Baez on “Energy, the Environment and What We Can Do”

12Jan12

John Baez who is currently speaking at EQuaLS5 will be giving a public talk on “Energy, the Environment and What We Can Do” on Friday January 13, 2012 at 3.15pm in Al-Khawarizmi Seminar Room, Mathematics Building, Universiti Putra Malaysia. The abstract of his talk is as follows:

Our heavy reliance on fossil fuels is causing two serious problems: global warming, and the decline of cheaply available oil reserves. Unfortunately the second problem will not cancel out the first. Each one individually seems extremely hard to solve, and taken together they demand a major worldwide effort starting now. After an overview of these problems, we turn to the question: what can we do about them?

Free admission – no registration fees but space is limited. All are welcome. Be early.

Baex "Energy, the Environment and What We Can Do"

Poster for Baez Public Talk


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Prof. Bakhadyr Khoussainov’s Talk at UPM

16May11

This is another announcement post:

The Laboratory of Computational Sciences & Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia will be organizing a Public Seminar on “A Journey From Computable to Automatic Structures” by Prof. Bakhadyr M. Khoussainov (Univ. of Auckland, New Zealand) on May 19, 2011 at 10.30 am in the Meeting Room of the Mathematics Department. This will be followed by an unoffcial meeting cum discussion between Prof. Khoussainov and laboratory members as well as interested researchers in the afternoon at 2.30pm.

Abstract for his talk:

In this talk, we introduce two classes of structures: computable structures and automatic structures. The goal is to move from computable to automatic structures with an eye towards decidability. We present several examples, theorems and discuss their motivations, proof and importance.


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Lectures on Functional Analysis by Prof. Abdumalik Rakhimov

09Feb11

This is an announcement post:

The Laboratory of Computational Sciences & Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia will be organizing a series of lectures on Functional Analysis given by Research Fellow, Prof. Abdumalik Rakhimov every Thursday at 4pm.

Prof. Abdumalik Rakhimov

The lecture begins tomorrow as shown below:

Lectures on Functional Analysis (Lecture 1)
by Prof. Abdumalik Rakhimov (Research Fellow, INSPEM, UPM)
Date: Thursday, February 10, 2011
Time: 4.00-5.00pm
Venue: Al-Khawarizmi Seminar Room, Maths Bldg, UPM

The lectures cover the following topics:

Section 1: Metric Spaces

  • Basic concepts, definitions and examples 
  • n-dimensional Euclidean space
  • Cauchy-Schwarz inequality
  • Holder’s inequality
  • Holder’s integral inequality
  • Minkowski’s integral inequality
  • C[a,b] all continuous function space
  • All bounded infinite sequences
  • l_2 space
  • l_p space
  • Continuous mappings and homeomorphisms
  • Isometric spaces
  • Closure of a set, limit points
  • Convergence and limits
  • Dense subsets, separable spaces
  • Closed sets, examples
  • Open sets
  • Open and closed sets on the real line
  • The Cantor set
  • Complete metric space, definitions and examples
  • The nested sphere theorem
  • Baire’s theorem
  • Completion of a metric space
  • Definition of a contraction mapping, fixed point theorem
  • Contraction mappings and differenetial equations
  • Contraction mappings and integral equations

Section 2: Topological Spaces

  • Definitions and examples of topological spaces
  • Comparison of topologies
  • Convergent sequences in a topological space
  • Axioms of separation
  • Continuous mappings, homeomorphisms
  • Compact topological spaces
  • Compactness in metric spaces, total boundedness
  • Relative compact subsets of a metric space

Section 3: Linear Spaces

  • Definition and examples of linear spaces
  • Linear dependence
  • Linear subspaces
  • Factor spaces
  • Linear functionals
  • Normed linear spaces, definitions and examples
  • Subspaces of a normed linear space
  • Euclidean spaces, scalar products, orthogonality and bases
  • Existence of an orthogonal basis, orthogonalization
  • Bessel’s inequality, closed orthogonal systems
  • Complete Euclidean spaces, Riesz-Fischer theorem
  • Hilbert space, isomorphism theorem
  • Subspaces of Hilbert space, orthogonal complements and direct sums
  • Topological linear spaces, definitions and examples

Section 4: Linear Functionals

  • Continuous linear functionals on a topological linear space
  • Continuous linear functionals on a normed linear space
  • Hahn-Banach theorem for a normed linear space
  • Conjugate space of  topological linear space
  • Weak topology in topological linear space

Section 5: Remarks on Measure and Integration

  • Measure of elementary sets
  • Lebesgue measure of plane sets
  • Measurable functions, basic properties
  • Simple functions, algebraic properties on measurable functions
  • Equialent functions
  • Almost everywhere convergence
  • Lebesgue integral, definition and basic properties
  • Some properties of the Lebesgue integral
  • Lebesgue integral vs Riemann integral
  • Some spaces of integrable functions

All are invited and no registration fee.


