Classes of functions subharmonic in ℝm and bounded on certain sets

Let Z_j be the Euclidean space of vectors \((z_{j,1,...,} z_{j_{j \cdot n_j + 1} } ), Z = \mathop \oplus \limits_{j = 1}^P Z_j\) . The function u: Z → ?₊, u ?0, is said to be logarithmically p-subharmonic if log u(z) is upper semicontinuous with respect to the totality of the variables and subharmonic or identically equal to ?∞ with respect to each z_j when the remaining ones are fixed. For such functions, with the growth estimate $$log u(z) \leqslant \delta \mathop \Pi \limits_{j = 1}^P (1 + |z_{j,n_j + 1} |) + N(\mathop {\sum\limits_{\mathop {1 \leqslant j \leqslant p}\limits_{} } {z_{j,k}^2 } }\limits_{1 \leqslant k \leqslant n_j } )^{1/2} + C; \delta ,N \geqslant 0, C \in \mathbb{R}$$ one proves theorems on equivalence of^∞) (L_q)-norms of their restrictions to \(X = \mathop \oplus \limits_{j = 1}^P (Z_{j,1} ,...,z_{j,n_j } )\) and to a relatively dense subset of it, generalizing the known Cartwright and Plancherel-Pólya results.     

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

Description of a Class of Solids in ℝ3

Let be a compact convex body such that for each parallel projection onto any plane no two opposite faces of Q are projected strictly inside the projection of the entire Q. Then Q is either a cone, or a frustum of a trihedral pyramid, or a prism (possibly with nonparallel bases). Bibliography: 1 title.     

On unbounded solutions of ergodic problems in ℝm for viscous Hamilton–Jacobi equations

In this article, we study ergodic problems in the whole space ?^m for viscous Hamilton–Jacobi equations in the case of locally Lipschitz continuous and coercive right-hand sides. We prove in particular the existence of a critical value λ* for which (i) the ergodic problem has solutions for all λ≤λ*, (ii) bounded from below solutions exist and are associated to λ*, (iii) such solutions are unique (up to an additive constant). We obtain these properties without additional assumptions in the superquadratic case, while, in the subquadratic one, we assume the right-hand side to behave like a power. These results are slight generalizations of analogous results by Ichihara but they are proved in the present paper by partial differential equation (pde) methods, contrarily to Ichihara who is using a combination of pde technics with probabilistic arguments.     

Sharp asymptotics of large deviations in ℝ d

Given a sequence {X₁}i=1,2,3,... of i.i.d. random variables taking values in ? ^d ,d≥2, letS _n =Σ _i=1 ⁿ X _t=1. For Λ a Borel set in ? ^d having smooth boundary, witha=inf_x∈ΛI(x) the minimal value of the large deviation rate functionI(x) over Λ, we find, under suitable hypotheses, asymptotic results asn→∞, of the form $$P(S_n \in n\Gamma ) = n^\gamma e^{ - na} (d_0 + o(1))$$ where the constant γ depends sensitively on the geometry of Λ and the dimensiond, and takes values ?∞<γ≤(d?2/2). For fixeda=inf_x∈ΛI(x), we construct examples having any specific γ in this range.     

On uniqueness of Frankl-Morawetz problem in ℝ2

For the equation $$L[u]: = K(y)u_{xx} + u_{yy} + r(x,y)u = f(x,y)$$ (K (y)?0 whenevery?0) inG, bounded by a piecewise smooth curveΓ ₀ fory>0 which intersects the liney=0 at the pointsA(?1, 0) andB(1, 0) and fory<0 by a smooth curveΓ ₁ throughA which meets the characteristic of (1) throughB at the pointP, the uniqueness of the Frankl-Morawetz problem is proved without assuming thatΓ ₁ is monotone.     

Affine differential geometry of surfaces in ℝ4

Employing the method of moving frames, i.e. Cartan's algorithm, we find a complete set of invariants for nondegenerate oriented surfacesM ² in ⁴ relative to the action of the general affine group on ⁴. The invariants found include a normal bundle, a quadratic form onM ² with values in the normal bundle, a symmetric connection onM ² and a connection on the normal bundle. Integrability conditions for these invariants are also determined. Geometric interpretations are given for the successive reductions to the bundle of affine frames overM ², obtained by using the method of moving frames, that lead to the aforementioned invariants. As applications of these results we study a class of surfaces known as harmonic surfaces, finding for them a complete set of invariants and their integrability conditions. Further applications involve the study of homogeneous surfaces; these are surfaces which are fixed by a group of affine transformations that act transitively on the surface. All homogeneous harmonic surfaces are determined.     

