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# Generalizations of the Nash Equilibrium Theorem in the KKM TheoryAcademic research paper on "Mathematics"

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## Academic research paper on topic "Generalizations of the Nash Equilibrium Theorem in the KKM Theory"

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 234706,23 pages doi:10.1155/2010/234706

Research Article

Generalizations of the Nash Equilibrium Theorem in the KKM Theory

Sehie Park12

1 The National Academy of Sciences, Seoul 137-044, Republic of Korea

2 Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea Correspondence should be addressed to Sehie Park, shpark@math.snu.ac.kr

Received 5 December 2009; Accepted 2 February 2010 Academic Editor: Anthony To Ming Lau

Copyright © 2010 Sehie Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. These results are compared with previously known cases for G-convex spaces. Consequently, our results unify and generalize most of previously known particular cases of the same nature. Finally, we add some detailed historical remarks on related topics.

1. Introduction

In 1928, John von Neumann found his celebrated minimax theorem [1] and, in 1937, his intersection lemma [2], which was intended to establish easily his minimax theorem and his theorem on optimal balanced growth paths. In 1941, Kakutani [3] obtained a fixed point theorem for multimaps, from which von Neumann's minimax theorem and intersection lemma were easily deduced.

In 1950, John Nash [4, 5] established his celebrated equilibrium theorem by applying the Brouwer or the Kakutani fixed point theorem. In 1952, Fan [6] and Glicksberg [7] extended Kakutani's theorem to locally convex Hausdorff topological vector spaces, and Fan generalized the von Neumann intersection lemma by applying his own fixed point theorem. In 1972, Himmelberg [8] obtained two generalizations of Fan's fixed point theorem [6] and applied them to generalize the von Neumann minimax theorem by following Kakutani's method in [3] .

In 1961, Ky Fan [9] obtained his KKM lemma and, in 1964 [10], applied it to another intersection theorem for a finite family of sets having convex sections. This was applied in 1966 [11] to a proof of the Nash equilibrium theorem. This is the origin of the application of the KKM theory to the Nash theorem. In 1969, Ma [12] extended Fan's intersection theorem

[10] to infinite families and applied it to an analytic formulation of Fan type and to the Nash theorem for arbitrary families.

Note that all of the above results are mainly concerned with convex subsets of topological vector spaces; see Granas [13]. Later, many authors tried to generalize them to various types of abstract convex spaces. The present author also extended them in our previous works [14-28] in various directions. In fact, the author had developed theory of generalized convex spaces (simply, G-convex spaces) related to the KKM theory and analytical fixed point theory. In the framework of G-convex spaces, we obtained some minimax theorems and the Nash equilibrium theorems in our previous works [17,18, 21, 22], based on coincidence theorems or intersection theorems for finite families of sets, and in [22] , based on continuous selection theorems for the Fan-Browder maps.

In our recent works [24-26], we studied the foundations of the KKM theory on abstract convex spaces. The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its "open" version. We noticed that many important results in the KKM theory are closely related to KKM spaces or spaces satisfying the partial KKM principle. Moreover, a number of such results are equivalent to each other.

On the other hand, some other authors studied particular types of KKM spaces and deduced some Nash-type equilibrium theorem from the corresponding partial KKM principle, for example, [17, 21, 29-33], explicitly, and many more in the literature, implicitly. Therefore, in order to avoid unnecessary repetitions for each particular type of KKM spaces, it would be necessary to state clearly them for spaces satisfying the partial KKM principle. This was simply done in [27].

In this paper, we study several stages of such developments from the KKM principle to the Nash theorem and related results within the frame of the KKM theory of abstract convex spaces. In fact, we clearly show that a sequence of statements from the partial KKM principle to the Nash equilibria can be obtained for any space satisfying the partial KKM principle. This unifies previously known several proper examples of such sequences for particular types of KKM spaces. More precisely, our aim in this paper is to obtain generalized forms of the KKM space versions of known results due to von Neumann, Sion, Nash, Fan, Ma, and many followers. These results are mainly obtained by (1) fixed point method, (2) continuous selection method, or (3) the KKM method. In this paper, we follow method (3) and will compare our results to corresponding ones already obtained by method (2).

In Section 2, we state basic facts and examples of abstract convex spaces in our previous works [24-26]. Section 3 deals with a characterization of the partial KKM principle and shows that such principle is equivalent to the generalized Fan-Browder fixed point theorem. In Section 4, we deduce a general Fan-type minimax inequality from the partial KKM principle. Section 5 deals with various von Neumann-Sion-type minimax theorems for abstract convex spaces.

In Section 6, a collective fixed point theorem is deduced as a generalization of the Fan-Browder fixed point theorem. Section 7 deals with the Fan-type intersection theorems for sets with convex sections in product abstract convex spaces satisfying the partial KKM principle. In Section 8, we deduce a Fan-type analytic alternative and its consequences. Section 9 is devoted to various generalizations of the Nash equilibrium theorem and their consequences. Finally, in Section 10, some known results related to the Nash theorem and historical remarks are added.

This paper is a revised and extended version of [22, 27] and a supplement to [24-26], where some other topics on abstract convex spaces can be found.

2. Abstract Convex Spaces and the KKM Spaces

Multimaps are also called simply maps. Let (D) denote the set of all nonempty finite subsets of a set D. Recall the following in [24-26].

Definition 2.1. An abstract convex space (E, D; r) consists of a topological space E, a nonempty set D, and a multimap r : (D) ^ E with nonempty values rA := r(A) for A e (D).

For any D' c D, the r-convex hull of D' is denoted and defined by

corD' := U{rA | A e (D')} c E. (2.1)

A subset X of E is called a r-convex subset of (E, D; r) relative to D' if for any N e (D') we have that rN c X, that is, corD' c X.

When D c E, the space is denoted by (E d D; r). In such case, a subset X of E is said to be r-convex if cor(X n D) c X; in other words, X is r-convex relative to D' := X n D. In case E = D, let (E; r) := (E,E; r).

Example 2.2. The following are known examples of abstract convex spaces.

(1) A triple (An d V;co) is given for the original KKM theorem [34], where An is the standard n-simplex, V is the set of its vertices {e^n=0, and co: (V) ^ An is the convex hull operation.

(2) A triple (X d D; r) is given, where X and D are subsets of a t.v.s. E such that co D c X and r := co. Fan's celebrated KKM lemma [9] is for (E d D;co).

(3) A convex space (X d D; r) is a triple where X is a subset of a vector space such that co D c X, and each rA is the convex hull of A e (D) equipped with the Euclidean topology. This concept generalizes the one due to Lassonde for X = D; see [35]. However he obtained several KKM-type theorems w.r.t. (X d D; r).

(4) A triple (X d D; r), is called an H-space if X is a topological space and r = {rA} is a family of contractible (or, more generally, ^-connected) subsets of X indexed by A e (D) such that rA c rB whenever A c B e (D).If D = X, then (X; r) := (X,X; r) is called a c-space by Horvath [36, 37].

(5) Hyperconvex metric spaces due to Aronszajn and Panitchpakdi are particular cases of c-spaces; see [37] .

(6) Hyperbolic spaces due to Reich and Shafrir [38] are also particular cases of c-spaces. This class of metric spaces contains all normed vector spaces, all Hadamard manifolds, the Hilbert ball with the hyperbolic metric, and others. Note that an arbitrary product of hyperbolic spaces is also hyperbolic.

(7) Any topological semilattice (X, <) with path-connected interval is introduced by Horvath and Llinares [39] .

(8) A generalized convex space or a G-convex space (X, D; r) due to Park is an abstract convex space such that for each A e (D) with the cardinality |A| = n + 1 there exists a continuous function : An ^ r(A) such that J e (A) implies that $a(Aj) c r(J). Here, Aj is the face of An corresponding to J e (A), that is, if A = {a0, a1,.. .,an} and J = {aio,aii,...,aik} c A,then Aj = co{efo,eh,...,eh}. For details, see references of [17, 21, 22, 40-42]. (9) A -space (X,D; {$A}Ae(D}) consists of a topological space X, a nonempty set D, and a family of continuous functions : An ^ X (that is, singular n-simplexes) for A e (D) with |A| = n + 1. Every $A-space can be made into a G-convex space; see [43]. Recently -spaces are called GFC-spaces in [44] and FC-spaces [43] or simplicial spaces [45] when X = D. (10) Suppose that X is a closed convex subset of a complete R-tree H, and for each A e (X), rA := convH(A), where convH(A) is the intersection of all closed convex subsets of H that contain A; see Kirk and Panyanak [46]. Then (H d X; r) is an abstract convex space. (11) A topological space X with a convexity in the sense of Horvath [47] is another example. (12) A B-space due to Briec and Horvath [30] is an abstract convex space. Note that each of (2)-(12) has a large number of concrete examples and that all examples (1)-(9) are G-convex spaces. Definition 2.3. Let (E, D; r) be an abstract convex space. If a multimap G : D ^ E satisfies Ta c G(A) := J G(y) VA e (D), (2.2) then G is called a KKM map. Definition 2.4. The partial KKM principle for an abstract convex space (E, D; r) is the statement that, for any closed-valued KKM map G : D ^ E, the family {G(y)}yeD has the finite intersection property. The KKM principle is the statement that the same property also holds for any open-valued KKM map. An abstract convex space is called a KKM space if it satisfies the KKM principle. In our recent works [24-26], we studied the foundations of the KKM theory on abstract convex spaces and noticed that many important results therein are related to the partial KKM principle. Example 2.5. We give examples of KKM spaces as follows. (1) Every G-convex space is a KKM space [18]. (2) A connected linearly ordered space (X, <) can be made into a KKM space [26]. (3) The extended long line L* is a KKM space (L*, D; r) with the ordinal space D := [0, Q]; see [26]. But L* is not a G-convex space. (4) For a closed convex subset X of a complete R-tree H, and rA := convH(A) for each A e (X), the triple (H d X; r) satisfies the partial KKM principle; see [46]. Later we found that (H d X; r) is a KKM space [48]. (5) Horvath's convex space (X; r) with the weak Van de Vel property is a KKM space, where rA := [[A]] for each A e (X);see [47, 48]. (6) A B-space due to Briec and Horvath [30] is a KKM space. Now we have the following diagram for triples (E, D; r): simplex convex subset of a t.v.s. Lassonde-type convex space H-space G-convex space ^^$A-space KKM space

space satisfying the partial KKM principle abstract convex space.

It is not known yet whether there is a space satisfying the partial KKM principle that is not a KKM space.

3. The KKM Principle and the Fan-Browder Map

Let (E, D; r) be an abstract convex space.

Recall the following equivalent form of [26, Theorem 8.2].

Theorem 3.1. Suppose that (E, D; r) satisfies the partial KKM principle and a map G : D ^ E satisfies the following.

(1.1) G is closed valued.

(1.2) G is a KKM map (i.e., rA c G(A) for all A e (D)).

(1.3) There exists a nonempty compact subset K of E such that one of the following holds:

(i) K = E,

(ii) k = n(G(z) | z e M} for some M e (D),

(iii) for each N e (D), there exists a compact r-convex subset LN of E relative to some D' c D such that N c D' and

Ln n p| G(z) c K. (3.1)

Then K n n{G(z) | z e D} / 0.

Remark 3.2. Conditions (i)—(iii) in (1.3) are called compactness conditions or coercivity conditions. In this paper, we mainly adopt simply (i), that is, (E, D; r) is compact. However, most of results can be reformulated to the ones adopting (ii) or (iii).

Definition 3.3. For a topological space X and an abstract convex space (E,D; r), a multimap T : X ^ E is called a O-map or a Fan-Browder map provided that there exists a multimap S : X ^ D satisfying the follwing:

(a) for each x e X, corS(x) c T(x) (i.e., N e (S(x)) implies that rN c T(x)),

(b) X = UzeMInt S-(z) for some M e (D).

Here, Int denotes the interior with respect to E and, for each z e D, S-(z) := {x e X | z e S(x)}.

There are several equivalent formulations of the partial KKM principle; see [26]. For example, it is equivalent to the Fan-Browder-type fixed point theorem as follows.

