Unlimited Accuracy Research Articles

Practical modelling of monopolar axonal stimulation

Three models describing monopolar electrical stimulation of unmyelinated axons in large, uniformly conducting volumes are presented. The first is a representation of the axon as a continuous and straight electric cable of finite length with sealed ends. It has a power-series solution for the steady state which after truncation beyond the fifth term provides an accurate reflection of the imposed membrane potential for all parts of the axon except, under certain conditions, near terminals and directly opposite a closely apposed electrode. At these points, the inclusion of higher-order terms improves the accuracy quite slowly — II terms are sometimes still unsatisfactory. The second is a steady-state solution, in terms of higher transcendental functions, for the point opposite the stimulating electrode in a similar but infinitely long cable. This turns out to be practical for estimating the membrane potential at the site of cathodal excitation under the majority of realistic geometrical and electrical parameters, and consequently complements the first method. It may be calculated with ease from mathematical tables. The third, a simulation of the cable by means of discrete electrical components (compartments), can give the complete distribution of membrane potential as a function of time with potentially unlimited accuracy. However, it takes special computer programs to set up and solve either the steady-state determinant or, if transients are desired, the time-dependent differential equation. Calculation of the voltage distribution with one of the analytical methods will often be faster for a reasonable level of accuracy, since their algorithms are independent of the total number of points introduced.

Quantum theory based on common relativity and fuzzy matrices as operators

Four-dimensional commutation relations and Hamiltonian dynamics with constraints are discussed. The invariant volume element |b|d3x/co in common relativity is employed to give a meaning to the invariant probability |ψI(xμ)|2 d4x in quantum theory. In fuzzy quantum mechanics, the co-ordinate and the momentum variables are “fuzzy continuous matrices” which can never be diagonalized. They have fuzzy values and fuzzy basic vectors which satisfy a new closure relation. Fuzzy matrices satisfy a new law of matrix multiplication. The values of a fuzzy dynamical variable cannot be experimentally measured with unlimited accuracy. In particular, a physically realizable wave packet must satisfy Δqmin ≈R≠0, whereR is a radical length, which characterizes the fuzziness of space at short distances. The modification of the superposition principle due to the radical lengthR≲10−18 cm is also discussed. The results do not contradict experiments.

The inverse problem of resistivity sounding

The electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two‐layer systems. This new representation is called the bilayer expansion for the Green’s function. It enables the forward problem of resistivity sounding to be solved very efficiently, even for complicated profiles. Also, the bilayer expansion facilitates the solution of the resistivity inverse problem: the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model. Specifically, I consider models comprising uniformly conducting layers of equal thickness; for a given finite data set a quadratic program can be used to find the best‐fitting model in this class for any specified thickness. As the thickness is reduced, models of this kind can approximate arbitrary profiles with unlimited accuracy. If there is a model that satisfies the data well, there are other models equally good or better whose variation takes place in an infinitesimally thin zone near the surface, below which there is a perfectly conducting region. This extraordinary class of solutions underscores the serious ambiguity in the interpretation of apparent resistivity data. It is evident that strong constraints from outside the electrical data set must be applied if reliable solutions are to be discovered. Previous work seems to have given a somewhat overly optimistic impression of the resolving abilities of this kind of data. I consider briefly a regularization technique designed to maximize the smoothness of models found with the bilayer inversion.

Four-dimensional symmetry from a broad viewpoint

Four-dimensional symmetry (from a broad viewpoint) actually allows for a new principle of universal probability, which dictates the distribution of quantized oscillators of all free fields, so that one can overcome divergence difficulties without upsetting unitarity. In order to gain a new perspective, we trace the origin of this principle in quantum mechanics. We find that it is related to the existence of a radical lengthR and the impossibility of measuring particle positions with unlimited accuracy. We formulate «R-inherent quantum mechanics» which incorporates these ideas. We derive a newR-inherent uncertainty relation which, in addition to the usual restriction, also implies that the ultimate position accuracy is Δq ult≈R rather than Δq ult=0. It indicates a further departure from the classical physics. Thep andq representations are connected by a new transform which reduces to the Fourier transform asR→0. The usual Hilbert space with symmetry betweenp andq representations is no longer adequate to be the mathematical foundation. It has to be the generalized to a «R-inherent Hilbert space» whose co-ordinates, in terms of the momentum wave functions, appear to be «curved» with the universal probability function as the «metric». In the «q-representation» we have the usual momentum−iJ∂/∂q and co-ordinateq. However, in thep-representation, we have the momentump and the co-ordinateq -iJ∂/∂q-iR p/2(p 2+m 2)1/2. This indicates a violation of theq-q symmetry. The theory indicates that a particle can never be confined in a volume much smaller than 4πR 3/3 and that matter cannot be squeezed into a singularity in space with infinite mass density by any force.

