4,654 research outputs found
Rota-Baxter algebras and new combinatorial identities
The word problem for an arbitrary associative Rota-Baxter algebra is solved.
This leads to a noncommutative generalization of the classical Spitzer
identities. Links to other combinatorial aspects, particularly of interest in
physics, are indicated.Comment: 8 pages, improved versio
Levy Processes and Quasi-Shuffle Algebras
We investigate the algebra of repeated integrals of semimartingales. We prove
that a minimal family of semimartingales generates a quasi-shuffle algebra. In
essence, to fulfill the minimality criterion, first, the family must be a
minimal generator of the algebra of repeated integrals generated by its
elements and by quadratic covariation processes recursively constructed from
the elements of the family. Second, recursively constructed quadratic
covariation processes may lie in the linear span of previously constructed ones
and of the family, but may not lie in the linear span of repeated integrals of
these. We prove that a finite family of independent Levy processes that have
finite moments generates a minimal family. Key to the proof are the Teugels
martingales and a strong orthogonalization of them. We conclude that a finite
family of independent Levy processes form a quasi-shuffle algebra. We discuss
important potential applications to constructing efficient numerical methods
for the strong approximation of stochastic differential equations driven by
Levy processes.Comment: 10 page
Flows and stochastic Taylor series in Ito calculus
For stochastic systems driven by continuous semimartingales an explicit
formula for the logarithm of the Ito flow map is given. A similar formula is
also obtained for solutions of linear matrix-valued SDEs driven by arbitrary
semimartingales. The computation relies on the lift to quasi-shuffle algebras
of formulas involving products of Ito integrals of semimartingales. Whereas the
Chen-Strichartz formula computing the logarithm of the Stratonovich flow map is
classically expanded as a formal sum indexed by permutations, the analogous
formula in Ito calculus is naturally indexed by surjections. This reflects the
change of algebraic background involved in the transition between the two
integration theories
Generalized shuffles related to Nijenhuis and TD-algebras
Shuffle and quasi-shuffle products are well-known in the mathematics
literature. They are intimately related to Loday's dendriform algebras, and
were extensively used to give explicit constructions of free commutative
Rota-Baxter algebras. In the literature there exist at least two other
Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called
TD-algebra. The explicit construction of the free unital commutative Nijenhuis
algebra uses a modified quasi-shuffle product, called the right-shift shuffle.
We show that another modification of the quasi-shuffle product, the so-called
left-shift shuffle, can be used to give an explicit construction of the free
unital commutative TD-algebra. We explore some basic properties of TD-operators
and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We
relate our construction to Loday's unital commutative dendriform trialgebras,
including the involutive case. The concept of Rota-Baxter, Nijenhuis and
TD-bialgebras is introduced at the end and we show that any commutative
bialgebra provides such objects.Comment: 20 pages, typos corrected, accepted for publication in Communications
in Algebr
Exponential renormalization
Moving beyond the classical additive and multiplicative approaches, we
present an "exponential" method for perturbative renormalization. Using Dyson's
identity for Green's functions as well as the link between the Faa di Bruno
Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the
composition of formal power series is analyzed. Eventually, we argue that the
new method has several attractive features and encompasses the BPHZ method. The
latter can be seen as a special case of the new procedure for renormalization
scheme maps with the Rota-Baxter property. To our best knowledge, although very
natural from group-theoretical and physical points of view, several ideas
introduced in the present paper seem to be new (besides the exponential method,
let us mention the notions of counterfactors and of order n bare coupling
constants).Comment: revised version; accepted for publication in Annales Henri Poincar
Involuntary Embarrassing Exposures in Online Social Networks: A Replication Study
In this study, we carry out a methodological replication of the research done by Choi et al. (2015) published in Information System Research. In the original study, the authors integrate the privacy and teasing literatures under a social exchange framework to understand online involuntary exposures. The original study was conducted on students from Southeast Asia. Our study uses a significantly larger sample of college students in the United States. Our replication results show that whereas most of the hypotheses supported by the original results on behavioral responses replicate with high consistency (8 out of 12 hypotheses), the results that deal with the effects of network commonality on perceived privacy invasion and perceived relationship bonding did not replicate (4 out of 12 hypotheses). These results could stem from a failed manipulation of network commonality. We look into the possible rationales for this and show what would be an effective manipulation in our context. Further, we expand the original study by testing an additional embarrassing scenario catered to our subject pool. The results suggest that perceived privacy invasion and perceived relationship bonding affect individual’s behavioral responses to embarrassing exposures
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