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Abstract: Schreier bases are introduced and used to show that skew polynomial rings are free ideal rings, i.e., rings whose one-sided ideals are free of unique rank, as well as to compute a rank of one-sided ideals together with a description of corresponding bases. The latter fact, a so-called Schreier-Lewin formula (Lewin Trans. Am. Math. Soc. 145, 455–465 1969), is a basic tool determining a module type of perfect localizations which reveal a close connection between classical Leavitt algebras, skew polynomial rings, and free associative algebras. PubDate: 2022-01-22
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Abstract: In this article, we give explicit minimal generators of the first syzygy of the Hibi ring for a planar distributive lattice in terms of sublattices. We also give a characterization when it is linearly related and derive an exact formula for the first Betti number of a planar distributive lattice. PubDate: 2022-01-18
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Abstract: In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gröbner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gröbner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange. PubDate: 2022-01-15
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Abstract: The purpose of this paper is to introduce a new kind of q −Stancu-Kantorovich type operators and study its various approximation properties. We establish some local direct theorems, e.g., Voronovskaja type asymptotic theorem, global approximation and an estimate of error by means of the Lipschitz type maximal function and the Peetre K-functional. We also consider a n th-order generalization of these operators and study its approximation properties. Next, we define a bivariate case of these operators and investigate the order of convergence by means of moduli of continuity and the elements of Lipschitz class. Furthermore, we consider the associated Generalized Boolean Sum (GBS) operators and examine the approximation degree for functions in a Bögel space. Some numerical examples to illustrate the convergence of these operators to certain functions are also given. PubDate: 2022-01-13
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Abstract: Let R be a prime ring of char(R)≠ 2, U its Utumi ring of quotients and center C = Z(U) its extended centroid, I a both sided ideal of R, f(x1,…,xn) a multilinear polynomial over C, that is noncentral-valued on R, F, G be two generalized derivations of R and d be a derivation of R. Let f(I) be the set of all evaluations of the multilinear polynomial f(x1,…,xn) in I. If @@@ for all u ∈ f(I), then all possible forms of the maps are determined. As an application of this result, we also study the commutator identity [F2(u)u,G2(v)v] = 0 for all u,v ∈ f(I), where F and G are two generalized derivations of R. PubDate: 2022-01-10
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Abstract: The degree excess function ðœ–(I; n) is the difference between the maximal generating degree d(In) of the n-th power of a homogeneous ideal I of a polynomial ring and p(I)n, where p(I) is the leading coefficient of the asymptotically linear function d(In). It is shown that any non-increasing numerical function can be realized as a degree excess function, and there is a monomial ideal I whose ðœ–(I; n) has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on ðœ–(I; n) is provided. As an application, it is shown that in the worst case, the so-called stability index of the Castelnuovo-Mumford regularity of a monomial ideal I must be at least an exponential function of the number of variables. PubDate: 2022-01-08
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Abstract: The aim of this paper is to introduce the notion of Lelong number and the log canonical thresholds of plurisubharmonic functions on analytic subsets A in an open subset Ω of \(\mathbb {C}^{n}\) . Next, we establish some results about the relationship between these quantities in the relation with the analyticity of A. PubDate: 2022-01-05
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Abstract: A Correction to this paper has been published: https://doi.org/10.1007/s40306-021-00447-w PubDate: 2021-12-01
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Abstract: The purpose of this paper is to establish the existence solutions of the complex Monge-Ampère type equation in plurifinely open subsets of \(\mathbb {C}^{n}\) . PubDate: 2021-12-01
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Abstract: We give a necessary and sufficient condition for an autonomous first-order AODE to have a rational liouvillian solution. We also give an algorithm to compute a rational liouvillian general solution if it exists. The algorithm is based on the rational parametrization of the corresponding algebraic curve of the first-order autonomous AODE and the existence of a rational liouvillian element over \(\mathbb {C}\) . When the corresponding algebraic curve is rational, this method covers the known cases of rational solutions and radical solutions. PubDate: 2021-12-01
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Abstract: Let \(R = \oplus _{n\in \mathbb {N}_{0}}R_{n}\) be a Noetherian homogeneous ring with irrelevant ideal \(R_{+} = \oplus _{n\in \mathbb {N}} R_{n}\) and with local base ring \((R_{0},\mathfrak {m}_{0})\) . Let M, N be two finitely generated \(\mathbb {Z}\) -graded R-modules. We show that the lengths of the graded components of various graded submodules and quotients of the i-th generalized local cohomology \(H^{i}_{R_{+}}(M, N)\) are anti-polynomial. Under some mild assumptions, the Artinianness of \(H^{i}_{R_{+}}(M, N)\) and the asymptotic behavior of the R0-modules \(H^{i}_{R_{+}}(M, N)_{n}\) for \(n\rightarrow -\infty \) in the range \(i\leq \inf \{i\in \mathbb {N}_{0} \vert \sharp \{n\vert \ell _{R_{0}}\) \((H^{i}_{ R_{+}}(M , N)_{n}) = \infty \}=\infty \}\) will be studied. Moreover, it has been proved that, if u is the least integer i for which \(H^{i}_{R_{+}}(M,N)\) is not Artinian and \(\mathfrak {q}_{0}\) is an \(\mathfrak {m}_{0}\) -primary ideal of R0, then \(H^{u}_{R_{+}}(M,N)/\mathfrak q_{0}H^{u}_{R_{+}}(M,\) N) is Artinian with Hilbert-Kirby polynomial of degree less than u. In particular, with M = R, we deduce the correspondent result for ordinary local cohomology module \(H^{i}_{R_{+}}(N)\) . PubDate: 2021-12-01
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Abstract: Describing the subgroup structure of a non-commutative division ring is the subject of an intensive study in the theory of division rings in particular, and of the theory of skew linear groups in general. This study is still so far to be complete. In the present paper, we study this problem for weakly locally finite division rings. Such division rings constitute a large class which strictly contains the class of locally finite division rings. PubDate: 2021-12-01
