The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases when the solution of the limit problem has a point of gradient catastrophe. Asymptotic solutions are found by using the Cole–Hopf transform. The integrals determining the asymptotic solutions correspond to the Lagrange singularities of type (A) and the boundary singularities of type (B). The behavior of the asymptotic solutions is described in terms of the weighted Sobolev spaces.
{"title":"Asymptotic Solutions of a Parabolic Equation Near Singular Points of A and B Types","authors":"S. Zakharov","doi":"10.15826/UMJ.2019.1.010","DOIUrl":"https://doi.org/10.15826/UMJ.2019.1.010","url":null,"abstract":"The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases when the solution of the limit problem has a point of gradient catastrophe. Asymptotic solutions are found by using the Cole–Hopf transform. The integrals determining the asymptotic solutions correspond to the Lagrange singularities of type (A) and the boundary singularities of type (B). The behavior of the asymptotic solutions is described in terms of the weighted Sobolev spaces.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43423821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The space of ultrafilters of a (pi)-system endowed with the topology of Wallman type is considered. The question on the supercompactness of this space is investigated. For this, the enveloping space of maximal linked systems with the corresponding topology of Wallman type is used. Necessary and sufficient conditions for the coincidence of the set of all ultrafilters of the initial (pi)-system and the set of all maximal linked systems for this (pi)-system are obtained. Specific variants of wide sense measurable spaces with this coincidence property are given.
{"title":"To a Question on the Supercompactness of Ultrafilter Spaces","authors":"A. Chentsov","doi":"10.15826/UMJ.2019.1.004","DOIUrl":"https://doi.org/10.15826/UMJ.2019.1.004","url":null,"abstract":"The space of ultrafilters of a (pi)-system endowed with the topology of Wallman type is considered. The question on the supercompactness of this space is investigated. For this, the enveloping space of maximal linked systems with the corresponding topology of Wallman type is used. Necessary and sufficient conditions for the coincidence of the set of all ultrafilters of the initial (pi)-system and the set of all maximal linked systems for this (pi)-system are obtained. Specific variants of wide sense measurable spaces with this coincidence property are given.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47342597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A ring (R) is called weakly invo-clean if any its element is the sum or the difference of an involution and an idempotent. For each commutative unital ring (R) and each abelian group (G), we find only in terms of (R), (G) and their sections a necessary and sufficient condition when the group ring (R[G]) is weakly invo-clean. Our established result parallels to that due to Danchev-McGovern published in J. Algebra (2015) and proved for weakly nil-clean rings.
{"title":"Commutative Weakly Invo–Clean Group Rings","authors":"P. Danchev","doi":"10.15826/UMJ.2019.1.005","DOIUrl":"https://doi.org/10.15826/UMJ.2019.1.005","url":null,"abstract":"A ring (R) is called weakly invo-clean if any its element is the sum or the difference of an involution and an idempotent. For each commutative unital ring (R) and each abelian group (G), we find only in terms of (R), (G) and their sections a necessary and sufficient condition when the group ring (R[G]) is weakly invo-clean. Our established result parallels to that due to Danchev-McGovern published in J. Algebra (2015) and proved for weakly nil-clean rings.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46296505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study an optimal control problem in infinite time, where the integrand does not depend explicitly on the state variable. A special case of such problem is the Ramsey optimal capital accumulation in centralized economy. To complete the optimality conditions of Pontryagin's maximum principle, so called transversality conditions of different types are used in the literature. Here, instead of a transversality condition, an additional maximum condition is considered.
{"title":"On Necessary Optimality Conditions for Ramsey-Type Problems","authors":"A. Belyakov","doi":"10.4213/proc22960","DOIUrl":"https://doi.org/10.4213/proc22960","url":null,"abstract":"We study an optimal control problem in infinite time, where the integrand does not depend explicitly on the state variable. A special case of such problem is the Ramsey optimal capital accumulation in centralized economy. To complete the optimality conditions of Pontryagin's maximum principle, so called transversality conditions of different types are used in the literature. Here, instead of a transversality condition, an additional maximum condition is considered.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49012142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we prove the existence and approximation of solutions of the initial value problems of nonlinear hybrid Caputo fractional integro-differential equations. The main tool employed here is the Dhage iteration principle in a partially ordered normed linear space. An example is also given to illustrate the main results.
{"title":"Approximating Solutions of Nonlinear Hybrid Caputo Fractional Integro-Differential Equations via Dhage Iteration Principle","authors":"A. Ardjouni, A. Djoudi","doi":"10.15826/UMJ.2019.1.001","DOIUrl":"https://doi.org/10.15826/UMJ.2019.1.001","url":null,"abstract":"In this article, we prove the existence and approximation of solutions of the initial value problems of nonlinear hybrid Caputo fractional integro-differential equations. The main tool employed here is the Dhage iteration principle in a partially ordered normed linear space. An example is also given to illustrate the main results.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41719144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Neumann boundary value problem (BVP) in a unit circle is discussed. For the solution of the Neumann BVP, we built a method employing series representation of given (2 pi)-periodic continuous boundary function by interpolating wavelets consisting of trigonometric polynomials. It is convenient to use the method due to the fact that such series is easy to extend to harmonic polynomials inside a circle. Moreover, coefficients of the series have an easy-to-calculate form. The representation by the interpolating wavelets is constructed by using an interpolation projection to subspaces of a multiresolution analysis with basis (2 pi)-periodic scaling functions (more exactly, their binary rational compressions and shifts). That functions were developed by Subbotin and Chernykh on the basis of Meyer-type wavelets. We will use three kinds of such functions, where two out of the three generates systems, which are orthogonal and simultaneous interpolating on uniform grids of the corresponding scale and the last one generates only interpolating on the same uniform grids system. As a result, using the interpolation property of wavelets mentioned above, we obtain the exact representation of the solution for the Neumann BVP by series of that wavelets and numerical bound of the approximation of solution by partial sum of such series.
