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## ‣ A confirmatory factor analysis of Attitudes Toward Mathematics Inventory (ATMI)

Fonte: Association of Mathematics Educators
Publicador: Association of Mathematics Educators

Tipo: Artigo de Revista Científica

Publicado em //2013
Português

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#Attitudes toward mathematics#Attitudes Toward Mathematics Inventory (ATMI)#confirmatory factor analysis#Australia

Students’ attitudes toward mathematics have been known to influence students’ participation, engagement, and achievement in mathematics. A variety of instruments have been developed to measure students’ attitudes toward mathematics for example Mathematics Attitude Scale (Aiken, 1974), Fennema-Sherman Mathematics Attitudes Scales (Fennema & Sherman, 1976), and Attitudes Toward Mathematics Inventory (ATMI) (Tapia & Marsh, 1996). The purpose of this paper is to report the validation of the ATMI instrument. It was administered to 699 Year 7 and 8 students in 14 schools in South Australia. The students responded on a five-point Likert scale. Confirmatory factor analysis (CFA) supported the original four-factor correlated structure based on several fit indices. The validation provided evidence that ATMI can be a viable scale to measure students’ attitudes toward mathematics in a South Australian context; Aysha Abdul Majeed, I Gusti Ngurah Darmawan, Peggy Lynch

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## ‣ An investigation into the nature of mathematics textbooks at junior cycle and their role in mathematics education

Fonte: University of Limerick
Publicador: University of Limerick

Tipo: info:eu-repo/semantics/doctoralThesis; all_ul_research; ul_published_reviewed; ul_theses_dissertations

Português

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peer-reviewed; This research study is aimed at improving the quality of the mathematics textbooks
available for junior cycle students. It is widely agreed that there is room for
improvement with regard to the quality of mathematics at both junior and senior
cycle level in Ireland. One such area which can be improved is the e ectiveness of
the resources available in both junior and senior cycle mathematics classrooms.
While the TIMSS report (Valverde et al., 2002) has explored textbooks on an
international scale, minimal research (minor role in TIMSS Report) has been
carried out on Irish mathematics textbooks. Considering the level of responsibility
shouldered by mathematics textbooks, there is an obvious gap in mathematics
education research.
The aim of this study is to investigate the quality of the mathematics textbooks
currently in use at junior secondary school level in Ireland. This is achieved by
investigating, extending and applying suitable methodological tools for textbook
analysis. Ultimately the aim of this research is to improve the quality of teaching
and learning of mathematics at junior cycle level which should feed directly into
improving the quality of mathematics at senior cycle. This will be achieved by
rst measuring the quality of the current junior cycle mathematics textbooks and
then highlighting the role of improved textbooks in students' conceptual under-
standing. At present in Ireland...

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## ‣ Applications of Differential Chains to Complex Analysis and Dynamics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/12/2010
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#Mathematics - Functional Analysis#Mathematics - Complex Variables#Mathematics - Differential Geometry#Mathematics - Dynamical Systems#28A75 (Primary), 30E20, 37C40 (Secondary)

This thesis is divided into three parts. In the first part, we give an
introduction to J. Harrison's theory of differential chains. In the second
part, we apply these tools to generalize the Cauchy theorems in complex
analysis. Instead of requiring a piecewise smooth path over which to integrate,
we can now do so over non- rectifiable curves and divergence-free vector fields
supported away from the singularities of the holomorphic function in question.
In the third part, we focus on applications to dynamics, in particular, flows
on compact Riemannian manifolds. We prove that the asymptotic cycles are
differential chains, and that for an ergodic measure, they are equal as
differential chains to the differential chain associated to the vector field
and the ergodic measure. The first part is expository, but the second and third
parts contain new results.

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## ‣ Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/11/2012
Português

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#Mathematics - Functional Analysis#Mathematical Physics#Mathematics - History and Overview#Mathematics - Logic#Mathematics - Operator Algebras#Primary 26E35, Secondary 03A05

