Time-dependent coating flows in a strip, Part I: The linearized problem
Avner
Friedman;
Juan
J. L.
Velázquez
2981-3074
Abstract: This work is concerned with time-dependent coating flow in a strip $0 < y < 1$. The Navier-Stokes equations are satisfied in the fluid region, the bottom substrate $y = 0$ is moving with fixed velocity $(U,0)$, and fluid is entering the strip through the upper boundary $y = 1$. The free boundary has the form $y = f(x,t)$ for $-\infty < x < R(t)$, where $R(t)$ is the moving contact point. Our objective is to prove that if the initial data are close to those of a stationary solution (the existence of such a solution was established by the authors in an earlier paper) then the time-dependent problem has a unique solution with smooth free boundary, at least for a small time interval. In this Part I we study the linearized problem, about the stationary solution, and obtain sharp estimates for the solution and its derivatives. These estimates will be used in Part II to establish existence and uniqueness for the full nonlinear problem.
Tetragonal curves, scrolls and $K3$ surfaces
James
N.
Brawner
3075-3091
Abstract: In this paper we establish a theorem which determines the invariants of a general hyperplane section of a rational normal scroll of arbitrary dimension. We then construct a complete intersection surface on a four-dimensional scroll and prove it is regular with a trivial dualizing sheaf. We determine the invariants for which the surface is nonsingular, and hence a $K3$ surface. A general hyperplane section of this surface is a tetragonal curve; we use the first theorem to determine for which tetragonal invariants such a construction is possible. In particular we show that for every genus $g\geq 7$ there is a tetragonal curve of genus $g$ that is a hyperplane section of a $K3$ surface. Conversely, if the tetragonal invariants are not sufficiently balanced, then the complete intersection must be singular. Finally we determine for which additional sets of invariants this construction gives a tetragonal curve as a hyperplane section of a singular canonically trivial surface, and discuss the connection with other recent results on canonically trivial surfaces.
The second variation of nonorientable minimal submanifolds
Marty
Ross
3093-3104
Abstract: Suppose $M$ is a complete nonorientable minimal submanifold of a Riemannian manifold $N$. We derive a second variation formula for the area of $M$ with respect to certain perturbations, giving a sufficient condition for the instability of $M$. Some simple applications are given: we show that the totally geodesic $\mathbb {R} \mathbb {P}^{2}$ is the only stable surface in $\mathbb {R} \mathbb {P}^{3}$, and we show the non-existence of stable nonorientable cones in $\mathbb {R}^{4}$. We reproduce and marginally extend some known results in the truly non-compact setting.
Matrix polynomials and the index problem for elliptic systems
B.
Rowley
3105-3148
Abstract: The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.'s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.'s.
An index formula for elliptic systems in the plane
B.
Rowley
3149-3179
Abstract: An index formula is proved for elliptic systems of P.D.E.'s with boundary values in a simply connected region $\Omega$ in the plane. Let $\mathcal {A}$ denote the elliptic operator and $\mathcal {B}$ the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function $\Delta ^{+}_{{\mathcal {B}}}$ defined on the unit cotangent bundle of $\partial \Omega$ was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function $\Delta ^{+}_{{\mathcal {B}}}$ have invertible values. In the present paper, the index of $({\mathcal {A}},{\mathcal {B}})$ is expressed in terms of the winding number of the determinant of $\Delta ^{+}_{{\mathcal {B}}}$.
An infinite dimensional Morse theory with applications
Wojciech
Kryszewski;
Andrzej
Szulkin
3181-3234
Abstract: In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.
The quantum analog of a symmetric pair: a construction in type $(C_n,A_1\times C_{n-1})$
Welleda
Baldoni;
Pierluigi
Möseneder
Frajria
3235-3276
Abstract: Let $\mathcal {I}$ be the ideal in the enveloping algebra of $\mathfrak {sp}(n,\mathbb C)$ generated by the maximal compact subalgebra of $\mathfrak {sp}(n-1,1)$. In this paper we construct an analog of $\mathcal I$ in the quantized enveloping algebra $\mbox {$\mathfrak {U}$}$ corresponding to a type $C_{n}$ diagram at generic $q$. We find generators for $\mathcal {I}$ and explicit bases for $\mbox {$\mathfrak {U}$}/\mathcal {I}$.
On the second adjunction mapping. The case of a $1$-dimensional image
Mauro
C.
Beltrametti;
Andrew
J.
Sommese
3277-3302
Abstract: Let $\widehat {L}$ be a very ample line bundle on an $n$-dimensional projective manifold $\widehat {X}$, i.e., assume that $\widehat {L}\approx i^*{\mathcal O}_{{\mathbb P}^{N}}(1)$ for some embedding $i:\widehat {X}\hookrightarrow {\mathbb P}^{N}$. In this article, a study is made of the meromorphic map, $\widehat {\varphi } : \widehat {X}\to \Sigma$, associated to $|K_{\widehat {X}}+(n-2)\widehat {L}|$ in the case when the Kodaira dimension of $K_{\widehat {X}}+(n-2)\widehat {L}$ is $\ge 3$, and $\widehat {\varphi }$ has a $1$-dimensional image. Assume for simplicity that $n=3$. The first main result of the paper shows that $\widehat \varphi$ is a morphism if either $h^0(K_{\widehat X}+\widehat L)\geq 7$ or $\kappa (\widehat {X})\geq 0$. The second main result of this paper shows that if $\kappa (\widehat X)\ge 0$, then the genus, $g(f)$, of a fiber, $f$, of the map induced by $\widehat \varphi$ on hyperplane sections is $\leq 6$. Moreover, if $h^0(K_{\widehat X}+\widehat L)\ge 21$ then $g(f)\leq 5$, a connected component $F$ of a general fiber of $\widehat \varphi$ is either a $K3$ surface or the blowing up at one point of a $K3$ surface, and $h^1({\mathcal O}_{\widehat X})\le 1$. Finally the structure of the finite to one part of the Remmert-Stein factorization of $\widehat \varphi$ is worked out.
