Covering properties and square principles

Covering matrices were used by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom and later by Sharon and Viale to investigate the impact of stationary reflection on the approachability ideal. In the course of this work, they isolated two reflection principles, CP and S, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of CP and S and variations on Jensen’s square principle. We prove that, for a regular cardinal λ > ω ₁, assuming large cardinals, □(λ, 2) is consistent with CP(λ, θ) for all θ with θ ⁺ < λ. We demonstrate how to force nice θ-covering matrices for λ which fail to satisfy CP and S. We investigate normal covering matrices, showing that, for a regular uncountable κ, □ _κ implies the existence of a normal ω-covering matrix for κ ⁺ but that cardinal arithmetic imposes limits on the existence of a normal θ-covering matrix for κ ⁺ when θ is uncountable. We introduce the notion of a good point for a covering matrix, in analogy with good points in PCF-theoretic scales. We develop the basic theory of these good points and use this to prove some non-existence results about covering matrices. Finally, we investigate certain increasing sequences of functions which arise from covering matrices and from PCF-theoretic considerations and show that a stationary reflection hypothesis places limits on the behavior of these sequences.     

On the asymptotic optimality of a solution of the euclidean problem of covering a graph by <Emphasis Type="Italic">m</Emphasis> nonadjacent cycles of maximum total weight

In the problem of covering an n-vertex graph by m cycles of maximum total weight, it is required to find a family of m vertex-nonadjacent cycles such that it covers all vertices of the graph and the total weight of edges in the cover is maximum. The paper presents an algorithm for approximately solving the problem of covering a graph in Euclidean d-space R^d by m nonadjacent cycles of maximum total weight. The algorithm has time complexity O(n³). An estimate of the accuracy of the algorithm depending on the parameters d, m, and n is substantiated; it is shown that if the dimension d of the space is fixed and the number of covering cycles is m = o(n), then the algorithm is asymptotically exact.     

Approximations and Mittag-Leffler conditions the applications

A classic result by Bass says that the class of all projective modules is covering if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules C, which is precovering and closed under direct limits, is covering, and asked whether the converse is true. We employ the tools developed in [18] and give a positive answer when C = A, or C is the class of all locally A^≤ω-free modules, where A is any class of modules fitting in a cotorsion pair (A, B) such that B is closed under direct limits. This setting includes all cotorsion pairs and classes of locally free modules arising in (infinite-dimensional) tilting theory. We also consider two particular applications: to pure-semisimple rings, and Artin algebras of infinite representation type.     

Parallel covering of isosceles triangles with squares

Suppose that T^h is an isosceles triangle with base length 1 and with height h. Let S be a square with a side parallel to the base of T^h and let {S_n} be a sequence of the homothetic copies of S. We first determine the bound of sums of areas of squares from the sequence {S_n} that permits a parallel covering of T^h. Then we show that the results about covering isosceles triangles are also true for acute or right triangles.     

The cubic Fermat equation and complex multiplication on the Deuring normal form

The arithmetic on elliptic curves in Deuring normal form is shown to be related to solutions of the Fermat equation 27X ³+27Y ³=X ³ Y ³. This arithmetic is used to give conditions for the existence of multipliers μ on supersingular elliptic curves in characteristic p for which μ ²=?3p. Together with an explicit factorization of a certain class equation, these conditions imply that the number of irreducible binomial quadratic factors (mod p) of the Legendre polynomial P _(p?e)/3(x) of degree (p?e)/3 is a simple linear function of the class number of the quadratic field \(\mathbb{Q}(\sqrt{-3p})\).     

Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

We present a simple and practical (1+ε)-approximation algorithm for the Fréchet distance between two polygonal curves in ? ^d . To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low-density polygonal curves.     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.     

Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles

We consider the m-Cycle Cover Problem of covering a complete undirected graph by m vertex-nonadjacent cycles of extremal total edge weight. The so-called TSP approach to the construction of an approximation algorithm for this problem with the use of a solution of the traveling salesman problem (TSP) is presented. Modifications of the algorithm for the Euclidean Max m-Cycle Cover Problem with deterministic instances (edge weights) in a multidimensional Euclidean space and the Random Min m-Cycle Cover Problem with random instances UNI(0,1) are analyzed. It is shown that both algorithms have time complexity O(n ³) and are asymptotically optimal for the number of covering cycles m = o(n) and \(m \leqslant \frac{{n^{1/3} }}{{\ln n}}\), respectively.     

Hochschild Cohomology of skew group rings and invariants

Let A be a k-algebra and G be a group acting on A. We show that G also acts on the Hochschild cohomology algebra HH ^⊙ (A) and that there is a monomorphism of rings HH ^⊙ (A) ^G →HH ^⊙ (A[G]). That allows us to show the existence of a monomorphism from HH ^⊙ (Ã) ^G into HH ^⊙ (A), where à is a Galois covering with group G.     

Non-generic unramified representations in metaplectic covering groups

Let G^(r) denote the metaplectic covering group of the linear algebraic group G. In this paper we study conditions on unramified representations of the group G^(r) not to have a nonzero Whittaker function. We state a general Conjecture about the possible unramified characters χ such that the unramified subrepresentation of \(Ind_{{B^{\left( r \right)}}}^{{G^{\left( r \right)}}}{X^{\delta _B^{1/2}}}\) will have no nonzero Whittaker function. We prove this Conjecture for the groups GL _n ^{( r)} with r ≥ n ? 1, and for the exceptional groups G ₂ ^{( r)} when r ≠ 2.     

