Armenian Journal of Mathematics
https://armjmath.sci.am/index.php/ajm
<p><strong>Editor in Chief - <a href="http://math.sci.am/user/anry-nersesyan">Anry Nersessian</a></strong> (Institute of Mathematics NAS, Armenia)</p> <p><strong> Deputy Editor - <a href="http://math.sci.am/user/rafayel-barkhudaryan">Rafayel Barkhudaryan</a></strong> (Institute of Mathematics NAS, Armenia)<strong><br /></strong></p> <p><strong>Managing Editor - <a href="http://math.sci.am/user/linda-khachatryan">Linda Khachatryan</a></strong> (Institute of Mathematics NAS, Armenia)<br />e-mail: <a href="mailto:ajm@instmath.sci.am">ajm@instmath.sci.am </a></p>National Academy of Sciences of ArmeniaenArmenian Journal of Mathematics1829-1163Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$
https://armjmath.sci.am/index.php/ajm/article/view/1014
<p>In this paper, we consider the general linear group $GL(2, 7)$ of $2 \times 2$ invertible matrices over the finite field of order $7$ and compute the unit group of the semisimple group algebra $\mathbb{F}_qGL(2,7)$, where $\mathbb{F}_q$ is a finite field. For the computation of the unit group, we need the Wedderburn decomposition of $\mathbb{F}_qGL(2,7)$, which is determined by first computing the Wedderburn decomposition of the group algebra $\mathbb{F}_q(PSL(3, 2)\rtimes C_2)$. Here $PSL(3,2)$ is the projective special linear group of degree 3 over a finite field of 2 elements.</p>
ArticlesUnit GroupGroup AlgebraGeneral Linear GroupWedderburn Decomposition16U6020C05Namatchivayam Umapathy SivaranjaniElumalai NandakumarGaurav MittalRajendra Kumar Sharma
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2024-03-132024-03-1311410.52737/18291163-2024.16.3-1-14Characterization of the Three-Variate Inverted Dirichlet Distributions
https://armjmath.sci.am/index.php/ajm/article/view/1061
<p>In this paper, we prove a characterization of three-variate inverted Dirichlet distributions by an independence property. The main technical challenge was a problem involving the solution of a related functional equation.</p>
ArticlesCharacterization of Probability DistributionsFunctional EquationIndependenceInverted Dirichlet DistributionTransformation62E1062H05Mohamed Ben Farah
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2023-12-142023-12-141910.52737/18291163-2023.15.12-1-9A Combinatorial Interpretation of the Padovan Generalized Polynomial Sequence
https://armjmath.sci.am/index.php/ajm/article/view/948
<p>We investigate a combinatorial interpretation of the Padovan polynomial sequence, also addressing its polynomial extensions. We thus include the Tridovan polynomial sequence, Tetradovan polynomial sequences, leading up to the Z-dovan polynomial generalization.</p>
ArticlesCombinatoricsGeneralizationPadovan Polynomial Sequence11B3711B39Renata Passos Machado VieiraFrancisco Regis Vieira AlvesPaula Maria Machado Cruz Catarino
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2023-11-282023-11-281910.52737/18291163-2023.15.11-1-9The Geometry of the Projective Action of $\text{SL}(3,\mathbb{R})$ from the Erlangen Perspective
https://armjmath.sci.am/index.php/ajm/article/view/908
<p>In this paper, we have investigated the projective action of the Lie group $\text{SL}(3,\mathbb{R})$ on the homogeneous space $\mathbb{RP}^2$. In particular, we have studied the action of the subgroups of $\text{SL}(3,\mathbb{R})$ on the non-degenerate conics in the space $\mathbb{RP}^2$. Using the Iwasawa decomposition of $\text{SL}(2,\mathbb{R})$, we demonstrate that the isotropy subgroup of the projective unit circle is isomorphic to $\text{PSL}(2,\mathbb{R})$ under certain conditions.</p>
ArticlesLie Group SL(3,R)Homogeneous SpaceConicsExponential MapIwasawa Decomposition57S2057S2551A0551H2022F30Debapriya BiswasIpsita Rajwar
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2024-01-112024-01-1112810.52737/18291163-2024.16.1-1-28On the Links between Miura Transformations of Bogoyavlensky Lattices and Inverse Spectral Problems for Band Operators
https://armjmath.sci.am/index.php/ajm/article/view/931
