The structure of an even liaison class
Giorgio
Bolondi;
Juan C.
Migliore
1-37
Abstract: We describe a structure called the Lazarsfeld-Rao property for even liaison classes in projective space. This property holds for many even liaison classes of curves in $ {{\mathbf{P}}^3}$. We give a procedure for showing that an even liaison class in codimension $2$ possesses this property, and we prove it for a family of even liaison classes in codimension $ 2$ in any $ {{\mathbf{P}}^n},\;n \geqslant 3$. However, we conjecture that it in fact holds for every even liaison class in codimension $ 2$, so we want to give consequences for an even liaison class that possesses this property. The main element in describing this structure is the notion of a basic double link. The Lazarsfeld-Rao property says that there exist minimal elements of the even liaison class and that any element of the even liaison class can be deformed to a curve obtained by a sequence of basic double links beginning with any minimal element. We show that there is a unique standard type of sequence for any given element of the even liaison class. As a result, we can express the even liaison class as a disjoint union of irreducible nonempty families parameterized by certain finite sequences of integers. The standard numerical invariants of the elements of any family can be computed from the associated sequence of integers. We apply this to surfaces in ${{\mathbf{P}}^4}$. Our main tool for these results is a deformation technique related to liaison in codimension $2$. We also study Schwartau's procedure of Liaison Addition in codimension $2$ from the point of view of vector bundles. Using this, we give a different sort of structure for an even liaison class with the Lazarsfeld-Rao property.
Generators for the bordism algebra of immersions
M. A.
Aguilar
39-51
Abstract: Let us denote by $ I(n,k)$ the group of bordism classes of immersions of closed smooth $n$-manifolds in closed smooth $ (n + k)$-manifolds $ (k > 0)$. We can make $I({\ast},k)$ into a graded algebra over the unoriented bordism ring. This algebra is polynomial. In this paper we give two sets of immersions which are polynomial generators.
On some limit theorems for continued fractions
Jorge D.
Samur
53-79
Abstract: As a consequence of previous results on mixing random variables, some functional limit theorems for quantities related to the continued fraction expansion of a random number in $ (0,1)$ are given.
Harmonic measure and radial projection
Donald E.
Marshall;
Carl
Sundberg
81-95
Abstract: Among all curves in the closed unit disk that meet every radius, there is one, ${\gamma _0}$, whose harmonic measure at the origin is minimal. We give an explicit description of ${\gamma _0}$ and compute its harmonic measure. We also give a quadratically convergent algorithm to compute the harmonic measure of one side of a rectangle at its center.
Remarks on approximation methods in degree theory
W.
Kryszewski;
B.
Przeradzki;
S.
Wereński
97-114
Abstract: An approximative approach to a generalized theory of the topological mapping degree is presented. Some new wide classes of operators acting in Banach spaces, which include $ A$-proper mappings of Petryshyn, are introduced and studied from the viewpoint of the homotopic properties of the topological degree. The results are applied in some existence aspects of abstract nonlinear equations.
The generalized Borel conjecture and strongly proper orders
Paul
Corazza
115-140
Abstract: The Borel Conjecture is the statement that $C = {[\mathbb{R}]^{ < {\omega _1}}}$, where $ C$ is the class of strong measure zero sets; it is known to be independent of ZFC. The Generalized Borel Conjecture is the statement that $C = {[\mathbb{R}]^{ < {\mathbf{c}}}}$. We show that this statement is also independent. The construction involves forcing with an $ {\omega _2}$-stage iteration of strongly proper orders; this latter class of orders is shown to include several well-known orders, such as Sacks and Silver forcing, and to be properly contained in the class of $\omega$-proper, $ {\omega ^\omega }$-bounding orders. The central lemma is the observation that A. W. Miller's proof that the statement $({\ast})$ "Every set of reals of power c can be mapped (uniformly) continuously onto $ [0,1]$" holds in the iterated Sacks model actually holds in several other models as well. As a result, we show for example that $ ({\ast})$ is not restricted by the presence of large universal measure zero $({{\text{U}}_0})$ sets (as it is by the presence of large $C$ sets). We also investigate the $ \sigma$-ideal $\mathcal{J} = \{ X \subset \mathbb{R}:X\;{\text{cannot be mapped uniformly continuously onto }}[0,1]\}$ and prove various consistency results concerning the relationships between $ \mathcal{J},\;{{\text{U}}_0}$, and AFC (where $\operatorname{AFC} = \{ X \subset \mathbb{R}:X\;{\text{is always first category\} }}$). These latter results partially answer two questions of J. Brown.
