某些收敛序列的凸面组合

IF 2 3区数学 Q1 MATHEMATICS, APPLIED Mathematical Methods in the Applied Sciences Pub Date : 2024-09-01 DOI:10.1002/mma.10463

Stevo Stević

{"title":"某些收敛序列的凸面组合","authors":"Stevo Stević","doi":"10.1002/mma.10463","DOIUrl":null,"url":null,"abstract":"We consider the convex combinations \n<math>\n <semantics>\n <mrow>\n <msubsup>\n <mrow>\n <mi>c</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n <mrow>\n <mi>α</mi>\n </mrow>\n </msubsup>\n <mo>:</mo>\n <mo>=</mo>\n <mo>(</mo>\n <mn>1</mn>\n <mo>−</mo>\n <mi>α</mi>\n <mo>)</mo>\n <msub>\n <mrow>\n <mi>a</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>+</mo>\n <mi>α</mi>\n <msub>\n <mrow>\n <mi>b</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>,</mo>\n <mspace></mspace>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℕ</mi>\n <mo>,</mo>\n <mspace></mspace>\n <mi>α</mi>\n <mo>∈</mo>\n <mo>[</mo>\n <mn>0,1</mn>\n <mo>]</mo>\n </mrow>\n <annotation>$$ {c}_n&amp;amp;#x0005E;{\\alpha}:&amp;amp;#x0003D; \\left(1-\\alpha \\right){a}_n&amp;amp;#x0002B;\\alpha {b}_n,n\\in \\mathrm{\\mathbb{N}},\\alpha \\in \\left[0,1\\right] $$</annotation>\n </semantics></math>, of a pair of sequences of real numbers \n<math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mo>(</mo>\n <msub>\n <mrow>\n <mi>a</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>)</mo>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℕ</mi>\n </mrow>\n </msub>\n </mrow>\n <annotation>$$ {\\left({a}_n\\right)}_{n\\in \\mathrm{\\mathbb{N}}} $$</annotation>\n </semantics></math> and \n<math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mo>(</mo>\n <msub>\n <mrow>\n <mi>b</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>)</mo>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℕ</mi>\n </mrow>\n </msub>\n </mrow>\n <annotation>$$ {\\left({b}_n\\right)}_{n\\in \\mathrm{\\mathbb{N}}} $$</annotation>\n </semantics></math> such that \n<math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mi>a</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>≤</mo>\n <msub>\n <mrow>\n <mi>b</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>,</mo>\n <mspace></mspace>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℕ</mi>\n </mrow>\n <annotation>$$ {a}_n\\le {b}_n,n\\in \\mathrm{\\mathbb{N}} $$</annotation>\n </semantics></math>, converging to \n<math>\n <semantics>\n <mrow>\n <mi>ln</mi>\n <mn>2</mn>\n </mrow>\n <annotation>$$ \\ln 2 $$</annotation>\n </semantics></math>, and study the location of the limit inside the intervals \n<math>\n <semantics>\n <mrow>\n <mo>[</mo>\n <msub>\n <mrow>\n <mi>a</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>b</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>]</mo>\n </mrow>\n <annotation>$$ \\left[{a}_n,{b}_n\\right] $$</annotation>\n </semantics></math>, for every \n<math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℕ</mi>\n </mrow>\n <annotation>$$ n\\in \\mathrm{\\mathbb{N}} $$</annotation>\n </semantics></math> or for sufficiently large \n<math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation>$$ n $$</annotation>\n </semantics></math>. We also investigate the same problem for the case of two corresponding sequences converging to \n<math>\n <semantics>\n <mrow>\n <mi>ln</mi>\n <mn>3</mn>\n </mrow>\n <annotation>$$ \\ln 3 $$</annotation>\n </semantics></math>. Among other results, we prove some, a bit, unexpected ones. Namely, for each \n<math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo>∈</mo>\n <mo>[</mo>\n <mn>0,1</mn>\n <mo>]</mo>\n </mrow>\n <annotation>$$ \\alpha \\in \\left[0,1\\right] $$</annotation>\n </semantics></math>, we determine the exact index \n<math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mi>n</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>∈</mo>\n <mi>ℕ</mi>\n </mrow>\n <annotation>$$ {n}_0\\in \\mathrm{\\mathbb{N}} $$</annotation>\n </semantics></math> at which the sequence \n<math>\n <semantics>\n <mrow>\n <msubsup>\n <mrow>\n <mi>c</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n <mrow>\n <mi>α</mi>\n </mrow>\n </msubsup>\n </mrow>\n <annotation>$$ {c}_n&amp;amp;#x0005E;{\\alpha } $$</annotation>\n </semantics></math> changes the monotonicity, and we also determine the type of the monotonicity. A number of interesting remarks are also presented.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 3","pages":"2819-2832"},"PeriodicalIF":2.0000,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Convex combinations of some convergent sequences\",\"authors\":\"Stevo Stević\",\"doi\":\"10.1002/mma.10463\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the convex combinations \\n<math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mrow>\\n <mi>c</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <mrow>\\n <mi>α</mi>\\n </mrow>\\n </msubsup>\\n <mo>:</mo>\\n <mo>=</mo>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mi>α</mi>\\n <mo>)</mo>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>+</mo>\\n <mi>α</mi>\\n <msub>\\n <mrow>\\n <mi>b</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>,</mo>\\n <mspace></mspace>\\n <mi>n</mi>\\n <mo>∈</mo>\\n <mi>ℕ</mi>\\n <mo>,</mo>\\n <mspace></mspace>\\n <mi>α</mi>\\n <mo>∈</mo>\\n <mo>[</mo>\\n <mn>0,1</mn>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$$ {c}_n&amp;amp;#x0005E;{\\\\alpha}:&amp;amp;#x0003D; \\\\left(1-\\\\alpha \\\\right){a}_n&amp;amp;#x0002B;\\\\alpha {b}_n,n\\\\in \\\\mathrm{\\\\mathbb{N}},\\\\alpha \\\\in \\\\left[0,1\\\\right] $$</annotation>\\n </semantics></math>, of a pair of sequences of real numbers \\n<math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n <mo>∈</mo>\\n <mi>ℕ</mi>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation>$$ {\\\\left({a}_n\\\\right)}_{n\\\\in \\\\mathrm{\\\\mathbb{N}}} $$</annotation>\\n </semantics></math> and \\n<math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mrow>\\n <mi>b</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n <mo>∈</mo>\\n <mi>ℕ</mi>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation>$$ {\\\\left({b}_n\\\\right)}_{n\\\\in \\\\mathrm{\\\\mathbb{N}}} $$</annotation>\\n </semantics></math> such that \\n<math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>≤</mo>\\n <msub>\\n <mrow>\\n <mi>b</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>,</mo>\\n <mspace></mspace>\\n <mi>n</mi>\\n <mo>∈</mo>\\n <mi>ℕ</mi>\\n </mrow>\\n <annotation>$$ {a}_n\\\\le {b}_n,n\\\\in \\\\mathrm{\\\\mathbb{N}} $$</annotation>\\n </semantics></math>, converging to \\n<math>\\n <semantics>\\n <mrow>\\n <mi>ln</mi>\\n <mn>2</mn>\\n </mrow>\\n <annotation>$$ \\\\ln 2 $$</annotation>\\n </semantics></math>, and study the location of the limit inside the intervals \\n<math>\\n <semantics>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mrow>\\n <mi>a</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mrow>\\n <mi>b</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$$ \\\\left[{a}_n,{b}_n\\\\right] $$</annotation>\\n </semantics></math>, for every \\n<math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>∈</mo>\\n <mi>ℕ</mi>\\n </mrow>\\n <annotation>$$ n\\\\in \\\\mathrm{\\\\mathbb{N}} $$</annotation>\\n </semantics></math> or for sufficiently large \\n<math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation>$$ n $$</annotation>\\n </semantics></math>. We also investigate the same problem for the case of two corresponding sequences converging to \\n<math>\\n <semantics>\\n <mrow>\\n <mi>ln</mi>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$$ \\\\ln 3 $$</annotation>\\n </semantics></math>. Among other results, we prove some, a bit, unexpected ones. Namely, for each \\n<math>\\n <semantics>\\n <mrow>\\n <mi>α</mi>\\n <mo>∈</mo>\\n <mo>[</mo>\\n <mn>0,1</mn>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$$ \\\\alpha \\\\in \\\\left[0,1\\\\right] $$</annotation>\\n </semantics></math>, we determine the exact index \\n<math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <mrow>\\n <mn>0</mn>\\n </mrow>\\n </msub>\\n <mo>∈</mo>\\n <mi>ℕ</mi>\\n </mrow>\\n <annotation>$$ {n}_0\\\\in \\\\mathrm{\\\\mathbb{N}} $$</annotation>\\n </semantics></math> at which the sequence \\n<math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mrow>\\n <mi>c</mi>\\n </mrow>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <mrow>\\n <mi>α</mi>\\n </mrow>\\n </msubsup>\\n </mrow>\\n <annotation>$$ {c}_n&amp;amp;#x0005E;{\\\\alpha } $$</annotation>\\n </semantics></math> changes the monotonicity, and we also determine the type of the monotonicity. 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引用次数: 0

