Theory Of Integers Research Articles

VeriEQL: Bounded Equivalence Verification for Complex SQL Queries with Integrity Constraints

The task of SQL query equivalence checking is important in various real-world applications (including query rewriting and automated grading) that involve complex queries with integrity constraints; yet, state-of-the-art techniques are very limited in their capability of reasoning about complex features (e.g., those that involve sorting, case statement, rich integrity constraints, etc.) in real-life queries. To the best of our knowledge, we propose the first SMT-based approach and its implementation, VeriEQL, capable of proving and disproving bounded equivalence of complex SQL queries. VeriEQL is based on a new logical encoding that models query semantics over symbolic tuples using the theory of integers with uninterpreted functions. It is simple yet highly practical -- our comprehensive evaluation on over 20,000 benchmarks shows that VeriEQL outperforms all state-of-the-art techniques by more than one order of magnitude in terms of the number of benchmarks that can be proved or disproved. VeriEQL can also generate counterexamples that facilitate many downstream tasks (such as finding serious bugs in systems like MySQL and Apache Calcite).

Two first principles of earth surface thermodynamics. mesoscopy, energy accumulation, and the branch point in boson–fermion transition

The author constructs his thermodynamics on the following two “first principles”: the partition theory of integers and the notion of Earth gravity. On the basis of number theory, equivalence classes in mesoscopy and soft condensates in the partition theory of integers are considered. The self-consistent equation obtained by the author on the basis of Gentile statistics is used to describe the effect of energy accumulation at themoment of transition of the boson branch of the partition of a number to the fermion branch. The branch point in the transition from bosons to fermions is interpreted as an analog of a jump of the spin.

Bounds of the repeated limit for the Bose–Einstein distribution and the construction of partition theory of integers

Bounds of the repeated limit for the Bose–Einstein distribution and the construction of partition theory of integers

Can fractional quantum Hall effect be due to the formation of coherent wave structures in a 2D electron gas?

A microscopic theory of integer and fractional quantum Hall effects is presented here. In quantum density wave representation of charged particles, it is shown that, in a two-dimensional electron gas coherent structures form under the low temperature and high density conditions. With a sufficiently high applied magnetic field, the combined [Formula: see text] particle quantum density wave exhibits collective periodic oscillations. As a result the corresponding quantum Hall voltage function shows a step-wise change in multiples of the ratio [Formula: see text]. At lower temperatures further subdivisions emerge in the Hall resistance, exhibiting the fractional quantum Hall effect.

On decidability of the theory with the transitive closure operator

In this paper we consider a theory of integers with the successor function, divisibility predicates, equality and a transitive closure operator. The order relation can be expressed in this theory using the transitive closure operator.

Real Number Graph Labellings with Distance Conditions

The theory of integer $\lambda$-labellings of a graph, introduced by Griggs and Yeh [J. R. Griggs and R. K.-C. Yeh, SIAM J. Discrete Math., 5 (1992), pp. 586-595], seeks to model efficient channel assignments for a network of transmitters. To prevent interference, labels for nearby vertices must be separated by specified amounts $k_i$ depending on the distance $i$, $1\le i\le p$. Here we expand the model to allow real number labels and separations. The main finding (D-Set Theorem) is that for any graph, possibly infinite, with maximum degree at most $\Delta$, there is a labelling of minimum span in which all of the labels have the form $\sum_{i=1}^p a_i k_i$, where the $a_i$'s are integers $\ge0$. We show that the minimum span is a continuous function of the $k_i$'s, and we conjecture that it is piecewise linear with finitely many pieces. Our stronger conjecture is that the coefficients $a_i$ can be bounded by a constant depending only on $\Delta$ and $p$. We offer results in strong support of the conjectures, and we give formulas for the minimum spans of several graphs with general conditions at distance two.

The theory of integer multiplication with order restricted to primes is decidable

AbstractWe show here that the first order theory of the positive integers equipped with multiplication remains decidable when one adds to the language the usual order restricted to the prime numbers. We see moreover that the complexity of the latter theory is a tower of exponentials, of height O(n).

A NEW APPROACH TO TEACHING DISCRETE MATHEMATICS

ABSTRACT We advocate teaching introductory discrete mathematics by first teaching equational propositional and predicate logic and then using it as tool (and not simply viewing it as an object of study) throughout the study of later topics — e.g., set theory, induction, relations, theory of integers, combinatorics, solving recurrence relations, and modern algebra. The in-depth (6–7 weeks) treatment of logic emphasizes rigorous proofs, strategies for developing proofs, and much practice, so that students develop a skill in formal manipulation. Care is taken to explain all concepts clearly and rigorously. Later topics are dealt with by viewing them as theories — extensions of the predicate calculus. The course should motivate the use of logic as a pervasive tool. It must enlighten, and not stifle and oppress. Our experience shows that this is possible. Anecdotal evidence shows that students come away with less fear of mathematics, proof, and notation, more confidence in their mathematical abilities, an appr...

Disorder in Josephson-junction arrays in a magnetic field.

