Showing papers on "Section (fiber bundle) published in 2007"

Rigidity of gradient Ricci Solitons

Abstract: We define a gradient Ricci soliton to be rigid if it is a flat bundle $% N\times_{\Gamma}\mathbb{R}^{k}$ where $N$ is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.

Bihermitian metrics on Del Pezzo surfaces

Abstract: From a hermitian metric on the anticanonical bundle on a Del Pezzo surface, and a holomorphic section of it, we construct a one-parameter family of bihermitian metrics (or equivalently generalized Kahler structures). The construction appears to be linked to noncommutative geometry.

Phase transitions in the Kuramoto model.

Abstract: We consider the Kuramoto model of phase oscillators with natural frequencies distributed according to a unimodal function with the plateau section in the middle representing the maximum and symmetric tails falling off predominantly as $\ensuremath{\mid}\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0}{\ensuremath{\mid}}^{m}$, $mg0$, in the vicinity of the flat region. It is found that the phase transition is of first order as long as there is a finite flat region and that in the vicinity of the critical coupling the following scaling law holds $r\ensuremath{-}{r}_{c}\ensuremath{\propto}(K\ensuremath{-}{K}_{c}{)}^{2∕(2m+3)}$, where $r$ is the order parameter and $K$ is the coupling strength of the interacting oscillators.

Hadron cancer therapy complex using nonscaling fixed field alternating gradient accelerator and gantry design

Abstract: Nonscaling fixed field alternating gradient (FFAG) rings for cancer hadron therapy offer reduced physical aperture and large dynamic aperture as compared to scaling FFAGs. The variation of tune with energy implies the crossing of resonances during acceleration. Our design avoids intrinsic resonances, although imperfection resonances must be crossed. We consider a system of three nonscaling FFAG rings for cancer therapy with 250 MeV protons and $400\text{ }\mathrm{MeV}/u$ carbon ions. Hadrons are accelerated in a common radio frequency quadrupole and linear accelerator, and injected into the FFAG rings at $v/c=0.1294$. ${\mathrm{H}}^{+}/{\mathrm{C}}^{6+}$ ions are accelerated in the two smaller/larger rings to 31 and $250\text{ }\text{ }\mathrm{MeV}/68.8$ and $400\text{ }\mathrm{MeV}/u$ kinetic energy, respectively. The lattices consist of doublet cells with a straight section for rf cavities. The gantry with triplet cells accepts the whole required momentum range at fixed field. This unique design uses either high-temperature superconductors or superconducting magnets reducing gantry magnet size and weight. Elements with a variable field at the beginning and at the end set the extracted beam at the correct position for a range of energies.

A variant of Davenport's constant

Abstract: Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of ]n[≔ {1, 2,…, n} such that elements of A are incongruent modulo p and non-zero modulo p. Let k ≥ D(G/|A| be any integer where D(G) denotes the well-known Davenport’s constant. In this article, we prove that for any sequence g1, g2,…, gk (not necessarily distinct) in G, one can always extract a subsequence \( g_{i_1 } ,g_{i_2 } , \ldots ,g_{i_\ell } \) with 1 ≤ l ≤ k such that $$ \sum\limits_{j = 1}^\ell {a_j g_{i_j } } = 0 in G, $$ where aj ∈ A for all j. We provide examples where this bound cannot be improved. Furthermore, for the cyclic groups, we prove some sharp results in this direction. In the last section, we explore the relation between this problem and a similar problem with prescribed length. The proof of Theorem 1 uses group-algebra techniques, while for the other theorems, we use elementary number theory techniques.

