Narrow Results

We present a method of construction of vector valued bivariate fractal interpolation functions on random grids in ℝ². Examples and applications are also included.

The sensitivity analysis for a class of hidden variable fractal interpolation functions (HVFIFs) and their moments is made in the work. Based on a vector valued iterated function system (IFS) determined, we introduce a perturbed IFS and investigate the relations between the two HVFIFs generated by the IFS determined and its perturbed IFS, respectively. An explicit expression for the difference between the two HVFIFs is presented, from which, we show that the HVFIFs are not sensitive to a small perturbation in IFSs. Furthermore, we compute the moment integrals of the HVFIFs and discuss the error of moments of the two HVFIFs. An upper estimate for the error is obtained.

This paper presents a method to construct nonlinear hidden variable fractal interpolation functions (FIFs) and their stability results. We ensure that the projections of attractors of vector-valued nonlinear iterated function systems (IFSs) constructed by Rakotch contractions and function vertical scaling factors are graphs of some continuous functions interpolating the given data. We also give an explicit example illustrating obtained results. Then, we get the stability results of the constructed FIFs in the case of the generalized interpolation data having small perturbations.

In this paper, we present the construction of new nonlinear recurrent hidden variable fractal interpolation surfaces (RHVFISs) with function vertical scaling factors. We use Rakotch’s fixed point theorem which is a generalization of Banach’s fixed point theorem to get new nonlinear fractal surfaces. We construct recurrent vector-valued iterated function systems (IFSs) with function vertical scaling factors on rectangular grids and generate flexible and diverse RHVFISs which are attractors of the IFSs. We also give an explicit example to show the effectiveness of obtained results.

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the effectiveness of our results.

We argue that it is logically possible to have a sort of both reality and locality in quantum mechanics. To demonstrate this, we construct a new quantitative model of hidden variables (HV's), dubbed solipsistic HV's, that interpolates between the orthodox no-HV interpretation and nonlocal Bohmian interpretation. In this model, the deterministic point-particle trajectories are associated only with the essential degrees of freedom of the observer, and not with the observed objects. In contrast with Bohmian HV's, nonlocality in solipsistic HV's can be substantially reduced down to microscopic distances inside the observer. Even if such HV's may look philosophically unappealing to many, the mere fact that they are logically possible deserves attention.

In this paper, we present a construction of hidden variable bivariate fractal interpolation functions (HVBFIFs) with function vertical scaling factors and estimate errors of HVBFIFs on perturbation of the function vertical scaling factor. We construct HVBFIFs on the basis of the iterated function system (IFS) with function vertical scaling factors. The perturbation of the function vertical scaling factors in the IFS causes a change in the HVBFIF. An upper estimation of the errors between the original HVBFIF and the perturbed HVBFIF is given.

In the 1920s when quantum mechanics was being established, there was a famous controversy between Bohr and Einstein about the statistical interpretation of quantum mechanics. The controversy was finally concluded by an emphatic victory of Bohr, since his statistical interpretation and the duality of wave and particle explained many experimental facts without any difficulty. The “Gedanken” experiment on two entangled particles (the EPR pair) presented by Einstein, Podolski and Rosen in the course of the controversy, however, has had a continuous influence for many years on quantum communication and quantum cryptography which are being developed at the present day.

The following sections are included:

• EPR pair

• Transmission of quantum states

• Einstein's locality principle in quantum mechanics

• Measurement of two correlated particles and hidden variable theorem
  • CHSH inequality
  • Classical correlation vs. quantal correlation: a case of nuclear fission

• EPR experiment by photon pairs

• Four-photon GHZ states

• Problems