Статті в журналах: "Kato square root problem"

1

Arlinskiĭ, Yury. "On the Kato square root problem." Journal of Mathematical Analysis and Applications 505, no. 2 (January 2022): 125501. http://dx.doi.org/10.1016/j.jmaa.2021.125501.

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2

Morris, Andrew J. "The Kato square root problem on submanifolds." Journal of the London Mathematical Society 86, no. 3 (August 10, 2012): 879–910. http://dx.doi.org/10.1112/jlms/jds039.

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3

Escauriaza, Luis, and Steve Hofmann. "Kato square root problem with unbounded leading coefficients." Proceedings of the American Mathematical Society 146, no. 12 (September 17, 2018): 5295–310. http://dx.doi.org/10.1090/proc/14224.

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4

AXELSSON, ANDREAS, STEPHEN KEITH, and ALAN McINTOSH. "THE KATO SQUARE ROOT PROBLEM FOR MIXED BOUNDARY VALUE PROBLEMS." Journal of the London Mathematical Society 74, no. 01 (August 2006): 113–30. http://dx.doi.org/10.1112/s0024610706022873.

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5

Bechtel, Sebastian, Moritz Egert, and Robert Haller-Dintelmann. "The Kato square root problem on locally uniform domains." Advances in Mathematics 375 (December 2020): 107410. http://dx.doi.org/10.1016/j.aim.2020.107410.

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6

Egert, Moritz, Robert Haller-Dintelmann, and Patrick Tolksdorf. "The Kato Square Root Problem for mixed boundary conditions." Journal of Functional Analysis 267, no. 5 (September 2014): 1419–61. http://dx.doi.org/10.1016/j.jfa.2014.06.003.

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7

Bandara, Lashi, and Alan McIntosh. "The Kato Square Root Problem on Vector Bundles with Generalised Bounded Geometry." Journal of Geometric Analysis 26, no. 1 (January 27, 2015): 428–62. http://dx.doi.org/10.1007/s12220-015-9557-y.

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8

Skubachevskii, A. L. "Elliptic differential‐difference operators with degeneration and the Kato square root problem." Mathematische Nachrichten 291, no. 17-18 (August 31, 2018): 2660–92. http://dx.doi.org/10.1002/mana.201700475.

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9

Zhang, Junqiang, Dachun Yang, and Sibei Yang. "Kato square root problem for degenerate elliptic operators on bounded Lipschitz domains." Journal of Differential Equations 353 (April 2023): 1–62. http://dx.doi.org/10.1016/j.jde.2022.12.039.

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10

Egert, M., R. Haller-Dintelmann, and P. Tolksdorf. "The Kato Square Root Problem follows from an extrapolation property of the Laplacian." Publicacions Matemàtiques 60 (July 1, 2016): 451–83. http://dx.doi.org/10.5565/publmat_60216_05.

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11

Auscher, P., A. McIntosh, and A. Nahmod. "The square root problem of Kato in one dimension, and first order elliptic systems." Indiana University Mathematics Journal 46, no. 3 (1997): 0. http://dx.doi.org/10.1512/iumj.1997.46.1423.

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12

Auscher, Pascal, Steve Hofmann, Alan McIntosh, and Philippe Tchamitchian. "The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $." Journal of Evolution Equations 1, no. 4 (December 2001): 361–85. http://dx.doi.org/10.1007/pl00001377.

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13

Auscher, Pascal, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph Tchamitchian. "The Solution of the Kato Square Root Problem for Second Order Elliptic Operators on \Bbb R n." Annals of Mathematics 156, no. 2 (September 2002): 633. http://dx.doi.org/10.2307/3597201.

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14

Muravnik, Andrey B. "Nonlocal problems and functional-differential equations: theoretical aspects and applications to mathematical modelling." Mathematical Modelling of Natural Phenomena 14, no. 6 (2019): 601. http://dx.doi.org/10.1051/mmnp/2019010.

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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.

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15

Chen, Li, José María Martell, and Cruz Prisuelos-Arribas. "Conical square functions for degenerate elliptic operators." Advances in Calculus of Variations 13, no. 1 (January 1, 2020): 75–113. http://dx.doi.org/10.1515/acv-2016-0062.

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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}}.

