Arithmetic Relations Research Articles

The reaching task: evidence for vector arithmetic in the motor system?

During a reaching task, the population vector is an encoding of direction based on cells with cosine response functions. Scaling the response by a magnitude factor produces a vector encoding, enabling vector arithmetic to be performed by the summation of firing rates. We show that the response properties of selected populations of cells in the primary motor cortex and area 5 can be explained in terms of arithmetic relationships among load, goal, and motor command vectors. Our computer simulations show good agreement with single-cell recording data.

Definability of Arithmetic Operations from the Order and a Random Relation1

For almost all binary relations R ⊆ N 2 the addition and multiplication on the set N of nonnegative integers (and hence all arithmetical relations) are first order definable in the structure (N; ⩽, R). The defining formulae can be chosen independently on R and the words “for almost all” mean “with probability 1” by a very natural probability measure.

Hecke eigenforms and representation numbers of arbitrary rank lattices

Introduction. Given a totally positive quadratic form Q over a totally real number field K, one can obtain a Hilbert modular form by restricting Q to a lattice L and forming the theta series attached to L; the Fourier coefficients of the theta series are the representation numbers of Q on L. The space of Hilbert modular forms generated by all theta series attached to lattices of the same weight, level and character is invariant under a subalgebra of the Hecke algebra, hence one can (in theory) diagonalize this space of modular forms with respect to an appropriate Hecke subalgebra and infer relations on the representation numbers of the lattices. In a previous paper the author found such relations by constructing eigenforms from theta series attached to lattices of even rank which are “nice” at dyadic primes; the purpose of this paper is to extend the previous results to all lattices. We begin by proving a lemma (Lemma 1.1) which allows us to remove the restriction regarding dyadic primes. Then using our previous work we find that associated to any even rank lattice L is a family of lattices famL which is partitioned into nuclear families (which are genera when the ground field is Q), and the averaged representation numbers of these nuclear families satisfy arithmetic relations (Theorem 1.2). In §2 we define “Fourier coefficients” attached to integral ideals for a half-integral weight Hilbert modular form. Then in analogy to the case K = Q, we describe the effect of the Hecke operators on these Fourier coefficients (Theorem 2.5). In §3 we use theta series attached to odd rank lattices to construct eigenforms for the Hecke operators; the results of §2 then give us arithmetic relations on the representation numbers of the odd rank lattices. When the ground field is Q, we may assume Q(L) ⊆ Z and then these relations may be stated as

Passion and Knowledge

Nothing that can be called thinking is formalized or formalizable; nor can it be likened to a mechanical process (Church's hypothesis). Rather, thinking sets into motion human imagination and passion.Having already written extensively on the imagination,' I will limit myself here to outlining its basic structure. At the two opposite poles of knowledge, as well as in its center, lies the creative power of the human being, that is, radical imagination. It is thanks to the imagination that the world is presented in this form and not in some other: it is imagination that creates axioms, postulates, and the fundamental patterns that subtend the structures of knowledge; finally, it is imagination that both furnishes the hypothetical models and idea-images of knowledge, and makes possible their potential development and/or insight into them. This imagination, however, both in itself and in its essential operations – which include its social forms, experienced on this level as the creation of an anonymous collective force – is neither formalized nor formalizable. Obviously, the imagination contains – just as does everything else that exists – a totalizing, identity-bearing dimension (which for brevity's sake we have elsewhere called ensidiqt,ce)2; but this dimension of the imagination is not the essential one, neither in its operations nor its results, no more than the arithmetic relations between tones are the essential element of a Bach fugue.

ON A GENERALIZATION OF FERMAT'S LITTLE THEOREM

We obtain a congruence type arithmetic relation on the set of all triples (G,H,P), where G is a finite group, H is a subgroup, and P is a representation of G by permutations. This relation becomes Fermat's Little Theorem in the case when G=Zp, H=1, and P is the regular representation of G.

An evaluation of bias in k-factor analysis

Randomization and simulation are used to detect bias in k-factor analysis. In nine previously published data sets there is strong evidence of bias. This may result from either non-independence of observations or the arithmetic relationship used to estimate k-factors, which can generate "spurious correlations". Randomization can be used to test for density dependence without bias. This procedure confirms the existence of densitydependent effects in 8 of the 9 populations and 11 of the 16 k-factors previously thought to have density-dependent effects.

Definability in terms of the successor function and the coprimeness predicate in the set of arbitrary integers

AbstractUsing coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate ⊥ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are {S, ⊥}-definable over Z. Applications to definability over Z and N are stated as corollaries of the main theorem.

Definability in terms of the successor function and the coprimeness predicate in the set of arbitrary integers

AbstractUsing coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate ⊥ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are {S, ⊥}-definable over Z. Applications to definability over Z and N are stated as corollaries of the main theorem.

