On MTL's First Decade

Three experiments used multiple methods—open-ended assessments, multiple-choice questionnaires, and interviews—to investigate the hypothesis that the development of students' understanding of the concept of real variable in algebra may be influenced in fundamental ways by their initial concept of number, which seems to be organized around the notion of natural number. In the first two experiments 91 secondary school students (ranging in age from 12.5 to 14.5 years) were asked to indicate numbers that could or could not be used to substitute literal symbols in algebraic expressions. The results showed that there was a strong tendency on the part of the students to interpret literal symbols to stand for natural numbers and a related tendency to consider the phenomenal sign of the algebraic expressions as their “real” sign. Similar findings were obtained in a third, individual interview study, conducted with tenth grade students. The results were interpreted to support the interpretation that there is a systematic natural number bias on students' substitutions of literal symbols in algebra.     

Ahiezer-Kac type Fredholm determinant asymptotics for convolution operators with rational symbols

Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied. The explicit asymptotic formulae obtained can be considered as a direct extension of the Ahiezer-Kac formula to symbols with real zeros.

     

Contraction semigroups of elliptic quadratic differential operators

We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time.     

Searle freed of every flaw

Strong Al presupposes (1) that Super-Searle (henceforth ‘Searle’) comes to know that the symbols he manipulates are meaningful, and (2) that there cannot be two or more semantical interpretations for the system of symbols that Searle manipulates such that the set of rules constitutes a language comprehension program for each interpretation. In this paper, I show that Strong Al is false and that presupposition #1 is false, on the assumption that presupposition #2 is true. The main argument of the paper constructs a second program, isomorphic to Searle’s, to show that if someone, say Dan, runs this isomorphic program, he cannot possibly come to know what its mentioned symbols mean because they do not mean anything to anybody. Since Dan and Searle do exactly the same thing, except that the symbols they manipulate are different, neither Dan nor Searle can possibly know whether the symbols they manipulate are meaningful (let alone what they mean, if they are meaningful). The remainder of the paper responds to an anticipated Strong Al rejoinder, which, I believe, is a necessary extension of Strong Al.     

The classical decision problem and partial functions

We consider the decision problem for sets of sentences of first-order logic when instead of interpreting function symbols as total functions over the universe of a model (henceforth referred to as the usual interpretation) we interpret them as partial functions.We consider only standard classes, which are certain sets of prenex sentences specified by restrictions on the prefix and on the numbers ofk-place predicate and function symbols for eachk1. Standard classes are introduced in [1] and it is proved there that the decision problem for any set of prenex sentences specified by such restrictions reduces to that for the standard classes.We solve the decision problem completely for standard classes with at least one function symbol and both with and without equality.This problem was suggested to me by my supervisor, Professor Yuri Gurevich who was confident that the results would be very similar to those for the usual interpretation and could be achieved by similar techniques.     

Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.     

Fredholm theory of Wiener-Hopf equations in terms of realization of their symbols

The Fredholm properties (index, kernel, image, etc.) of Wiener-Hopf integral operators are described in terms of realization of the symbol for a class of matrix symbols that are analytic on the real line but not at infinity. The realizations are given in terms of exponentially dichotomous operators. The results obtained give a complete analogue of the earlier results for rational symbols.     

Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.     

An introduction to the perplex number system

The perplex number system is a generalization of the abstract logical relationships among electrical particles. The inferential logic of the new number system is homologous to the inferential logic of the progression of the atomic numbers. An electrical progression is defined categorically as a sequence of objects with teridentities. Each identity infers corresponding values of an integer, units and a correspondence relation between each unit and its integer. Thus, in this logical system, each perplex numeral contains an exact internal representational structure; it carries an internal message. This structure is a labeled bipartite graph that is homologous to the internal electrical structure of a chemical atom. The formal logical operations are conjunctions and disjunctions. Combinations of conjunctions and disjunctions compose the spatiality of objects. Conjunctions may include the middle term of pairs of propositions with a common term, thereby creating new information. The perplex numerals are used as a universal source of diagrams.The perplex number system, as an abstract generalization of concrete objects and processes, constitutes a new exact notation for chemistry without invoking alchemical symbols. Practical applications of the number system to concrete objects (chemical elements, simple ions and molecules, and the perplex isomers, ethanol and dimethyl ether) are given. In conjunction with the real number system, the relationships between the perplex number system and scientific theories of concrete systems (thermodynamics, intra-molecular dynamics, molecular biology and individual medicine) are described.     