<p>We consider semi-infinite and finite Bogoyavlensky lattices<br />$$<br />\overset\cdot a_i =a_i\left(\prod_{j=1}^{p}a_{i+j}-\prod_{j=1}^{p}a_{i-j}\right),<br />$$<br />$$<br />\overset\cdot b_i = b_i\left(\sum_{j=1}^{p}<br />b_{i+j}-\sum_{j=1}^{p}b_{i-j}\right),<br />$$<br />for some $p\ge 1$, and Miura-like transformations between these systems, defined for $p\ge 2$. Both lattices are integrable (via Lax pair formalism) by the inverse spectral problem method for band operators, i.e., operators generated by band matrices. The key role in this method is played by the moments of the Weyl matrix of the corresponding band operator and their evolution in time. We find a description of the above-mentioned transformations in terms of these moments and apply this result to study finite Bogoyavlensky lattices and, in particular, their first integrals.</p>
ArticlesDifference OperatorsInverse Spectral ProblemsNonlinear LatticesMiura Transformations47B3637K1037K15Andrey Osipov
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2024-02-132024-02-1312810.52737/18291163-2024.16.2-1-28On the Distance Spectrum and Distance-Based Topological Indices of Central Vertex-Edge Join of Three Graphs
https://armjmath.sci.am/index.php/ajm/article/view/946
<p>In this paper, we introduce a new graph operation based on a central graph called central vertex-edge join (denoted by $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$) and then determine the distance spectrum of $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$ in terms of the adjacency spectra of regular graphs $G_1$, $G_2$ and $G_3$ when $G_1$ is triangle-free. As a consequence of this result, we construct new families of non-D-cospectral D-equienergetic graphs. Moreover, we determine bounds for the distance spectral radius and distance energy of the central vertex-edge join of three regular graphs. In addition, we provide its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, Wiener index. Using these results, we calculate the topological indices of the organic compounds Methylcyclobutane $(C_5H_{10})$ and Spirohexane $(C_6H_{10})$.</p>
ArticlesDistance MatrixDistance EigenvaluesDistance Equienergetic GraphsTopological Indices05C5005C7605C12T. HarithaA.V. Chithra
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2023-10-102023-10-1011610.52737/18291163-2023.15.10-1-16A Mesh-Free Algorithm to Solve an Inverse Source Problem for Degenerate Two-Dimensional Parabolic Equation from Final Observations
https://armjmath.sci.am/index.php/ajm/article/view/826
<p>The main purpose of this work is to propose a new network architecture model for deep learning applied to solve an inverse source problem for a two-dimensional degenerate parabolic equation from final observations with degeneracy occurring anywhere in the spatial domain.</p>
ArticlesDeep LearningHeat EquationInverse ProblemDegenerate Two-Dimensional EquationOptimization15A2947A5234A3893C2060J7035K0535K65Khalid Atifi
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2023-05-202023-05-2011110.52737/18291163-2023.15.8-1-11Nonlocal Solvability of the Cauchy Problem for a System with Negative Functions of the Variable $t$
https://armjmath.sci.am/index.php/ajm/article/view/793
<p>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 $t$.</p>
ArticlesFirst-Order Partial Differential EquationsCauchy ProblemAdditional Argument MethodGlobal Estimates35F5035F5535A01Marina Dontsova
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2023-03-222023-03-2211010.52737/18291163-2023.15.4-1-10Optimality of the Least Sum of Logarithms in the Problem of Matching Map Recovery in the Presence of Noise and Outliers
https://armjmath.sci.am/index.php/ajm/article/view/900
<p>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.</p>
ArticlesMultidimensional StatisticsVector MatchingSeparation Rate62H99Tigran GalstyanArshak Minasyan
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2023-03-302023-03-301910.52737/18291163-2023.15.5-1-9A Note on Location of the Zeros of Quaternionic Polynomials
https://armjmath.sci.am/index.php/ajm/article/view/854
<p>The purpose of this paper is to investigate the extensions of the classical Eneström-Kakeya theorem and its various generalizations concerning the distribution of zeros of polynomials from the complex to the quaternionic setting. Using a maximum modulus theorem and the zero set structure in the recently published theory of regular functions and polynomials of a quaternionic variable, we construct new bounds of the Eneström-Kakeya type for the zeros of these polynomials. The obtained results for this subclass of polynomials and slice regular functions give generalizations of a number of results previously reported in the relevant literature.</p>