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16

Musa, Bashir, Nasser Yimen, Sani Isah Abba, Humphrey Hugh Adun, and Mustafa Dagbasi. "Multi-State Load Demand Forecasting Using Hybridized Support Vector Regression Integrated with Optimal Design of Off-Grid Energy Systems—A Metaheuristic Approach." Processes 9, no. 7 (July 5, 2021): 1166. http://dx.doi.org/10.3390/pr9071166.

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The prediction accuracy of support vector regression (SVR) is highly influenced by a kernel function. However, its performance suffers on large datasets, and this could be attributed to the computational limitations of kernel learning. To tackle this problem, this paper combines SVR with the emerging Harris hawks optimization (HHO) and particle swarm optimization (PSO) algorithms to form two hybrid SVR algorithms, SVR-HHO and SVR-PSO. Both the two proposed algorithms and traditional SVR were applied to load forecasting in four different states of Nigeria. The correlation coefficient (R), coefficient of determination (R2), mean square error (MSE), root mean square error (RMSE), and mean absolute percentage error (MAPE) were used as indicators to evaluate the prediction accuracy of the algorithms. The results reveal that there is an increase in performance for both SVR-HHO and SVR-PSO over traditional SVR. SVR-HHO has the highest R2 values of 0.9951, 0.8963, 0.9951, and 0.9313, the lowest MSE values of 0.0002, 0.0070, 0.0002, and 0.0080, and the lowest MAPE values of 0.1311, 0.1452, 0.0599, and 0.1817, respectively, for Kano, Abuja, Niger, and Lagos State. The results of SVR-HHO also prove more advantageous over SVR-PSO in all the states concerning load forecasting skills. This paper also designed a hybrid renewable energy system (HRES) that consists of solar photovoltaic (PV) panels, wind turbines, and batteries. As inputs, the system used solar radiation, temperature, wind speed, and the predicted load demands by SVR-HHO in all the states. The system was optimized by using the PSO algorithm to obtain the optimal configuration of the HRES that will satisfy all constraints at the minimum cost.

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17

ELST, A. F. M. ter, and Derek W. ROBINSON. "On Kato's square root problem." Hokkaido Mathematical Journal 26, no. 2 (February 1997): 365–76. http://dx.doi.org/10.14492/hokmj/1351257970.

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18

Zhang, Xu, Xue Tian, Chen Wang, and Tao Li. "Multiple-Square-Root Minimization Problem." American Journal of Industrial and Business Management 07, no. 07 (2017): 927–43. http://dx.doi.org/10.4236/ajibm.2017.77066.

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19

Konstantin, Tiffany, Indah Setyawati Tantular, Alpha Fardah Athiyyah, and Lynda Rossyanti. "Hubungan Karakteristik Sosiodemografi dengan Status Gizi Siswa Sekolah Dasar." JI-KES (Jurnal Ilmu Kesehatan) 3, no. 2 (February 29, 2020): 46–50. http://dx.doi.org/10.33006/ji-kes.v3i2.135.

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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

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20

Ladioktaviagusdi, Rangga Kurniawan, and Raden Sumiharto. "Implementasi Kontrol Model Prediksi Berbasis ANFIS Pada Mesin Penghasil Uap Air." IJEIS (Indonesian Journal of Electronics and Instrumentation Systems) 5, no. 1 (May 1, 2015): 99. http://dx.doi.org/10.22146/ijeis.7158.

