1. 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: 32; Prenosov: 10
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2. 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: 79; Prenosov: 28
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3. Singular $p$-biharmonic problem with the Hardy potentialAmor Drissi, Abdeljabbar Ghanmi, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: The aim of this paper is to study existence results for a singular problem involving the $p$-biharmonic operator and the Hardy potential. More precisely, by combining monotonicity arguments with the variational method, the existence of solutions is established. By using the Nehari manifold method, the multiplicity of solutions is proved. An example is also given to illustrate the importance of these results.
Ključne besede: p-biharmonic equation, variational methods, existence of solutions, Hardy potential, Nehari manifold, fibering map
Objavljeno v DiRROS: 08.07.2024; Ogledov: 105; Prenosov: 45
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5. The powerful class of groupsPrimož Moravec, 2024, izvirni znanstveni članek Povzetek: Pro-$p$ groups of finite powerful class are studied. We prove that these are $p$-adic analytic, and further describe their structure when their powerful class is small. It is also shown that there are only finitely many finite $p$-groups of fixed coclass and powerful class.
Ključne besede: finite p-groups, powerful class, pro-p groups
Objavljeno v DiRROS: 02.07.2024; Ogledov: 88; Prenosov: 57
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6. The truncated moment problem on reducible cubic curves I : Parabolic and circular type relationsSeonguk Yoo, Aljaž Zalar, 2024, izvirni znanstveni članek Povzetek: In this article we study the bivariate truncated moment problem (TMP) of degree $2k$ on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved by the second author (2022), while the degree 6 cases by the first author (2017). Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle, and a union of a line and a parabola. In both cases we also determine the number of atoms in a minimal representing measure.
Ključne besede: truncated moment problems, K-moment problems, K-representing measure, minimal measure, moment matrix extensions
Objavljeno v DiRROS: 18.06.2024; Ogledov: 109; Prenosov: 65
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7. Extending multivariate sub-quasi-copulasDamjana Kokol-Bukovšek, Tomaž Košir, Blaž Mojškerc, Matjaž Omladič, 2024, izvirni znanstveni članek Povzetek: In this paper, we introduce patchwork constructions for multivariate quasi-copulas. These results appear to be new since the kind of approach has been limited to either copulas or only bivariate quasi-copulas so far. It seems that the multivariate case is much more involved, since we are able to prove that some of the known methods of bivariate constructions cannot be extended to higher dimensions. Our main result is to present the necessary and sufficient conditions both on the patch and the values of it for the desired multivariate quasi-copula to exist. We also give all possible solutions.
Ključne besede: mathematics, multivariate analysis
Objavljeno v DiRROS: 18.06.2024; Ogledov: 124; Prenosov: 71
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8. Best possible upper bounds on the restrained domination number of cubic graphsBoštjan Brešar, Michael A. Henning, 2024, izvirni znanstveni članek Povzetek: A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G - S$ obtained by removing all vertices in $S$ is isolate-free. The domination number $\gamma(G)$ and the restrained domination number $\gamma_{r}(G)$ are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of $G$. Let $G$ be a cubic graph of order $n$. A classical result of Reed [Combin. Probab. Comput. 5 (1996), 277-295] states that $\gamma(G) \le \frac{3}{8}n$, and this bound is best possible. To determine a best possible upper bound on the restrained domination number of $G$ is more challenging, and we prove that $\gamma_{r}(G) \le \frac{2}{5}n$.
Ključne besede: domination, restrained domination, cubic graphs
Objavljeno v DiRROS: 18.06.2024; Ogledov: 108; Prenosov: 74
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9. Generalized noncooperative Schrödinger-Kirchhoff-type systems in ${\mathbb R}^N$Nabil Chems Eddine, Dušan Repovš, 2024, izvirni znanstveni članek Povzetek: We consider a class of noncooperative Schrödinger-Kirchhof-type system, which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration-compactness principle for weighted-variable exponent Sobolev spaces and the principle of symmetric criticality of Krawcewicz and Marzantowicz.
Ključne besede: concentration–compactness principle, critical points theory, critical Sobolev exponents, generalized capillary operator, limit index theory, p-Laplacian, p(x)-Laplacian, Palais–Smale condition, Schrödinger-Kirchhoff-type problems, weighted exponent spaces
Objavljeno v DiRROS: 17.06.2024; Ogledov: 121; Prenosov: 73
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10. Relaxations and exact solutions to Quantum Max Cut via the algebraic structure of swap operatorsAdam Bene Watts, Anirban Chowdhury, Aidan Epperly, J. William Helton, Igor Klep, 2024, izvirni znanstveni članek Povzetek: The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance $10^{-7}$) on all QMC instances with uniform edge weights on graphs with at most 8 vertices. The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.
Ključne besede: Quantum Max Cut, swap operators, noncommutative polynomials, symmetric group, Gröbner bases
Objavljeno v DiRROS: 04.06.2024; Ogledov: 144; Prenosov: 79
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