Note on surjective polynomial operators Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2017, Том 19, Выпуск 4, С. 70-75

YffK 517.9

A NOTE ON SURJECTIVE POLYNOMIAL OPERATORS1

M. Saburov

A linear Markov chain is a discrete time stochastic process whose transitions depend only on the current state of the process. A nonlinear Markov chain is a discrete time stochastic process whose transitions may depend on both the current state and the current distribution of the process. These processes arise naturally in the study of the limit behavior of a large number of weakly interacting Markov processes. The nonlinear Markov processes were introduced by McKean and have been extensively studied in the context of nonlinear Chapman-Kolmogorov equations as well as nonlinear Fokker-Planck equations. The nonlinear Markov chain over a finite state space can be identified by a continuous mapping (a nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex) of the finite state space and by a family of transition matrices depending on occupation probability distributions of states. Particularly, a linear Markov operator is a linear operator associated with a square stochastic matrix. It is well-known that a linear Markov operator is a surjection of the simplex if and only if it is a bijection. The similar problem was open for a nonlinear Markov operator associated with a stochastic hyper-matrix. We solve it in this paper. Namely, we show that a nonlinear Markov operator associated with a stochastic hyper-matrix is a surjection of the simplex if and only if it is a permutation of the Lotka-Volterra operator.

Mathematics Subject Classification 2010: 47H60, 47N10.

Key words: Stochastic hyper-matrix, polynomial operator, Lotka-Volterra operator.

1. Introduction

Let Im := {1, • • • ,m} be a finite set, a := Im \ a be a complement of a subset a C Im, and |a| be the number of its elements. Suppose that Rm is equipped with the Zi-norm ||x||i := m=1 IxkI where x = (xi, • • • , xm) £ Rm and (e^}ieim stands for the standard basis. We say that x ^ 0 (respectively x > 0) if Xi ^ 0 (respectively Xi > 0) for all i £ Im. Let Sm-i = (x £ Rm : ||x||i = 1,x ^ 0} be the (m — 1)-dimensional standard simplex. An element of the simplex Sm-i is called a stochastic vector. For a stochastic vector x £ Sm-i, we set supp(x) = (i £ Im : xi > 0},null(x) = (i £ Im : xi = 0}. We define a face ra = TOnv(ei}iea of the simp lex Sm-i wher e a C Im and co nv(A) is the convex hull of a set A. Let int ra = (x £ ra : supp(x) = a} and dra = ra \int ra be respectively the relative interior and boundary of the face ra.

Recall that a square matrix P = (pij)m-=i is called non-negative, written P ^ 0, if p^ ^ 0 for all i £ Im. A square matrix P = (pij)mnj=i is called stochastic if each row p^ = (pii,..., pim) is a stochastic vector for all i £ Im. Let L : Sm-i ^ Sm-i be a linear operator (a Markov operator) associated with a square stochastic matrix P = (pij)m-=i, i. e.,

m

L (x) = xP = xiPi-

i=i

© 2017 Saburov M.

1 This work has been partially supported by the MOHE grant FRGS14-141-0382. The author also thanks to the Junior Associate Scheme, Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, where this paper was written, for the invitation and hospitality.

It is easy to see that the linear operator L : Sm-1 ^ Sm-1 jg a surjeetion if and only if it is a bijeetion. Indeed, the straightforward calculation shows that if L : Sm-1 ^ Sm-1 is a surjeetion then for each i there exists j such that L-1(ei) = ej where L-1(ei) is a preimage of the vertex ej of the simp lex Sm-1. Consequently, surjective linear operators of the simplex are only permutation operators.

Recently, the similar problem for a quadratic operator (a nonlinear Markov operator [6]) associated with a cubic stochastic matrix was solved in the paper [5]. In general, the convexity of the quadratic operators is strongly tied up with the nonlinear optimization problems [1, 2, 4, 7j and is not an easy problem [8]. In this paper, we provide a criterion for surjectivity of polynomial operators associated with stochastic hyper-matrices.

2. Polynomial Operators Associated with Stochastic Hyper-Matrices

Let P = (pi1...ik )m jk=1 be a ft-order m-dimensional hyper-matrix. We define the following vectors and matrices

Pil...ik-1^ = {pil--ik-l1 ...ik-1 m) , Pii...ik-2" = (pil—ik-2j'i)j,1=1>

for any i1,___, ik-1 G Im. In what follows, we denote i[1:i] := i1 ... i\ for index.

A hyper-matrix P = (pil...ik)m ik=1 is called non-negative and written P ^ 0 if Pi[l.k-1]• ^ 0 for all i1,..., ife-1 G Im. A hyper-matrix P = (pil...ik)m ¿k=1 is called stochastic if each vector Pi[1.k-1]^ is stochastic for all i1;... ,ik-1 G Im.

