Pub Date : 2023-03-30DOI: 10.52737/18291163-2023.15.5-1-9
T. Galstyan, A. Minasyan
We consider the problem of estimating the matching map between two sets of feature-vectors observed in a noisy environment and contaminated by outliers. It was already known in the literature that in the outlier-free setting, the least sum of squares (LSS) and the least sum of logarithms (LSL) are both minimax-rate-optimal. It has been recently proved that the optimality properties of the LSS continue to hold in the case the data sets contain outliers. In this work, we show that the same is true for the LSL as well. Therefore, LSL has the same desirable properties as the LSS, and, in addition, it is minimax-rate-optimal in the outlier-free setting with heteroscedastic noise.
{"title":"Optimality of the Least Sum of Logarithms in the Problem of Matching Map Recovery in the Presence of Noise and Outliers","authors":"T. Galstyan, A. Minasyan","doi":"10.52737/18291163-2023.15.5-1-9","DOIUrl":"https://doi.org/10.52737/18291163-2023.15.5-1-9","url":null,"abstract":"We consider the problem of estimating the matching map between two sets of feature-vectors observed in a noisy environment and contaminated by outliers. It was already known in the literature that in the outlier-free setting, the least sum of squares (LSS) and the least sum of logarithms (LSL) are both minimax-rate-optimal. It has been recently proved that the optimality properties of the LSS continue to hold in the case the data sets contain outliers. In this work, we show that the same is true for the LSL as well. Therefore, LSL has the same desirable properties as the LSS, and, in addition, it is minimax-rate-optimal in the outlier-free setting with heteroscedastic noise.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47585131","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}
Pub Date : 2023-03-22DOI: 10.52737/18291163-2023.15.4-1-10
M. Dontsova
We obtain sufficient conditions for the existence and uniqueness of a local solution of the Cauchy problem for a quasilinear system with negative functions of the variable $t$ and show that the solution has the same $x$-smoothness as the initial function. We also obtain sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem for a quasilinear system with negative functions of the variable $t$.
{"title":"Nonlocal Solvability of the Cauchy Problem for a System with Negative Functions of the Variable $t$","authors":"M. Dontsova","doi":"10.52737/18291163-2023.15.4-1-10","DOIUrl":"https://doi.org/10.52737/18291163-2023.15.4-1-10","url":null,"abstract":"We obtain sufficient conditions for the existence and uniqueness of a local solution of the Cauchy problem for a quasilinear system with negative functions of the variable $t$ and show that the solution has the same $x$-smoothness as the initial function. We also obtain sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem for a quasilinear system with negative functions of the variable $t$.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49620976","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}
Pub Date : 2023-03-10DOI: 10.52737/18291163-2023.15.3-1-33
Hamza Brahim Boulares, D. Drihem, Wafa Hebbache
This paper is concerned with the boundedness properties of singular integral operators on variable weak Herz spaces and variable weak Herz-type Hardy spaces. Allowing our parameters to vary from point to point will raise extra difficulties, which, in general, are overcome by imposing regularity assumptions on these exponents, either at the origin or at infinity. Our results cover the classical results on weak Herz-type Hardy spaces with fixed exponents.
{"title":"Weak Type Estimate of Singular Integral Operators on Variable Weak Herz-Type Hardy Spaces","authors":"Hamza Brahim Boulares, D. Drihem, Wafa Hebbache","doi":"10.52737/18291163-2023.15.3-1-33","DOIUrl":"https://doi.org/10.52737/18291163-2023.15.3-1-33","url":null,"abstract":"This paper is concerned with the boundedness properties of singular integral operators on variable weak Herz spaces and variable weak Herz-type Hardy spaces. Allowing our parameters to vary from point to point will raise extra difficulties, which, in general, are overcome by imposing regularity assumptions on these exponents, either at the origin or at infinity. Our results cover the classical results on weak Herz-type Hardy spaces with fixed exponents.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44705523","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}
Pub Date : 2023-02-08DOI: 10.52737/18291163-2023.15.2-1-8
Bikash Barman, Kukil Kalpa Rajkhowa
In this paper, we relate some properties of non-comaximal graph of ideals of a commutative ring with identity with the properties of the ring.
本文将具有恒等交换环的非极大理想图的一些性质与环的性质联系起来。
{"title":"On Non-Comaximal Graphs of Ideals of Commutative Rings","authors":"Bikash Barman, Kukil Kalpa Rajkhowa","doi":"10.52737/18291163-2023.15.2-1-8","DOIUrl":"https://doi.org/10.52737/18291163-2023.15.2-1-8","url":null,"abstract":"In this paper, we relate some properties of non-comaximal graph of ideals of a commutative ring with identity with the properties of the ring.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48971317","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}
Pub Date : 2023-01-26DOI: 10.52737/18291163-2023.15.1-1-15
S. Vassallo
Sharp upper and lower bounds for the isoperimetric deficit of triangles or parallelograms with the minimal annulus of radii R and r are given.
给出了半径为R和R的最小环空三角形或平行四边形等周亏缺的尖锐上界和下界。
{"title":"On the Minimal Annulus of Triangles and Parallelograms","authors":"S. Vassallo","doi":"10.52737/18291163-2023.15.1-1-15","DOIUrl":"https://doi.org/10.52737/18291163-2023.15.1-1-15","url":null,"abstract":"Sharp upper and lower bounds for the isoperimetric deficit of triangles or parallelograms with the minimal annulus of radii R and r are given.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47248030","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}
Pub Date : 2022-12-21DOI: 10.52737/18291163-2022.14.15-1-16
A. S. Silva
In this paper, we focus on the existence of solutions to a fractional boundary value problem at resonance. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin.
