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PubMed2003

Use of the mathematical principle of inversion in young children.

An important issue in the development of mathematical cognition is the extent to which children use and understand fundamental mathematical concepts. We examined whether young children successfully use the principle of inversion and, if so, whether they do so based on qualitative identity, length, or quantity. Twenty-four preschool children and 24 children in Grade 1 were presented with three-term inversion problems (e.g., 3+2-2) and standard problems of similar magnitude (e.g., 2+4-3). Problems were presented in three conditions to determine whether children used inversion at all and, if so, whether their decisions were based on quantitative or nonquantitative features of the problems. Both preschool and Grade 1 children showed evidence of using inversion in a fully quantitative manner, indicating that this principle is available in some form prior to extensive formal instruction in arithmetic.

Child↗
PubMed1996

Nuclear medicine and mathematics.

The purpose of this review is not to present a comprehensive description of all the mathematical tools used in nuclear medicine, but to emphasize the importance of the mathematical method in nuclear medicine and to elucidate some of the mathematical concepts currently used. We can distinguish three different areas in which mathematical support has been offered to nuclear medicine: physiology, methodology and data processing. Nevertheless, the boundaries between these areas can be indistinct. It is impossible in a single article to give even an idea of the extent and complexity of the procedures currently used in nuclear medicine, such as image processing, reconstruction from projections and artificial intelligence. These disciplines do not belong to nuclear medicine: they are already branches of engineering, and my interest will reside simply in revealing a little of the elegance and the fantastic potential of these new "allies" of nuclear medicine. In this review the mathematics of physiological interpretation and methodology are considered together in the same section. General aspects of data-processing methods, including image processing and artificial intelligence, are briefly analysed. The mathematical tools that are most often used to assist the interpretation of biological phenomena in nuclear medicine are considered; these include convolution and deconvolution methods, Fourier analysis, factorial analysis and neural networking.

Artificial Intelligence↗
PubMed1997

On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity.

We introduce the mathematical concept of multifractality and describe various multifractal spectra for dynamical systems, including spectra for dimensions and spectra for entropies. We support the study by providing some physical motivation and describing several nontrivial examples. Among them are subshifts of finite type and one-dimensional Markov maps. An essential part of the article is devoted to the concept of multifractal rigidity. In particular, we use the multifractal spectra to obtain a "physical" classification of dynamical systems. For a class of Markov maps, we show that, if the multifractal spectra for dimensions of two maps coincide, then the maps are differentiably equivalent. (c) 1997 American Institute of Physics.

Journal Article↗
PubMedDate not supplied

World hypotheses and the evolution of integrative medicine: combining categorical diagnoses and cause-effect interventions with whole systems research and nonvisualizable (seemingly "impossible") healing.

It has been proposed that to understand (1) the evolution of science and medicine, and (2) the integration of conventional, complementary and alternative medicine, it is essential to consider at least eight universal implicit meta-cognitive hypotheses. It has been suggested that these implicit "world" hypotheses can be applied in every discipline of science. The present paper reviews the eight world hypotheses and proposes an additional hypothesis, termed the nonvisualizable or "Nth" world hypothesis (adopting the mathematical concept of "N"; eg, as in N dimensional space). Drawing on contemporary mathematics and quantum physics, we propose that certain theories and data-by their inherent nature-can not be visualized, and therefore may seem "unimaginable" and "impossible" (if not "unbelievable"), even though they are real. Certain seemingly anomalous observations in mind-body and energy medicine, including areas historically labeled as parapsychology or spiritual energy healing, often elicit strongly skeptical and dismissive reactions. We propose that these skeptical and dismissive reactions to purportedly impossible (yet logical) theories and seemingly unbelievable (yet replicable) data can be tempered when the Nth world hypothesis is understood and incorporated. Integrity in evidence-based science and medicine may require that scientists and nonscientists alike develop comfort and humility in accepting the human mind's restricted ability to envision and imagine certain nonvisualizable-yet fundamental and real-concepts and effects, as illustrated in contemporary physics and complementary and alternative medicine.

Clinical Medicine↗
PubMed1988

Can quantum-bumps in photoreceptors be reconstructed from noise-data?