We examine some of Connes' criticisms of Robinson's infinitesimals starting
in 1995. Connes sought to exploit the Solovay model S as ammunition against
non-standard analysis, but the model tends to boomerang, undercutting Connes'
own earlier work in functional analysis. Connes described the hyperreals as
both a "virtual theory" and a "chimera", yet acknowledged that his argument
relies on the transfer principle. We analyze Connes' "dart-throwing" thought
experiment, but reach an opposite conclusion. In S, all definable sets of reals
are Lebesgue measurable, suggesting that Connes views a theory as being
"virtual" if it is not definable in a suitable model of ZFC. If so, Connes'
claim that a theory of the hyperreals is "virtual" is refuted by the existence
of a definable model of the hyperreal field due to Kanovei and Shelah. Free
ultrafilters aren't definable, yet Connes exploited such ultrafilters both in
his own earlier work on the classification of factors in the 1970s and 80s, and
in his Noncommutative Geometry, raising the question whether the latter may not
be vulnerable to Connes' criticism of virtuality. We analyze the philosophical
underpinnings of Connes' argument based on Goedel's incompleteness theorem, and
detect an apparent circularity in Connes' logic. We document the reliance on
non-constructive foundational material...

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## ‣ Analysis in J_2

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Logic#Mathematics - Functional Analysis#Mathematics - General Mathematics#Mathematics - History and Overview

This is an expository paper in which I explain how core mathematics,
particularly abstract analysis, can be developed within a concrete countable
set J_2 (the second set in Jensen's constructible hierarchy). The implication,
well-known to proof theorists but probably not to most mainstream
mathematicians, is that ordinary mathematical practice does not require an
enigmatic metaphysical universe of sets. I go further and argue that J_2 is a
superior setting for normal mathematics because it is free of irrelevant
set-theoretic pathologies and permits stronger formulations of existence
results.; Comment: 31 pages

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## ‣ Meaning in Classical Mathematics: Is it at Odds with Intuitionism?

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/10/2011
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#Mathematics - Logic#Mathematics - Classical Analysis and ODEs#Mathematics - History and Overview#01A85, 26E35, 03A05, 97A20, 97C30

We examine the classical/intuitionist divide, and how it reflects on modern
theories of infinitesimals. When leading intuitionist Heyting announced that
"the creation of non-standard analysis is a standard model of important
mathematical research", he was fully aware that he was breaking ranks with
Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a
comparative textual analysis of three of Bishop's texts, we analyze the
ideological and/or pedagogical nature of his objections to infinitesimals a la
Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop,
published as part of his Crisis lecture, in reality was never uttered in front
of an audience. We compare the realist and the anti-realist intuitionist
narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and
Tennant. Variational principles are important physical applications, currently
lacking a constructive framework. We examine the case of the Hawking-Penrose
singularity theorem, already analyzed by Hellman in the context of the
Quine-Putnam indispensability thesis.; Comment: 88 pages, to appear in Intellectica 56 (2011), no. 2

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## ‣ A cognitive analysis of Cauchy's conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/01/2014
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In this paper we use theoretical frameworks from mathematics education and
cognitive psychology to analyse Cauchy's ideas of function, continuity, limit
and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the
development of mathematical thinking from human perception and action into more
sophisticated forms of reasoning and proof, offering different insights from
those afforded by historical or mathematical analyses. It highlights the
conceptual power of Cauchy's vision and the fundamental change involved in
passing from the dynamic variability of the calculus to the modern
set-theoretic formulation of mathematical analysis. This offers a re-evaluation
of the relationship between the natural geometry and algebra of elementary
calculus that continues to be used in applied mathematics, and the formal set
theory of mathematical analysis that develops in pure mathematics and evolves
into the logical development of non-standard analysis using infinitesimal
concepts. It suggests that educational theories developed to evaluate student
learning are themselves based on the conceptions of the experts who formulate
them. It encourages us to reflect on the principles that we use to analyse the
developing mathematical thinking of students...

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## ‣ Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Representation Theory#Mathematical Physics#Mathematics - Classical Analysis and ODEs#Mathematics - Combinatorics#Mathematics - Probability

The infinite-dimensional unitary group U(infinity) is the inductive limit of
growing compact unitary groups U(N). In this paper we solve a problem of
harmonic analysis on U(infinity) stated in the previous paper math/0109193. The
problem consists in computing spectral decomposition for a remarkable
4-parameter family of characters of U(infinity). These characters generate
representations which should be viewed as analogs of nonexisting regular
representation of U(infinity).
The spectral decomposition of a character of U(infinity) is described by the
spectral measure which lives on an infinite-dimensional space Omega of
indecomposable characters. The key idea which allows us to solve the problem is
to embed Omega into the space of point configurations on the real line without
2 points. This turns the spectral measure into a stochastic point process on
the real line. The main result of the paper is a complete description of the
processes corresponding to our concrete family of characters. We prove that
each of the processes is a determinantal point process. That is, its
correlation functions have determinantal form with a certain kernel. Our
kernels have a special `integrable' form and are expressed through the Gauss
hypergeometric function.
In simpler situations of harmonic analysis on infinite symmetric group and
harmonic analysis of unitarily invariant measures on infinite hermitian
matrices similar results were obtained in our papers math/9810015...