Interpretation of fuzzy data by means of fuzzy rules with applications to speech recognition

Recently, a speech recognition methodology has been proposed which has as one of its main principles the explicit assumption of intrinsic uncertainty of the data (speech signals) and inexactness of the knowledge (acoustic phonetic, etc…) available to interpret them. The main problem presented by this methodology is that of parsing ‘fuzzy data’ by means of ‘fuzzy rules’. To solve this problem, an appropriate fuzzy parsing and interpretation scheme has been proposed. It assumes the data to be represented as strings of ‘fuzzy symbols’, defined as fuzzy sets over the appropriate set of categories, and knowledge as finite-state networks with the arcs labelled by fuzzy symbols of the same type. A formal presentation of this scheme is the main topic of this paper. Included is a brief discussion of the application to Automatic Speech Recognition, and a summary of some results obtained from an implementation example.     

Process convergence for the complexity of Radix Selection on Markov sources

A fundamental algorithm for selecting ranks from a finite subset of an ordered set is Radix Selection. This algorithm requires the data to be given as strings of symbols over an ordered alphabet, e.g., binary expansions of real numbers. Its complexity is measured by the number of symbols that have to be read. In this paper the model of independent data identically generated from a Markov chain is considered.The complexity is studied as a stochastic process indexed by the set of infinite strings over the given alphabet. The orders of mean and variance of the complexity and, after normalization, a limit theorem with a centered Gaussian process as limit are derived. This implies an analysis for two standard models for the ranks: uniformly chosen ranks, also called grand averages, and the worst case rank complexities which are of interest in computer science.For uniform data and the asymmetric Bernoulli model (i.e. memoryless sources), we also find weak convergence for the normalized process of complexities when indexed by the ranks while for more general Markov sources these processes are not tight under the standard normalizations.     

Student-Invented Numeration Systems: Pattern-Analysis and Mathematical Understanding

Thirteen fifth graders were given an assignment to invent their own numeration systems, following a unit on bases and a look at early events in the history of numbers. The task presented options that required the students to make decisions (such as whether to use a base, which base to use, design of symbols, etc.), and build a rationale for the elements of their system. Analyses of patterns embedded in their invented systems provided an assessment of student understanding of numeration. The progression of more and less complex thinking related to the student's choice of a base other than 10, consistency of logic throughout the system in words and symbols, rationale for change, and perception of real life examples that would change if the system was adopted. The invention task is presented as another way to make connections.     

Geometric quantization for proper moment maps: the Vergne conjecture

We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.     

Octonionic geometry

We extend vector formalism by including it in the algebra of split octonions, which we treat as the universal algebra to describe physical signals. The new geometrical interpretation of the products of octonionic basis units is presented. Eight real parameters of octonions are interpreted as the space-time coordinates, momentum and energy. In our approach the two fundamental constants, c and have the geometrical meaning and appear from the condition of positive definiteness of the octonion norm. We connect the property of non-associativity with the time irreversibility and fundamental probabilities in physics.     

Extension of the Ahiezer–Kac Determinant Formula to the Case of Real-Valued Symbols with Two Real Zeros

The Fredholm determinant asymptotics for self-adjoint convolution operators on finite intervals with real symbols vanishing on the real axis is studied. Explicit formulae are obtained in the case where the symbol satisfies the generalized zero index condition and has only two simple zeros of analytic type. These formulae are direct extensions of the Ahiezer–Kac–Szegö limit theorem which, in particular, take into account the oscillating character of the asymptotics.     

Interval efficiency measures in data envelopment analysis with imprecise data

The conventional data envelopment analysis (DEA) measures the relative efficiencies of a set of decision making units (DMUs) with exact values of inputs and outputs. For imprecise data, i.e., mixtures of interval data and ordinal data, some methods have been developed to calculate the upper bound of the efficiency scores. This paper constructs a pair of two-level mathematical programming models, whose objective values represent the lower bound and upper bound of the efficiency scores, respectively. Based on the concept of productive efficiency and the application of a variable substitution technique, the pair of two-level nonlinear programs is transformed to a pair of ordinary one-level linear programs. Solving the associated pairs of linear programs produces the efficiency intervals of all DMUs. An illustrative example verifies the idea of this paper. A real case is also provided to give some interpretation of the interval efficiency. Interval efficiency not only describes the real situation in better detail; psychologically, it also eases the tension of the DMUs being evaluated as well as the persons conducting the evaluation.     

Axioms for a theory of semantic equivalence

Gödel-type semantic completeness theorems are established for a formal theory of semantic equivalence based on L.A. Zadeh's concept of a linguistic variable. The linguistics that is employed allows for the expression of propositions such as “it is not the case that ‘young’ is semantically equivalent with ‘not old’”, or, in symbols (young(x) ≅ ∼old(x)).The result is a two-leveled semantics which, at the lower level, is a multivalent interpretation of fuzzy assertions (e.g., ∼old(x)) in terms of fuzzy subsets of a given universe and, at the upper level, is a two-valued (classical) interpretation based on the fact that two closed fuzzy assertions either do or do not have the same truth value. The main proof is of the Henkin variety, employing adaptations of the algebraic methods found in Rasiowa [9] and Rasiowa and Sikorski [10].