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Abstract: In this paper, we introduce the concept of ARA transform in q-calculus namely q-ARA transform and establish some properties. Furthermore, several propositions concerned with the properties of q-ARA transform are explored. We also give some applications of q-ARA transform for solving some ordinary and partial differential equations with initial and boundary values problems. PubDate: 2022-01-12
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Abstract: Let \(A=\oplus _{\alpha \in G}A_{\alpha }\) be a commutative ring with unity graded by an arbitrary grading commutative monoid G, E be a graded A-module and \(R=A\propto E\) the graded trivial extension. In this paper, in the second section, we improve some results on graded trivial extension and we give some new ones and the theme throughout is how homogeneous properties are related to those of A and E. Then, in another section, we introduce and study the notions of graded-v-coherent, graded-quasi-coherent and graded-finite conductor rings, then we study their transfer in the graded trivial extension. PubDate: 2022-01-07
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Abstract: In this paper, we prove some new results on operational second order differential equations of elliptic type with general Robin boundary conditions in a non-commutative framework. The study is developed in Hölder spaces under some natural assumptions generalizing those in [4]. We give necessary and sufficient conditions on the data to obtain a unique strict solution satisfying the maximal regularity property, see Theorems 1 and 2. This work completes the one given in [4] and [12]. PubDate: 2022-01-04
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Abstract: We propose a new method to estimate plant diversity with Rényi and Rao indexes through the so called High Order Singular Value Decomposition (HOSVD) of tensors. Starting from NASA multi-spectral images we evaluate diversity and we compare original diversity estimates with those realized via the HOSVD compression methods for big data. Our strategy turns out to be extremely powerful in terms of memory storage and precision of the outcome. The obtained results are so promising that we can support the efficiency of our method in the ecological framework. PubDate: 2021-12-01
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Abstract: Let a function f belongs to the Lebesgue class \(L^p({{\mathbb {R}}})\) , \(1\le p\le 2\) , and let \({\widehat{f}}\) be the Fourier transform of f. The classical theorem of E. Titchmarsh states that if the function f belongs to the Lipschitz class \(Lip(\alpha ,p; {{\mathbb {R}}})\) , \(0<\alpha \le 1\) , then \({{\widehat{f}}}\) belongs to the Lebesgue classes \(L^r({{\mathbb {R}}})\) for \(\frac{p}{p+\alpha p-1}< r\le \frac{p}{p-1}\) . In this paper we generalize this result for the case when the function f belongs to the Nikol’skiĭ class \(H_p^\alpha ({{\mathbb {R}}})\) , \(1\le p\le 2\) , \(\alpha >0\) . PubDate: 2021-12-01
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Abstract: We present a number of existence results for some general classes of monotone nonexpansive mappings in partially ordered hyperbolic metric spaces. We also give some examples to show the generality of the mappings considered herein. PubDate: 2021-11-23
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Abstract: In this paper, we study on the existence of transcendental entire solutions of certain non-linear generalized delay-differential equations. In this respect we have improved a recent result of Wang et al. (Turk J Math 43:941–954, 2019). Also at the time of investigating the solutions of shift equations we have improved as well as extended an earlier result due to Latreuch (Mediterr J Math 14:1–16, 2017). A handful number of examples have been exhibited by us relevant to the content of the paper to show that each case as demonstrated in the conclusions of the theorems actually occurs. At last we raise a question for future investigations. PubDate: 2021-11-09
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Abstract: We generalise the surface-complexity of closed 3-manifolds to the compact case. This generalisation preserves the (natural generalisation of the) properties holding in the closed case: the surface-complexity on compact 3-manifolds is a natural number measuring how much the manifolds are complicated, it is subadditive under connected sum and it is finite-to-one on \(\mathbb {P}^2\) -irreducible and boundary-irreducible manifolds without essential annuli and Möbius strips. Moreover, for these manifolds, it equals the minimal number of cubes in an ideal cubulation of the manifold, except for a finite number of cases. We will also give estimations of the surface-complexity by means of ideal triangulations and Matveev complexity. PubDate: 2021-09-27 DOI: 10.1007/s40574-021-00308-2