Theorem 3.4 (see [26]). An abstract convex space (E,D; r) satisfies the partial KKM principle if and only if any O-map T : E ^ E has a fixed point x0 e E, that is, x0 e T (xo).

The following is known.

Lemma 3.5. Let {(Xi,Di; r^}^ be any family of abstract convex spaces. Let X := nieIXi be equipped with the product topology and D = nieIDi. For each i e I, let ni : D ^ Di be the projection. For each A e (D), define r(A) := nieIri(ni(A)). Then (X,D; r) is an abstract convex space.

Let {(Xi, Di; ri) }ieI be a family of G-convex spaces. Then (X, D; r) is a G-convex space.

It is not known yet whether this holds for KKM spaces.

From now on, for simplicity, we are mainly concerned with compact abstract convex spaces (E; r) satisfying the partial KKM principle. For example, any compact G-convex space, any compact H-space, or any compact convex space is such a space.

4. The Fan-Type Minimax Inequalities

Recall that an extended real-valued function f : X ^ R, where X is a topological space, is lower [resp., upper] semicontinuous (l.s.c.) (resp., u.s.c.) if {x e X | f (x) > r} (resp., {x e X | f (x) < r}) is open for each r e R.

For an abstract convex space (E d D; r), an extended real-valued function f : E ^ R is said to be quasiconcave (resp., quasiconvex) if {x e E | f (x) > r} (resp., {x e E | f (x) < r}) is r-convex for each r e R.

From the partial KKM principle we can deduce a very general version of the Ky Fan minimax inequality as follows.

Theorem 4.1. Let (X, D; r) be an abstract convex space satisfying the partial KKM principle, f : D x X ^ R,g : X x X ^ R extended real functions, and y e R such that

(3.1) for each z e D, {y e X | f (z,y) < y} is closed,

(3.2) for each y e X, cor{z e D | f (z,y) >j}c{x e X | g(x,y) > j},

(3.3) the compactness condition (1.3) holds for G(z) := {y e X | f (z,y) < y}.

Then either (i) there exists a x e X such that f (z,x) < y for all z e D or (ii) there exists an x0 e X such that g(xo,xo) > j-

Proof. Let G : D ^ X be a map defined by G(z) := {y e X | f (z,y) < j} for z e D. Then each G(z) is closed by (3.1).

Case (i): G is a KKM map.

By Theorem 3.1, we have f)zeD G(z) / 0. Hence, there exists a x e X such that x e G(z) for all z e D, that is, f (z, x) < j for all z e D. Case (ii): G is not a KKM map.

Then there exists N e (D) such that rN/UzeNG(z). Hence there exists an xo e rN such that x0 / G(z) for each z e N, or equivalently f (z,x0) > j for each z e N. Since {z e D | f (z,x0) > j} contains N, by (3.2), we have x0 e rN c {x e X | g(x,x0) > y}, and hence, g(xo,xo) >y. □

Corollary 4.2. Under the hypothesis of Theorem 4.1, let j := supxeXg(x, x). Then

infsupf (z,y) < supg(x,x). (4.1)

yeX zeD xeX

Example 4.3. (1) For a compact convex subset X = D of a t.v.s. and f = g, if f (-,y) is quasiconcave, then (3.2) holds; and if f (x, ■) is l.s.c., then (3.1) holds. Therefore, Corollary 4.2 generalizes the Ky Fan minimax inequality [49].

(2) For a convex space X = D and f = g, Corollary 4.2 reduces to Cho et al. [50, Theorem 9].

(3) There is a very large number of generalizations of the Fan minimax inequality for convex spaces, H-spaces, G-convex spaces, and others. These would be particular forms of Corollary 4.2. For example, see Park [18, Theorem 11], where (X, D; r) is a G-convex space.

(4) Some particular versions of Corollary 4.2 were given in [27].

5. The von Neumann-Sion-Type Minimax Theorems

Let (X; ri) and (Y; r2) be abstract convex spaces. For their product, as in the Lemma 3.5 we can define Txxy(A) := ri(n(A)) x ^(.^(A)) for A e (X x Y).

Theorem 5.1. Let (E; r) := (X x Y; rXxY) be the product abstract convex space, and let f,s,t,g : X x Y ^ R be four functions, then

p := infsupf (x,y), v := supmfg(x,y). (5.1)

yeYxeX xeXyeY

Suppose that

(4.1) f (x,y) < s(x,y) < t(x,y) < g(x,y) for each (x,y) e X x Y,

(4.2) for each r < p and y e Y, {x e X | s(x,y) > r} is r1-convex; for each r > v and x e X, {y e Y | t(x,y) <r} is r2-convex,

(4.3) for each r >v, there exists a finite set {xi}r=1 c X such that

Y = y Int {y e Y | f(xuy)>r}, (5.2)

(4.4) for each r < p, there exists a finite set {yj }n=i c Y such that

X = y Int{x e X | g(x,yj)<r}. (5.3)

If (E; r) satisfies the partial KKM principle, then

p = infsupf x y) < ¡mpmfgtx y) = v. (5.4)

yeYxeX xeXVeY

Proof. Suppose that there exists a real c such that

v = supinfg(x,y) < c < infsupf (x,y) = p. (5.5)

xeXyeY yeY xeX

For the abstract convex space

(E, D; r) := (x x Y, {{xuyj )}..; w), (5.6)

define two maps S : E ^ D,T : E ^ E by

S {xi,y-j) := Int {x e X | g(x,yj) < c} x Int {y e Y | f(xiry) > c), (5.7)

T(x,y) := {x e X | s(x,y) > c) x {y e Y | t(x,y) < c], (5.8)

for (xi,yj) e D and (x,y) e E, respectively. Then each T(x,y) is nonempty and T-convex and E is covered by a finite number of open sets S~(xi,yj)'s. Moreover,

S(x,y) c {{^Vj) | g(x,yj) < c> f(xuy) > c

_ _ _ _ (5.9)

c { (x,y) | s(x,y) > c,t(x,y) < c) c T(x,y).

This implies that corS(x,y) c T(x,y) for all (x,y) e E. Then T is a ®-map. Therefore, by Theorem 3.4, we have (xo,yo) e X x Y such that (xo,yo) e T(x0,y0). Therefore, c < s(x0,y0) < t(x0,y0) < c, a contradiction. □

Example 5.2. For convex spaces X, Y, and f = s = t = g, Theorem 5.1 reduces to that by Cho etal. [50, Theorem 8].

Corollary 5.3. Let (X; r1) and (Y; r2) be compact abstract convex spaces, let (E; r) := (X x Y; rXxY) be the product abstract convex space, and let f,g : X x Y ^ R be functions satisfying the following:

(1) f (x,y) < s(x,y) < t(x,y) < g(x,y) for each (x,y) e X x Y,

(2) for each x e X,f (x, ■) is l.s.c. and t(x, ■) is quasiconvex on Y,

(3) for each y e Y, s(■, y) is quasiconcave and g(■, y) is u.s.c. on X.

If (E; r) satisfies the partial KKM principle, then

minsupf (X y < maxinfg (x y. (5.10)

yeY xeX xeX yeY

Proof. Note that y ^ sup^Xf (x,y) is l.s.c. on Y and x ^ infyeYg(x,y) is u.s.c. on X. Therefore, both sides of the inequality exist. Then all the requirements of Theorem 5.1 are satisfied. □

Example 5.4. (1) Particular or slightly different versions of Corollary 5.3 are obtained by Liu [51], Granas [13, Theoremes 3.1 et 3.2], and Shih and Tan [52, Theorem 4] for convex subsets of t.v.s.

(2) For f = s,g = t, Corollary 5.3 reduces to [27, Theorem 3].

For the case f = s = t = g, Corollary 5.3 reduces to the following.

Corollary 5.5 (see [27]). Let (X;ri) and (Y;r2) be compact abstract convex spaces and let f : X x Y ^ R bean extended real function such that

(1) for each x e X,f (x, ■) is l.s.c. and quasiconvex on Y,

(2) for each y e Y,f (■,y) is u.s.c. and quasiconcave on X.

If (X x Y; rXxY) satisfies the partial KKM principle, then (i) f has a saddle point (x0, y0) e X x Y, ( ii) one has

max min f fx, y) = min max f fx, y). (5.11)

xeX yeYJ y J' yeY xeXJ y J'

Example 5.6. We list historically well-known particular forms of Corollary 5.5 in chronological order as follows.

(1) von Neumann [1], Kakutani [3]. X and Y are compact convex subsets of Euclidean spaces and f is continuous.

(2) Nikaido [53]. Euclidean spaces above are replaced by Hausdorff topological vector spaces, and f is continuous in each variable.

(3) Sion [54]. X and Y are compact convex subsets of topological vector spaces in Corollary 5.5.

(4) Komiya [55, Theorem 3]. X and Y are compact convex spaces in the sense of Komiya.

(5) Horvath [36, Proposition 5.2]. X and Y are c-spaces with Y being compact and without assuming the compactness of X.

In these two examples, Hausdorffness of Y is assumed since they used the partition of unity argument.

(6) Bielawski [29, Theorem (4.13)]. X and Y are compact spaces having certain simplicial convexities.

(7) Park [17, Theorem 5]. X and Y are G-convex spaces.

In 1999, we deduced the following von Neumann-Sion type minimax theorem for G-convex spaces based on a continuous selection theorem:

Theorem 5.7 (see [17]). Let (X, ri) and (Y,r2) be G-convex spaces, Y Hausdorff compact, f : X x Y ^ R an extended real function, and p := supxeX infyeYf (x,y). Suppose that

(5.1) f (x, ■) is l.s.c. on Y and {y e Y | f (x,y) < r} is r2-convexfor each x e X and r > p,

(5.2)f (-,y) is u.s.c. on X and {x e X | f (x,y) > r} is r 1-convex for each y e Y and r > p.

sup minf (x,y) = min supf (x,y). (5.12)

xeX yeY yeY xeX

Example 5.8. (1) Komiya [55, Theorem 3]. X and Y are compact convex spaces in the sense of Komiya.

(2) Slightly different form of Theorem 5.7 can be seen in [17] with different proof.

6. Collective Fixed Point Theorems

We have the following collective fixed point theorem.

Theorem 6.1. Let {(Xi; ri) }"=1 be a finite family of compact abstract convex spaces such that (X; r) = (I! n=1Xi; r) satisfies the partial KKM principle, and for each i, T : X ^ X{ is a O-map. Then there

exists a point x e X such that x e T(x) := Yin=1Ti(x), that is, Xi = ni(x) e Ti(x) for each i = 1,2,... ,n.

Proof. Let Si : X ^ Xi be the companion map corresponding to the ®-map Ti. Define S : X ^ X by

S(x) := Si(x) for each x e X. (6.1)

We show that T is a ®-map with the companion map S. In fact, we have

x e S~(y) ^^ y e S(x) ^^ yi e Si(x) for each i ^^ x e S-(y) for each i, (6.2)

where y = {y1,...,yn}. Since each S'-(yi) is open, we have

(a) for each y e X, S~(y) = f)i=1 S-(yi) is open. Note that

M e (S(x)) n(M) e (Si(x)) Ti(xi(M)) c Ti(x), (6.3)

and hence,

Tm ^ri(ni(M)) c]JTi(x) = T(x). (6.4)

i=1 i=1

Therefore, we have

(b) for each x e X,M e (S(x)) implies that Tm c T(x).

Moreover, let x e X. Since Si : X ^ Xi is the companion map corresponding to the

®-map Ti, for each i, there exists j = j(i) such that

x e S-( yj ^ yirj e Si(x) ^ y e Si(x) = S(x) ^ x e S~( y), (6.5)

where y := (yi ,j(i),... ,yn,j(n)). Since X is compact, we have

(c) X = UzeM S-(z) for some M e (X).

Since (X; r) satisfies the partial KKM principle, by Theorem 3.4, the ®-map T has a fixed point. □

Example 6.2. (1) If n = 1, X is a convex space, and S = T, then Theorem 6.1 reduces to the well-known Fan-Browder fixed point theorem; see Park [56].

(2) For the case n = 1, Theorem 6.1 for a convex space X was obtained by Ben-El-Mechaiekh et al. [69, Theorem 1] and Simons [57, Theorem 4.3]. This was extended by many authors; see Park [56] .

