Matrix zeros of polynomials

The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analogue to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley-Hamilton theorem.

TENSOR PRODUCTS OF STEINBERG ALGEBRAS

We prove that $A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$ if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R}\otimes L_{3,R}$ and $L_{2,R}\otimes L_{2,R}$ . In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $\ast$ -isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\not \cong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ (as $\ast$ -rings).

Section problems for configuration spaces of surfaces

In this paper, we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of [Formula: see text] ordered points on a surface [Formula: see text] of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it? More precisely, let PConf[Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points in [Formula: see text]. Let [Formula: see text] be the map given by [Formula: see text]. We classify all continuous sections of [Formula: see text] up to homotopy by proving the following: (1) If [Formula: see text] and [Formula: see text], any section of [Formula: see text] is either “adding a point at infinity” or “adding a point near [Formula: see text]”. (We define these two terms in Sec. 2.1; whether we can define “adding a point near [Formula: see text]” or “adding a point at infinity” depends in a delicate way on properties of [Formula: see text].) (2) If [Formula: see text] a [Formula: see text]-sphere and [Formula: see text], any section of [Formula: see text] is “adding a point near [Formula: see text]”; if [Formula: see text] and [Formula: see text], the bundle [Formula: see text] does not have a section. (We define this term in Sec. 3.2). (3) If [Formula: see text] a surface of genus [Formula: see text] and for [Formula: see text], we give an easy proof of [D. L. Gonçalves and J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl. Algebra 182 (2003) 33–64, Theorem 2] that the bundle [Formula: see text] does not have a section.

Operator Valued Measures as Multipliers of L1(I,X) with order Convolution

Let I = (0;∞) with the usual topology and product as max multiplication. Then I becomes a locally compact topo- logical semigroup. Let X be a Banach Space. Let L1(I;X) be the Banach space of X-valued measurable functions f such that ,we define It turns out that ƒ ∗ g ∈ L1(I;X) and L1(I;X) becomes an L1(I)-Banach module. A bounded linear operator T on L1(I;X) is called a multiplier of L1(I;X) if T(f ∗ g) = f ∗ Tg for all f ∈ L1(I) and g ∈ L1(I;X). We characterize the multipliers of L1(I;X) in terms of operator valued measures with point-wise finite variation and give an easy proof of some results of Tewari[12].

Coarse amenability versus paracompactness

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using partitions of unity. In this paper we go deeper into divulging analogies between coarse amenability and paracompactness. In particular, we define a new coarse analog of paracompactness modeled on the defining characteristics of expanders. That analog gives an easy proof of three categories of spaces being coarsely non-amenable: expander sequences, graph spaces with girth approaching infinity, and unions of powers of a finite nontrivial group.

SCROLLS AND HYPERBOLICITY

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in ℙ3of degree d ≥ 5. We also show that there exist Kobayashi hyperbolic surfaces in ℙ3of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Our method yields also some lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d ≥ 15 in ℙ4.