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Mathematics

*Communications in Mathematics*—created in 2009—is the continuation of the journal *Acta Mathematica et Informatica Universitatis Ostraviensis*, founded in 1993, which became *Acta Mathematica Universitatis Ostraviensis* in 2003. *Communications in Mathematics* publishes research and review articles in English in all areas of pure and applied mathematics, mathematical physics and computer science.

- Director of publication: Jan Lata
- Editor-in-chief: Ivan Kaygorodov
- Medium: electronic
- Frequency: annual
- Date created: 2009
- Date of publication on Episciences: 2021
- eISSN: 2336-1298
- Subjects: mathematics
- Languages of publication: English

- Review process: single blind peer review
- CC BY-SA 4.0 licence

- Publisher: University of Ostrava
- Address: Dvořákova 7, 701 03 Ostrava
- Country: Czech Republic

- Contact: cm AT episciences.org

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive scalar products. To each rook placement $D$ and each map $\xi$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one can naturally assign the coadjoint orbit $\Omega_{D,\xi}$ in the dual space $\mathfrak{n}^*$. By definition, $\Omega_{D,\xi}$ is the orbit of $f_{D,\xi}$, where $f_{D,\xi}$ is the sum of root covectors $e_{\alpha}^*$ multiplied by $\xi(\alpha)$, $\alpha\in D$. (In fact, almost all coadjoint orbits studied at the moment have such a form for certain $D$ and $\xi$.) It follows from the results of Andr\`e that if $\xi_1$ and $\xi_2$ are distinct maps from $D$ to $\mathbb{C}^{\times}$ then $\Omega_{D,\xi_1}$ and $\Omega_{D,\xi_2}$ do not coincide for classical root systems $\Phi$. We prove that this is true if $\Phi$ is of type $G_2$, or if $\Phi$ is of type $F_4$ and $D$ is orthogonal.

Ignatev, Mikhail V.

March 22, 2023

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Let $\mathds{k}$ be a real quadratic number field. Denote by $\mathrm{Cl}_2(\mathds{k})$ its $2$-class group and by $\mathds{k}_2^{(1)}$ (resp. $\mathds{k}_2^{(2)}$) its first (resp. second) Hilbert $2$-class field. The aim of this paper is to study, for a real quadratic number field whose discriminant is divisible by one prime number congruent to $3$ modulo 4, the metacyclicity of $G=\mathrm{Gal}(\mathds{k}_2^{(2)}/\mathds{k})$ and the cyclicity of $\mathrm{Gal}(\mathds{k}_2^{(2)}/\mathds{k}_2^{(1)})$ whenever the rank of $\mathrm{Cl}_2(\mathds{k})$ is $2$, and the $4$-rank of $\mathrm{Cl}_2(\mathds{k})$ is $1$.

Azizi, A

March 01, 2023

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Social Sciences and Humanities

*Journal of Philosophical Economics* is a social science journal founded in 2007. Published by Editura ASE, *J. Philos. Econ.* publishes one volume per year containing articles and book reviews in German, English or French.

- Director of publication: Dr. Simona Bușoi
- Editor-in-chief: Valentin Cojanu
- Medium: print and electronic
- Frequency: annual
- Date created: 2007
- Date of publication on Episciences: 2021
- eISSN: 1844-8208
- Subjects: Social sciences, Philosophy, Economic theory
- Languages of publication: English, French, German

- Review process: single blind peer review
- CC BY-NC-SA 4.0 licence

- Publisher: Editura ASE
- Address: Academia de Studii Economice Bucuresti, Piata Romana 6, 010374 Bucuresti
- Country: Romania

- Contact: jpe AT episciences.org

Review of Lehmann, P.-J., Liberalism and Capitalism Today, London, Hoboken: ISTE-Wiley, 2021, 214 pp, ISBN 978-1-78630-689-0

Văduva, Ionuț

February 13, 2023

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The philosophy of the Enlightenment and political thought of modernity found tough opposition in the Roman Catholic Church. Liberalism was associated with Free Masons and revolutionary intent. Nonetheless, liberalism and political economy stimulated some theoretical analysis and specific theoretical positions in terms of social philosophy and social economics by the Church. This paper presents an analysis of encyclical letters and other papal documents, as well as the writings of other Catholic scholars, to elaborate on the theoretical points used to contrast liberalism. Compromises, as well as turning points in the evolution of the Catholic position, are investigated. Lastly, the epistemological and historical reasons for the affinity of Roman Catholicism with ethical liberalism and the limits of this similarity are discussed. 1. Liberal and Catholic, an Italian drama

Solari, Stefano

February 13, 2023

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Informatics and Applied Mathematics

Established in 2021, *TheoretiCS* covers all areas of Theoretical Computer Science (TCS). A joint effort of the TCS community, the journal is published by the TheoretiCS Foundation, a non-profit organisation based in Germany. The journal publishes research papers in English on an ongoing basis, including revised and extended versions of conference papers.

- Director of publication: Thomas Schwentick
- Editors-in-chief: Javier Esparza ; Uri Zwick
- Medium: electronic
- Frequency: continuous
- Date created: 2021
- Date of publication on Episciences: 2021
- eISSN: 2751-4838
- Subjects: Theoretical Computer Science
- Language of publication: English

- Review process: single blind peer review
- CC BY 4.0 licence

- Publisher: TheoretiCS Foundation e.V.
- Address: c/o Thomas Schwentick – Schillingstr. 23, D-44139 Dortmund
- Country: Germany

- Contact: theoretics AT episciences.org

Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of some fixed Hamming weight $k\in [q,n-q]$ must also vanish at all points in $\{0,1\}^n$ of weight $k + q$. This lemma was used by Heged\H{u}s (2009) to give a solution to \emph{Galvin's problem}, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrube\v{s}, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-$2$ threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Heged\H{u}s's lemma. Informally, this version says that if a polynomial of degree $o(q)$ vanishes at most points of weight $k$, then it vanishes at many points of weight $k+q$. We prove this lemma and give three different applications.

Srinivasan, Srikanth

March 01, 2023

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We consider fixpoint algorithms for two-player games on graphs with $\omega$-regular winning conditions, where the environment is constrained by a strong transition fairness assumption. Strong transition fairness is a widely occurring special case of strong fairness, which requires that any execution is strongly fair with respect to a specified set of live edges: whenever the source vertex of a live edge is visited infinitely often along a play, the edge itself is traversed infinitely often along the play as well. We show that, surprisingly, strong transition fairness retains the algorithmic characteristics of the fixpoint algorithms for $\omega$-regular games -- the new algorithms have the same alternation depth as the classical algorithms but invoke a new type of predecessor operator. For Rabin games with $k$ pairs, the complexity of the new algorithm is $O(n^{k+2}k!)$ symbolic steps, which is independent of the number of live edges in the strong transition fairness assumption. Further, we show that GR(1) specifications with strong transition fairness assumptions can be solved with a 3-nested fixpoint algorithm, same as the usual algorithm. In contrast, strong fairness necessarily requires increasing the alternation depth depending on the number of fairness assumptions. We get symbolic algorithms for (generalized) Rabin, parity and GR(1) objectives under strong transition fairness assumptions as well as a direct symbolic algorithm for qualitative winning in stochastic $\omega$-regular games that runs in $O(n^{k+2}k!)$ symbolic steps, improving the state of the art. Finally, we have implemented a BDD-based synthesis engine based on our algorithm. We show on a set of synthetic and real benchmarks that our algorithm is scalable, parallelizable, and outperforms previous algorithms by orders of magnitude.

Banerjee, Tamajit

February 24, 2023

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