scholarly journals Bivariate C1 cubic spline space over a nonuniform type-2 triangulation and its subspaces with boundary conditions

2005 ◽  
Vol 49 (11-12) ◽  
pp. 1853-1865 ◽  
Author(s):  
Huan-Wen Liu ◽  
Don Hong ◽  
Dun-Qian Cao
Keyword(s):  
Boundary Conditions ◽  
Cubic Spline ◽  
Spline Space

On the Dimension of Bivariate C1 Cubic Spline Space with Homogeneous Boundary Conditions over a CT Triangulation

2011 ◽  
Vol 50-51 ◽  
pp. 488-492
Author(s):  
Dian Xuan Gong ◽  
Feng Gong Lang
Keyword(s):  
Boundary Conditions ◽  
Computer Aided Design ◽  
Numerical Approximation ◽  
Cubic Spline ◽  
Spline Space ◽  
Singular Boundary ◽  
Piecewise Polynomial ◽  
Nite Element ◽  
Bivariate Spline ◽  
Aided Design

A bivariate spline is a piecewise polynomial with some smoothness de ned on a parti- tion. In this paper, we mainly study the dimensions of bivariate C1 cubic spline spaces S1;0 3 (CT ) and S1;1 3 (CT ) with homogeneous boundary conditions over CT by using interpolating technique, where CT stands for a CT triangulation. The dimensions are related with the numbers of the inter vertices and the singular boundary vertices. The results of this paper can be applied in many elds such as the nite element method for partial di erential equation, computer aided design, numerical approximation, and so on.

scholarly journals I am a rock; I am an island: Implications of avoidant attachment for communal coping in adults with type 2 diabetes

2019 ◽  
Vol 36 (11-12) ◽  
pp. 3711-3732
Author(s):  
Meredith Van Vleet ◽  
Vicki S. Helgeson
Keyword(s):  
Type 2 Diabetes ◽  
Chronic Illness ◽  
Boundary Conditions ◽  
Relationship Quality ◽  
Current Investigation ◽  
Heterosexual Couples ◽  
Communal Coping ◽  
Diabetes Outcomes ◽  
Avoidant Attachment

Accumulating evidence indicates that communal coping is beneficial for individuals with chronic illness. The current investigation examined attachment as a moderator of the effects of communal coping in a sample of persons with type 2 diabetes. We hypothesized that patient communal coping would be associated with higher relationship quality, lower distress, and better diabetes outcomes for patients low in avoidant attachment, but it would not be beneficial for patients high in avoidant attachment. Patient communal coping was coded from videotaped interactions in which 86 heterosexual couples discussed difficulties managing diabetes. The results indicated that patient communal coping was beneficial when avoidant attachment was low. When avoidant attachment was high, patient communal coping was related to lower relationship quality and higher distress and was unrelated to diabetes outcomes. This work sheds light on potential boundary conditions of communal coping’s benefits, which will be important to consider in future communal coping interventions.

On splitting for cubic spline wavelets with four zero moments on an interval

2021 ◽  
pp. 72-87
Author(s):  
Борис Михайлович Шумилов
Keyword(s):  
Cubic Spline ◽  
Spline Space ◽  
Dirichlet Boundary Conditions ◽  
Second Derivative ◽  
Algebraic Equations ◽  
Splitting Algorithm ◽  
Spline Wavelets ◽  
Wavelet Space ◽  
Linear Algebraic Equations ◽  
The Stability

В пространстве кубических сплайнов построены вейвлеты, удовлетворяющие однородным граничным условиям Дирихле и обнулению первых четырех моментов. Получены неявные соотношения, связывающие сплайн-коэффициенты разложения на начальном уровне со сплайн-коэффициентами и вейвлет-коэффициентами на вложенном уровне ленточной системой линейных алгебраических уравнений с невырожденной матрицей. После расщепления на четные и нечетные уравнения матрица преобразования имеет пять (вместо трех в случае двух нулевых моментов) диагоналей. Доказано наличие строгого диагонального доминирования по столбцам. Для сравнения использованы вейвлеты с двумя нулевыми моментами и интерполяционные кубические сплайновые вейвлеты. Результаты численных экспериментов показывают, что схема с четырьмя нулевыми моментами точнее при аппроксимации функций, но грубее при аппроксимации второй производной. The article examines the problem of constructing a splitting algorithm for cubic spline wavelets. First, a cubic spline space is constructed for splines with homogeneous Dirichlet boundary conditions. Then, using the first four zero moments, the corresponding wavelet space is constructed. The resulting space consists of cubic spline wavelets that satisfy the orthogonality conditions for all thirddegree polynomials. The originality of the research lies in obtaining implicit relations connecting the coefficients of the spline expansion at the initial level with the spline coefficients and wavelet coefficients at the embedded level by a band system of linear algebraic equations with a nondegenerate matrix. Excluding the even rows of the system, the resulting transformation algorithm is obtained as a solution to a sequence of band systems of linear algebraic equations with five (instead of three in the case of two zero moments) diagonals. The presence of strict diagonal dominance over the columns is proved, which confirms the stability of the computational process. For comparison, we adopt the results of calculations using wavelets orthogonal to first-degree polynomials and interpolating cubic spline wavelets with the property of the best mean-square approximation of the second derivative of the function being approximated. The results of numerical experiments show that the scheme with four zero moments is more accurate in the approximation of functions, but becomes inferior in accuracy to the approximation of the second derivative.