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## ‣ A quantization of the harmonic analysis on the infinite-dimensional unitary group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Representation Theory#Mathematics - Classical Analysis and ODEs#Mathematics - Combinatorics#Mathematics - Probability

The present work stemmed from the study of the problem of harmonic analysis
on the infinite-dimensional unitary group U(\infty). That problem consisted in
the decomposition of a certain 4-parameter family of unitary representations,
which replace the nonexisting two-sided regular representation (Olshanski, J.
Funct. Anal., 2003, arXiv:0109193). The required decomposition is governed by
certain probability measures on an infinite-dimensional space \Omega, which is
a dual object to U(\infty). A way to describe those measures is to convert them
into determinantal point processes on the real line, it turned out that their
correlation kernels are computable in explicit form --- they admit a closed
expression in terms of the Gauss hypergeometric function 2-F-1 (Borodin and
Olshanski, Ann. Math., 2005, arXiv:0109194).
In the present work we describe a (nonevident) q-discretization of the whole
construction. This leads us to a new family of determinantal point processes.
We reveal its connection with an exotic finite system of q-discrete orthogonal
polynomials --- the so-called pseudo big q-Jacobi polynomials. The new point
processes live on a double q-lattice and we show that their correlation kernels
are expressed through the basic hypergeometric function 2-\phi-1.
A crucial novel ingredient of our approach is an extended version G of the
Gelfand-Tsetlin graph (the conventional graph describes the Gelfand-Tsetlin
branching rule for irreducible representations of unitary groups). We find the
q-boundary of G...

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## ‣ Geometry and Analysis of Dirichlet forms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/08/2012
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#Mathematics - Classical Analysis and ODEs#Mathematics - Functional Analysis#Mathematics - Metric Geometry#Mathematics - Probability

Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$
and $d$ the associated intrinsic distance. Assume that the topology induced by
$d$ coincides with the original topology on $ X$, and that $X$ is compact,
satisfies a doubling property and supports a weak $(1, 2)$-Poincar\'e
inequality. We first discuss the (non-)coincidence of the intrinsic length
structure and the gradient structure. Under the further assumption that the
Ricci curvature of $X$ is bounded from below in the sense of
Lott-Sturm-Villani, the following are shown to be equivalent:
(i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr
U_\infty$,
(ii) $\mathscr E$ satisfies the Newtonian property,
(iii) the intrinsic length structure coincides with the gradient structure.
Moreover, for the standard (resistance) Dirichlet form on the Sierpinski
gasket equipped with the Kusuoka measure, we identify the intrinsic length
structure with the measurable Riemannian and the gradient structures. We also
apply the above results to the (coarse) Ricci curvatures and asymptotics of the
gradient of the heat kernel.; Comment: Advance in Mathematics, to appear,51pp

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## ‣ Applications of Convex Analysis within Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Functional Analysis#Mathematics - Optimization and Control#Primary 47N10, 90C25, Secondary 47H05, 47A06, 47B65

In this paper, we study convex analysis and its theoretical applications. We
first apply important tools of convex analysis to Optimization and to Analysis.
We then show various deep applications of convex analysis and especially
infimal convolution in Monotone Operator Theory. Among other things, we
recapture the Minty surjectivity theorem in Hilbert space, and present a new
proof of the sum theorem in reflexive spaces. More technically, we also discuss
autoconjugate representers for maximally monotone operators. Finally, we
consider various other applications in mathematical analysis.; Comment: 38 pages, minor revision incorporating referees' comments

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## ‣ An invitation to harmonic analysis associated with semigroups of operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/04/2013
Português

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#Mathematics - Classical Analysis and ODEs#Mathematics - Functional Analysis#Mathematics - Operator Algebras

This article is an introduction to our recent work in harmonic analysis
associated with semigroups of operators, in the effort of finding a
noncommutative Calder\'on-Zygmund theory for von Neumann algebras. The
classical CZ theory has been traditionally developed on metric measure spaces
satisfying additional regularity properties. In the lack of such metrics -or
with very little information on the metric- Markov semigroups of operators
appear to be the right substitutes of classical metric/geometric tools in
harmonic analysis. Our approach is particularly useful in the noncommutative
setting but it is also valid in classical/commutative frameworks.; Comment: To appear in Proc. 9th Int. Conf. Harmonic Analysis and PDE's. El
Escorial, 2012

