3,649 research outputs found
Algebraic Proofs over Noncommutative Formulas
AbstractWe study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a Cook–Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in Buss et al. (1997) and Grigoriev and Hirsch (2003). We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas (PC over ordered formulas, for short). Given some fixed linear order on variables, an arithmetic formula is ordered if for each of its product gates the left subformula contains only variables that are less-than or equal, according to the linear order, than the variables in the right subformula of the gate. We show that PC over ordered formulas (when the base field is of zero characteristic) is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR), and admits polynomial-size refutations for the pigeonhole principle and Tseitinʼs formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas (Nisan, 1991).The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs
The Lorentzian distance formula in noncommutative geometry
For almost twenty years, a search for a Lorentzian version of the well-known
Connes' distance formula has been undertaken. Several authors have contributed
to this search, providing important milestones, and the time has now come to
put those elements together in order to get a valid and functional formula.
This paper presents a historical review of the construction and the proof of a
Lorentzian distance formula suitable for noncommutative geometry.Comment: 16 pages, final form, few references adde
Grothendieck duality made simple
It has long been accepted that the foundations of Grothendieck duality are
complicated. This has changed recently. By "Grothendieck duality" we mean what,
in the old literature, used to go by the name "coherent duality". This isn't to
be confused with what is nowadays called "Verdier duality", and used to pass as
"-adic duality".Comment: Revised to incorporate improvements suggested by a few people, most
notably an anonymous refere
Algebraic properties of Manin matrices 1
We study a class of matrices with noncommutative entries, which were first
considered by Yu. I. Manin in 1988 in relation with quantum group theory. They
are defined as "noncommutative endomorphisms" of a polynomial algebra. More
explicitly their defining conditions read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). The basic claim
is that despite noncommutativity many theorems of linear algebra hold true for
Manin matrices in a form identical to that of the commutative case. Moreover in
some examples the converse is also true. The present paper gives a complete
list and detailed proofs of algebraic properties of Manin matrices known up to
the moment; many of them are new. In particular we present the formulation in
terms of matrix (Leningrad) notations; provide complete proofs that an inverse
to a M.m. is again a M.m. and for the Schur formula for the determinant of a
block matrix; we generalize the noncommutative Cauchy-Binet formulas discovered
recently [arXiv:0809.3516], which includes the classical Capelli and related
identities. We also discuss many other properties, such as the Cramer formula
for the inverse matrix, the Cayley-Hamilton theorem, Newton and
MacMahon-Wronski identities, Plucker relations, Sylvester's theorem, the
Lagrange-Desnanot-Lewis Caroll formula, the Weinstein-Aronszajn formula, some
multiplicativity properties for the determinant, relations with
quasideterminants, calculation of the determinant via Gauss decomposition,
conjugation to the second normal (Frobenius) form, and so on and so forth. We
refer to [arXiv:0711.2236] for some applications.Comment: 80 page
Noncommutative Koszul Algebras from Combinatorial Topology
Associated to any uniform finite layered graph Gamma there is a
noncommutative graded quadratic algebra A(Gamma) given by a construction due to
Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras
are Koszul. Unfortunately, a mistake in the literature states that all such
algebras are Koszul. That is not the case and the theorem was recently
retracted. We analyze the Koszul property of these algebras for two large
classes of graphs associated to finite regular CW complexes, X. Our methods are
primarily topological. We solve the Koszul problem by introducing new
cohomology groups H_X(n,k), generalizing the usual cohomology groups H^n(X).
Along with several other results, our methods give a new and primarily
topological proof of a result of Serconek and Wilson and of Piontkovski.Comment: 22 pages, 1 figur
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
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