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Abstract: Abstract We investigate differential geometric properties of a parabolic point of a surface in the Euclidean three space. We introduce the contact cylindrical surface which is a cylindrical surface having a degenerate contact type with the original surface at a parabolic point. Furthermore, we show that such a contact property gives a characterization to the \(\mathcal {A}\) -singularity of the orthogonal projection of a surface from the asymptotic direction. PubDate: 2024-06-25 DOI: 10.1007/s40598-024-00251-y
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Abstract: Abstract Linear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence \({F}_{n}={F}_{n-1}+{F}_{n-2}\) , have been intensely studied over millennia and yet still hide interesting undiscovered mathematics. Such a recursion was used by Apéry in his proof of the irrationality of \(\zeta \left(3\right)\) , which was later named the Apéry constant. Apéry’s proof used a specific linear recursion that contained integer polynomials (polynomially recursive) and formed a continued fraction; such formulas are called polynomial continued fractions (PCFs). Similar polynomial recursions can be used to prove the irrationality of other fundamental constants such as \(\pi\) and \(e\) . More generally, the sequences generated by polynomial recursions form Diophantine approximations, which are ubiquitous in different areas of mathematics such as number theory and combinatorics. However, in general it is not known which polynomial recursions create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we present general conclusions and conjectures about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as \(\pi\) , \(e\) , \(\zeta \left(3\right)\) , and the Catalan constant. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new conjectures about Diophantine approximations based on PCFs. Looking forward, our findings could motivate a search for a wider theory on sequences created by any linear recursions with integer coefficients. Such results can help the development of systematic algorithms for finding Diophantine approximations of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., \(\zeta \left(5\right)\) ). PubDate: 2024-06-18 DOI: 10.1007/s40598-024-00250-z
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Abstract: Abstract We investigate the question of how many subgroups of a finite group are not in its Chermak–Delgado lattice. The Chermak–Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasolă and Tărnăuceanu (Bull Aust Math Soc 107(3):451–455, 2023) asked how many subgroups are not in the Chermak–Delgado lattice and classified all groups with two or less subgroups not in the Chermak–Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak–Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak–Delgado lattice is nilpotent. In this vein, we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak–Delgado lattice is \(S_3\) . PubDate: 2024-06-01 DOI: 10.1007/s40598-023-00237-2
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Abstract: Abstract The existence of a conjugate point on the volume-preserving diffeomorphism group of a compact Riemannian manifold M is related to the Lagrangian stability of a solution of the incompressible Euler equation on M. The Misiołek curvature is a reasonable criterion for the existence of a conjugate point on the volume-preserving diffeomorphism group corresponding to a stationary solution of the incompressible Euler equation. In this article, we introduce a class of stationary solutions on an arbitrary Riemannian manifold whose behavior is nice with respect to the Misiołek curvature and give a positivity result of the Misiołek curvature for solutions belonging to this class. Moreover, we also show the existence of a conjugate point in the three-dimensional ellipsoid case as its corollary. PubDate: 2024-06-01 DOI: 10.1007/s40598-023-00238-1
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Abstract: Abstract The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes. In Pukhlikov and Khovanskiĭ (Algebra i Analiz 4(4):188–216, 1992), Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over \({\mathbb {C}}\) can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley–Reisner algebras. In Sankaran and Uma (Comment Math Helv 78(3):540–554, 2003), Sankaran and Uma generalized the “Stanley–Reisner description” to the case of toric bundles, i.e., equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion–Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety G/B. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties G/B previously obtained by Kaveh (J Lie Theory 21(2):263–283, 2011). PubDate: 2024-06-01 DOI: 10.1007/s40598-023-00233-6
