This class contains any other known class that possesses this property and contains weights of exponential growth. In particular, it is strictly larger than Ap. The second problem that we consider is the Kato square root problem for divergence form elliptic operators with potential V : R n → C. This is the equivalence statement (L + V ) 1 2 u L2(Rn) ' k∇ukL2(Rn) + V 1 2 u L2(Rn) , where L + V := −div (A∇) + V and the perturbation A is an L ∞ complex matrix-valued function satisfying an ellipticity condition. One possible path to a solution for this problem is by proving square function estimates for perturbations of associated non-homogeneous Dirac-type operators. At present, there is no general method to obtain such square function estimates other than for potentials bounded both from above and below (cf. [10]). We develop such a method by adapting the homogeneous framework introduced by A. Axelsson, S. Keith and A. McIntosh in their seminal paper [11]. Two distinct approaches will be considered when adapting this framework. The second such approach will yield a satisfying solution to the potential dependent Kato problem for a large class of potentials. This class will include any potential V with range contained in some sector of angle ωV ∈ [0, π 2 ) and for which |V | belongs either to the reverse H¨older class RH2 in any dimension or L n 2 (R n ) for n > 4.

The connection between quadratic estimates and the existence of a bounded holomorphic functional calculus of an operator provides a framework for applying harmonic analysis to the theory of differential operators. This is a generalization of the connection between Littlewood--Paley--Stein estimates and the functional calculus provided by the Fourier transform. We use the former approach in this thesis to study first-order differential operators on Riemannian manifolds. The theory developed is local in the sense that it does not depend on the spectrum of the operator in a neighbourhood of the origin. When we apply harmonic analysis to obtain estimates, the local theory only requires that we do so up to a finite scale. This allows us to consider manifolds with exponential volume growth in situations where the global theory requires polynomial volume growth. A holomorphic functional calculus is constructed for operators on a reflexive Banach space that are bisectorial except possibly in a neighbourhood of the origin. We prove that this functional calculus is bounded if and only if certain local quadratic estimates hold. For operators with spectrum in a neighbourhood of the origin, the results are weaker than those for bisectorial operators. For operators with a spectral gap in a neighbourhood of the origin, the results are stronger. In each case, however, local quadratic estimates are a more appropriate tool than standard quadratic estimates for establishing that the functional calculus is bounded. This theory allows us to define local Hardy spaces of differential forms that are adapted to a class of first-order differential operators on a complete Riemannian manifold with at most exponential volume growth. The local geometric Riesz transform associated with the Hodge--Dirac operator is bounded on these spaces provided that a certain condition on the exponential growth of the manifold is satisfied. A characterisation of these spaces in terms of local molecules is also obtained. These results can be viewed as the localisation of those for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ. Finally, we introduce a class of first-order differential operators that act on the trivial bundle over a complete Riemannian manifold with at most exponential volume growth and on which a local Poincar\'{e} inequality holds. A local quadratic estimate is established for certain perturbations of these operators. As an application, we solve the Kato square root problem for divergence form operators on complete Riemannian manifolds with Ricci curvature bounded below that are embedded in Euclidean space with a uniformly bounded second fundamental form. This is based on the framework for Dirac type operators that was introduced by Axelsson, Keith and McIntosh.