Cone In R3 Research Articles

On a faithful representation of Sturmian morphisms

The set of morphisms mapping any Sturmian sequence to a Sturmian sequence forms together with composition the so-called monoid of Sturm. For this monoid, we define a faithful representation by (3×3)-matrices with integer entries. We find three convex cones in R3 and show that a matrix R∈Sl(Z,3) is a matrix representing a Sturmian morphism if the three cones are invariant under multiplication by R or R−1. This property offers a new tool to study Sturmian sequences. We provide alternative proofs of four known results on Sturmian sequences fixed by a primitive morphism and a new result concerning the square root of a Sturmian sequence.

Uniqueness property for 2-dimensional minimal cones in $\mathbb{R}^3$

In this article we treat two closely related problems: 1) the upper semicontinuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the 2-dimensional minimal cones in R3 . Given an open set Ω ⊂ Rn, a closed set E ⊂ Ω is said to be Almgren minimal of dimension d in Ω if it minimizes the d-Hausdorff measure among all its Lipschitz deformations in Ω. We say that a d-dimensional minimal set E in an open set Ω admits upper semi-continuity if, whenever {fn(E)}n is a sequence of deformations of E in Ω that converges to a set F, then we have Hd(F) ≥ lim supn Hd(fn(E)). This guarantees in particular that E minimizes the d-Hausdorff measure, not only among all its deformations, but also among limits of its deformations. As proved in [19], when several 2-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known 2-dimensional minimal cones in Rn to generate new families of minimal cones by taking their almost orthogonal unions. The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].

Stable s-minimal cones in ℝ3 are flat for s ~ 1

Abstract We prove that half spaces are the only stable nonlocal s-minimal cones in ℝ 3 {\mathbb{R}^{3}} , for s ∈ ( 0 , 1 ) {s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from s = 1 {s=1} . In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

Why Do All Triangles Form a Triangle?

We provide a geometric explanation, based on symmetry, for why the moduli space of all triangles up to similarity is itself a triangle. Symmetries occur because the lengths of the sides define triples in ℝ3 so are acted on by the symmetric group 핊3, which is isomorphic to the symmetry group 픻3 of an equilateral triangle. The moduli space for triangles is a fundamental domain for the action of 픻3 on an equilateral triangle in ℝ3 determined by all triangles with unit perimeter and is chosen from a subdivision into six congruent triangles. Isosceles and equilateral triangles occupy special locations determined by their symmetries. The sides of a right triangle lie on one of three double cones in ℝ3, and those of unit perimeter lie on a segment of a hyperbola in the moduli space.

The associated families of semi-homogeneous complete hyperbolic affine spheres

Hildebrand classified all semi-homogeneous cones in ℝ3 and computed their corresponding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand's examples. The generic member of these affine spheres is given by Weierstrass ▪, ζ and σ functions. In general any regular convex cone in ℝ3 has a natural associated S1 -family of such cones, which deserves further studies.

Mean curvature self-shrinkers of high genus: Non-compact examples

Abstract We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent’s torus in 1989. The surfaces exist for any sufficiently large prescribed genus g, and are non-compact with one end. Each has 4 ⁢ g + 4 {4g+4} symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry. Each is at infinity asymptotic to the cone in ℝ 3 {\mathbb{R}^{3}} over a 2 ⁢ π / ( g + 1 ) {2\pi/(g+1)} -periodic graph on an equator of the unit sphere 𝕊 2 ⊆ ℝ 3 {\mathbb{S}^{2}\subseteq\mathbb{R}^{3}} , with the shape of a periodically “wobbling sheet”. This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions. The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein–Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted Hölder spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.

Geometric Tomography of Convex Cones

The parallel X-ray of a convex set K⊂ℝn in a direction u is the function that associates to each line l, parallel to u, the length of K∩l. The problem of finding a set of directions such that the corresponding X-rays distinguish any two convex bodies has been widely studied in geometric tomography. In this paper we are interested in the restriction of this problem to convex cones, and we are motivated by some applications of this case to the covariogram problem. We prove that the determination of a cone by parallel X-rays is equivalent to the determination of its sections from a different type of tomographic data (namely, point X-rays of a suitable order). We prove some new results for the corresponding problem which imply, for instance, that convex polyhedral cones in ℝ3 are determined by parallel X-rays in certain sets of two or three directions. The obtained results are optimal.

