Examples In Dimensions Research Articles

Topologically transitive homeomorphisms of quotients of tori

This paper considers the following question: for what finite subgroups $G\subset GL(n, \mathbb Z)$, does there exist an element $A\in GL(n, \mathbb Z)$ inducing a topologically transitive homeomorphism of $T^n$/$G$ We show that for $n = 2$ and 3, the only possibility is $G =\{\pm I\}$. Curiously, in higher dimension the structure is less restrictive. We give a variety of examples in dimension 4. Nevertheless, we show that in dimension $\geq 4$, there are relatively few irreducible examples.

Yang-baxter operators arising from (co)algebra structures

We construct parametrized Yang-Baxter operators from algebra struc-tures, transfer the theory to coalgebras, and find the cases where such operators arising from (co) algebras are isomorphic. We give examples in dimension 2.

Algebraic structures and invariant manifolds of differential systems

Algebraic tools are applied to find integrability properties of ODEs. Bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations, since all equations in this class are related to a canonical quadratic differential system: the Lotka–Volterra system. These algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given. Their subalgebras are associated to nonlinear invariant manifolds in the phase space. These manifolds are calculated explicitly. More general algebraic invariant surfaces are also obtained by combining a theorem of Walcher and the Lotka–Volterra canonical form. Applications are given for Lorenz model, Lotka, May–Leonard, and Rikitake systems.

On the p-rank of an abelian variety and its endomorphism algebra

Let $A$ be an abelian variety defined over a finite field. In this paper, we discuss the relationship between the $p$-rank of $A$, $r(A)$, and its endomorphism algebra, $\operatorname{End}^0(A)$. As is well known, $\operatorname{End}^0(A)$ determines $r(A)$ when $A$ is an elliptic curve. We show that, under some conditions, the value of $r(A)$ and the structure of $\operatorname{End}^0(A)$ are related. For example, if the center of $\operatorname{End}^0(A)$ is an abelian extension of $\Bbb Q$, then $A$ is ordinary if and only if $\operatorname{End}^0(A)$ is a commutative field. Nevertheless, we give an example in dimension 3 which shows that the algebra $\operatorname{End}^0(A)$ does not determine the value $r(A)$.

Faster Evaluation of Multidimensional Integrals

In a recent paper Keister proposed two quadrature rules as alternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasi-Monte Carlo method with generalized Faure points is very effective for a variety of high dimensional integrals occurng in mathematical finance. In this paper we report test results of this method on Keister's examples of dimension 9 and 25, and also for examples of dimension 60, 80 and 100. For the 25 dimensional integral we achieved accuracy of 0.01 with less than 500 points while the two methods tested by Keister used more than 220,000 points. In all of our tests, for n sample points we obtained an empirical convergence rate proportional to n^{-1} rather than the n^{-1/2} of Monte Carlo.

AFFINE STRUCTURES ON NILMANIFOLDS

We investigate the existence of affine structures on nilmanifolds Γ\G in the case where the Lie algebra g of the Lie group G is filiform nilpotent of dimension less or equal to 11. Here we obtain examples of nilmanifolds without any affine structure in dimensions 10, 11. These are new counterexamples to the Milnor conjecture. So far examples in dimension 11 were known where the proof is complicated, see [5] and [4]. Using certain 2-cocycles we realize the filiform Lie algebras as deformation algebras from a standard graded filiform algebra. Thus we study the affine algebraic variety of complex filiform nilpotent Lie algebra structures of a given dimension ≤11. This approach simplifies the calculations, and the counterexamples in dimension 10 are less complicated than the known ones. We also obtain results for the minimal dimension µ(g) of a faithful g-module for these filiform Lie algebras g.

Examples of Domains with Non-Compact Automorphism Groups

We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any Reinhardt domain. We give an analogous example in dimension two, but the domain fails to be smooth at one boundary point---indeed it is not in any Lipschitz class at the exceptional point.

