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Approximation to the Derivatives of a Function Defined on a Simplex under Lagrangian Interpolation 拉格朗日插值法下简约上定义的函数的导数近似值
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010012
N. V. Baidakova, Yu. N. Subbotin
{"title":"Approximation to the Derivatives of a Function Defined on a Simplex under Lagrangian Interpolation","authors":"N. V. Baidakova, Yu. N. Subbotin","doi":"10.1134/s0001434624010012","DOIUrl":"https://doi.org/10.1134/s0001434624010012","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> New upper bounds are found in the problem of approximation to the <span>(k)</span>th derivatives of a function of <span>(d)</span> variables defined on a simplex by the derivatives of an algebraic polynomial of degree at most <span>(n)</span> (<span>(0le kle n)</span>) interpolating the values of the function at equidistant nodes of the simplex. The estimates are obtained in terms of the diameter of the simplex, the angular characteristic introduced in the paper, the dimension <span>(d)</span>, the degree <span>(n)</span> of the polynomial, and the order <span>(k)</span> of the derivative to be estimated and do not contain unknown parameters. These estimates are compared with those most frequently used in the literature. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Periodic Gibbs Measures and Their Extremality for the HC-Blume–Capel Model in the Case of a Wand with a Chemical Potential on a Cayley Tree 凯利树上具有化学势的魔杖情况下 HC 布朗-卡佩尔模型的周期吉布斯量及其极端性
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010085
N. M. Khatamov
{"title":"Periodic Gibbs Measures and Their Extremality for the HC-Blume–Capel Model in the Case of a Wand with a Chemical Potential on a Cayley Tree","authors":"N. M. Khatamov","doi":"10.1134/s0001434624010085","DOIUrl":"https://doi.org/10.1134/s0001434624010085","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Periodic Gibbs measures for the HC-Blume–Capel model with a chemical potential with parameters <span>((theta,eta))</span> on a Cayley tree in the case of a wand graph are studied. We prove that in this case for <span>(theta^3leeta)</span> there exist precisely three periodic Gibbs measures, all of which are translation-invariant, while for <span>(theta^3&gt;eta)</span> there exist precisely three periodic Gibbs measures, one of which is translation-invari The (non)extremality of these measures is also studied. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"93 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Strong Convexity Supporting Condition and the Lipschitz Condition for the Metric Projection 公设投影的强凸支持条件和 Lipschitz 条件
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010152
M. V. Balashov, K. Z. Biglov
{"title":"The Strong Convexity Supporting Condition and the Lipschitz Condition for the Metric Projection","authors":"M. V. Balashov, K. Z. Biglov","doi":"10.1134/s0001434624010152","DOIUrl":"https://doi.org/10.1134/s0001434624010152","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We prove that the strong convexity supporting condition is typical for an arbitrary convex compact set in <span>(mathbb R^n)</span>. It is shown that, in a certain sense for almost all points, the metric projection onto a convex compact set satisfies the Lipschitz condition with Lipschitz constant strictly less than 1. This condition characterizes the strong convexity supporting condition. The linear convergence of the alternating projection method for a convex compact set with the strong convexity supporting condition and for a proximally smooth set is proved under a certain relation between the constant in the strong convexity supporting condition and the proximal smoothness constant. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"19 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Well-Posedness of an Initial–Boundary Value Problem for a Degenerate Heat Equation 论退化热方程初始边界值问题的良好拟合
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010188
A. R. Zainullov, K. B. Sabitov
{"title":"On the Well-Posedness of an Initial–Boundary Value Problem for a Degenerate Heat Equation","authors":"A. R. Zainullov, K. B. Sabitov","doi":"10.1134/s0001434624010188","DOIUrl":"https://doi.org/10.1134/s0001434624010188","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The well-posedness of an initial–boundary value problem for a model parabolic equation with two power-law degeneration lines is studied. Two initial–boundary value problems depending on the degeneracy exponents are stated, and uniqueness and existence theorems are proved. The solutions of these problems are constructed in closed form. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"57 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bernstein Inequality for the Riesz Derivative of Order $$0 统一规范中指数型全函数的 $$0<alpha<1$$ 阶里兹衍生物的伯恩斯坦不等式
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s000143462401019x
A. O. Leont’eva
{"title":"Bernstein Inequality for the Riesz Derivative of Order $$0<alpha<1$$ of Entire Functions of Exponential Type in the Uniform Norm","authors":"A. O. Leont’eva","doi":"10.1134/s000143462401019x","DOIUrl":"https://doi.org/10.1134/s000143462401019x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider Bernstein’s inequality for the Riesz derivative of order <span>(0&lt;alpha&lt;1)</span> of entire functions of exponential type in the uniform norm on the real line. For this operator, the corresponding interpolation formula is obtained; this formula has nonequidistant nodes. Using this formula, the sharp Bernstein inequality is obtained for all <span>(0&lt;alpha&lt;1)</span>; namely, the extremal entire function and the sharp constant are written out. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"52 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Implementation of Linear Boolean Functions by Self-Correcting Circuits of Unreliable Logic Gates 用不可靠逻辑门的自校正电路实现线性布尔函数
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010073
K. A. Popkov