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INSPEM GeoGebra Workshop, Dec. 16, 2010

15Dec10

Just to announce that Laboratory of Innovations in Mathematics Education, Institute for Mathematical Research, Universiti Putra Malaysia will be organizing the following workshop on GeoGebra:

Workshop on Applications of Geogebra in Mathematics
Date: December 16, 2010 (Thursday)
Time: 10am-12 noon
Venue: Al-Khaitam Laboratory, Mathematics Building, UPM


Speaker: Zsolt Lavicza, Faculty of Education, University of Cambridge

Abstract:
GeoGebra is rapidly gaining popularity in the teaching and learning of mathematics around the world. Currently, GeoGebra is translated to 52 languages, used in 190 countries, and downloaded by approximately 300,000 users in each month. The use of technology is slowly becoming a substantial part of today’s education. Although due to the increased accessibility of affordable computing technologies in the 1980s and 90s it was predicted that computers would become rapidly integrated into mathematics teaching and learning (Kaput, 1992), technology uptake in schools has been considerably slow. The current expansion of technology use took a new unconventional direction: a bottom‐up, community‐based collaborative development, catalysed by Internet‐based communities and increasingly available community‐developed software packages.

During the past decades it has been demonstrated that a large number of enthusiasts can alter conventional thinking and models of development and innovation. The success of open source projects like Linux, Firefox, Moodle, and Wikipedia shows that collaboration and sharing can produce valuable resources in a variety of areas of life. While working on the open‐source project GeoGebra we are witnessing the emergence of an enthusiastic international community around the software. GeoGebra has grown from a small student project to an international organisation. Due to the involvement of thousands of volunteers we hope that it will offer a new way to teach mathematics around the world and contribute to the education of students.

Note: Limited to 30 places only.

For Further Details, Please Contact:
Ms. Fara Nadia Zainuddin
Phone: 603-8946 7594
Email: faranadia@putra.upm.edu.my

This workshop is followed by  a seminar on MATHEMATICAL INSTUITIONS HIDDEN IN THE TEXTBOOK: USING E-TEXTBOOK OF SCHOOTEN (1646) AND JAPANESE TEXTBOOK by Masami Isoda (University of Tsukuba, Japan) at 2.30pm in the Al-Khawarizmi Seminar Room.


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Phase Spaces and Symplectic Geometry

03Dec10

These are notes from my Monday nite Musings on October 25, 2010.

What are phase spaces? There are many different usages of this term but generally they can be said to be simply state spaces. Then the next question is what are states? A good brief discussion can be found in Isham’s nifty book. Enough to say that states are essentially the optimal information regarding a physical system and this is very much model-dependent. It’s optimal in the sense that it is not superfluous but enough to generate other information of the system. Note that in general, states are often described by the “properties” of a system and they are necessarily expressed in numbers. Most obvious case is the ON-OFF state but this often given the numerical representation 1, 0. Pushing to the extremes, one can talk for example about the “mental states” of happy, sad or indifferent which certainly have no obvious numerical representations. The advantage of using numbers is the properties that go with them e.g. ordering, performing calculus. A good high-school example is the state of an ideal gas, which can be any two of the observables pressure P, volume V, and temperature T, all of which are related by the following equation of state:

\cfrac{PV}{T} = \textrm{constant}\quad.

In fact, in many cases, equations of state are actually sought for, to help define the state of the physical system. Another important property of the states is the fact that the knowledge of one state at an initial time t_0 allows access of information of states at other times t. This is often possible through equations of motion.

Let us consider the familiar Newtonian mechanics with the celebrated second law for the equation of motion:

\underline{F} = \cfrac{d\underline{p}}{dt}\quad,

best written in this form

\cfrac{d\underline{r}}{dt} = \cfrac{\underline{p}}{m} = \cfrac{\partial}{\partial\underline{p}}\left(\cfrac{\underline{p}^2}{2m}\right)\quad ;

\cfrac{d\underline{p}}{dt} = - \underline{\nabla} V(\underline{r}) = - \cfrac{\partial}{\partial\underline{r}} V(\underline{r})\quad .

The classical mechanical state is now the pair (\underline{r}, \underline{p}), which can be solved for at any time t.

Alternatively, one could also use the Lagrangian mechanics formalism with Lagrangian function

L = \frac{1}{2} m\dot{\underline{r}}\cdot\dot{\underline{r}} -V(\underline{r})

with equations of motion

\cfrac{d\underline{\dot{r}}}{dt} = \cfrac{\partial L}{\partial\underline{r}} = - \cfrac{\partial V}{\partial\underline{r}}\quad ;

\cfrac{d\underline{r}}{dt} = \cfrac{\partial L}{\partial\dot{\underline{r}}} = \cfrac{\partial}{\partial\dot{\underline{r}}} \left(\frac{1}{2} m\dot{\underline{r}}\cdot\dot{\underline{r}}\right)\quad .

