# 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.

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**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 ⊥ .