Self-organized, tubular hybrid vascular tissue composed of vascular cells and collagen for low-pressure-loaded venous system

A tubular, hierarchically structured hybrid vascular tissue composed of vascular cells and collagen was prepared. First, a cold mixed solution of bovine aortic smooth muscle cells (SMCs) and Type I collagen was poured into a tubular glass mold composed of a mandrel and a sheath (example of dimensions: inner diameter, 1.5 mm; outer diameter, 7 mm; length, 7 cm). Upon incubation at 37°C, an SMC-incorporated collagenous gel was formed. After the sheath was removed, the resulting fragile tissue, when cultured in medium, thinned in a time-dependent manner to form an opaque, dense tissue. Higher SMC seeding density and lower initial collagen concentration induced more rapid and prominent shrinkage of the tissue. Morphologic investigation showed that over time, bipolarly elongated SMCs and collagen fiber bundles became positioned around the mandrel. Both components became circumferentially oriented. When the mandrel was removed, a tubular hybrid medial tissue was formed. A hybrid vascular tissue with a hierarchical structure was constructed by seeding endothelial cells onto the inner surface of the hybrid medial tissue. Prepared tissues tolerated luminal pressures as great as 100 mmHg and mechanical stress applied during an anastomotic procedure. This method allowed us to prepare a tubular hybrid medial tissue of predetermined size (inner diameter, wall thickness, and length) by selecting appropriate mold design, initial collagen concentration, and SMC seeding density. Such hybrid vascular tissues may provide physiological functions when implanted into the venous system.

Function Fields of Pfister Neighbors

A quadratic form Q is called a special Pfister neighbor if Q is similar to a form of the shape P0 ⊥ aP1, where P0 is Pfister, a ∈ k*, and P1 is a nonzero subform of P0. The Pfister form P0 ⊥ aP0, which is uniquely determined by Q, is called the associated Pfister form of Q. If P is an anisotropic Pfister form of dimension > 8, then every subform Q of P of codimension ≤ 4 is a special Pfister neighbor; and there exists an example with dim P = 16 and codim Q = 5 which is not special. Special Pfister neighbors of the same dimension and with the same associated Pfister form define the same function field, but there exists an example in dimension 5 which shows that such forms need not be similar.

Self-organized, tubular hybrid vascular tissue composed of vascular cells and collagen for low-pressure-loaded venous system

A tubular, hierarchically structured hybrid vascular tissue composed of vascular cells and collagen was prepared. First, a cold mixed solution of bovine aortic smooth muscle cells (SMCs) and Type I collagen was poured into a tubular glass mold composed of a mandrel and a sheath (example of dimensions: inner diameter, 1.5 mm; outer diameter, 7 mm; length, 7 cm). Upon incubation at 37°C, an SMC-incorporated collagenous gel was formed. After the sheath was removed, the resulting fragile tissue, when cultured in medium, thinned in a time-dependent manner to form an opaque, dense tissue. Higher SMC seeding density and lower initial collagen concentration induced more rapid and prominent shrinkage of the tissue. Morphologic investigation showed that over time, bipolarly elongated SMCs and collagen fiber bundles became positioned around the mandrel. Both components became circumferentially oriented. When the mandrel was removed, a tubular hybrid medial tissue was formed. A hybrid vascular tissue with a hierarchical structure was constructed by seeding endothelial cells onto the inner surface of the hybrid medial tissue. Prepared tissues tolerated luminal pressures as great as 100 mmHg and mechanical stress applied during an anastomotic procedure. This method allowed us to prepare a tubular hybrid medial tissue of predetermined size (inner diameter, wall thickness, and length) by selecting appropriate mold design, initial collagen concentration, and SMC seeding density. Such hybrid vascular tissues may provide physiological functions when implanted into the venous system.

Self-organized, tubular hybrid vascular tissue composed of vascular cells and collagen for low-pressure-loaded venous system.