A generalisation of the matroid lift construction
Geoff
Whittle
141-159
Abstract: This paper introduces a general matroid-theoretic construction which includes, as special cases, elementary lifts of matroids and bias matroids of biased graphs. To perform the construction on a matroid $M$, it is necessary (but not sufficient) to have a submodular function inducing $ M$. Elementary lifts are obtained when the submodular function chosen is the rank function of $M$. We define what is meant by a $ k$-induced matroid. These matroids simultaneously generalise matroids of graphs, transversal matroids and Dilworth truncations. They are induced by a particularly natural class of submodular functions. The effect of the above construction on $k$-induced matroids using these natural submodular functions is studied. Results on minors of $k$-induced matroids and the matroids obtained from them using the construction are given.
On a problem of S. Mazur
László
Székelyhidi
161-164
Abstract: In this work a generalization of Mazur's problem concerning the continuity of linear functionals is given.
On the variety of plane curves of degree $d$ with $\delta$ nodes and $\kappa$ cusps
Pyung-Lyun
Kang
165-192
Abstract: Let ${{\mathbf{P}}^N}$ be the projective space which parametrizes all plane curves of degree $d$ and $V(d,\delta ,\kappa )$ the subvariety of ${{\mathbf{P}}^N}$ consisting of all reduced and irreducible plane curves of degree $d$ with $\delta$ nodes and $\kappa$ cusps as their only singularities. In this paper we prove that $V(d,\delta ,\kappa )$ is irreducible if $\kappa \leqslant 3$, except possibly when $\kappa = 3$ and $d = 5$ or $6$.
Quantization of curvature of harmonic two-spheres in Grassmann manifolds
Yunbo
Zheng
193-214
Abstract: Various pinching theorems for curvature of minimal two-spheres in Grassmann manifolds have been proved. In particular, we show that when the curvature is large, then the minimal map from ${S^2}$ into $G(m,N)$ must be either holomorphic or antiholomorphic. Also, minimal two-spheres of curvature $\kappa \geqslant 2$ in $ G(2,4)$ have been classified.
Complemented subspaces of products of Banach spaces
Paweł
Domański;
Augustyn
Ortyński
215-231
Abstract: It is proved that: (i) every complemented subspace in an infinite product of ${L_1}$-predual Banach spaces $\prod\nolimits_{i \in I} {{X_i}}$ is isomorphic to $Z \times {{\mathbf{K}}^\mathfrak{m}}$, where $ \dim {\mathbf{K}} = 1,\;\mathfrak{m} \leqslant \operatorname{card} I$ and $ Z$ is isomorphic to a complemented subspace of $ \prod\nolimits_{i \in J} {{X_i},\;J \subseteq I,\;Z}$ contains an isomorphic cop[ill] of $c_0^{\operatorname{card} J}$; (ii) every injective lcs (in particular, Fréchet) is of the form $Z \times {{\mathbf{K}}^\mathfrak{m}},\;\dim {\mathbf{K}} = 1$, where $Z$ has a fundamental family of seminorms of the cardinality $\tau$ and $Z$ contains an isomorphic copy of $l_\infty ^\mathfrak{r}$ (this is a generalization of Lindenstrauss' theorem on injective Banach spaces); (iii) whenever $ X \simeq {l_p},\;1 \leqslant p \leqslant \infty$, or $X \simeq {c_0}$, then every complemented subspace in a power $ {X^\mathfrak{m}}$ ( $\mathfrak{m}$ is an arbitrary cardinal number) is isomorphic to $ {X^\mathfrak{r}} \times {{\mathbf{K}}^\mathfrak{s}},\;\mathfrak{r} + s \leqslant \mathfrak{m}$ (a generalization of the results due to Lindenstrauss and Pełczyński for $ \mathfrak{m} = 1$).
