By Branislav Kisacanin

ISBN-10: 0306459671

ISBN-13: 9780306459672

A steady advent to the hugely refined international of discrete arithmetic, *Mathematical difficulties and Proofs* provides subject matters starting from basic definitions and theorems to complex themes -- reminiscent of cardinal numbers, producing features, homes of Fibonacci numbers, and Euclidean set of rules. this wonderful primer illustrates greater than one hundred fifty suggestions and proofs, completely defined in transparent language. The beneficiant historic references and anecdotes interspersed through the textual content create attention-grabbing intermissions that will gasoline readers' eagerness to inquire additional concerning the subject matters and a few of our best mathematicians. the writer publications readers via the method of fixing enigmatic proofs and difficulties, and assists them in making the transition from challenge fixing to theorem proving.

right away a considered necessary textual content and an relaxing learn, *Mathematical* *Problems and Proofs* is a superb entrée to discrete arithmetic for complicated scholars attracted to arithmetic, engineering, and technology.

**Read or Download Mathematical Problems and Proofs: Combinatorics, Number Theory, and Geometry PDF**

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**Additional info for Mathematical Problems and Proofs: Combinatorics, Number Theory, and Geometry**

**Example text**

Let E be a vector space and C C E a nonempty convex set. Let G : C --+ 2c be a set-valued map such that G* : C --+ 2c is not a KKM-map. Then: (i) there exists a point wEe with w E conv(Gw), (ii) if G has convex values, then G has a fixed point. PROOF. Because (i)=>(ii), we need only establish (i). As, by assumption, G* is not a KKM-map, there is w E conv{xl , ... , xn} for some Xl, ... , Xn E C such that n n n i=l i=l i=l Consequently, Xi E Gw for each i E [n], and so w E conv(Gw). o Before we give a few examples of KKM-maps, we recall some definitions.

C . 1) Let (X , d) be complete, Xo E X and r > 0. Let T : K(xo, r) -> CB(X) be an a-contractive map (0 ::; a < 1) such that d(Txo, xo) < (1 - a)r. Prove: T has a fixed point. [Define by induction a sequence {Yn} in K(xo , r) such that Yn +l E T(Yn) and d(Yn+I' Yn) ::; a n (1 - a)r. Then, using the estimate d(Yn+p , Yn) ::; [1 + a + a 2 + ... + a P- I Ja n (1- a)r, show that {Yn} is a Cauchy sequence and hence converges to a c ertain Yo E K(xo,r) . 2) Let (X, d) be complete, U C X open , and {Ht : U -> CB(X)} an a-contractive family of maps which are fixed point free on the boundary au.

9). , X f--t f(x, y) is concave for each y E Y and y f--t f(x, y) is convex for each X E X). Then the map G : X x Y --+ 2xxY defined by G(x,y) = {(x',y') E Ex x E y I f(x,y') - f(x',y):s O} 39 §3. Results Based on Convexity is strongly KKM. Indeed, (x, y) E G(x, y) for each (x, y) E X x Y, and because (x,y) f-+ f(x,y') - f(x' , y) is concave, the cofibers G*(x',y') = {(x,y) E X x Y I f(x , y') - f(x',y) > O} of G are convex. (ii) Let C be a convex subset of a vector space E and 9 : ex C -+ R be a function such that: (a) g(x, x) :s: 0 for each x E C , (b) x f-+ g( x, y) is quasi-concave on C for each y E C.

### Mathematical Problems and Proofs: Combinatorics, Number Theory, and Geometry by Branislav Kisacanin

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