We have already the following collective fixed point theorem for arbitrary family of G-convex spaces.

Theorem 6.3 (see [40]). Let {(X^ rf) }ieJ be a family of compact Hausdorff G-convex spaces, X = YlieIXi, and for each i e I, let Ti : X ^ Xi be a O-map. Then there exists a point x e X such that x e T(x) := nieITi(x), that is, xi = ni(x) e Ti(x) for each i e I.

Example 6.4. In case when (Xi; ri) are all H-spaces, Theorem 6.3 reduces to Tarafdar [58, Theorem2.3]. This is applied to sets with H-convex sections [58, Theorem3.1] and to existence of equilibrium point of an abstract economy [58, Theorem 4.1 and Corollary 4.1]. These results also can be extended to G-convex spaces and we will not repeat then here.

Remark 6.5. Each of Theorems 6.1, 7.1, 8.1, 9.1, and 9.4, respectively, in this paper is based on the KKM method and concerns with finite families of abstract convex spaces such that their product satisfies the partial KKM principle. Each of them has a corresponding Theorems 6.3, 7.3, 8.3, 9.2 and 9.6, respectively, based on continuous selection method for infinite families of Hausdorff G-convex spaces. Note that for finite families the Hausdorffness is redundant in these corresponding theorems.

7. Intersection Theorems for Sets with Convex Sections

In our previous work [17], from a G-convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem for n subsets of a cartesian product of n compact G-convex spaces. This was applied to obtain a von Neumann-sion-type minimax theorem and a Nash-type equilibrium theorem for G-convex spaces.

In the present section, we generalize the abovementioned intersection theorem to product abstract convex spaces satisfying the partial KKM principle.

The collective fixed point theorem in Section 6 can be reformulated to a generalization of various Fan-type intersection theorems for sets with convex sections as follows. Let {Xi}ieI be a family of sets, and let i e I be fixed. Let

X = ^Xj, Xi = n Xj. (7.1)

jeI jeI\{i}

If xi e Xi and j e I \ {i}, then let xj denote the jth coordinate of xi. If xi e Xi and xi e Xi, then let [xi,xi] e X be defined as follows: its ith coordinate is xi and for j / i the j th coordinate is xj. Therefore, any x e X can be expressed as x = [xi,xi] for any i e I, where xi denotes the projection of x in Xi.

Theorem 7.1. Let {(Xi; ri)}f=1 be a finite family of compact abstract convex spaces such that (X; r) = (11 n=iX r) satisfies the partial KKM principle and, for each i, let Ai and Bi be subsets of X satisfying the following.

(7.1) For each xi e Xi, 0 = cori Bi(xi) c Ai(xi) := {yi e Xi | [xi,yi] e Ai}.

(7.2) For each yi e Xi,Bi(yi) := {xi e Xi | [x\yi] e Bi} is open in Xi. Then nn=1 Ai = 0.

Proof. We apply Theorem 6.1 with multimaps Si,Ti : X ^ Xi given by Si(x) := Bi(xi) and Ti(x) := Ai(xi) for each x e X. Then for each i we have the following.

(a) For each x e X, we have 0 / cori Si(x) c Ti(x).

(b) For each yi e Xi, we have

x e S-(y^ yi e Si(x) = B^x^ [xi,y^ e B{ c Xi x Xi = X. (7.2)

Hence,

S-(yi) = {x = [x^x] e X | ^ e B(yl),xi e XJ = Bi(yi) x Xi. (7.3)

Note that S-(yi) is open in X = Xi x Xi and that Ti is a ®-map. Therefore, by Theorem 6.1, there exists x e X such that xi e Ti(x) = Ai(xi) for all i. Hence x = [xi,xi] e f)"=1 Ai / 0. □

Example 7.2. For convex spaces Xi, particular forms of Theorem 7.1 have appeared as follows:

(1) Fan [10, Theeoreme 1]. Ai = Bi for all i.

(2) Fan [11, Theorem 1']. n = 2 and A{ = Bi for i = 1,2.

From these results, Fan [11] deduced an analytic formulation, fixed point theorems, extension theorems of monotone sets, and extension theorems for invariant vector subspaces. For particular types of G-convex spaces, Theorem 7.1 was known as follows.

(3) Bielawski [29, Proposition (4.12) and Theorem (4.15)]. Xi have the finitely local convexity.

(4) Kirk et al. [32, Theorem 5.2]. Xi are hyperconvex metric spaces.

(5) Park [17, Theorem4], [18, Theorem 19]. In [17], from a G-convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem for n subsets of a cartesian product of n compact G-convex spaces. This was applied to obtain a von Neumann-Sion-type minimax theorem and a Nash-type equilibrium theorem for G-convex spaces.

(6) Park [27, Theorem 4]. We gave a different proof.

In [22], a collective fixed point theorem was reformulated to a generalization of various Fan-type intersection theorems for arbitrary number of sets with convex sections as follows.

Theorem 7.3 (see [22]). Let {(Xi; ri) }ieI be a family ofHausdorff compact G-convex spaces and, for each i e I, let Ai and Bi be subsets of X = n ieIXi satisfying the following.

(7.1)' For each xi e Xi, 0 = cori Bi(xi) c Ai(xi) := {yi e Xi | [xi,yi] e Ai}.(7.2)' For each yi e Xi, Bi(yi) := {xi e Xi | [xi,yi] e Bi} is open in Xi. Then nteiAi = 0.

Example 7.4. For convex subsets Xi of topological vector spaces, particular forms of Theorem 7.3 have appeared as follows.

(1) Ma [12, Theorem 2]. The case Ai = Bi for all i e I with a different proof is given.

(2) Chang [59, Theorem 4.2] obtained Theorem 7.3 with a different proof. The author also obtained a noncompact version of Theorem 7.3 as in [59, Theorem 4.3].

(3) Park [19, Theorem 4.2]. Xi are convex spaces.

Note that if I is finite in Theorem 7.3, the Hausdorffness is redundant by Theorem 7.1.

8. The Fan-Type Analytic Alternatives

From the intersection Theorem 7.1, we can deduce the following equivalent form of a generalized Fan-type minimax inequality or analytic alternative. Our method is based on that of Fan [9,10] and Ma [12].

Theorem 8.1. Let {(Xi; ri)}JL1 be a finite family of compact abstract convex spaces such that (X; r) = (11 n=1Xi; r) satisfies the partial KKM principle and, for each i, let fi,gi : X = X1 x Xi ^ R be real functions satisfying

(8.1) fi(x) < gi(x) for each x e X,

(8.2) for each xj e X1, xi ^ gi [x\ xi] is quasiconcave on Xi,

(8.3) for each xi e Xi,xi ^ fi[xi,xi] is l.s.c. on Xj.

Let {ii}"=1 be a family of real numbers. Then either (a) there exist an i and an xi e Xi such that

fijx^yi] < ti Vyi e Xi,

or (b) there exists an x e X such that

gi(x) >ti Vi = 1,2,..

Proof. Suppose that (a) does not hold, that is, for any i and any xj e X1, there exists an xi e Xi such that fi[xi,xi] > ti. Let

Ai := {x e X | gi(x) > ti}, Bi = {x e X | fi(x) > ti}

for each i. Then

(1) for each xj e Xj, 0 = Bj(xj) c Aj(xj),

(2) for each xj e X1, Ai(xi) is ^-convex,

(3) for each yi e Xi, Bi(yi) is open in Xj.

Therefore, by Theorem 7.1, there exists an x e f)"=i Ai. This is equivalent to (b). □

Example 8.2. Fan [9, Theoreme 2], [10, Theorem3]. Xi are convex subsets of t.v.s., and fi = gi for all i. From this, fan [9, 10] deduced Sion's minimax theorem [54], the Tychonoff fixed point theorem, solutions to systems of convex inequalities, extremum problems for matrices, and a theorem of Hardy-Littlewood-Polya.

From the intersection Theorem 7.3, we can deduce the following equivalent form of a generalized Fan-type minimax inequality or analytic alternative.

Theorem 8.3 (see [22]). Let {(x,; }ieI be a family of compact Hausdorff G-convex spaces and, for each i e I, let fi,gi : X = X* x X* ^ R be real functions as in Theorem 8.1. Then the conclusion of Theorem 8.1 holds.

Example 8.4. (1) Ma [12, Theorem 3]. Xi are convex subsets of t.v.s. and fi = gi for all i e I. (2) Park [19, Theorem 8.1]. Xi are convex spaces.

Remark 8.5. (1) We obtained Theorem 8.1 from Theorem 7.1. As was pointed out by Fan [9] for his case, we can deduce Theorem 7.1 from Theorem 8.1 by considering the characteristic functions of the sets Ai and Bi.

( 2) The conclusion of Theorems 8.1 and 8.3 can be stated as follows

minsup fi\xi,xi\>ti Vi, (8.4)

xieXiXi, eXi, L J

then (b) holds; see Fan [9,10].

(3) For I = (1,2}, Theorems 8.1 and 8.3 imply the Fan minimax inequality.

9. The Nash-Type Equilibrium Theorems

From Theorem 8.1, we obtain the following form of the Nash-Fan-type equilibrium theorems in [27] with different proofs.

Theorem 9.1. Let {(Xi; r¿) }"=1 be a finite family of compact abstract convex spaces such that (X; r) = (IIn=1Xi; r) satisfies the partial KKM principle and, for each i, let fi,gi : X = X1 x Xi ^ R be real functions such that

(9.0)fi(x) < gi(x) for each x e X,

(9.1) for each x1 e X1, xi ^ gi[xi,xi] is quasiconcave on Xi,

(9.2) for each x1 e X1, xi ^ fi[xi,xi] is u.s.c. on Xi,

(9.3) for each xi e Xi, x1 ^ fi[xi,xi] is l.s.c. on X1. Then there exists a point x e X such that

gi(x) > maxfi fx1,yÁ Vi = 1,2,...,n. (9.1)

yieXi L J

Proof. Since each Xi is compact, by (9.2), for any s > 0, ti := maxyieXifi[xi,yi] - e exists for all x1 e X1 and all i. Hence Theorem 8.1(a) does not hold. Then by Theorem 8.1(b), there exists an x e E such that gi(x) > ti = maxyieXifi[xi,yi] - e for all i e I. Since e > 0 is arbitrary, the conclusion follows. □

This is not comparable to the following generalized Nash-Ma type theorem:

Theorem 9.2 (see [22]). Let {(Xi; ri) }ieI be a family of compact Hausdorff G-convex spaces and, for each i e I, let fi,gi : X = X1 x Xi ^ R be real functions satisfying (9.0)-(9.3). Then there exists a point x e X such that

gi(x) > maxfi fx1, yÁ Vi e I. (9.2)

yieXi L J

Example 9.3. Park [19, Theorem 8.2]. Xi are convex spaces.

From Theorem 9.1 for fi = gi, we obtain the following form of the Nash-Fan-type equilibrium theorem for abstract convex spaces.

Theorem 9.4. Let {(Xi; ri) }"=1 be a finite family of compact abstract convex spaces such that (X; r) = d! i=1Xi; r) satisfies the partial KKM principle and, for each i e I, let fi: X ^ R be a function such that

(10.1) for each xi e Xi, xi ^ fi[xi,xi] is quasiconcave on Xi,

(10.2) for each xi e Xi, xi ^ fi[xi,xi] is u.s.c. on Xi,

(10.3) for each xi e Xi, xi ^ fi[xi,xi] is l.s.c. on Xi. Then there exists a point x e X such that

fi(x) = maxfi \xi, Vil Vi = 1,2,..., n. (9.3)

yieXi L J

Example 9.5. For continuous functions fi, a number of particular forms of Theorem 9.4 have appeared for convex subsets Xi of Hausdorff topological vector spaces as follows:

(1) Nash [5, Theorem 1] where Xi are subsets of Euclidean spaces,

(2) Nikaido and Isoda [60, Theorem 3.2],

(3) Fan [10, Theorem 4],

(4) Tanetal. [61, Theorem 2.1].

For particular types of G-convex spaces Xi and continuous functions fi, particular forms of Theorem 9.4 have appeared as follows.