A tubular, hierarchically structured hybrid vascular tissue composed of vascular cells and collagen was prepared. First, a cold mixed solution of bovine aortic smooth muscle cells (SMCs) and Type I collagen was poured into a tubular glass mold composed of a mandrel and a sheath (example of dimensions: inner diameter, 1.5 mm; outer diameter, 7 mm; length, 7 cm). Upon incubation at 37 degrees C, an SMC-incorporated collagenous gel was formed. After the sheath was removed, the resulting fragile tissue, when cultured in medium, thinned in a time-dependent manner to form an opaque, dense tissue. Higher SMC seeding density and lower initial collagen concentration induced more rapid and prominent shrinkage of the tissue. Morphologic investigation showed that over time, bipolarly elongated SMCs and collagen fiber bundles became positioned around the mandrel. Both components became circumferentially oriented. When the mandrel was removed, a tubular hybrid medial tissue was formed. A hybrid vascular tissue with a hierarchical structure was constructed by seeding endothelial cells onto the inner surface of the hybrid medial tissue. Prepared tissues tolerated luminal pressures as great as 100 mmHg and mechanical stress applied during an anastomotic procedure. This method allowed us to prepare a tubular hybrid medial tissue of predetermined size (inner diameter, wall thickness, and length) by selecting appropriate mold design, initial collagen concentration, and SMC seeding density. Such hybrid vascular tissues may provide physiological functions when implanted into the venous system.

W-Methods with Automatic Partitioning by Krylov Techniques for Large Stiff Systems

Solving the stage equations in W-methods approximately by a Krylov process may be interpreted as an automatic partitioning method, where some of the components are integrated by an implicit scheme whereas others are treated by an explicit Runge–Kutta method. The authors consider an implementation which uses only one family of Krylov spaces for all stages, introducing an additional error of the size of the discretization error. Two main results for the linear autonomous case show that (i) the method stays at the asymptotic limit solution under mild restrictions, which may be enforced in the numerical computation, and (ii) the dimensions of the Krylov spaces need only be slightly larger than the number of “fast” solution components with nonnegligible contribution to the local solution. Numerical examples of dimensions 20 to 302 indicate that the method may perform well even for Krylov spaces with low dimension.

The Niagara Therapy Bench for Children

VIBRATING benches are currently in use at Tudor House Nursery, Liverpool, to facilitate treatment of children under five years old who have cerebral palsy. Niagara therapy benches are robustly constructed and covered in flame retardant and easily cleaned vinyl. Rubber feet reduce transmitted vibration and prevent sliding. The vibration they provide is of variable frequency between 15-30 Hz, and has the physiological effect of reducing spasticity through muscle relaxation. The frequency is controlled by a switch sited underneath the bench, out of reach of little fingers. A separate hand-held unit houses the on/off control. The benches are 5 0 cm long. Their height and width are determined by the size of the users, and the manufacturer will supply to suit individual needs. The height should allow the children to sit comfortably with both feet flat on the floor, wi th hips and knees at right angles. As the children usually sit astride the benches, careful consideration must be given to their width, as excessive hip abduction would lead to increased flexion at the hips. To give an example of dimensions, one of the benches currently in use with children aged three to four years, is 18 cm high and 18 c m wide. The benches have now been in use at Tudor House for 2 ?h years, and have the following benefits:

A Martin Boundary in the Plane

Let $E$ be an open connected subset of Euclidean space, with a Green function, and let $\lambda$ be harmonic measure on the Martin boundary $\Delta$ of $E$. We will show that, except for a $\lambda \otimes \lambda$-null set of $(x,y) \in {\Delta ^2}$, $x$ is an entrance point for Brownian motion conditioned to leave $E$ at $y$. R. S. Martin gave examples in dimension $3$ or higher, for which there exist minimal accessible Martin boundary points $x \ne y$ for which this condition fails. We will give a similar example in dimension $2$.

Completely solvable gauge-field equations in dimension greater than four

This paper deals with analogues, in higher dimensions, of the self-dual Yang-Mills equations in dim4. Requiring that they be the integrability conditions for a linear system leads to a procedure for constructing their solutions. A partial classification of such equations is given, and three examples in dimension eight are studied in detail.