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AbstrakMesin penghasil uap air mempunyai permasalahan pada parameter keluaran berupa dependent variabel, oleh karena itu diperlukan suatu cara untuk mendapatkan perilaku sistem dari mesin penghasil uap air yang direpresentasikan dengan model dari plant tersebut.Implementasi kontrol model prediksi pada penelitian ini direpresentasikan dengan menggunakan simulasi, sehingga objek yang dikontrol berupa mesin virtual penghasil uap air.Dalam penelitan ini digunakan perangkat lunak MATLAB sebagai mesin virtual penghasil uap air dan juga untuk melakukan komputasi ANFIS, sedangkan perangkat lunak LabVIEW digunakan sebagai representasi dari ruang kontrol.Dari hasil penelitian, didapatkan parameter terbaik untuk masing-masing ANFIS yang dijadikan sebagai unit model pada kontrol model prediksi, yaitu dengan menggunakan historical data ke-4 yang berjumlah 800 data, dengan rasio presentase pembelajaran untuk training data dan checking data pada masing-masing ANFIS untuk setiap struktur model secara berurutan sebesar 90% dan 10%, kecuali rasio presentase untuk ANFIS pada parameter aliran air secara berurutan sebesar 80% dan 20%. Hasil validasi RMSE (Root Mean Square Error) dengan melakukan pengujian terhadap 100 data, didapatkan nilai sebagai berikut: aliran air=1.9941, tekanan air=48.0236, aliran udara=604.0621, tekanan bahan bakar=0.7087, temperatur bahan bakar=18.6594, O2 content=0.9591, tekanan uap air=76.1557, kualitas uap air=3.9734 dan aliran uap air=264.9173. Kata kunci— Mesin penghasil uap air, kontrol model prediksi, pemodelan, ANFIS, MATLAB, LabVIEW AbstractSteam generator has problem such as dependent variable on the output parameters, therefore it is needed a way to get system behavior of steam generator which is represented by model of the plant. Implementation of the model predictive control in this research was represented by using simulation, so the object that was controlled was virtual steam generator. In this research was used MATLAB software as a virtual steam generator and also for computes ANFIS, whereas the LabVIEW software was used as a representation of control room.From the research, it was found the best parameters for each ANFIS that was used as a model unit in the model predictive control, that was by using historical data 4th as much as 800 datas, the percentage ratio of learning for training data and checking data on each ANFIS for each model structures sequentially by 90% and 10%, except the percentage ratio for ANFIS on water flow parameter sequentially by 80% and 20%. The results of validation RMSE (Root Mean Square Error) by testing for 100 datas, it was obtained values as follows: water flow=1.9941, water pressure=48.0236, air flow=604.0621, fuel gas pressure=0.7087, fuel gas temperature=18.6594, O2 content=0.9591, steam pressure=76.1557, steam quality=3.9734 and steam flow=264.9173. Keywords— Steam generator, model predictive control, modeling, ANFIS, MATLAB, LabVIEW

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21

Journé, J. L. "Remarks on Kato's square-root problem." Publicacions Matemàtiques 35 (January 1, 1991): 299–321. http://dx.doi.org/10.5565/publmat_35191_16.

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22

Roh, Dongyoung, and Sang Geun Hahn. "The square root Diffie–Hellman problem." Designs, Codes and Cryptography 62, no. 2 (April 12, 2011): 179–87. http://dx.doi.org/10.1007/s10623-011-9503-3.

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23

Diagana, Toka. "An application to Kato's square root problem." International Journal of Mathematics and Mathematical Sciences 29, no. 3 (2002): 179–81. http://dx.doi.org/10.1155/s0161171202007706.

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We find all complex potentialsQsuch that the general Schrödinger operator onℝn, given byL=−Δ+Q, whereΔis the Laplace differential operator, verifies the well-known Kato's square problem. As an application, we will consider the case whereQ∈Lloc1(Ω).

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24

Hytönen, Tuomas, Alan McIntosh, and Pierre Portal. "Kato's square root problem in Banach spaces." Journal of Functional Analysis 254, no. 3 (February 2008): 675–726. http://dx.doi.org/10.1016/j.jfa.2007.10.006.

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25

Grosser-Clarkson, Dana L. "The Root of the Problem." Mathematics Teacher 109, no. 2 (September 2015): 98–102. http://dx.doi.org/10.5951/mathteacher.109.2.0098.

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26

Chen, Yan Ping, Yong Ding та Steve Hofmann. "The commutator of the Kato square root for second order elliptic operators on ℝ n". Acta Mathematica Sinica, English Series 32, № 10 (15 вересня 2016): 1121–44. http://dx.doi.org/10.1007/s10114-016-5719-5.

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27

Lin, Lan, and Yixun Lin. "Square-root rule of two-dimensional bandwidth problem." RAIRO - Theoretical Informatics and Applications 45, no. 4 (September 22, 2011): 399–411. http://dx.doi.org/10.1051/ita/2011120.

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28

Jonker, Harm J. J., Maarten van Reeuwijk, Peter P. Sullivan, and Edward G. Patton. "On the scaling of shear-driven entrainment: a DNS study." Journal of Fluid Mechanics 732 (August 30, 2013): 150–65. http://dx.doi.org/10.1017/jfm.2013.394.