The tail behaviour of a random sum of subexponential random variables and vectors

Abstract: Let $\left\{ X,X_{i},i=1,2,...\right\} $ denote independent positive random variables having common distribution function (d.f.) F(x) and, independent of X, let ν denote an integer valued random variable. Using X 0=0, the random sum Z=∑ i=0 ν X i has d.f. $G(x)=\sum_{n=0}^{\infty }\Pr\{ u =n\}F^{n\ast }(x)$ where F n∗(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each e>0 we can find a constant K such that the inequality $$ 1-F^{n\ast }(x)\leq K(1+\varepsilon )^{n}(1-F(x))\, , \qquad n\geq 1,x\geq 0 \, , $$ holds. When F is subexponential and E(1 +e) ν <∞, it is a standard result in risk theory that G(x) satisfies $$ 1 - G{\left( x \right)} \sim E{\left( u \right)}{\left( {1 - F{\left( x \right)}} \right)},\,\,x \to \infty \,\,{\left( * \right)} $$ In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F. Stam (Adv. Appl. Prob. 5:308–327, 1973) considered the case where $ \overline{F}(x)=1-F(x)$ is regularly varying with index –α. He proved that if α>1 and $E{\left( { u ^{{\alpha + \varepsilon }} } \right)} < \infty $ , then relation (*) holds. For 0<α<1, it is sufficient that Eν<∞. In this paper we consider the case where $\overline{F}(x)$ is an O-regularly varying subexponential function. If the lower Matuszewska index $\beta (\overline{F})<-1$ , then the condition ${\text{E}}{\left( { u ^{{{\left| {\beta {\left( {\overline{F} } \right)}} \right|} + 1 + \varepsilon }} } \right)} < \infty$ is sufficient for (*). If $\beta (\overline{F} )>-1$ , then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio $\overline{F^{n\ast }}(x)/\overline{F} (x)$ . In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n≥2, the ratio $\overline{F^{n\ast }}(x)/\overline{F}(x)\uparrow n$ as x↑∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).

Classification of Fano manifolds containing a negative divisor isomorphic to projective space

Abstract: We classify n-dimensional complex Fano manifolds X (n ≥ 3) containing a divisor E isomorphic to \({\mathbb{P}^{n-1}}\) such that deg NE/X is strictly negative. Our main tool is the extremal contraction theory together with numerical arguments on intersection numbers of divisors on X. In the last section, we consider, more generally, Fano manifolds X containing a prime divisor with Picard number one, and show that the Picard number of such X is less than or equal to three.

Harmonic Almost Contact Structures

Abstract: An almost contact metric structure is parametrized by a section σ of an associated homogeneous fibre bundle, and conditions for σ to be a harmonic section, and a harmonic map, are studied These involve the characteristic vector field ξ, and the almost complex structure in the contact subbundle Several examples are given where the harmonic section equations for σ reduce to those for ξ, regarded as a section of the unit tangent bundle These include trans-Sasakian structures On the other hand, there are examples where ξ is harmonic but σ is not a harmonic section Many examples arise by considering hypersurfaces of almost Hermitian manifolds, with the induced almost contact structure, and comparing the harmonic section equations for both structures

The braid groups of the projective plane and the Fadell–Neuwirth short exact sequence

Abstract: We study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \to P_m(RP^2 \ {x_1,...,x_n} \to P_{n+m}(RP^2) \stackrel{p_{\ast}}{\to} P_n(RP^2) \to 1$, where $n\geq 2$ and $m\geq 1$, and $p_{\ast}$ is the homomorphism which corresponds geometrically to forgetting the last $m$ strings. This problem is equivalent to that of the existence of a section for the associated fibration $p: F_{n+m}(RP^2) \to F_n(RP^2)$ of configuration spaces. Van Buskirk proved in 1966 that $p$ and $p_{\ast}$ admit a section if $n=2$ and $m=1$. Our main result in this paper is to prove that there is no section if $n\geq 3$. As a corollary, it follows that $n=2$ and $m=1$ are the only values for which a section exists. As part of the proof, we derive a presentation of $P_n(RP^2)$: this appears to be the first time that such a presentation has been given in the literature.

