A Simple Proof of the Classification of Normal Toeplitz Matrices

We give an easy proof to show that every complex normal Toeplitz matrix is classified as either of type I or of type II. Instead of difference equations on elements in the matrix used in past studies, polynomial equations with coefficients of elements are used. In a similar fashion, we show that a real normal Toeplitz matrix must be one of four types: symmetric, skew-symmetric, circulant, or skew-circulant. Here we use trigonometric polynomials in the complex case and algebraic polynomials in the real case.Comment: 5 page

Factorization of number into prime numbers viewed as decay of particle into elementary particles conserving energy

Number theory is considered, by proposing quantum mechanical models and string-like models at zero and finite temperatures, where the factorization of number into prime numbers is viewed as the decay of particle into elementary particles conserving energy. In these models, energy of a particle labeled by an integer $n$ is assumed or derived to being proportional to $\ln n$. The one-loop vacuum amplitudes, the free energies and the partition functions at finite temperature of the string-like models are estimated and compared with the zeta functions. The $SL(2, {\bf Z})$ modular symmetry, being manifest in the free energies is broken down to the additive symmetry of integers, ${\bf Z}_{+}$, after interactions are turned on. In the dynamical model existing behind the zeta function, prepared are the fields labeled by prime numbers. On the other hand the fields in our models are labeled, not by prime numbers but by integers. Nevertheless, we can understand whether a number is prime or not prime by the decay rate, namely by the corresponding particle can decay or can not decay through interactions conserving energy. Among the models proposed, the supersymmetric string-like model has the merit of that the zero point energies are cancelled and the energy levels may be stable against radiative corrections.Comment: 16 pages, no figure

Will income inequality cause a middle-income trap in Asia? Bruegel Working Paper 2013/06, 10 October 2013

The Asian economy is expected to realise favourable growth during the first half of this century, but there is no guarantee. There is a discussion about a ‘middle-income trap’, which refers to a country that has realised rapid growth to become a middle-income country but is unable to grow further. A middle-income trap could occur not only if there is a delay in shifting the economy toward a productivity-driven structure, but also if there is a worsening of income distribution.We consider this in line with the theories of development economics and through a quantitative analysis. The relationship between income inequality and the trap can be explained by the Kuznets hypothesis and the basic-needs approach. Our quantitative analysis supports the Kuznets hypothesis, and indicates that,although a low-income country can accelerate its economic growth with the worsening of income distribution as an engine, a middle income country would experience a decreasing growth rate if it fails to narrow the income gap between the top and bottom income groups. The results also show that the basic-needs approach is also applicable in practice, and imply that the improvement of access to secondary education is important. A sensitivity analysis for three Asian upper-middle-income countries(China, Malaysia and Thailand) also shows that the situation related to a middle-income trap is worse than average in China and Malaysia. These two countries, according to the result of the sensitivity analysis, should urgently improve access to secondary education and should implement income redistribution measures to develop high-tech industries, before their demographic dividends expire. Income redistribution includes the narrowing of rural urban income disparities, benefits to low-income individuals, direct income transfers, vouchers or free provision of education and health-care, and so on, but none of these are simple to implement

Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link

There is a question asking whether a handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link. This question for the case of a trivial surface-link is affirmatively answered. That is, a handle-irreducible summand of every stably trivial surface-link is only a trivial 2-link. By combining this result with an old result of F. Hosowaka and the author that every surface-knot with infinite cyclic fundamental group is a stably trivial surface-knot, it is concluded that every surface-knot with infinite cyclic fundamental group is a trivial (i.e., an unknotted) surface-knot.Comment: This is the part I on "Ribbonness of a stable-ribbon surface-link

Magic Wavelength for Hydrogen 1S-2S Transition

The magic wavelength for an optical lattice for hydrogen atoms that cancels the lowest order AC Stark shift of the 1S-2S transition is calculated to be 513 nm. The magnitude of AC Stark shift $\Delta E=-1.19$ kHz/(10kW/cm$^2$) and the slope $d\Delta E/d\nu = -27.7$ Hz/(GHz $\cdot$ 10 kW/cm$^2$) at the magic wavelength suggests that a stable and narrow linewidth trapping laser is necessary to achieve a deep enough optical lattice to confine hydrogen atoms in a way that gives a small enough light shift for the precision spectroscopy of the 1S-2S transition.Comment: 5 pages, 2 figure

On a Morelli type expression of cohomology classes of torus orbifolds

Let X be a complete toric variety of dimension n and \del the fan in a lattice N associated to X. For each cone \sigma of \del there corresponds an orbit closure V(\sigma) of the action of complex torus on X. The homology classes {[V(\sigma)]| \dim \sigma=k} form a set of specified generators of H_{n-k}(X,Q). Then any x\in H_{n-k}(X,Q) can be written in the form x=\sum_{\sigma\in\del_X, \dim\sigma=k}\mu(x,\sigma)[V(\sigma)]. A question occurs whether there is some canonical way to express \mu(x,\sigma). Morelli gave an answer when X is non-singular and at least for x= \T_{n-k}(X) the Todd class of X. However his answer takes coefficients in the field of rational functions of degree 0 on the Grassmann manifold G_{n-k+1}(N_Q) of (n-k+1)-planes in N_Q. His proof uses Baum-Bott's residue formula for holomorphic foliations applied to the action of complex torus on X. On the other hand there appeared several attempts for generalizing non-singular toric varieties in topological contexts. Such generalized manifolds of dimension 2n acted on by a compact n dimensional torus T are called by the names quasi-toric manifolds, torus manifolds, toric manifolds, toric origami manifolds, topological toric manifolds and so on. Similarly torus orbifold can be considered. To a torus orbifold $X$ a simplicial set \del_X called multi-fan of X is associated. A question occurs whether a similar expression to Morelli's formula holds for torus orbifolds. It will be shown the answer is yes in this case too at least when the rational cohomology ring H^*(X)_Q is generated by H^2(X)_Q. Under this assumption the equivariant cohomology ring with rational coefficients H^*_T(X,Q) is isomorphic to H^*_T(\del_X,Q), the face ring of the multi-fan \del_X, and the proof is carried out on H^*_T(\del_X,Q) by using completely combinatorial terms.Comment: This is an updated version of "On a Morelli type expression of cohomology classes of toric varieties" arXiv:1007.204