A Number Game—Summing Consecutive Positive Integers

One of the classroom activities that my students and I enjoy is finding expressions that relate the digits of the page number of an assignment. For example, when we are working on page 283, the students might notice that the power of 2 that produces 8 is 3. All arithmetic facts and relationships are fair to use. Phone numbers and license plates also furnish some interesting possibilities.

Extending Prolog to Provide Better Support for Design Code Expert Systems

The CSIR03 Division of Building Research (DBR) has built a number of text‐based and graphics‐based design code expert systems in PROLOG. Each of these systems involves several thousand lines of PROLOG and stretches to the limit and the capacity of the IBM ATs on which they run. We have found PROLOG to be a very powerful language for expressing the complex rules and relationships found in building design codes and related applications. Nevertheless, it has a number of weaknesses, particularly in expressing procedural knowledge and arithmetic relationships, and in implementing complex tables and user interfaces. We have designed and implemented extensions to PROLOG to overcome these difficulties. These include new built‐in functions to perform functions that are difficult to code in PROLOG, an easy way to add further built‐in functions, a meta‐interpreter to keep track of intermediate results, sets of routines to support user input and explanation, and a definite clause grammar to implement a more syntactically pleasant language for writing rules. These extensions are being used to develop a larger and much more complete version of the WINDLOADER expert system, and have greatly facilitated this task.

On possible uses of traditional angolan sand drawings in the mathematics classroom

Both industrialised and Third World countries see themselves confronted with the need to ‘multi-culturalise’ their mathematics curricula. Following a brief description of the drawing tradition of the Tchokwe people (Angola), some possible uses of their pictograms in the mathematics classroom are suggested. The examples given in this paper range from the study of arithmetical relationships, progressions, symmetry, similarity, and Euler graphs to the determination of the greatest common divisor of two natural numbers.

Aid, the Public Sector and the Market in Less Developed Countries

The article in this JOURNAL by Mosley et al. (I987) concludes its analysis of aid effectiveness as demonstrating '... inability of development aid over more than twenty years to provide a net increment to overall growth in the Third World' (p. 636). This note purports to show that this conclusion, which is ostensibly 'distressing' as the authors claim, is not warranted. The section of the study which is the basis of the conclusion cited and the subject of this note is a cross-section regression analysis, full details of which are shown in Mosley et al. Table 3. The note is concerned with the relation between the dependent variable specified as the growth rate of real GDP per cent per annum (hereafter GDP) and aid specified as a percentage of GDP (hereafter Aid).' The Aid results, with only two significant positive coefficients in one subsample, certainly give no basis for claiming aid-effectiveness in terms of GDP growth. The intention in this note is to interpret the results which are assumed to be valid. This is made possible by the fact that the authors have helpfully provided the full input set of mean values derived from the raw data. From this it is possible to calculate the GDP/Aid relationship for each country for the second and third period. The procedure for deriving these relationships is as follows. For each country the mean values of the time series of GDP and Aid are extracted from the data for I 960-70 and for I 970-80. Using I 960-70 as the base, the signs of the change in the levels of the GDP and Aid mean values between periods are recorded. This is repeated for I980-4 with I970-80 mean values as the base. The relationship between the two signs is an indication of the effect of Aid on GDP in the period in which the change in levels is effected. Thus in the case of an increase in Aid being combined with a decrease in GDP the latter must have fallen relatively to the former and aid-effectiveness is negative; Mutatis mutandis, for the other three sign combinations, thus generating the appropriate GDP/Aid relationship for each country in each period. A frequency distribution of the four sign-combinations, relating to the full sample, is presented in Table I. Since the signs of the changes in the levels of GDP and Aid are derived by making explicit the arithmetic relationship implicit in the data they form a valid base from which to draw conclusions without requiring any test of

A boundedness theorem in ID1(W)

In this paper we prove a boundedness theorem in the theory ID1(W). This answers a question asked by Feferman, for example in [3]. The background is the following.Let A[X, x] be an X-positive formula arithmetic in X. The theory ID1(PA) is an extension of Peano arithmetic PA by the following axioms:for arbitrary formulas F; PA is a constant for the least fixed point of A[X, x]. Set-theoretically, PA can be defined by recursion on the ordinals as follows:where is the first nonrecursive ordinal.Now let a ≺ b be the arithmetic relation which expresses that the recursive tree coded by a is a proper subtree of the tree coded by b, and defineThe least fixed point of Tree[X, x] is the set PTree of all well-founded recursive trees. We write W or Wα for PTree or , respectively. Since W is complete we have for all α < . If we define for each element a ∈ W its inductive norm ∣a∣ by ∣a∣≔ min{ξ: a ∈ Wξ}, then we have = {∣a∣: a ∈ W} and the elements of W can be used as codes for the ordinals less than .Assume that B[X, x] is an X-positive formula arithmetic in X with the only free variables X and x, and assume that QB is a relation that satisfiesIf we definethen we obviously have PB = IB.

PROLOG: Arithmetic relations

PROLOG: Arithmetic relations

Laws governing degeneration of the genetic code.