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Abstract: Let \(\mathcal X\) be a Banach space and let T be a bounded linear operator on \(\mathcal {X}\) . We denote by S(T) the set of all complex \(\lambda \in \mathcal {C}\) such that T does not have the single-valued extension property. In this paper it is shown that if MC is a 2 × 2 upper triangular operator matrix acting on the Banach space \(\mathcal {X} \oplus \mathcal {Y}\) , then the passage from σLD(A) ∪ σLD(B) to σLD(MC) is accomplished by removing certain open subsets of σd(A) ∩ σLD(B) from the former, that is, there is the equality σLD(A) ∪ σLD(B) = σLD(MC) ∪ℵ, where ℵ is the union of certain of the holes in σLD(MC) which happen to be subsets of σd(A) ∩ σLD(B). Generalized Weyl’s theorem and generalized Browder’s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how generalized Weyl’ theorem, generalized Browder’s theorem, generalized a-Weyl’s theorem and generalized a-Browder’s theorem survive for 2 × 2 upper triangular operator matrices on the Banach space. PubDate: 2021-12-01
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Abstract: Important mathematical models in science and technology are based on first-order symmetric hyperbolic systems of differential equations whose solutions must satisfy certain constraints. When the models are restricted to bounded domains, the problem of well-posed, constraint-preserving boundary conditions arises naturally. However, for numerical solutions, finding such boundary conditions may represent just a step in the right direction. Including the constraints as dynamical variables of a larger, unconstrained system associated to the original one could provide better numerical results, as the constraints are kept under control during evolution. One of the main goals of this work is to investigate this idea in the case of constrained constant-coefficient first-order symmetric hyperbolic systems of differential equations subject to maximal nonnegative boundary conditions. PubDate: 2021-12-01
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Abstract: We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain \({\varOmega }\subset \mathbb {R}^{N} (N \geq 2)\) . We prove the existence of entropy solutions avoiding sign condition and coercivity on the lower order terms. PubDate: 2021-12-01
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Abstract: This paper studies the solvability of Cauchy problems for fractional evolution equations with uncertainty. By using a new approach based on the concept of non-compactness measure and the principle of condensing mappings in the spaces without linearity, we prove the existence of C0 −solutions without assuming the Lipschitz continuity of the function on the right-hand side. The present results extend previous results when external forces are always required to satisfy some kinds of generalized Lipschitz conditions. Moreover, the principle of condensing mappings for fuzzy-valued functions or set-valued functions found as an application of noncompactness measure is a useful result when studying dynamical systems containing uncertainties. PubDate: 2021-12-01
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Abstract: The algebra of polynomials in \(\mathbb {R}[x]\) which are bounded on a semi-algebraic set determined by a polynomial inequality f(x) ≤ r with f(0) = 0 is studied and the case when it is generated by a finite set of monomials is discussed. A large class of polynomials which are asymptotic to finitely many monomials (including nondegenerate polynomials) is introduced and the algebra of polynomials bounded on [f ≤ r] can be determined by a cone and is independent on r > 0, where f belongs to this class. Note that the set of all polynomials whose supports lie in a given closed convex cone in the first quadrant forms an algebra generated by a finite set of monomials. In other cases, we can give upper and lower bounds of the algebra via outer normal cones of the faces of the Newton polyhedron. As a consequence, some sufficient conditions which ensure that the algebra under consideration is generated by finitely many monomials is given. PubDate: 2021-12-01
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Abstract: Let R be a polynomial ring over a field and let I ⊂ R be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of I. The obtained formula depends only on the number of variables of R, the minimal number of generators of I, and the degree of the syzygies of I. Applying results from Busé et al. (Proc. Lond. Math. Soc. 121(4):743–787, 2020) we get a formula for the j-multiplicity of I and an effective method to study a rational map determined by a minimal set of generators of I. PubDate: 2021-12-01
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Abstract: In this paper, we propose a method for determining all minimal representations of a face of a polyhedron defined by a system of linear inequalities. Main difficulties for determining prime and minimal representations of a face are that the deletion of one redundant constraint can change the redundancy of other constraints and the number of descriptor index pairs for the face can be huge. To reduce computational efforts in finding all minimal representations of a face, we prove and use properties that deleting strongly redundant constraints does not change the redundancy of other constraints and all minimal representations of a face can be found only in the set of all prime representations of the face corresponding to the maximal descriptor index set for it. The proposed method is based on a top-down search strategy, is easy to implement, and has many computational advantages. Based on minimal representations of a face, a reduction of degeneracy degrees of the face and ideas to improve some known methods for finding all maximal efficient faces in multiple objective linear programming are presented. Numerical examples are given to illustrate the method. PubDate: 2021-12-01
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Abstract: Let v be a weight on a domain D in a metrizable locally convex space E and F be a complete locally convex space. Denote by Hv(D, F) the weighted space of F-valued holomorphic functions on D satisfying that v.f is bounded, Av(D) subspace of \(H_{v}(D, \mathbb C)\) with the unit ball is compact for the open-compact topology. The aim of this paper is to study linearization theorems and approximation properties in several different topologies for weighted spaces Av(D, F) of functions f ∈ Hv(D, F) such that u ∘ f ∈ Av(D) for every continuous linear functional u on F. PubDate: 2021-12-01
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