164 research outputs found
Divided difference operators in equivariant -theory
Let be a compact connected Lie group with a maximal torus . Let ,
be --algebras. We define certain divided difference
operators on Kasparov's -equivariant -group and show that
is a direct summand of . More precisely, a
-equivariant -class is -equivariant if and only if it is annihilated
by an ideal of divided difference operators. This result is a generalization of
work done by Atiyah, Harada, Landweber and Sjamaar.Comment: 27 pages. arXiv admin note: text overlap with arXiv:0906.1629 by
other author
Derivations on four dimensional genetic Volterra algebra
In this paper, we describe all derivations on four dimensional genetic
Volterra algebras. We show that any local derivation is a derivation of the
algebra. It is a positive answer to a conjecture made by Ganikhodzhaev,
Mukhamedov, Pirnapasov and Qaralleh
K-theory of weight varieties
Let be a compact torus and a Hamiltonian -space. We give
a new proof of the -theoretic analogue of the Kirwan surjectivity theorem in
symplectic geometry by using the equivariant version of the Kirwan map
introduced in one of R. Goldin's papers. We compute the kernel of this
equivariant Kirwan map, and hence give a computation of the kernel of the
Kirwan map. As an application, we find the presentation of the kernel of the
Kirwan map for the -equivariant -theory of flag varieties where
is a compact, connected and simply-connected Lie group. In the last section, we
find explicit formulae for the -theory of weight varieties.Comment: 16 page
Determinants containing powers of polynomial sequences
We derive identities for the determinants of matrices whose entries are
(rising) powers of (products of) polynomials that satisfy a recurrence
relation. In particular, these results cover the cases for Fibonacci
polynomials, Lucas polynomials and certain orthogonal polynomials. These
identities naturally generalize the determinant identities obtained by Alfred,
Carlitz, Prodinger, Tangboonduangjit and Thanatipanonda.Comment: 12 page
A remark on commutative subalgebras of Grassmann algebra
Let and . We show that there exists maximal commutative
subalgebras (with respect to inclusion) of dimension less that
A Probabilistic Two-Pile Game
We consider a game with two piles, in which two players take turn to add
or chips (, are not necessarily positive) randomly and independently
to their respective piles. The player who collects chips first wins the
game. We derive general formulas for , the probability of the second
player winning the game by collecting chips first and show the calculation
for the cases = and . The latter case was asked
by Wong and Xu \cite{WX}. At the end, we derive the general formula for
, the probability of the second player winning the game by
collecting chips before the first player collects chips.Comment: 14 page
Outlier Detection in High Dimensional Data
High-dimensional data poses unique challenges in outlier detection process.
Most of the existing algorithms fail to properly address the issues stemming
from a large number of features. In particular, outlier detection algorithms
perform poorly on data set of small size with a large number of features. In
this paper, we propose a novel outlier detection algorithm based on principal
component analysis and kernel density estimation. The proposed method is
designed to address the challenges of dealing with high-dimensional data by
projecting the original data onto a smaller space and using the innate
structure of the data to calculate anomaly scores for each data point.
Numerical experiments on synthetic and real-life data show that our method
performs well on high-dimensional data. In particular, the proposed method
outperforms the benchmark methods as measured by the -score. Our method
also produces better-than-average execution times compared to the benchmark
methods
On Horadam quaternions by using matrix method
In this paper, we give several matrix representations for the Horadam
quaternions. We derive several identities related to these quaternions by using
the matrix method. Since quaternion multiplication is not commutative, some of
our results are non-commutative analogues of the well known identities for the
Fibonacci-like integer sequences. Lastly, we derive some binomial-sum
identities for the Horadam quaternions as an application of the matrix method
Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences
In this paper, we consider a generalization of Horadam sequence fwng which is
defined by the recurrence relation wn = x(n)wn-1+ cwn-2; where x(n) = a if n is
even, x(n) = b if n is odd with arbitrary initial conditions w0;w1 and nonzero
real numbers a, b and c. As a special case, by taking initial conditions 0, 1
and 2, b we define the sequences {un} and {vn}, respectively. The main purpose
of this study is to derive some basic properties of the sequences {un}, {vn}
and {wn} by using matrix approach
Game of Pure Chance with Restricted Boundary
We consider various probabilistic games with piles for one player or two
players. In each round of the game, a player randomly chooses to add or
chips to his pile under the condition that and are not necessarily
positive. If a player has a negative number of chips after making his play,
then the number of chips he collects will stay at and the game will
continue. All the games we considered satisfy these rules. The game ends when
one collects chips for the first time. Each player is allowed to start with
chips where . We consider various cases of including the
pairs and in particular. We investigate the probability
generating functions of the number of turns required to end the games. We
derive interesting recurrence relations for the sequences of such functions in
and write these generating functions as rational functions. As an
application, we derive other statistics for the games which include the average
number of turns required to end the game and other higher moments.Comment: 19 page
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