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Dec 2006

Volume 47, Issue 12, Articles (12xxxx)

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Bayesian analog of Gleason’s theorem

Thomas Marlow

J. Math. Phys. 47, 122101 (2006); http://dx.doi.org/10.1063/1.2390658 (12 pages)

Online Publication Date: 5 December 2006

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We introduce a novel notion of probability within quantum history theories and give a Gleasonesque proof for these assignments. This involves introducing a tentative novel axiom of probability. We also discuss how we are to interpret these generalized probabilities as partially ordered notions of preference, and we introduce a tentative generalized notion of Shannon entropy. A Bayesian approach to probability theory is adopted throughout; thus the axioms we use will be minimal criteria of rationality rather than ad hoc mathematical axioms.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.50.Cw Probability theory
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Pauli-Hamiltonian in the presence of minimal lengths

Khireddine Nouicer

J. Math. Phys. 47, 122102 (2006); http://dx.doi.org/10.1063/1.2393151 (11 pages) | Cited 7 times

Online Publication Date: 5 December 2006

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We construct the Pauli-Hamiltonian on a space where the position and momentum operators obey generalized commutation relations leading to the appearance of a minimal length. Using the momentum space representation we determine exactly the energy eigenvalues and eigenfunctions for a charged particle of spin half moving under the action of a constant magnetic field. The thermal properties of the system in the regime of high temperatures are also investigated, showing a magnetic behavior in terms of the minimal length.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Fd Algebraic methods
02.10.Ud Linear algebra

On bipartite pure-state entanglement structure in terms of disentanglement

Fedor Herbut

J. Math. Phys. 47, 122103 (2006); http://dx.doi.org/10.1063/1.2375035 (19 pages) | Cited 2 times

Online Publication Date: 6 December 2006

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Schrödinger’s disentanglement [E. Schrödinger, Proc. Cambridge Philos. Soc. 31, 555 (1935)] , i.e., remote state decomposition, as a physical way to study entanglement, is carried one step further with respect to previous work in investigating the qualitative side of entanglement in any bipartite state vector. Remote measurement (or, equivalently, remote orthogonal state decomposition) from previous work is generalized to remote linearly independent complete state decomposition both in the nonselective and the selective versions. The results are displayed in terms of commutative square diagrams, which show the power and beauty of the physical meaning of the (antiunitary) correlation operator inherent in the given bipartite state vector. This operator, together with the subsystem states (reduced density operators), constitutes the so-called correlated subsystem picture. It is the central part of the antilinear representation of a bipartite state vector, and it is a kind of core of its entanglement structure. The generalization of previously elaborated disentanglement expounded in this article is a synthesis of the antilinear representation of bipartite state vectors, which is reviewed, and the relevant results of [Cassinelli et al., J. Math. Anal. Appl. 210, 472 (1997)] in mathematical analysis, which are summed up. Linearly independent bases (finite or infinite) are shown to be almost as useful in some quantum mechanical studies as orthonormal ones. Finally, it is shown that linearly independent remote pure-state preparation carries the highest probability of occurrence. This singles out linearly independent remote influence from all possible ones.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods
02.10.Ud Linear algebra

Shape invariance through Crum transformation

José Orlando Organista, Marek Nowakowski, and H. C. Rosu

J. Math. Phys. 47, 122104 (2006); http://dx.doi.org/10.1063/1.2397556 (19 pages) | Cited 3 times

Online Publication Date: 7 December 2006

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We show in a rigorous way that Crum’s result regarding the equal eigenvalue spectrum of Sturm-Liouville problems can be obtained iteratively by successive Darboux transformations. Furthermore, it can be shown that all neighboring Darboux-transformed potentials of higher order, uk and uk+1, satisfy the condition of shape invariance provided the original potential u does so. Based on this result, we prove that under the condition of shape invariance, the nth iteration of the original Sturm-Liouville problem defined solely through the shape invariance is equal to the nth Crum transformation.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Fd Algebraic methods

Parametric-time coherent states for the generalized MIC-Kepler system

Nuri Ünal

J. Math. Phys. 47, 122105 (2006); http://dx.doi.org/10.1063/1.2399362 (15 pages) | Cited 2 times

Online Publication Date: 14 December 2006

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In this study, we construct the parametric-time coherent states for the negative energy states of the generalized MIC-Kepler system, in which a charged particle is in a monopole vector potential, a Coulomb potential, and a Bohm-Aharonov potantial. We transform the system into four isotropic harmonic oscillators and construct the parametric-time coherent states for these oscillators. Finally, we compactify these states into the physical time coherent states for the generalized MIC-Kepler system.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
03.65.Fd Algebraic methods
03.65.Ta Foundations of quantum mechanics; measurement theory