Photovoltaic oscillations due to edge-magnetoplasmon modes in a very high-mobility two-dimensional electron gas

Abstract: Using very high-mobility $\mathrm{Ga}\mathrm{As}∕{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ two-dimensional electron Hall bar samples, we have experimentally studied the photoresistance and/or photovoltaic oscillations induced by microwave irradiation in the regime where both $1∕B$ and $B$-periodic oscillations can be observed. In the frequency range between 27 and $130\phantom{\rule{0.3em}{0ex}}\mathrm{GHz}$, we found that these two types of oscillations are decoupled from each other, consistent with the respective models that $1∕B$ oscillations occur in bulk while the $B$ oscillations occur along the edges of the Hall bars. In contrast to the original report of this phenomenon [I. V. Kukushkin et al., Phys. Rev. Lett. 92, 236803 (2004)], the periodicity of the $B$ oscillations in our samples is found to be independent of $L$, the length of the Hall bar section between voltage measuring leads.

Observables IV: The presheaf perspective

Abstract: In this fourth of our series of papers on observables we show that one can associate to each von Neumann algebra R a pair of isomorphic presheaves, the upper presheaf O^{+}_{R} and the lower presheaf O^{-}_{R}, on the category of abelian von Neumann subalgebras of R. Each $A \in R_{sa}$ induces a global section of O^{+}_{R} and of O^{-}_{R} respectively. We call them \emph {contextual observables}. But we show that, in general, not every global section of these presheaves arises in this way. Moreover, we discuss states of a von Neumann algebra in the presheaf context.

Uppers to zero and semistar operations in polynomial rings

Abstract: Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Pr\"ufer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott \cite[Section 2]{hmm} in the star operation setting. Moreover, we show that $D$ is a Pr\"ufer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Pr\"ufer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain $D$ (Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$.

The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

Abstract: We study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \to P_m(RP^2 {x_1,...,x_n} \to P_{n+m}(RP^2) \stackrel{p_{\ast}}{\to} P_n(RP^2) \to 1$, where $n\geq 2$ and $m\geq 1$, and $p_{\ast}$ is the homomorphism which corresponds geometrically to forgetting the last $m$ strings. This problem is equivalent to that of the existence of a section for the associated fibration $p: F_{n+m}(RP^2) \to F_n(RP^2)$ of configuration spaces. Van Buskirk proved in 1966 that $p$ and $p_{\ast}$ admit a section if $n=2$ and $m=1$. Our main result in this paper is to prove that there is no section if $n\geq 3$. As a corollary, it follows that $n=2$ and $m=1$ are the only values for which a section exists. As part of the proof, we derive a presentation of $P_n(RP^2)$: this appears to be the first time that such a presentation has been given in the literature.

X=M Theorem: Fermionic formulas and rigged configurations under review

Abstract: We give a review of the current status of the X=M conjecture. Here X stands for the one-dimensional configuration sum and M for the corresponding fermionic formula. There are three main versions of this conjecture: the unrestricted, the classically restricted and the level-restricted version. We discuss all three versions and illustrate the methods of proof with many examples for type A_{n-1}^{(1)}. In particular, the combinatorial approach via crystal bases and rigged configurations is discussed. Each section ends with a conglomeration of open problems.

Rips complexes and covers in the uniform category

Abstract: James \cite{Jam} introduced uniform covering maps as an analog of covering maps in the topological category. Subsequently Berestovskii and Plaut \cite{BP3} introduced a theory of covers for uniform spaces generalizing their results for topological groups \cite{BP1}-\cite{BP2}. Their main concepts are discrete actions and pro-discrete actions, respectively. In case of pro-discrete actions Berestovskii and Plaut provided an analog of the universal covering space and their theory works well for the so-called coverable spaces. As will be seen in Section \ref{SECTION-Comparison}, \cite{BP3} generalizes only regular covering maps in topology and pro-discrete actions may not be preserved by compositions. In this paper we redefine the uniform covering maps and we generalize pro-discrete actions using Rips complexes and the chain lifting property. We expand the concept of generalized paths of Krasinkiewicz and Minc \cite{KraMin}.

Holomorphic geometric models for representations of $C^*$-algebras

Abstract: Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of this technique to dilation theory of completely positive maps are explored and the critical role of complexified homogeneous spaces in connection with the Stinespring dilations is pointed out. The general results are further illustrated by a discussion of several specific topics, including similarity orbits of representations of amenable Banach algebras, similarity orbits of conditional expectations, geometric models of representations of Cuntz algebras, the relationship to endomorphisms of ${\mathcal B}({\mathcal H})$, and non-commutative stochastic analysis.

