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Abstract: Abstract Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined. PubDate: 2022-04-26

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Abstract: Abstract In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties. PubDate: 2022-04-22

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Abstract: Abstract We prove various mathematical aspects of the quantitative uncertainty principles, including Donoho–Stark’s uncertainty principle and a variant of Benedicks theorem for Lions transform. PubDate: 2022-04-16

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Abstract: Abstract The fundamental theorem of affine geometry says that if a self-bijection f of an affine space of dimenion n over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then f of the expected type, namely f is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type. PubDate: 2022-03-24

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Abstract: Abstract A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of bicritical circle maps, which are \(C^3\) circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377, 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies \(C^{1+\alpha }\) rigidity for real-analytic bicritical circle maps with rotation number of bounded type (Corollary 1.1). PubDate: 2022-03-03 DOI: 10.1007/s40598-022-00199-x

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Abstract: Abstract A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree. PubDate: 2022-02-22 DOI: 10.1007/s40598-022-00197-z

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Abstract: Abstract We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called catastrophe, since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra, we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself. PubDate: 2022-02-18 DOI: 10.1007/s40598-021-00193-9

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Abstract: Abstract We prove an \({{\mathbb {F}}}_p\) -Selberg integral formula, in which the \({{\mathbb {F}}}_p\) -Selberg integral is an element of the finite field \({{\mathbb {F}}}_p\) with odd prime number p of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo p. PubDate: 2022-01-24 DOI: 10.1007/s40598-021-00191-x

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Abstract: Abstract When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in a neighborhood of a singular point' The present paper, of a survey nature, presents a research program around this question. A way to answer is to use normal forms. However, there are large classes of dynamical systems for which the change of coordinates to a normal form diverges. In this paper, we discuss the case of singularities for which the normalizing transformation is k-summable, thus allowing to provide moduli spaces. We explain the common geometric features of these singularities, and show that the study of their unfoldings allows understanding both the singularities themselves, and the geometric obstructions to convergence of the normalizing transformations. We also present some moduli spaces for generic k-parameter families unfolding such singularities. PubDate: 2022-01-20 DOI: 10.1007/s40598-021-00196-6

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Abstract: Abstract This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021. PubDate: 2022-01-17 DOI: 10.1007/s40598-022-00198-y

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Abstract: Abstract We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or \(\tau \) -model), or as three permutations on the set of half-edges (rotation system or \(\sigma \) -model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality. PubDate: 2022-01-03 DOI: 10.1007/s40598-021-00194-8

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Abstract: Abstract We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory. PubDate: 2021-12-20 DOI: 10.1007/s40598-021-00195-7

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Abstract: Abstract This article is devoted to the geometry of billiard trajectories in a regular polygon and geodesics on the surface of a regular polyhedron. Main results are formulated as conjectures based on ample computer experimentation. PubDate: 2021-12-01 DOI: 10.1007/s40598-020-00170-8

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Abstract: Abstract We consider an embedded n-dimensional compact complex manifold in \(n+d\) dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of \(C_n\) in \(M_{n+d}\) is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in \(M_{n+d}\) having \(C_n\) as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors condition arising from solutions to their cohomological equations. PubDate: 2021-10-15 DOI: 10.1007/s40598-021-00192-w

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Abstract: Abstract Let \({\mathcal {M}}\) denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote \(k^{\times }\) , and a uniformizer we denote \(\pi \) . In this paper, we consider the map \(T_{v}: {\mathcal {M}} \rightarrow {\mathcal {M}}\) defined by $$\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$$ where b(x) denotes the equivalence class to which \(\frac{\pi ^{v(x)}}{x}\) belongs in \(k^{\times }\) . We show that \(T_v\) preserves Haar measure \(\mu \) on the compact abelian topological group \({\mathcal {M}}\) . Let \({\mathcal {B}}\) denote the Haar \(\sigma \) -algebra on \({\mathcal {M}}\) . We show the natural extension of the dynamical system \(({\mathcal {M}}, {\mathcal {B}}, \mu , T_v)\) is Bernoulli and has entropy \(\frac{\#( k)}{\#( k^{\times })}\log (\#( k))\) . The first of these two properties is used to study the average behaviour of the convergents arising from \(T_v\) . Here for a finite set A its cardinality has been denoted by \(\# (A)\) . In the case \(K = {\mathbb {Q}}_p\) , i.e. the field of p-adic numbers, the map \(T_v\) reduces to the well-studied continued fraction map due to Schneider. PubDate: 2021-10-15 DOI: 10.1007/s40598-021-00190-y

