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Register a new userGeneralized iterated-sums signatures
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Added by
Nikolas Tapia
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Kiraly and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.
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Driven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of (real-valued) time series that are not easily available using existing signature-type objects.
We give an abstract categorical treatment of Plonka sums and products using lax and oplax morphisms of monads. Plonka sums were originally defined as operations on algebras of regular theories. Their arities are sup-semilattices. It turns out that even more general operations are available on the categories of algebras of semi-analytic monads. Their arities are the categories of the regular polynomials over any sup-semilattice, i.e. any algebra for the terminal semi-analytic monad. We also show that similar operations can be defined on any category of algebras of any analytic monad. This time we can allow the arities to be the categories of linear polynomials over any commutative monoid, i.e. any algebra for the terminal analytic monad. There are also dual operations of Plonka products. They can be defined on Kleisli categories of commutative monads.
In this paper we deal with the problem of computing the sum of the $k$-th powers of all the elements of the matrix ring $mathbb{M}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case $R=mathbb{Z}/nmathbb{Z}$ and give some results that compute the value of this sum if $R$ is an arbitrary finite commutative ring $R$ for many values of $k$ and $d$. Finally, based on computational evidence and using some technical results proved in the paper we conjecture that the sum of the $k$-th powers of all the elements of the matrix ring $mathbb{M}_d(R)$ is always $0$ unless $d=2$, $textrm{card}(R) equiv 2 pmod 4$, $1<kequiv -1,0,1 pmod 6$ and the only element $ein R setminus {0}$ such that $2e =0$ is idempotent, in which case the sum is $textrm{diag}(e,e)$.
Motivated by work of Coxeter (1957), we study a class of algebras associated to Coxeter groups, which we term generalized nil-Coxeter algebras. We construct the first finite-dimensional examples other than usual nil-Coxeter algebras; these form a $2$-parameter type $A$ family that we term $NC_A(n,d)$. We explore the combinatorial properties of these algebras, including the Coxeter word basis, length function, maximal words, and their connection to Khovanovs categorification of the Weyl algebra. Our broader motivation arises from complex reflection groups and the Broue-Malle-Rouquier freeness conjecture (1998). With generic Hecke algebras over real and complex groups in mind, we show that the first finite-dimensional examples $NC_A(n,d)$ are in fact the only ones, outside of the usual nil-Coxeter algebras. The proofs use a diagrammatic calculus akin to crystal theory.
In this paper, we introduce two new generalized inverses of matrices, namely, the $bra{i}{m}$-core inverse and the $pare{j}{m}$-core inverse. The $bra{i}{m}$-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler cite{BT} and the core-EP inverse defined by Manjunatha Prasad and Mohana cite{MM}. The $pare{j}{m}$-core inverse of a complex matrix extends the notions of the core inverse and the ${rm DMP}$-inverse defined by Malik and Thome cite{MT}. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers.
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