1. On G-Drazin partial order in ringsGregor Dolinar, Bojan Kuzma, Janko Marovt, Dijana Mosić, 2024, izvirni znanstveni članek Povzetek: We extend the concept of a G-Drazin inverse from the set $M_n$ of all $n \times n$ complex matrices to the set ${\cal R}^D$ of all Drazin invertible elements in a ring ${\cal R}$ with identity. We also generalize a partial order induced by G-Drazin inverses from $M_n$ to the set of all regular elements in ${\cal R}^D$, study its properties, compare it to known partial orders, and generalize some known results. Ključne besede: Drazin inverse, core-nilpotent decomposition, partial order, annihilator, ring Objavljeno v DiRROS: 05.09.2024; Ogledov: 71; Prenosov: 38 Celotno besedilo (516,17 KB) Gradivo ima več datotek! Več... |
2. On the Gromov hyperbolicity of the minimal metricMatteo Fiacchi, 2024, izvirni znanstveni članek Povzetek: In this paper we study the hyperbolicity in the sense of Gromov of domains in $\mathbb{R}^d$ $(d\geq3)$ with respect to the minimal metric introduced by Forstnerič and Kalaj. In particular, we prove that every bounded strongly minimally convex domain is Gromov hyperbolic and its Gromov compactification is equivalent to its Euclidean closure. Moreover, we prove that the boundary of a Gromov hyperbolic convex domain does not contain non-trivial conformal harmonic disks. Finally, we study the relation between the minimal metric and the Hilbert metric in convex domains. Ključne besede: minimal surfaces, minimal metric, hyperbolic domain, Gromov hyperbolicity, convex domain, Hilbert metric Objavljeno v DiRROS: 05.09.2024; Ogledov: 49; Prenosov: 24 Celotno besedilo (330,70 KB) Gradivo ima več datotek! Več... |
3. Application of a metric for complex polynomials to bounded modification of planar Pythagorean-hodograph curvesRida A. M. T. Farouki, Marjetka Knez, Vito Vitrih, Emil Žagar, 2025, izvirni znanstveni članek Povzetek: By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed. Ključne besede: complex polynomials, inner product, norm, metric, Pythagorean-hodograph curves, bounded modification Objavljeno v DiRROS: 04.09.2024; Ogledov: 63; Prenosov: 657 Celotno besedilo (3,60 MB) Gradivo ima več datotek! Več... |
4. Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian productsJing Tian, Sandi Klavžar, 2024, izvirni znanstveni članek Povzetek: If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $\mu_{\rm t}(G)$ of $G$. Graphs with $\mu_{\rm t}(G) = 0$ are characterized as the graphs in which no vertex is the central vertex of a convex $P_3$. The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, $\mu_{\rm t}(K_n\,\square\, K_m) = \max\{n,m\}$ and $\mu_{\rm t}(T\,\square\, H) = \mu_{\rm t}(T)\mu_{\rm t}(H)$, where $T$ is a tree and $H$ an arbitrary graph. It is also demonstrated that $\mu_{\rm t}(G\,\square\, H)$ can be arbitrary larger than $\mu_{\rm t}(G)\mu_{\rm t}(H)$. Ključne besede: mutual-visibility set, total mutual-visibility set, bypass vertex, Cartesian product of graphs, trees Objavljeno v DiRROS: 26.08.2024; Ogledov: 124; Prenosov: 61 Celotno besedilo (184,44 KB) Gradivo ima več datotek! Več... |
5. Persistent homology with selective Rips complexes detects geodesic circlesŽiga Virk, 2024, izvirni znanstveni članek Povzetek: This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions, we show that such a loop either terminates a 1- dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes. Ključne besede: simple closed geodesic, Rips complex, persistent homology, local winding number Objavljeno v DiRROS: 26.08.2024; Ogledov: 119; Prenosov: 64 Celotno besedilo (906,54 KB) Gradivo ima več datotek! Več... |
6. Complexity of 2-rainbow total domination problemTadeja Kraner Šumenjak, Aleksandra Tepeh, 2024, izvirni znanstveni članek Povzetek: In this paper,we extend the findings of recent studies on $k$-rainbow total domination by placing our focus on its computational complexity aspects. We show that the problem of determining whether a graph has a $2$-rainbow total dominating function of a given weight is NP-complete. This complexity result holds even when restricted to planar graphs. Along the way tight bounds for the $k$-rainbow total domination number of rooted product graphs are established. In addition, we obtain the closed formula for the $k$-rainbow total domination number of the corona product $G ∗ H$, provided that $H$ has enough vertices. Ključne besede: domination, rainbow domination, rooted product, NP-complete Objavljeno v DiRROS: 26.08.2024; Ogledov: 130; Prenosov: 62 Celotno besedilo (391,20 KB) Gradivo ima več datotek! Več... |
7. Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencilsDaniel Kressner, Bor Plestenjak, 2024, izvirni znanstveni članek Povzetek: The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the ▫$\delta$▫-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution. Ključne besede: singular pencil, singular generalized eigenvalue problem, eigenvalue condition number, randomized numerical method, random matrices Objavljeno v DiRROS: 26.08.2024; Ogledov: 125; Prenosov: 65 Celotno besedilo (659,18 KB) Gradivo ima več datotek! Več... |
8. Pairs of fixed points for a class of operators on Hilbert spacesAbdelhak Mokhtari, Kamel Saoudi, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results. Ključne besede: Hilbert spaces, potential operators, genus, fixed point theorem, boundary value problem Objavljeno v DiRROS: 26.08.2024; Ogledov: 138; Prenosov: 60 Celotno besedilo (317,15 KB) Gradivo ima več datotek! Več... |
9. The truncated moment problem on curves $y = q(x)$ and $yx^\ell = 1$Aljaž Zalar, 2024, izvirni znanstveni članek Povzetek: In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form $y = q(x), q(x) \in \mathbb{R} [x], \deg q ≥ 3$ and $yx^\ell = 1, \ell \in \mathbb{N}$ \ $\{1\}$. For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety $y = x^3$. Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety $y = x^3$. Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: 10.1007/s00020-021-02628-6.; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. doi.org/10.1016/j.laa.2022.05.008.; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: 10.1016/j.jmaa.2022. 126563.], and improve Fialkow’s bound to $\deg q − 1$ (resp. $\ell + 1$) for curves $y = q(x)$ (resp. $yx^\ell = 1$). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $k \deg q$ (resp. $k(\ell+ 1)$) for curves $y = q(x)$ (resp. $yx^\ell = 1$) for even degree sequences, while for odd ones they are $k \deg q − \bigl \lceil \frac{\deg q}{2} \bigr \rceil$ (resp. $k(\ell + 1) − \bigl \lfloor \frac{\ell}{2} \bigr \rfloor + 1$) for curves $y = q(x)$ (resp. $yx^\ell = 1$). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves $y = x^\ell, \ell = 2, 3,$ and add a new solvability condition to the even degree case on the curve $y = x^2$. Ključne besede: truncated moment problems, K-moment problems, K-representing measure, minimal measure, moment matrix extensions, positivstellensatz, linear matrix inequality Objavljeno v DiRROS: 25.07.2024; Ogledov: 207; Prenosov: 171 Celotno besedilo (3,50 MB) Gradivo ima več datotek! Več... |
10. Embedded complex curves in the affine planeAntonio Alarcón, Franc Forstnerič, 2024, izvirni znanstveni članek Povzetek: This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces in the affine plane ${\mathbb C}^2$ satisfying interpolation and hitting conditions. We also show that in every compact Riemann surface there is a Cantor set whose complement admits a proper holomorphic embedding in ${\mathbb C}^2$. The focal point is a lemma saying the following. Given a compact bordered Riemann surface, $M$, a closed discrete subset $E$ of its interior ${\mathring M}=M\setminus bM$, a compact subset $K\subset {\mathring M}\setminus E$ without holes in $\mathring M$, and a ${\cal C}^1$ embedding $f: M\hookrightarrow \mathbb C^2$ which is holomorphic in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic embedding $F: bM\hookrightarrow {\mathbb C}^2$ which maps $E\cup bM$ out of a given ball and satisfies some interpolation conditions. Ključne besede: Riemann surfaces, complex curves, complete holomorphic embedding Objavljeno v DiRROS: 15.07.2024; Ogledov: 203; Prenosov: 119 Celotno besedilo (579,03 KB) Gradivo ima več datotek! Več... |