Time-of-arrival probabilities and quantum measurements

Charis Anastopoulos and Ntina Savvidou

J. Math. Phys. 47, 122106 (2006); http://dx.doi.org/10.1063/1.2399085 (29 pages) | Cited 14 times

Online Publication Date: 18 December 2006

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In this paper we study the construction of probability densities for time of arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time of arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of the Schrödinger’s equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit—there are, however, problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the time-of-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Ge Solutions of wave equations: bound states
03.65.Yz Decoherence; open systems; quantum statistical methods
02.50.Cw Probability theory
02.50.Ey Stochastic processes

Hudson’s theorem for finite-dimensional quantum systems

D. Gross

J. Math. Phys. 47, 122107 (2006); http://dx.doi.org/10.1063/1.2393152 (25 pages) | Cited 15 times

Online Publication Date: 19 December 2006

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We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.
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03.65.Fd Algebraic methods
03.65.Ta Foundations of quantum mechanics; measurement theory
02.30.Nw Fourier analysis
02.30.Uu Integral transforms
02.50.Ng Distribution theory and Monte Carlo studies

Fractional supersymmetry and hierarchy of shape invariant potentials

M. Daoud and M. R. Kibler

J. Math. Phys. 47, 122108 (2006); http://dx.doi.org/10.1063/1.2401711 (11 pages) | Cited 12 times

Online Publication Date: 28 December 2006

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Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.
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11.30.Pb Supersymmetry
03.65.Fd Algebraic methods
02.10.De Algebraic structures and number theory

Propagator for finite range potentials

Ilaria Cacciari and Paolo Moretti

J. Math. Phys. 47, 122109 (2006); http://dx.doi.org/10.1063/1.2401728 (8 pages) | Cited 5 times

Online Publication Date: 28 December 2006

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The Schrödinger equation in integral form is applied to the one-dimensional scattering problem in the case of a general finite range, nonsingular potential. A simple expression for the Laplace transform of the transmission propagator is obtained in terms of the associated Fredholm determinant, by means of matrix methods; the particular form of the kernel and the peculiar aspects of the transmission problem play an important role. The application to an array of delta potentials is shown.
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03.65.Ge Solutions of wave equations: bound states
03.65.Nk Scattering theory
03.65.Fd Algebraic methods
02.30.Rz Integral equations
02.30.Uu Integral transforms
02.10.Yn Matrix theory
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Relativistic quaternionic wave equation

Charles Schwartz

J. Math. Phys. 47, 122301 (2006); http://dx.doi.org/10.1063/1.2397555 (13 pages) | Cited 4 times

Online Publication Date: 1 December 2006

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We study a one-component quaternionic wave equation which is relativistically covariant. Bilinear forms include a conserved four-vector current and an antisymmetric second rank tensor. Waves propagate within the light cone and there is a conserved quantity which looks like helicity. The principle of superposition is retained in a slightly altered manner. External potentials can be introduced in a way that allows for gauge invariance. There are some results for scattering theory and for two-particle wave functions as well as the beginnings of second quantization. However, we are unable to find a suitable Lagrangian or an energy-momentum tensor.
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03.65.Ge Solutions of wave equations: bound states
03.65.Pm Relativistic wave equations
11.15.-q Gauge field theories

Generating loop graphs via Hopf algebra in quantum field theory

Ângela Mestre and Robert Oeckl

J. Math. Phys. 47, 122302 (2006); http://dx.doi.org/10.1063/1.2390657 (14 pages) | Cited 2 times

Online Publication Date: 5 December 2006

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We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected n-point functions. The recursion proceeds by loop order and vertex number.
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11.10.-z Field theory
02.10.Ox Combinatorics; graph theory