{"title":"Implementation of Linear Boolean Functions by Self-Correcting Circuits of Unreliable Logic Gates","authors":"K. A. Popkov","doi":"10.1134/s0001434624010073","DOIUrl":"https://doi.org/10.1134/s0001434624010073","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We prove that if a Boolean function essentially depends on at least two variables, then it cannot be implemented by a circuit that consists of unreliable gates with at most two inputs each and is self-correcting with respect to at least some faults of an arbitrary number of gates. In view of the previous results, it suffices to establish this fact for linear functions. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"32 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Two Contrasting Examples of Multidimensional Differential Systems with Lyapunov Extreme Instability 具有 Lyapunov 极端不稳定性的多维微分系统的两个对比实例
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010036
A. A. Bondarev
{"title":"Two Contrasting Examples of Multidimensional Differential Systems with Lyapunov Extreme Instability","authors":"A. A. Bondarev","doi":"10.1134/s0001434624010036","DOIUrl":"https://doi.org/10.1134/s0001434624010036","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Using specific examples, we constructively show that, in dimensions greater than <span>(1)</span>, the Lyapunov extreme instability of a differential system, i.e., the property that the phase curves of all nonzero solutions starting sufficiently close to zero leave any prescribed compact set, does not imply that these solutions go arbitrarily far away from zero in the sense of Perron or in the upper-limit sense as <span>(ttoinfty)</span>. Namely, we construct two Lyapunov extremely unstable systems such that all solutions of the first system tend to zero, while the solutions of the second system are divided into two types: all nonzero solutions starting in the closed unit ball tend to infinity in norm, and all the other solutions tend to zero. Further, both systems constructed in the paper have zero first approximation along the zero solution. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"126 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extremal Interpolation in the Mean in the Space $$L_1(mathbb R)$$ with Overlapping Averaging Intervals 具有重叠平均区间的 $$L_1(mathbb R)$$ 空间中平均值的极值插值法
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010097
V. T. Shevaldin
{"title":"Extremal Interpolation in the Mean in the Space $$L_1(mathbb R)$$ with Overlapping Averaging Intervals","authors":"V. T. Shevaldin","doi":"10.1134/s0001434624010097","DOIUrl":"https://doi.org/10.1134/s0001434624010097","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> On a uniform grid on the real axis <span>(mathbb R)</span>, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space <span>(L_1(mathbb R))</span> of two-way real sequences with the least value of the norm of a linear formally s This problem is considered for the class of sequences for which the generalized finite differences of order <span>(n)</span> corresponding to the operator <span>(mathcal L_n)</span> are bounded in the space <span>(l_1)</span>. In this paper, the least value of the norm is calculated exactly if the grid step <span>(h)</span> and the averaging step <span>(h_1)</span> of the function to be interpolated in the mean are related by the inequalities <span>(h&lt;h_1le 2h)</span>. The paper is a continuation of the research by Yu. N. Subbotin and the author in this problem, initiated by Yu. N. Subbotin in 1965. The result obtained is new, in particular, for the <span>(n)</span>-times differentiation operator <span>(mathcal L_n(D)=D^n)</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"29 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140882728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Rational Spline Solutions of Differential Equations with Singularities in the Coefficients of the Derivatives 论微分方程的有理样条解法与微分系数的奇异性
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010061
V. G. Magomedova, A.-R. K. Ramazanov
{"title":"On Rational Spline Solutions of Differential Equations with Singularities in the Coefficients of the Derivatives","authors":"V. G. Magomedova, A.-R. K. Ramazanov","doi":"10.1134/s0001434624010061","DOIUrl":"https://doi.org/10.1134/s0001434624010061","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For one generalization of the Riemann differential equation, we obtain sufficient conditions for the approximability by twice continuously differentiable rational interpolation spline functions. To solve the corresponding boundary value problem numerically, a tridiagonal system of linear algebraic equations is constructed and conditions on the coefficients of the differential equation are found guaranteeing the uniqueness of the solution of such Estimates of the deviation of the discrete solution of the boundary value problem from the exact solution on a grid are presented. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"59 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Solutions of the One-Dimensional Goldshtik Problem 论一维高兹提克问题的解决方案
IF 0.6 4区 数学
Mathematical Notes Pub Date : 2024-04-22 DOI: 10.1134/s0001434624010024
O. V. Baskov, D. K. Potapov
{"title":"On Solutions of the One-Dimensional Goldshtik Problem","authors":"O. V. Baskov, D. K. Potapov","doi":"10.1134/s0001434624010024","DOIUrl":"https://doi.org/10.1134/s0001434624010024","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A one-dimensional analog of the Goldshtik mathematical model for separated flows in an incompressible fluid is considered. The model is a boundary value problem for a second-order ordinary differential equation with discontinuous right-hand side. Some properties of the solutions of the problem, as well as the properties of the energy functional for different values of vorticity, are established. An approximate solution of the boundary value problem under study is found using the shooting method. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"8 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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