Quantum chemistry by random walk: Method of successive corrections

The random-walk method of solving the Schrödinger equation is reformulated to allow the direct calculation of the difference δ between a true wavefunction ψ and a trial wave-function ψ 0. For a trial wavefunction from any source the difference δ may be calculated and used to correct the trial wavefunction. Successive calculations offer the possibility of further corrections and wavefunctions of unlimited accuracy. The calculation of δ is illustrated for the cases of the particle-in-a-box and the hydrogen atom. Energies are determined directly from the random-walk calculations and indirectly from computation of the expectation values for the corrected wavefunctions.

Antenna pattern synthesis based on optimization in a probabilistic sense

Antenna optimization theory is usually formulated on the premise that all parameters could be adjusted with unlimited accuracy. Since no practical antenna can be built with absolute precision, only optimization in some probabilistic sense of a random performance index appears to be meaningful. The method also generalizes and gives meaning to the concept of regularization used by others. An approximate solution to the probability distribution of the random performance index is obtained. From this several possible senses of optimization may be defined. Several examples in the synthesis of a two-dimensional secant pattern are studied in detail. In general, it may be impractical to build an array with deterministically optimum currents but also next to impossible to even compute the currents accurately. A detailed numerical analysis is given to show possible sources of errors.

A set of procedures making real arithmetic of unlimited accuracy possible within ALGOL 60

The facilities to perform floating point arithmetic of unlimited accuracy and to continuously control the round-off error are often found desirable. The paper describes how the authors have implemented this feature using the ALGOL 60 procedure concept.

The Structure of Space and the Formalism of Relativistic Quantum Theory. I

The structure of a Euclidean space can be approached, with an unlimited accuracy, by a part of a maximally ordered finite linear space. Accordingly, all the physical theories based on the space-time continuum can also be considered in such a finite space-time. The finiteness of the underlying space makes also some new kinds of theories possible. Among them is a purely group theoretical formalism of relativistic quantum theory, including a free-particle theory as well as a group formalism of interaction of particles. The free-particle formalism of a finite space-time is considered here (Part I). An essential difference in comparison with the formalism of continuous space-time is that there is, as a consequence of the relations of Euclidicity to be imposed on observable 4-vectors, a nontrivial spectrum of momentum, mass, and energy in a finite geometry.

Six-Digit Fraction Conversion from Decimal to Octal with Unlimited Accuracy

Six-Digit Fraction Conversion from Decimal to Octal with Unlimited Accuracy

Remarks on the measurability of electromagnetic fields

Indeterminacy relations are derived, which express the facts, that only 4-dimensional averages of electromagnetic fields are measurable and that also an average cannot be measured with unlimited accuracy simultaneously with the position of the averaging region in space-time continuum. Essentially in showing this is the use of test bodies, which obey Heisenbergs indeterminacy relations and possess a finite extension, finite mass and finite electric charge.

SCIENCE, COMMON SENSE AND DECENCY*

We must stress that the article of Dr. Langmuir deals with two entirely different worlds, the physico-mathematical world of exact sciences, where the aristotelian principles do not apply fundamentally; and the human world, still ruled by antiquated aristotelian methods, which are in the main responsible for the present-day tragedies and confusion. It is not enough to contemplate the limitations of the methods of exact sciences. Our work is introductory to that new world in which physico-mathematical methods can be generalized to human problems, where those limitations will be non-existent. --Excerpted from the Foreword by Alfred Korzybski UP TO THE beginning of the present century one of the main goals of science was to discover natural laws. This was usually accomplished by making experiments under carefully controlled conditions and observing the results. Most experiments when repeated under identical conditions gave the same results. The scientist, through his own experiments, or from previous knowledge based on the work of others, usually developed some theory or explanation of the results of his experiments. In the beginning this might be a mere guess or hypothesis which he would proceed to test by new types of experiments. If a satisfactory theory is obtained which seems in accord with all the data and with other known facts, the solution or goal of the investigation was considered to have been reached. A satisfactory theory should make possible the prediction of new relationships or the forecasting of the results of new experiments under different conditions. The usefulness of the theory lies just in its ability to predict the results of future experiments. The extraordinary accomplishments of the great mathematical physicists in applying Newton's laws to the motions of the heavenly bodies gave scientists of more than a century ago the conviction that all natural phenomena were determined by accurate relations between cause and effect. If the positions, the velocities and the masses of the heavenly bodies were given it was possible to predict with nearly unlimited accuracy the position of the bodies at any future time. The idea of causation, or a necessary relation of cause and effect, has long been embedded in the minds of men. The recognized responsibility of the criminal for his acts, the belief of the value of education, and thousands of words in our language all show how implicitly we believe in cause and effect. The teachings of classical science, that is, the science up to 1900, all seem to reinforce this idea of causation for all phenomena. Philosophers, considering many fields other than science, were divided in their opinions. Many went so far as to believe that everything was absolutely fixed by the initial conditions of the universe and that free will or choice was impossible. Others thought that cause and effect relations were mere illusions. From the viewpoint of the early classical scientist, the proper field for science was unlimited. Given sufficient knowledge, all natural phenomena, even human affairs, could be predicted with certainty. Ampere, for example, stated that if he were given the positions and velocities of all the atoms in the universe it should be possible theoretically to determine the whole future history of the universe. Practically, of course, such predictions would be impossible because we could never hope to get the necessary knowledge nor the time to carry out such elaborate calculations. A little later, scientists developed the kinetic theory of gases according to which the molecules of a gas are moving with high velocity and are continually colliding with one another. They found that the behavior of gases could be understood only by considering the average motions of the individual molecules. The particular motion of a single molecule was of practically no importance. They were thus taught the value of statistical methods, like those which insurance companies now use to calculate the probable number of its policy holders that will die within a year. …