Double shock fronts for hyperbolic systems of conservation laws in multidimensional space
An Ton
Bui;
De Ning
Li
233-250
Abstract: The existence of a unique double shock front for hyperbolic systems of conservation laws in several space variables is established, extending an earlier result of Metivier. An example of a double shock wave arising from physical applications is given.
$*$-differential identities of prime rings with involution
Chen-Lian
Chuang
251-279
Abstract: Main Theorem. Let $R$ be a prime ring with involution $ ^{\ast}$. Suppose that $\phi (x_i^{{\Delta _j}},{(x_i^{{\Delta _j}})^{\ast}}) = 0$ is a $ {\ast}$-differential identity for $R$, where $ {\Delta _j}$ are distinct regular words of derivations in a basis $ M$ with respect to a linear order $<$ on $M$. Then $\phi ({z_{ij}},z_{ij}^{\ast}) = 0$ is a ${\ast}$-generalized identity for $ R$, where $ {z_{ij}}$ are distinct indeterminates. Along with the Main Theorem above, we also prove the following: Proposition 1. Suppose that $^{\ast}$ is of the second kind and that $ C$ is infinite. Then $R$ is special. Proposition 2. Suppose that ${S_W}(V) \subseteq R \subseteq {L_W}(V)$. Then $Q$, the two-sided quotient ring of $ R$, is equal to $ {L_W}(V)$. Proposition 3 (Density theorem). Suppose that $ {}_DV$ and $ {W_D}$ are dual spaces with respect to the nondegenerate bilinear form $(,)$. Let $ {v_1}, \ldots ,{v_s},\;v_s^\prime , \ldots ,v_s^\prime \in V$ and ${u_1}, \ldots ,{u_t},\;u_1^\prime , \ldots ,u_t^\prime \in W$ be such that $\{ {v_1}, \ldots ,{v_s}\} $ is $ D$-independent in $ V$ and $\{ {u_1}, \ldots ,{u_t}\}$ is $D$-independent in $W$. Then there exists $a \in {S_W}(V)$ such that ${v_i}a = v_i^\prime \,(i = 1, \ldots ,s)$ and ${a^{\ast}}{u_j} = u_j^\prime \,(j = 1, \ldots ,t)$ if and only if $(v_i',{u_j}) = ({v_i},u_j')$ for $i = 1, \ldots ,s$ and $j = 1, \ldots ,t$. Proposition 4. Suppose that $R$ is a prime ring with involution $ ^{\ast}$ and that $ f$ is a $ {\ast}$-generalized polynomial. If $f$ vanishes on a nonzero ideal of $ R$, than $ f$ vanishes on $ Q$, the two-sided quotient ring of $R$.
Some explicit cases of the Selberg trace formula for vector valued functions
Jeffrey
Stopple
281-293
Abstract: The trace formula for $ SL(2,{\mathbf{Z}})$ can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation $\pi$. This paper makes this version explicit for the class of representations which can be realized as representations of the finite group $PSL(2,{\mathbf{Z}}/q)$ for some prime $ q$. The body of the paper is devoted to computing, for the singular representations $ \pi$, the determinant of the scattering matrix $ \Phi (s,\pi )$ on which the applications depend. The first application is a version of the Roelcke-Selberg conjecture. This follows from known results once the scattering matrix is given. The study of representations of $SL(2,{\mathbf{Z}})$ in finite-dimensional vector spaces of (scalar-valued) holomorphic forms dates back to Hecke. Similar problems can be studied for vector spaces of Maass wave forms, with fixed level $ q$ and eigenvalue $ \lambda$. One would like to decompose the natural representation of $SL(2,{\mathbf{Z}})$ in this space, and count the multiplicities of its irreducible components. The eigenvalue estimate obtained for vector-valued forms is equivalent to an asymptotic count, as $\lambda \to \infty$, of these multiplicities.