Packing Cones and Their Negatives in Space

Let C be a cone in R3 whose base B is a planar convex body in a horizontal plane π and whose tip is a point v ∉ π. Let C be a packing formed by translates of C and -C in R3. We exhibit an explicit constant c > 0 such that the density of any such C is smaller than 1 - c, answering a question of Wlodek Kuperberg.

A new example of a uniformly Levi degenerate hypersurface in C3

We present a homogeneous real analytic hypersurface in C3, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube ΓC over the two-dimensional real cone in R3 is also homogeneous and has a seven-dimensional CR automorphism group. However, our example isnot biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of ΓC are, in contrast, noncommutative.

A new example of a uniformly Levi degenerate hypersurface in C3

We present a homogeneous real analytic hypersurface in C3, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube ΓC over the two-dimensional real cone in R3 is also homogeneous and has a seven-dimensional CR automorphism group. However, our example is not biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of ΓC are, in contrast, noncommutative.

Three-Dimensional Competitive Lotka--Volterra Systems with no Periodic Orbits

The following conjecture of M. L. Zeeman is proved. If three interacting species modeled by a competitive Lotka--Volterra system can each resist invasion at carrying capacity, then there can be no coexistence of the species. Indeed, two of the species are driven to extinction. It is also proved that in the other extreme, if none of the species can resist invasion from either of the others, then there is stable coexistence of at least two of the species. In this case, if the system has a fixed point in the interior of the positive cone in R3 , then that fixed point is globally asymptotically stable, representing stable coexistence of all three species. Otherwise, there is a globally asymptotically stable fixed point in one of the coordinate planes of R3 , representing stable coexistence of two of the species.

Every stationary polyhedral set in Rnis area minimizing under diffeomorphisms

There are infinitely many minimal cones in R3. Of these only three are area minimizing under Lipschitz maps: the plane; the three half planes meeting along their common boundary line at an angle of 120 degrees; the cone over the one-skeleton of the regular tetrahedron (see [T]). A recent work of Lawlor and Morgan [LM] has produced a generalization to higher dimensional cones in # n. Namely, the hypercone over the (n — 2)-skeleton of the regular simplex in Rn has the least area among all surfaces separating the (n — l)-dimensional faces of the simplex. Consequently this cone is area minimizing under Lipschitz maps. Moreover, Brakke hats proved that the hypercone over the (n - 2)-skeleton of the cube in Rn is area minimizing under Lipschitz maps when and only when n > 4 [B]. In this paper we prove that every stationary polyhedral set in Rn is area minimizing under diffeomorphisms leaving the boundary fixed. Therefore if we only consider competing surfaces of diffeomorphic images of a minimal cone C in i?3, C has the least area. Hence in R3 all minimal cones are stable. We wish to thank Frank Morgan for helpful comments on the extension of the main theorem. 1. Terminology.

Projectionally exposed cones in R3

Programming problems may be classified, on the basis of the objective function and types of constraints, as linear, nonlinear, discrete, integer, Boolean, etc. These programming problems represent special cases of the following more general abstract convex programming problem (ACPP): Find min{ƒ(x):g(x)∈−K, x∈Ω} , where Ω⊆R n is convex, K is a convex cone, and f, g are convex functions. Characterizations of optimality to the ACPP are of paramount importance in the investigation of optimization problems. A cone K in R n is called projectionally exposed if for each face F of K there exists a projection P F of R n such that P F ( K) = F. In particular, it has been shown that when the constraint function g of the ACPP takes values in a projectionally exposed cone, then certain multipliers, associated with optimality, may be chosen from a smaller set. The projectionally exposed cones of R 3 are completely characterized in this paper.

On the restriction of the Fourier transform to a conical surface

Let Γ \Gamma be the surface of a circular cone in R 3 {{\mathbf {R}}^3} . We show that if 1 ⩽ p > 4 / 3 1 \leqslant p > 4/3 , 1 / q = 3 ( 1 − 1 / p ) 1/q = 3(1 - 1/p) and f ∈ L p ( R 3 ) f \in {L^p}({{\mathbf {R}}^3}) , then the Fourier transform of f f belongs to L q ( Γ , d σ ) {L^q}(\Gamma ,d\sigma ) for a certain natural measure σ \sigma on Γ \Gamma . Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint p = 4 / 3 p = 4/3 , with logarithmic growth of the bound as the thickness of the annulus tends to zero.