Renaissance notions of number and magnitude

In the 16th and 17th centuries the classical Greek notions of (discrete) number and (continuous) magnitude (preserved in medieval Latin translations of Euclid's Elements) underwent a major transformation that turned them into continuous but measurable magnitudes. This article studies the changes introduced in the classical notions of number and magnitude by three influential Renaissance editions of Euclid's Elements. Besides providing evidence of earlier discussions preparing notions and arguments eventually introduced in Simon Stevin's Arithmétique of 1585, these editions document the role abacus algebra and Renaissance views on the history of mathematics played in bridging the gulf between discrete numbers and continuous magnitudes.     

Decimal periods and their tables: A German research topic (1765–1801)

At the beginning of the 18th century, several mathematicians noted regularities in the decimal expansions of common fractions. Rules of thumb were set up, but it was only from 1760 onward that the first attempts to try to establish a coherent theory of periodic decimal fractions appeared. J.H. Lambert was the first to devote two essays to the topic, but his colleagues at the Berlin Academy, J. III Bernoulli and J.L. Lagrange, also spent time on the problem. Apart from the theoretical side of the question, the applications (factoring, irrationality proofs, and computational advantages), as well as the tabulation of decimal periods, aroused considerable interest, especially among Lambert's correspondents, C.F. Hindenburg and I. Wolfram. Finally, in 1797–1801, the young C.F. Gauss, informed of these developments, based the whole theory on firm number-theoretic foundations, thereby solving most of the open problems left by the mathematicians before him.     

Vector least-squares solutions for coupled singular matrix equations

The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian's work and Lev-Ari's results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case.     

The Grand Astrologerʼs platform and ramp: Four problems in solid geometry from Wang Xiaotongʼs ‘Continuation of ancient mathematics’ (7th century AD)

Wang Xiaotong?s Jigu suanjing is primarily concerned with problems in solid and plane geometry leading to cubic equations which are to be solved numerically by the Chinese variant of Horner?s method. The problems in solid geometry give the volume of a solid and certain constraints on its dimensions, and the dimensions are required; we translate and analyze four of these. Three are solved using dissections, while one is solved using reasoning about calculations with very little recourse to geometrical considerations. The problems in Wang Xiaotong?s text cannot be seen as practical problems in themselves, but they introduce mathematical methods which would have been useful to administrators in organizing labor forces for public works.     

An early Chinese derivation of the volume of a pyramid: Liu Hui,third century A.D.

The Chiu-chang suan-shu (“Arithmetic in nine chapters”) is a mathematical book of the late first century A.D. It gives practical problems and states algorithms for their solution, with no explanation. A commentary attributed to Liu Hui, of the third century A.D., gives an explanation of each algorithm; these explanations satisfy many of the criteria for what we would call a proof. In this article Liu Hui's explanation of the formula for the volume of a particular kind of pyramid is translated and discussed.     

The introduction of logarithms into Spain

During the first half of the 17th century, logarithms were taught by some professors in Spain, but knowledge of this subject remained scanty until the publication of Architectura civil by Juan Caramuel (1678) and especially of Trigonometria española by José Zaragoza (1672). Logarithms were considered only as an aid for computation up to the second half of the 18th century. Only when the infinitesimal calculus became more widely spread in Spanish mathematics, analytical interpretations of logarithms were also taken into account in books such as Elementos de matemáticas by Benito Bails (1776).     

q-Analogs of some congruences involving Catalan numbers

We present some variations on the Greene–Krammer?s identity which involve q-Catalan numbers. Our method reveals an intriguing analogy between these new identities and some congruences modulo a prime.     

Dual subgradient algorithms for large-scale nonsmooth learning problems

“Classical” First Order (FO) algorithms of convex optimization, such as Mirror Descent algorithm or Nesterov’s optimal algorithm of smooth convex optimization, are well known to have optimal (theoretical) complexity estimates which do not depend on the problem dimension. However, to attain the optimality, the domain of the problem should admit a “good proximal setup”. The latter essentially means that (1) the problem domain should satisfy certain geometric conditions of “favorable geometry”, and (2) the practical use of these methods is conditioned by our ability to compute at a moderate cost proximal transformation at each iteration. More often than not these two conditions are satisfied in optimization problems arising in computational learning, what explains why proximal type FO methods recently became methods of choice when solving various learning problems. Yet, they meet their limits in several important problems such as multi-task learning with large number of tasks, where the problem domain does not exhibit favorable geometry, and learning and matrix completion problems with nuclear norm constraint, when the numerical cost of computing proximal transformation becomes prohibitive in large-scale problems. We propose a novel approach to solving nonsmooth optimization problems arising in learning applications where Fenchel-type representation of the objective function is available. The approach is based on applying FO algorithms to the dual problem and using the accuracy certificates supplied by the method to recover the primal solution. While suboptimal in terms of accuracy guaranties, the proposed approach does not rely upon “good proximal setup” for the primal problem but requires the problem domain to admit a Linear Optimization oracle—the ability to efficiently maximize a linear form on the domain of the primal problem.     