Generalizing circles over algebraic extensions

Abstract: This paper deals with a family of spatial rational curves that were introduced by Andradas, Recio and Sendra, under the name of hypercircles, as an algorithmic cornerstone tool in the context of improving the rational parametrization (simplifying the coefficients of the rational functions, when possible) of algebraic varieties. A real circle can be defined as the image of the real axis under a Moebius transformation in the complex field. Likewise, and roughly speaking, a hypercircle can be defined as the image of a line ("the ${\mathbb{K}}$-axis") in a $n$-degree finite algebraic extension $\mathbb{K}(\alpha)\thickapprox\mathbb{K}^n$ under the transformation $\frac{at+b}{ct+d}:\mathbb{K}(\alpha)\to\mathbb{K}(\alpha)$. The aim of this article is to extend, to the case of hypercircles, some of the specific properties of circles. We show that hypercircles are precisely, via $\mathbb{K}$-projective transformations, the rational normal curve of a suitable degree. We also obtain a complete description of the points at infinity of these curves (generalizing the cyclic structure at infinity of circles). We characterize hypercircles as those curves of degree equal to the dimension of the ambient affine space and with infinitely many ${\mathbb{K}}$-rational points, passing through these points at infinity. Moreover, we give explicit formulae for the parametrization and implicitation of hypercircles. Besides the intrinsic interest of this very special family of curves, the understanding of its properties has a direct application to the simplification of parametrizations problem, as shown in the last section.

Complete Ricci-flat K\"ahler metrics on the canonical bundles of toric Fano manifolds

Abstract: We prove the existence of a complete Ricci-flat Kahler metric on the total space of the canonical line bundle $K_M$ of a toric Fano manifold $M$ as an application of the existence result of toric Sasaki-Einstein metrics by H.Ono, G.Wang and the author. We also prove the existence of a complete scalar-flat Kahler metric on the toric Kahler cone manifold (minus the apex) constructed from a toric diagram with a constant height, and in particular there exists a complete scalar-flat Kahler metric on the total space of $K_M$ minus the zero section. These are part of results obtained by extending Calabi ansatz developed by A.Hwang and M.Singer to Sasakian $\eta$-Einstein manifolds.

On 2-Fold Covers of Graphs

Abstract: A regular covering projection $\p\colon \tX \to X$ of connected graphs is $G$-admissible if $G$ lifts along $\p$. Denote by $\tG$ the lifted group, and let $\CT(\p)$ be the group of covering transformations. The projection is called $G$-split whenever the extension $\CT(\p) \to \tG \to G$ splits. In this paper, split 2-covers are considered. Supposing that $G$ is transitive on $X$, a $G$-split cover is said to be $G$-split-transitive if all complements $\bG \cong G$ of $\CT(\p)$ within $\tG$ are transitive on $\tX$; it is said to be $G$-split-sectional whenever for each complement $\bG$ there exists a $\bG$-invariant section of $\p$; and it is called $G$-split-mixed otherwise. It is shown, when $G$ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. For cubic symmetric graphs split 2-cover are necessarily cannonical double covers when $G$ is 1- or 4-regular. In all other cases, that is, if $G$ is $s$-regular, $s=2,3$ or 5, a necessary and sufficient condition for the existence of a transitive complement $\bG$ is given, and an infinite family of split-transitive 2-covers based on the alternating groups of the form $A_{12k+10}$ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group $G$ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed.

Space charge compensation studies of hydrogen ion beams in a drift section

Abstract: Demands of high intensity beams are increasing for material science and engineering applications. To control the high intensity beam dynamics, the nonlinear space charge forces, produced by the beam and the residual plasma at low energy, have to be carefully studied. Interactions between the beam and the residual gas tend to neutralize the beam space charge. The temporal and spatial evolutions of these interactions complicate the beam dynamics, even into a drift section. A 1.5D ($xy$ for motion, $r$ for space charge) PIC code, CARTAGO, has been written in order to simulate the beam dynamics in a low energy beam transport line including the space charge compensation effects. This paper relates the structure and the numerical methods used in the code and shows studies of the space charge compensation regime for ${\mathrm{H}}^{+}$ and ${\mathrm{H}}^{\ensuremath{-}}$ beams in a drift section.