Endomorphisms on half-sided modular inclusions

Rolf Dyre Svegstrup

J. Math. Phys. 47, 122303 (2006); http://dx.doi.org/10.1063/1.2393147 (22 pages)

Online Publication Date: 7 December 2006

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In algebraic quantum field theory we consider nets of von Neumann algebras indexed over regions of the space time. Wiesbrock [“Conformal quantum field theory and half-sided modular inclusions of von Neumann algebras,” Commun. Math. Phys. 158, 537–543 (1993)] has shown that strongly additive nets of von Neumann algebras on the circle are in correspondence with standard half-sided modular inclusions. We show that a finite index endomorphism on a half-sided modular inclusion extends to a finite index endomorphism on the corresponding net of von Neumann algebras on the circle. Moreover, we present another approach to encoding endomorphisms on nets of von Neumann algebras on the circle into half-sided modular inclusions. There is a natural way to associate a weight to a Möbius covariant endomorphism. The properties of this weight have been studied by Bertozzini et al. [“Covariant sectors with infinite dimension and positivity of the energy,” Commun. Math. Phys. 193, 471–492 (1998)] . In this paper we show the converse, namely, how to associate a Möbius covariant endomorphism to a given weight under certain assumptions, thus obtaining a correspondence between a class of weights on a half-sided modular inclusion and a subclass of the Möbius covariant endomorphisms on the associated net of von Neumann algebras. This allows us to treat Möbius covariant endomorphisms in terms of weights on half-sided modular inclusions. As our aim is to provide a framework for treating endomorphisms on nets of von Neumann algebras in terms of the apparently simpler objects of weights on half-sided modular inclusions, we lastly give some basic results for manipulations with such weights.
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11.10.-z Field theory
02.10.-v Logic, set theory, and algebra

Minimal redefinition of the OSV ensemble

Shahrokh Parvizi and Alireza Tavanfar

J. Math. Phys. 47, 122304 (2006); http://dx.doi.org/10.1063/1.2393149 (20 pages)

Online Publication Date: 13 December 2006

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In the interesting conjecture, ZBH = |Ztop|2, proposed by Ooguri, Strominger, and Vafa (OSV), the black hole ensemble is a mixed ensemble. So if working in the complex polarization, the black hole degeneracy of states as obtained from the ensemble inverse-Laplace integration, generically receives prefactors that do not respect the electric-magnetic duality. One way to handle this, as claimed recently, is working instead of the complex polarization in the real polarization. The other idea would be imposing nontrivial measures for the ensemble sum in the complex polarization. We address this problem in the complex polarization, which is canonical, and upon a redefinition of the OSV ensemble with variables as numerous as the electric potentials, show that for restoring the symmetry no non-Euclidean measure is needed. In detail, applying the electric-magnetic duality as a constraint governing the proper definition of the ensemble variables, we rewrite the OSV free energy as a function of new variables that are combinations of the electric potentials and the black hole charges. Subsequently the Legendre transformation, which bridges between the entropy and the black hole free energy in terms of these variables, points to a generalized ensemble that is well behaved in the complex polarization. In this context, we will consider all the cases of relevance: small and large black holes, with or without D6-brane charge. For the case of vanishing D6-brane, the new ensemble is purely canonical and the electric-magnetic duality is restored exactly, leading to proper results for the black hole degeneracy of states to all orders in an asymptotic expansion. For more general cases as well, the construction does the job as far as the violation of the duality by the corresponding OSV result is restricted to a prefactor. In the case of black holes with nonvanishing D6-brane charge, in a concrete example, we shall show that there are cases where the duality violation goes beyond this restriction and imposing nontrivial measures is incapable of restoring the duality.
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04.70.Dy Quantum aspects of black holes, evaporation, thermodynamics
05.70.Ce Thermodynamic functions and equations of state
11.27.+d Extended classical solutions; cosmic strings, domain walls, texture
97.60.Lf Black holes
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Fractional Israel layers

J. P. Krisch

J. Math. Phys. 47, 122501 (2006); http://dx.doi.org/10.1063/1.2390660 (12 pages) | Cited 1 time

Online Publication Date: 22 December 2006

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A fractional Lie derivative, valid in the thin shell limit, is developed. The nonlocal nature of the fractional derivative allows the inclusion of shell thickness in the stress energy description of zero thickness Israel layers. The method is applied to several examples.
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04.20.Cv Fundamental problems and general formalism
02.50.-r Probability theory, stochastic processes, and statistics
05.40.Fb Random walks and Levy flights
05.60.-k Transport processes
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Infinitely many periodic orbits for the rhomboidal five-body problem

Montserrat Corbera and Jaume Llibre

J. Math. Phys. 47, 122701 (2006); http://dx.doi.org/10.1063/1.2378617 (13 pages)

Online Publication Date: 8 December 2006

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We prove the existence of infinitely many symmetric periodic orbits for a regularized rhomboidal five-body problem with four small masses placed at the vertices of a rhombus centered in the fifth mass. The main tool for proving the existence of such periodic orbits is the analytic continuation method of Poincaré together with the symmetries of the problem.
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45.50.Jf Few- and many-body systems
02.30.Hq Ordinary differential equations