The method of reconstructing quantum bumps in photoreceptor cells from noise data by making use of shot noise theory is critically reviewed. The application of this method produces results irrespective of whether the conditions for reconstructing bumps by the method are satisfied or not and even irrespective of whether at high stimulus intensities quantum bumps exist or not. We argue that at high intensities the concept of quantum bumps indeed becomes physically meaningless and degenerates to a purely mathematical concept. In order to investigate the meaning of the results of the reconstruction method, we submit it to a test model for which bumps and single channel opening events can be evaluated analytically. By comparing the analytical results of the test model with that of the reconstruction method applied to the test model we find: (1) even at low intensities, the reconstructed bump values deviate from the analytical results by up to an order of magnitude due to the variability of the bumps, (2) at high intensities, the reconstruction method produces single channel opening events rather than anything like a quantum bump. We also find, however, that there is no continuous transition from a bump at low intensities to a single channel event at high intensities.

Animals↗
PubMed1994

Nucleic acid structure analysis. Mathematics for local Cartesian and helical structure parameters that are truly comparable between structures.

Analyzing nucleic acid structures in a comparable manner has become increasingly important as the number of solved structures has increased. This paper presents the concepts, mathematics, theorems, and proofs that form the basis of a new program to analyze three-dimensional DNA and RNA structures. The approach taken here provides numerical data in accordance with guidelines set at a 1988 EMBO workshop. Mathematical definitions are provided for all local structural parameters described in the guidelines. The definitions satisfy the guideline requirements while preserving the original physical intuition of the parameters. In particular, the rotational parameters are true rotations based on a simple physical model (net rotation at constant angular velocity), not Euler angles or angles between vectors and planes as is the case with other approaches. As a result, the mathematical definitions are symmetrical with the property that a 5 degrees tilt is the same as a 5 degrees roll and a 5 degrees twist, except that the rotations take place about different axes. In other approaches, a 5 degrees tilt can mean a different amount of net rotation than a 5 degrees roll or a 5 degrees twist. A second unique feature of the mathematics is that it explicitly incorporates the concept of a pivot point, which is the point about which a base in a base-pair rotates as it buckles, propeller twists, and opens. Pivot points enable one to model the physical motion of bases more accurately. As a result, they greatly reduce and/or eliminate the statistical correlations between rotational and translational parameters that arise as mathematically induced artifacts in other approaches. This paper, together with the statistical analysis in the companion paper for determining the locations of the pivot points, provides everything needed to understand the output of the program as it relates to individual structures.

Base Composition↗
PubMed2004

3-D physical models of mitosis (with asters) and cytokinesis.

First, we define new concepts of Life Objects, Informative Objects and Virtual Objects, Discrete Chromosome Rings (DCR); we introduce a mathematical concept of meridian plane (MP) in a three dimensional (3-D) cylindrical coordinate system (CCS). Based on these concepts, classic mechanics, classic electromagnetism and published biological data, we develop our 3-D physical models of natural and normal mitosis (with asters) and cytokinesis, for animal cells in M phase. We propose following hypotheses: Chromosomes Exclusion: No normally and naturally replicated chromosomes can occupy the same nucleus without growing sizes of the nucleus and the cell. Spontaneous and strong electromagnetic fields (EMF) forces among chromosomes, centrosomes and microtubules split the nucleus and separate the two sets of sister chromatids when they are strong enough. Nuclei Exclusion: No normally and naturally doubled nuclei can occupy the same cell if the doubled size of nuclei is not far smaller than size of the cell. The spontaneous and strong EMF forces in protoplasm (or cortex), separate two sets of chromosomes, spindles and poles, drive contractile proteins to the equator in cell cortex, and continue to guide and to transport free charged objects until complete the cytokinesis. Centrioles Exclusion: No naturally and normally doubled centrioles can occupy the same centrosome. The spontaneous and strong repulsive EMF forces are the primary cause for the exclusions. The principles of our models are also applied to mitosis and cytokinesis for lower plant cells, to that of multiple nuclei or mutant chromosomes, and to meiosis, for both animal cells and lower plant cells.

