CM defect and Hilbert functions of monomial curves

In this article we consider a semigroup ring R = KGamma] of a numerical semigroup Gamma and study the Cohen- Macaulayness of the associated graded ring G(Gamma) := gr(m), (R) := circle plus(n is an element of N) m(n)/m(n+1) and the behaviour of the Hilbert function H-R of R. We define a certain (finite) subset B(Gamma) subset of F and prove that G(Gamma) is Cohen-Macaulay if and only if B(Gamma) = empty set. Therefore the subset B(Gamma) is called the Cohen-Macaulay defect of G(Gamma). Further, we prove that if the degree sequence of elements of the standard basis of is non-decreasing, then B(F) = empty set and hence G(Gamma) is Cohen-Macaulay. We consider a class of numerical semigroups Gamma = Sigma(3)(i=0) Nm(i) generated by 4 elements m(0), m(1), m(2), m(3) such that m(1) + m(2) = mo m3-so called ``balanced semigroups''. We study the structure of the Cohen-Macaulay defect B(Gamma) of Gamma and particularly we give an estimate on the cardinality |B(Gamma, r)| for every r is an element of N. We use these estimates to prove that the Hilbert function of R is non-decreasing. Further, we prove that every balanced ``unitary'' semigroup Gamma is ``2-good'' and is not ``1-good'', in particular, in this case, c(r) is not Cohen-Macaulay. We consider a certain special subclass of balanced semigroups Gamma. For this subclass we try to determine the Cohen-Macaulay defect B(Gamma) using the explicit description of the standard basis of Gamma; in particular, we prove that these balanced semigroups are 2-good and determine when exactly G(Gamma) is Cohen-Macaulay. (C) 2011 Published by Elsevier B.V.

A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems

Even though dynamic programming offers an optimal control solution in a state feedback form, the method is overwhelmed by computational and storage requirements. Approximate dynamic programming implemented with an Adaptive Critic (AC) neural network structure has evolved as a powerful alternative technique that obviates the need for excessive computations and storage requirements in solving optimal control problems. In this paper, an improvement to the AC architecture, called the �Single Network Adaptive Critic (SNAC)� is presented. This approach is applicable to a wide class of nonlinear systems where the optimal control (stationary) equation can be explicitly expressed in terms of the state and costate variables. The selection of this terminology is guided by the fact that it eliminates the use of one neural network (namely the action network) that is part of a typical dual network AC setup. As a consequence, the SNAC architecture offers three potential advantages: a simpler architecture, lesser computational load and elimination of the approximation error associated with the eliminated network. In order to demonstrate these benefits and the control synthesis technique using SNAC, two problems have been solved with the AC and SNAC approaches and their computational performances are compared. One of these problems is a real-life Micro-Electro-Mechanical-system (MEMS) problem, which demonstrates that the SNAC technique is applicable to complex engineering systems.

Anomalous X-ray scattering study of local structures in the superionic conducting class (CuI)(0.3)(Cu2O)(0.35)(MoO3)(0.35)

The anomalous X-ray scattering (AXS) method using Cu and Mo K absorption edges has been employed for obtaining the local structural information of superionic conducting glass having the composition (CuI)(0.3)(Cu2O)(0.35)(MoO3)(0.35). The possible atomic arrangements in near-neighbor region of this glass were estimated by coupling the results with the least-squares analysis so as to reproduce two differential intensity profiles for Cu and Mo as well as the ordinary scattering profile. The coordination number of oxygen around Mo is found to be 6.1 at the distance of 0.187 nm. This implies that the MoO6 octahedral unit is a more probable structural entity in the glass rather than MoO4 tetrahedra which has been proposed based on infrared spectroscopy. The pre-peak shoulder observed at about 10 nm(-1) may be attributed to density fluctuation originating from the MoO6 octahedral units connected with the corner sharing linkage, in which the correlation length is about 0.8 nm. The value of the coordination number of I- around Cu+ is estimated as 4.3 at 0.261 nm, suggesting an arrangement similar to that in molten CuI.

