Mathematical Tapas: Volume 1 (for Undergraduates) by Jean-Baptiste Hiriart-Urruty

By Jean-Baptiste Hiriart-Urruty

This publication features a number of routines (called “tapas”) at undergraduate point, ordinarily from the fields of actual research, calculus, matrices, convexity, and optimization.
Most of the issues awarded listed here are non-standard and a few require wide wisdom of other mathematical matters that allows you to be solved. the writer offers a few tricks and (partial) solutions and in addition places those rigorously selected routines into context, provides details on their origins, and reviews on attainable extensions. With stars marking the degrees of hassle, those tapas exhibit or end up anything attention-grabbing, problem the reader to unravel and research, and will have mind-blowing results.

This first quantity of Mathematical Tapas will attract mathematicians, encouraged undergraduate scholars from science-based parts, and people ordinarily drawn to mathematics.

Show description

Read Online or Download Mathematical Tapas: Volume 1 (for Undergraduates) PDF

Similar linear books

Mengentheoretische Topologie

Eine verständliche und vollständige Einführung in die Mengentheoretische Topologie, die als Begleittext zu einer Vorlesung, aber auch zum Selbststudium für Studenten ab dem three. Semester bestens geeignet ist. Zahlreiche Aufgaben ermöglichen ein systematisches Erlernen des Stoffes, wobei Lösungshinweise bzw.

Combinatorial and Graph-Theoretical Problems in Linear Algebra

This IMA quantity in arithmetic and its functions COMBINATORIAL AND GRAPH-THEORETICAL difficulties IN LINEAR ALGEBRA relies at the court cases of a workshop that used to be a vital part of the 1991-92 IMA software on "Applied Linear Algebra. " we're thankful to Richard Brualdi, George Cybenko, Alan George, Gene Golub, Mitchell Luskin, and Paul Van Dooren for making plans and enforcing the year-long software.

Linear Algebra and Matrix Theory

This revision of a widely known textual content contains extra subtle mathematical fabric. a brand new part on purposes presents an creation to the fashionable therapy of calculus of numerous variables, and the concept that of duality gets elevated assurance. Notations were replaced to correspond to extra present utilization.

Extra info for Mathematical Tapas: Volume 1 (for Undergraduates)

Sample text

A Show that I(r) is minimized at one, and only one, point r∗ . 37 146. Impossible mean-value constraints Let μ ∈ [0, 1] . We look for continuous functions f : [0, 1] → R satisfying the following three mean-value constraints: 1 f (x) dx = 1; (1) xf (x) dx = μ; (2) x2 f (x) dx = μ2 . (3) 0 1 0 1 0 What are they? 147. Let f : R → R be a continuously differentiable function. We define g : R2 → R as follows: f (x) − f (y) g(x, y) = if x = y; g(x, x) = f (x). x−y 1◦ ) Is g a continuous function? 2◦ ) We suppose that f is twice continuously differentiable on R.

C2n+1 (f ) = c−(2n+1) (f ) = 0 for all n = 1, 2, ... 3◦ ) Deduce the solutions of the posed problem. A general “product breaking” formula for the cosine function 109. We know that 1 [cos(a + b) + cos(a − b)] 2 1 = [cos(a + b) + cos(−a − b) + cos(a − b) + cos(−a + b)] 4 (since cosine is an even function). cos a cos b = (1) (2) We intend to generalize these formulas. Prove that, given real numbers a1 , a2 , ... , an (n cos a1 cos a2 ... cos an = = 1 2n 2), we have: cos(ε1 a1 + ε2 a2 + ... + εn an ) (3) εi =+1 or −1 1 2n−1 cos(a1 + ε2 a2 + ...

184. A sufficient condition for positive definiteness of A using an auxiliary positive definite B Let A and B be two symmetric real (n, n) matrices. Prove that 185. ⎞ B positive definite ⎠ ⇒ (A positive definite) . ⎝ and AB + BA positive definite ⎛ (1) An original characterization of positive definiteness Let A ∈ Mn (R) be symmetric and invertible. Let H denote a vector space of Rn , not reduced to the extreme cases {0} or Rn (to avoid trivialities). The orthogonal vector space of H is denoted by H ⊥ .

Download PDF sample

Rated 4.30 of 5 – based on 16 votes