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AbstractThe deepening of a shear-driven turbulent layer penetrating into a stably stratified quiescent layer is studied using direct numerical simulation (DNS). The simulation design mimics the classical laboratory experiments by Kato & Phillips (J. Fluid Mech., vol. 37, 1969, pp. 643–655) in that it starts with linear stratification and applies a constant shear stress at the lower boundary, but avoids sidewall and rotation effects inherent in the original experiment. It is found that the layers universally deepen as a function of the square root of time, independent of the initial stratification and the Reynolds number of the simulations, provided that the Reynolds number is large enough. Consistent with this finding, the dimensionless entrainment velocity varies with the bulk Richardson number as$R{i}^{- 1/ 2} $. In addition, it is observed that all cases evolve in a self-similar fashion. A self-similarity analysis of the conservation equations shows that only a square root growth law is consistent with self-similar behaviour.

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29

Chen, Yanping, Yong Ding та Kai Zhu. "Weighted Norm Inequalities for Commutators of the Kato Square Root of Second Order Elliptic Operators on ℝn". Acta Mathematica Scientia 42, № 4 (квітень 2022): 1310–32. http://dx.doi.org/10.1007/s10473-022-0404-5.

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30

Nyström, K. "Square function estimates and the Kato problem for second order parabolic operators in Rn+1." Advances in Mathematics 293 (April 2016): 1–36. http://dx.doi.org/10.1016/j.aim.2016.02.006.

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31

Duan, Xue-Feng, Cun-Yun Wang, and Chun-Mei Li. "Newton’s method for solving the tensor square root problem." Applied Mathematics Letters 98 (December 2019): 57–62. http://dx.doi.org/10.1016/j.aml.2019.05.031.

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32

Psiaki, Mark L. "The Square Root Information Increment Ensemble Filter." Monthly Weather Review 144, no. 12 (November 9, 2016): 4667–86. http://dx.doi.org/10.1175/mwr-d-15-0295.1.

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Abstract A new type of ensemble filter is developed, one that stores and updates its state information in an efficient square root information filter form. It addresses two shortcomings of conventional ensemble Kalman filters: the coarse characterization of random forecast model error effects and the overly optimistic approximation of the estimation error statistics. The new filter uses an assumed a priori covariance approximation that is full rank but sparse, possibly with a dense low-rank increment. This matrix can be used to develop a nominal square root information equation for the system state and uncertainty. The measurements are used to develop an additional low-rank square root information equation. New algorithms provide forecasts and analyses of these increments at a computational cost comparable to that of existing ensemble Kalman filters. Model error effects are implicit in the a priori covariance time history, thereby obviating one of the reasons for including an inflation operation. The use of an a priori full-rank covariance allows the analysis operations to improve the state estimate without the need for a localization adjustment. This new filter exhibited worse performance than a typical covariance square root ensemble Kalman filter when operating on the Lorenz-96 problem in a chaotic regime. It excelled on a version of the Lorenz-96 problem where nonlinearities in the forecast model were weak, where the state vector uncertainty lay predominantly in a small subspace, and where the observations were spatially sparse. Such a problem might be representative of ionospheric space weather data assimilation where forcing variability can dominate the state uncertainty and where remote sensing data coverage can be sparse.

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33

Lee, Sang Hoon, and Jasang Yoon. "The Square Root Problem and Aluthge transforms of weighted shifts." Mathematische Nachrichten 290, no. 17-18 (July 31, 2017): 2925–33. http://dx.doi.org/10.1002/mana.201600302.

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34

Guerry, Marie-Anne. "On the Embedding Problem for Discrete-Time Markov Chains." Journal of Applied Probability 50, no. 4 (December 2013): 918–30. http://dx.doi.org/10.1239/jap/1389370090.

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When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable.In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (kxk) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.

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35

Guerry, Marie-Anne. "On the Embedding Problem for Discrete-Time Markov Chains." Journal of Applied Probability 50, no. 04 (December 2013): 918–30. http://dx.doi.org/10.1017/s002190020001370x.

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When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable. In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (k x k) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.

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36

Huang, Yulong, Yonggang Zhang, Ning Li, and Lin Zhao. "Improved square-root cubature information filter." Transactions of the Institute of Measurement and Control 39, no. 4 (July 22, 2016): 579–88. http://dx.doi.org/10.1177/0142331215608428.