HIGHER-ORDER PARAMETER UNIFORM CONVERGENT SCHEMES FOR ROBIN TYPE REACTION-DIFFUSION PROBLEMS USING ADAPTIVELY GENERATED GRID

2012 ◽  
Vol 09 (04) ◽  
pp. 1250052 ◽  
Author(s):  
PRATIBHAMOY DAS ◽  
SRINIVASAN NATESAN
Keyword(s):  
Boundary Conditions ◽  
Reaction Diffusion ◽  
Perturbation Parameter ◽  
Cubic Spline ◽  
Diffusion Problem ◽  
Central Difference ◽  
Monitor Function ◽  
Type Boundary ◽  
Robin Boundary ◽  
Interior Points

In this article, a singularly perturbed reaction-diffusion problem with Robin boundary conditions, is considered. In general, the solution of this problem possesses boundary layers at both the ends of the domain. To solve this problem, we propose a numerical scheme, involving the cubic spline scheme for boundary conditions and the classical central difference scheme for the differential equation (DE) at the interior points. The grid is generated by the equidistribution of a positive monitor function. It has been proved that classical forward–backward approximation for mixed type boundary conditions, gives first-order convergence, whereas our proposed cubic spline scheme provides second-order accuracy independent of the perturbation parameter. Numerical experiments have been provided to validate the theoretical results.

scholarly journals Glycemic Control and Risk of Sepsis and Subsequent Mortality in Type 2 Diabetes

2021 ◽  
Author(s):  
Anca Balintescu ◽  
Marcus Lind ◽  
Mikael Andersson Franko ◽  
Anders Oldner ◽  
Maria Cronhjort ◽  
...  
Keyword(s):  
Type 2 Diabetes ◽  
Regression Analysis ◽  
Cox Regression ◽  
Cubic Spline ◽  
Hazard Ratio ◽  
Adjusted Hazard Ratio ◽  
Cox Regression Analysis ◽  
Risk Of Death ◽  
Increased Risk

<b>Objective</b> <p>To investigate the nature of<b> </b>the relationship between HbA1c and sepsis among individuals with type 2 diabetes and to assess the association of sepsis and all-cause mortality in such patients.<b></b></p> <p><b>Research design and methods</b></p> <p>We included 502,871 individuals with type 2 diabetes recorded in the Swedish National Diabetes Register and used multivariable Cox regression and restricted cubic spline analyses to assess the association between time-updated HbA1c values and sepsis occurrence between January 1, 2005 and December 31, 2015. The association between sepsis and death was examined using multivariable Cox regression analysis.</p> <p><b>Result</b></p> <p>Overall, 14,534 (2.9%) patients developed sepsis during the study period. On multivariable Cox regression analysis, compared with an HbA1c of 48-52 mmol/mol (6.5-6.9%), the adjusted hazard ratio for sepsis was 1.15 (95% CI 1.07-1.24) for HbA1c <43 mmol/mol (6.1%); 0.93 (0.87-0.99) for HbA1c 53-62 mmol/mol (7.0-7.8%); 1.05 (0.97-1.13) for HbA1c 63-72 mmol/mol (7.9-8.7%); 1.14 (1.04-1.25) for HbA1c 73-82 mmol/mol (8.8-9.7%); and 1.52 (1.37-1.68) for HbA1c >82 mmol/mol (9.7%). In the cubic spline model, a reduction of the adjusted risk was observed within the lower HbA1c range until 53 mmol/mol (7.0%), with a hazard ratio of 0.78 (0.73-0.82) per standard deviation, and increased thereafter (P for non-linearity <0.001). As compared to patients without sepsis, the adjusted hazard ratio for death among patients with sepsis was 4.16 (4.03-4.30).</p> <p><b>Conclusions</b></p> <p>In a nationwide cohort of individuals with type 2 diabetes, we found a U-shaped association between HbA1c and sepsis and a four-fold increased risk of death among those developing sepsis. </p>

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