Showing papers in "Czechoslovak Mathematical Journal in 2022"

A convex treatment of numerical radius inequalities

Abstract: In this article, we prove an inner product inequality for Hilbert space operators. This inequality, then, is utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining new forms that generalize and extend some well known results in the literature, with an application to the newly defined generalized numerical radius. We emphasize that the approach followed in this article is different from the approaches used in the literature to obtain the refined versions.

Local cohomology, cofiniteness and homological functors of modules

Abstract: Let I be an ideal of a commutative Noetherian ring R . It is shown that the R -modules H I j ( M ) are I -cofinite for all finitely generated R -modules M and all j ∈ ℕ 0 if and only if the R -modules Ext R i ( N ,H I j ( M )) and Tor i R ( N, H I j ( M )) are I -cofinite for all finitely generated R -modules M, N and all integers i, j ∈ ℕ 0 .

Wiener index of graphs with fixed number of pendant or cut-vertices

Abstract: The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.

On the quasi-periodic p-adic Ruban continued fractions

Abstract: We study a family of quasi periodic p -adic Ruban continued fractions in the p -adic field ℚ p and we give a criterion of a quadratic or transcendental p -adic number which based on the p -adic version of the subspace theorem due to Schlickewei.

Classification of ideals of 8-dimensional Radford Hopf algebra

Abstract: Let Hm,n be the mn2-dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. We give the classification of all ideals of 8-dimensional Radford Hopf algebra H2,2 by generators.

Carleson measures for weighted harmonic mixed norm spaces on bounded domains in ℝn

Abstract: We study weighted mixed norm spaces of harmonic functions defined on smoothly bounded domains in ℝ n . Our principal result is a characterization of Carleson measures for these spaces. First, we obtain an equivalence of norms on these spaces. Then we give a necessary and sufficient condition for the embedding of the weighted harmonic mixed norm space into the corresponding mixed norm space.

On the $r$-free values of the polynomial $x^2 + y^2 + z^2 +k$

Abstract: Let k be a fixed integer. We study the asymptotic formula of R(H, r, k), which is the number of positive integer solutions x, y, z greater than or equal to 1 and less than or equal to H such that the polynomial x^2+y^2+z^2+k is r-free. We obtained the asymptotic formula of R(H, r, k) for all r greater than or equal to 2. Our result is new even in the case r = 2. We proved that R(H, 2, k) = ckH^3 + O(H^(9/4+epsilon)), where ck>0 is a constant depending on k. This improves upon the error term O(H^(7/3+epsilon)) obtained by Zhou and Ding.

On hyper-Zagreb index conditions for hamiltonicity of graphs

Abstract: During the last decade, several research groups have published results on sufficient conditions for the hamiltonicity of graphs by using some topological indices. We mainly study hyper-Zagreb index and some hamiltonian properties. We give some sufficient conditions for graphs to be traceable, hamiltonian or Hamilton-connected in terms of their hyper-Zagreb indices. In addition, we also use the hyper-Zagreb index of the complement of a graph to present a sufficient condition for it to be Hamilton-connected.

Semi n-ideals of commutative rings

Abstract: Let R be a commutative ring with identity. A proper ideal I is said to be an n-ideal of R if for a, b ∈ R, ab ∈ I and a∉0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a otin \sqrt 0 $$\end{document} imply b ∈ I. We give a new generalization of the concept of n-ideals by defining a proper ideal I of R to be a semi n-ideal if whenever a ∈ R is such that a2 ∈ I, then a∈0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in \sqrt 0 $$\end{document} or a ∈ I. We give some examples of semi n-ideal and investigate semi n-ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new class of ideals. Moreover, we prove that every proper ideal of a zero dimensional general ZPI-ring R is a semi n-ideal if and only if R is a UN-ring or R ≌ F1 × F2 × … × Fk, where Fi is a field for i = 1,…, k. Finally, for a ring homomorphism f: R → S and an ideal J of S, we study some forms of a semi n-ideal of the amalgamation R ⋈fJ of R with S along J with respect to f.

Local cohomology, cofiniteness and homological functors of modules

Abstract: Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules HIj(M) are I-cofinite for all finitely generated R-modules M and all j ∈ ℕ0 if and only if the R-modules ExtRi (N,HIj(M)) and ToriR (N, HIj(M)) are I-cofinite for all finitely generated R-modules M, N and all integers i, j ∈ ℕ0.

On $k$-free numbers over Beatty sequences

Abstract: In this paper, we consider $k$-free numbers over Beatty sequences. New results are given.

On a family of elliptic curves of rank at least 2

Abstract: Let Cm:y2 = x3 − m2x + p2q2 be a family of elliptic curves over ℚ, where m is a positive integer and p, q are distinct odd primes. We study the torsion part and the rank of Cm(ℚ). More specifically, we prove that the torsion subgroup of Cm(ℚ) is trivial and the ℚ-rank of this family is at least 2, whenever m ≢ 0 (mod 3), m ≢ 0 (mod 4) and m ≡ 2 (mod 64) with neither p nor q dividing m.

A convex treatment of numerical radius inequalities

Abstract: We prove an inner product inequality for Hilbert space operators. This inequality will be utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining new forms that generalize and extend some well known results in the literature, with an application to the newly defined generalized numerical radius. We emphasize that the approach followed in this article is different from the approaches used in the literature to obtain such versions.