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## ‣ A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - History and Overview#Mathematics - Classical Analysis and ODEs#Mathematics - Logic#01A85, 26E35, 03A05, 97A20, 97C30

We analyze the developments in mathematical rigor from the viewpoint of a
Burgessian critique of nominalistic reconstructions. We apply such a critique
to the reconstruction of infinitesimal analysis accomplished through the
efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's
foundational work associated with the work of Boyer and Grabiner; and to
Bishop's constructivist reconstruction of classical analysis. We examine the
effects of a nominalist disposition on historiography, teaching, and research.; Comment: 57 pages; 3 figures. Corrected misprints

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## ‣ Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/07/2015
Português

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#Mathematics - Functional Analysis#Mathematics - Classical Analysis and ODEs#Mathematics - Operator Algebras#Mathematics - Representation Theory#Primary 47L60, 46N30, 46N50, 42C15, 65R10, Secondary 46N20, 22E70,
31A15, 58J65, 81S25

We study two classes of extension problems, and their interconnections: (i)
Extension of positive definite (p.d.) continuous functions defined on subsets
in locally compact groups $G$; (ii) In case of Lie groups, representations of
the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian
operators acting in a reproducing kernel Hilbert space (RKHS)
$\mathscr{H}_{F}$.
Why extensions? In science, experimentalists frequently gather spectral data
in cases when the observed data is limited, for example limited by the
precision of instruments; or on account of a variety of other limiting external
factors. Given this fact of life, it is both an art and a science to still
produce solid conclusions from restricted or limited data. In a general sense,
our monograph deals with the mathematics of extending some such given partial
data-sets obtained from experiments. More specifically, we are concerned with
the problems of extending available partial information, obtained, for example,
from sampling. In our case, the limited information is a restriction, and the
extension in turn is the full positive definite function (in a dual variable);
so an extension if available will be an everywhere defined generating function
for the exact probability distribution which reflects the data; if it were
fully available. Such extensions of local information (in the form of positive
definite functions) will in turn furnish us with spectral information. In this
form...

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## ‣ Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Optimization and Control#Computer Science - Systems and Control#Mathematics - Classical Analysis and ODEs#Mathematics - Dynamical Systems

Stability analysis and control of linear impulsive systems is addressed in a
hybrid framework, through the use of continuous-time time-varying discontinuous
Lyapunov functions. Necessary and sufficient conditions for stability of
impulsive systems with periodic impulses are first provided in order to set up
the main ideas. Extensions to stability of aperiodic systems under minimum,
maximum and ranged dwell-times are then derived. By exploiting further the
particular structure of the stability conditions, the results are
non-conservatively extended to quadratic stability analysis of linear uncertain
impulsive systems. These stability criteria are, in turn, losslessly extended
to stabilization using a particular, yet broad enough, class of state-feedback
controllers, providing then a convex solution to the open problem of robust
dwell-time stabilization of impulsive systems using hybrid stability criteria.
Relying finally on the representability of sampled-data systems as impulsive
systems, the problems of robust stability analysis and robust stabilization of
periodic and aperiodic uncertain sampled-data systems are straightforwardly
solved using the same ideas. Several examples are discussed in order to show
the effectiveness and reduced complexity of the proposed approach.; Comment: 12 pages...

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## ‣ Fundaments of Quaternionic Clifford Analysis III: Fischer Decomposition in Symplectic Harmonic Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/04/2014
Português

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#Mathematics - Classical Analysis and ODEs#Mathematics - Analysis of PDEs#Mathematics - Complex Variables#35J05 31B05 30G35 22E60

In the framework of quaternionic Clifford analysis in Euclidean space
$\mathbb{R}^{4p}$, which constitutes a refinement of Euclidean and Hermitian
Clifford analysis, the Fischer decomposition of the space of complex valued
polynomials is obtained in terms of spaces of so--called (adjoint) symplectic
spherical harmonics, which are irreducible modules for the symplectic group
Sp$(p)$. Its Howe dual partner is determined to be $\mathfrak{sl}(2,\mathbb{C})
\oplus \mathfrak{sl}(2,\mathbb{C}) = \mathfrak{so}(4,\mathbb{C})$.

