The socle series of a Leavitt path algebra

We investigate the ascending Loewy socle series of Leavitt path algebras L _K (E) for an arbitrary graph E and field K. We classify those graphs E for which L _K (E) = S _λ for some element S _λ of the Loewy socle series. We then show that for any ordinal λ there exists a graph E so that the Loewy length of L _K (E) is λ. Moreover, λ ≤ ω ₁ (the first uncountable ordinal) if E is a row-finite graph.     

The Hilbert number of a class of differential equations

The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form \[\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)\] defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$.     

The Clifford algebra of differential forms

This paper reviews Clifford algebras in mathematics and in theoretical physics. In particular, the little-known differential form realization is constructed in detail for the four-dimensional Minkowski space. This setting is then used to describe spinors as differential forms, and to solve the Klein-Gordon and Kähler-Dirac equations. The approach of this paper, in obtaining the solutions directly in terms of differential forms, is much more elegant and concise than the traditional explicit matrix methods. A theorem given here differentiates between the two real forms of the Dirac algebra by showing that spin can be accommodated in only one of them.     

Positive integrable elements relative to a left Hilbert algebra

Let $\text{U}$ be an achieved left Hilbert algebra. Let $\text{η∈D}{}^{\mathrm{?}}$ be an element such that π′(η) is a positive operator. Then, following M. A. Rieffel, η is called integrable if sup{(η|e)e∈U and ee^?e²} < + ∞. It is shown that η is integrable if and only if there is an element ζ∈D^flat; such π′(ζ) is positive and ζ is a square root of η in an appropriate sense. This is shown to be a generalization of Godement's well known theorem on the existence of a convolution square root for a continuous square-integrable positive-definite function on a locally compact group. An “integral” and an “L¹-norm” are then defined on the linear span of the positive integrable elements and the completion of this space, denoted by L¹($\text{U}$), is shown to be the predual of $\text{l}$($\text{U}$). “Godement's theorem” is then used to investigate square-integrable representations of $\text{U}$.     

Rearrangements of series in a Hilbert space

A theorem on rearrangements of numerical series, proved by Agnew, is extended to series in a Hilbert space. A complete proof is given of Orlicz's theorem on unconditionally convergent series in a Hilbert space.Translated from Matematicheskie Zametki, Vol. 12, No. 3, pp. 275–280, September, 1972.     

On the free implicative semilattice extension of a Hilbert algebra

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.     

The Hilbert series of prime PI rings

We study the Hilbert series of finitely generated prime PI algebras. We show that given such an algebraA there exists some finite dimensional subspaceV ofA which contains 1 _A and generatesA as an algebra such that the Hilbert series ofA with respect to the vector spaceV is a rational function.     

Noncommutative local algebra

The main purpose of this work is to introduce the first notions of noncommutative algebraic geometry — the spectrum of an abelian category, localizations at points of the spectrum, canonical topologies, supports, associated points etc. — and to study their basic properties.     

Prolongations in differential algebra

We develop the theory of higher prolongations of algebraic varieties over fields in arbitrary characteristic with commuting Hasse-Schmidt derivations. Prolongations were introduced by Buium in the context of fields of characteristic 0 with a single derivation. Inspired by work of Vojta, we give a new construction of higher prolongations in a more general context. Generalizing a result of Buium in characteristic 0, we prove that these prolongations are represented by a certain functor, which shows that they can be viewed as ‘twisted jet spaces.’ We give a new proof of a theorem of Moosa, Pillay and Scanlon that the prolongation functor and jet space functor commute. We also prove that the m-th prolongation and m-th jet space of a variety are differentially isomorphic by showing that their infinite prolongations are isomorphic as schemes.     

Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement V₁∪V₂∪?∪V_m⊆V is an intersection I₁∩I₂∩?∩I_m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I₁I₂?I_m without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.     

Tor gives the inverse to the Hilbert function of a graded algebra

The Hilbert generating function of Tor^U(A, A) gives the inverse of the Hilbert generating function of U where U is a graded A algebra.