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In this paper, a theoretical comparison between existing the sigma-point information filter (SPIF) framework and the unscented information filter (UIF) framework is presented. It is shown that the SPIF framework is identical to the sigma-point Kalman filter (SPKF). However, the UIF framework is not identical to the classical SPKF due to the neglect of one-step prediction errors of measurements in the calculation of state estimation error covariance matrix. Thus SPIF framework is more reasonable as compared with UIF framework. According to the theoretical comparison, an improved cubature information filter (CIF) is derived based on the superior SPIF framework. Square-root CIF (SRCIF) is also developed to improve the numerical accuracy and stability of the proposed CIF. The proposed SRCIF is applied to a target tracking problem with large sampling interval and high turn rate, and its performance is compared with the existing SRCIF. The results show that the proposed SRCIF is more reliable and stable as compared with the existing SRCIF. Note that it is impractical for information filters in large-scale applications due to the enormous computational complexity of large-scale matrix inversion, and advanced techniques need to be further considered.

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37

Duan, Yaqiong, and Shujun Lian. "Smoothing Approximation to the Square-Root Exact Penalty Function." Journal of Systems Science and Information 4, no. 1 (February 25, 2016): 87–96. http://dx.doi.org/10.1515/jssi-2016-0087.

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AbstractIn this paper, smoothing approximation to the square-root exact penalty functions is devised for inequality constrained optimization. It is shown that an approximately optimal solution of the smoothed penalty problem is an approximately optimal solution of the original problem. An algorithm based on the new smoothed penalty functions is proposed and shown to be convergent under mild conditions. Three numerical examples show that the algorithm is efficient.

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38

Wandji Nyamsi, W., B. Espinar, P. Blanc, and L. Wald. "Estimating the photosynthetically active radiation under clear skies by means of a new approach." Advances in Science and Research 12, no. 1 (February 18, 2015): 5–10. http://dx.doi.org/10.5194/asr-12-5-2015.

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Abstract. The k-distribution method and the correlated-k approximation of Kato et al. (1999) is a computationally efficient approach originally designed for calculations of the broadband solar radiation by dividing the solar spectrum in 32 specific spectral bands from 240 to 4606 nm. This paper describes a technique for an accurate assessment of the photosynthetically active radiation (PAR) from 400 to 700 nm at ground level, under clear-sky conditions using twelve of these spectral bands. It is validated against detailed spectral calculations of the PAR made by the radiative transfer model libRadtran. For the direct and global PAR irradiance, the bias is −0.4 W m−2 (−0.2%) and −4 W m−2 (−1.3%) and the root mean square error is 1.8 W m−2 (0.7%) and 4.5 W m−2 (1.5%). For the direct and global Photosynthetic Photon Flux Density, the biases are of about +10.3 μmol m−2 s−1 (+0.8%) and 1.9 μmol m−2 s−1 (−0.1%) respectively, and the root mean square error is 11.4 μmol m−2 s−1 (0.9%) and 4.0 μmol m−2 s−1 (0.3%). The correlation coefficient is greater than 0.99. This technique provides much better results than two state-of-the-art empirical methods computing the daily mean of PAR from the daily mean of broadband irradiance.

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39

Ajiev, Sergey S. "Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I." Journal of the Australian Mathematical Society 83, no. 3 (December 2007): 297–326. http://dx.doi.org/10.1017/s1446788700037940.

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AbstractSeveral rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from L2 to Lp are established. As consequences, we obtain some embedding theorems, quadratic estimates and Littlewood–Paley theorems in terms of this calculus in Lebesgue spaces. Some further generalizations, utilised in Part II devoted to applications, which include the Kato square root model, are discussed. We use resolvent approach and show the irrelevance of the semigroup one. Auxiliary results include a high order counterpart of the Hilbert identity, the derivation of new forms of ‘off-diagonal’ estimates, and the study of the structure of the model in Lebesgue spaces and its interpolation properties. In particular, some coercivity conditions for forms in Banach spaces are used as a substitution of the ellipticity ones. Attention is devoted to the relations between the properties of perturbed and unperturbed generalized Dirac operators. We do not use any stability results.

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40

Abdelhedi, Wael, and Hichem Chtioui. "On a Nirenberg-type problem involving the square root of the Laplacian." Journal of Functional Analysis 265, no. 11 (December 2013): 2937–55. http://dx.doi.org/10.1016/j.jfa.2013.08.005.

