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

Radiometric correction of LANDSAT data

The author has identified the following significant results. The six independent sensors of the multispectral band scanner are supposed to be identical; however, in actual practice, they may have different gain settings and offset factors, which result in the effect known as stripping (black lines at regular intervals) of the imagery. A simple two parameter method to correct the gain settings and offset factors of each of the sensors with respect to one sensor, taken as reference, was developed. This method assumes: (1) the response of a detector varies linearly with the radiance of radiation received, and (2) the means, as well as the standard deviations, of a reasonably large number of pixels, in a given wavelength band, are equal for each of the detectors for the radiometrically corrected data

Hidden Extra U(1) at the Electroweak/TeV Scale

We propose a simple extension of the Standard Model (SM) by adding an extra U(1) symmetry which is hidden from the SM sector. Such a hidden U(1) has not been considered before, and its existence at the TeV scale can be explored at the LHC. This hidden U(1) does not couple directly to the SM particles, and couples only to new SU(2)_L singlet exotic quarks and singlet Higgs bosons, and is broken at the TeV scale. The dominant signals at the high energy hadron colliders are multi lepton and multi b-jet final states with or without missing energy. We calculate the signal rates as well as the corresponding Standard Model background for these final states. A very distinctive signal is 6 high p_T b-jets in the final state with no missing energy. For a wide range of the exotic quarks masses the signals are observable above the background at the LHC.Comment: 19 pages, 5 figure

Evidence for the Collective Nature of the Reentrant Integer Quantum Hall States of the Second Landau Level

We report an unexpected sharp peak in the temperature dependence of the magnetoresistance of the reentrant integer quantum Hall states in the second Landau level. This peak defines the onset temperature of these states. We find that in different spin branches the onset temperatures of the reentrant states scale with the Coulomb energy. This scaling provides direct evidence that Coulomb interactions play an important role in the formation of these reentrant states evincing their collective nature

Particle-hole Asymmetry of Fractional Quantum Hall States in the Second Landau Level of a Two-dimensional Hole System

We report the first unambiguous observation of a fractional quantum Hall state in the Landau level of a two-dimensional hole sample at the filling factor $\nu=8/3$. We identified this state by a quantized Hall resistance and an activated temperature dependence of the longitudinal resistance and found an energy gap of 40 mK. To our surprise the particle-hole conjugate state at filling factor $\nu=7/3$ in our sample does not develop down to 6.9 mK. This observation is contrary to that in electron samples in which the 7/3 state is typically more stable than the 8/3 state. We present evidence that the asymmetry between the 7/3 and 8/3 states in our hole sample is due to Landau level mixing

Integrated Electronic Transport and Thermometry at milliKelvin Temperatures and in Strong Magnetic Fields

We fabricated a He-3 immersion cell for transport measurements of semiconductor nanostructures at ultra low temperatures and in strong magnetic fields. We have a new scheme of field-independent thermometry based on quartz tuning fork Helium-3 viscometry which monitors the local temperature of the sample's environment in real time. The operation and measurement circuitry of the quartz viscometer is described in detail. We provide evidence that the temperature of two-dimensional electron gas confined to a GaAs quantum well follows the temperature of the quartz viscometer down to 4mK

Studies on mechanical behavior of glass epoxy composites with induced defects and correlations with NDT characterization parameters

Degradation in mechanical properties (compression, flexural and ILS) of glass epoxy composite laminates with induced defects (simulating delaminations) was studied. The defects were characterized by using the A-scan ultrasonic pulse-echo technique. The two A-scan parameters, viz. the back wall echo amplitude and time of flight, were followed respectively, as functions of the defect size (diameter) and its location in the laminate thickness. The mechanical properties of laminates were evaluated destructively for different defect sizes and locations. An attempt has been made to express the mechanical properties in terms of the two NDT scan parameters characterizing the defects and empirical equations presente