Solution structure of BTK-2, a novel hK(v)1.1 inhibiting scorpion toxin, from the eastern Indian scorpion Mesobuthus tamulus

The three dimensional structure of a 32 residue three disulfide scorpion toxin, BTK-2, from the Indian red scorpion Mesobuthus tamulus has been determined using isotope edited solution NMR methods. Samples for structural and electrophysiological studies were prepared using recombinant DNA methods. Electrophysiological studies show that the peptide is active against hK(v)1.1 channels. The structure of BTK-2 was determined using 373 distance restraints from NOE data, 66 dihedral angle restraints from NOE, chemical shift and scalar coupling data, 6 constraints based on disulfide linkages and 8 constraints based on hydrogen bonds. The root mean square deviation (r.m.s.d) about the averaged co-ordinates of the backbone (N, C-alpha, C') and all heavy atoms are 0.81 +/- 0.23 angstrom and 1.51 +/- 0.29 angstrom respectively. The backbone dihedral angles (phi and psi) for all residues occupy the favorable and allowed regions of the Ramachandran map. The three dimensional structure of BTK-2 is composed of three well defined secondary structural regions that constitute the alpha-beta-beta, structural motif. Comparisons between the structure of BTK-2 and other closely related scorpion toxins pointed towards distinct differences in surface properties that provide insights into the structure-function relationships among this important class of voltage-gated potassium channel inhibiting peptides. (C) 2011 Elsevier B.V. All rights reserved.

A characteristic based numerical method with tracking for nonlinear wave equations

Many physical problems can be modeled by scalar, first-order, nonlinear, hyperbolic, partial differential equations (PDEs). The solutions to these PDEs often contain shock and rarefaction waves, where the solution becomes discontinuous or has a discontinuous derivative. One can encounter difficulties using traditional finite difference methods to solve these equations. In this paper, we introduce a numerical method for solving first-order scalar wave equations. The method involves solving ordinary differential equations (ODEs) to advance the solution along the characteristics and to propagate the characteristics in time. Shocks are created when characteristics cross, and the shocks are then propagated by applying analytical jump conditions. New characteristics are inserted in spreading rarefaction fans. New characteristics are also inserted when values on adjacent characteristics lie on opposite sides of an inflection point of a nonconvex flux function, Solutions along characteristics are propagated using a standard fourth-order Runge-Kutta ODE solver. Shocks waves are kept perfectly sharp. In addition, shock locations and velocities are determined without analyzing smeared profiles or taking numerical derivatives. In order to test the numerical method, we study analytically a particular class of nonlinear hyperbolic PDEs, deriving closed form solutions for certain special initial data. We also find bounded, smooth, self-similar solutions using group theoretic methods. The numerical method is validated against these analytical results. In addition, we compare the errors in our method with those using the Lax-Wendroff method for both convex and nonconvex flux functions. Finally, we apply the method to solve a PDE with a convex flux function describing the development of a thin liquid film on a horizontally rotating disk and a PDE with a nonconvex flux function, arising in a problem concerning flow in an underground reservoir.

Adaptive optimal tuning of a general class of stable LTI systems with restricted inputs

The problem addressed is one of model reference adaptive control (MRAC) of asymptotically stable plants of unknown order with zeros located anywhere in the s-plane except at the origin. The reference model is also asymptotically stable and lacking zero(s) at s = 0. The control law is to be specified only in terms of the inputs to and outputs of the plant and the reference model. For inputs from a class of functions that approach a non-zero constant, the problem is formulated in an optimal control framework. By successive refinements of the sub-optimal laws proposed here, two schemes are finally design-ed. These schemes are characterized by boundedness, convergence and optimality. Simplicity and total time-domain implementation are the additional striking features. Simulations to demonstrate the efficacy of the control schemes are presented.