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Abstract: An analogue of the Mukai map \(m_g: {\mathcal {P}}_g \rightarrow {\mathcal {M}}_g\) is studied for the moduli \({\mathcal {R}}_{g, \ell }\) of genus g curves C with a level \(\ell \) structure. Let \({\mathcal {P}}^{\perp }_{g, \ell }\) be the moduli space of 4-tuples \((S, {\mathcal {L}}, {\mathcal {E}}, C)\) so that \((S, {\mathcal {L}})\) is a polarized K3 surface of genus g, \({\mathcal {E}}\) is orthogonal to \({\mathcal {L}}\) in \({{\,\mathrm{Pic}\,}}S\) and defines a standard degree \(\ell \) K3 cyclic cover of S, \(C \in \vert {\mathcal {L}} \vert \) . We say that \((S, {\mathcal {L}}, {\mathcal {E}})\) is a level \(\ell \) K3 surface. These exist for \(\ell \le 8\) and their families are known. We define a level \(\ell \) Mukai map \(r_{g, \ell }: {\mathcal {P}}^{\perp }_{g, \ell } \rightarrow {\mathcal {R}}_{g, \ell }\) , induced by the assignment of \((S, {\mathcal {L}}, {\mathcal {E}}, C)\) to \( (C, {\mathcal {E}} \otimes {\mathcal {O}}_C)\) . We investigate a curious possible analogy between \(m_g\) and \(r_{g, \ell }\) , that is, the failure of the maximal rank of \(r_{g, \ell }\) for \(g = g_{\ell } \pm 1\) , where \(g_{\ell }\) is the value of g such that \(\dim {\mathcal {P}}^{\perp }_{g, \ell } = \dim {\mathcal {R}}_{g,\ell }\) . This is proven here for \(\ell = 3\) . As a related open problem we discuss Fano threefolds whose hyperplane sections are level \(\ell \) K3 surfaces and their classification. PubDate: 2021-09-21 DOI: 10.1007/s40574-021-00306-4
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Abstract: Let A be a positive bounded linear operator on a Hilbert space \(\left( { {\mathscr {H}}},\left\langle .,.\right\rangle \right) \) . The semi-inner product \(\left\langle x,y\right\rangle _{A}:=\left\langle Ax,y\right\rangle \) , x, y \(\in \) \({{\mathscr {H}}}\) , induces a semi-norm \(\left\ .\right\ _{A}\) on \({{\mathscr {H}}}\) . Let \(\omega _{A}\) \(\left( T\right) \) denote the A -numerical radius of an operator T in semi-Hilbertian space \(\left( { {\mathscr {H}}},\left\ .\right\ _{A}\right) \) . Our aim in this paper is to give new inequalities of A-numerical radius of operators in semi-Hilbertian spaces. In particular, we show that $$\begin{aligned} \omega _{A}^{n}\left( T\right) \le \frac{1}{2^{n-1}}\omega _{A}\left( T^{n}\right) +\left\ T\right\ _{A}\displaystyle \sum _{p=1}^{n-1}\frac{ 1}{2^{p}}\omega _{A}^{n-p-1}\left( T\right) \left\ T^{p}\right\ _{A}, \end{aligned}$$ for all \(n=2,3,\ldots \) Further, an extension of some inequalities of bounded linear operators on a Hilbert space due to Dragomir (Inequalities for the numerical radius of linear operators in Hilbert spaces. Springer briefs in mathematics, Springer, Berlin, 2013; Tamkang J Math 39(1):1–7, 2008) and Kittaneh et al. (Linear Algebra Appl 471:46–53, 2015) are proved on a semi-Hilbert space and some more related results are also obtained. PubDate: 2021-09-20 DOI: 10.1007/s40574-021-00307-3
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Abstract: In this paper, we prove a strengthening of the generic vanishing result in characteristic \(p>0\) given in Hacon and Patakfalvi (Am J Math 138(4):963–998, 2016). As a consequence of this result, we show that irreducible \(\Theta \) divisors are strongly F-regular and we prove a related result for pluri-theta divisors. PubDate: 2021-09-06 DOI: 10.1007/s40574-021-00304-6
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Abstract: The aim of this note is to find those commutative rings over which an exact analogue of the structure theory of injective modules over commutative noetherian rings holds for weak-injective modules, i.e. for modules M satisfying \(\mathop {\mathrm{Ext}}\nolimits _R^1(A,M)=0\) for all modules A of weak dimension \(\le 1\) . We will show that, surprisingly, but a very few commutative rings R possess the property that their weak-injective modules admit (up to isomorphism) unique decompositions into direct sums of indecomposable modules each of which is the injective or the weak-injective envelope of a cyclic module of the form \(R/{\mathsf {p}} \) with a prime ideal \({\mathsf {p}} \) . PubDate: 2021-09-01 DOI: 10.1007/s40574-021-00283-8