Algebra in Roth,Faulhaber, and Descartes

Descartes' “multiplicative” theory of equations in the Géométrie (1637) systematically treats equations as polynomials set equal to zero, bringing out relations between equations, roots, and polynomial factors. We here consider this theory as a response to Peter Roth's suggestions in Arithmetica Philosophica (1608), notably in his “seventh-degree” problem set. These specimens of arithmetic-masterly problem design develop skills with multiplicative and other degree-independent techniques. The challenges were fine-tuned by introducing errors disguised as printing errors. During Descartes' visit to Germany in 1619–1622, he probably worked with Johann Faulhaber (1580–1635) on these problems; they are discussed in Faulhaber's Miracula Arithmetica (1622), which also looks forward to fuller publication, probably by Descartes.     

Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem

It may seem odd that Abel, a protagonist of Cauchy's new rigor, spoke of “exceptions” when he criticized Cauchy's theorem on the continuity of sums of continuous functions. However, when interpreted contextually, exceptions appear as both valid and viable entities in the early 19th century. First, Abel's use of the term “exception” and the role of the exception in his binomial paper is documented and analyzed. Second, it is suggested how Abel may have acquainted himself with the exception and his use of it in a process denoted critical revision is discussed. Finally, an interpretation of Abel's exception is given that identifies it as a representative example of a more general transition in the understanding of mathematical objects that took place during the period. With this interpretation, exceptions find their place in a fundamental transition during the early 19th century from a formal approach to analysis toward a more conceptual one.     

Lower bounds for nonlinear assignment problems using many body interactions

This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O(N^2(α+β−1)) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problems With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's.     

On Mascheroni's La geometria del compasso at the beginning of the 19th century

Lorenzo Mascheroni's 1797 work La geometria del compasso, which develops a geometry based solely on compass constructions, is considered by the author as stepping back behind the “demarcation line” of Euclidean geometry. In this work Mascheroni emphasizes the practical aspects of this geometry over a theoretical approach. A century later, in 1899, David Hilbert and his student Michael Feldblum proposed a totally different approach – algebraic and axiomatic – concerning geometric constructions based on various instruments. Taking into account that, at the end of the 18th century, straightedge geometry was also developed, one may ask what happened to the image of instrument-based geometry during the 19th century? By focusing on Mascheroni's book and its reception, this article aims to examine the various views and conceptions of mathematicians with respect to this geometry.     

A stable approach to Newton's method for general mathematical programming problems inR n

The usual approach to Newton's method for mathematical programming problems with equality constraints leads to the solution of linear systems ofn +m equations inn +m unknowns, wheren is the dimension of the space andm is the number of constraints. Moreover, these linear systems are never positive definite. It is our feeling that this approach is somewhat artificial, since in the unconstrained case the linear systems are very often positive definite. With this in mind, we present an alternate Newton-like approach for the constrained problem in which all the linear systems are of order less than or equal ton. Furthermore, when the Hessian of the Lagrangian at the solution is positive definite (a situation frequently occurring), all our systems will be positive definite. Hence, in all cases, our Newton-like method offers greater numerical stability. We demonstrate that the convergence properties of this Newton-like method are superior to those of the standard approach to Newton's method. The operation count for the new method using Gaussian elimination is of the same order as the operation count for the standard method. However, if the Hessian of the Lagrangian at the solution is positive definite and we use Cholesky decomposition, then the order of the operation count for the new method is half that for the standard approach to Newton's method. This theory is generalized to problems with both equality and inequality constraints.     

An application of Lemke's method to a class of Markov decision problems

This paper presents an application of Lemke's method to a class of Markov decision problems, appearing in the optimal stopping problems, and other well-known optimization problems. We consider a special case of the Markov decision problems with finitely many states, where the agent can choose one of the alternatives; getting a fixed reward immediately or paying the penalty for one term. We show that the problem can be reduced to a linear complementarity problem that can be solved by Lemke's method with the number of iterations less than the number of states. The reduced linear complementarity problem does not necessarily satisfy the copositive-plus condition. Nevertheless we show that the Lemke's method succeeds in solving the problem by proving that the problem satisfies a necessary and sufficient condition for the extended Lemke's method to compute a solution in the piecewise linear complementarity problem.     

On the dual nature of partial theta functions and Appell–Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell–Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell–Lerch sums. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers–Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral q-series with mixed mock modular behaviour. In subsequent work we see that our bilateral series are well-suited for computing radial limits of Ramanujan's mock theta functions.     

False position in Leonardo of Pisa's Liber Abbaci

We examine the rhetorical methods of Leonardo of Pisa in his exposition of single false position in Liber Abbaci. For example, Leonardo makes extensive use of formulaic phrases in his solutions. Some of these formulas also seem to indicate whether a particular solution needs further justification. Although he prefers proofs in terms of the pseudo-Euclidean canon of al-Khwārizmī, sometimes such proof eludes Leonardo and he resorts instead to justification by experiment. We also look at the extent to which using symbolic representations might distort our view of Leonardo's thinking.