How far can we go with Amitsur’s theorem in differential polynomial rings?

A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.     

Functional Identities and Rings of Quotients

The fundamental theorem on functional identities states that a prime ring R with \(\deg (R)\ge d\) is a d-free subset of its maximal left ring of quotients Q _{m l} (R). We consider the question whether the same conclusion holds for symmetric rings of quotients. This indeed turns out to be the case for the maximal symmetric ring of quotients Q _{m s} (R), but not for the symmetric Martindale ring of quotients Q _s (R). We show, however, that if the maps from the basic functional identities have their ranges in R, then the maps from their standard solutions have their ranges in Q _s (R). We actually prove a more general theorem which implies both aforementioned results. Its proof is somewhat shorter and more compact than the standard proof used for establishing d-freeness in various situations.     

Inner and outer approximation of convex sets using alignment

We show that there exists, for each closed bounded convex set C in the Euclidean plane with nonempty interior, a quadrangle Q having the following two properties. Its sides support C at the vertices of a rectangle r and at least three of the vertices of Q lie on the boundary of a rectangle R that is a dilation of r with ratio 2. We will prove that this implies that quadrangle Q is contained in rectangle R and that, consequently, the inner approximation r of C has an area of at least half the area of the outer approximation Q of C. The proof makes use of alignment or Schüttelung, an operation on convex sets.     

On a theorem of Halin

This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G, with \(\aleph _0 \le |{\text {Aut}}(G)| < 2^{\aleph _0}\) and subdegree-finite automorphism group, has a finite set F of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called distinguishing, is also provided. To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof.     

Computational Complexity of the Vertex Cover Problem in the Class of Planar Triangulations

We study the computational complexity of the vertex cover problem in the class of planar graphs (planar triangulations) admitting a plane representation whose faces are triangles. It is shown that the problem is strongly NP-hard in the class of 4-connected planar triangulations in which the degrees of vertices are of order O(log n), where n is the number of vertices, and in the class of plane 4-connected Delaunay triangulations based on the Minkowski triangular distance. A pair of vertices in such a triangulation is adjacent if and only if there is an equilateral triangle ?(p, λ) with p ∈ R² and λ > 0 whose interior does not contain triangulation vertices and whose boundary contains this pair of vertices and only it, where ?(p, λ) = p + λ? = {x ∈ R²: x = p + λa, a ∈ ?}; here ? is the equilateral triangle with unit sides such that its barycenter is the origin and one of the vertices belongs to the negative y-axis. Keywords: computational complexity, Delaunay triangulation, Delaunay TD-triangulation.     

On a class of partial planes related to biplanes

In this note we consider partial planes in which for each element x (point or line) there exists a unique opposite element or antipode x* which cannot be joined to x or has no intersection with x. We also require the existence of a triangle. Such partial planes will be called antipodal planes. We are mainly interested in the subclass of regular antipodal planes satisfying: p I L implies p* I L* for all points p and lines L. We shall provide a free construction of infinite regular antipodal planes. The objects thus constructed are not free objects in the usual sense since between antipodal planes there do not exist proper homomorphisms. On the other hand, regular antipodal planes do have a canonical homomorphic image which is a biplane (cf. Payne, J Comb Theory A 12:268–282, 1972). Regular antipodal planes can be coordinatized by certain algebraic systems in a similar way as projective planes are coordinatized by ternary rings. Again by a free construction, we shall provide examples satisfying a configuration theorem comparable to the Fano condition with fixed line at infinity.     

On homeomorphisms of the plane which have no fixed points

Homeomorphisms of the planeR ² onto itself are studied, subject to the restriction that they should preserve the sense of orientation and have no fixed points. The results of this investigation are then applied to the problem of determining which homeomorphisms can be embedded in flows, i.e., in one-parameter subgroups of the full homeomorphism group of the plane. A “free mapping” ofR ² onto itself is defined to be a homeomorphismT, without fixed points, such thatC∩TC=0 impliesC∩T ⁿ C=0 for alln≠0 wheneverC is a compact connected subset ofR ². Free mappings turn out to be just those homeomorphisms ofR ² onto itself that preserve orientation and have no fixed points. A fundamental property of free mappingsis the fact that ifT is a free mapping andA is any compact subset ofR ² then\(\mathop U\limits_{ - \infty }^{ + \infty } T^n A\) does not meet some unbounded connected subsetB ofR ². The proof of this theorem is lengthy, and will appear elsewhere. The theorem can be weakened by adding the extra assumption thatT be embedded in a flow; the proof of this weakened version is much easier, and is included in the present article. It is found that for an arbitrary free mappingT there exists a natural partition of the plane into a collection of “fundamental regions”, with the property that ifT is embedded in a flow then each of the fundamental regions is invariant under the flow. An example is given of a free mapping whose fundamental regions are bad enough so that the mapping cannot be embedded in a flow. It is proved, on the other hand, that if a free mappingT has just one fundamental region thenT is equivalent to a translation, i.e., there is a homeomorphismU ofR ² onto itself such thatUTU ^?1 is just the translation(x, y)→(x+1, y). Indeed,T is equivalent to a translation if and only ifT has just one fundamental region.     