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Abstract: Abstract In this paper, the normality of a family of meromorphic functions is deduced from the normality of a given family. Precisely, we have proved: Let \({\mathcal {F}}\) and \({\mathcal {G}}\) be two families of meromorphic functions on a domain D, and \(a,\ b,\ c\) be three finite complex numbers such that \(a\ne 0\) and \(b\ne c\) . Suppose that \({\mathcal {G}}\) is normal in D such that no sequence in \({\mathcal {G}}\) converges locally uniformly to infinity in D. If \(n\ge 3\) and for each function \(f\in {\mathcal {F}}\) there exists \(g\in {\mathcal {G}}\) such that \(f^{'}-af^{n}\) and \(g^{'}-ag^{n}\) partially share the values b and c, then \({\mathcal {F}}\) is normal in D. Further, examples are given to establish the sharpness of the result. PubDate: 2024-06-01 DOI: 10.1007/s40598-023-00236-3
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Abstract: Abstract We prove that the space of affine, transversal at infinity, nonsingular real cubic surfaces has 15 connected components. We give a topological criterion to distinguish them and show also how these 15 components are adjacent to each other via wall-crossing. PubDate: 2024-06-01 DOI: 10.1007/s40598-023-00231-8
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Abstract: Abstract A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the pressure at any point. Such solutions are called Gavrilov flows. We describe the local structure of Gavrilov flows in terms of the geometry of isobaric hypersurfaces. In the 3D case, we obtain a system of PDEs for axisymmetric Gavrilov flows and find consistency conditions for the system. Two numerical examples of axisymmetric Gavrilov flows are presented: with pressure function periodic in the axial direction, and with isobaric surfaces diffeomorphic to the torus. PubDate: 2024-06-01 DOI: 10.1007/s40598-023-00234-5
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Abstract: Abstract Let A denote the cylinder \({\mathbb {R}} \times S^1\) or the band \({\mathbb {R}} \times I\) , where I stands for the closed interval. We consider 2-moderate immersions of closed curves (“doodles”) and compact surfaces (“blobs”) in A, up to cobordisms that also are 2-moderate immersions in \(A \times [0, 1]\) of surfaces and solids. By definition, the 2-moderate immersions of curves and surfaces do not have tangencies of order \(\ge 3\) to the fibers of the obvious projections \(A \rightarrow S^1\) , \(A \times [0, 1] \rightarrow S^1 \times [0, 1]\) or \(A \rightarrow I\) , \(A \times [0, 1] \rightarrow I \times [0, 1]\) . These bordisms come in different flavors: in particular, we consider one flavor based on regular embeddings of doodles and blobs in A. We compute the bordisms of regular embeddings and construct many invariants that distinguish between the bordisms of immersions and embeddings. In the case of oriented doodles on \(A= {\mathbb {R}} \times I\) , our computations of 2-moderate immersion bordisms \(\textbf{OC}^{\textsf{imm}}_{\mathsf {moderate \le 2}}(A)\) are near complete: we show that they can be described by an exact sequence of abelian groups $$\begin{aligned} 0 \rightarrow {\textbf{K}} \rightarrow \textbf{OC}^{\textsf{imm}}_{\mathsf {moderate \le 2}}(A)\big /\textbf{OC}^{\textsf{emb}}_{\mathsf {moderate \le 2}}(A) {\mathop {\longrightarrow }\limits ^{{\mathcal {I}} \rho }} {\mathbb {Z}} \times {\mathbb {Z}} \rightarrow 0, \end{aligned}$$ where \(\textbf{OC}^{\textsf{emb}}_{\mathsf {moderate \le 2}}(A) \approx {\mathbb {Z}} \times {\mathbb {Z}}\) , the epimorphism \({\mathcal {I}} \rho \) counts different types of crossings of immersed doodles, and the kernel \({\textbf{K}}\) contains the group \(({\mathbb {Z}})^\infty \) whose generators are described explicitly. PubDate: 2024-05-16 DOI: 10.1007/s40598-024-00249-6
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Abstract: Abstract We study the existence of points on a compact oriented surface at which a symmetric bilinear two-tensor field is conformal to a Riemannian metric. We give applications to the existence of conformal points of surface diffeomorphisms and vector fields. PubDate: 2024-04-12 DOI: 10.1007/s40598-024-00248-7
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Abstract: Abstract As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra \(\mathfrak g_2\) . In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence. PubDate: 2024-04-04 DOI: 10.1007/s40598-024-00247-8
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Abstract: Abstract We prove that the Euler characteristic of a collapsing Alexandrov space (in particular, a Riemannian manifold) is equal to the sum of the products of the Euler characteristics with compact support of the strata of the limit space and the Euler characteristics of the fibers over the strata. This was conjectured by Semyon Alesker. PubDate: 2024-03-18 DOI: 10.1007/s40598-024-00246-9