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41

Tan, Jinggang. "The Brezis–Nirenberg type problem involving the square root of the Laplacian." Calculus of Variations and Partial Differential Equations 42, no. 1-2 (October 12, 2010): 21–41. http://dx.doi.org/10.1007/s00526-010-0378-3.

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42

Martínez-Zaldívar, F. J., A. M. Vidal-Maciá, and D. Giménez. "Heterogeneous pipelined square-root Kalman Filter algorithm for the MMSE-OSIC problem." Journal of Supercomputing 58, no. 2 (November 7, 2009): 235–43. http://dx.doi.org/10.1007/s11227-009-0354-x.

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43

Raanes, Patrick Nima, Alberto Carrassi, and Laurent Bertino. "Extending the Square Root Method to Account for Additive Forecast Noise in Ensemble Methods." Monthly Weather Review 143, no. 10 (October 1, 2015): 3857–73. http://dx.doi.org/10.1175/mwr-d-14-00375.1.

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Abstract A square root approach is considered for the problem of accounting for model noise in the forecast step of the ensemble Kalman filter (EnKF) and related algorithms. The primary aim is to replace the method of simulated, pseudo-random additive so as to eliminate the associated sampling errors. The core method is based on the analysis step of ensemble square root filters, and consists in the deterministic computation of a transform matrix. The theoretical advantages regarding dynamical consistency are surveyed, applying equally well to the square root method in the analysis step. A fundamental problem due to the limited size of the ensemble subspace is discussed, and novel solutions that complement the core method are suggested and studied. Benchmarks from twin experiments with simple, low-order dynamics indicate improved performance over standard approaches such as additive, simulated noise, and multiplicative inflation.

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44

Ristow, Dietrich, and Thomas Rühl. "3-D implicit finite‐difference migration by multiway splitting." GEOPHYSICS 62, no. 2 (March 1997): 554–67. http://dx.doi.org/10.1190/1.1444165.

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We show that 3-D implicit finite‐difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome. The basic idea is as follows. We approximate the 3-D square root operator by a sequence of 2-D operators in three, four, or six directions to solve the azimuth symmetry problem. Each 2-D square root operator is then approximated by a sequence of implicit 2-D operators to improve steep dip accuracy. This sequence contains some unknown coefficients, which are calculated by a Taylor expansion technique or by an optimization technique. In the Taylor expansion method, the square root and its approximation are expanded into power series. By comparing the terms, the unknown coefficients are calculated. The more 2-D finite‐difference operators for cascading are taken and the more directions for downward continuation are chosen, the more terms from power series can be compared to obtain a higher‐degree migration operator with better circular symmetry. In the second method, optimized coefficients are calculated by an optimization procedure whereby a variation of all unknown coefficients is performed, in such a way that both the sum of all deviations between the correct square root and its approximation and the sum of all deviations from azimuth symmetry are minimized. A mathematical criterion for azimuth symmetry has been defined and incorporated into the opfimization procedure.

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45

Zhang, Yu-feng, Qi-xun Zhou, Ju-zhong Zhang, Yi Jiang, and Kai Wang. "A FastSLAM Algorithm Based on Nonlinear Adaptive Square Root Unscented Kalman Filter." Mathematical Problems in Engineering 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/4197635.

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For fast simultaneous localization and mapping (FastSLAM) problem, to solve the problems of particle degradation, the error introduced by linearization and inconsistency of traditional algorithm, an improved algorithm is described in the paper. In order to improve the accuracy and reliability of algorithm which is applied in the system with lower measurement frequency, a new decomposition strategy is adopted for a posteriori estimation. In proposed decomposition strategy, the problem of solving a 3-dimensional state vector and N 2-dimensional state vectors in traditional FastSLAM algorithm is transformed to the problem of solving N 5-dimensional state vectors. Furthermore, a nonlinear adaptive square root unscented Kalman filter (NASRUKF) is used to replace the particle filter and Kalman filter employed by traditional algorithm to reduce the model linearization error and avoid solving Jacobian matrices. Finally, the proposed algorithm is experimentally verified by vehicle in indoor environment. The results prove that the positioning accuracy of proposed FastSLAM algorithm is less than 1 cm and the azimuth angle error is 0.5 degrees.

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46

Hopkins, David. "Dropping plates." Mathematical Gazette 106, no. 566 (June 22, 2022): 193–205. http://dx.doi.org/10.1017/mag.2022.59.