Perturbed block circulant matrices and their application to the wavelet method of chaotic control

Jonq Juang, Chin-Lung Li, and Jing-Wei Chang

J. Math. Phys. 47, 122702 (2006); http://dx.doi.org/10.1063/1.2400828 (11 pages) | Cited 3 times

Online Publication Date: 26 December 2006

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Controlling chaos via wavelet transform was proposed by Wei et al. [Phys. Rev. Lett. 89, 284103.1–284103.4 (2002) ]. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue λ2(α,β) of the (wavelet) transformed coupling matrix C(α,β) for each α and β. Here β is a mixed boundary constant and α is a scalar factor. In particular, β = 1 (0) gives the nearest neighbor coupling with periodic (Neumann) boundary conditions. In this paper, we obtain two main results. First, the reduced eigenvalue problem for C(α,0) is completely solved. Some partial results for the reduced eigenvalue problem of C(α,β) are also obtained. Second, we are then able to understand behavior of λ2(α,0) and λ2(α,1) for any wavelet dimension j ∊ math and block dimension n ∊ math. Our results complete and strengthen the work of Shieh et al. [J. Math. Phys. 47, 082701.1–082701.10 (2006) ] and Juang and Li [J. Math. Phys. 47, 072704.1–072704.16 (2006) ].
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05.45.Gg Control of chaos, applications of chaos
05.45.Xt Synchronization; coupled oscillators
02.10.Ud Linear algebra
02.10.Yn Matrix theory
02.30.Uu Integral transforms
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Existence and uniqueness for the reflection and transmission problem in stratified electromagnetic media

Alessia Berti

J. Math. Phys. 47, 122901 (2006); http://dx.doi.org/10.1063/1.2401751 (14 pages)

Online Publication Date: 20 December 2006

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The reflection-transmission problem of time-harmonic waves in a stratified electromagnetic medium is investigated. The waves are sent from upward or downward with oblique incidence. By means of the energy flux, up-going and down-going waves are distinguished and the reflection and transmission matrices are introduced. When the solid occupies a strip between two homogeneous media, the existence and uniqueness of the reflected and transmitted waves are proved. The same conclusions are obtained for a dielectric without memory extended in the whole space.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation

Green functions for wave propagation on a five-dimensional manifold and the associated gauge fields generated by a uniformly moving point source

I. Aharonovich and L. P. Horwitz

J. Math. Phys. 47, 122902 (2006); http://dx.doi.org/10.1063/1.2401692 (26 pages) | Cited 7 times

Online Publication Date: 29 December 2006

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Gauge fields associated with the manifestly covariant dynamics of particles in (3,1) space time are five dimensional (5D). We provide solutions of the classical 5D gauge field equations in both (4,1) and (3,2) flat space-time metrics for the simple example of a uniformly moving point source. Green functions for the 5D field equations are obtained, which are consistent with the solutions for uniform motion obtained directly from the field equations with free asymptotic conditions.
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11.15.-q Gauge field theories
04.20.Gz Spacetime topology, causal structure, spinor structure
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New class mathN of statistical models: Transfer matrix eigenstates, chain Hamiltonians, factorizable S matrix

B. Abdesselam and A. Chakrabarti

J. Math. Phys. 47, 123301 (2006); http://dx.doi.org/10.1063/1.2374882 (27 pages) | Cited 1 time

Online Publication Date: 6 December 2006

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Statistical models corresponding to a new class of braid matrices (mathN;N ≥ 3) presented in a previous paper are studied. Indices labeling states spanning the Nr dimensional base space of T(r)(θ), the rth order transfer matrix are so chosen that the operators W (the sum of the state labels) and (CP) (the circular permutation of state labels) commute with T(r)(θ). This drastically simplifies the construction of eigenstates, reducing it to solutions of relatively small number of simultaneous linear equations. Roots of unity play a crucial role. Thus for diagonalizing the 81 dimensional space for N = 3, r = 4, one has to solve a maximal set of five linear equations. A supplementary symmetry relates invariant subspaces pairwise [W = (r,Nr) and so on] so that only one of each pair needs study. The case N = 3 is studied fully for r = (1,2,3,4). Basic aspects for all (N,r) are discussed. Full exploitation of such symmetries lead to a formalism quite different from, possibly generalized, algebraic Bethe ansatz. Chain Hamiltonians are studied. The specific types of spin flips they induce and propagate are pointed out. The inverse Cayley transform of the YB matrix giving the potential leading to factorizable S matrix is constructed explicitly for N = 3 as also the full set of mathtt relations. Perspectives are discussed in a final section.
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05.20.-y Classical statistical mechanics
02.10.Yn Matrix theory
02.10.Ud Linear algebra
02.30.Zz Inverse problems
02.30.Uu Integral transforms

