Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComputations of volumes in five candidates elections
51
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We describe several analytical results obtained in five candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality, negative plurality and Borda voting, including their runof
rate research
Read More
We describe several experimental results obtained in four candidates social choice elections. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes.
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $xin (X,Delta)$ satisfies the ACC if the coefficients of $Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.
In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the non-empty subsets $Asubset {1,..,n}$, is it possible to construct a body $Tsubset mathbb{R}^n$ such that $x_A=|T_A|$ where $|T_A|$ is the $|A|$-dimensional volume of the projection of $T$ onto the subspace spanned by the axes in $A$? As it is more convenient to take logarithms we denote by $psi_n$ the set of all vectors $x$ for which there is a body $T$ such that $x_A=log |T_A|$ for all $A$. Bollobas and Thomason showed that $psi_n$ is contained in the polyhedral cone defined by the class of `uniform cover inequalities. Tan and Zeng conjectured that the convex hull $DeclareMathOperator{conv}{conv}$ $conv(psi_n)$ is equal to the cone given by the uniform cover inequalities. We prove that this conjecture is `nearly right: the closed convex hull $overline{conv}(psi_n)$ is equal to the cone given by the uniform cover inequalities. However, perhaps surprisingly, we also show that $conv (psi_n)$ is not closed for $nge 4$, thus disproving the conjecture.
We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikovs theory of generalized permutohedra.
We generalize valuations on polyhedral cones to valuations on fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion-restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion-restriction proof of a result of Klivans-Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these intrinsic indicators yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