(5) Bielawski [29, Theorem 4.16]. Xi have the finitely local convexity.

(6) Kirk et al. [32, Theorem 5.3]. Xi are hyperconvex metric spaces.

(7) Park [17, Theorem 6.1], [18, Theorem 20]. Xi are G-convex spaces.

(8) Park [21, Theorem 4.7]. A variant of Theorem 9.4 is under the hypothesis that (X; r) is a compact G-convex space and f\,...,fn : X ^ R are continuous functions.

(9) Gonzalez et al. [31]. Each Xi is a compact, sequentially compact L-space and each fi is continuous as in 8.

(10) Briec and Horvath [30, Theorem 3.2]. Each Xi is a compact B-convex set and each fi is continuous as in 8.

From Theorem 9.2, we obtain the following generalization of the Nash-Ma-type equilibrium theorem for G-convex spaces.

Theorem 9.6 (see [22]). Let {(Xi; ri) }ieI be a family of compact Hausdorff G-convex spaces and, for each i e I, let fi : X ^ R be a function satisfying conditions (10.1M10.3). Then there exists a point x e X such that

fi(x) = maxfi \xi, Vil Vi e I. (9.4)

yieXi L J

Example 9.7. For continuous functions fi and for convex subsets Xi of Hausdorff topological vector spaces, Theorem 9.6 was due to Ma [12, Theorem 4].

The point x in the conclusion of Theorems 9.4 or 9.6 is called a Nash equilibrium. This concept is a natural extension of the local maxima and the saddle point as follows. In case I is a singleton, we obtain the following.

Corollary 9.8. Let X be a closed bounded convex subset of a reflexive Banach space E and f : X ^ R a quasiconcave u.s.c. function. Then f attains its maximum on X, that is, there exists an x e X such that f (x) > f (x) for all x e X.

Proof. Let E be equipped with the weak topology. Then, by the Hahn-Banach theorem, f is still u.s.c. because f is quasiconcave, and X is still closed. Being bounded, X is contained in some closed ball which is weakly compact. Since any closed subset of a compact set is compact, X is (weakly) compact. Now, by Theorem 9.4 for a single family, we have the conclusion. □

Corollary 9.8 is due to Mazur and Schauder in 1936. Some generalized forms of Corollary 9.8 were known by Park et al. [62, 63].

For I = (1,2}, Theorem 9.4 reduces to Corollary 5.5 as follows.

Proof of Corollary 5.5 from Theorem 9.4. Let fi(x,y) := -f (x,y) and f2(x,y) := f (x,y). Then all of the requirements of Theorem 9.4 are satisfied. Therefore, by Theorem 9.4, there exists a point (x0,y0) e X x Y such that

fi( x0,yci) = maxfi(xc,y), fi(xo,yc) = maxf2(x,y0). (9.5)

yeY xeX

Therefore, we have

-f (xo,yo) = fi(xc,yc) > fi(xo,y) = -f(xc,y) Vy e Y, (9.6)

f(xc,yc) = fi{xc,yc) > fi{x,yc) = f{x,yc) Vx e X. (9.7)

f{x,yc) < f{xc,yc) < f(xc,y) V(x,y) e X x Y. (9.8)

Therefore

max/(x,yo) < f(xo,yo) < minf (xo,y). (9.9)

xeX yeX

This implies that

minmaxf (x,y) < f(x0,y0) < maxminf x,y). (9.10)

yeX xeX xeX yeX

On the other hand, we have trivially

minf(x, v) <maxf(x, v), (9.11)

and hence

maxminf (x,v) < minmaxf (x,v)- (9.12)

xeX yx ^ J/ v^X x(XJ y J'

Therefore, we have the conclusion. □

10. Historical Remarks on Related Results

(I) As we have seen in Sections 1-3, we have three methods in our subject as follows:

(1) fixed point method—applications of the Kakutani theorem and its various generalizations (e.g., acyclic-valued multimaps, admissible maps, or better admissible maps in the sense of Park); see [3-8,10,12,14-16,19, 20, 23, 28, 42,53, 64-68] and others,

(2) continuous selection method—applications of the fact that Fan-Browder-type maps have continuous selections under certain assumptions like Hausdorffness and compactness of relevant spaces; see [17, 22,36,39, 40,58, 69, 70] and others,

(3) the KKM method—as for the Sion theorem, direct applications of the KKM theorem, or its equivalents like the Fan-Browder fixed point theorem for which we do not need the Hausdorffness; see [9, 11, 17, 21, 24-27, 30, 31, 33, 35-37, 39, 43, 45, 4750,54,55,57,59, 71, 72] and others.

For Case (1), we will study it elsewhere and, in this paper, we are mainly concerned with Cases (2) and (3).

(II) An upper semicontinuous (u.s.c.) multimap with nonempty compact convex values is called a Kakutani map. The Fan-Glicksberg theorem was extended by Himmelberg [8] in 1972 for compact Kakutani maps instead of assuming compactness of domains. In 1990, Lassonde [67] extended the Himmelberg theorem to multimaps factorizable by Kakutani maps through convex sets in Hausdorff topological vector spaces. Moreover, Lassonde applied his theorem to game theory and obtained a von Neumann-type intersection theorem for finite number of sets and a Nash-type equilibrium theorem comparable to Debreu's social equilibrium existence theorem [66].

Fixed point theorems extending the Kakutani theorem can be applied to particular forms of results in this paper. Since such extended theorems usually assume Hausdorffness and certain (abstract) local convexity of the related space, their applicability is restrictive.

(III) In 1946, the Kakutani theorem was extended for acyclic maps by Eilenberg and Montgomery [73]. This result was applied by Debreu [66] to the social equilibrium existence theorem and related results.

(IV) Since 1996 [72], many authors have published some results of the present paper for hyperconvex metric spaces. For example, Kirk et al. in 2000 [32] established the KKM theorem, its equivalent formulations, fixed point theorems, and the Nash theorem for hyperconvex metric spaces. However, already in 1993, Horvath [37] found that hyperconvex metric spaces are a particular type of c-spaces.

(V) In 1998 [16], an acyclic version of the social equilibrium existence theorem of Debreu is obtained. This is applied to deduce acyclic versions of theorems on saddle points, minimax theorems, and the following Nash equilibrium theorem.

Corollary 10.1 (see [16]). Let {Xj}"=1 be a family of acyclic polyhedra, X = YXi=\Xi, and for each i, fi : X ^ R a continuous function such that

(0) for each xi e Xi and each a e R, the set

(VI) In the present paper, for abstract convex spaces, we notice that the partial KKM principle ^ the Fan-Browder fixed point theorem ^ the Nash equilibrium theorem, with or without additional intermediate steps. This procedure can be called "from the KKM principle to the Nash equilibria" (simply, "K to N"); see [27].

In 1999 [17], we obtained a "K to N" for G-convex spaces. These results extended and unified a number of known results for particular types of G-convex spaces; see also [1821, 42]. Therefore, the procedure also holds for Lassonde type-convex spaces, Horvath's c-space, hyperconvex metric spaces, and others.

(VII) In 2000 [20] and 2002 [23], we applied our fixed point theorem for compact compositions of acyclic maps on admissible (in the sense of Klee) convex subsets of a t.v.s. to obtain a cyclic coincidence theorem for acyclic maps, generalized von Neumann-type intersection theorems, the Nash type equilibrium theorems, and the von Neumann minimax theorem.

The following examples are generalized forms of quasi equilibrium theorem or social equilibrium existence theorems which directly imply generalizations of the Nash-Ma-type equilibrium existence theorem.

Theorem 10.2 (see [20]). Let {Xi }i=1 be a family of convex sets, each in a t.v.s. Ei, Ki a nonempty compact subset of Xi, Si : X ^ Ki a closed map, and fi,gi : X = Xi x Xi ^ R u.s.c. functions for each i.

Suppose that, for each i,

(i) gi(x) < fi(x) for each x e X,

(ii) the function Mi defined on X by

(10.1)

is empty or acyclic.

Then there exists a point a e X such that

(10.2)

Mi(x) = max gi(xl,y for x e X

(10.3)

is l.s.c., and

(in) for each x e X, the set

[y e Si(x) | fi(xl,y) > Mt(x)} (10.4)

is acyclic.

If X is admissible in E = TVj=iEj, then there exists an x e K such that, for each i,

xi e Si(x), f^xi,x^ > gi(xi/y^ Vy e Si(x). (10.5)

Theorem 10.3 (see [23]). Let X0 be a topological space and let (Xi}"=1 be a family of convex sets, each in a t.v.s. Ei. For each i = 0,1,...,n, let Si : Xi ^ Xi be a closed map with compact values, and fi/gi : X = n n=0Xi ^ R u.s.c. real-valued functions. Suppose that, for each i,

(i) gi(x) < fi(x) for each x e X,

(ii) the function Mi : Xi ^ R defined by

Mdx0 = max gdxi,yA for xi e Xi (10.6)

V ' yieSi(xi) V '

is l.s.c., and

(iii) for each xi e Xi, the set

[xi e Sifx1) I fift/x^ > Mi(xi)} (10.7)

is acyclic.

If X0 is admissible in E0 = Tl^Ej and if all the maps Si are compact except possibly Sn and Sn is u.s.c., then there exists an equilibrium point x e X, that is,

xi e Si[xl), fi(x) > max gi{xl,y) Vi e Zn+1. (10.8)

V ' yie Si(xi) v '

The following are particular results of ours for G-convex spaces or KKM spaces. (VIII) In 2001 [22] , we obtained generalized forms of the von Neumann-Sion-type minimax theorem, the Fan-Ma intersection theorem, the Fan-Ma type analytic alternative, and the Nash-Ma-equilibrium theorem for G-convex spaces. In [22], all G-convex spaces were assumed to be Hausdorff because the results are based on a selection theorem in [40], where Hausdorffness was indispensable.

As we have seen already in Theorems 7.3, 8.3, 9.2, and 9.6, results following the selection theorem in [40] hold for arbitrary family of Hausdorff G-convex spaces. But corresponding results following the partial KKM principle hold for finite families of abstract convex spaces. At present, we have no way to unify these three cases, namely, (1) fixed point method, (2) selection method, and (3) KKM method.

(IX) In 2001, for any topological semilattice (X, <) with path-connected interval introduced by Horvath and Llinares [39], the KKM theorem, the Fan-Browder theorem, and the Nash theorem are shown by Luo [33]. Note that such semilattice is known to be a G-convex space.

(X) Cain and Gonzalez [71] considered relationship among some subclasses of the class of G-convex spaces and introduced a subclass of the so-called L-spaces. In 2007, Gonzalez et al. [31] repeated to show that G-convex spaces and L-spaces satisfy the partial KKM principle. They added that L-spaces satisfy the properties of the Fan type minimax inequality, Fan-Browder-type fixed point, and the Nash-type equilibrium. All of such results are already known for more general G-convex spaces.

(XI) In 2008, Kulpa and Szymanski [45] introduced a series of theorems called Infimum Principles in simplicial spaces. As for applications, they derive fixed point theorems due to Schauder, Tychnoff, Kakutani, and Fan-Browder: minimax theorems, the Nash equilibrium theorem, the Gale-Nikaido-Debreu theorem, and the Ky Fan minimax inequality. Their study is based on and utilizes the techniques of simplicial structure and the Fan-Browder map. Recall that for any abstract convex spaces satisfying abstract KKM principle we can deduce such classical theorems without using any Infimum Principles. Moreover, we note that the newly defined L*-spaces in [45] are particular types of abstract convex spaces satisfying the abstract KKM principle.

(XII) In 2008, for B-spaces, Briec and Horvath [30] showed that some theorems mentioned in this paper hold, that is, Fan-Browder fixed point theorem, Himmelbergtype (in fact, Browder-type and Kakutani-type) fixed point theorems, Fan type minimax inequality, existence of Nash equilibria, and others. Note that B-spaces are KKM spaces [30, Corollary 2.2], and their authors depend the Pelegs type multiple KKM theorem.

(XIII) Finally, recall that there are several hundred published works on the KKM theory and we can cover only a part of them. For more historical background for the related fixed point theory and for more involved or related results in this paper, see the references [24-27, 41, 43,48] and the literature therein.