Animals↗
PubMed1987

Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem.

In this paper basic mathematical and physical concepts of the biomagnetic inverse problem are reviewed with some new approaches. The forward problem is discussed for both homogeneous and inhomogeneous media. Geselowitz' formulae and a surface integral equation are presented to handle a piecewise homogeneous conductor. The special cases of a spherically symmetric conductor and a horizontally layered medium are discussed in detail. The non-uniqueness of the solution of the magnetic inverse problem is discussed and the difficulty caused by the contribution of the electric potential to the magnetic field outside the conductor is studied. As practical methods of solving the inverse problem, a weighted least-squares search with confidence limits and the method of minimum norm estimate are discussed.

Animals↗
PubMedDate not supplied

[What do you say, agnostic?].

This short novel is the story of the perplexity of a statistician travelling for more than a decade in the quicksands of quantitative psychopathology, from the nonvalidated concepts of the DSM-III to their validation by the Dexamethasone Suppression Test whose validity is ignored, from the "mathematical" concepts of Lacan to items suppressed from the AMDP-System, such as autism, because of a lack of agreement of the psychiatrists on objective criteria. The author concludes with a note of hope: comprehensive scales like the AMDP, combined with biological predictors, should facilitate the discovery of new psychotropic drugs, more specific for certain syndromes, hence contributing to new classificatory models.

Humans↗
PubMed2004

Familial aggregation patterns in mathematical ability.

Mathematical talent is an asset in modern society both at an individual and a societal level. Environmental factors such as quality of mathematics education undoubtedly affect an individual's performance, and there is some evidence that genetic factors also may play a role. The current study was performed to investigate the feasibility of undertaking genetics studies on mathematical ability. Because the etiology of low ability in mathematics is likely to be multifactorial and heterogeneous, we evaluated families ascertained through a proband with high mathematical performance in grade 7 on the SAT to eliminate, to some degree, adverse environmental factors. Families of sex-matched probands, selected for high verbal performance on the SAT, served as the comparison group. We evaluated a number of proxy measures for their usefulness in the study of clustering of mathematical talent. Given the difficulty of testing mathematics performance across developmental ages, especially with the added complexity of decreasing exposure to formal mathematics concepts post schooling, we also devised a semiquantitative scale that incorporated educational, occupational, and avocational information as a surrogate for an academic mathematics measure. Whereas several proxy measures showed no evidence of a genetic basis, we found that the semiquantitative scale of mathematical talent showed strong evidence of a genetic basis, with a differential response as a function of the performance measure used to select the proband. This observation suggests that there may be a genetic basis to specific mathematical talent, and that specific, as opposed to proxy, investigative measures that are designed to measure such talent in family members could be of benefit for this purpose.

Aptitude↗
PubMed2005

The challenge of computer mathematics.

Progress in the foundations of mathematics has made it possible to formulate all thinkable mathematical concepts, algorithms and proofs in one language and in an impeccable way. This is not in spite of, but partially based on the famous results of Gödel and Turing. In this way statements are about mathematical objects and algorithms, proofs show the correctness of statements and computations, and computations are dealing with objects and proofs. Interactive computer systems for a full integration of defining, computing and proving are based on this. The human defines concepts, constructs algorithms and provides proofs, while the machine checks that the definitions are well formed and the proofs and computations are correct. Results formalized so far demonstrate the feasibility of this 'computer mathematics'. Also there are very good applications. The challenge is to make the systems more mathematician-friendly, by building libraries and tools. The eventual goal is to help humans to learn, develop, communicate, referee and apply mathematics.

Algorithms↗
PubMed2000

Middle-school students of United Arab Emirates: effects of heterogeneous small group work on attitudes toward mathematics.

This study compared the effects of two instructional strategies, small heterogeneous cooperative learning experience versus lecture and discussion, on students' attitudes toward mathematics. 54 boys and 57 girls in Grade 8 of four middle-school mathematics classes participated. Two classes (57 students) were taught using a cooperative learning method and the other two classes (54 students) were taught using traditional lecture and discussion. Differences between attitudes of boys and girls were also investigated and discussed in the light of Arabic culture. The results suggested that cooperative learning might be a valuable method with which to teach mathematics concepts to boys.