A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel

Abstract: In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on B\"ar and Pf\"affle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on the space of piecewise geodesic paths which is the pullback by a natural section of Mathai and Quillen's Thom form of a bundle over this space. In the case of closed paths, the bundle is the tangent space to the space of geodesic paths, and the integral of this form passes in the limit to the supertrace of the heat kernel.

On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group

Abstract: The following inequality \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that \cat X\le \lceil\frac{\dim X-1}{2}\rceil+cd(\pi_1(X)) for every complex $X$ with $cd(\pi_1(X))\le 2$.

Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence

Abstract: The aim of this paper is to clarify the relation between the following objects: $ (a) $ rank 1 projective modules (ideals) over the first Weyl algebra $ A_1(\C)$; $ (b) $ simple modules over deformed preprojective algebras $ \Pi_{\lambda}(Q) $ introduced by Crawley-Boevey and Holland; and $ (c) $ simple modules over the rational Cherednik algebras $ H_{0,c}(S_n) $ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on $\A$-modules over $ A_1 $ to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $ \C^2/\Gamma $, where $ \Gamma $ is a finite cyclic subgroup of $ \SL(2, \C) $.

The diffeomorphism group of a K3 surface and Nielsen realization

Abstract: The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of B(pi_0 Diff(M)) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.

Vector bundles on degenerations of elliptic curves and Yang-Baxter equations

Abstract: In this paper we introduce the notion of a geometric associative r-matrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. We also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstrass cubic curves.

Skew Hopf algebras, irreducible extensions and the pi-method

Abstract: To a depth two extension A | B, we associate the dual bialgebroids S := \End {}_BA_B and T := (A \o_B A)^B over the centralizer R=C_A(B). In the set-up where R is a subalgebra of B, which is quite common, two nondegenerate pairings of S and T will define an anti-automorphism \tau of the algebra S. Making use of a two-sided depth two structure, we prove that \tau is an antipode and S is a Hopf algebroid of a type we call skew Hopf algebra. A final section discusses how \tau and the nondegenerate pairings generalize to modules via the pi-method for depth two, and a certain derived mapping of cochain complexes is nullhomotopic.

Canonical Melnikov theory for diffeomorphisms

Abstract: We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincar\'{e} and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect the transverse intersection of stable and unstable manifolds and relate this intersection to the set of zeros of the Melnikov displacement.

An Algebra of Deformation Quantization for Star-Exponentials on Complex Symplectic Manifolds

Abstract: The cotangent bundle T*X to a complex manifold X is classically endowed with the sheaf of k-algebras \({\mathcal{W}_{T*X}}\) of deformation quantization, where k := \({\mathcal{W}_{\{pt\}}}\) is a subfield of \({\mathbb{C}[[\hbar, \hbar^{-1}]}\) . Here, we construct a new sheaf of k-algebras \({\mathcal{W}^t_{T*X}}\) which contains \({\mathcal{W}_{T*X}}\) as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If P is any section of order zero of \({\mathcal{W}_{T*X}}\) , we show that \({{\rm exp}(t\hbar^{-1} P)}\) is well defined in \({\mathcal{W}^t_{T*X}}\) .

The determination of convex bodies from derivatives of section functions

Abstract: It is known that non-symmetric convex bodies generally cannot be characterized by the volumes of hyperplane sections through one interior point Falconer and Gardner, however, independently proved that volumes of hyperplane sections through two different interior points determine the body uniquely We prove that if −1 < q < n − 1 is not an integer, then the derivatives of the order q at zero of parallel section functions at one interior point completely characterize convex bodies in \({\mathbb{R}^{n}}\) If 0 ≤ q < n − 1 is an integer then one needs the derivatives of order q at two different interior points (except for the case where q = n − 2, q odd), generalizing the results of Falconer and Gardner