References

[1] J. von Neumann, "Zur theorie der gesellschaftsspiele," Mathematische Annalen, vol. 100, no. 1, pp. 295-320,1928.

[2] J. von Neumann, "Uber ein okonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes," Ergebnisse eines Mathematischen Kolloquiums, vol. 8, pp. 73-83,1937.

[3] S. Kakutani, "A generalization of Brouwer's fixed point theorem," Duke Mathematical Journal, vol. 8, pp. 457-459, 1941.

[4] J. F. Nash Jr., "Equilibrium points in N-person games," Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 48-49,1950.

[5] J. Nash, "Non-cooperative games," Annals of Mathematics, vol. 54, pp. 286-295,1951.

[6] K. Fan, "Fixed-point and minimax theorems in locally convex topological linear spaces," Proceedings of the National Academy of Sciences of the United States of America, vol. 38, pp. 121-126,1952.

[7] I. L. Glicksberg, "A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points," Proceedings of the American Mathematical Society, vol. 3, pp. 170-174,1952.

[8] C. J. Himmelberg, "Fixed points of compact multifunctions," Journal of Mathematical Analysis and Applications, vol. 38, pp. 205-207,1972.

[9] K. Fan, "A generalization of Tychonoff's fixed point theorem," Mathematische Annalen, vol. 142, pp. 305-310,1961.

[10] K. Fan, "Sur un theoreme minimax," Comptes Rendus des Seances de l'Academie des Sciences. Serie I, vol. 259, pp. 3925-3928,1964.

[11] K. Fan, "Applications of a theorem concerning sets with convex sections," Mathematische Annalen, vol. 163, pp. 189-203,1966.

T. Ma, "On sets with convex sections," Journal of Mathematical Analysis and Applications, vol. 27, pp. 413-416,1969.

A. Granas, "Sur quelques méthodes topologiques en analyse convexe," in Méthodes Topologiques en Analyse Convexe, vol. 110, pp. 11-77, Presses Universite de Montreal, Montreal, Canada, 1990.

A. Idzik and S. Park, "Leray-Schauder type theorems and equilibrium existence theorems," in Differential Inclusions and Optimal Control, vol. 2 of Lecture Notes in Nonlinear Analysis, pp. 191-197, University in Torun, Torun, Poland, 1998.

S. Park, "Applications of the Idzik fixed point theorem," Nonlinear Functional Analysis and Its Applications, vol. 1, pp. 21-56,1996.

S. Park, "Remarks on a social equilibrium existence theorem of G. Debreu," Applied Mathematics Letters, vol. 11, no. 5, pp. 51-54,1998.

S. Park, "Minimax theorems and the Nash equilibria on generalized convex spaces," in NLA98: Convex Analysis and Chaos, vol. 1 of Josai Mathematical Monographs, pp. 33-45, Josai University, Sakado, Japan, 1999.

S. Park, "Elements of the KKM theory for generalized convex spaces," The Korean Journal of Computational & Applied Mathematics, vol. 7, no. 1, pp. 1-28, 2000.

S. Park, "Fixed points, intersection theorems, variational inequalities, and equilibrium theorems," International Journal of Mathematics and Mathematical Sciences, vol. 24, no. 2, pp. 73-93, 2000. S. Park, "Acyclic versions of the von Neumann and Nash equilibrium theorems," Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 83-91, 2000.

S. Park, "New topological versions of the Fan-Browder fixed point theorem," Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 1, pp. 595-606, 2001.

S. Park, "Generalizations of the Nash equilibrium theorem on generalized convex spaces," Journal of the Korean Mathematical Society, vol. 38, no. 4, pp. 697-709, 2001.

S. Park, "Remarks on acyclic versions of generalized von Neumann and Nash equilibrium theorems," Applied Mathematics Letters, vol. 15, no. 5, pp. 641-647, 2002.

S. Park, "Elements of the KKM theory on abstract convex spaces," Journal of the Korean Mathematical Society, vol. 45, no. 1, pp. 1-27, 2008.

S. Park, "Equilibrium existence theorems in KKM spaces," Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4352-5364, 2008.

S. Park, "New foundations of the KKM theory," Journal of Nonlinear and Convex Analysis, vol. 9, no. 3, pp. 331-350, 2008.

S. Park, "From the KKM principle to the Nash equilibria," International Journal of Mathematics and Statistics, vol. 6, no. S10, pp. 77-88, 2010.

S. Park and J. A. Park, "The Idzik type quasivariational inequalities and noncompact optimization problems," Colloquium Mathematicum, vol. 71, no. 2, pp. 287-295,1996.

R. Bielawski, "Simplicial convexity and its applications," Journal of Mathematical Analysis and Applications, vol. 127, no. 1, pp. 155-171,1987.

W. Briec and C. Horvath, "Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and B-convexity," Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 188199, 2008.

L. Gonzalez, S. Kilmer, and J. Rebaza, "From a KKM theorem to Nash equilibria in L-spaces," Topology and Its Applications, vol. 155, no. 3, pp. 165-170, 2007.

W. A. Kirk, B. Sims, and G. X.-Z. Yuan, "The Knaster-Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications," Nonlinear Analysis: Theory, Methods & Applications, vol. 39, no. 5, pp. 611-627, 2000.

Q. Luo, "KKM and Nash equilibria type theorems in topological ordered spaces," Journal of Mathematical Analysis and Applications, vol. 264, no. 2, pp. 262-269, 2001.

B. Knaster, K. Kuratowski, and S. Mazurkiewicz, "Ein beweis des fixpunktsatzes fur n-dimensionale simplexe," Fundamenta Mathematicae, vol. 14, pp. 132-137,1929.

M. Lassonde, "On the use of KKM multifunctions in fixed point theory and related topics," Journal of Mathematical Analysis and Applications, vol. 97, no. 1, pp. 151-201,1983.

C. D. Horvath, "Contractibility and generalized convexity," Journal of Mathematical Analysis and Applications, vol. 156, no. 2, pp. 341-357,1991.

C. D. Horvath, "Extension and selection theorems in topological spaces with a generalized convexity structure," Toulouse Annales Faculte des Sciences, vol. 2, no. 2, pp. 253-269,1993. S. Reich and I. Shafrir, "Nonexpansive iterations in hyperbolic spaces," Nonlinear Analysis: Theory, Methods & Applications, vol. 15, no. 6, pp. 537-558,1990.

[39] C. D. Horvath and J. V. Llinares Ciscar, "Maximal elements and fixed points for binary relations on topological ordered spaces," Journal of Mathematical Economics, vol. 25, no. 3, pp. 291-306,1996.

[40] S. Park, "Continuous selection theorems in generalized convex spaces," Numerical Functional Analysis and Optimization, vol. 20, no. 5-6, pp. 567-583,1999.

[41] S. Park, "Ninety years of the Brouwer fixed point theorem," Vietnam Journal of Mathematics, vol. 27, no. 3, pp. 187-222,1999.

[42] S. Park, "Fixed points of better admissible maps on generalized convex spaces," Journal of the Korean Mathematical Society, vol. 37, no. 6, pp. 885-899, 2000.

[43] S. Park, "Generalized convex spaces, L-spaces, and FC-spaces," Journal of Global Optimization, vol. 45, no. 2, pp. 203-210, 2009.

[44] P. Q. Khanh, N. H. Quan, and J.-C. Yao, "Generalized KKM-type theorems in GFC-spaces and applications," Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1227-1234, 2009.

[45] W. Kulpa and A. Szymanski, "Applications of general infimum principles to fixed-point theory and game theory," Set-Valued Analysis, vol. 16, no. 4, pp. 375-398, 2008.

[46] W. A. Kirk and B. Panyanak, "Best approximation in R-trees," Numerical Functional Analysis and Optimization, vol. 28, no. 5-6, pp. 681-690, 2007.

[47] C. D. Horvath, "Topological convexities, selections and fixed points," Topology and Its Applications, vol. 155, no. 8, pp. 830-850, 2008.

[48] S. Park, "Remarks on the partial KKM principle," Nonlinear Analysis Forum, vol. 14, pp. 51-62, 2009.

[49] K. Fan, "A minimax inequality and applications," in Inequalities, III, O. Shisha, Ed., pp. 103-113, Academic Press, New York, NY, USA, 1972.

[50] Y. J. Cho, J. K. Kim, and B. Y. Lee, "Remarks on KKM maps and applications," The Journal of Advanced Research in Applied Mathematics, vol. 1, no. 1, pp. 1-8, 2009.

[51] F. C. Liu, "A note on the von Neumann-Sion minimax principle," Bulletin of the Institute of Mathematics. Academia Sinica, vol. 6, no. 2, part 2, pp. 517-524,1978.

[52] M. H. Shih and K.-K. Tan, "Noncompact sets with convex sections. II," Journal of Mathematical Analysis and Applications, vol. 120, no. 1, pp. 264-270,1986.

[53] H. Nikaido, "On von Neumann's minimax theorem," Pacific Journal of Mathematics, vol. 4, pp. 65-72, 1954.

[54] M. Sion, "On general minimax theorems," Pacific Journal ofMathematics, vol. 8, pp. 171-176, 1958.

[55] H. Komiya, "Convexity on a topological space," Fundamenta Mathematicae, vol. 111, no. 2, pp. 107-113, 1981.

[56] S. Park, "Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps," Journal of the Korean Mathematical Society, vol. 31, no. 3, pp. 493-519,1994.

[57] S. Simons, "Two-function minimax theorems and variational inequalities for functions on compact and noncompact sets, with some comments on fixed-point theorems," in Nonlinear Functional Analysis and Its Applications, Part 2, vol. 45 of Proceedings of Symposia in Pure Mathematics, pp. 377-392, American Mathematical Society, Providence, RI, USA, 1986.

[58] E. Tarafdar, "Fixed point theorems in H-spaces and equilibrium points of abstract economies," Journal of Australian Mathematical Society, vol. 53, no. 2, pp. 252-260,1992.

[59] S. Y. Chang, "A generalization of KKM principle and its applications," Soochow Journal of Mathematics, vol. 15, no. 1, pp. 7-17,1989.

[60] H. Nikaido and K. Isoda, "Note on non-cooperative convex games," Pacific Journal of Mathematics, vol. 5, pp. 807-815, 1955.

[61] K.-K. Tan, J. Yu, and X.-Z. Yuan, "Existence theorems of Nash equilibria for non-cooperative N-person games," International Journal of Game Theory, vol. 24, no. 3, pp. 217-222,1995.

[62] S. Park, "Variational inequalities and extremal principles," Journal of the Korean Mathematical Society, vol. 28, no. 1, pp. 45-56,1991.

[63] S. Park and S. K. Kim, "On generalized extremal principles," Bulletin of the Korean Mathematical Society, vol. 27, no. 1, pp. 49-52,1990.

[64] H. F. Bohnenblust and S. Karlin, "On a theorem of Ville," in Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, pp. 155-160, Princeton University Press, Princeton, NJ, USA, 1950.

[65] G. B. Dantzig, "Constructive proof of the min-max theorem," Pacific Journal of Mathematics, vol. 6, pp. 25-33, 1956.

[66] G. Debreu, "A social equilibrium existence theorem," Proceedings of the National Academy of Sciences of the United States of America, vol. 38, pp. 886-893,1952.

[67] M. Lassonde, "Fixed points for Kakutani factorizable multifunctions," Journal of Mathematical Analysis and Applications, vol. 152, no. 1, pp. 46-60,1990.

[68] J. P. Torres-Martinez, "Fixed points as Nash equilibria," Fixed Point Theory and Applications, vol. 2006, Article ID 36135,4 pages, 2006.

[69] H. Ben-El-Mechaiekh, P. Deguire, and A. Granas, "Une alternative non linéaire en analyse convexe et applications," Comptes Rendus des Seances de lAcademie des Sciences. Série I, vol. 295, no. 3, pp. 257-259, 1982.

[70] F. E. Browder, "The fixed point theory of multi-valued mappings in topological vector spaces," Mathematische Annalen, vol. 177, pp. 283-301,1968.

