896 research outputs found
Unimodality for free multiplicative convolution with free normal distributions on the unit circle
We study unimodality for free multiplicative convolution with free normal
distributions on the unit circle. We give four results on
unimodality for : (1) if is a symmetric unimodal
distribution on the unit circle then so is at any time
; (2) if is a symmetric distribution on supported on
for some , then is unimodal for sufficiently large
; (3) is not unimodal at any time ,
where is the equally weighted Bernoulli distribution on ;
(4) is not freely strongly unimodal for sufficiently small .
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
-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
The diamond rule for multi-loop Feynman diagrams
An important aspect of improving perturbative predictions in high energy
physics is efficiently reducing dimensionally regularised Feynman integrals
through integration by parts (IBP) relations. The well-known triangle rule has
been used to achieve simple reduction schemes. In this work we introduce an
extensible, multi-loop version of the triangle rule, which we refer to as the
diamond rule. Such a structure appears frequently in higher-loop calculations.
We derive an explicit solution for the recursion, which prevents spurious poles
in intermediate steps of the computations. Applications for massless propagator
type diagrams at three, four, and five loops are discussed
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