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Abstract: Abstract A \(d^{\{n\}}\) -cage \(\mathsf K\) is the union of n groups of hyperplanes in \(\mathbb P^n\) , each group containing d members. The hyperplanes from the distinct groups are in general position, thus producing \(d^n\) points where hyperplanes from all groups intersect. These points are called the nodes of \(\mathsf K\) . We study the combinatorics of nodes that impose independent conditions on the varieties \(X \subset \mathbb P^n\) containing them. We prove that if X, given by homogeneous polynomials of degrees \(\le d\) , contains the points from such a special set \(\mathsf A\) of nodes, then it contains all the nodes of \(\mathsf K\) . Such a variety X is very special: in particular, X is a complete intersection. PubDate: 2021-09-30 DOI: 10.1007/s40598-021-00189-5

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Abstract: Abstract Chord diagrams and 4-term relations were introduced by Vassiliev in the late 1980. Various constructions of weight systems are known, and each of such constructions gives rise to a knot invariant. In particular, weight systems may be constructed from Lie algebras as well as from the so-called 4-invariants of graphs. A Chmutov–Lando theorem states that the value of the weight system constructed from the Lie algebra \(\mathfrak {sl}_2\) on a chord diagram depends on the intersection graph of the diagram, rather than the diagram itself. This inspired a question due to Lando about whether it is possible to extend the weight system \(\mathfrak {sl}_2\) to a graph invariant satisfying the four term relations for graphs. We show that for all graphs with up to 8 vertices such an extension exists and is unique, thus answering in affirmative to Lando’s question for small graphs. PubDate: 2021-09-06 DOI: 10.1007/s40598-021-00187-7

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Abstract: Abstract We consider a pair of games where two players alternately select previously unselected elements of \(\mathbb {Z}_n\) given a particular starting element. On each turn, the player either adds or multiplies the element they selected to the result of the previous turn. In one game, the first player wins if the final result is 0; in the other, the second player wins if the final result is 0. We determine which player has the winning strategy for both games except for the latter game with nonzero starting element when \(n \in \{2p,4p\}\) for some odd prime p. PubDate: 2021-08-18 DOI: 10.1007/s40598-021-00185-9

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Abstract: Abstract We classify global surfaces of section for flows on 3-manifolds defining Seifert fibrations. We discuss branched coverings—one way or the other—between surfaces of section for the Hopf flow and those for any other Seifert fibration of the 3-sphere, and we relate these surfaces of section to algebraic curves in weighted complex projective planes. PubDate: 2021-08-05 DOI: 10.1007/s40598-021-00184-w

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Abstract: Abstract We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e., a surface \(Z\subset {\mathbb {R}}{\mathrm {P}}^3\) defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group O(4) by change of variables. Such invariant distributions are completely described by one parameter \(\lambda \in [0,1]\) and as a function of this parameter the expected number of real lines equals: $$\begin{aligned} E_\lambda =\frac{9(8\lambda ^2+(1-\lambda )^2)}{2\lambda ^2+(1-\lambda )^2}\left( \frac{2\lambda ^2}{8\lambda ^2+(1-\lambda )^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8\lambda ^2+(1-\lambda )^2}{20\lambda ^2+(1-\lambda )^2}}\right) . \end{aligned}$$ This result generalizes previous results by Basu et al. (Math Ann 374(3–4):1773–1810, 2019) for the case of a Kostlan polynomial, which corresponds to \(\lambda =\frac{1}{3}\) and for which \(E_{\frac{1}{3}}=6\sqrt{2}-3.\) Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case \(\lambda =1\) and for which \(E_1=24\sqrt{\frac{2}{5}}-3\) . PubDate: 2021-07-02 DOI: 10.1007/s40598-021-00182-y