Planar Posets,Dimension, Breadth and the Number of Minimal Elements

In recent years, researchers have shown renewed interest in combinatorial properties of posets determined by geometric properties of its order diagram and topological properties of its cover graph. In most cases, the roots for the problems being studied today can be traced back to the 1970’s, and sometimes even earlier. In this paper, we study the problem of bounding the dimension of a planar poset in terms of the number of minimal elements, where the starting point is the 1977 theorem of Trotter and Moore asserting that the dimension of a planar poset with a single minimal element is at most 3. By carefully analyzing and then refining the details of this argument, we are able to show that the dimension of a planar poset with t minimal elements is at most 2t + 1. This bound is tight for t = 1 and t = 2. But for t ≥ 3, we are only able to show that there exist planar posets with t minimal elements having dimension t + 3. Our lower bound construction can be modified in ways that have immediate connections to the following challenging conjecture: For every d ≥ 2, there is an integer f(d) so that if P is a planar poset with dim(P) ≥ f(d), then P contains a standard example of dimension d. To date, the best known examples only showed that the function f, if it exists, satisfies f(d) ≥ d + 2. Here, we show that lim _d→∞ f(d)/d ≥ 2.     

The Nöther and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nöther type theorems for C ^µ piecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible C ^µ piecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the C ^µ piecewise algebraic curve is established.     

Dimension and Matchings in Comparability and Incomparability Graphs

We develop some new inequalities for the dimension of a finite poset. These inequalities are then used to bound dimension in terms of the maximum size of matchings. We prove that if the dimension of P is d and d=3, then there is a matching of size d in the comparability graph of P. There is no analogue of this result for cover graphs, as we show that there is a poset P of dimension d for which the maximum matching in the cover graph of P has size \(O(\log d)\). On the other hand, there is a dual result in which the role of chains and antichains is reversed, as we show that there is also a matching of size d in the incomparability graph of P. The proof of the result for comparability graphs has elements in common with Perles’ proof of Dilworth’s theorem. Either result has the following theorem of Hiraguchi as an immediate corollary: \(\dim (P)\le |P|/2\) when |P|=4.     

Two operator representations for the trivariate <Emphasis Type="Italic">q</Emphasis>-polynomials and Hahn polynomials

In this paper, we introduce a trivariate q-polynomials \(F_n(x,y,z;q)\) as a general form of Hahn polynomials \(\psi _n^{(a)}(x|q)\) and \(\psi _n^{(a)}(x,y|q)\). We represent \(F_n(x,y,z;q)\) by two operators: the homogeneous q-shift operator \(L(b\theta _{xy})\) given by Saad and Sukhi (Appl Math Comput 215:4332–4339, 2010), and the Cauchy companion operator \(E(a,b;\theta )\) given by Chen (q-Difference Operator and Basic Hypergeometric Series, 2009) to derive the generating function, symmetric property, Mehler’s formula, Rogers formula, another Roger-type formula, linearization formula, and an extended Rogers formula for the trivariate q-polynomials. Then, we give the corresponding formulas for our new definitions of Hahn polynomials \(\psi _n^{(a)}(x|q)\) and \(\psi _n^{(a)}(x,y|q)\) by representing Hahn polynomials by the operators \(L(b\theta _{xy})\) and \(E(a,b;\theta )\), and by a special substitution in the trivariate q-polynomials \(F_n(x,y,z;q)\).     

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

As shown in Aldaz (Bull. Lond. Math. Soc. 39:203–208, 2007), the lowest constants appearing in the weak type (1, 1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this result to a wider class of radial measures and to some values of p > 1. Furthermore, we improve the previously known bounds for p = 1. Roughly speaking, whenever \({p\in (1, 1.03]}\), if μ is defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c _p,d,μ in the weak type (p, p) inequalities satisfy c _p,d,μ ≥ 1.005 ^d for all d sufficiently large. We also show that exponential increase of the best constants occurs for certain families of doubling measures, and for arbitrarily high values of p.     

Spectral multipliers on 2-step groups: topological versus homogeneous dimension

Let G be a 2-step stratified group of topological dimension d and homogeneous dimension Q. Let \({\mathcal{L}}\) be a homogeneous sub-Laplacian on G. By a theorem due to Christ and to Mauceri and Meda, an operator of the form \({F(\mathcal{L})}\) is of weak type (1, 1) and bounded on L ^p (G) for all p ∈ (1, ∞) whenever the multiplier F satisfies a scale-invariant smoothness condition of order s > Q/2. It is known that, for several 2-step groups and sub-Laplacians, the threshold Q/2 in the smoothness condition is not sharp and in many cases it is possible to push it down to d/2. Here we show that, for all 2-step groups and sub-Laplacians, the sharp threshold is strictly less than Q/2, but not less than d/2.     