ArticlesQuaternionic PolynomialZerosEneström-Kakeya theorem30E1030G3516K20Irfan Ahmad WaniAdil Hussain
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2023-04-272023-04-2711210.52737/10.52737/18291163-2023.15.7-1-12A New Family of Number Sequences: Leonardo-Alwyn Numbers
https://armjmath.sci.am/index.php/ajm/article/view/763
<p>In this study, we define a new type of number sequence called Leonardo-Alwyn sequence. We obtain the Binet formula, generating function and some relations for these numbers. Moreover, we give the matrix representation of the Leonardo-Alwyn numbers.</p>
ArticlesLeonardo NumberLeonardo-Alwyn NumberJohn-Edouard NumberErnst Number11B3711B8315B36Hasan Gökbaş
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2023-04-132023-04-1311310.52737/18291163-2023.15.6-1-13On Non-Comaximal Graphs of Ideals of Commutative Rings
https://armjmath.sci.am/index.php/ajm/article/view/762
<p>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.</p>
ArticlesArtinian RingNon-Comaximal GraphMinimal Ideal05C2516D1013C99Bikash BarmanKukil Kalpa Rajkhowa
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2023-02-082023-02-081810.52737/18291163-2023.15.2-1-8Generalized Rational Evaluation Subgroups of the Inclusion between Complex Projective Spaces
https://armjmath.sci.am/index.php/ajm/article/view/851
<p>We use a model of mapping spaces to compute the generalized rational Gottlieb groups of the inclusion $i_{n,k}: \mathbb{C}P^n \hookrightarrow \mathbb{C}P^{n+k}$ between complex projective spaces.</p>
ArticlesMapping Space$L_{\infty}$ AlgebraGottlieb Groups55P6254C35Jean-Baptiste Gatsinzi
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2023-09-052023-09-051610.52737/18291163-2023.15.9-1-6Acceleration of Convergence of Fourier Series Using the Phenomenon of Over-Convergence
https://armjmath.sci.am/index.php/ajm/article/view/803
<p>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.</p>
ArticlesFourier SeriesAcceleration of ConvergenceParametric BiorthogonalizationSpectral MethodsOver-Convergence Phenomenon42A1642A2042A2434B05Anry Nersessian
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2022-12-122022-12-1213110.52737/18291163-2022.14.14-1-31Existence of Solutions for a Fractional Boundary Value Problem at Resonance
https://armjmath.sci.am/index.php/ajm/article/view/839
<p>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.</p>
ArticlesFractional Differential EquationsBoundary Value ProblemCaputo DerivativeCoincidence Degree TheoryResonance34A0834B1526A3347H11Anabela S. Silva
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2022-12-212022-12-2111610.52737/18291163-2022.14.15-1-16Weak Type Estimate of Singular Integral Operators on Variable Weak Herz-Type Hardy Spaces
https://armjmath.sci.am/index.php/ajm/article/view/694
<p>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.</p>
ArticlesHerz-Type Hardy SpaceWeak Herz SpacesWeak Herz-Type Hardy SpacesAtomVariable ExponentSingular Intergral Operators42B2042B3546E30Hamza Brahim BoularesDouadi DrihemWafa Hebbache
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2023-03-102023-03-1013310.52737/18291163-2023.15.3-1-33On the Minimal Annulus of Triangles and Parallelograms
https://armjmath.sci.am/index.php/ajm/article/view/817
<p>Sharp upper and lower bounds for the isoperimetric deficit of triangles or parallelograms with the minimal annulus of radii $R$ and $r$ are given.</p>
ArticlesIsoperimetric InequalityMinimal AnnulusBonnesen InequalityFavard Inequality52B6051M2552A1052A4052A38Salvatore Vassallo
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2023-01-262023-01-2611510.52737/18291163-2023.15.1-1-15Bifurcation Analysis of a Piecewise Smooth Map with Two Asymptotes
https://armjmath.sci.am/index.php/ajm/article/view/632
<p>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.</p>
ArticlesDiscontinuous SystemDivergenceFlip BifurcationPeriodic Orbit37C8337G1537N99Roya Makrooni
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2022-10-022022-10-0211210.52737/18291163-2022.14.13-1-12Hypergeometric connections between balancing polynomials and Chebyshev polynomials of first and second kinds
https://armjmath.sci.am/index.php/ajm/article/view/626
<p>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.</p>