Kernel of locally nilpotent $R$-derivations of $R[X,Y]$
S.
M.
Bhatwadekar;
Amartya
K.
Dutta
3303-3319
Abstract: In this paper we study the kernel of a non-zero locally nilpotent $R$-derivation of the polynomial ring $R[X,Y]$ over a noetherian integral domain $R$ containing a field of characteristic zero. We show that if $R$ is normal then the kernel has a graded $R$-algebra structure isomorphic to the symbolic Rees algebra of an unmixed ideal of height one in $R$, and, conversely, the symbolic Rees algebra of any unmixed height one ideal in $R$ can be embedded in $R[X,Y]$ as the kernel of a locally nilpotent $R$-derivation of $R[X,Y]$. We also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general.
Strict definiteness of integrals via complete monotonicity of derivatives
L.
Mattner
3321-3342
Abstract: Let $k$ be a nonnegative integer and let $\varphi : (0,\infty ) \rightarrow \Bbb R$ be a $C^\infty$ function with $(-)^k\cdot \varphi ^{(k)}$ completely monotone and not constant. If $\sigma \neq 0$ is a signed measure on any euclidean space $\Bbb R^d$, with vanishing moments up to order $k-1$, then the integral $\int _{\Bbb R^d} \int _{\Bbb R^d} \varphi ( \|x-y\|^2 ) \, d\sigma (x) d\sigma (y)$ is strictly positive whenever it exists. For general $d$ no larger class of continuous functions $\varphi$ seems to admit the same conclusion. Examples and applications are indicated. A section on ''bilinear integrability'' might be of independent interest.
Extensions of modules over Weyl algebras
S.
C.
Coutinho
3343-3352
Abstract: In this paper we calculate some $\mathrm {Ext}$ groups of singular modules over the complex Weyl algebra $A_{n}$. In particular we determine conditions under which $\mathrm {Ext}$ is an infinite dimensional vector space when $n =2$ or $3$.
Hecke algebras, $U_qsl_n$, and the Donald-Flanigan conjecture for $S_n$
Murray
Gerstenhaber;
Mary
E.
Schaps
3353-3371
Abstract: The Donald-Flanigan conjecture asserts that the integral group ring $\mathbb {Z}G$ of a finite group $G$ can be deformed to an algebra $A$ over the power series ring $\mathbb {Z}[[t]]$ with underlying module $\mathbb {Z}G[[t]]$ such that if $p$ is any prime dividing $\#G$ then $A\otimes _{\mathbb {Z}[[t]]}\overline {\mathbb {F}_{p}((t))}$ is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of $\mathbb {C}G.$ We prove this for $G = S_{n}$ using the natural representation of its Hecke algebra $\mathcal {H}$ by quantum Yang-Baxter matrices to show that over $\mathbb {Z}[q]$ localized at the multiplicatively closed set generated by $q$ and all $i_{q^{2}} = 1+q^{2} + q^{4} + \dots + q^{2(i-1)}, i = 1,2,\dots , n$, the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case ($q=1$), from those of Young.
Kaehler structures on $K_{\mathbf C}/(P,P)$
Meng-Kiat
Chuah
3373-3390
Abstract: Let $K$ be a compact connected semi-simple Lie group, let $G = K_{\mathbf C}$, and let $G = KAN$ be an Iwasawa decomposition. To a given $K$-invariant Kaehler structure $\omega$ on $G/N$, there corresponds a pre-quantum line bundle ${\mathbf L}$ on $G/N$. Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections ${\mathcal O}({\mathbf L})$ as a $K$-representation space. We defined a $K$-invariant $L^2$-structure on ${\mathcal O}({\mathbf L})$, and let $H_\omega \subset {\mathcal O}({\mathbf L})$ denote the space of square-integrable holomorphic sections. Then $H_\omega$ is a unitary $K$-representation space, but not all unitary irreducible $K$-representations occur as subrepresentations of $H_\omega$. This paper serves as a continuation of that work, by generalizing the space considered. Let $B$ be a Borel subgroup containing $N$, with commutator subgroup $(B,B)=N$. Instead of working with $G/N = G/(B,B)$, we consider $G/(P,P)$, for all parabolic subgroups $P$ containing $B$. We carry out a similar construction, and recover in $H_\omega$ the unitary irreducible $K$-representations previously missing. As a result, we use these holomorphic sections to construct a model for $K$: a unitary $K$-representation in which every irreducible $K$-representation occurs with multiplicity one.
Extreme points in triangular UHF algebras
Timothy
D.
Hudson;
Elias
G.
Katsoulis;
David
R.
Larson
3391-3400
Abstract: We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points.
Matrix extensions and eigenvalue completions, the generic case
William
Helton;
Joachim
Rosenthal;
Xiaochang
Wang
3401-3408
Abstract: In this paper we provide new necessary and sufficient conditions for the so-called eigenvalue completion problem.