Frobenius nonclassicality of Fermat curves with respect to cubics

For Fermat curves F: aX ⁿ + bY ⁿ = Z ⁿ defined over F _q , we establish necessary and sufficient conditions for F to be F _q -Frobenius nonclassical with respect to the linear system of plane cubics. In the new F _q -Frobenius nonclassical cases, we determine explicit formulas for the number N _q (F) of F _q -rational points on F. For the remaining Fermat curves, nice upper bounds for N _q (F) are immediately given by the Stöhr–Voloch Theory.     

Baum-Bott indices for curves of singularities

Let F be a holomorphic foliation on Pⁿ by curves such that the components of its singular locus are curves C_i and points p_j. We compute the Baum-Bott indices BB_φ(F, C_i) in terms of the main invariants of F and C_i. We also determine the sum of the BB_φ(F, p_i) in terms of the same invariants.When φ corresponds to the determinant, the latter result generalizes, from special to all holomorphic foliations, a formula for the number of isolated singularities of F, counted with multiplicities.     

On the gonality sequence of smooth curves

Let C be a smooth curve of genus g. For each positive integer r the r-gonality d _r (C) of C is the minimal integer t such that there is \({L\in {\rm Pic}^t(C)}\) with h ⁰(C, L) = r + 1. Here we use nodal plane curves to construct several smooth curves C with d ₂(C)/2 < d ₃(C)/3, i.e., for which a slope inequality fails.     

Quadratic differentials (A(z − a)(z − b)/(z − c)2) dz2 and algebraic Cauchy transform

We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A(z ? a)(z ? b)×(z ? c)^?2 dz². More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.     

Relative Gorenstein injective covers with respect to a semidualizing module

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to C which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every G _C -injective module G, the character module G ⁺ is G _C -flat, then the class \(\mathcal{GI}_{C}(R)\cap\mathcal{A}_C(R)\) is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class \(\mathcal{GI}_{C}(R)\cap\mathcal{A}_C(R)\) is covering.     

Steepest descent curves of convex functions on surfaces of constant curvature

Let S be a complete surface of constant curvature K = ±1, i.e., S ² or л ², and Ω ? S a bounded convex subset. If S = S ², assume also diameter(Ω) < π/2. It is proved that the length of any steepest descent curve of a quasi-convex function in Ω is less than or equal to the perimeter of Ω. This upper bound is actually proved for the class of G-curves, a family of curves that naturally includes all steepest descent curves. In case S = S ², the existence of G-curves, whose length is equal to the perimeter of their convex hull, is also proved, showing that the above estimate is indeed optimal. The results generalize theorems by Manselli and Pucci on steepest descent curves in the Euclidean plane.     

Variations on a theorem of Arhangel’skii and Pytkeev

We identify continuous real-valued functions on a Tychonoff space X with their (closed) graphs thus allowing for C(X) to naturally inherit the lower Vietoris topology from the ambient hyperspace. We then calculate a bitopological version of tightness using the weak Lindelöf numbers of finite powers of X. We also characterize bitopological versions of countable fan and strong fan tightness of the point-open topology with respect to the lower Vietoris topology on C(X) in terms of suitable covering properties of the powers X ⁿ formulated using the language of S ₁ and S _fin selection principles.     

Invariant ordering on the simply connected covering of the Shilov boundary of a symmetric domain

The Shilov boundary of a symmetric domain D = G/K of tube type has the form G/P, where P is a maximal parabolic subgroup of the group G. We prove that the simply connected covering of the Shilov boundary possesses a unique (up to inversion) invariant ordering, which is induced by the continuous invariant ordering on the simply connected covering of G and can readily be described in terms of the corresponding Jordan algebra.     

Lower Bounds for the Degree of a Branched Covering of a Manifold

The problem of finding new lower bounds for the degree of a branched covering of a manifold in terms of the cohomology rings of this manifold is considered. This problem is close to M. Gromov’s problem on the domination of manifolds, but it is more delicate. Any branched (finite-sheeted) covering of manifolds is a domination, but not vice versa (even up to homotopy). The theory and applications of the classical notion of the group transfer and of the notion of transfer for branched coverings are developed on the basis of the theory of n-homomorphisms of graded algebras.The main result is a lemma imposing conditions on a relationship between the multiplicative cohomology structures of the total space and the base of n-sheeted branched coverings of manifolds and, more generally, of Smith–Dold n-fold branched coverings. As a corollary, it is shown that the least degree n of a branched covering of the N-torus T ^N over the product of k 2-spheres and one (N ? 2k)-sphere for N ≥ 4k + 2 satisfies the inequality n ≥ N ? 2k, while the Berstein–Edmonds well-known 1978 estimate gives only n ≥ N/(k + 1).     

A characterization of the Ejiri torus in <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

We conjecture that a Willmore torus having Willmore functional between 2π ² and 2π ² \(\sqrt 3 \) is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S ⁵ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S ⁿ by reducing them into elastic curves in S ³, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S ⁵ attains the minimum 2π ² \(\sqrt 3 \), which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S ⁵. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S ³. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.