Note that in this case, the classical mechanical state is the pair (\underline{r},\dot{\underline{r}}) and the “symplectic nature” is no longer obvious or available directly (To resolve one needs to define the energy function in addition to the Lagrangian function).

What does one mean by the “symplectic nature” in the above? Let us rewrite our earlier equation of motion in matrix form:

\cfrac{d}{dt} \left( \begin{matrix} \underline{r}\\ \underline {p} \end{matrix}\right) = \left( \begin{matrix}{\cfrac{\partial}{\partial\underline{p}} (\underline{p}\cdot\underline{p})/(2m)}\\ - \cfrac{\partial}{\partial\underline{r}} V(\underline{r}) \end{matrix} \right)\quad .

If H= (\underline{p}\cdot\underline{p})/(2m) + V(\underline{r}), then can write

\cfrac{d}{dt} \left( \begin{matrix} \underline{r}\\ \underline{p} \end{matrix}\right) = \left( \begin{matrix} \cfrac{\partial H}{\partial\underline{p}}\\ - \cfrac{\partial H}{\partial\underline{r}} \end{matrix}\right) = \left( \begin{matrix} 0&I\\-I&0 \end{matrix}\right) \left(\begin{matrix} \cfrac{\partial H}{\partial\underline{r}}\\ \cfrac{\partial H}{\partial\underline{p}} \end{matrix} \right)\quad .

Note that the square matrix J on the RHS is important in matching the \underline{r} and \underline{p}-components on both sides and getting the right signs. The matrix J can be called symplectic form – a more technical version will be given later. Here, in fact, if one assumes that the phase space vectors are in place, then J encodes  (in a way) the equations of motion.

Generally, the matrices M satisfying M^t JM= J are called symplectic matrices in general. From the condition, we also have that symplectic matrices have the property \det M = \pm 1.The matrix J itself obeys this condition by noting J^t = -J. An important case is when M is a transformation matrix, say, for the transformation (\underline{r}, \underline{p})\rightarrow (\underline{R},\underline{P}). One obvious example of such transformation is that arising from simple translations of \underline{r} and \underline{p} – these are simply linear ones. We can have however interesting nonlinear ones.

Consider transformation Q = \frac{1}{2}(q^2 + p^2) and P=\tan^{-1}(p/q) for the harmonic oscillator. Then

\cfrac{\partial Q}{\partial q} = q\quad\quad ;\quad\quad \cfrac{\partial Q}{\partial p} = p\ .

\tan P = \cfrac{p}{q}=\cfrac{p/\sqrt{p^2+ q^2}}{q/\sqrt{p^2 + q^2}}\equiv\cfrac{\sin P}{\cos P}\ ;

\Rightarrow\quad\cfrac{\partial P}{\partial q}= -\cfrac{p}{q^2\sec^2 P}= -\cfrac{p}{p^2 + q^2}\ .

\Rightarrow\quad\cfrac{\partial P}{\partial p}= \cfrac{1}{q\sec^2 P}=\cfrac{q}{p^2 + q^2}\ .

One easily checks that

M = \left( \begin{matrix} q&p\\ \cfrac{-p}{q^2 + p^2}&\cfrac{q}{q^2 + p^2}\end{matrix} \right)

satisfies the symplectic condition M^t J M = J and \det M = 1. This is in fact the transformation from the position representation to Fock or number representation.

So far \underline{r},\ \underline{p} belong to vector spaces. In general however, \underline{r} can belong to a nonlinear space Q. To enable the advantage of the earlier linear structure, one creates vectors on Q by taking tangent vectors to points in Q . In some local neighbourhood, in fact, one can set up local coordinates, making Q looks like \mathbb{R}^n. We can thus defne vectors at q\in Q by

v= v^i \cfrac{\partial}{\partial x^i}\in T_q Q\ .

Einstein summation convention implied there. Note that one can easily check that vector space operations hold for such objects. So at each point q, we have tangent space T_q Q, which is a vector space. The collection of all such vector spaces at all points give rise to the tangent bundle TQ=\bigcup_{q\in Q} T_q Q.

Dual to vectors are covectors which are really linear functionals of vectors, mapping vectors into the reals. Thus one can form the cotangent space T_q ^* Q, dual to T_q Q. A covector is written as l = l_i\ dx^i\ \in\ T_q^* Q. The duality can be established from defining the contraction of their bases:

\left\langle dx^i\ ,\ \cfrac{\partial}{\partial x^j} \right\rangle_q = \delta^i_j\ .

Note that the space T_q^* Q is a vector space and by the same token, the collection of all such vector spaces form the cotangent bundle T^* Q=\bigcup_{q\in Q} T_q^* Q. From here, one can initiate the tensor industry, forming much richer structure through the use of tensor product \otimes (will be skipped here).

Of interest here, is a special tensor product of covectors, that is antisymmetric. This is best demonstrated from the definition of a wedge product between two bases of covector:

dx^i\wedge dx^j = dx^i \otimes dx^j - dx^j\otimes dx^i = - dx^j\wedge dx^i\ .