[71] G. L. Cain Jr. and L. Gonzalez, "The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities," Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 563-571, 2008.

[72] M. A. Khamsi, "KKM and Ky Fan theorems in hyperconvex metric spaces," Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 298-306,1996.

[73] S. Eilenberg and D. Montgomery, "Fixed point theorems for multi-valued transformations," American Journal ofMathematics, vol. 68, pp. 214-222, 1946.

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text:
# MPI topic: Functional parallelism

##### Experimental html version of downloadable textbook, see http://www.tacc.utexas.edu/~eijkhout/istc/istc.html
2.2 : Starting and running MPI processes
2.2.2 : Initialization / finalization
2.2.2.1 : Aborting an MPI run
2.2.2.2 : Testing the initialized/finalized status
2.2.2.3 : Information about the run
2.2.2.4 : Commandline arguments
2.3 : Processor identification
2.3.1 : Processor name
2.3.2 : Communicators
2.3.3 : Process and communicator properties: rank and size
2.4 : Functional parallelism
2.5 : Distributed computing and distributed data
2.6 : Review questions

# 2 MPI topic: Functional parallelism

## 2.1 The SPMD model

crumb trail: > mpi-functional > The SPMD model

MPI programs conform largely to the SPMD model, where each processor runs the same executable. This running executable we call a process .

When MPI was first written, 20 years ago, it was clear what a processor was: it was what was in a computer on someone's desk, or in a rack. If this computer was part of a networked cluster, you called it a node . So if you ran an MPI program, each node would have one MPI process;

figure~ 2.1 . You could of course run more than one process, using the time slicing of the OS , but that would give you no extra performance.

These days the situation is more complicated. You can still talk about a node in a cluster, but now a node can contain more than one processor chip (sometimes called a socket ), and each processor chip probably has multiple cores .

Figure~ 2.2 shows how you could explore this using a mix of MPI between the nodes, and a shared memory programming system on the nodes.

However, since each core can act like an independent processor, you can also have multiple MPI processes per node. To MPI, the cores look like the old completely separate processors. This is the pure MPI' model of figure~ 2.3 , which we will use in most of this part of the book. (Hybrid computing will be discussed in chapter~ Hybrid computing .)

This is somewhat confusing: the old processors needed MPI programming, because they were physically separated. The cores on a modern processor, on the other hand, share the same memory, and even some caches. In its basic mode MPI seems to ignore all of this: each core receives an MPI process and the programmer writes the same send/receive call no matter where the other process is located. In fact, you can't immediately see whether two cores are on the same node or different nodes. Of course, on the implementation level MPI uses a different communication mechanism depending on whether cores are on the same socket or on different nodes, so you don't have to worry about lack of efficiency.

Remark

In some rare cases you may want to run in an MPMD mode, rather than SPMD . This can be achieved either on the OS level (see section~ 14.9.4 ), using options of the mpiexec mechanism, or you can use MPI's built-in process management; chapter~ MPI topic: Process management . Like I~said, this concerns only rare cases.

## 2.2 Starting and running MPI processes

crumb trail: > mpi-functional > Starting and running MPI processes

The SPMD model may be initially confusing. Even though there is only a single source, compiled into a single executable, the parallel run comprises a number of independently started MPI processes (see section~ 1.3 for the mechanism).

The following exercises are designed to give you an intuition for this one-source-many-processes setup. In the first exercise you will see that the mechanism for starting MPI programs starts up independent copies. There is nothing in the source that says and now you become parallel'.

The following exercise demonstrates this point.

Exercise

Write a hello world' program, without any MPI in it, and run it in parallel with mpiexec or your local equivalent. Explain the output.

(On TACC machines such as stampede, use ibrun , no processor count.)

\skeleton{hello}

This exercise is illustrated in figure  2.4 .

crumb trail: > mpi-functional > Starting and running MPI processes > Headers

If you use MPI commands in a program file, be sure to include the proper header file, mpi.h or mpif.h .

#include "mpi.h" // for C
#include "mpif.h" ! for Fortran

For Fortran90 , many MPI installations also have an MPI module, so you can write

use mpi     ! pre 3.0
use mpi_f08 ! 3.0 standard

The internals of these files can be different between MPI installations, so you can not compile one file against one mpi.h file and another file, even with the same compiler on the same machine, against a different MPI.

Python note

It's easiest to

from mpi4py import MPI

MPL note

To compile MPL programs, add a line

#include <mpl/mpl.hpp>

to your file. End of MPL note

### 2.2.2 Initialization / finalization

Every (useful) MPI program has to start with MPI initialization through a call to

C:
int MPI_Init(int *argc, char ***argv)

Fortran:
MPI_Init(ierror)
INTEGER, OPTIONAL, INTENT(OUT) :: ierror

MPI_Init , and have
C:
int MPI_Finalize(void)

Fortran:
MPI_Finalize(ierror)
INTEGER, OPTIONAL, INTENT(OUT) :: ierror

MPI_Finalize to finish the use of MPI in your program. The init call is different between the various languages.

In C, you can pass argc and argv , the arguments of a C language main program:

int main(int argc,char **argv) {
....
return 0;
}

(It is allowed to pass NULL for these arguments.)

Fortran (before 2008) lacks this commandline argument handling, so MPI_Init lacks those arguments.

Python note

There are no initialize and finalize calls: the import statement performs the initialization.

MPL note

There is no initialization or finalize call.

Initialization is done at the first mpl::environment method call, such as comm_world .

End of MPL note

This may look a bit like declaring this is the parallel part of a program', but that's not true: again, the whole code is executed multiple times in parallel.

Exercise

Add the commands MPI_Init and MPI_Finalize to your code. Put three different print statements in your code: one before the init, one between init and finalize, and one after the finalize. Again explain the output.

Run your program on a large scale, using a batch job. Where does the output go? Experiment with

MY_MPIRUN_OPTIONS="-prepend-rank" ibrun yourprogram

Remark

For hybrid MPI-plus-threads programming there is also a call MPI_Init_thread . For that, see section  45.2 .

#### 2.2.2.1 Aborting an MPI run

Apart from MPI_Finalize , which signals a successful conclusion of the MPI run, an abnormal end to a run can be forced by

Synopsis:
int MPI_Abort(MPI_Comm comm, int errorcode)

Input Parameters
comm : communicator of tasks to abort
errorcode : error code to return to invoking environment

Python:
MPI.Comm.Abort(self, int errorcode=0)

MPL:
void mpl::communicator::abort ( int ) const

MPI_Abort . This stop execution on all processes associated with the communicator, but many implementations will terminate all processes. The value parameter is returned to the environment.

#### 2.2.2.2 Testing the initialized/finalized status

The commandline arguments argc and argv are only guaranteed to be passed to process zero, so the best way to pass commandline information is by a broadcast (section  3.3.3 ).

There are a few commands, such as MPI_Get_processor_name , that are allowed before MPI_Init .

If MPI is used in a library, MPI can have already been initialized in a main program. For this reason, one can test where MPI_Init has been called with

C:
int MPI_Initialized(int *flag)

Fortran:
MPI_Initialized(flag, ierror)
LOGICAL, INTENT(OUT) :: flag
INTEGER, OPTIONAL, INTENT(OUT) :: ierror

MPI_Initialized .

You can test whether MPI_Finalize has been called with

C:
int MPI_Finalized( int *flag )

Fortran:
MPI_Finalized(flag, ierror)
LOGICAL, INTENT(OUT) :: flag
INTEGER, OPTIONAL, INTENT(OUT) :: ierror

MPI_Finalized .

#### 2.2.2.3 Information about the run

Once MPI has been initialized, the MPI_INFO_ENV object contains a number of key/value pairs describing run-specific information; see section  14.1.1.1 .

#### 2.2.2.4 Commandline arguments

The MPI_Init routines takes a reference to argc and argv for the following reason: the MPI_Init calls filters out the arguments to mpirun or mpiexec , thereby lowering the value of argc and elimitating some of the argv arguments.

On the other hand, the commandline arguments that are meant for mpiexec wind up in the MPI_INFO_ENV object as a set of key/value pairs; see section  14.1.1 .

## 2.3 Processor identification

crumb trail: > mpi-functional > Processor identification

Since all processes in an MPI job are instantiations of the same executable, you'd think that they all execute the exact same instructions, which would not be terribly useful. You will now learn how to distinguish processes from each other, so that together they can start doing useful work.

### 2.3.1 Processor name

crumb trail: > mpi-functional > Processor identification > Processor name

In the following exercise you will print out the hostname of each MPI process with

C:
int MPI_Get_processor_name(char *name, int *resultlen)
name : buffer char[MPI_MAX_PROCESSOR_NAME]

Fortran:
MPI_Get_processor_name(name, resultlen, ierror)
CHARACTER(LEN=MPI_MAX_PROCESSOR_NAME), INTENT(OUT) :: name
INTEGER, INTENT(OUT) :: resultlen
INTEGER, OPTIONAL, INTENT(OUT) :: ierror

Python:
MPI.Get_processor_name()

MPI_Get_processor_name as a first way of distinguishing between processes. This routine has a character buffer argument, which needs to be allocated by you. The length of the buffer is also passed, and on return that parameter has the actually used length. The maximum needed length is MPI_MAX_PROCESSOR_NAME .
// procname.c
int name_length = MPI_MAX_PROCESSOR_NAME;
char proc_name[name_length];
MPI_Get_processor_name(proc_name,&name_length);
printf("This process is running on node <<%s>>\n",proc_name);

(Underallocating the buffer will not lead to a runtime error.)

Exercise

Use the command MPI_Get_processor_name . Confirm that you are able to run a program that uses two different nodes.

TACC nodes have a hostname cRRR-CNN , where RRR is the rack number, C is the chassis number in the rack, and NN is the node number within the chassis. Communication is faster inside a rack than between racks!

MPL note

The processor_name returning a std::string :

std::string mpl::environment::processor_name ();

End of MPL note

### 2.3.2 Communicators

crumb trail: > mpi-functional > Processor identification > Communicators

First we need to introduce the concept of communicator , which is an abstract description of a group of processes. For now you only need to know about the existence of the MPI_Comm data type, and that there is a pre-defined communicator MPI_COMM_WORLD which describes all the processes involved in your parallel run.

In the procedural languages C, a communicator is a variable that is passed to most routines:

#include <mpi.h>
MPI_Comm comm = MPI_COMM_WORLD;
MPI_Send( /* stuff */ comm );

Fortran note

In Fortran, pre-2008 a communicator was an opaque handle , stored in an Integer . With Fortran 2008 , communicators are derived types:

use mpi_f098
Type(MPI_Comm} :: comm = MPI_COMM_WORLD
call MPI_Send( ... comm )

Python note

In object-oriented languages, a communicator is an object , and rather than passing it to routines, the routines are often methods of the communicator object:

from mpi4py import MPI
comm = MPI.COMM_WORLD
comm.Send( buffer, target )

MPL note

The naive way of declaring a communicator would be:

// commrank.cxx
mpl::communicator comm_world =
mpl::environment::comm_world();

calling the predefined environment method comm_world

However, if the variable will always correspond to the world communicator, it is better to make it const and declare it to be a reference:

const mpl::communicator &comm_world =
mpl::environment::comm_world();

End of MPL note

MPL note

The communicator class has its copy operator deleted; however, copy initialization exists:

// commcompare.cxx
const mpl::communicator &comm =
mpl::environment::comm_world();
cout << "same: " << boolalpha << (comm==comm) << endl;

mpl::communicator copy =
mpl::environment::comm_world();
cout << "copy: " << boolalpha << (comm==copy) << endl;

mpl::communicator init = comm;
cout << "init: " << boolalpha << (init==comm) << endl;

(This outputs true/false/false respectively.)

The copy initializer performs an MPI_Comm_dup . End of MPL note

MPL note

Pass communicators by reference to avoid communicator duplication:

// commpass.cxx
// BAD! this does a MPI_Comm_dup.
void comm_val( const mpl::communicator comm );

// correct!
void comm_ref( const mpl::communicator &comm );

End of MPL note

You will learn much more about communicators in chapter  MPI topic: Communicators .