A note on a class of singular integro-differential equations

A simplified analysis is employed to handle a class of singular integro-differential equations for their solutions

Reduced Order Suboptimal Control Design for a Class of Nonlinear Distributed Parameter Systems Using POD and /spl thetas/-D Techniques

A new computational tool is presented in this paper for suboptimal control design of a class of nonlinear distributed parameter systems. First proper orthogonal decomposition based problem-oriented basis functions are designed, which are then used in a Galerkin projection to come up with a low-order lumped parameter approximation. Next, a suboptimal controller is designed using the emerging /spl thetas/-D technique for lumped parameter systems. This time domain sub-optimal control solution is then mapped back to the distributed domain using the same basis functions, which essentially leads to a closed form solution for the controller in a state feedback form. Numerical results for a real-life nonlinear temperature control problem indicate that the proposed method holds promise as a good suboptimal control design technique for distributed parameter systems.

An optimal dynamic inversion approach for controlling a class of one-dimensional nonlinear distributed parameter systems

Combining the principles of dynamic inversion and optimization theory, a new approach is presented for stable control of a class of one-dimensional nonlinear distributed parameter systems, assuming the availability a continuous actuator in the spatial domain. Unlike the existing approximate-then-design and design-then-approximate techniques, here there is no need of any approximation either of the system dynamics or of the resulting controller. Rather, the control synthesis approach is fairly straight-forward and simple. The controller formulation has more elegance because we can prove the convergence of the controller to its steady state value. To demonstrate the potential of the proposed technique, a real-life temperature control problem for a heat transfer application is solved. It has been demonstrated that a desired temperature profile can be achieved starting from any arbitrary initial temperature profile.

Single-molecule thermodynamics: the heat distribution function of a charged particle in a static magnetic field

Interest in the applicability of fluctuation theorems to the thermodynamics of single molecules in external potentials has recently led to calculations of the work and total entropy distributions of Brownian oscillators in static and time-dependent electromagnetic fields. These calculations, which are based on solutions to a Smoluchowski equation, are not easily extended to a consideration of the other thermodynamic quantity of interest in such systems-the heat exchanges of the particle alone-because of the nonlinear dependence of the heat on a particle's stochastic trajectory. In this paper, we show that a path integral approach provides an exact expression for the distribution of the heat fluctuations of a charged Brownian oscillator in a static magnetic field. This approach is an extension of a similar path integral approach applied earlier by our group to the calculation of the heat distribution function of a trapped Brownian particle, which was found, in the limit of long times, to be consistent with experimental data on the thermal interactions of single micron-sized colloids in a viscous solvent.

Lossless and lossy image compression using Boolean function minimization

A novel approach for lossless as well as lossy compression of monochrome images using Boolean minimization is proposed. The image is split into bit planes. Each bit plane is divided into windows or blocks of variable size. Each block is transformed into a Boolean switching function in cubical form, treating the pixel values as output of the function. Compression is performed by minimizing these switching functions using ESPRESSO, a cube based two level function minimizer. The minimized cubes are encoded using a code set which satisfies the prefix property. Our technique of lossless compression involves linear prediction as a preprocessing step and has compression ratio comparable to that of JPEG lossless compression technique. Our lossy compression technique involves reducing the number of bit planes as a preprocessing step which incurs minimal loss in the information of the image. The bit planes that remain after preprocessing are compressed using our lossless compression technique based on Boolean minimization. Qualitatively one cannot visually distinguish between the original image and the lossy image and the value of mean square error is kept low. For mean square error value close to that of JPEG lossy compression technique, our method gives better compression ratio. The compression scheme is relatively slower while the decompression time is comparable to that of JPEG.