Contracted with the basis of vectors gives

dx^i\wedge dx^j\ \left(\cfrac{\partial}{\partial x^k}\ ,\ \cfrac{\partial}{\partial x^l}\right)=\ \delta^i_k \delta^j_l - \delta^j_k\delta^i_l\ .

The product dx^i\wedge dx^j forms the basis for the space of two-forms \Lambda^2(Q) which are really antisymmetric tensors of covariant rank two. One can generalize this further by defining the space of k-forms \Lambda^k(Q) such that

\Lambda^k(Q)\ni\chi :\ T_q Q\times \cdots\textrm{k times}\cdots\times T_q Q\ \rightarrow\ \mathbb{R}\ .

We will use these k-forms later to define  a special structure for phase space.

We return now to our constructed phase spaces. In the Lagrangian formalism, the phase space is the tangent bundle TQ that can be coordinatized by \{ x^i, \cdot{x}^i\}. In the Hamiltonian formalism, it is cotangent bundle T^*Q that can be coordinatized by \{ x^i,p_i\}. The use of Hamiltonian formalism is preferred due to its inherent symplectic nature. On the fibres of T^* Q, we have the covectors p_i dx^i. It is sometimes useful to take p_i as the fibre coordinate function

p_i\equiv\cfrac{\partial}{\partial x^i}\ ,

analogous to the coordinate function x^i(q) = q^i on the configuration space. This is so from the following action on the fibre

p_i(l) =\cfrac{\partial}{\partial x^i}\ (l_j dx^i) = l_j\ \delta_i^j = l_i\ .

It is interesting to note that this coordinate function is reminiscent of the quantum momentum operator

\hat{p}_i = - i\hbar\cfrac{\partial}{\partial x^i}\ ,

though we are not sure whether there is any deep meaning in this.

Going back to Hamilton’s equations

\cfrac{d}{dt}\ x^i = \cfrac{\partial H}{\partial p_i}\quad\quad ;\quad\quad \cfrac{d}{dt}\ p_i = - \cfrac{\partial H}{\partial x^i}\ ,

one can generalise these to any observables f= f(x^i,p_j):

\cfrac{df}{dt} = \cfrac{\partial f}{\partial x^i}\cfrac{dx^i}{dt}\ +\ \cfrac{\partial f}{\partial p_i}\cfrac{dp_i}{dt} = \cfrac{\partial f}{\partial x^i}\cfrac{\partial H}{\partial p_i}\ -\ \cfrac{\partial f}{\partial p_i}\cfrac{\partial H}{\partial x^i}\equiv \{ f, H\}\ .

What we have just defined is the Poisson bracket structure:

\{ f,g\} = \cfrac{\partial f}{\partial x^i}\cfrac{\partial g}{\partial p_i}\ -\ \cfrac{\partial f}{\partial p_i}\cfrac{\partial g}{\partial x^i}\ .

Note that in this case, it is the Poisson bracket encodes the equation of motion instead of the symplectic matrix J.

So far we treated the phase space T^* Q as a manifold and have not define any structures on it. We can do so now by writing df/dt = \{ f, H\} as

\left( \cfrac{\partial H}{\partial p_i} \cfrac{\partial}{\partial x^i} - \cfrac{\partial H}{\partial x^i} \cfrac{\partial}{\partial p_i}\right) f\ \equiv\ \xi_H f\ .

Note that \partial/ \partial x^i,\ \partial/\partial p_i are basis vectors on T^* Q. Thus, \xi_H belongs to the tangent space T_{(x,p)}(T^* Q) (the tangent space to the cotangent bundle). Generalizing, we have in this tangent space

\cfrac{\partial f}{\partial p_i}\cfrac{\partial}{\partial x^i} - \cfrac{\partial f}{\partial x^i}\cfrac{\partial}{\partial p_i}\ =\ \xi_f\ .

Next, we dualize. Suppose l\in T^*_{(x,p)}(T^*Q), then we can write:

l = l_i^{(x)} dx^i + l^i_{(p)} dp_i\ .

Contracting with \xi_f gives

\xi_f (l) = l(\xi_f) = \cfrac{\partial f}{\partial p_i}\ l_i^{(x)} - \cfrac{\partial f}{\partial x^i}\ l^i_{(p)}\ .

Consider now a two-form on T^* Q  i.e.

\omega = \sum_i dx^i\wedge dp_i \in \Lambda^2 (T^* Q)\ .

This two-form is known as the symplectic form on the phase space T^* Q. Note that if one contracts it with a vector field, one has

\omega(\xi_f) = \cfrac{\partial f}{\partial p_i}\ dp_i + \cfrac{\partial f}{\partial x^i}\ dx^i = df

which is in T^*_{(x,p)} (T^*Q). Further contracting with another vector field gives

\omega(\xi_f , \xi_g) = - \cfrac{\partial f}{\partial p_i}\ \cfrac{\partial g}{\partial x^i} + \cfrac{\partial f}{\partial x^i}\ \cfrac{\partial g}{\partial p_i}

which is precisely the Poisson bracket \{ f, g \}! Thus the chain of concepts from “equations of motion” to “Poisson bracket” to “symplectic form”.