### 2.3.3 Process and communicator properties: rank and size

To distinguish between processes in a communicator, MPI provides two calls

1. Semantics:
MPI_COMM_SIZE(comm, size)
IN comm: communicator (handle)
OUT size: number of processes in the group of comm (integer)

C:
int MPI_Comm_size(MPI_Comm comm, int *size)

Fortran:
MPI_Comm_size(comm, size, ierror)
TYPE(MPI_Comm), INTENT(IN) :: comm
INTEGER, INTENT(OUT) :: size
INTEGER, OPTIONAL, INTENT(OUT) :: ierror

Python:
MPI.Comm.Get_size(self)

MPL:
int mpl::communicator::size ( ) const

MPI_Comm_size reports how many processes there are in all; and
2. Semantics:
MPI_COMM_RANK(comm, rank)
IN comm: communicator (handle)
OUT rank: rank of the calling process in group of comm (integer)

C:
int MPI_Comm_rank(MPI_Comm comm, int *rank)

Fortran:
MPI_Comm_rank(comm, rank, ierror)
TYPE(MPI_Comm), INTENT(IN) :: comm
INTEGER, INTENT(OUT) :: rank
INTEGER, OPTIONAL, INTENT(OUT) :: ierror

Python:
MPI.Comm.Get_rank(self)

MPL:
int mpl::communicator::rank ( ) const

MPI_Comm_rank states what the number of the process is that calls this routine.

If every process executes the MPI_Comm_size call, they all get the same result, namely the total number of processes in your run. On the other hand, if every process executes MPI_Comm_rank , they all get a different result, namely their own unique number, an integer in the range from zero to the number of processes minus 1. See figure  2.5 .

In other words, each process can find out I am process 5 out of a total of 20'.

Exercise

Write a program where each process prints out a message reporting its number, and how many processes there are:

Hello from process 2 out of 5!

Write a second version of this program, where each process opens a unique file and writes to it. On some clusters this may not be advisable if you have large numbers of processors, since it can overload the file system. \skeleton{commrank}

Exercise

Write a program where only the process with number zero reports on how many processes there are in total.

In object-oriented approaches to MPI, that is, mpi4py and MPL , the MPI_Comm_rank and MPI_Comm_size routines are methods of the communicator class:

Python note

Rank and size are methods of the communicator object. Note that their names are slightly different from the MPI standard names.

procid = comm.Get_rank()
nprocs = comm.Get_size()

MPL note The rank of a process (by mpl::communicator:: rank the size of a communicator (by mpl::communicator:: size

are both methods of the communicator

const mpl::communicator &comm_world =
mpl::environment::comm_world();
int procid = comm_world.rank();
int nprocs = comm_world.size();

End of MPL note

## 2.4 Functional parallelism

crumb trail: > mpi-functional > Functional parallelism

Now that processes can distinguish themselves from each other, they can decide to engage in different activities. In an extreme case you could have a code that looks like

// climate simulation:
if (procid==0)
earth_model();
else if (procid=1)
sea_model();
else
air_model();

Practice is a little more complicated than this. But we will start exploring this notion of processes deciding on their activity based on their process number.

Being able to tell processes apart is already enough to write some applications, without knowing any other MPI. We will look at a simple parallel search algorithm: based on its rank, a processor can find its section of a search space. For instance, in Monte Carlo codes a large number of random samples is generated and some computation performed on each. (This particular example requires each MPI process to run an independent random number generator, which is not entirely trivial.)

Exercise

Is the number $N=2,000,000,111$ prime? Let each process test a disjoint set of integers, and print out any factor they find. You don't have to test all integers $<N$: any factor is at most $\sqrt N\approx 45,200$.

(Hint: i%0 probably gives a runtime error.)

Can you find more than one solution? \skeleton{prime}

Remark

Normally, we expect parallel algorithms to be faster than sequential. Now consider the above exercise. Suppose the number we are testing is divisible by some small prime number, but every process has a large block of numbers to test. In that case the sequential algorithm would have been faster than the parallel one. Food for thought.

As another example, in Boolean satisfiability problems a number of points in a search space needs to be evaluated. Knowing a process's rank is enough to let it generate its own portion of the search space. The computation of the Mandelbrot set can also be considered as a case of functional parallelism. However, the image that is constructed is data that needs to be kept on one processor, which breaks the symmetry of the parallel run.

Of course, at the end of a functionally parallel run you need to summarize the results, for instance printing out some total. The mechanisms for that you will learn next.

## 2.5 Distributed computing and distributed data

crumb trail: > mpi-functional > Distributed computing and distributed data

One reason for using MPI is that sometimes you need to work on a single object, say a vector or a matrix, with a data size larger than can fit in the memory of a single processor. With distributed memory, each processor then gets a part of the whole data structure and only works on that.

So let's say we have a large array, and we want to distribute the data over the processors. That means that, with p processes and n  elements per processor, we have a total of $\mathtt{n}\cdot\mathtt{p}$ elements.

We sometimes say that data is the local part of a distributed array with a total size of $\mathtt{n}\cdot\mathtt{p}$ elements. However, this array only exists conceptually: each processor has an array with lowest index zero, and you have to translate that yourself to an index in the global array. In other words, you have to write your code in such a way that it acts like you're working with a large array that is distributed over the processors, while actually manipulating only the local arrays on the processors.

Your typical code then looks like

int myfirst = .....;
for (int ilocal=0; ilocal<nlocal; ilocal++) {
int iglobal = myfirst+ilocal;
array[ilocal] = f(iglobal);
}

Exercise

Allocate on each process an array:

int my_ints[10];

and fill it so that process 0 has the integers $0\cdots 9$, process 1 has $10\cdots 19$, et cetera.

It may be hard to print the output in a non-messy way.

If the array size is not perfectly divisible by the number of processors, we have to come up with a division that is uneven, but not too much. You could for instance, write

int Nglobal, // is something large
Nlocal = Nglobal/ntids,
excess = Nglobal%ntids;
if (mytid==ntids-1)
Nlocal += excess;

Exercise

Argue that this strategy is not optimal. Can you come up with a better distribution? Load balancing is further discussed in  Eijkhout:IntroHPC .

## 2.6 Review questions

crumb trail: > mpi-functional > Review questions

For all true/false questions, if you answer that a statement is false, give a one-line explanation.

Exercise

True or false: mpicc is a compiler.

Exercise

T/F?

1. In C, the result of MPI_Comm_rank is a number from zero to number-of-processes-minus-one, inclusive.
2. In Fortran, the result of MPI_Comm_rank is a number from one to number-of-processes, inclusive.

Exercise

What is the function of a hostfile?

\begin{pcse}

Exercise

An architecture is called symmetric' or `uniform' if the relation between any pair of processes is essentially the same. In what way are MPI processes run on stampede symmetric; in what way not?

\end{pcse}

format: plain


text:
# A New Stat! bbFIP

A few weeks ago I began dabbling with a stat that combined extremely well hit balls and walks plus extremely poorly hit balls and strike outs. In other words, automatic successful at bats versus automatic failed at bats. The aim here is to blend the line between Defense Independent Pitching, which largely ignores BABIP, and the newer, more nuanced approach towards judging batted ball quality.

When I announced this plan on twitter, Tom Tango pointed me towards an article he had written all the way back in 2010, which accomplished something very similar. Tom used line drives, ground balls, infield fly balls and outfield fly ball designations to create two separate categories of plate appearance value, which he called bigs and smalls.

$\mathrm{Bigs}=\left(\mathrm{BB}+\mathrm{LD}\right)-\left(K+\mathrm{iFB}\right)$ $\mathrm{Smalls}=\left(\mathrm{oFB}+\mathrm{GB}\right)$

The Bigs were given a weight of 11, and the smalls a weight of 3. Sum the two and add a constant (whatever number is required to bring the league average figure up to the league average ERA).

$bb{FIP}_{Tango}=11×Bigs+3×Smalls+C$

After reading this, I set out to create my own up to date version of bbFIP using Statcast derived numbers.

In the following weeks I have gone through many iterations of this stat, using many different starting points. I tried using hit probabilities, and while the results were okay, they weren't on par with what I was expecting. I have since moved to xOBA values, which has helped quite a bit. I toyed with including or excluding walks, hit by pitch, strike outs, home runs, medium hit balls, and many other factors. In the end, I came up with the following formula.

$\mathrm{OUTs}=\frac{.77×W+.17×K-.98×BB-.69×HBP-1.52×S-2.52×sHR}{PA}$

where W = weak contact (xOBA ≤ .245), S = strong contact (xOBA ≥ .634), and sHR = strong home runs (xHR% ≥ .55).

This gives you what I call "OUTs".  You can then convert this to bbFIP by multiplying by -11 and adding a constant, which is roughly 5.4.

Strikeouts have an xOBA value of 0, that means the strikeout is double counted in this formula. The first time it is given a value of .77, and the second time it is given a value of .17, so together it has a total value of .94.

The same is true for walks, hit by pitch, and home runs. They have a total value of 2.50, 2.21, and 4.04, respectively.

I will continue to test and iterate, so I may stumble upon superior combinations of stats over time. However, I've already ran through nearly 400 combinations and these have generated the best results.

# Why use bbFIP?

I've looked at every pitcher over the past 2+ seasons who threw at least 20 innings in one season. Of these 1380 pitchers (many are double or triple counted), FIP had an MSE of 1.027, and bbFIP .968. In other words, for roughly two thirds of pitchers, bbFIP was within .968 runs of their ERA, while FIP was within 1.027 runs.

You can consult the table below for the in season results.

Year w-kFIP FIP bbFIP
2015 1.45 1.02 0.93
2016 1.54 0.96 0.93
2017 2.02 1.13 1.08

And the Year to Year results.

Year w-kFIP FIP bbFIP
15-16 1.80 2.98 2.10
15-17 2.71 3.92 3.23
16-17 2.27 2.84 2.72

In all but one case, bbFIP is superior. The case where it isn't? Comparing this season to last. That is an odd one, I'm not sure I can explain it. Maybe you can? Either way, bbFIP appears to have less error over time, and the error is, for the most part, less than a run. In the past two seasons, it has hovered around .93 of a run, although the first half of this season is up towards 1.08. The 2015 to 2016 numbers and 2015 to 2017 numbers are both promising, and perhaps by the end of the season the 2016-2017 numbers will fall into place as well.

In my original post, I made an error in the numbers.  This error was caused by sloppiness, in which I accidentally combined two groups of data that should not have been combined. I was able to recreate the error, so I know exactly what I did wrong. I have reran the numbers, this time without that sloppiness, and I have updated this post with those results. I have also included walks minus strikeouts FIP.  Which follows a simple formula.

${FIP}_{w-k}=\frac{BB+HBP-K}{PA}+C$

In terms of predictive ability, bbFIP might be somewhere between w-kFIP and FIP. Simplicity still wins the day.

However, I still believe this stat is a worthy of your attention, and I am curious how it fares as time goes on.

I know that FIP was originally developed to be easy to calculate. You just take a few numbers multiply by integers, add them up, and divide by innings pitched. You just needed a sheet of paper to do the long division, or a simple calculator. This version of bbFIP is not as easy. That is unfortunate. In principle it isn't very difficult, just multiplication, addition, and a bit of division. However, calculating the xOBA and xHR values cannot be done by hand. And they aren't available publicly, although you could use the xwOBA which might give similar results.

In my spreadsheets I have supplied bbFIP for all pitchers, and OUTs for all batters. You can easily convert from OUTs to bbFIP by multiplying by -11 and adding 5.4-ish. Tell me if you'd rather have bbFIP for batters.

Finally, here are the league average OUTs and bbFIP Constants.

Year OUTs bbFIP C
2015 .124 5.343
2016 .106 5.384
2017 .096 5.415


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### College Physics (Urone)

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# A high-quality diamond may be quite clear and colourless, transmitting all visible wavelengths with little absorption. Explain how it can sparkle with flashes of brilliant colour when illuminated by white light

However, the cause of the brilliant color flashes observed in the diamond is due to the fact that the diamond refracts each color in white light differently; thus, refraction is the cause of those brilliant colors.

See the step by step solution

## Step 1: Definition of wavelengths

The distance between two successive crests or troughs of a wave is defined as its wavelength. It is measured in the wave's direction.

