Affine Mirković-Vilonen polytopes

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra \(\mathfrak{g}\) has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given \(\mathfrak{g}\) , there is a limit crystal, usually denoted by B(?∞), which contains all the other crystals. When \(\mathfrak{g}\) is finite dimensional, a convex polytope, called the Mirkovi?-Vilonen polytope, can be associated to each element in B(?∞). This polytope sits in the dual space of a Cartan subalgebra of \(\mathfrak{g}\) , and its edges are parallel to the roots of \(\mathfrak{g}\) . In this paper, we generalize this construction to the case where \(\mathfrak{g}\) is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(?∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as  \(\mathfrak{g}\) . The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category \(\Lambda \text {\upshape -}\mathrm {mod}\) . The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.     

The smallest Randić index for trees

The general Randi? index R _α (G) is the sum of the weight d(u)d(v) ^α over all edges uv of a graph G, where α is a real number and d(u) is the degree of the vertex u of G. In this paper, for any real number α?≠?0, the first three minimum general Randi? indices among trees are determined, and the corresponding extremal trees are characterized.     

On a question of H. Kraljević

In this paper, assuming a certain set-theoretic hypothesis, a positive answer is given to a question of H. Kraljevi, namely it is shown that there exists a Lebesgue measurable subsetA of the real line such that the set {c R: A + cA contains an interval} is nonmeasurable. Here the setA + cA = {a + ca: a, a A}. Two other results about sets of the formA + cA are presented.     

A quadratic programming approach to the Randić index

Let G(k, n) be the set of connected graphs without multiple edges or loops which have n vertices and the minimum degree of vertices is k. The Randi? index χ = χ(G) of a graph G   is defined by χ(G)=_∑(uv)(_δuδv)^-1/2$\chi (G)={\sum }_{(\mathit{uv})}({\delta }_u{\delta }_v{)}^{-1/2}$, where δ_u is the degree of vertex u and the summation extends over all edges (uv) of G. Caporossi et al. [G. Caporossi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs IV: Chemical trees with extremal connectivity index, Computers and Chemistry 23 (1999) 469–477] proposed the use of linear programming as one of the tools for finding the extremal graphs. In this paper we introduce a new approach based on quadratic programming for finding the extremal graphs in G(k, n) for this index. We found the extremal graphs or gave good bounds for this index when the number n_k of vertices of degree k is between n − k and n. We also tried to find the graphs for which the Randi? index attained its minimum value with given k (k ? n/2) and n. We have solved this problem partially, that is, we have showed that the extremal graphs must have the number n_k of vertices of degree k less or equal n − k and the number of vertices of degree n − 1 less or equal k.     

Rakić duality principle in the almost Hermitian geometry

Raki? duality principle turns out to be one of the crucial steps in proving Osserman conjecture. Basically, it claims that if ${\mathcal{R}}$ is an Osserman algebraic curvature tensor and X and Y are unit vectors, then Y is an eigenvector of the Jacobi operator ${\mathcal{R}(\cdot, X)X}$ if and only if X is an unit eigenvector of ${\mathcal{R}(\cdot, Y)Y}$ with the same eigenvalue. We prove necessary and sufficient conditions for certain almost Hermitian manifolds, the so called AH ₃-manifolds, to have pointwise constant holomorphic curvature and pointwise constant antiholomorphic sectional curvature. It turns out that for this class of almost Hermitian manifolds these conditions are directly connected to the duality principle.     

Extremal trees with given degree sequence for the Randić index

The Randi? index of a graph G is the sum of $((d(u))(d(v)){)}^{\alpha }$ over all edges $\mathit{uv}$ of G, where $d(v)$ denotes the degree of $v$ in G, $\alpha \ne 0$. When $\alpha =1$, it is the weight of a graph. Delorme, Favaron, and Rautenbach characterized the trees with a given degree sequence with maximum weight, where the question of finding the tree that minimizes the weight is left open. In this note, we characterize the extremal trees with given degree sequence for the Randi? index, thus answering the same question for weight. We also provide an algorithm to construct such trees.     

Conjugated tricyclic graphs with the maximum zeroth-order general Randić index

Let G be a simple connected graph and α be a given real number. The zeroth-order general Randi? index of G is defined as ⁰ R _α (G)=∑ _v∈V(G)[d _G (v)] ^α , where d _G (v) denotes the degree of the vertex v of G. In this paper, for any α>2, we give sharp upper bounds of the zeroth-order general Randi? index ⁰ R _α of all conjugated tricyclic graphs with 2m vertices.     

The Tverberg-Vrećica problem and the combinatorial geometry on vector bundles

It is shown that many classical and many new combinatorial geometric results about finite sets of points inR ^d , specially the theorems of Tverberg type, can be generalized to the case of vector bundles, where they become combinatorial geometric statements about finite families of continuous cross-sections. The well known Tverberg-Vrećica conjecture is interpreted as a result of this type and its partial solution is obtained with the aid of the parametrized, ideal-valued, cohomological index theory. In the same spirit, classical “nonembeddability” and “coincidence” results like have higher dimensional analogues. A new ingredient is that the coincidence condition is often interpreted as the existence of a common affinek-dimensional transversal, which reduces to the classical case fork=0. Supported in part by the Ministry for Science and Technology of Serbia, Grant 04M03.     

Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture

We prove the Mirkovi?–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves.     

Properties of a Subclass of Avakumović Functions and Their Generalized Inverses

We study properties of a subclass of ORV functions introduced by Avakumovi and provide their applications for the strong law of large numbers for renewal processes.     

Inequality of Petrović and Giaccardi for Convex Function of Higher Order

In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrovis inequality.     

Fixed Points of Sequence of Ćirić Generalized Contractions of Perov Type

Perov used the concept of vector valued metric space and obtained a Banach type fixed point theorem on such a complete generalized metric space. In this article, we study fixed point results for the new extensions of sequence of ?iri? generalized contractions on cone metric space, and we give some generalized versions of the fixed point theorem of Perov. The theory is illustrated with some examples. It is worth mentioning that the main result in this paper could not be derived from ?iri?’s result by the scalarization method, and hence indeed improves many recent results in cone metric spaces.