Recall earlier the idea of canonical transformations (eliminating dependence on the particular coordinates of phase space) preserving the equations of motion. This is now replaced by symplectomorphisms that preserve the symplectic form, providing the needed generalization to a more abstract geometry. A manifold M with a symplectic form \omega is called a symplectic manifold (M,\omega). The study of symplectic manifolds is called symplectic geometry and it is the modern geometric language for classical mechanics. As such, it is also the starting point for several geometric approaches to quantization where classical systems are converted to quantum ones.


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Prof. Basyaruddin’s Inaugural Lecture

28Oct10

Since I cannot find any proper announcement of the following inaugural lecture anywhere on the web, I thought I do one over here. Prof. Mohd. Basyaruddin Abdul Rahman, a chemist who has interests in computational chemistry and biology, will be giving his inaugural lecture on Friday, October 29, 2010 at 9.30 am in the Experimental Theatre of UPM. The lecture is entitled, “From Genomics to Mathematical Biology – A Chemist’s Perspective“. He is also the Director of the Structural Biology Research Centre in Malaysia Genome Institute. Below is the poster announcement.

Prof. Basya will also be one of the invited speakers in our International Conference on Mathematical and Computational Biology 2011. We hope that we will be able to work together in the Institute for Mathematical Research in the near future.


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Isao Harada’s Visit & Talk@UPM

27Oct10

I have been informed by Dr. Mahmudur-Rahman (our new condensed matter theorist at UPM, finally) that Prof. Isao Harada will be visiting him/UPM on the 2-3 Novmber 2010. He will be giving a talk on the Nov 2, 2010 at 3pm in Lecture Room 304 besides the UPM Physics Department General Office. The talk is on “High Energy X-Ray Spectroscopy of Strongly Correlated Electron Systems”. Please help publicise this to interested colleagues and friends. We need the crowd. Below are the details:

High Energy X-Ray Spectroscopy of Strongly Correlated Electron Systems

by
Prof. Dr. Isao HARADA
Okayama University
3-2-2 Tsushima-naka, Kita-ku, Okayama 700-8530, Japan

Time and Date: 3pm, November 2, 2010
Venue: Lecture Room 304, Physics Department, UPM

Abstract: 

X-ray Absorption Spectroscopy (XAS) and Magnetic Circular Dichroism (XMCD) have become a powerful technique for studying electronic and magnetic properties of magnetic materials with the aid of a recent development of X-ray light sources. These experiments give us important information on an inner shell electron excitation to the valence electron states, which is element specified as well as shell specified. To analyze these useful data, we need a sophisticated theory for XAS and XMCD.

In this talk, I will start with an introduction of XAS and XMCD, and extend the talk to a recent progress of our study: XAS and XMCD of valence mixed systems of rare-earth compounds in high magnetic fields. Strong magnetic fields induce a valence transition, which have recently been observed in \textrm{EuNi}_2(\textrm{Si}_{0.18}\textrm{Ge}_{0.82})_2 at the L-edges of \textrm{Eu}. We have investigated theoretically this phenomenon based on an effective model, taking into account the hybridization between a localized 4f electron and an itinerant 5d conduction electron. As a result, we show this is a transition from the \textrm{Eu}^{3+} nonmagnetic phase to the \textrm{Eu}^{2+} magnetic phase controlled by high magnetic fields.

Professor Dr. Isao Harada received his B.Sc.in physics, M.Sc. and Ph.D. in theoretical condensed matter physics from Osaka University in 1967, 1969 and 1972 respectively. He started his career as an Assistant Professor in Kobe University. He moved to Okayama University in 1990 to be an Associate Professor before taking up an appointment as a Professor in Okayama University. He has retired in March 2010, and is a Professor Emeritus as well as the Director of the Educational Center for Science in Okayama University. He was also a Visiting Assistant Professor at Ohio University in 1977-1978, Research Associate (RA) at Honnover University in 1987-1988, RA at Hahn-Meitner Institute, Berlin in 2001, Visiting Professor (VP) at the University of Louis Pasteur in Strasbourg in 2007, and VP at the University of Cergy-Pontoise in Paris in 2009.

The scientific activity of Prof. Harada covers the fields of theoretical condensed matter physics including the statistical physics for low-dimensional quantum spin systems, the surface science of magnetic materials, nonlinear excitation, soliton, one-dimensional magnets, and high energy spectroscopy of magnetic materials. He has been always interested not only in mathematical formulation of physics but also in applications of mathematical models to real substances. Of notable contribution is his excellent theoretical contribution to the magnetic circular dichroism for magnetic materials in high magnetic fields. Recently, his interests are also focused on the education of natural sciences for young students. For instance, he was a leader of the Japan-team for International Physics Olympiad and has tried to establish a new concept of scientific education for talented young students.


