A remark on the size of πq (Sn)

Classical results of homotopy theory are used to prove that dim for all primes p. Simultaneously we obtain bounds on the order of the p-torsion subgroup of _q(Sⁿ).partially supported by Studienstiftung des deutschen Volkes     

On the Merkulov Construction of A∞-(Co)Algebras

Ukrainian Mathematical Journal - The aim of this short note is to complete some aspects of a theorem proved by S. Merkulov in [Int. Math. Res. Not. IMRN, 3, 153–167 (1999)] (Theorem 3.4), as...     

Universal Associative Envelopes of (n+ 1)-Dimensional n-Lie Algebras

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie (Filippov) algebras. More generally, for n even and any (n + 1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map L → U(L) is injective. We use noncommutative Gröbner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.     

On the Various Realizations of the Basic Representation of An−1(1) and the Combinatorics of Partitions

The infinite dimensional Lie algebra l _n = A _n–1 ⁽¹⁾ can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l _n corresponding to all partitions of n can be described very simply by using combinatorial constructions.     

On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.     

On the maximum size of Erdős-Ko-Rado sets in H(2d+1, q^2)

Erd?s-Ko-Rado sets in finite classical polar spaces are sets of generators that intersect pairwise non-trivially. We improve the known upper bound for Erd?s-Ko-Rado sets in \(H(2d+1, q^2)\) for \(d>2\) and \(d\) even from approximately \(q^{d^2+d}\) to \(q^{d^2+1}.\)      

On the lattice structure of the set of stable matchings for a many-to-one model ∗

For the many-to-one matching model with firms having substitutable and q-separable preferences we propose two very natural binary operations that together with the unanimous partial ordering of the workers endow the set of stable matchings with a lattice structure. We also exhibit examples in which, under this restricted domain of firms' preferences, the classical binary operations may not even be matching     

On the Evaluation of the Tutte Polynomial at the Points (1, –1) and (2, –1)

Motivated by the identity t (K _n+2; 1, –1) = t (K _n ; 2, –1), where t(G; x, y) is the Tutte polynomial of a graph G, we search for graphs G having the property that there is a pair of vertices u, v such that t(G; 1, –1) = t(G – {u, v}; 2, –1). Our main result gives a sufficient condition for a graph to have this property; moreover, it describes the graphs for which the property still holds when each vertex is replaced by a clique or a coclique of arbitrary order. In particular, we show that the property holds for the class of threshold graphs, a well-studied class of perfect graphs.     

On the solution set of an equation of the type f(t, Φ(u(t)) = 0

We investigate the solution set Γ of an equation of the type f(t, Φ(u(t)) = 0, where Φ is a linear homeomorphism from a topological vector space X onto L ¹(T) and f: T×R → R is a Carathéodory function. More precisely, we characterize the property of Γ of intersecting each closed hyperplane of X.     

On the Integral Curvature of Curve-(III)

In this paper we prove the following theorem.It is a generalization of Tenchel's theorem on theintegral curvature of curve.Theorem.If 1 is the length of a curve C=AB and φ is the angle between the tangent vectors ofC at A,B,then the integral curvature of C     

A characterization of convex TU games by means of the Mas-Colell bargaining set (à la Shimomura)

Within the class of zero-monotonic games, we prove that a cooperative game with transferable utility is convex if and only if the core of the game coincides with the Mas-Colell bargaining set (à la Shimomura, in Int J Game Theory 26:283–302, 1997).     

On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $?M:={\beta }_{\mathrm{s}}^{?}M?M$ of the semialgebraic Stone–Cěch compactification ${\beta }_{\mathrm{s}}^{?}M$ of a semialgebraic set $M?{\mathbb{R}}^m$ in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of ${\beta }_{\mathrm{s}}^{?}M$ that admit a metrizable neighborhood in ${\beta }_{\mathrm{s}}^{?}M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_{\mathrm{s}}^{?}M}({\stackrel{￣}{M}}_{\le 1})?{\stackrel{￣}{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of M and ${\stackrel{￣}{M}}_{\le 1}$ is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets $\stackrel{?}{?}M$ and $\stackrel{?}{?}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ?M and that the differences $?M?\stackrel{?}{?}M$ and $\stackrel{?}{?}M?\stackrel{?}{?}M$ are also dense subsets of ?M. It holds moreover that all the points of $\stackrel{?}{?}M$ have countable systems of neighborhoods in ${\beta }_{\mathrm{s}}^{?}M$.     

On a minimal set of generators for the invariants of 3 × 3 matrices

Let K be a field of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.     

On the reduction (mod 1) of completely monotone functions (0, ∞)

Summary The Euler-Maclaurin summation formula and its harmonic analysis (Poisson) are applied to the case of functions which are completely monotone on an open half-line. What thus results is a curious class of Fourier series, which can be determined explicitly and which represent completely monotone functions on the first half of the period. A by-product is the complete monotony (on the first half-period) of the Bernoulli functions, whether the index is integral or fractional.