Bernstein-Bézier techniques for divergence of polynomial spline vector fields in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Bernstein-Bézier techniques for analyzing polynomial spline fields in n variables and their divergence are developed. Dimension and a minimal determining set for continuous piecewise divergence-free spline fields on the Alfeld split of a simplex in ? ⁿ are obtained using the new techniques, as well as the dimension formula for continuous piecewise divergence-free splines on the Alfeld refinement of an arbitrary simplicial partition in ? ⁿ .     

Nonsingularity,positive definiteness,and positive invertibility under fixed-point data rounding

For a real square matrix A and an integer d ? 0, let A _(d) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A _(d) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number α(d), computed solely from A _(d) (not from A), such that the following alternative holdsif d > α(d), then nonsingularity (positive definiteness, positive invertibility) of A _(d) implies the same property for A if d < α(d) and A _(d) is nonsingular (positive definite, positive invertible), then there exists a matrix A′ with A′_(d) = A _(d) which does not have the respective property.For nonsingularity and positive definiteness the formula for α(d) is the same and involves computation of the NP-hard norm ‖ · ‖_∞,1; for positive invertibility α(d) is given by an easily computable formula.     

On sets of directions determined by subsets of ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Given E ? ? ^d , d ≥ 2, define

$D(E) \equiv \left\{ {{{x - y} \over {\left| {x - y} \right|}}:x,y \in E} \right\} \subset {S^{d - 1}}$

to be the set of directions determined by E. We prove that if the Hausdorff dimension of E is greater than d ? 1, then σ(D(E)) > 0, where σ denotes the surface measure on S ^d?1. In the process, we prove some tight upper and lower bounds for the maximal function associated with the Radon-Nikodym derivative of the natural measure on D. This result is sharp, since the conclusion fails to hold if E is a (d ? 1)-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir ([22, 23]) on directions determined by finite subsets of ? ^d . We also discuss the case when the Hausdorff dimension of E is precisely d ? 1, where some interesting counter-examples have been obtained by Simon and Solomyak ([25]) in the planar case. In response to the conjecture stated in this paper, T. Orponen and T. Sahlsten ([20]) have recently proved that if the Hausdorff dimension of E equals d ? 1 and E is rectifiable and is not contained in a hyper-pane, the Lebesgue measure of the set of directions is still positive. Finally, we show that our continuous results can be used to recover and, in some cases, improve the exponents for the corresponding results in the discrete setting for large classes of finite point sets. In particular, we prove that a finite point set P ? ? ^d , d ≥ 3, satisfying a certain discrete energy condition (Definition 3.1) determines ? #P distinct directions.

     

On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator L _r of arbitrary order r with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from C to C) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator L ₃ = D(D ² ? β ²) (β > 0), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.     

Continuous reducibility and dimension of metric spaces

If (X, d) is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space (X, d) of positive dimension, there are uncountably many Borel subsets of (X, d) that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \((X,\tau )\) is called the Wadge quasi-order for \((X,\tau )\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \((X,\tau )\), is a well-quasiorder if and only if \((X,\tau )\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs.     

Polynomials with bounds and numerical approximation

We discuss the generation of polynomials with two bounds —an upper bound and a lower bound— on compact sets \(\mathcal C=[0,1]^{d}\subset \mathbb {R}^{d}\) in view on numerical approximation and scientific computing. The presentation is restricted to d = 1,2. We show that a composition formula based on a weighted 4-squares Euler identity generates all such polynomials in dimension d=1. It yields a new algebraic or compositional approach to the classical problem related to polynomials with minimal uniform norm. The theoretical generalization to the multivariate case is discussed in dimension d=2 by means of the 8-squares Degen identity and the connection with quaternions algebras is made explicit. Numerical results in dimension d=1 illustrate the potentialities of this approach and some implementation details are provided.     