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Research Opportunity with Teo Lee Peng

22Oct10

I’m posting this letter from Assoc. Prof. Dr. Teo Lee Peng

Dear All,
 
A position of full time Research Assistant (RA) position is available in the Department of Applied Mathematics, University of Nottingham Malaysia Campus. This is a contract position and the candidate is expected to register for a degree of MPhil or PhD. The applicant is required to have at least a good honours bachelor degree, preferably second class upper and above, good command of English and preference will be given to candidate who has strong background in mathematics and electromagnetics. Candidates that have knowledge in numerical methods will be at added advantage.
 
The successful RA may receive complete/partial tuition fees waiver for his/her study and a monthly stipend of RM1500 under the project funding.
S/he is expected to assist academic staff with teaching, research and administration duties up to a maximum commitment of 200 hours per academic year.
Interested candidates please email full resume to Dr. Teo Lee Peng at LeePeng.Teo@nottingham.edu.my. Any further enquiry is welcomed.
 
Thank you.
 
 
Regards,
Dr. Teo Lee Peng
Associate Professor
Department of Applied Mathematics
Faculty of Engineering,
University of Nottingham Malaysia Campus,
Jalan Broga,
43500 Semenyih,
Selangor, Malaysia
 
Tel: +603-89248356
Fax:+603-89248017


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Impact Factors and Citations in Mathematics

30Aug10

All last week, I was occupied with a report that tells about achievements and future directions of the laboratory I’m heading for the moment. Naturally, the main issue is publications and among the performance indicators sought was Cumulative Impact Factor (CIF) obtained by summing impact factors of journals where the lab/institute publishes in. Impact factor IF_A(T)  in year T  is a measure of how many citations that a particular journal A gets within a two-year period. Its precise formula is as follows:

IF_A(T) = \cfrac{C_A(T-1, T-2)}{P_A(T-1,T-2)}\quad ,

where C_A(T-1,T-2)  is the number of citations for articles published in A  from the years T-1, T-2, while P_A(T-1,T-2)  is the number of articles published in A  in the years T-1, T-2. By looking at CIF, some indication is given on the “quality” productivity of the lab. Unfortunately IF itself is a rather crude measure whose real significance is not very clear and it is probably unfair when one does comparison across disciplines.

Just to show that there is great variability between disciplines, I list down the highest impact factor for a few different disciplines below for comparison:

 From just the above (random) listing, one can see that mathematics appear at the bottom of the list and it is no wonder that mathematicians here are screaming when they are directly compared with other disciplines. The complain is of course shouted elsewhere too – a report was in fact written by Robert Adler, John Ewing (chair) and Peter Taylor for International Mathematical Union warning on the misuse of Citation Statistics. One of the keypoints stated is that the use of (a) number to replace the subjective peer review of evaluating research/researchers belies the subjectivity in its interpretation (more here for summary). Why should a number (faithfully) represents the multidimensional character of research? At best, citation indices (which include impact factors) are one-dimensional projections onto some properties which aren’t even clear in the first place. Just to take a simple example, that given on pages 10-12 of the report. Suppose journal A has impact factor \bar{a} which is greater than journal B of impact factor \bar{b} i.e. \bar{a} >\bar{b}. The statement that a paper in A will be cited more than a paper in B, can be more than 50% wrong (calculate \sum_{a<b} P(a) P(b) a) renders the impact factor useless in ranking a paper. A more appropriate measure is to follow tha actual citation number that each paper has.

Some would like to suggest that one should rescale the highest impact factor of mathematics journal to that of other disciplines but really this wouldn’t be fair either; certainly there are properties that other discplines have that mathematics doesn’t (and vice versa, of course). Even mathematically, one knows that direct comparison can’t be made since the distribution of citation data for each discipline may be altogether different. There is one interesting result made by Henk F. Moed in his paper “Citation Analysis of Scientific Journals and Journal Impact Measures”, Current Science 89 (12) (2005) 1990-1996. By observing that review journals are often cited more than normal journals, Moed suggested to incorporate such difference within a discipline and calculate what he calls a normalized impact measure:

\cfrac{n_r c_r + n_a c_a}{n_r \bar{c}_r + n_a \bar{c}_a}

where n_r, n_a are respectively the number of review articles and the number of ordinary articles that the journal has with their corresponding actual citations c_r, c_a per document while \bar{c}_r, \bar{c}_a are the citations per document within the whole discipline that the journal belongs to. Here is a plot from the paper of the normalized impact measure versus the JCR impact factor fro two differing disciplines of mathematics and biochemistry & microbiology.

Normalized vs JCR-like impact measure

Observe the similar ranges of the values of the normalized impact measures for both disciplines. I have yet to understand how this actually works but such comparability seems encouraging. Moed apparently has a book entitled “Citation Analysis in Research Evaluation” (Springer, 2010) which may be worth getting.