We consider the effects of disorder in Josephson-junction arrays with an average flux quanta per plaquette ${f}_{0}$. We show that for ${f}_{0}=n+\frac{1}{2}$, positional disorder decouples the excitations responsible for the destruction of superconductivity from other excitations so that near the critical field ${f}_{c}$, the superconducting properties are described by the theory of integer ${f}_{0}$, leading to the same critical field.

Number theory and the real world

Number theory has been considered since time immemorial to be the very paradigm of pure (some would say useless) mathematics. According to Carl Friedrich Gauss, the "Princeps Mathemat icorum", "mathematics is the queen of sc iences--and number theory is the queen of mathemat ics" . In fact, in Chinese the very name of mathematics is Number Science. What could be more beautiful than a deep, satisfying relation between whole numbers? (One is almost tempted to call them wholesome numbers.) Indeed, it is hard to come up with a more appropriate designation than their learned name: the integers, meaning the "untouched ones". How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex numbers, whose first names virtually exude unsavory involvement with the complex realities of everyday life! Yet the theory of integers can provide totally unexpected answers to real-world problems. In fact, discrete mathematics is taking on an ever more important role. If nothing else, the advent of the digital computer and digital communication has seen to that. But even earlier, in physics, the emergence of quantum mechanics and discrete elementary particles put a premium on the methods and, indeed, the spirit of discrete mathematics. In mathematics proper, Hermann Minkowski, in the preface to his introductory book on number theory, Diophantische Approximationen, published in 1907 (the year he gave special relativity its proper four-dimensional clothing in preparation for its journey into general covariance and cosmology) expressed his conviction that the "deepest interrelationships in analysis are of an arithmetical nature". Or, on another occasion: "The primary source (Urquell) of all of mathematics are the integers". Yet much of our schooling concentrates on analysis and other branches of continuum mathematics to the virtual exclusion of number theory, group theory, combinatorics and graph theory. As an illustration, at a recent symposium on information theory, the author met several young women, formally trained in mathematics and working in the field of primality testing, who, in all their studies up to the Ph.D., had not heard a single lecture on number theory! Or, to give an earlier example, when Werner Heisenberg discovered "matrix" mechanics in 1925, he didn't know what a matrix was (Max Born had to tell him), and neither Heisenberg nor Born knew what to make of the appearance of matrices in the context of the atom. (David Hilbert is reported to have told them to go look for a differential equation with the same eigenvalues, if that would make them happier. They did not follow Hilbert's well-meant advice and thereby may have missed discovering the Schr6dinger wave equation.) Integers have repeatedly played a crucial role in the

Number theory in music, speech, and acoustics

Number theory has been considered since time immemorial to be the very paradigm of pure (some would say useless) mathematics. Yet the theory of integers can provide totally unexpected answers to real‐world problems. In acoustics, these range from the distribution of normal modes in rooms (a problem the author encountered in his thesis) and the construction of new musical scales to the optimum diffusion of sound (to improve acoustic quality in concert halls or to boost noise abatement). Even the making of precision measurements in physical acoustics and the design of low peak‐factor waveforms (for farther‐reaching sonar and better‐sounding speech synthesizers) can profit from number theoretic relationships. These as well as a few applications to other fields will be discussed.

Solvability of the theory of integers with addition, order, and multiplication by an arbitrary number

It is known that the arithmetic of natural and integer numbers is unsolvable. Even the universal theory of integers with addition and multiplication is unsolvable. It is proved herein that an elementary theory of integers with addition, order, and multiplication by one arbitrary number is solvable and multiplication by the power of one number is unsolvable. For a certain n, the universal theory of integers with addition and n multiplications by an arbitrary number is also unsolvable.

A system of formal logic without an analogue to the curry W operator

The system of formal logic to be presented in this paper has the following main properties:(1). The Mengenlehre paradoxes do not arise in it.(2). It seems to be free from other contradictions.(3). It does not resort to a theory of types.(4). It uses negation and material implication and does not weaken the principle of excluded middle.(5). It contains no analogue to the Curry W operator; consequently, given a propositional function ϕ, a propositional function Ψ cannot always be found such that, for all x, ϕ(x, x) and Ψ(x) are equivalent propositions.(6). It employs a universal quantifier, G, such that (x)f(x), (x)f(x, x), (x)f(x, x, x), …, would be respectively expressed thus, G(f, f), G(f, f, f), G(f, f, f, f), ….(7). It employs Schönfinkel's functional notation, so that f(x1, x2, …, xn) will be written as (( … ((fx1)x2) …)xn) and abbreviated to (fx1x2… xn) and, when no ambiguity results, to fx1x2…xn.(8). A theory of integers and real numbers cannot, apparently, be deduced from it.(9). A form of the axiom of choice is provable in it.

Book Review: Elements of the Theory of Integers

Book Review: Elements of the Theory of Integers

Elements of the Theory of Integers.

Elements of the Theory of Integers.

A Theory of Integers, in Relation to the Iteration of Algebraic Functions

A Theory of Integers, in Relation to the Iteration of Algebraic Functions

Book Review: Elements of the Theory of Integers

Book Review: Elements of the Theory of Integers

Elements of the Theory of Integers. By J. Bowden, Ph.D. Pp. x, 258. (Macmillan.)

Elements of the Theory of Integers. By J. Bowden, Ph.D. Pp. x, 258. (Macmillan.)