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Abstract: We study the regularity properties of Green’s function \(G_T({\mathbf {x}},t {\mathbf {x}}_0,t_0)\) associated to Robin’s problem for a class of second order operators on \(\Omega \times ]-T,T[,\) \(\Omega \subset {\mathbb {R}}^n\) a bounded open set with regular boundary, including the hyperbolic heat equation for anisotropic non homogeneous bodies with constant thermal properties as a particular case. We show that for every source point \(({\mathbf {x}}_0,t_0)\) and every \(k\in {\mathbb {N}}\) there is an open set \({\mathcal {O}}_k\subset \Omega \times ]-T,T[\) such that \(G_T({\mathbf {x}},t {\mathbf {x}}_0,t_0)\) is k times differentiable with continuity on \({\mathcal {O}}_k\) and \(\mu \bigl ((\Omega \times ]-T,T[)\backslash {\mathcal {O}}_k\bigr )=0.\) PubDate: 2021-09-01 DOI: 10.1007/s40574-020-00264-3
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Abstract: The singular set of a viscosity solution to a Hamilton–Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on \(\mathbb {R}^2\) , two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861–885, 2016]. PubDate: 2021-09-01 DOI: 10.1007/s40574-021-00279-4
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Abstract: Let D be an integral domain with quotient field K and X an indeterminate over K. The set \(\mathrm {Int}(D):=\{f\in K[X]: f(D)\subseteq D\}\) , of integer-valued polynomials over D, is known to be a ring. The purpose of this note is to calculate the Krull dimension of rings between D[X] and \(\mathrm {Int}(D)\) when D is either locally essential or t-locally Noetherian. By the way, we find a problem in the proof of the first main result of Fontana and Kabbaj (Proc Am Math Soc 132:2529–2535) and then we will recover it. Also, we give some examples to illustrate our main result. PubDate: 2021-09-01 DOI: 10.1007/s40574-021-00281-w
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Abstract: We show that in a noetherian commutative unital topological algebra, the prime ideals associated with a closed ideal as well as its isolated primary components are closed. We obtain a version of the closed graph theorem. An example of a noetherian (even principal) commutative unital semi-simple and incomplete normed algebra whose each ideal is closed is also given. PubDate: 2021-09-01 DOI: 10.1007/s40574-020-00274-1
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Abstract: The Kuramoto–Velarde equation describes slow space-time variations of disturbances at interfaces, diffusion–reaction fronts and plasma instability fronts. It also describes Benard–Marangoni cells that occur when there is large surface tension on the interface in a microgravity environment. Under appropriate assumption on the initial data, of the time T, and the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation. PubDate: 2021-07-20 DOI: 10.1007/s40574-021-00303-7
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Abstract: In this paper, we present several numerical radius and norm inequalities for sum of Hilbert space operators. These inequalities improve some earlier related inequalities. For an operator T ∈ B(H), we prove that $$ {\omega}^{2} \left( T \right) \le \frac{1}{2}\omega \left( {T^{2} } \right) + \frac{1}{{2\sqrt 2 }}\omega \left( {\left T \right ^{2} + i\left {T^{\ast} } \right ^{2} } \right) .$$ PubDate: 2021-07-07 DOI: 10.1007/s40574-021-00289-2
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Abstract: Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group. PubDate: 2021-06-28 DOI: 10.1007/s40574-021-00301-9
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Abstract: An inequality proved firstly by Remak and then generalized by Friedman shows that there are only finitely many number fields with a fixed signature and whose regulator is less than a prescribed bound. Using this inequality, Astudillo, Diaz y Diaz, Friedman and Ramirez-Raposo succeeded to detect all fields with small regulators having degree less or equal than 7. In this paper we show that a certain upper bound for a suitable polynomial, if true, can improve Remak–Friedman’s inequality and allows a classification for some signatures in degree 8 and better results in degree 5 and 7. The validity of the conjectured upper bound is extensively discussed. PubDate: 2021-06-22 DOI: 10.1007/s40574-021-00298-1
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