ArticlesBalancing polynomialsChebyshev polynomials of the first kindChebyshev polynomials of the second kindHypergeometric functions11B3711B3933C0533C45Adikanda BeheraPrasanta Kumar Ray
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2022-09-232022-09-2312010.52737/18291163-2022.14.12-1-20Geometry associated with the $\text{SL}(3,\mathbb{R})$ action on homogeneous space using the Erlangen program
https://armjmath.sci.am/index.php/ajm/article/view/659
<p>We investigate the action of the Lie group $\text{SL}(3,\mathbb{R})$ on the two-dimensional homogeneous space. All the one-parameter subgroups (up to conjugacy) of $\text{SL}(3,\mathbb{R})$ are considered. We discuss the orbits and curvatures of these one-parameter subgroups. We also classify these subgroups in terms of fixed points.</p>
ArticlesLie Group $\text{SL}(3,\mathbb{R})$Homogeneous SpaceIwasawa DecompositionOne-Parameter SubgroupsGroup ActionDerived RepresentationOrbitCurvatureFixed Point57S2057S2551A0522F30Debapriya BiswasIpsita Rajwar
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2022-08-162022-08-1611510.52737/18291163-2022.14.11-1-15Generalization of an Eneström-Kakeya type theorem to the quaternions
https://armjmath.sci.am/index.php/ajm/article/view/651
<p>The well-known Eneström-Kakeya theorem states that polynomial $p(z)=\sum_{\nu =0}^n a_\nu z^\nu$, where $0\leq a_0\leq a_1\leq \cdots\leq a_n$, has all of its (complex) zeros in $|z|\leq 1$. Many generalizations of this result exist in the literature. In this paper, we extend one such result to the quaternionic setting and state one of the possible corollaries.</p>
ArticlesLocation of Zeros of a PolynomialEneström-Kakeya TheoremQuaternionic Polynomial30E1016K20Robert B. GardnerMariah Taylor
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2022-06-292022-06-291810.52737/18291163-2022.14.9-1-8Groups whose derived subgroup is not supplemented by any proper subgroup
https://armjmath.sci.am/index.php/ajm/article/view/718
<p>In this paper, we introduce two new classes of groups that are described as weakly nilpotent and weakly solvable groups. A group $G$ is weakly nilpotent if its derived subgroup does not have a supplement except $G$ and a group $G$ is weakly solvable if its derived subgroup does not have a normal supplement except $G$. We present some examples and counter-examples for these groups and characterize a finitely generated weakly nilpotent group. Moreover, we characterize the nilpotent and solvable groups in terms of weakly nilpotent and weakly solvable groups. Finally, we prove that if $F$ is a free group of rank $n$ such that every normal subgroup of $F$ has rank $n$, then $F$ is weakly solvable.</p>
ArticlesDerived subgroupSupplementFrat(G)nFrat(G)Weakly nilpotent groupsWeakly solvable groupsFree groups20F1420F19Shiv NarainSunil KumarGaurav MittalSandeep Kumar
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2022-07-082022-07-0811310.52737/18291163-2022.14.10-1-13On the invertibility of one integral operator
https://armjmath.sci.am/index.php/ajm/article/view/715
<p>The present paper considers an integral operator defined on the entire real axis, which differs from the Hilbert transform with terms where kernels are constructed using integral exponential functions. The considered operator has similar properties with respect to the Hilbert transform. The form of the inverse operator is obtained.</p>
ArticlesIntegral operatorexponential integral function$\mathcal{L}$-Wiener-Hopf operator47G1047B35Grigor Kirakosyan
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2022-04-302022-04-3011010.52737/18291163-2022.14.6-1-10Description of random fields by systems of conditional distributions
https://armjmath.sci.am/index.php/ajm/article/view/666
<p>In this paper, we consider the direct and inverse problems of the description of lattice positive random fields by various systems of finite-dimensional (as well as one-point) probability distributions parameterized by boundary conditions. In the majority of cases, we provide necessary and sufficient conditions for the system to be a conditional distribution of a (unique) random field. The exception is Dobrushin-type systems for which only sufficient conditions are known. Also, we discuss possible applications of the considered systems.</p>
ArticlesRandom fieldconditional distributionspecificationMarkov random field60G6060E0560J99Linda Khachatryan