On the minimum size of binary codes with length 2<Emphasis Type="Italic">R</Emphasis> + 4 and covering radius <Emphasis Type="Italic">R</Emphasis>

The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.     

Nodal sets of smooth functions with finite vanishing order and <Emphasis Type="Italic">p</Emphasis>-sweepouts

We show that on a compact Riemannian manifold (M, g), nodal sets of linear combinations of any \(p+1\) smooth functions form an admissible p-sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a result, we obtain a new proof of the Gromov, Guth, Marques–Neves upper bounds on the min–max p-widths of M. We also prove that close to a point at which a smooth function on \(\mathbb {R}^{n+1}\) vanishes to order k, its nodal set is contained in the union of \(k\,W^{1,p}\) graphs for some \(p > 1\). This implies that the nodal set is locally countably n-rectifiable and has locally finite \(\mathcal {H}^n\) measure, facts which also follow from a previous result of Bär. Finally, we prove the continuity of the Hausdorff measure of nodal sets under heat flow.     

Structure of Abelian rings

Let R be a ring with identity. We use J(R); G(R); and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R; respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ? J(R) of R; that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2; 3; 4; and 5 orbits under the left regular action on X(R) by G(R): For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R); then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.     

Spherical means in annular regions in the <Emphasis Type="Italic">n</Emphasis>-dimensional real hyperbolic spaces

Let Z _r,R be the class of all continuous functions f on the annulus Ann(r, R) in the real hyperbolic space \(\mathbb B^n\) with spherical means M _s f(x)?=?0, whenever s?>?0 and \(x\in\mathbb B^n\) are such that the sphere S _s (x)???Ann(r, R) and \(B_r(o)\subseteq B_s(x).\) In this article, we give a characterization for functions in Z _r,R . In the case R?=?∞, this result gives a new proof of Helgason’s support theorem for spherical means in the real hyperbolic spaces.     

Antipodal covers of distance-transitive generalized polygons

In an earlier paper, the first two authors found all distance-regular antipodal covers of all known primitive distance-transitive graphs of diameter at least 3 with one possible exception. That remaining case is resolved here with the proof that a primitive and distance-transitive collinearity graph of a finite generalized 2d-gon with \(d\ge 3\) has no distance-regular antipodal cover of diameter 2d.     

Maximal regularity of the spatially periodic stokes operator and application to nematic liquid crystal flows

We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal L^p-regularity of the periodic Laplace and Stokes operators and a local-intime existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group \(G: = \mathbb{R}^{n - 1} \times \mathbb{R}/L\mathbb{Z}\) to obtain an R-bound for the resolvent estimate. Then, Weis’ theorem connecting R-boundedness of the resolvent with maximal L^p regularity of a sectorial operator applies.     

The spectral excess theorem for distance-regular graphs having distance-<Emphasis Type="Italic">d</Emphasis> graph with fewer distinct eigenvalues

Let \(\Gamma \) be a distance-regular graph with diameter d and Kneser graph \(K=\Gamma _d\), the distance-d graph of \(\Gamma \). We say that \(\Gamma \) is partially antipodal when K has fewer distinct eigenvalues than \(\Gamma \). In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues) and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with \(d+1\) distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a more general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.     

Local reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(? ₁, ..., ? _N ) generated by finitely many compactly supported functions ? ₁, ..., ? _N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(? ₁, ..., ? _N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(?) generated by a compactly supported refinable function ?, we prove that for almost all \((x_0, x_1)\in [0,1]^2\), any signal in V(?) can be locally reconstructed from its samples from \(\{x_0, x_1\}+{\mathbb Z}\) with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(?) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.     

Affine embeddings of Cantor sets and dimension of <Emphasis Type="Italic">αβ</Emphasis>-sets

Let E,F ? R^d be two self-similar sets, and suppose that F can be affinely embedded into E. Under the assumption that E is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between the contraction ratios of E and F. This gives a partial affirmative answer to Conjecture 1.2 in [9]. The proof is based on our study of the boxcounting dimension of a class of multi-rotation invariant sets on the unit circle, including the αβ-sets initially studied by Engelking and Katznelson.     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

A sharp eigenvalue theorem for fractional elliptic equations

The invisibility graph I(X) of a set X ? R ^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number ω(I(X)), the chromatic number χ(I(X)) and the convexity number γ(X), which is the minimum number of convex subsets of X that cover X.We settle a conjecture of Matou?ek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also find sets X in R⁵ with χ(X) = 2, but γ(X) arbitrarily large.