Landau diamagnetism revisited

The problem of diamagnetism, solved by Landau, continues to pose fascinating issues which have relevance even today. These issues relate to inherent quantum nature of the problem, the role of boundary and dissipation, the meaning of thermodynamic limits, and above all, the quantum-classical crossover occasioned by environment-induced decoherence. The Landau Diamagnetism provides a unique paradigm for discussing these issues, the significance of which are far-reaching. Our central result is a remarkable one as it connects the mean orbital magnetic moment, a thermodynamic property, with the electrical resistivity, which characterizes transport properties of materials.Comment: 4 pages, 1 figur

Strings, Junctions and Stability

Identification of string junction states of pure SU(2) Seiberg-Witten theory as B-branes wrapped on a Calabi-Yau manifold in the geometric engineering limit is discussed. The wrapped branes are known to correspond to objects in the bounded derived category of coherent sheaves on the projective line \cp{1} in this limit. We identify the pronged strings with triangles in the underlying triangulated category using Pi-stability. The spiral strings in the weak coupling region are interpreted as certain projective resolutions of the invertible sheaves. We discuss transitions between the spiral strings and junctions using the grade introduced for Pi-stability through the central charges of the corresponding objects.Comment: 15 pages, LaTeX; references added. typos correcte

Birefringence analysis of multilayer leaky cladding optical fibre

We analyse a multilayer leaky cladding (MLC) fibre using the finite element method and study the effect of the MLC on the bending loss and birefringence of two types of structures: (i) a circular core large-mode-area structure and (ii) an elliptical-small-core structure. In a large-mode-area structure, we verify that the multilayer leaky cladding strongly discriminates against higher order modes to achieve single-mode operation, the fibre shows negligible birefringence, and the bending loss of the fibre is low for bending radii larger than 10 cm. In the elliptical-small-core structure we show that the MLC reduces the birefringence of the fibre. This prevents the structure from becoming birefringent in case of any departures from circular geometry. The study should be useful in the designs of MLC fibres for various applications including high power amplifiers, gain flattening of fibre amplifiers and dispersion compensation.Comment: 18 page

On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

Recently, Gupta et.al. [GKKS2013] proved that over Q any $n^{O(1)}$-variate and $n$-degree polynomial in VP can also be computed by a depth three $\Sigma\Pi\Sigma$ circuit of size $2^{O(\sqrt{n}\log^{3/2}n)}$. Over fixed-size finite fields, Grigoriev and Karpinski proved that any $\Sigma\Pi\Sigma$ circuit that computes $Det_n$ (or $Perm_n$) must be of size $2^{\Omega(n)}$ [GK1998]. In this paper, we prove that over fixed-size finite fields, any $\Sigma\Pi\Sigma$ circuit for computing the iterated matrix multiplication polynomial of $n$ generic matrices of size $n\times n$, must be of size $2^{\Omega(n\log n)}$. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the $n^{O(1)}$-variate and $n$-degree polynomials in VP by depth 3 circuits of size $2^{o(n\log n)}$. The result [GK1998] can only rule out such a possibility for depth 3 circuits of size $2^{o(n)}$. We also give an example of an explicit polynomial ($NW_{n,\epsilon}(X)$) in VNP (not known to be in VP), for which any $\Sigma\Pi\Sigma$ circuit computing it (over fixed-size fields) must be of size $2^{\Omega(n\log n)}$. The polynomial we consider is constructed from the combinatorial design. An interesting feature of this result is that we get the first examples of two polynomials (one in VP and one in VNP) such that they have provably stronger circuit size lower bounds than Permanent in a reasonably strong model of computation. Next, we prove that any depth 4 $\Sigma\Pi^{[O(\sqrt{n})]}\Sigma\Pi^{[\sqrt{n}]}$ circuit computing $NW_{n,\epsilon}(X)$ (over any field) must be of size $2^{\Omega(\sqrt{n}\log n)}$. To the best of our knowledge, the polynomial $NW_{n,\epsilon}(X)$ is the first example of an explicit polynomial in VNP such that it requires $2^{\Omega(\sqrt{n}\log n)}$ size depth four circuits, but no known matching upper bound

The sphere packing problem in dimension 24

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.Comment: 17 page

Theoretical studies on structural and decay properties of Z=119Z=119 superheavy nuclei

In this manuscript, we analyze the structural properties of $Z=119$ superheavy nuclei in the mass range of 284 $\le$ A $\le$ 375 within the framework of deformed relativistic mean field theory (RMF) and calculate the binding energy, radii, quadrupole deformation parameter, separation energies and density profile. Further, a competition between possible decay modes such as $\alpha-$decay, $\beta-$decay and spontaneous fission (SF) of the isotopic chain of $Z=119$ superheavy nuclei under study is systematically analyzed within self-consistent relativistic mean field model. Moreover, our analysis confirmed that $\alpha-$decay is restricted within the mass range 284 $\leq$ A $\leq$ 296 and thus being the dominant decay channel in this mass range. However, for the mass range 297 $\leq$ A $\leq$ 375 the nuclei are unable to survive fission and hence SF is the principal mode of decay for these isotopes. There is no possibility of $\beta-$decay for the considered isotopic chain. In addition, we forecasted the mode of decay $^{284-296}$119 as one $\alpha$ chain from $^{284}$119 and $^{296}$119, two consistent $\alpha$ chains from $^{285}$119 and $^{295}$119, three consistent $\alpha$ chains from $^{286}$119 and $^{294}$119, four consistent alpha chains from $^{287}$119, six consistent alpha chains from $^{288-293}$119. Also from our analysis we inferred that for the isotopes $^{264-266,269}$Bh both $\alpha$ decay and SF are equally competent and can decay via either of these two modes. Thus, such studies can be of great significance to the experimentalists in very near future for synthesizing $Z=119$ superheavy nuclei.Comment: 14 pages, 6 figures. arXiv admin note: text overlap with arXiv:1611.00232, arXiv:1704.0315