{"title":"Existence of Solutions for a Fractional Boundary Value Problem at Resonance","authors":"A. S. Silva","doi":"10.52737/18291163-2022.14.15-1-16","DOIUrl":"https://doi.org/10.52737/18291163-2022.14.15-1-16","url":null,"abstract":"In this paper, we focus on the existence of solutions to a fractional boundary value problem at resonance. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48530477","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}
Pub Date : 2022-12-12DOI: 10.52737/18291163-2022.14.14-1-31
A. Nersessian
In recent publications of the author, the phenomenon of over-convergence was discovered, and a spectral method has been presented for accelerating the convergence of truncated Fourier series for smooth functions. On this basis, a certain parametric system that is biorthogonal to the corresponding segment of the Fourier system turned out to be unusually effective. This article reconsiders some approaches and makes some adjustments to previous publications. As a result, two improved schemes for the recovery of a function based on a finite set of its Fourier coefficients are proposed. Numerical experiments confirm a significant increase in the efficiency of corresponding algorithms in typical classes of smooth functions. In conclusion, some prospects for the development and generalization of the above approaches are discussed.
{"title":"Acceleration of Convergence of Fourier Series Using the Phenomenon of Over-Convergence","authors":"A. Nersessian","doi":"10.52737/18291163-2022.14.14-1-31","DOIUrl":"https://doi.org/10.52737/18291163-2022.14.14-1-31","url":null,"abstract":"In recent publications of the author, the phenomenon of over-convergence was discovered, and a spectral method has been presented for accelerating the convergence of truncated Fourier series for smooth functions. On this basis, a certain parametric system that is biorthogonal to the corresponding segment of the Fourier system turned out to be unusually effective. This article reconsiders some approaches and makes some adjustments to previous publications. As a result, two improved schemes for the recovery of a function based on a finite set of its Fourier coefficients are proposed. Numerical experiments confirm a significant increase in the efficiency of corresponding algorithms in typical classes of smooth functions. In conclusion, some prospects for the development and generalization of the above approaches are discussed.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48232410","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}
Pub Date : 2022-10-02DOI: 10.52737/18291163-2022.14.13-1-12
Roya Makrooni
In this paper, we consider a discontinuous piecewise smooth system involving four parameters and two asymptotes, recently introduced as a model in engineering sciences. We classify and investigate its bifurcation behaviour. A local bifurcation analysis of the system in the range of parameters which has not been studied so far is undertaken and then supported by numerical computations. This reveals the existence of a flip bifurcation depends on the power singularity. Moreover, we state that a set of positive measure of points with divergent dynamic behaviour exists.
{"title":"Bifurcation Analysis of a Piecewise Smooth Map with Two Asymptotes","authors":"Roya Makrooni","doi":"10.52737/18291163-2022.14.13-1-12","DOIUrl":"https://doi.org/10.52737/18291163-2022.14.13-1-12","url":null,"abstract":"In this paper, we consider a discontinuous piecewise smooth system involving four parameters and two asymptotes, recently introduced as a model in engineering sciences. We classify and investigate its bifurcation behaviour. A local bifurcation analysis of the system in the range of parameters which has not been studied so far is undertaken and then supported by numerical computations. This reveals the existence of a flip bifurcation depends on the power singularity. Moreover, we state that a set of positive measure of points with divergent dynamic behaviour exists.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49217590","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}
Pub Date : 2022-09-23DOI: 10.52737/18291163-2022.14.12-1-20
A. Behera, P. Ray
In the present study, we find several connections between balancing polynomials and the Chebyshev polynomials of the first and second kinds. The Chebyshev polynomials of the first and second kinds are expressed as the sum of two terms of balancing polynomials with hypergeometric coefficients. As an inversion, the balancing polynomials are also expressed as the sum of two terms of the Chebyshev polynomials of the first kind and the Chebyshev polynomials of the second kind with hypergeometric coefficients.
{"title":"Hypergeometric connections between balancing polynomials and Chebyshev polynomials of first and second kinds","authors":"A. Behera, P. Ray","doi":"10.52737/18291163-2022.14.12-1-20","DOIUrl":"https://doi.org/10.52737/18291163-2022.14.12-1-20","url":null,"abstract":"In the present study, we find several connections between balancing polynomials and the Chebyshev polynomials of the first and second kinds. The Chebyshev polynomials of the first and second kinds are expressed as the sum of two terms of balancing polynomials with hypergeometric coefficients. As an inversion, the balancing polynomials are also expressed as the sum of two terms of the Chebyshev polynomials of the first kind and the Chebyshev polynomials of the second kind with hypergeometric coefficients.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44813588","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}
Pub Date : 2022-08-16DOI: 10.52737/18291163-2022.14.11-1-15
D. Biswas, Ipsita Rajwar
We investigate the action of the Lie group SL(3,R) on the two-dimensional homogeneous space. All the one-parameter subgroups (up to conjugacy) of SL(3,R) are considered. We discuss the orbits and curvatures of these one-parameter subgroups. We also classify these subgroups in terms of fixed points.
{"title":"Geometry associated with the SL(3,R) action on homogeneous space using the Erlangen program","authors":"D. Biswas, Ipsita Rajwar","doi":"10.52737/18291163-2022.14.11-1-15","DOIUrl":"https://doi.org/10.52737/18291163-2022.14.11-1-15","url":null,"abstract":"We investigate the action of the Lie group SL(3,R) on the two-dimensional homogeneous space. All the one-parameter subgroups (up to conjugacy) of SL(3,R) are considered. We discuss the orbits and curvatures of these one-parameter subgroups. We also classify these subgroups in terms of fixed points.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41313251","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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