{"title":"Harmonic Interpolating Wavelets in Neumann Boundary Value Problem in a Circle","authors":"Dmitry A. Yamkovoi","doi":"10.15826/UMJ.2019.1.009","DOIUrl":"https://doi.org/10.15826/UMJ.2019.1.009","url":null,"abstract":"The Neumann boundary value problem (BVP) in a unit circle is discussed. For the solution of the Neumann BVP, we built a method employing series representation of given (2 pi)-periodic continuous boundary function by interpolating wavelets consisting of trigonometric polynomials. It is convenient to use the method due to the fact that such series is easy to extend to harmonic polynomials inside a circle. Moreover, coefficients of the series have an easy-to-calculate form. The representation by the interpolating wavelets is constructed by using an interpolation projection to subspaces of a multiresolution analysis with basis (2 pi)-periodic scaling functions (more exactly, their binary rational compressions and shifts). That functions were developed by Subbotin and Chernykh on the basis of Meyer-type wavelets. We will use three kinds of such functions, where two out of the three generates systems, which are orthogonal and simultaneous interpolating on uniform grids of the corresponding scale and the last one generates only interpolating on the same uniform grids system. As a result, using the interpolation property of wavelets mentioned above, we obtain the exact representation of the solution for the Neumann BVP by series of that wavelets and numerical bound of the approximation of solution by partial sum of such series.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44547258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An asymptotic model of an arterial bifurcation is presented. We propose a simple approximate method of calculation of the pressure drop matrix. The entries of this matrix are included in the modified transmission conditions, which were introduced earlier by Kozlov and Nazarov, and which give better approximation of 3D flow by 1D flow near a bifurcation of an artery as compared to the classical Kirchhoff conditions. The present modeling takes into account the heuristic Murrey cubic law.
{"title":"A Mathematical Model of an Arterial Bifurcation","authors":"G. Zavorokhin","doi":"10.15826/UMJ.2019.1.011","DOIUrl":"https://doi.org/10.15826/UMJ.2019.1.011","url":null,"abstract":"An asymptotic model of an arterial bifurcation is presented. We propose a simple approximate method of calculation of the pressure drop matrix. The entries of this matrix are included in the modified transmission conditions, which were introduced earlier by Kozlov and Nazarov, and which give better approximation of 3D flow by 1D flow near a bifurcation of an artery as compared to the classical Kirchhoff conditions. The present modeling takes into account the heuristic Murrey cubic law.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43962702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap., 14, no.1 (1985), 38–42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit.
{"title":"IDENTITIES IN BRANDT SEMIGROUPS, REVISITED","authors":"Mikhail Volkov","doi":"10.15826/umj.2019.2.008","DOIUrl":"https://doi.org/10.15826/umj.2019.2.008","url":null,"abstract":"We present a new proof for the main claim made in the author's paper \"On the identity bases of Brandt semigroups\" (Ural. Gos. Univ. Mat. Zap., 14, no.1 (1985), 38–42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43028998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The time-optimal problem for the controllable equation of heat conductivity in a rod is considered. By means of the Fourier expansion, the problem reduced to a countable system of one-dimensional control systems with a combined constraint joining control parameters in one relation. In order to improve the time of a suboptimal control constructed by F.L. Chernous’ko, a method of grouping coupled terms of the Fourier expansion of a control function is applied, and a synthesis of the improved suboptimal control is obtained in an explicit form.
{"title":"On the Chernous'ko Time-Optimal Problem for the Equation of Heat Conductivity in a Rod","authors":"A. Azamov, Jasurbek A. Bakhramov, O. S. Akhmedov","doi":"10.15826/UMJ.2019.1.002","DOIUrl":"https://doi.org/10.15826/UMJ.2019.1.002","url":null,"abstract":"The time-optimal problem for the controllable equation of heat conductivity in a rod is considered. By means of the Fourier expansion, the problem reduced to a countable system of one-dimensional control systems with a combined constraint joining control parameters in one relation. In order to improve the time of a suboptimal control constructed by F.L. Chernous’ko, a method of grouping coupled terms of the Fourier expansion of a control function is applied, and a synthesis of the improved suboptimal control is obtained in an explicit form.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67264332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cavalcanti, Fatori, and Ma, Attractors for wave equations with degenerate memory, J. Differential Equations (2016). Under some additional assumptions, we show the existence of a compact absorbing set. This result provides further regularity for the global attractor and shows that it consists of regular solutions.
{"title":"Regular Global Attractors for Wave Equations with Degenerate Memory","authors":"J. Shomberg","doi":"10.15826/UMJ.2019.1.007","DOIUrl":"https://doi.org/10.15826/UMJ.2019.1.007","url":null,"abstract":"We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cavalcanti, Fatori, and Ma, Attractors for wave equations with degenerate memory, J. Differential Equations (2016). Under some additional assumptions, we show the existence of a compact absorbing set. This result provides further regularity for the global attractor and shows that it consists of regular solutions.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67264467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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