The potential-Ramsey number of Kn and Kt−k

Abstract: A nonincreasing sequence π = (d1,…, dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π. Given two graphs G1 and G2, A. Busch et al. (2014) introduced the potential-Ramsey number of G1 and G2, denoted by rpot(G1, G2), as the smallest nonnegative integer m such that for every m-term graphic sequence π, there is a realization G of π with G1 ⊆ G or with G2⊆G¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_2} \subseteq \overline G $$\end{document}, where G¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline G $$\end{document} is the complement of G. For t ≽ 2 and 0⩽k⩽⌊t2⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant k\leqslant \left\lfloor {{t \over 2}} \right\rfloor $$\end{document}, let Kt−k be the graph obtained from Kt by deleting k independent edges. We determine rpot (Kn, Kt−k) for t⩾3,1⩽k⩽⌊t2⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\geqslant 3,1\leqslant k\leqslant \left\lfloor {{t \over 2}} \right\rfloor $$\end{document} and n⩾⌈2k⌉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant \left\lceil {\sqrt {2k}} \right\rceil $$\end{document}, which gives the complete solution to a result in J. Z. Du, J. H. Yin (2021).

Weak n-injective and weak n-fat modules

Abstract: We introduce and study the concepts of weak n-injective and weak n-flat modules in terms of super finitely presented modules whose projective dimension is at most n, which generalize the n-FP-injective and n-flat modules. We show that the class of all weak n-injective R-modules is injectively resolving, whereas that of weak n-flat right R-modules is projectively resolving and the class of weak n-injective (or weak n-flat) modules together with its left (or right) orthogonal class forms a hereditary (or perfect hereditary) cotorsion theory.

On some combinatorial properties of generalized commutative Jacobsthal quaternions and generalized commutative Jacobsthal-Lucas quaternions

Abstract: We study generalized commutative Jacobsthal quaternions and generalized commutative Jacobsthal-Lucas quaternions. We present some properties of these quaternions and the relations between the generalized commutative Jacobsthal quaternions and generalized commutative Jacobsthal-Lucas quaternions.

Möbius metric in sector domains

Abstract: The M\"obius metric $\delta_G$ is studied in the cases where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the M\"obius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the M\"obius metric and its connection to the hyperbolic metric in polygon domains.

Discretization of prime counting functions, convexity and the Riemann hypothesis

Abstract: We study tails of prime counting functions. Our approach leads to representations with a main term and an error term for the asymptotic size of each tail. It is further shown that the main term is of a specific shape and can be written discretely as a sum involving probabilities of certain events belonging to a perturbed binomial distribution. The limitations of the error term in our representation give us equivalent conditions for various forms of the Riemann hypothesis, for classical type zero-free regions in the case of the Riemann zeta function and the size of semigroups of integers in the sense of Beurling. Inspired by the works of Panaitopol, asymptotic companions pertaining to the magnitude of specific prime counting functions are obtained in terms of harmonic numbers, hyperharmonic numbers and the number of indecomposable permutations. By introducing the notion of asymptotic convexity and fusing it with a nice generalization of an inequality of Ramanujan due to Hassani, we arrive at a curious asymptotic inequality for the classical prime counting function at any convex combination of its arguments and further show that quotients arising from prime counting functions of progressions furnish examples of asymptotically convex, but not convex functions.

Weak n-injective and weak n-fat modules

Abstract: We introduce and study the concepts of weak n -injective and weak n -flat modules in terms of super finitely presented modules whose projective dimension is at most n , which generalize the n -FP-injective and n -flat modules. We show that the class of all weak n -injective R -modules is injectively resolving, whereas that of weak n -flat right R -modules is projectively resolving and the class of weak n -injective (or weak n -flat) modules together with its left (or right) orthogonal class forms a hereditary (or perfect hereditary) cotorsion theory.

The extremal irregularity of connected graphs with given number of pendant vertices

Abstract: The irregularity of a graph G = (V, E) is defined as the sum of imbalances ∣du − dv∣ over all edges uv ∈ E, where du denotes the degree of the vertex u in G. This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with n vertices and p pendant vertices (1 ⩽ p ⩽ n − 1), and characterize the corresponding extremal graphs.

Stable Tubes in Extriangulated Categories

Abstract: Let χ be a semibrick in an extriangulated category. If χ is a τ-semibrick, then the Auslander-Reiten quiver Γ(ℱ(X))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ({\cal F}({\cal X}))$$\end{document} of the filtration subcategory ℱ(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal F}({\cal X})$$\end{document} generated by χ is ℤA∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{ZA}_\infty}$$\end{document}. If χ = {Xi}i=1t is a τ-cycle semibrick, then Γ(ℱ(X))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ({\cal F}({\cal X}))$$\end{document} is ℤA∞/τA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{ZA}_\infty}/{\tau _{\mathbb{A}}}$$\end{document}.

On quasi n-ideals of commutative rings

Abstract: Let R be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of n-ideals and the class of (2, n)-ideals. A proper ideal I of R is said to be a quasi n-ideal if I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt I $$\end{document} is an n-ideal of R. Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the n-ideals, the quasi primary ideals, the (2, n)-ideals and the pr-ideals. Moreover, we use the quasi n-ideals to characterize some kind of rings. Finally, we investigate quasi n-ideals under various contexts of constructions such as direct product, power series, idealization, and amalgamation of a ring along an ideal.

Isomorphisms between graded Frobenius algebras constructed from twisted superpotentials

Abstract: In order to distinguish the connected graded Frobenius algebras determined by different twisted superpotentials, we introduce the nondegeneracy of twisted superpotentials. We give the sufficient and necessary condition for connected graded Frobenius algebras determined by two nondegenerate twisted superpotentials to be isomorphic. As an application, we classify the connected ℤ-graded Frobenius algebra of length 3, whose dimension of the degree 1 is 2.