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Abstract: The concept of Hopfian modules has been extensively studied in the literature. In this paper, we introduce and study the notion of \(\delta \) -weakly Hopfian modules. The class of \(\delta \) -weakly Hopfian modules lies properly between the class of Hopfian modules and the class of weakly Hopfian modules. It is shown that over a ring R, every quasi-projective R-module is \(\delta \) -weakly Hopfian iff \(\delta (R)=J(R)\) . We prove that any weak duo module, with zero radical is \(\delta \) -weakly Hopfian. Let M be a module such that M satisfies ascending chain conditions on \(\delta \) -small submodules. Then it is shown that M is \(\delta \) -weakly Hopfian. Some other properties of \(\delta \) -weakly Hopfian modules are also obtained with examples. PubDate: 2022-01-08

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Abstract: A model for thermal convection with generalized friction is investigated. It is shown that the linear instability threshold is the same as the global stability one. In addition, decay of the energy in the \(L^2\) norm is shown for the perturbation velocity and temperature fields. However, due to the presence of the generalized friction we establish exponential decay in the \(L^{\beta +1}\) norm for the perturbation temperature, where \(\beta >1\) . PubDate: 2021-12-17

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Abstract: It is shown that the solution to the boundary - initial value problem for a Kelvin–Voigt fluid of order one depends continuously upon the Kelvin–Voigt parameters, the viscosity, and the viscoelastic coefficients. Convergence of a solution is also shown. PubDate: 2021-11-22 DOI: 10.1007/s11565-021-00381-7

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Abstract: Our study in this paper is focused on mixed equilibrium problems as well as combination with and fixed point problems in real Hilbert spaces. We introduce a new extension of a skew-symmetric bi-functions, called generalized skew-symmetric bi-functions. Two iterative methods based on the subgradient extragradient method is presented and their weak convergence theorems are proved under some mild assumptions. Further, we discuss some numerical examples to demonstrate the applicability of the iterative algorithm. The methods and analysis is new and extend several related result in the literature. PubDate: 2021-11-01 DOI: 10.1007/s11565-021-00380-8

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Abstract: We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on \((0,T]\times {\mathbb {R}}^n\) in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in x with their x-derivatives and t-derivative of order \(\text {O}(t^{-\delta }),\delta \in [0,1),\) and \(\text {O}(t^{-1})\) respectively. This type of singular behavior allows coefficients to be either oscillatory or logarithmically bounded at \(t=0\) . We use the Planck function associated with the metric to subdivide the extended phase space and define an appropriate generalized parameter dependent symbol class. We report that the solution experiences a finite loss in the Sobolev space index in relation to the initial datum defined in the Sobolev space tailored to the metric. Our analysis suggests that an infinite loss is quite expected when the order of singularity of the first time derivative of the leading coefficients exceeds \(O(t^{-1})\) . We confirm this by providing a counterexample. Further, using the \(L^1\) integrability of the logarithmic singularity in t and the global properties of the operator with respect to x, we derive an anisotropic cone condition in our setting. PubDate: 2021-10-27 DOI: 10.1007/s11565-021-00378-2

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Abstract: In this paper, by using the generalized symmetric difference \(\Delta _{h}^{m}\) of order m and step \(h>0\) , we obtain a generalization of Titchmarsh’s Theorem for deformed Hankel transform. PubDate: 2021-10-11 DOI: 10.1007/s11565-021-00379-1

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Abstract: Let \({\mathfrak {T}}={\mathfrak {T}}_3(\mathrm {R}_i, \mathrm {M}_{ij})\) be a triangular 3-matrix ring. In the present paper, we study of multiplicative Lie triple derivation on triangular 3-matrix rings and prove that every multiplicative Lie triple derivation on triangular 3-matrix rings can be written as a sum of an additive derivation and a center valued map vanishing at each second commutator. PubDate: 2021-09-25 DOI: 10.1007/s11565-021-00374-6

