Symbolic language in early modern mathematics: The Algebra of Pierre Hérigone (1580–1643)

The creation of a formal mathematical language was fundamental to making mathematics algebraic. A landmark in this process was the publication of In artem analyticem isagoge by François Viète (1540–1603) in 1591. This work was diffused through many other algebra texts, as in the section entitled Algebra in the Cursus mathematicus (Paris, 1634, 1637, 1642; second edition 1644) by Pierre Hérigone (1580–1643). The aim of this paper is to analyze several features of Hérigone's Algebra. Hérigone was one of the first mathematicians to consider that symbolic language might be used as a universal language for dealing with pure and mixed mathematics. We show that, although Hérigone generally used Viète's statements, his notation, presentation style, and procedures in his algebraic proofs were quite different from Viète's. In addition, we emphasize how Hérigone handled algebraic operations and geometrical procedures by making use of propositions from Euclid's Elements formulated in symbolic language.     

Corrigendum to “A note on the Loewenstein–Prelec theory of intertemporal choice” [Math. Social Sci. 52 (1) (2006) 99–108]

In a critique of the Loewenstein and Prelec [Loewenstein G., Prelec D., 1992. Anomalies in intertemporal choice: Evidence and an interpretation. The Quarterly Journal of Economics 107, 573–597] theory of intertemporal choice, [al-Nowaihi, A., Dhami, S., 2006. A note on the Loewenstein–Prelec theory of intertemporal choice. Mathematical Social Sciences 52, 99–108] point out four errors. One of the alleged errors was that the elasticity of the value function in prospect theory is decreasing. But it is in fact increasing. We provide a correction and a formal proof. As a corollary, we show that the elasticity of the value function is bounded between zero and one. Nevertheless, all the remaining points in [al-Nowaihi, A., Dhami, S., 2006. A note on the Loewenstein–Prelec theory of intertemporal choice. Mathematical Social Sciences 52, 99–108] remain valid     

The decline of Cayley's invariant theory (1863–1895)

From the 1860s the German symbolic approach to invariant theory was in ascendancy. This article discusses the of work Arthur Cayley (1821–1895) and his reaction to this new line of enquiry. The symbolic method is outlined and compared with Cayley's viewpoint in which the calculation and exhibition of invariants and covariants were of primary importance. Cayley's Law and Gordan's finiteness theorem, two principal results in the theory, are discussed. Also covered is J. J. Sylvester's Fundamental Postulate, which both reveals the character of the English empirical approach to invariant theory and illustrates its inherent weakness. The article examines the background to Cayley's final three memoirs on quantics, his last work in invariant theory, and it makes use of correspondence with his friend Sylvester.     

On extensions of some Flugede–Putnam type theorems involving (p,k)‐quasihyponormal,spectral, and dominant operators

A Hilbert space operator S is called (p, k)‐quasihyponormal if S *^k ((S *S)^p – (SS *)^p )S^k ≥ 0 for an integer k ≥ 1 and 0 < p ≤ 1. In the present note, we consider (p, k)‐quasihyponormal operator S ∈ B (H) such that SX = XT for some X ∈ B (K,H) and prove the Fuglede–Putnam type theorems when the adjoint of T ∈ B (K) is either (p, k)‐quasihyponormal or dominant or a spectral operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on unsolvable systems of max–min (fuzzy) equations

For unsolvable systems of linear equations of the form Ax=b over the max–min (fuzzy) algebra

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we propose an efficient method for finding a Chebychev-best approximation of the matrix in the set

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.     

The convergence of fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation

In the present paper, we analyze a second-order in time fully discrete finite element method for the BBM equation. The discretization in space is based on the standard Galerkin method, for the time discretization the Crank–Nicolson scheme is used. We also prove the convergence of a linearized Galerkin modification scheme.     

Postscript: “Endomorphism spectra of graphs” [Discrete Mathematics 109 (1992) 45–57]

We answer some questions about the existence of graphs with a given endotype posed in Böttcher and Knauer (Discrete Math. 109 (1992) 45–57). All definitions and needed terminology can be found there and for brevity they will not be repeated here. All results have been obtained by computer calculations.     

Erratum to: “The Gromov norm of the product of two surfaces” [Topology 44 (2005) 321–339]

We report an error in the proof of Lemma 2.7 of the original article. This invalidates one direction of our main theorem.     

The crossing number of Cm × Cn is as conjectured for n ≥ m(m + 1)

It has been long conjectured that the crossing number of C_m × C_n is (m?2)n, for all m, n such that n ≥ m ≥ 3. In this paper, it is shown that if n ≥ m(m + 1) and m ≥ 3, then this conjecture holds. That is, the crossing number of C_m × C_n is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and further developed by Adamsson and Richter. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 53–72, 2004     

Erratum to: “The Schrödinger–Maxwell system with Dirac mass” [Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (5) (2007) 773–793]

We correct the proof of [G.M. Coclite, H. Holden, The Schrödinger–Maxwell system with Dirac mass, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (5) (2007) 773–793, Lemma 4.1].     

Disjoint 3‐Cycles in Tournaments: A Proof of The Bermond–Thomassen Conjecture for Tournaments

We prove that every tournament with minimum out‐degree at least contains k disjoint 3‐cycles. This provides additional support for the conjecture by Bermond and Thomassen that every digraph D of minimum out‐degree contains k vertex disjoint cycles. We also prove that for every , when k is large enough, every tournament with minimum out‐degree at least contains k disjoint cycles. The linear factor 1.5 is best possible as shown by the regular tournaments.     

The existence of solutions for an impulsive fractional coupled system of (p,q)‐Laplacian type without the Ambrosetti‐Rabinowitz condition

In this article, based on the variational approach, the existence of at least one nontrivial solution is studied for (p, q)‐Laplacian type impulsive fractional differential equations involving Riemann‐Liouville derivatives. Without the usual Ambrosetti‐Rabinowitz condition, the nonlinearity f in the paper is considered under some suitable assumptions.     

The convergence of q ‐Bernstein polynomials (0 < q < 1) in the complex plane

The paper focuses at the estimates for the rate of convergence of the q ‐Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of analytic continuation of the limit function and an elaboration of the theorem by Wang and Meng is presented (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Necessary optimality conditions for fractional action‐like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β)

We derive Euler–Lagrange‐type equations for fractional action‐like integrals of the calculus of variations which depend on the Riemann–Liouville derivatives of order (α, β), α>0, β>0, recently introduced by Cresson. Some interesting consequences are obtained and discussed. Copyright © 2007 John Wiley & Sons, Ltd.