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A friend of mine [1] mentioned a problem to me, which he was told had an interesting solution involving an unexpected square root. I have not seen this problem described elsewhere, so I have carried out my own analysis, which I will present here. In fact the solution involves not only square roots, but also higher roots … and a logarithm.

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47

Vabishchevich, Petr N. "NUMERICAL SOLVING UNSTEADY SPACE-FRACTIONAL PROBLEMS WITH THE SQUARE ROOT OF AN ELLIPTIC OPERATOR." Mathematical Modelling and Analysis 21, no. 2 (March 18, 2016): 220–38. http://dx.doi.org/10.3846/13926292.2016.1147000.

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An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two- level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.

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48

Alexandrov, D. V., A. P. Malygin, and I. V. Alexandrova. "Solidification of leads: approximate solutions of non-linear problem." Annals of Glaciology 44 (2006): 118–22. http://dx.doi.org/10.3189/172756406781811213.

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AbstractWe present new analytical results relating to the growth and evolution of sea ice. It is noteworthy that thin sea ice plays a central role in the surface heat and mass balance of the Arctic Ocean. In order to describe these balances, we analyze highly resolved temperature data taken through the air/sea/ice interface during the transition from an ice-free to an ice-covered Arctic Ocean surface. Our detailed analysis of the field data is based on the classical model of a mushy layer, which is modified in order to obtain analytical solutions in explicit form (so, for example, ice thickness and growth rate, temperature distributions, conductive and latent heat fluxes are determined). Furthermore, we find that the sea-ice growth is not simply a square-root function of time. It depends on the temperature variations in the atmosphere and lies between two square-root functions of time for the maximum and minimum temperatures found during observations. The theory under consideration is in good agreement with observations.

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49

Yan, L., and L. Zhao. "AN APPROACH ON ADVANCED UNSCENTED KALMAN FILTER FROM MOBILE ROBOT-SLAM." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLIII-B4-2020 (August 25, 2020): 381–89. http://dx.doi.org/10.5194/isprs-archives-xliii-b4-2020-381-2020.

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Abstract. In the past 30 years, Kalman filter is a classical method to solve the problem of simultaneous localization and mapping (SLAM) of mobile robots. Extended Kalman filter (EKF) and unscented Kalman filter (UKF) are derived from Kalman filter. Extended Kalman filter (EKF) overcomes the nonlinear problem that Kalman filter cannot solve. However, when it is strongly nonlinear, EKF violates the assumption of local linearity, and EKF algorithm may make the filter diverge. Secondly, because the Jacobian matrix is needed in the online processing of EKF, its tedious calculation process makes the implementation of this method relatively difficult. Unscented Kalman filter (UKF) uses nonlinear model directly, avoids operation of Jacobian matrix of complex nonlinear function, and ensures the general adaptability of nonlinear system. In this paper, based on the square root unscented Kalman filter, sigma points are selected according to the square-root decomposition of prior covariance, and then weighted mean and covariance are calculated. The quaternion is used to represent the attitude, and the quaternion vector is converted to the rotation space for matrix operation. Comparing the robot poses estimated based on the square root traceless Kalman filter (QSR-UKF), square root traceless Kalman filter (SR-UKF), and extended Kalman filter (EKF), the simulation results show the QSR-UKF proposed in this paper is effective.

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50

Chen, Ke, Hongkai Wang, Zhangchi Ying, Chengxin Zhang, and Jiaqi Wang. "Online cleaning method of power grid energy anomaly data based on improved random forest." Journal of Physics: Conference Series 2108, no. 1 (November 1, 2021): 012067. http://dx.doi.org/10.1088/1742-6596/2108/1/012067.

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Abstract Aiming at the problem of high root mean square error of traditional power grid energy anomaly data online cleaning, a power grid energy anomaly data online cleaning method based on improved random forest is designed. Firstly, an outlier data recognition model of isolated forest is designed to identify outliers in the data. Secondly, an improved random forest regression model is established to improve the adaptability of random forest to mixed abnormal data, and the data trend is fitted and predicted. Finally, the improved random forest data cleaning method is used to compensate the missing data after removing the mixed abnormal data, so as to clean the abnormal energy data of the power grid. The experimental results show that when the amount of power grid energy anomaly data increases, the cleaning root mean square error of the experimental group is significantly lower than that of the control group. The method in this paper solves the problem of high root-mean-square error in the online cleaning of abnormal data of traditional grid energy.

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