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Abstract: Let P(z) be a polynomial of degree at most n. We consider an operator N, which carries a polynomial P(z) into $$\begin{aligned} N[P](z):=\sum \limits _{j=0}^{m}\lambda _j\bigg (\frac{nz}{2}\bigg )^j\frac{P^{(j)}(z)}{j!}, \end{aligned}$$ where \(\lambda _0,\lambda _1,\ldots ,\lambda _m\) are such that all the zeros of $$\begin{aligned} u(z)=\sum \limits _{j=0}^{m}\left( {\begin{array}{c}n\\ j\end{array}}\right) \lambda _jz^j \end{aligned}$$ lie in the half plane $$\begin{aligned} z \le \bigg z-\frac{n}{2}\bigg . \end{aligned}$$ In this paper, we estimate the minimum and maximum modulii of N[P(z)] on \( z =1\) with restrictions on the zeros of P(z) and thereby obtain compact generalizations of some well known polynomial inequalities. PubDate: 2021-09-20 DOI: 10.1007/s11565-021-00375-5

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Abstract: Let \(Cb_n\) be the group of basis conjugating automorphisms of a free group \(\mathbb {F}_n\) , and \(C_n\) the group of conjugating automorphisms of \(\mathbb {F}_n\) . Valerij G. Bardakov has constructed representations of \(Cb_n\) , \(C_n\) in the groups \(GL_n(\mathbb {Z}[{t_1}^{\pm 1}, \ldots ,{t_n}^{\pm 1}])\) and in \(GL_n(\mathbb {Z}[{t}^{\pm 1}])\) respectively, where \(t_1, \ldots , t_n, t\) are indeterminate variables. We show that these representations are reducible and we determine the irreducible components of the representations in \(GL_n(\mathbb {C})\) , which are obtained by giving values to the variables above. Next, we consider the tensor product of the representations of \(Cb_n\) , \(C_n\) and study their irreduciblity in the case \(n=3\) . PubDate: 2021-09-18 DOI: 10.1007/s11565-021-00376-4

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Abstract: We propose a two-parameter derivative-free family of methods with memory of convergence order 1.84 for finding the real roots of nonlinear equations. The new methods require only one function evaluation per iteration, so efficiency index is also 1.84. The process is carried out by approximating the derivative in Newton’s iteration using general quadratic equation \(\alpha u^2+\beta v^2+\alpha _1 u+\beta _1 v+\delta =0\) in terms of coefficients \(\alpha , \beta \) . Various options of \(\alpha , \beta \) correspond to various quadratic forms viz. circle, ellipse, hyperbola and parabola. The application of new methods is validated on Kepler’s problem, Isentropic supersonic flow problem, L-C-R circuit problem and Population growth problem. In addition, a comparison of the performance of new methods with existing methods of same nature is also presented to check the consistency. PubDate: 2021-09-13 DOI: 10.1007/s11565-021-00377-3

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Abstract: In this paper, we study the long-time dynamics of Navier–Stokes equations with a fractional operator in the memory term and external forces $$\begin{aligned} \partial _tu-\nu \Delta u +\displaystyle \int _{0}^{\infty }g(s)(-\Delta )^\alpha u(t-s)\,\mathrm{d}s +(u\cdot \nabla )u+\nabla p=\epsilon f(x) \end{aligned}$$ on a bounded domain \(\Omega \) in \(\mathbb {R}^2\) with smooth boundary. We establish the existence of global attractor for the associated dynamical system. We prove the continuity of global attractors with respect to forcing parameter \(\epsilon \) in a residual dense set. Moreover, the upper-semicontinuity with respect to parameter \(\epsilon \) of global attractors is shown. PubDate: 2021-08-21 DOI: 10.1007/s11565-021-00373-7

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Abstract: A well known approximation for the displacement of two immiscible fluids in a porous medium is the Hele-Shaw model. In experiments it was observed that a liquid with variable viscosity, introduced between the two initial fluids, can minimize the Saffman-Taylor instability. In some works an attempt was made to replace the variable viscosity liquid with a sequence of several immiscible liquids with constant viscosities. We prove that the linear stability analysis of this multi-layer Hele-Shaw model leads us to an ill-posed problem. PubDate: 2021-08-11 DOI: 10.1007/s11565-021-00371-9

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Abstract: In this paper, we use the fixed point theory to obtain the existence and uniqueness of solutions for a class of nonlinear fractional differential equations. Two examples are given to illustrate this work. PubDate: 2021-08-07 DOI: 10.1007/s11565-021-00372-8

