How Many People Can the Earth Support? By Joel E. Cohen. Norton: 1995. Pp. 532. $30, £22.50. UK publication date, 27 March.

Book of numbers

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

THAT the foundations of mathematics are A important is a proposition which will find few opponents, for the science of mathematics is commonly regarded as man's securest intellectual possession. What constitutes these foundations is a subject on which agreement has not been reached. There are, however, three main directions into which the body of modern research has branched, namely, the logistic, the intuitionistic and the formalistic theories. Broadly speaking, the logistic theory regards mathematics as a branch of logic, the intuitionistic theory regards the theorems of mathematics as having actual significance, the formalistic theory regards mathe matics as devoid of meaning per se. The most powerful exponent of the formalistic attitude is Hilbert, and the present volume is the first part of a systematic exposition of mathematical foundations from the point of view of the school of thought which he has founded. This is probably the most important book on mathe-matical foundations which has appeared since Whitehead and Russell's “Principia Mathematical Grundlagen der Mathematik
 Band 1. Von D. Hilbert und P. Bernays. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriick-sichtigung der Anwendungsgebiete, herausgegeben von R. Courant, Band 40.) Pp. xii + 471. (Berlin: Julius Springer, 1934.) 37.80 gold marks.

Grundlagen der Mathematik

Does geometry constitute a core set of intuitions present in all humans, regardless of their language or schooling? We used two nonverbal tests to probe the conceptual primitives of geometry in the Mundurukú, an isolated Amazonian indigene group. Mundurukú children and adults spontaneously made use of basic geometric concepts such as points, lines, parallelism, or right angles to detect intruders in simple pictures, and they used distance, angle, and sense relationships in geometrical maps to locate hidden objects. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

Core Knowledge of Geometry in an Amazonian Indigene Group@@@Mastering the Geometry of the Jungle (Or Doin' What Comes Naturally)

A history of vector analysis: The evolution of the idea of a vectorial system: by Michael J. Crowe. 270 pages, diagrams, 6 x 9 in. Notre Dame, Indiana, University of Notre Dame Press, 1967.

Fully measurable grand Lebesgue spaces

Anatriello and Fiorenza (J Math Anal Appl 422:783–797, 2015) introduced the fully measurable grand Lebesgue spaces on the interval $$(0,1)\subset \mathbb R$$
 , which contain some known Banach spaces of functions, among which there are the classical and the grand Lebesgue spaces, and the $$EXP_\alpha $$
 spaces $$(\alpha >0)$$
 . In this paper we introduce the weighted fully measurable grand Lebesgue spaces and we prove the boundedness of the Hardy–Littlewood maximal function. Namely, let $$\begin{aligned} \Vert f\Vert _ {p[\cdot ],\delta (\cdot ), w}={{\mathrm{ess\,sup}}}_{x\in (0,1)} \left( \int _0^1 (\delta (x)f(t))^{p(x)} w(t)\mathrm{dt}\right) ^{\frac{1}{p(x)}}, \end{aligned}$$
 where w is a weight, $$0<\delta (\cdot )\le 1\le p(\cdot )<\infty $$
 , we show that if $$\displaystyle {p^+}=\Vert p\Vert _\infty <+\infty $$
 , the inequality $$\begin{aligned}\Vert Mf\Vert _{p[\cdot ],\delta (\cdot ),w} \le c\Vert f\Vert _{p[\cdot ],\delta (\cdot ),w} \end{aligned}$$
 holds with some constant c independent of f if and only if the weight w belongs to the Muckenhoupt class $$A_{p^+}$$
 .

Weighted fully measurable grand Lebesgue spaces and the maximal theorem

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid The results also extend similar properties of Fibonacci-like sequences

Tribonacci-like sequences and generalized Pascal's pyramids

The norm of the grand Lebesgue spaces is defined through the supremum of Lebesgue norms, balanced by an infinitesimal factor. In this paper we consider the spaces defined by a norm with an analogous expression, where Lebesgue norms are replaced by grand Lebesgue norms. Without the use of interpolation theory, we prove an iteration-type theorem, and we establish that the new norm is again equivalent to the norm of grand Lebesgue spaces. We prove that the expression involved satisfy the axioms of Banach Function Spaces, and we find explicit values of the constants of the equivalence. Analogous results are proved for small Lebesgue spaces.

Giuseppina Anatriello

Papers

Fully measurable grand Lebesgue spaces

Weighted fully measurable grand Lebesgue spaces and the maximal theorem

Tribonacci-like sequences and generalized Pascal's pyramids

Iterated grand and small Lebesgue spaces

Fully measurable small Lebesgue spaces