Statistical approximation by positive linear operators on modular spaces

In this paper, we investigate the problem of statistical approximation to a function by means of positive linear operators defined on a modular space. Especially, in order to get more powerful results than the classical aspects we mainly use the concept of statistical convergence. A non-trivial application is also presented.     

Pointwise Comparison of PCM and Σ Δ Quantization

A quantitative comparison of Pulse Code Modulation (PCM) and Sigma–Delta (Σ Δ) quantization methods is made in the setting of finite frames. Frames allow for redundant, overcomplete signal decompositions. PCM and Σ Δ are two industry-standard quantization methods, and the setting of finite frames is appropriate for a host of modern applications. Previous results for this comparison are known for upper error bounds, where Σ Δ performs better in the setting of frames, as opposed to orthonormal bases, where PCM is optimal. We answer the following question: For which signals x is the PCM error, that is, the norm of the difference between x and its PCM approximant, less than the Σ Δ error? We prove that, typically, in the setting of frames, Σ Δ outperforms PCM, but not always.     

A link between combinatorial designs and three-weight linear codes

In this paper, we show that partial geometric designs can be constructed from certain three-weight linear codes, almost bent functions and ternary weakly regular bent functions. In particular, we show that existence of a family of partial geometric difference sets is equivalent to existence of a certain family of three-weight linear codes. We also provide a link between ternary weakly regular bent functions, three-weight linear codes and partial geometric difference sets.     

Statistical convergence of double-complex Picard integral operators

In this work, we study the statistical approximation properties of the double-complex Picard integral operators. We also show that our statistical approach is more applicable than the classical one.     

A principle of linear covariance for quantum mechanics and the electronic structure theory of molecules and other atom clusters

Quantum mechanical equations should look the same in any non-orthonormal or orthonormal basis frame when properly formulated so as to be fully covariant under the largest group indicated. Non-orthonormal frames are crucial especially for the quantum theory of chemistry. Various methods such as valence-bond, localized orbital, molecular orbital, etc. result from a single formulation using the principle of linear covariance which is stated, proved, and exemplified. Molecular quantities with the full inclusion of overlaps are derived with the same ease as without overlaps.     

Recognition-mediated assembly of nanoparticles into micellar structures with diblock copolymers

Polystyrene-based diblock copolymers, featuring diaminotriazine functionality on one of the blocks were used to assemble complementary uracil-functionalized nanoparticles into micellar aggregates. The size of these self-assembled aggregates was controlled by block length, as determined in solution (using dynamic light scattering), and in thin films (using transmission electron microscopy).     

Directed graphs of structurally stable potential energy surfaces representing a-priori reaction pathways

An intrinsic property of potential energy surfaces (PES) that holds within the adiabatic approximation is established: its structural stability.We derive the condition that ensures this property: There cannot be any integral curve of the gradient field of the PES that connects two classical transition state configurations without passing through another critical configuration in between.Under this situation, we can establish a one-to-one correspondence: a whole class of adiabatic PES defining one reaction mechanism is associated to a directed graph. Thus, the problem of finding a-priori pathways involving a given number m of chemical species narrows down to a classifying certain directed graphs with m sinks. The combinatorial method is derived in this paper.Detailed examples on a-priori pathways for degenerate thermal rearrangements and on 1-2 hydrocarbon shifts are worked out and found in agreement with experimental evidence.     

Constancy maximization based weight optimization in high dimensional model representation

This work focuses on the weight function optimization in high dimensional model representation (HDMR) via constancy maximization. There are a lot of circumstances where HDMR’s weight function becomes completely flexible in its factors. The univariate coordinate changes which can be constructed to produce nonnegative factors in the integrands of HDMR component, are perhaps the most important ones of these cases. Here, the weight function is considered as the square of a linear combination of certain basis functions spanning an appropriately chosen Hilbert space. Then, the coefficients of these linear combinations are determined to maximize the HDMR’s constant term contribution to the function. Although the resulting equations are nonlinear we could have been able to approximate the solutions by using recently proven fluctuationlessness theorem on matrix representations appearing in the equations.     

Numerical simulation of minimum B‐jumps at abrupt drops

A numerical simulation of minimum B‐jumps in horizontal rectangular channels having an abrupt drop is given. Gradually varied, steady, supercritical flow is assumed as the initial condition. An unsteady flow is created by increasing the downstream depth. One‐dimensional, unsteady Saint‐Venant equations are solved by using the MacCormack and the dissipative two–four explicit finite difference schemes. The steady flow solution is obtained by treating the time variable as an iteration parameter and letting the solution converge to the steady state. The abrupt drop is treated as an interior boundary and solved by the method of characteristics. The results are compared with experimental and analytical studies. Copyright © 2007 John Wiley & Sons, Ltd.