Critical earthquake input power spectral density function models for engineering structures

The problem of determining optimal power spectral density models for earthquake excitation which satisfy constraints on total average power, zero crossing rate and which produce the highest response variance in a given linear system is considered. The solution to this problem is obtained using linear programming methods. The resulting solutions are shown to display a highly deterministic structure and, therefore, fail to capture the stochastic nature of the input. A modification to the definition of critical excitation is proposed which takes into account the entropy rate as a measure of uncertainty in the earthquake loads. The resulting problem is solved using calculus of variations and also within linear programming framework. Illustrative examples on specifying seismic inputs for a nuclear power plant and a tall earth dam are considered and the resulting solutions are shown to be realistic.

Perturbative growth of cosmological clustering .2. The two-point correlation

We use the BBGKY hierarchy equations to calculate, perturbatively, the lowest order nonlinear correction to the two-point correlation and the pair velocity for Gaussian initial conditions in a critical density matter-dominated cosmological model. We compare our results with the results obtained using the hydrodynamic equations that neglect pressure and find that the two match, indicating that there are no effects of multistreaming at this order of perturbation. We analytically study the effect of small scales on the large scales by calculating the nonlinear correction for a Dirac delta function initial two-point correlation. We find that the induced two-point correlation has a x(-6) behavior at large separations. We have considered a class of initial conditions where the initial power spectrum at small k has the form k(n) with 0 < n less than or equal to 3 and have numerically calculated the nonlinear correction to the two-point correlation, its average over a sphere and the pair velocity over a large dynamical range. We find that at small separations the effect of the nonlinear term is to enhance the clustering, whereas at intermediate scales it can act to either increase or decrease the clustering. At large scales we find a simple formula that gives a very good fit for the nonlinear correction in terms of the initial function. This formula explicitly exhibits the influence of small scales on large scales and because of this coupling the perturbative treatment breaks down at large scales much before one would expect it to if the nonlinearity were local in real space. We physically interpret this formula in terms of a simple diffusion process. We have also investigated the case n = 0, and we find that it differs from the other cases in certain respects. We investigate a recently proposed scaling property of gravitational clustering, and we find that the lowest order nonlinear terms cause deviations from the scaling relations that are strictly valid in the linear regime. The approximate validity of these relations in the nonlinear regime in l(T)-body simulations cannot be understood at this order of evolution.

Dielectric function and optical conductivity of TiOx (0.8

Reflection electron energy-loss spectra are reported for the family of compounds TiOx over the entire homogeneity range (0.8 < a: < 1.3). The spectra exhibit a plasmon feature on the low-energy side, while several interband transitions are prominent at higher energies. The real and imaginary parts of dielectric functions and optical conductivity for these compounds are determined using the Kramers-Kronig analysis. The results exhibit systematic behavior with varying oxygen stoichiometry.

Engineered dimer interface mutants of triosephosphate isomerase: the role of inter-subunit interactions in enzyme function and stability

The role of inter-subunit interactions in maintaining optimal catalytic activity in triosephosphate isomerase (TIM) has been probed, using the Plasmodium falciparum enzyme as a model. Examination of subunit interface contacts in the crystal structures suggests that residue 75 (Thr, conserved) and residue 13 (Cys, variable) make the largest number of inter-subunit contacts. The mutants Cys13Asp (C13D) and Cys13Glu (C13E) have been constructed and display significant reduction in catalytic activity when compared with wild-type (WT) enzyme (similar to 7.4-fold decrease in k(cat) for the C13D and similar to 3.3-fold for the C13E mutants). Analytical gel filtration demonstrates that the C13D mutant dissociates at concentrations < 1.25 mu M, whereas the WT and the C13E enzymes retain the dimeric structure. The order of stability of the mutants in the presence of chemical denaturants, like urea and guanidium chloride, is WT > Cys13Glu > Cys13Asp. Irreversible thermal precipitation temperatures follow the same order as well. Modeling studies establish that the Cys13Asp mutation is likely to cause a significantly greater structural perturbation than Cys13Glu. Analysis of sequence and structural data for TIMs from diverse sources suggests that residues 13 and 82 form a pair of proximal sites, in which a limited number of residue pairs may be accommodated.