Adolescent↗
PubMed2004

Failure to construct and transfer correct representations across probability problems.

Previous studies carried out on "purely random" situations (with dice or poker chips) show the difficulties encountered by people in such situations, however simple they may be. In fact, in this type of situation, prior knowledge guides spontaneous representations, and the "errors" observed could be explained by the activation of "implicit models" which form the basis of erroneous representations. 42 statistically naïve undergraduates were given several variants of a probability problem on which errors are common. In a learning phase, subjects were given four problems involving geometric figures which were pairwise related by complementarity and equivalence relations. In a subsequent transfer phase, they were given a fifth problem involving poker chips, which was structurally isomorphic to the fourth geometric-figures problem. The findings show that people do not realize the relations between problems, and that transfer occurred only for the subset of subjects who performed correctly on the training problems of the learning phase. These results appear to have some significant implications in teaching mathematical concepts.

Adolescent↗
PubMed2004

Influence of the spatial beam profile on hard tissue ablation, part II: pulse energy and energy density distribution in simple beams.

When calculating applied flux densities in practice, the beam profile of a laser is often erroneously assumed to be homogeneous. In addition, there is usually no consistency in the choice of a suitable measuring method for determining the beam diameter. This failure to observe the inhomogeneous intensity distribution within the beam cross-section, combined with the imprecise knowledge of the beam diameter, leads to flux densities being stated that represent mean values at best. The present paper gives definitions for the flux densities of simple, radially symmetrical beam cross-sections, taking the top-hat and Gaussian profiles as examples. In connection with the inhomogeneous energy distribution in the Gaussian beam, a concept of integral and local energy density is discussed, which differs from the customary definition of the energy density as a constant. Also presented are the consequences of the mathematical concepts in terms of measurement, giving particular consideration to the case where the energy density as the measured variable matches the integral energy density. The significance of the integral and local energy density for hard-tissue ablation is described, based on the practical example of the ablation of dental hard substance. The central result is that the integral flux density is directly accessible as a measured variable, while the effect on the tissue is determined by the local flux density. If the form of the beam is known, the integral flux density can be converted into the local flux density.

Dental Enamel↗
PubMed2005

Hidden Markov Models, grammars, and biology: a tutorial.

Biological sequences and structures have been modelled using various machine learning techniques and abstract mathematical concepts. This article surveys methods using Hidden Markov Model and functional grammars for this purpose. We provide a formal introduction to Hidden Markov Model and grammars, stressing on a comprehensive mathematical description of the methods and their natural continuity. The basic algorithms and their application to analyzing biological sequences and modelling structures of bio-molecules like proteins and nucleic acids are discussed. A comparison of the different approaches is discussed, and possible areas of work and problems are highlighted. Related databases and softwares, available on the internet, are also mentioned.

Algorithms↗
PubMed1986

Acalculia following a dominant-hemisphere subcortical infarct.

A 60-year-old, right-handed woman experienced persistent impairment of calculating ability following a subcortical infarct involving the head of the left caudate nucleus, the anterior superior putamen, and the anterior limb of the internal capsule extending superiorly into the periventricular white matter. Acalculia resulted from defects of numerical syntax, the loss of ability to manipulate mathematical concepts, and impaired working memory.

Aphasia↗
PubMed2002

Analysis and the hierarchy of nature in eighteenth-century chemistry.

What was the impact of Lavoisier's new elementary chemical analysis on the conception and practice of chemistry in the vegetable kingdom at the end of the eighteenth century? I examine how this elementary analysis relates both to more traditional plant analysis and to philosophical and mathematical concepts of analysis of current in the Enlightenment. Thus I explore the relationship between algebra, Condillac's philosophy and Lavoisier's chemical system as well as comparing Lavoisier's analytical approach to those of his predecessors, such as Baumè and Bucquet. With reference to the aims of vegetable analysis, I show how the dominance of elementary analysis devalued a tradition that sought to isolate immediate principles (plant extracts), marginalizing the chemical practices of many doctors and pharmacists in the context of the new chemistry in France.

Chemistry, Analytic↗