Arnold Mathematical Journal最新文献

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).

We characterize the actions of compact tori on smooth closed manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds, the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments, we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections, we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza–Klein model of Dirac’s monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.

We generalize the construction of a toric variety associated with an integer convex polyhedron to construct generalized analytic varieties associated with polyhedra with not necessarily rational vertices. For germs of generalized analytic functions with a given Newton polyhedron (Gamma ), the generalized analytic variety associated with (Gamma ) provides a stratified resolution of singularities of these functions; also ensuring a full resolution for almost all of them. Thus, this constructive and elementary approach replaces the non-effective previous proof of this result based on consecutive blow-ups.

We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e., with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.

We develop a technique for calculating the cohomology groups of spaces of complex parametric knots in ({{mathbb {C}}}^k), (k ge 3), and obtain these groups of low dimensions.

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.

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).

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 (ane 0) and (bne c). Suppose that ({mathcal {G}}) is normal in D such that no sequence in ({mathcal {G}}) converges locally uniformly to infinity in D. If (nge 3) and for each function (fin {mathcal {F}}) there exists (gin {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.

We investigate the effect of an (varepsilon )-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric d-tensor f, a norm (leftVert cdot rightVert ) on the space of symmetric d-tensors, and (varepsilon >0) are given. What is the smallest symmetric tensor rank in the (varepsilon )-neighborhood of f? In other words, what is the symmetric tensor rank of f after a clever (varepsilon )-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind, we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.

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).