摘要

我们考虑一对实数序列的凸组合，使得，收敛于，并研究在区间内的极限位置，对于每个或对于足够大的。我们还研究了收敛于 , 的两个相应序列的相同问题。在其他结果中，我们证明了一些有点出乎意料的结果。也就是说，对于每个，我们确定了序列改变单调性的确切指数，并且还确定了单调性的类型。我们还提出了一些有趣的评论。

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Convex combinations of some convergent sequences

We consider the convex combinations $c_{n}^{α} : = (1 - α) a_{n} + α b_{n}, n \in ℕ, α \in [0,1]$ , of a pair of sequences of real numbers ${(a_{n})}_{n \in ℕ}$ and ${(b_{n})}_{n \in ℕ}$ such that $a_{n} \leq b_{n}, n \in ℕ$ , converging to $\ln 2$ , and study the location of the limit inside the intervals $[a_{n}, b_{n}]$ , for every $n \in ℕ$ or for sufficiently large $n$ . We also investigate the same problem for the case of two corresponding sequences converging to $\ln 3$ . Among other results, we prove some, a bit, unexpected ones. Namely, for each $α \in [0,1]$ , we determine the exact index $n_{0} \in ℕ$ at which the sequence $c_{n}^{α}$ changes the monotonicity, and we also determine the type of the monotonicity. A number of interesting remarks are also presented.

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.