Realization of the level two standard ${\rm sl}(2k+1,{\bf C})\sp \sim$-modules
Kailash C.
Misra
295-309
Abstract: In this paper we study the level two standard modules for the affine special linear Lie algebras. In particular, we give the vertex operator realizations of all level two standard modules for the affine special linear Lie algebras of odd rank.
Additive cohomology operations
Jeanne
Duflot
311-325
Abstract: The bigraded group $ \{ {H_i}({\Sigma _n},{\mathbf{Z}}/p)\}$ becomes a Hopf algebra, if multiplication is induced by restriction, and comultiplication is induced by transfer. Using Steenrod's method of considering elements of this bigraded group as $ \bmod {\text{ - }}p$ cohomology operations, the primitives of this Hopf algebra correspond to additive cohomology operations. In this paper we use the results known about the homology and cohomology of the symmetric groups and the operations they induce in $ \bmod {\text{ - }}p$ cohomology to write down two (additive) bases of the bigraded vector space of primitives of the above Hopf algebra.
On the dissipative evolution equations associated with the Zakharov-Shabat system with a quadratic spectral parameter
Jyh-Hao
Lee
327-336
Abstract: In this paper we derive some results for the Zakharov-Shabat system of the form $ dm/dx = {z^2}[J,m] + (zQ + P)m$; $J$ is diagonal and skew-Hermitian $ [8,10,12]$. Following the idea of R. Beals and R. R. Coifman, we estimate the wedge products of the columns of $m$ by ${L^2}$-norm of the potential $(Q,P)\,[4]$. By this result we have the global existence of the dissipative evolution equations associated with this spectral problem if the generic initial data $(Q(x,0),\,P(x,0)) = ({Q_0},{P_0})$ is of Schwartz class.
Extendible sets in Peano arithmetic
Stuart T.
Smith
337-367
Abstract: Let $\mathcal{A}$ be a structure and let $ U$ be a subset of $\vert\mathcal{A}\vert$. We say $U$ is extendible if whenever $\mathcal{B}$ is an elementary extension of $\mathcal{A}$, there is a $V \subseteq \vert\mathcal{B}\vert$ such that $(\mathcal{A},U) \prec (\mathcal{B},V)$. Our main results are: If $ \mathcal{M}$ is a countable model of Peano arithmetic and $U$ is a subset of $\vert\mathcal{M}\vert$, then $U$ is extendible iff $ U$ is parametrically definable in $ \mathcal{M}$. Also, the cofinally extendible subsets of $\vert\mathcal{M}\vert$ are exactly the inductive subsets of $ \vert\mathcal{M}\vert$. The end extendible subsets of $\vert\mathcal{M}\vert$ are not completely characterized, but we show that if $ \mathcal{N}$ is a model of Peano arithmetic of arbitrary cardinality and $ U$ is any bounded subset of $\mathcal{N}$, then $U$ is end extendible.
The intrinsic conformal structure and Gauss map of a light-like hypersurface in Minkowski space
Marek
Kossowski
369-383
Abstract: We begin by pointing out two subtleties in the global properties of hypersurfaces in Minkowski space which inherit a uniformly degenerate metric (i.e., the existence of global space-like sections and the notion of an icon; see Appendices 1 and 2). We then construct a Gauss map for such hypersurfaces and an intrinsic invariant. This leads us to results concerning light-like hypersurfaces which parallel known results concerning surfaces in Euclidean space.