Science, Common Sense and Decency

We must stress that the article of Dr. Langmuir deals with two entirely different worlds, the physico-mathematical world of exact sciences, where the aristotelian principles do not apply fundamentally; and the human world, still ruled by antiquated aristotelian methods, which are in the main responsible for the present-day tragedies and confusion. It is not enough to contemplate the limitations of the methods of exact sciences. Our work is introductory to that new world in which physico-mathematical methods can be generalized to human problems, where those limitations will be non-existent. --Excerpted from the Foreword by Alfred Korzybski UP TO THE beginning of the present century one of the main goals of science was to discover natural laws. This was usually accomplished by making experiments under carefully controlled conditions and observing the results. Most experiments when repeated under identical conditions gave the same results. The scientist, through his own experiments, or from previous knowledge based on the work of others, usually developed some theory or explanation of the results of his experiments. In the beginning this might be a mere guess or hypothesis which he would proceed to test by new types of experiments. If a satisfactory theory is obtained which seems in accord with all the data and with other known facts, the solution or goal of the investigation was considered to have been reached. A satisfactory theory should make possible the prediction of new relationships or the forecasting of the results of new experiments under different conditions. The usefulness of the theory lies just in its ability to predict the results of future experiments. The extraordinary accomplishments of the great mathematical physicists in applying Newton's laws to the motions of the heavenly bodies gave scientists of more than a century ago the conviction that all natural phenomena were determined by accurate relations between cause and effect. If the positions, the velocities and the masses of the heavenly bodies were given it was possible to predict with nearly unlimited accuracy the position of the bodies at any future time. The idea of causation, or a necessary relation of cause and effect, has long been embedded in the minds of men. The recognized responsibility of the criminal for his acts, the belief of the value of education, and thousands of words in our language all show how implicitly we believe in cause and effect. The teachings of classical science, that is, the science up to 1900, all seem to reinforce this idea of causation for all phenomena. Philosophers, considering many fields other than science, were divided in their opinions. Many went so far as to believe that everything was absolutely fixed by the initial conditions of the universe and that free will or choice was impossible. Others thought that cause and effect relations were mere illusions. From the viewpoint of the early classical scientist, the proper field for science was unlimited. Given sufficient knowledge, all natural phenomena, even human affairs, could be predicted with certainty. Ampere, for example, stated that if he were given the positions and velocities of all the atoms in the universe it should be possible theoretically to determine the whole future history of the universe. Practically, of course, such predictions would be impossible because we could never hope to get the necessary knowledge nor the time to carry out such elaborate calculations. A little later, scientists developed the kinetic theory of gases according to which the molecules of a gas are moving with high velocity and are continually colliding with one another. They found that the behavior of gases could be understood only by considering the average motions of the individual molecules. The particular motion of a single molecule was of practically no importance. They were thus taught the value of statistical methods, like those which insurance companies now use to calculate the probable number of its policy holders that will die within a year. …

Quick and Easy Methods of Calculating A Simple Explanation of the Theory and Use of the Slide-Rule, Logarithms, &c

THE author makes his title, “Quick and Easy Methods of Calculating”—at least, that is all that is in large print on the title-page; but the binder calls it, on the outside of the book, “The Slide-Rule.” The binder is right. The author gives a very short account of some methods of shortened arithmetic, in which he points out that it is unnecessary to work out the results of an observation with very great or unlimited accuracy when the observation itself is subject to well-known possible errors. He might have traced the connection between the desired accuracy of the arithmetic and the probable accuracy of the observation as dependant on its form, but he has not. There is a simple non-algebraical and very clear explanation of logarithms; then the real object of the book, an explanation of the slide-rule, follows. As in all explanations of the slide-rule that are published, however clear and obvious they may be to the user of the slide-rule, there is, of necessity perhaps, such an amount of detail and of rule as to possibly scare any would-be user of tnis invaluable instrument with the fear that he could not hope to remember it all. The writer of this notice has always felt that this difficulty can only be overcome by half an hour's personal explanation, in which case a book becomes unnecessary; however, for those who cannot meet with this personal assistance, the little book before us is clear, logical and accurate. A great number of examples, mainly derived from the engineering laboratory, are given, which serve both to show the great scope of the slide-rule and as exercises in its use. Quick and Easy Methods of Calculating. A Simple Explanation of the Theory and Use of the Slide-Rule, Logarithms, &c. With numerous Examples worked out by Robert Gordon Blaine., &c. (London: E. and F. N. Spon, Ltd. New York: Spon and Chamberlain, 1898.)