Precipitous ideals and Σ 1 4 setssets

We prove under the assumption of the existence of a measurable, cardinal and precipitous ideal onw ¹ that every Σ ₁ ³ set is Lebesgue measurable, has the Baire property and is either countable or contians a perfect subset. We get similar results for Σ ₁ ⁴ sets, if we add the additional assumptions of C. H. and that carries a normal precipitous ideal.     

4-Dimensional Frobenius manifolds and Painleve’ VI

A Frobenius manifold has tri-Hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We study the case of the lowest nontrivial dimension \(n=4\) and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painlevé \(\hbox {VI}\mu \) equation. Since the solutions of this equation are known to parametrize semisimple Frobenius manifolds of dimension \(n=3\) , this leads to an explicit procedure mapping 3-dimensional Frobenius structures of 4-dimensional ones, and giving all tri-Hamiltonian structures in four dimensions. We illustrate the construction by computing two examples in the framework of Frobenius structures on Hurwitz spaces.     

(4, k)-Factor-Critical Graphs and Toughness

WorksupportedbytheNSFCandtheNationalEducationCommissionDoctoralFoundationofChian.WorksupportedbyNaturalSciencesandEngineeringResearchCouncilofCanadaundergrantOGP0122059andUniversityCollegeoftheCaribooScholarlyActivityCrant.Thegraphsconsideredinthispaperaresimpleandfinite.LetGbegraphwithvertexsetV(G)andedgesetE(G).Ak-factorofGisaspanningsubgrapliFofGsuchthatdF(x)=kforeachx6V(G),wheredF(x)denotesthedegreeofxinF.ThetoughnessofG,t(G),isdefinedasthefollowing:wherew(G--S)isthenumbe…     

Invariants and Bonnet-type theorem for surfaces in ℝ4

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.     

Willmore Surfaces of ?4 and the Whitney Sphere

We make a contribution to the study of Willmore surfaces infour-dimensional Euclidean space ⁴ by making useof the identification of ⁴ with two-dimensionalcomplex Euclidean space ². We prove that theWhitney sphere is the only Willmore Lagrangian surface of genus zero in⁴ and establish some existence and uniquenessresults about Willmore Lagrangian tori in ^{4 2}.     

On 4-flattening theorems and the curves of Carathéodory, Barner and Segre

We discuss three classes of closed curves in the Euclidean space $\mathbb{R}^{3}$ which have non-vanishing curvature and at least 4 flattenings (points at which the torsion vanishes). Calling these classes (de.ned below) Barner, Segre and Carathéodory, we prove that Barner $\subset$ (Segre $\cap$ Carathéodory). We also prove that (Segre)\ (Segre $\cap$ Carathéodory) and (Carathéodory)\(Segre $\cap$ Carathéodory) are open sets in the space of closed smooth curves with the C-topology. Finally, we define a class of closed curves containing the class of Segre curves and -based on contact topology considerations, as the Huygens principle- we establish the conjecture that any curve of our class has at least 4 flattenings.     

Majorization, 4G Theorem and Schrödinger perturbations

Schrödinger perturbations of transition densities by singular potentials may fail to be comparable with the original transition density. For instance, this is so for the transition density of a subordinator perturbed by any time-independent unbounded potential. In order to estimate such perturbations, it is convenient to use an auxiliary transition density as a majorant and the 4G inequality for the original transition density and the majorant. We prove the 4G inequality for the 1/2-stable and inverse Gaussian subordinators, discuss the corresponding class of admissible potentials and indicate estimates for the resulting transition densities of Schrödinger operators. The connection of the transition densities to their generators is made via the weak-type notion of fundamental solution.     

The Willmore functional and the containment problem in R~4

Let∑be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional∫∑H2dσ. This bound is an invariant involving the area of∑, the volume and Minkowski quermassintegrals of the convex body that∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.     

A New 4 × 4 AKNS Spectral Problem and Its Associated Integrable Decomposition of the AKNS Equation

A new approach to construct a new 4×4 matrix spectral problem from a normal 2×2 matrix spectral problem is presented.AKNS spectral problem is discussed as an example.The isospectral evolution equation of the new 4×4 matrix spectral problem is nothing but the famous AKNS equation hierarchy.With the aid of the binary nonlino earization method,the authors get new integrable decompositions of the AKNS equation. In this process,the r-matrix is used to get the result.     

The Permutations 123p 4…p m and 321p 4…p m are Wilf-Equivalent

Write p ₁, p ₂…p _m for the permutation matrix δ _{pi, j} . Let S _n (M) be the set of n×n permutation matrices which do not contain the m×m permutation matrix M as a submatrix. In [7] Simion and Schmidt show bijectively that |S _n (123) |=|S _n (213) |. In [9] this was generalised to a bijection between S _n (12 p ₃…p _m ) and S _n (21 p ₃…p _m ). In the present paper we obtain a bijection between S _n (123 p ₄…p _m ) and S _n (321 p ₄…p _m ). Revised: March 24, 1999     

Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: III. Moves 1 ↔ 5 and Related Structures

We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. We first write formulas for moves 33 and 24 based on the results in our two previous works and then study moves 15 in detail. Based on this, we obtain the formula for a four-dimensional manifold invariant. As an example, we present a detailed calculation of our invariant for the sphere S ⁴; in particular, the complex does turn out to be acyclic.     

N = 4 Superconformal Algebra in Curved Space and Pseudo-Hyper-Kähler Geometry

We construct the representation of the small N=4 superconformal algebra in curved space under the minimal interaction assumption. We find that the structure relations of the algebra are satisfied within our assumption in the background of the metric of a pseudo-hyper-Kähler manifold.     

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori, Ⅱ

We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S~2×S~1 or RP~3#RP~3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni(2014) in an earlier paper.     

On the 2- and 4-dissections of the Rogers–Ramanujan functions

Basil Gordon showed that if the two Rogers–Ramanujan functions are expanded as series, in each case half the coefficients are almost always even. In order to do this, he gave formulae modulo 2 for the 2-dissections of each of the Rogers–Ramanujan functions. In this paper we give the exact 2- and 4-dissections. We then show how our 2-dissections lead to Gordon’s formulae, we go on to give examples of what he calls “linear zero congruences modulo 2” , and finally we give a simple proof of Gordon’s main result that \(g(2n+1)\) and \(h(2n)\) are almost always even.     

Skew-commutator relations and Gröbner-Shirshov basis of quantum group of type F 4

We give a Grobner-Shirshov basis of quantum group of type F4 by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel- Hall algebras of type F4 by using an ＇inductive＇ method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew- commutator relations; instead, we compute the ＇easier＇ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ＇deduce＇ others from these ＇easier＇ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal CrSbner-Shirshov basis of the positive part of the quantum group of type F4. Dually, we get a Grobner-Shirshov basis of the negative part of the quantum group of type F4. And finally, we give a Gr6bner-Shirshov basis for the whole quantum group of type F4.     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

On Sullivan��s conjecture on cycles in 4-free and 5-free digraphs

For a simple digraph G, let β(G) be the size of the smallest subset X■E(G) such that G-X has no directed cycles, and let γ(G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that β(G) ≤ 0.3819γ(G) if G is a 4-free digraph, and β(G) ≤ 0.2679γ(G) if G is a 5-free digraph. These improve the results of Sullivan in 2008.     

On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of ϑ4

Faulhuber and Steinerberger conjectured that the logarithmic derivative of ${?}_4$ has the property that $y^2{?}_4^{\prime }(y)/{?}_4(y)$ is strictly decreasing and strictly convex. In this small note, we prove this conjecture.