Localization at low temperature and infrared bounds

Volker Bach and Jacob Schach Møller

J. Math. Phys. 47, 123302 (2006); http://dx.doi.org/10.1063/1.2364180 (12 pages)

Online Publication Date: 12 December 2006

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We consider a class of classical lattice spin systems, with mathn-valued spins and two-body interactions. Our main result states that the associated Gibbs measure localizes in certain cylindrical neighborhoods of the global minima of the unperturbed Hamiltonian. As an application we establish existence of a first order phase transition at low temperature, for a reflection positive mexican hat model on mathd, d ≥ 3, with a nonferromagnetic interaction.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.10.Ab Logic and set theory

Feynman cycles in the Bose gas

Daniel Ueltschi

J. Math. Phys. 47, 123303 (2006); http://dx.doi.org/10.1063/1.2383008 (15 pages) | Cited 9 times

Online Publication Date: 29 December 2006

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We study the lengths of the cycles formed by trajectories in the Feynman-Kac representation of the Bose gas. We discuss the occurrence of infinite cycles and their relation to Bose-Einstein condensation.
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05.30.Jp Boson systems
03.75.Hh Static properties of condensates; thermodynamical, statistical, and structural properties
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Final state problem for Korteweg–de Vries type equations

Nakao Hayashi and Pavel I. Naumkin

J. Math. Phys. 47, 123501 (2006); http://dx.doi.org/10.1063/1.2374883 (16 pages)

Online Publication Date: 4 December 2006

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We study the final state problem for the Korteweg–de Vries type equations: ut−1/ρ∣∂x∣ρ−1ux = λu2ux, (t,x) ∊ R+×R,∥u(t)−FS(t)∥L2→0 as t→∞, where λ ∊ R, the function FS(t) we call a final state, defined by the final data u+. We show that there does not exist a nontrivial solution of this equation in the case of FS(t) = U(t)u+, where U(t) is the free evolution group of this equation. We construct the modified wave operator for the Korteweg–de Vries type equations under the conditions that the final data u+ arc real-valued functions and the Fourier transform math+(ξ) vanishes at the origin.
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02.30.Nw Fourier analysis
02.30.Uu Integral transforms
02.30.Sa Functional analysis
02.30.Tb Operator theory

Spectral theory of neutron transport semigroups with partly elastic collision operators

Mohammed Sbihi

J. Math. Phys. 47, 123502 (2006); http://dx.doi.org/10.1063/1.2397557 (12 pages)

Online Publication Date: 6 December 2006

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This paper deals with spectral properties of a class of neutron transport equations involving partly elastic collision operators introduced by Larsen and Zweifel [J. Math. Phys. 15, 1987–1997 (1974) ]. In particular, estimates of the essential type of associated semigroups are given.
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28.20.Gd Neutron transport: diffusion and moderation
02.30.Jr Partial differential equations
02.30.Tb Operator theory

Existence and computation of optimally localized coherent states

Matthias Holschneider and Gerd Teschke

J. Math. Phys. 47, 123503 (2006); http://dx.doi.org/10.1063/1.2375031 (12 pages)

Online Publication Date: 6 December 2006

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This paper is concerned with localization properties of coherent states. Instead of classical uncertainty relations we consider “generalized” localization quantities. This is done by introducing measures on the reproducing kernel. In this context we may prove the existence of optimally localized states. Moreover, we provide a numerical scheme for deriving them.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Fd Algebraic methods
02.10.Ud Linear algebra

Geometric equivalence of Clifford algebras

David M. Botman and William P. Joyce

J. Math. Phys. 47, 123504 (2006); http://dx.doi.org/10.1063/1.2375037 (10 pages)

Online Publication Date: 7 December 2006

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We motivate a notion of geometric equivalence that is not the usual notion of algebraic equivalence (or isomorphism of Clifford algebra). Using this definition tilting to the opposite metric is a geometric equivalence in contrast to such algebraic equivalences as Cℓ(3,0) ≅ Cℓ(1,2) which are not geometric. We define and discuss the classification of partitioned Clifford algebra and the geometric equivalence of Dirac formulations.
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02.10.-v Logic, set theory, and algebra
02.40.-k Geometry, differential geometry, and topology
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