Embedding Tn-like continua in Euclidean space

Many authors have been concerned with embedding ∏-like continua in Rⁿ where ∏ is some collection of polyhedra or manifolds. A similar concern has been embedding ∏-like continua in Rⁿ up to shape. In this paper we prove two main theorems. Theorem: If n ? 2 and X is Tⁿ-like, then X embeds in R²ⁿ. This result was conjectured by McCord for the case H¹(X) finitely generated and proved by McCord for the case that H¹(X) = 0 using a theorem of Isbell. The second theorem is a shape embedding theorem. Theorem: If X is Tⁿ-like, then X embeds in Rⁿ⁺² up to shape. This theorem is proved by showing that an n-dimensional compact connected abelian topological group embeds in Rⁿ⁺². Any Tⁿ-like continuum is shape equivalent to a k-dimensional compact connected abelian topological group for some 0 ? k ? n.     

The Zygmund Morse-Sard Theorem

The classical Morse-Sard Theorem says that the set of critical values off:R ^n+k →R ⁿ has Lebesgue measure zero iff ∈C ^k+1. We show theC ^k+1 smoothness requirement can be weakened toC ^k+Zygmund. This is corollary to the following theorem: For integersn >m >r > 0, lets = (n ?r)/(m ?r); iff:R ⁿ →R ^m belongs to the Lipschitz class Λ _s andE is a set of rankr forf, thenf(E) has measure zero.     

L p-Boundedness for fractional oscillatory integral operator with rough kernel

In this paper we give Lp-bound edness for the operator Tu defined bywhere P(x,y) is a real nontrivial polynomial on Rn×Rn,Ωis homogeneous of degree zero,Ω∈Lq(Sn-1),q>1/(1-μ) and b(r)∈BV(R+),The result can be regarded as an improvement of F.Ricci and E.M.Stein's result for fractional oscillatory integral operator with smoothness kernel.     

Localization in fine potential theory and uniform approximation by subharmonic functions

It is shown how the cone $\text{l}$(U) of superharmonic functions ?0 on an open set U in $\text{R}$ⁿ, n ? 3, can be recovered from the cone $\text{l}$ of superharmonic functions ?0 on the whole of $\text{R}$ⁿ by a process involving the operator of localization associated with U. Actually we treat the more general case where U is open in the Cartan-Brelot fine topology on $\text{R}$ⁿ. As an application we obtain a new proof of a theorem of J. Bliedtner and W. Hansen on uniform approximation by continuous subharmonic functions in open sets containing a given compact set K in $\text{R}$ⁿ.     

On certain functional equation related to two-sided centralizers

Let m ≥ 0, n ≥ 0 be fixed integers with m + n ≠ 0 and let R be a prime ring with char(R) = 0 or m + n + 1 ≤ char(R) ≠ 2. Suppose that there exists an additive mapping T : R → R satisfying the relation 2T(x ^m+n+1) = x ^m T(x) x ⁿ  + x ⁿ T(x)x ^m for all ${x\in R}$ . In this case T is a two-sided centralizer.     

Perron's method for second order hyperbolic PDE. Part II

We continue part I of this paper. Here in part II, comparison principles are proved for non-smooth sub and super solutions (with non-smooth Cauchy data) of semilinear hyperbolic PDE in compact regular domains of R⁺×R⁺ when n?3.     

Estimates for harmonic conjugate functions and Riesz-system

In this paper we generalize a theorem of Hardy and Littlewood concerning the growth of conjugate harmonic functions. Moreover, we obtain an estimate for Riesz systems in the upper half-space R ₊ ⁿ⁺¹ of R ⁿ⁺¹.     

Factorizations of Möbius Gyrogroups

In this paper we consider a M?bius gyrogroup on a real Hilbert space (of finite or infinite dimension) and we obtain its factorization by gyrosubgroups and subgroups. It is shown that there is a duality relation between the quotient spaces and the orbits obtained. As an example we will present the factorization of the M?bius gyrogroup of the unit ball in \mathbbRⁿ{\mathbb{R}}^{n} linked to the proper Lorentz group Spin⁺(1, n).     

On generalized Jordan 1-derivation in rings

Let n ⩾ 1 be a fixed integer and let R be an (n + 1)!-torsion free 1-ring with identity element e. If F, d:R → R are two additive mappings satisfying F(xⁿ⁺¹) = F(x)(x¹)ⁿ + xd(x)(x¹)ⁿ⁻¹ + x²d(x)(x¹)ⁿ⁻²+ ⋯ +xⁿd(x) for all x ∈ R, then d is a Jordan 1-derivation and F is a generalized Jordan 1-derivation on R.     