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2022-06-032022-06-0314010.52737/18291163-2022.14.8-1-40A generalization of connectedness via ideals
https://armjmath.sci.am/index.php/ajm/article/view/628
<p>In this paper, we define and study the $\diamond$-connected spaces as a generalization of the connectedness, and thus of the Ekici-Noiri and Modak-Noiri notions, through ideals.</p>
ArticlesConnectednessideal topological spaces54A1054A0554A20Raúl Pachón
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2022-05-132022-05-1311810.52737/18291163-2022.14.7-1-18Evaluation subgroups of a map and the rationalized $G$-sequence
https://armjmath.sci.am/index.php/ajm/article/view/576
<p>In this paper, we determine, in terms of the Sullivan models, rational evaluation subgroups of the inclusion $ \mathbb{C} P(n)\hookrightarrow \mathbb{C} P(n+k) $ between complex projective spaces and, more generally, the $ G $-sequence of the homotopy monomorphism $ \iota: X\hookrightarrow Y $ between simply connected formal homogeneous spaces for which $ \pi_{\ast}(Y)\otimes \mathbb{Q}$ is finite-dimensional.</p> <p> </p> <p><strong>Editorial Board's note.</strong><em> We inform our readers that J.-B. Gatsinzi mentions in his work, published in Armen. J. Math. <a href="https://doi.org/10.52737/18291163-2023.15.9-1-6">vol. 15, no. 9, 2023</a>, that O. Maphane's paper contains a mistake. According to Gatsinzi, his Corollary 1 corrects Theorem 2.2 of the current paper. We contacted Maphane on this issue, and he agreed with Gatzinzi's statement.</em></p>
ArticlesEvaluation subgroupsGottlieb group$G$-sequence55P6254C35Oteng Maphane
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2023-09-162023-09-1611010.52737/18291163-2022.14.2-1-10Exponential decay for a strain gradient porous thermoelasticity with second sound
https://armjmath.sci.am/index.php/ajm/article/view/525
<p>In this paper, we consider a strain gradient porous elastic bar subjected to a thermal disturbance modelled by Cattaneo's law for heat conduction. We use the semigroup approach to prove the existence of a unique weak solution. Although the thermal dissipation induced by the second sound thermoelasticity is weaker than that caused by the classical heat conduction, we prove that the solution decays exponentially.</p>
Articlesstrain gradientthermoelasticitysecond soundexponential decay35B4035G4635Q7474F0593D20Afaf AhmimaAbdelfeteh Fareh
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2022-03-032022-03-0312310.52737/18291163-2022.14.3-1-23Annular region containing all the zeros of lacunary-type polynomials
https://armjmath.sci.am/index.php/ajm/article/view/611
<p>In this paper, we find the annular region containing all the zeros of lacunary-type polynomials, whose coefficients are subjected to certain restrictions.</p>
ArticlesLacunary-type polynomialEneström-Kakeya theorem30A0130C1030C15Ashish KumarZahid ManzoorBashir Ahmad Zargar
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2022-03-042022-03-041910.52737/18291163-2022.14.4-1-9Controlled generalized fusion frame in the tensor product of Hilbert spaces
https://armjmath.sci.am/index.php/ajm/article/view/592
<p>We present controlled by operators generalized fusion frame in the tensor product of Hilbert spaces and discuss some of its properties. We also describe the frame operator for a pair of controlled $g$-fusion Bessel sequences in the tensor product of Hilbert spaces.</p>
ArticlesPrasenjit GhoshTapas Kumar Samanta
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2021-12-282021-12-2811810.52737/18291163-2021.13.13-1-18Integral representation of one class of entire functions
https://armjmath.sci.am/index.php/ajm/article/view/596
<p>In this paper, we study an integral representation of one class of entire functions. Conditions for the existence of this representation in terms of certain solutions of some differential equations are found. We obtain asymptotic estimates of entire functions from the considered class of functions. We also give examples of entire functions from this class.</p>
ArticlesPaley-Wiener theorementire function of exponential typeOrdinary Differential EquationsSchwarz inequalityasymptotic estimate30D1030D2030E1030E1542A1042B1044A0544A15Ruslan Khats'
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2022-02-102022-02-101910.52737/18291163-2022.14.1-1-9