Estimates of C m-Capacity of Compact Sets in ℝ N

For a given homogeneous elliptic partial differential operator L with constant complex coefficients, the Banach space V of distributions in $\mathbb{R}^N $ and a compact set X in $\mathbb{R}^N $ , we study the quantity $\lambda _{V,L} (X)$ equal to the distance in V from the class of functions f ₀ satisfying the equation Lf ₀ = 1 in a neighborhood of X (depending on f ₀) to the solution space of the equation Lf= 0 in the neighborhoods of X. For V=BC ^m , we obtain upper and lower bounds for $\lambda _{V,L} (X)$ in terms of the metric properties of the set X, which allows us to obtain estimates for $\lambda _{V,L} (X)$ for a wide class of spaces V.     

On the Construction of Some New Harmonic Maps from ℝ m to ℍ m

In this note, we construct some new harmonic maps from ℝ ^m to ℍ ^m via symmetric methods. Supportedb y NSF Postdoctoral Research Fellowship, China, and MCSEC.     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

The limit set of trajectory in quasi-homogeneous system in ℝ3

In this article, we study the limit set of trajectory in three-dimensional quasi-homogeneous system. For the trajectory which is close to a singular curve, we show that either it approaches a fixed point or infinity, or it oscillates. Moreover, an oscillating example is given. The behaviour of trajectory which is near a closed orbit is also studied. At the end, we classify the integral manifolds of the system with isolated singularities.     

Long-term analysis of degenerate parabolic equations in ℝ N

Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space R^N is considered. By using -trajectories methods, we proved that weak solutions generated by degenerate equations possess an (L_U² (R^N), L_loc² (R^N))-global attractor. Moreover, the upper bounds of the Kolmogorov ε-entropy for such global attractor are also obtained.     

Set of ambiguous points for functions in ℝ n

It is known that an arbitrary function in the open unit disk can have at most countable set of ambiguous points. Point ζ on the unit circle is an ambiguous point of a function if there exist two Jordan arcs, lying in the unit ball, except the endpoint ζ, such that cluster sets of function along these arcs are disjoint. We investigate whether it is possible to modify the notion of ambiguous point to keep the analogous result true for functions defined in the n-dimensional Euclidean unit ball.     

Self-packing of centrally symmetric convex bodies in ℝ2

LetB be a compact convex body symmetric around0 in ℝ² which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radiusρ(m,B) is the smallestt such thatt B can be packed withm translates of the interior ofB. Form≤6 we show that the self-packing radiusρ(m,B)=1+2/α(m,B) whereα(m,B) is the Minkowski length of the side of the largest equilateralm-gon inscribed inB (measured in the Minkowski metric determined byB). We showρ(6,B)=ρ(7,B)=3 for allB, and determine most of the largest and smallest values ofρ(m,B) form≤7. For allm we have

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Lines of curvature on surfaces immersed in ℝ4

The differential equation of thelines of curvature for immersions of surfaces into ⁴ is established. It is shown that, for a class of generic immersions of a surface into ⁴ in theC ^r -topology,r4, all of the umbilic points are locally topologically stable. This type of umbilic points is described.Dedicated to the memory of Ricardo MañéPart of this work was supported by CNPq-IMPAResearch supported by grant N. 049633GM Universidad de Santiago, Chile.     

The Geometry of Polygons in ℝ5 and Quaternions

We consider the moduli space _r of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between _r and a weighted quotient of n-fold products of the quaternionic projective line ¹ by the diagonal PSL(2, )-action. We explore the relation between _r and the fixed point set of an anti-symplectic involution on a GIT quotient Gr_ℂ(2, 4) ⁿ /SL(4, ℂ). We generalize the Gel'fand—MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space _r as a quotient of a subspace in the quaternionic Grassmannian Gr_ℍ(2, n) by the action of the group Sp(1) ⁿ . We also give analogues of the Gel'fand—Tsetlin coordinates on the space of quaternionic Hermitean marices and briefly describe generalized action—angle coordinates on _r .