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Abstract: We show that a Leray–Hopf weak solution to the three-dimensional Navier–Stokes the initial value problem is regular in (0, T] if \(\Vert \nabla u_0^+\Vert _2\) (or \(\Vert \nabla u_0^-\Vert _2\) ) for initial value \(u_0\) and \(\max \{\frac{d{\mathcal H}}{dt},0\}\) (or \(\max \{-\frac{d{\mathcal H}}{dt},0\}\) ) are suitably small depending on the initial kinetic energy and viscosity, where \(u_0^+=\int _0^{\infty } dE_\lambda u_0\) , \(u_0^-=\int _{-\infty }^0 dE_\lambda u_0\) , \(\{E_\lambda \}_{\lambda \in {\mathbb R}}\) is the spectral resolution of the \(\mathrm{curl}\) operator and \({\mathcal H}\equiv \int _{\mathbb R^3}u\cdot \mathrm{curl} u\,dx\) is the helicity of the fluid flow. The results suggest that the helicity change rate rather than the magnitude of the helicity itself affects regularity of the viscous incompressible flows. More precisely, an initially regular viscous incompressible flow with suitably small positive or negative maximal helical component does not lose its regularity as long as the total helical behavior of the flow with respect to time is not decreasing, or even weakened at a moderate rate in accordance with the initial kinetic energy and viscosity. PubDate: 2021-07-31 DOI: 10.1007/s11565-021-00370-w

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Abstract: A nonzero sum of a unit and a nilpotent element in a ring is called a fine element. This is a study of rings in which every nonzero nilpotent is fine, which we call NF rings. PubDate: 2021-07-06 DOI: 10.1007/s11565-021-00369-3

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Abstract: Let \(X\subset \mathbb {P}^3\) be an integral and non-degenerate curve. We say that \(q\in \mathbb {P}^3\setminus X\) has X-rank 3 if there is no line \(L\subset \mathbb {P}^3\) such that \(q\in L\) and \(\#(L\cap X)\ge 2\) . We prove that for all hyperelliptic curves of genus \(g\ge 5\) there is a degree \(g+3\) embedding \(X\subset \mathbb {P}^3\) with exactly \(2g+2\) points with X-rank 3 and another embedding without points with X-rank 3 but with exactly \(2g+2\) points \(q\in \mathbb {P}^3\) such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed \(q\in \mathbb {P}^3\setminus X\) . For all integers \(g\ge 5\) , \(x\ge 0\) we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, \(\deg (X) = x+g+3\) and having at least \(x+2g+2\) points of X-rank 3. PubDate: 2021-07-01 DOI: 10.1007/s11565-021-00368-4

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Abstract: In this paper, we deal with the asymptotic modeling of the behavior of a reinforced rectangular plate with a thin layer of high thermal conductivity. We focus on a thermo-elastic model described by a set of nonlinear time dependent partial differential equations, accounting for nonlinear mechanical and nonlinear thermal coupling. Our aim is to model this junction and reproduce the effect of the thin body by means of approximate boundary conditions, obtained by an asymptotic analysis with respect to the thickness of this latter. The study is carried out for two kinds of layers: Stiff and soft. Different limit behaviors occur according to the rigidity or the softness of this latter. PubDate: 2021-06-12 DOI: 10.1007/s11565-021-00364-8

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Abstract: We investigate the space-time decay properties for strong solutions to the 3D Navier–Stokes equations with power-law type nonlinear viscous fluid. Based on the temporal decay results for this equation, we find that one can obtain decay estimate of the weight for the velocity field u by direct weighted energy estimate. PubDate: 2021-05-31 DOI: 10.1007/s11565-021-00367-5

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Abstract: In this paper we consider wave equation with logarithmic nonlinearity and distributive internal delay. We establish the local existence result using the semigroup theory whereas the global existence by the well-depth method. Finally, under suitable assumptions on the weight of the delay and that of frictional damping, we prove that the system is exponentially stable. PubDate: 2021-05-26 DOI: 10.1007/s11565-021-00366-6

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Abstract: In this paper, a modification of Szász-Kantorovich type operators based on Brenke-type polynomials is introduced, and the convergence properties of the proposed operators with the help of Korovkin’s theorem are discussed. The order of convergence of these operators with the aid of classical and second-order modulus of continuity is studied. A Voronovkaja-type theorem is also established. Lastly, the rate of convergence and error estimation of these operators compared with the existing operators with the help of some graphs and tables using Mathematica. PubDate: 2021-05-22 DOI: 10.1007/s11565-021-00365-7