Abstract:
In this paper we establish Minkowski inequality and Brunn--Minkowski inequality for $p$-quermassintegral differences of convex bodies. Further, we give Minkowski inequality and Brunn--Minkowski inequality for quermassintegral differences of mixed projection bodies.

Abstract:
In this paper, we first introduce a new concept of {\it dual quermassintegral sum function} of two star bodies and establish Minkowski's type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov--Fenchel inequality and the Brunn--Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak's mixed prosection bodies inequalities.

Abstract:
In this paper we study duality for evaluation codes on intersections of d hypersurfaces with given d-dimensional Newton polytopes, so called toric complete intersection codes. In particular, we give a condition for such a code to be quasi-self-dual. In the case of d=2 it reduces to a combinatorial condition on the Newton polygons. This allows us to give an explicit construction of dual and quasi-self-dual toric complete intersection codes. We provide a list of examples over the field of 16 elements.

Abstract:
“Dual contouring” approaches provide an alternative to standard Marching Cubes (MC) method to extract and approximate an isosurface from trivariate data given on a volumetric mesh. These dual approaches solve some of the problems encountered by the MC methods. We present a simple method based on the MC method and the ray intersection technique to compute isosurface points in the cell interior. One of the advantages of our method is that it does not require us to use Hermite interpolation scheme, unlike other dual contouring methods. We perform a complete analysis of all possible configurations to generate a look-up table for all configurations. We use thelook-up table to optimize the ray-intersection method to obtain minimum number of points necessarily sufficient for defining topologically correct isosurfaces in all possible configurations. Isosurface points are connected using a simple strategy.

Abstract:
The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci; some explicit formulas for the Chern and Segre classes of Schur bundles with applications to enumerative geometry; flag degeneracy loci; fundamental classes, diagonals and Gysin maps; intersection rings of G/P and formulas for isotropic degeneracy loci; numerically positive polynomials for ample vector bundles. Apart of surveyed results, the paper contains also some new results as well as some new proofs of earlier ones: how to compute the fundamental class of a subvariety from the class of the diagonal of the ambient space; how to compute the class of the relative diagonal using Gysin maps; a new formula for pushing forward Schur's Q- polynomials in Grassmannian bundles; a new formula for the total Chern class of a Schur bundle; another proof of Schubert's and Giambelli's enumeration of complete quadrics; an operator proof of the Jacobi-Trudi formula; a Schur complex proof of the Giambelli-Thom-Porteous formula.

Abstract:
We Investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. Two types of Caputo fractional differences are introduced, one of them (dual one) is defined so that it obeys the investigated dual identities. The relation between Rieamnn and Caputo fractional differences is investigated and the delta and nabla discrete Mittag-Leffler functions are confirmed by solving Caputo type linear fractional difference equations. A nabla integration by parts formula is obtained for Caputo fractional differences as well.

Abstract:
The intersection body of a ball is again a ball. So, the unit ball $B_d \subset \R^d$ is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach-Mazur distance.We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to $B_d$ in Banach-Mazur distance converge to $B_d$ in Banach-Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of $B_d$.

Abstract:
It is proved that for a symmetric convex body K in R^n, if for some tau > 0, |K cap (x+tau K)| depends on ||x||_K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies ls are studied.

Abstract:
We formulate a strong positivity conjecture on characters afforded by the Alvis-Curtis dual of the intersection cohomology of Deligne-Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent $\ell$-blocks of finite reductive groups.

Abstract:
Let $\{\alpha\}$ and $\{\beta\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional K\"ahler manifold $X$ such that the difference of intersection numbers $\{\alpha\}^n - n\,\{\alpha\}^{n-1}.\,\{\beta\}$ is positive. We solve in a number of special but rather inclusive cases the quantitative part of Demailly's Transcendental Morse Inequalities Conjecture for this context predicting the lower bound $\{\alpha\}^n-n\,\{\alpha\}^{n-1}.\,\{\beta\}$ for the volume of the difference class $\{\alpha-\beta\}$. We completely solved the qualitative part in an earlier work. We also give general lower bounds for the volume of $\{\alpha-\beta\}$ and show that the self-intersection number $\{\alpha-\beta\}^n$ is always bounded below by $\{\alpha\}^n-n\,\{\alpha\}^{n-1}.\,\{\beta\}$. We also describe and estimate the relative psef and nef thresholds of $\{\alpha\}$ with respect to $\{\beta\}$ and relate them to the volume of $\{\alpha-\beta\}$. Finally, broadening the scope beyond the K\"ahler realm, we propose a conjecture relating the balanced and the Gauduchon cones of $\partial\bar\partial$-manifolds which, if proved to hold, would imply the existence of a balanced metric on any $\partial\bar\partial$-manifold.