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## ‣ Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Representation Theory#Mathematical Physics#Mathematics - Classical Analysis and ODEs#Mathematics - Complex Variables#Mathematics - Functional Analysis#43A85, 22E46, 53C35, 32A25, 43A90, 33C05, 33E20, 15A15

Consider the pseidounitary group $G=U(p,q)$ and its compact subgroup
$K=U(p)$. We construct an explicit unitary intertwining operator from the
tensor product of a holomorphic representation and a antiholomorphic
representation of $G$ to the space $L^2(G/K)$. This implies the existense of a
canonical action of the group $G\times G$ in $L^2(G/K)$. We also give a survey
of analysis of Berezin kernels and their relations with special functions.; Comment: 46 pages; misprints are corrected; addendum on pseudoriemannian
symmetric spaces is added

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## ‣ Harmonic analysis on quantum tori

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/06/2012
Português

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#Mathematics - Operator Algebras#Mathematics - Classical Analysis and ODEs#Mathematics - Functional Analysis#Primary: 46L50, 46L07, Secondary: 58L34, 43A55

This paper is devoted to the study of harmonic analysis on quantum tori. We
consider several summation methods on these tori, including the square Fej\'er
means, square and circular Poisson means, and Bochner-Riesz means. We first
establish the maximal inequalities for these means, then obtain the
corresponding pointwise convergence theorems. In particular, we prove the
noncommutative analogue of the classical Stein theorem on Bochner-Riesz means.
The second part of the paper deals with Fourier multipliers on quantum tori. We
prove that the completely bounded $L_p$ Fourier multipliers on a quantum torus
are exactly those on the classical torus of the same dimension. Finally, we
present the Littlewood-Paley theory associated with the circular Poisson
semigroup on quantum tori. We show that the Hardy spaces in this setting
possess the usual properties of Hardy spaces, as one can expect. These include
the quantum torus analogue of Fefferman's $\mathrm{H}_1$-BMO duality theorem
and interpolation theorems. Our analysis is based on the recent developments of
noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein
theory.

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## ‣ Harmonic and Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/01/2015
Português

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#Mathematics - Functional Analysis#Mathematics - Analysis of PDEs#Mathematics - Classical Analysis and ODEs

We use methods of harmonic analysis and group representation theory to study
the spectral properties of the abstract parabolic operator $\mathscr L =
-d/dt+A$ in homogeneous function spaces. We provide sufficient conditions for
invertibility of such operators in terms of the spectral properties of the
operator $A$ and the semigroup generated by $A$. We introduce a homogeneous
space of functions with absolutely summable spectrum and prove a generalization
of the Gearhart-Pr\"uss Theorem for such spaces. We use the results to prove
existence and uniqueness of solutions of a certain class of non-linear
equations.

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## ‣ Algebraic analysis on scalar generalized Verma modules of Heisenberg parabolic type I.: $A_n$-series

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/02/2015
Português

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#Mathematics - Representation Theory#Mathematical Physics#Mathematics - Analysis of PDEs#Mathematics - Differential Geometry#Mathematics - Functional Analysis#53A30, 22E47, 33C45, 58J70

In the present article, we combine some techniques in the harmonic analysis
together with the geometric approach given by modules over sheaves of rings of
twisted differential operators ($\mathcal{D}$-modules), and reformulate the
composition series and branching problems for objects in the
Bernstein-Gelfand-Gelfand parabolic category $\mathcal{O}^\mathfrak{p}$
geometrically realized on certain orbits in the generalized flag manifolds. The
general framework is then applied to the scalar generalized Verma modules
supported on the closed Schubert cell of the generalized flag manifold $G/P$
for $G={\rm SL}(n+2,\mathbb{C})$ and $P$ the Heisenberg parabolic subgroup, and
the algebraic analysis gives a complete classification of
$\mathfrak{g}'_r$-singular vectors for all
$\mathfrak{g}'_r=\mathfrak{sl}(n-r+2,\mathbb{C})\,\subset\,
\mathfrak{g}=\mathfrak{sl}(n+2,\mathbb{C})$, $n-r > 2$. A consequence of our
results is that we classify ${\rm SL}(n-r+2,\mathbb{C})$-covariant differential
operators acting on homogeneous line bundles over the complexification of the
odd dimensional CR-sphere $S^{2n+1}$ and valued in homogeneous vector bundles
over the complexification of the CR-subspheres $S^{2(n-r)+1}$.

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