Perhaps the other point of objection that is often raised and worth getting into is the source of the citation data. It is altogether well-known that the use of SCI journals only misses out other forms of research out ut such as books, chapters in books, and policy reports and are often raised by social scientists. In a different twist, it was pointed out that it is rather surprising that majority of researchers and research evaluators tend to rely very much the services of a commercial company from the United States and the journals from there. It is so much glaring that Moed in his report “Bibliometric Rankings of World Universities” showed that the US universities are highly overrepresented in the top ranks of world universities based on citation impacts. Many have criticise this biasness towards US journals and US universities in the rankings (see e.g. Kaltenborn and Kuhn). Charlton & Andras for example, in their article, “Evaluating Universities using Simple Scientometric Research Output Metrics: Total Citation Counts per Universities for a Retrospective Seven Year Rolling Sample“, Science & Public Policy 34 (8) (2007) 555-563, have opted out to include both ISI (leaning towards US) and Scopus (leaning towards Europe) in their database sources for better evaluating research. In this regards, it is perhaps worth noting that our local universities have decided to go mostly for ISI in research evaluation, very much to the drive for impact factor. While one can see the short term benefits of this, one should be aware of the dangers lurking and should note that one is running counter to multi-facted assessment suggested by many in the literature.


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Signs in Einstein’s Equation

13Aug10

Wow, months passed since I’ve moved Equatorial Frequencies to wordpress. Well, here’s a long overdue post that I meant to do right after EQuaLS4 but I left thinking about it until very recently. During that event, my student Hassan presented our work with Pradhan and showed the following form of Einstein’s equation (actually it’s the mixed tensor variant)

G_{ab}=-\kappa T_{ab}

A question was raised regarding the negative sign (unfortunately I wasn’t there) since many books use

G_{ab}=\kappa T_{ab}

There was a slight confusion then, but really the whole thing is very much dependent on how we define things. You might be surprised by this since  one negative sign can make a whole lot of difference in its physics. Well there are different sign conventions followed by physicists but ultimately (and I’ve checked) they all boiled down to the same physics. Almost immediately after the event, Hassan showed me the notes by M. Haehnelt at DAMTP, Cambridge, where a table of different sign conventions made by different authors is shown (and he himself adopted a different sign convention for some objects). I decided to follow up this study of signs and show that all results point to the same physics. I included all the references mentioned by Haehnelt (d’Inverno, Rindler, Morris-Thorne-Wheeler, Weinberg) including Haehnelt himself, all the standard text books (Schutz, Stephani, Wald, Hawking & Ellis) including the new ones by Jim Hartle and Sean Carroll, the textbook by Hervik (who was here) and peculiar ones that I use like Carmeli and deSabbata & Gasperini.

I will use d’Inverno as the standard reference here since I have always recommended this book to students for learning general relativity due to its friendly approach. As such, I will use an overbar for all quantities defined in d’Inverno and later relate them with corresponding quantities from the other books (without the overbar).

First and foremost is the signature of the metric adopted by the books. Many books adopted the (-,+,+,+) signature (Morris-Thorne-Wheeler, Weinberg, Hervik, Hartle, Schutz, Carroll,, Stephani, Hawking-Ellis). This signature can be thought of as a natural extension from the Euclidean signature i.e. for example in the flat case

ds^2 = dx^2 + dy^2 + dz^2 \longrightarrow ds^2= -dt^2+ dx^2 + dy^2 + dz^2

The other convention is to pick the signature (+,-,-,-) and this is adopted by d’Inverno, Rindler, Haehnelt and Carmeli. The signature is perhaps considered natural when one talks about time-like events for which

ds^2 = dt^2 -dx^2 -dy^2-dz^2>0

Now there are authors like de Sabbata & Gasperini who puts the time coordinate as the fourth coordinate and not the zeroth coordinate for which the signature will look like (-,-,-,+). But I will consider this is equivalent to (+,-,-,-) since it is simply a permutation. Note however some care is needed when shuffling indices of tensors to make tensors of different permutation of signature look equivalent. Thus, now if we denote g for the metric of the first signature and \bar{g} for the metric of the second signature (of d’ Inverno), then

g_{ab} = -\bar{g}_{ab}\quad .

The following quantities are defined in the same way in (almost) all books. Namely, the Christoffel symbols

\Gamma^a_{bc}= g^{ad}(\partial_b g_{dc} + \partial_c g_{bd} - \partial_d g_{bc})\equiv \bar{\Gamma}^a_{bc}\quad ,

the Ricci tensor R_{ab} = R^c_{acb}\equiv \bar{R}_{ab} and the Ricci scalar R = g^{ab} R_{ab}\equiv \bar{R}.

Do note that Rindler uses a different definition for the Ricci tensor, namely R_{ab}=R^c_{abc}. From symmetries of the Riemann tensor, one finds that

R_{ab}=-\bar{R}_{ab}\quad .