Perron's method for second order hyperbolic PDE. Part III

We continue Part I of this paper. Here, in Part III, comparison principles are proved for nonsmooth sub and super solutions (with nonsmooth Cauchy data) of semilinear hyperbolic PDE in compact regular domains of R⁺×R⁺ when n?3, and existence of a nonsmooth solution to the nonsmooth Cauchy problem is proved by a Perron-like method.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

Tiling R 5 by Crosses

An n-dimensional cross comprises 2n+1 unit cubes: the center cube and reflections in all its faces. It is well known that there is a tiling of R ⁿ by crosses for all n. AlBdaiwi and the first author proved that if 2n+1 is not a prime then there are $2^{\aleph_{0}}$ non-congruent regular (= face-to-face) tilings of R ⁿ by crosses, while there is a unique tiling of R ⁿ by crosses for n=2,3. They conjectured that this is always the case if 2n+1 is a prime. To support the conjecture we prove in this paper that also for R ⁵ there is a unique regular, and no non-regular, tiling by crosses. So there is a unique tiling of R ³ by crosses, there are $2^{\aleph_{0}}$ tilings of R ⁴, but for R ⁵ there is again only one tiling by crosses. We guess that this result goes against our intuition that suggests ‘the higher the dimension of the space, the more freedom we get’.     

Positively regular homeomorphisms of Euclidean spaces

A homeomorphism of Rⁿ onto itself is called positively regular (or EC⁺) iff its family of non-negative iterates is pointwise equicontinuous. For EC⁺ homeomorphism of Rⁿ such that some point of Rⁿ has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rⁿ→Rⁿ has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rⁿ/M→Rⁿ/M is conjugate to a contraction on Rⁿ.Theorem 2. In Rⁿ,n≠4,5,M compact iff there existsa disk D such that h(D)?IntD.Theorem 3. In R², either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied.     

Embeddability of multiple cones

The main result of this paper is that if X is a Peano continuum such that its nth cone Cn(X) embeds into Rn+2 then X embeds into S². This solves a problem proposed by W. Rosicki.     

Hyperbolic Volume of Manifolds with Geodesic Boundary and Orthospectra

In this paper we describe a function F _n : R ₊ → R ₊ such that for any hyperbolic n-manifold M with totally geodesic boundary ${\partial M \neq \emptyset}In this paper we describe a function F _n : R ₊ → R ₊ such that for any hyperbolic n-manifold M with totally geodesic boundary ?M 1 ?{\partial M \neq \emptyset} , the volume of M is equal to the sum of the values of F _n on the orthospectrum of M. We derive an integral formula for F _n in terms of elementary functions. We use this to give a lower bound for the volume of a hyperbolic n-manifold with totally geodesic boundary in terms of the area of the boundary.     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

Conjugate connections and Radon's theorem in affine differential geometry

For a given nondegenerate hypersurfaceM ⁿ in affine space ? ⁿ⁺¹ there exist an affine connection ?, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=?h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ? and a nondegenerate metrich on a differentiable manifoldM ⁿ such that ?h is totally symmetric and satisfies the apolarity condition relative toh, canM ⁿ be locally immersed in ? ⁿ⁺¹ in such a way that (?,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ?. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (?,h) is that the conjugate connection of ? relative toh is projectively flat.     

The maximum number of disjoint pairs in a family of subsets

Let ? be a family of 2 ⁿ⁺¹ subsets of a 2n-element set. Then the number of disjoint pairs in ? is bounded by (1+o(1))2²ⁿ . This proves an old conjecture of Erdös. Let ? be a family of 2^1/(k+1)+δ)n subsets of ann-element set. Then the number of containments in ? is bounded by (1-1/k+o(1))( ₂ ^|?| ). This verifies a conjecture of Daykin and Erdös. A similar Erdös-Stone type result is proved for the maximum number of disjoint pairs in a family of subsets.     

Embedding and unsolvability theorems for modular lattices

LetR be a nontrivial ring with 1 and δ a cardinal. Let,L(R, δ) denote the lattice of submodules of a free unitaryR-module on δ generators. Let ? be the variety of modular lattices. A lattice isR-representable if embeddable in the lattice of submodules of someR-module; ?(R) denotes the quasivariety of allR-representable lattices. Let ω denote aleph-null, and let a (m, n) presentation havem generators andn relations,m, n≤ω. THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable word problem for any quasivarietyV,V ? ?, such thatL(R, ω) is inV. THEOREM. IfL is a denumerable sublattice ofL(R, δ), then it is embeddable in some sublatticeK ofL(R,δ^*) having five generators, where δ^*=δ for infinite δ and δ^*=4δ(m+1) if δ is finite andL has a set ofm generators. THEOREM. The free ?(R)-lattice on ω generators is embeddable in the free ?(R)-lattice on five generators. THEOREM. IfL has an (m, n), ?(R)-presentation for denumerablem and finiten, thenL is embeddable in someK having a (5, 1) ?(R)-presentation.     

Order-constrained uniform approximations to the exponential based on restricted rationals

LetR _n/m(z∶γ)=P _n(z∶γ)/(1?γz) ^m be a rational approximation to exp (z),z ∈C, of ordern for all real positiveγ. In this paper we show there exists exactly one value ofγ in each of min(n+1,m) interpolation intervals such that the uniform error overR ^? is at a local minimum.