Sharp Weak Type Inequalities for Fractional Integral Operators

Suppose that d≥1 is an integer, α∈(0,d) is a fixed parameter and let I _α be the fractional integral operator associated with d-dimensional Walsh-Fourier series on (0,1] ^d . Let p, q be arbitrary numbers satisfying the conditions 1≤p<d/α and 1/q=1/p?α/d. We determine the optimal constant K, which depends on α, d and p, such that for any f∈L ^p ((0,1] ^d ) we have

$$ ||I_{\alpha } f||_{L^{q,\infty }((0,1]^{d})}\leq K||f||_{L^{p}((0,1]^{d})}. $$

In fact, we shall prove this inequality in the more general context of probability spaces equipped with a regular tree-like structures. This allows us to obtain this result also for non-integer dimension. The proof exploits a certain modification of the so-called Bellman function method and appropriate interpolation-type arguments. We also present a sharp weighted weak-type bound for I _α , which can be regarded as a version of the Muckenhoupt-Wheeden conjecture for fractional integral operators.

     

Sub-Riemannian distance in the Lie groups SU(2) and SO(3)

We calculate distances between arbitrary elements of the Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics ρ and d. In computing distances for the second metric, we substantially use the fact that the canonical two-sheeted covering epimorphism Ω of SU(2) onto SO(3) is a submetry and a local isometry in the metrics ρ and d. Despite the fact that the proof uses previously known formulas for geodesics starting at the unity, F. Klein’s formula for Ω, trigonometric functions, and the conventional differential calculus of functions of one real variable, we focus attention on a careful application of these simple tools in order to avoid the mistakes made in previously published mathematical works in this area.     

Dimensions of spline spaces over non-rectangular T-meshes

Spline spaces over rectangular T-meshes have been discussed in many papers. In this paper, we consider spline spaces over non-rectangular T-meshes. The dimension formulae of spline spaces over special simply connected T-meshes have been obtained. For T-meshes with holes, we discover a new type of dimension instability. We construct a relationship between the dimension of the spline space over a T-mesh \(\mathcal {T}\) with holes and the dimension of the spline space over a simply connected T-mesh associated with \(\mathcal {T}\).     

On the Classification of Almost Square-Free Modular Categories

Let \(\mathcal {C}\) be a modular category of Frobenius-Perron dimension d q ⁿ , where q >?2 is a prime number and d is a square-free integer. We show that \(\mathcal {C}\) must be integral and nilpotent and therefore group-theoretical. In the case where q =?2, we describe the structure of \(\mathcal {C}\) in terms of equivariantizations of group-crossed braided fusion categories.     

Picard dimension of signed radial Kato measures

The Picard dimension dimμ of a signed local Kato measure μ on the punctured unit ball in R^d, d ≥ 2, is the cardinal number of the set of extremal rays of the convex cone of all continuous solutions u ≥ 0 of the time-independent SchrSdinger equation Δu -- uμ = 0 on the punctured ball 0 〈 ｜｜x｜｜ 〈 1, with vanishing boundary values on the sphere ｜｜x｜｜ = 1. Using potential theory associated with the Schrodinger operator we prove, in this paper, that the dimμ for a signed radial Kato measure is 0, 1 or ＋∞. In particular, we obtain the Picard dimension of locally Holder continuous functions P proved by Nakai and Tada by other methods.     

On the sum of the L1 influences of bounded functions

Let f: {-1, 1}ⁿ → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L₁ influence of f can also be bounded as a function of d. Ba?kurs and Bavarian answered this question in the affirmative, providing a bound of O(d³) for general functions and O(d²) for homogeneous functions. We improve on their results by providing a bound of d² for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.     

On the Hausdorff Dimension of Continuous Functions Belonging to Hölder and Besov Spaces on Fractal <Emphasis Type="Italic">d</Emphasis>-Sets

The Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability p≥1) on fractal d-sets is studied. Denoting by s∈(0,1] the smoothness parameter, the sharp upper bound min{d+1?s,d/s} is obtained. In particular, when passing from d≥s to d<s there is a change of behaviour from d+1?s to d/s which implies that even highly nonsmooth functions defined on cubes in ? ⁿ have not so rough graphs when restricted to, say, rarefied fractals.     

On the unimprovability of full-memory strategies in the risk minimization problem

We continue the study of approximation properties of local exponential splines on a uniform grid with step h > 0 corresponding to a linear differential operator L with constant coefficients and real pairwise different roots of the characteristic polynomial (such splines were constructed by E.V. Strelkova and V.T. Shevaldin). We find order estimates as h → 0 for the error of approximation of certain Sobolev classes of functions by splines of the described type that are exact on the kernel of the operator L.