Unimodality for free multiplicative convolution with free normal distributions on the unit circle

We study unimodality for free multiplicative convolution with free normal distributions $\{\lambda_t\}_{t>0}$ on the unit circle. We give four results on unimodality for $\mu\boxtimes\lambda_t$: (1) if $\mu$ is a symmetric unimodal distribution on the unit circle then so is $\mu\boxtimes \lambda_t$ at any time $t>0$; (2) if $\mu$ is a symmetric distribution on $\mathbb{T}$ supported on $\{e^{i\theta}: \theta \in [-\varphi,\varphi]\}$ for some $\varphi \in (0,\pi/2)$, then $\mu \boxtimes \lambda_t$ is unimodal for sufficiently large $t>0$; (3) ${\bf b} \boxtimes \lambda_t$ is not unimodal at any time $t>0$, where ${\bf b}$ is the equally weighted Bernoulli distribution on $\{1,-1\}$; (4) $\lambda_t$ is not freely strongly unimodal for sufficiently small $t>0$. Moreover, we study unimodality for classical multiplicative convolution (with Poisson kernels), which is useful in proving the above four results.Comment: 19 pages, 4 figure

How to solve the cake-cutting problem in sublinear time

In this paper, we show algorithms for solving the cake-cutting problem in sublinear-time. More specifically, we preassign (simple) fair portions to o(n) players in o(n)-time, and minimize the damage to the rest of the players. All currently known algorithms require Omega(n)-time, even when assigning a portion to just one player, and it is nontrivial to revise these algorithms to run in $o(n)$-time since many of the remaining players, who have not been asked any queries, may not be satisfied with the remaining cake. To challenge this problem, we begin by providing a framework for solving the cake-cutting problem in sublinear-time. Generally speaking, solving a problem in sublinear-time requires the use of approximations. However, in our framework, we introduce the concept of "eps n-victims," which means that eps n players (victims) may not get fair portions, where 0< eps =< 1 is an arbitrary constant. In our framework, an algorithm consists of the following two parts: In the first (Preassigning) part, it distributes fair portions to r < n players in o(n)-time. In the second (Completion) part, it distributes fair portions to the remaining n-r players except for the eps n victims in poly}(n)-time. There are two variations on the r players in the first part. Specifically, whether they can or cannot be designated. We will then present algorithms in this framework. In particular, an O(r/eps)-time algorithm for r =< eps n/127 undesignated players with eps n-victims, and an O~(r^2/eps)-time algorithm for r =< eps e^{{sqrt{ln{n}}}/{7}} designated players and eps =< 1/e with eps n-victims are presented.Comment: 15 pages, no figur

Adequate bases of phase space master integrals for gg→hgg \to h at NNLO and beyond

We study master integrals needed to compute the Higgs boson production cross section via gluon fusion in the infinite top quark mass limit, using a canonical form of differential equations for master integrals, recently identified by Henn, which makes their solution possible in a straightforward algebraic way. We apply the known criteria to derive such a suitable basis for all the phase space master integrals in afore mentioned process at next-to-next-to-leading order in QCD and demonstrate that the method is applicable to next-to-next-to-next-to-leading order as well by solving a non-planar topology. Furthermore, we discuss in great detail how to find an adequate basis using practical examples. Special emphasis is devoted to master integrals which are coupled by their differential equations.Comment: 33 pages, 6 figure