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Abstract: Abstract In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, \(X=T^{*}{\text {Gr}}(k,n)\) . This integral representation can be used to compute the \(\hbar \rightarrow \infty \) limit of the vertex function, where \(\hbar \) denotes the equivariant parameter of a torus acting on X by dilating the cotangent fibers. We show that in this limit, the integral turns into the standard mirror integral representation of the A-series of the Grassmannian \({\text {Gr}}(k,n)\) with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong. PubDate: 2024-03-15 DOI: 10.1007/s40598-024-00245-w
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Abstract: Abstract We use spinal open books to construct contact manifolds with infinitely many different Weinstein fillings in any odd dimension \(> 1,\) which were previously unknown for dimensions equal to \(4n+1.\) The argument does not involve understanding factorizations in the symplectic mapping class group. PubDate: 2024-03-01 DOI: 10.1007/s40598-024-00244-x
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Abstract: Abstract We establish the existence of stationary solutions for certain systems of reaction–diffusion-type equations in the corresponding \(H^{2}\) spaces. Our method relies on the fixed point theorem when the elliptic problem contains second-order differential operators with and without the Fredholm property, which may depend on the outcome of the competition between the natality and the mortality rates involved in the equations of the systems. PubDate: 2024-03-01 DOI: 10.1007/s40598-023-00225-6
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Abstract: Abstract For the calculation of Springer numbers (of root systems) of type \(B_n\) and \(D_n\) , Arnold introduced a signed analogue of alternating permutations, called \(\beta _n\) -snakes, and derived recurrence relations for enumerating the \(\beta _n\) -snakes starting with k. The results are presented in the form of double triangular arrays ( \(v_{n,k}\) ) of integers, \(1\le k \le n\) . An Arnold family is a sequence of sets of such objects as \(\beta _n\) -snakes that are counted by \((v_{n,k})\) . As a refinement of Arnold’s result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of \(\tan x\) and \(\sec x\) , established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations. PubDate: 2024-03-01 DOI: 10.1007/s40598-023-00230-9
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Abstract: Abstract In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated with cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky’s cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper, we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework, we completely determine the red numbers and unrestricted red numbers for all finite mutation type of quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general. PubDate: 2024-03-01 DOI: 10.1007/s40598-023-00226-5
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Abstract: Abstract The aim of this paper is to study in details the regular holonomic \(D-\) module introduced in Barlet (Math Z 302 \(n^03\) : 1627–1655, 2022 arXiv:1911.09347 [math]) whose local solutions outside the polar hyper-surface \(\{\Delta (\sigma ).\sigma _k = 0 \}\) are given by the local system generated by the power \(\lambda \) of the local branches of the multivalued function which is the root of the universal degree k equation \(z^k + \sum _{h=1}^k (-1)^h\sigma _hz^{k-h} = 0 \) . We show that for \(\lambda \in \mathbb {C} {\setminus } \mathbb {Z}\) this D-module is the minimal extension of the holomorphic vector bundle with an integrable meromorphic connection with a simple pole which is its restriction on the open set \(\{\sigma _k\Delta (\sigma ) \not = 0\}\) . We then study the structure of these D-modules in the cases where \(\lambda = 0, 1, -1\) which are a little more complicated, but which are sufficient to determine the structure of all these D-modules when \(\lambda \) is in \(\mathbb {Z}\) . As an application we show how these results allow to compute, for instance, the Taylor expansion of the root near \(-1\) of the equation: $$\begin{aligned} z^k + \sum _{h=-1}^k (-1)^h\sigma _hz^{k-h} - (-1)^k = 0. \end{aligned}$$ near \(z^k - (-1)^k = 0\) . PubDate: 2024-03-01 DOI: 10.1007/s40598-023-00229-2
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Abstract: Abstract We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences, some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions. PubDate: 2024-03-01 DOI: 10.1007/s40598-023-00227-4
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Abstract: Abstract We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let A be the positive symmetric matrix taking the outer ellipsoid to the inner one. If \({\text {tr}}\, A = 1\) , there exists a bijection between the orthogonal group O(n) and the set of such labeled simplices. Similarly, if \({\text {tr}}\, A^2 = 1\) , there are families of parallelotopes and of cross polytopes, also indexed by O(n). PubDate: 2024-01-22 DOI: 10.1007/s40598-023-00243-4
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