The laws governing degeneration of the genetic code are discussed below. Of fundamental importance in this context is the classification of the amino acids into groups on the basis of the physicochemical behaviour of their residues. From this, it is possible to formulate arithmetic relationships between the number of amino acids in the same group and the number of coding triplets. It is found that the degeneration of the genetic code obeys certain laws, the reasons for this being related to the number and the qualitative properties of the amino acids and triplets. The fact that the three bases of a coding triplet have different priorities must also be a critical factor.

Epidemiologic evaluation of screening for risk factors: application to genetic screening.

To assess the usefulness of screening for risk factors, we derived arithmetic relationships between screening parameters (sensitivity, specificity, and positive predictive value PPV) and risk factor frequency, disease frequency and relative risk. We evaluated these relationships in the special case of genetic markers and disease susceptibility. It can be shown that even in the face of very large relative risks, sensitivity and positive predictive value are affected by the relative magnitude of disease and genetic marker frequencies. When the genetic marker is less frequent than the disease, PPV increases with increasing relative risk but sensitivity remains low. When the genetic marker is more frequent than the disease, sensitivity increases with increasing relative risk but PPV remains low. When marker and disease frequencies are equal, both PPV and sensitivity increase with increasing relative risks, but very high relative risks (greater than 100) have to be obtained for rare diseases. Depending on the goals of the screening program, these relationships can be used to predict the relative magnitudes of false positives (low PPV) and false negatives (low sensitivity). This approach can be generalized to evaluate nongenetic risk factors in screening programs as well.

Distributed Lags in Local Responses to Fluctuations in Unemployment

In this paper the sources of distributed lags, involving both positive and negative lead and lag terms, in the relationship between local and national unemployment series are analysed. First, the elements of labour supply and demand from which local unemployment changes result are considered, with emphasis on the role of induced labour migration. Second, the consequences of the arithmetical relationship between local and national unemployment series are examined. It is shown that the national-local unemployment relationship should be nonlinear and the lag distribution should be dependent on the level of unemployment. More explicit recognition of causal relationships is necessary if substantive conclusions are to be drawn from statistical analyses of lag structures.

Misconceptions of chemical equilibrium

Those propositions deemed necessary for an understanding of chemical equilibrium and Le Chatelier's Principle were defined by the investigators and validated. Thirty, Year 12 Western Australian chemistry students (17 years of age) who had studied chemical equilibrium were interviewed and students’ responses were coded into various categories of misconception that had been identified. The most significant misconceptions revealed by the study were: The rate of the forward reaction increases with time from the mixing of the reactants until equilibrium is established; a simple arithmetic relationship exists between the concentrations of reactants and products at equilibrium; and when a system is at equilibrium and a change is made in the conditions, the rate of the favoured reaction increases but the rate of the other reaction decreases. Students’ prior experience of reactions that proceed to completion appears to have influenced their conception of equilibrium reactions; many students failed to discriminate clearly between the characteristics of completion reactions and reversible reactions.

Does Horton's law of branch length apply to open branching systems?

Horton analysis has been applied to colonies of mycelial fungi and the mean branch lengths of branches per branch order tend to form a direct arithmetical, not geometric, series with branch order. Data for two botanical trees also show the arithmetical relationship. It is argued, from simulation models, that for branching systems showing outward unconstrained growth an arithmetical relationship results when linear growth rate of the elements is constant, but that a geometric relationship may result when linear growth rate shows a progressive decline.

Modulation of proliferation in epidermal keratinocyte cultures by lowered oxygen tension

The modulation of proliferation and differentiation in primary epidermal keratinocyte cultures by lowered gas phase oxygen tensions was studied. Neonatal mouse epidermal keratinocyte cultures were grown in an Heraeus type B 5060 EK/O2 incubator in oxygen tensions between 5% and 15% (within the physiologic range); the oxygen tension of ambient air being 21%. Cell morphology was studied using histochemical stains and electron microscopy. Differentiation was assessed using autoradiography of SDS PAGE gels of six serially extracted cell protein fractions with [3H]leucine as a marker. Autoradiographs using [14C]glucosamine and 32Pi as markers were also assessed as a measure of other cell functions. Proliferation was studied using autoradiography of [3H]thymidine ([3H]TdR) pulse-labeled cultures and [3H]TdR incorporation into isolated DNA fractions. The results of these studies showed that lowering the oxygen tension in the gas phase reversibly inhibited cell proliferation. There was a direct arithmetic relationship between the proliferative rate of the cultures and the oxygen tension. No change in differentiation as defined by [3H]leucine indexing of protein synthesis was seen. Other markers of cell function, such as [14C]glucosamine glycosylation and [32P] phosphorylation of proteins were also unchanged. These results suggest that oxygen tension regulates only proliferation in epidermal keratinocytes. This epidermal response is well adapted to its role in the healing wound, and is an example of a tissue-specific modification of a regulatory function.