I purposely mention the Ricci tensor first before the Riemann tensor since there are two definitions available in the literature. The common one, used by d’Inverno (and almost many others), is

\bar{R}^a_{bcd}=\partial_c\Gamma^a_{bd} - \partial_d\Gamma^a_{bc} +\Gamma^a_{ec}\Gamma^e_{bd} - \Gamma^a_{ed}\Gamma^e_{bc}\quad .

Weinberg and Haehnelt however use

R^a_{bcd}=\partial_d\Gamma^a_{bc} - \partial_c\Gamma^a_{bd} +\Gamma^a_{ed}\Gamma^e_{bc} - \Gamma^a_{ec}\Gamma^e_{bd}\quad ,

which implies R^a_{bcd} = -\bar{R}^a_{bcd}. It also implies R_{ab} = -\bar{R}_{ab} and R=-\bar{R} for the case of Weinberg and Haehnelt.

Next, the Einstein tensor, whose usual definition is \bar{G}_{ab}=\bar{R}_{ab}-\frac{1}{2}\bar{g}_{ab}\bar{R}. Haehnelt differs from the rest in his definition by putting G_{ab}=\frac{1}{2}g_{ab} R- R_{ab} but because his ‘negative’ definition of the Riemann tensor, one has G_{ab}=\bar{G}_{ab} (from double negative). Weinberg’s case is also the same; there is double negative arising from the different signature of the metric, which I will explain below.

Despite that everyone uses the same definition of the Ricci tensor (apart from Rindler), there is an implicit sign difference involved when different signature is used. Under the transformation of metric \bar{g}_{ab}\rightarrow g_{ab}=-\bar{g}_{ab} to one of different signature, I claim \bar{R}_{ab}\rightarrow R_{ab}=-\bar{R}_{ab}. This is definitely true for Einstein manifolds, where R_{ab} = k g_{ab}. For the general case, I’m afraid, I do not know how to show this apart from writing explicitly the equations out. However, note that it is this crucial sign difference that later shows us that we have the same physics in the Einstein’s field equations throughout irrespective of the sign conventions taken.

To write out the field equations, one has to pick out one stress-energy tensor T_{ab}. For our purposes, we will pick that of the perfect fluid. The one used by d’Inverno (and the rest that uses signature (+,-,-,-) is

T_{ab} = (\rho + p) u_a u_b - p g_{ab}\quad .

Those using signature (-,+,+,+) will use

T_{ab}=(\rho + p)u_a u_b + p g_{ab}\quad .

Do note the difference in sign in front of the second term. Now we can come to the field equations. Whatever sign taken in G_{ab}=\pm\kappa T_{ab}, the sign also persists in its contracted form G^a_a=\kappa T^a_a. The physics that arise can be checked easily in this contracted form. The contracted Einstein tensor gives

G^a_a = R^a_a -\frac{1}{2}\delta^a_a R=-R\quad .

For the perfect fluid stress-energy tensor T_{ab}=(\rho+p)u_a u_b\pm p g_{ab}, the contracted form is

T^a_a= (\rho+p)u^a u_a \pm p \delta^a_a=\pm(\rho-3p)\quad .

Let us now consider the field equation \bar{G}_{ab} = +\kappa \bar{T}_{ab}, its contracted form is

-\bar{R}=\kappa (\rho-3p)\quad .

If the physics is the same in whatever conventions are taken mentioned in this post, they should reproduce this equation. Let us now consider the case given by Weinberg (which is followed by our paper) where G_{ab}=-\kappa T_{ab}; since Weinberg uses the negative form of Riemann tensor (see remark earlier), his use of Ricci tensor (and hence Einstein tensor) equals that of d’Inverno. But now he uses the plus sign in the stress energy tensor, and thus

-R = -\kappa(3p-\rho)\quad\Rightarrow\quad -\bar{R} =\kappa(\rho-3p)

which is equivalent to the one by d’Inverno. Rindler too has G_{ab}=-\kappa T_{ab}. But now his Ricci tensor is R_{ab} = -\bar{R}_{ab} and hence G_{ab}=-\bar{G}_{ab} whose negative sign cancels the one on the right hand side of the field equation. Again it reproduces the same equation.

Let us now move to those who use (-,+,+,+) signature but has field equation G_{ab} =\kappa T_{ab} and T_{ab}=(\rho+p)u_a u_b + p g_{ab} (most authors use this). Note that in this case, the Einstein tensor is G_{ab}=-\bar{G}_{ab}. Writing out the contracted field equation:

-R=\kappa(3p-\rho)\quad\Rightarrow\quad +\bar{R}=\kappa(3p-\rho)

which is again the same equation. This should be comforting for us to know that all conventions give the same physics!

The lesson here is important that one does not simply write equations down without taking notice the convention for each step. For otherwise, one could really be doing a different physics.


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