## Step 2: Explanation

The way the diamond is cut is what makes it sparkle; a mined diamond is a transparent rock that does not sparkle; the way the craftsman shapes the diamond is what makes it sparkle.

Whereas white is made up of many colors, each of which has a different refractive index inside the diamond. As a result, each of those colors is refracted at a different angle, and a perfectly cut diamond functions similarly to a prism, trapping those different colors and allowing you to see the sparkle.

However, the cause of the brilliant color flashes observed in the diamond is due to the fact that the diamond refracts each color in white light differently; thus, refraction is the cause of those brilliant colors.


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# Number of permutations if every element may appear in certain distance from its initial position

Suppose i have an $n-$elements array. I want to count number of permutations for which element $a_i$ allowed to appear in range $i-k, \dots, i+k,$ so $2k+1$ positions available after permutation has happenned.

I got a rough bound $n$ elements at most $2k+1$ positions for every one. Thus, the bound is $n^{2k+1}.$ And this is an upper bound, but i need lower one as well. The best is exact formula.

EDIT: A lower bound is $(k!)^{n/k}.$

• You should have a look at oeis.org/A002524 which deals especially with the case $k=1$ but also discusses and has links to material on the general case. Maybe you could follow up on what you find there, and then come back here to post an answer summarizing what you've learned. – Gerry Myerson Nov 10 '14 at 11:40
• Had a chance to check that link? – Gerry Myerson Nov 11 '14 at 11:49
• @GerryMyerson, sorry, for my particular problem exact answer wasn't necessary, and doc about n-case not so simple for me to read it fast. – Yola Nov 11 '14 at 19:33
• OK. When you get a chance, then, I hope you'll write something up and post it as an answer. – Gerry Myerson Nov 12 '14 at 22:18


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# Multivariate outliers

#### mariap

##### New Member
Hi everyone!

I have a question about detecting multivariate outliers in a correlation analysis. I'm calculating Spearman correlations between 5 variables (in a within-subject design), and I was suggested to calculate Mahalanobis distance for all 5 variables for detecting multivariate outliers. However, each correlation is a two-dimensional space, so it might have different outliers with respect to the initial five-dimensional space, if I understand correctly. But detecting outliers for each correlation separately also doesn't seem like a good idea.
Could someone please clarify this? Thanks!

regards,

#### jkp3

##### New Member
Hi Maria,

You are supposed to screen for both univariate and multivariate outliers. There are different recommendations depending on your sample sizes, but usually for univariate outliers look at the raw data and compute the Z scores (through the analyse and descriptive statistics tab). Anything that is above +/- 3.29 (depending on sample size) should be counted as a univariate outlier (for a reference see Ghasemi & Zahediasl, 2012). Once you've dealt with them you run multivariate outlier analysis using Mahalanobis distances (if you're unsure of whether you will have outliers or not; see Zijlstra et al., 2010 for a justification) across all 5 variables. Again, you run it on your raw data rather than on the correlations themselves, and your exclusion value will be determined by having 5 variables. So you don't need to look at it separately, as having 5 is taken into account with the analysis anyway.

Hope that helps.


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# Exothermic vs. Endothermic and K

##### Table of Contents

An exothermic reaction occurs when the temperature of a system increases due to the evolution of heat. This heat is released into the surroundings, resulting in an overall negative quantity for the heat of reaction ($$q_{rxn} < 0$$). An endothermic reaction occurs when the temperature of an isolated system decreases while the surroundings of a non-isolated system gains heat. Endothermic reactions result in an overall positive heat of reaction ($$q_{rxn} > 0$$).

### Exothermic Reactions

Exothermic and endothermic reactions cause energy level differences and therefore differences in enthalpy ($$ΔH$$), the sum of all potential and kinetic energies. ΔH is determined by the system, not the surrounding environment in a reaction. A system that releases heat to the surroundings, an exothermic reaction, has a negative ΔH by convention, because the enthalpy of the products is lower than the enthalpy of the reactants of the system (Figure 1).

$C(s) + O_{2\;(g)} \rightarrow CO_{2\; (g)} \tag{ΔH = –393.5 kJ}$

$H_{2\;(g)} + 1/2 O_{2\;(g)} \rightarrow H_2O_{(l)} \tag{ΔH = –285.8 kJ}$

The enthalpies of these reactions are less than zero, and are therefore exothermic reactions.

### Endothermic Reactions

A system of reactants that absorbs heat from the surroundings in an endothermic reaction has a positive $$ΔH$$, because the enthalpy of the products is higher than the enthalpy of the reactants of the system.

$N_{2\;(g)} + O_{2\;(g)} \rightarrow 2NO_{(g)} \tag{ΔH = +180.5 kJ > 0}$

$C(s) + 2S_{(s)} \rightarrow CS_{2\;(l)} \tag{ΔH = +92.0 kJ > 0}$

Because the enthalpies of these reactions are greater than zero, they are endothermic reactions.

### The Equilibrium Constant K

The equilibrium constant (K) defines the relationship among the concentrations of chemical substances involved in a reaction at equilibrium. The Le Châtelier's principle states that if a stress, such as changing temperature, pressure, or concentration, is inflicted on an equilibrium reaction, the reaction will shift to restore the equilibrium. For exothermic and endothermic reactions, this added stress is a change in temperature. The equilibrium constant shows how far the reaction will progress at a specific temperature by determining the ratio of products to reactions using equilibrium concentrations.

The equilibrium expression for the following equation

$aA + bB \rightleftharpoons cC + dD$

is given below:

$K_c = \dfrac{[C]^c[D]^d}{[A]^a[B]^b}$

where

• Kc is the equilibrium constant
• [A], [B], [C], [D] are concentrations
• a, b, c, and d are the stoichiometric coefficients of the balanced equation

#### Exothermic Reactions

• If Kc decreases with an increase in temperature, the reaction shifts to the left.
• If Kc increases with a decreases in temperature, the reaction to shifts to the right.

#### Endothermic Reactions

• If Kc increases with an increase in temperature, the reaction to shifts to the right.
• If Kc decreases with a decrease in temperature, the reaction to shifts to the left.

### K Values

If the products dominate in a reaction, the value for K is greater than 1. The larger the K value, the more the reaction will tend toward the right and thus to completion.

• If K=1, neither the reactants nor the products are favored. Note that this is not the same as both being favored.
• If the reactants dominate in a reaction, then K< 1. The smaller the K value, the more the reaction will tend toward the left.

Example: The Haber Process

Suppose that the following reaction is at equilibrium and that the concentration of N2 is 2 M, the concentration of H2 is 4 M, and the concentration of NH3 is 3 M. What is the value of Kc

$N_2 + 3H_2 \rightleftharpoons 2NH_3$

The coefficients and the concentrations are plugged into the Kc equation to calculate its value.

$\begin{eqnarray} K_c &=& \dfrac{[NH_3]^2}{[N_2]^1[H_2]^3} \\ &=& \dfrac{[3]^2}{[2]^1[4]^3} \\ &=& \dfrac{9}{32} \\ &=& 0.3 \end{eqnarray}$

### Practice Problems

1. Determine the equilibrium constant, K, for the following chemical reaction at equilibrium if the molar concentrations of the molecules are 0.20 M H2, 0.10 M NO, 0.20 M H2O, and 0.10M N2

$2H_2 (g) + 2NO (g) \rightleftharpoons 2H_2O (g) + N_2 (g)$

2. In the previous equation, in which way will the equilibrium shift?

3. In the following reaction, the temperature is increased and the K value decreases from 0.75 to 0.55. Is this an exothermic or endothermic reaction?

$N_2 (g) + 3H_2 \rightleftharpoons 2NH_3 (g)$

4. In the following reaction, in which direction will the equilibrium shift if there is an increase in temperature and the enthalpy of reaction is given such that ΔH = -92.5 kJ?

$PCl_3(g) + Cl_2(g) \rightleftharpoons PCl_5(g) \tag{ΔH = -92.5 kJ}$

### Solutions

1. Using the equilibrium constant equation and plugging in the concentration values of each molecule:

$\begin{eqnarray} K_c &=& \dfrac{[C]^c[D]^d}{[A]^a[B]^b} \\ &=& 0.2 \end{eqnarray}$

2. Because Kc = 0.2, which is less than 1, the reaction will shift to the left.

3. Because the K value decreases with an increase in temperature, the reaction is an exothermic reaction.

4. In the initial reaction, the energy given off is negative and thus the reaction is exothermic. However, an increase in temperature allows the system to absorb energy and thus favor an endothermic reaction; the equilibrium will shift to the left.

### References

1. Petrucci, Harwood, Herring, Madura. General Chemistry Principles & Modern Applications. Prentice Hall. New Jersey, 2007

### Contributors

• Alyson Salmon, Nikita Patel (UCD), Deepak Nallur (UCD)

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# Electric Field Calculation given three point charges?

1. Feb 18, 2014

### P1nkButt3rflys

1. The problem statement, all variables and given/known data

Three point charges are located in an equilateral triangle with side 0.330 cm, where q1 = +8.30 µC, q2 = −8.20 µC, and q3 = +1.90 nC.

A) What is the electric field at the location of q3? Enter the x component, then the y component.

B) What is the force on q3? Enter the x component, then the y component.

DIAGRAM:
http://www.learning.physics.dal.ca/dalphysicslib/Graphics/Gtype51/threechrg1.gif

2. Relevant equations
|Fe|= (k*|q1|*|q2|)/r^2
|E|=(k*|Q|)/r^2

k = 8.99*10^9 (N*m^2)/C^2
Fe = Electrostatic force between a pair of charges
r = charge separation
q1, q2 = charge magnitudes
E = electric field
Q = charged object

3. The attempt at a solution

I've attempted this many ways and have been incorrect each time. I know the angles at the three points of the triangle will be 60deg. Knowing this and that the sides of the triangle are 0.330cm in length I calculated "r" -> the distance in respect to X and Y from q1 to q3 and from q2 to q3.

I got r(1-3x) = 0.165cm and r(1-3y) = 0.286cm and r(2-3x) = 0.165cm and r(2-3y) = 0.286cm.

Any help would be greatly appreciated. Even to just the first portion would help a lot.

Last edited: Feb 18, 2014
2. Feb 18, 2014

### jackarms

Careful -- to calculate the components of the force, you can't use the components of the separation. This is a pretty common mistake, I think, and not very intuitive, but you have to calculate the field vector using the full separation between the charges, and then find the components from that vector.

The difference comes from the formula for electric field. Say for finding the y component of the field, using just the y component of the separation looks like this:
$$E_{y} = k\frac{q_{1}q_{2}}{(rsin\theta)^{2}}$$
Whereas finding the component of the whole field vector looks like this:
$$E_{y} = k\frac{q_{1}q_{2}}{r^{2}}sin\theta$$

3. Feb 19, 2014

### P1nkButt3rflys

Hi jackarms! Thanks for the help but I'm still very confused. I attempted another solution, again it's wrong.. I'll post my attempt and maybe someone can give me some guidance. I'm not sure if I'm on the right track or completely off in the wrong direction.

[IMAGE REMOVED]

Thank you!

Last edited: Feb 19, 2014
4. Feb 19, 2014

### P1nkButt3rflys

So not only did I use some wrong values in my previous calculation, after reading through my textbook further I approached it differently and got the correct answer for Part A :)

I'm not sure what to do for Part B, but am going to attempt it and post my work.

5. Feb 19, 2014

### P1nkButt3rflys

Attempted B, to find the force on q3 and it is incorrect.

Any suggestions:

6. Feb 19, 2014

### jackarms

All the calculations look right for both the field and the force, so I think one must be wrong. For the field, I'm not sure about the 7.15 x 107 value - are you sure that's right?

Also, an easy way to go from field to force is to just use the definition of field: $E = \frac{F}{q}$
It's just nice to save some work.

7. Feb 19, 2014

### P1nkButt3rflys

Thanks for the suggestion! I used your equation and got the same result, well, almost. The force for x was the same, the force for y was 0.134 N, so I must have had an issue with my sig figs. I entered both answers and they are correct!

Thank you for your help!!