1,480 research outputs found
Some recent results and open problems on sets of lengths of Krull monoids with finite class group
Some of the fundamental notions related to sets of lengths of Krull monoids
with finite class group are discussed, and a survey of recent results is given.
These include the elasticity and related notions, the set of distances, and the
structure theorem for sets of lengths. Several open problems are mentioned
Inverse zero-sum problems II
Let be an additive finite abelian group. A sequence over is called a
minimal zero-sum sequence if the sum of its terms is zero and no proper
subsequence has this property. Davenport's constant of is the maximum of
the lengths of the minimal zero-sum sequences over . Its value is well-known
for groups of rank two. We investigate the structure of minimal zero-sum
sequences of maximal length for groups of rank two. Assuming a well-supported
conjecture on this problem for groups of the form , we
determine the structure of these sequences for groups of rank two. Combining
our result and partial results on this conjecture, yields unconditional results
for certain groups of rank two.Comment: new version contains results related to Davenport's constant only;
other results will be described separatel
A characterization of class groups via sets of lengths
Let be a Krull monoid with class group such that every class contains
a prime divisor. Then every nonunit can be written as a finite
product of irreducible elements. If , with
irreducibles , then is called the length of the
factorization and the set of all possible is called the set
of lengths of . It is well-known that the system depends only on the class group . In the present
paper we study the inverse question asking whether or not the system is characteristic for the class group. Consider a further Krull monoid
with class group such that every class contains a prime divisor and
suppose that . We show that, if one of the
groups and is finite and has rank at most two, then and are
isomorphic (apart from two well-known pairings).Comment: The current version is close to the one to appear in J. Korean Math.
Soc., yet it contains a detailed proof of Proposition 2.4. The content of
Chapter 4 of the first version had been split off and is presented in ' A
characterization of Krull monoids for which sets of lengths are (almost)
arithmetical progressions' by the same authors (see hal-01976941 and
arXiv:1901.03506
Rubidium metaborate, Rb3B3O6
Rubidium metaborate, Rb3B3O6, was obtained by the reaction of Rb2CO3 and BN using a radiofrequency furnace at a maximum reaction temperature of 1173 K. The crystal structure has been determined by single-crystal X-ray diffraction. The space group is , with all atoms positioned on a twofold axis (Wyckoff site 18e). The ionic compound is isotypic with Na3B3O6, K3B3O6 and Cs3B3O6
Brownian motion in a truncated Weyl chamber
We examine the non-exit probability of a multidimensional Brownian motion
from a growing truncated Weyl chamber. Different regimes are identified
according to the growth speed, ranging from polynomial decay over
stretched-exponential to exponential decay. Furthermore we derive associated
large deviation principles for the empirical measure of the properly rescaled
and transformed Brownian motion as the dimension grows to infinity. Our main
tool is an explicit eigenvalue expansion for the transition probabilities
before exiting the truncated Weyl chamber
The system of sets of lengths in Krull monoids under set addition
Let be a Krull monoid with class group and suppose that each class
contains a prime divisor. Then every element has a factorization into
irreducible elements, and the set of all possible factorization
lengths is the set of lengths of . We consider the system of all sets of lengths, and we characterize
(in terms of the class group ) when is additively closed
under set addition.Comment: Revista Matem{\'a}tica Iberoamericana, to appea
Random walks conditioned to stay in Weyl chambers of type C and D
We construct the conditional versions of a multidimensional random walk given
that it does not leave the Weyl chambers of type C and of type D, respectively,
in terms of a Doob h-transform. Furthermore, we prove functional limit theorems
for the rescaled random walks. This is an extension of recent work by
Eichelsbacher and Koenig who studied the analogous conditioning for the Weyl
chamber of type A. Our proof follows recent work by Denisov and Wachtel who
used martingale properties and a strong approximation of random walks by
Brownian motion. Therefore, we are able to keep minimal moment assumptions.
Finally, we present an alternate function that is amenable to an h-transform in
the Weyl chamber of type C.Comment: 12 pages, submitted to EC
On the Exact Solution of the Multi-Period Portfolio Choice Problem for an Exponential Utility under Return Predictability
In this paper we derive the exact solution of the multi-period portfolio
choice problem for an exponential utility function under return predictability.
It is assumed that the asset returns depend on predictable variables and that
the joint random process of the asset returns and the predictable variables
follow a vector autoregressive process. We prove that the optimal portfolio
weights depend on the covariance matrices of the next two periods and the
conditional mean vector of the next period. The case without predictable
variables and the case of independent asset returns are partial cases of our
solution. Furthermore, we provide an empirical study where the cumulative
empirical distribution function of the investor's wealth is calculated using
the exact solution. It is compared with the investment strategy obtained under
the additional assumption that the asset returns are independently distributed.Comment: 16 pages, 2 figure
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