Large weight code words in projective space codes

AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points

Graded persistence diagrams and persistence landscapes

We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram correspond to the local maxima and minima, respectively, of the k-th persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between k-th graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the k-th graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams.Comment: accepted for publication in Discrete and Computational Geometr

The Complexity of Orbits of Computably Enumerable Sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, \E, such that the question of membership in this orbit is $\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of \E is \wock +1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of \E; for all finite $\alpha \geq 9$, there is a properly $\Delta^0_\alpha$ orbit (from the proof). A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi

Magnetic compensation in the bimetallic oxalates and the cerium volume collapse

In this thesis the author reports his collaborative efforts on two distinct areas of research that has been conducted. The first part of the thesis pertains to the author and his collaborators research on a particular class of organic magnets called the bimetallic oxalates. The main theme of this research was to predict magnetic compensation (magnetization reversal) in unsynthesized bimetallic oxalate structures, motivated by experiments which showed that Fe(II)Fe(III) exhibited magnetic compensation. In addition it was known that a large amount of anisotropy was present in the bimetallic oxalate structure which resulted from the intermediate oxalate molecules between the transition metal ions which would drastically change the angular momentum of the transition metals. Consequently, because of the large anisotropy, we predicted that, if neutron diffraction measurements were performed on these materials, a spin-wave gap would exist of the order of 7.8 meV. The second half of this thesis is devoted to the author\u27s and his collaborators\u27 research on the cerium volume collapse. Until 2004 the collapse was largely believed to be understood as the result of Kondo screening of the local moment in cerium. However in 2004 it was realized that, in addition to a large Kondo effect driving the cerium volume collapse, the phonon frequency was very different between the large and small volume phases, and consequently the change in phonon frequency was the direct result of large electron-phonon correlations. This upset the apple cart of Kondo correlation being solely responsible for the volume collapse in cerium, and the change in phonon frequency must be accounted for to accurately describe the cerium volume collapse. To this end the author and his collaborators\u27 developed a model which would include both of the correlations (Kondo and phononic) in the volume collapse. To analyze this model we used Dynamical Mean Field Theory in conjunction with Continuous Time Quantum Monte Carlo. What we found in our simulations was that the small volume Kondo phase was drastically influenced by the presence of the electron-phonon correlations

Nanoclay syntactic foam composites

Syntactic foams are composite materials in which the matrix phase is reinforced with hollow particles called microballoons. They possess low moisture absorption, low thermal conductivity and high damage tolerance because of their compositions. Traditionally, syntactic foams are used in many high strength applications such as in aerospace and marine industries, thus there is a need to achieve both high compressive strength and high fracture strain with minimal increase in density. This research studies the effect of nanoclay on the high strain rate mechanical properties of syntactic foams. Nanoclay reinforced syntactic foams are fabricated by adding 1, 2 and 5% volume fraction of Nanomer I.30E nanoclay in syntactic foams having 10, 30 and 60% microballoon volume fraction. Transmission electron microscopy is performed to determine the dispersion of nanoclay in matrix. To compare the effect of nanoclay, plain syntactic foams without nanoclay are fabricated with same microballoon volume fraction. Two types of glass microballoons, S22 and K46, having different wall thickness are used in plain and nanoclay syntactic foams. High strain rate tests using split Hopkinson pressure bar (SHPB)apparatus are conducted on all types of plain and nanoclay syntactic foams and dynamic strength and modulus values are calculated. Also, quasi-static tests are conducted using MTS-810 machine and results are compared with dynamic SHPB results. The results demonstrated the importance of strain rate, nanoclay volume fraction and microballoon wall thickness in determination of syntactic foam properties. It is found that inclusion of 1% nanoclay gives the optimum enhancement in strength and modulus of nanoclay syntactic foams at all three microballoon volume fractions. The behavior of strength and modulus dependence on nanoclay volume fraction is found to be similar in both composite foams having S22 and K46 microballoons. Specimens exhibited higher strength and modulus at high strain rate than at lower strain rates. Based on stress-strain behavior of composite foams, energy absorption is also calculated. It is found that thicker walled microballoons (K46) composite foams showed higher strength, modulus and energy absorption than those with thin walled (S22) microballoons. Scanning electron microscopy is performed to study the fracture behavior under different loading rates