We define a polynomial operator P : Sm-1 ^ Sm-1 associated with ft-order m-dimensional stochastic hyper-matrix P = (pil...ik)™ ¿k=1 as follows

m m

P(x) = ^ ^ ^ Xil ■■■Xi k-l p«[l.k-l]^ (!) ¿1=1 ik-l=1

for any x G Sm-1. It is easy to check that

P(x) = xPx (2)

where

mm

Px = ^ • • ^ ^ Xj1 ■ ■ ■ Xjk-2 Pj[1.k-2] •• = (Pjl(x)) m1=1 ¿1 = 1 ¿k-2 =1

is a square stochastic matrix for any x G Sm-1. Due to the matrix form (2), the polynomial operator P : Sm-1 ^ Sm-1 associated with ft-order m-dimensional stochastic hyper-matrix P is a nonlinear Markov operator (see [6]). Unlike the classical Markov chain, the nonlinear

Px

x

Throughout this paper, without loss of generality, we assume that

pjl...jk-l • = pMl)...Mk-l)^

for any i1,..., ik-1 G Im and any permut ation n of the s et Ik-1. We also assume that m ^ k. We need the following auxiliary results.

Proposition 2.1 [6j. The following statements hold:

(i) supp(p(x)) = U supp(pi[l.k-l]^); j[l.k-l] esupp(x)

(ii) null(p(x)) = n null(pi[l.k-l]^);

j[l.k-l] esupp(x)

(iii) P(intra) c intr^ where ft = (J supp(pi[1:k-1]^);

¿[l.k-1] €«

(iv) P(int ra) c int rg if and only if P(x(0)) g int rg for some x(0) g int ra.

An absorbing state played an important role in the theory of the classical Markov chains. Analogously, the concept of absorbing sets for nonlinear Markov chains was introduced in the paper [6].

definition 2.1 [6]. a subset a c Im is called absorbing if one has that

Pi null(pj[l.k-l]0-

[l.k-1] €«

It is clear that a C Im is an absorbing set if and only if

U suPP(p»[l.k-l]^).

[l.k-1] €«

The following result presents an insight of an absorbing set. Proposition 2.2 [6j. The following statements are equivalent:

(i) a C Im

(ii) One has that P(int ra) C intra;

(iii) One has that P (x(0)) G mt ra for some x(0) G int ra.

Proposition 2.3. If any subset a C Im with |a| ^ ft — 1 is absorbing then so are all

Im

< Suppose that any subset a C Im with |a| ^ k — 1 is absorbing. It means that supp(pj[1.k-1]^) C a for any i1, • • • ,ik-1 G a. In particular, the sets a° = {if, • • • ,ik-1} and ft° = {j°} are absorbing for the given indices i\, • • • ,ik-1 ,j° G Im (the repetition of indices is allowed). We then obtain that supp(pj°_j°^) = {j°} and supp(pj°_j° C {i1, • • • ik-1} = a° for any given indices ij, ••• ,ik-1 , j ° G Im (the repetition of indeces is allowed). Hence, for any ft C Im one has that

U supp(pi[l.k-l]^) = U supp(pj„^) U U supp(pi[l.k-l]^) = ft. ¿[l.k-l] eg jeg ¿v=iM

It means that ft is an absorbing subset. This completes the proof. >

Lemma 2.1. If any subset a C Im with |a| ^ ft — 1 is absorbing then the polynomial operator P : Sm-1 ^ Sm-1 is a surjection.

< Due to Propositions 2.2 and 2.3, the polynomial operator P : Sm-1 ^ Sm-1 maps each face of the simplex Sm-1 into itself. It is well-known in algebraic topology that any continuous

Sm-1

Sm-1 (see Lemma 1, [5j). This completes the proof. >

3. Surjective Polynomial Operators vs Lotka-Volterra Operators

We recall a definition of Lotka-Volterra operators (see [5]).

Definition 3.1. A polynomial operator P : Sm-1 ^ Sm-1 is called the Lotka-Volterra operator if supp(pi[1:k-1]^) C {i1, ■ ■ ■, ik-1} for any i1,. ■ ■, ik-1 G Im.

We provide a criterion for the Lotka-Volterra operator in terms of absorbing sets.

Lemma 3.1. The following statements are equivalent:

(i) The polynomial operator P : Sm-i ^ Sm-i is the Lotka-Volterra operator;

(ii) Any subset a C Im with |a| ^ k — 1 is absorbing;

(iii) One has that P-i(int ra) = int ra for any subset a C Im with |a| ^ k — 1.

Remark 3.1. We always assume int ra := ra for the sub set a C Im wit h |a| = 1.

< We prove the following implieations (i) ^ (ii) ^ (iii) ^ (i).

(i) ^ (ii) : Let P : Sm-i ^ Sm-i be the Lotka-Volterra operator. We then have that supp(pi[1.k_1]^) C (ii,..., ifc_i} for any ii,..., ifc_i £ Im (the repetition of indices is allowed). Particularly, supp(pj„j^) = (j} for any j £ Im. Hence, for any a C Im with |a| ^ k — 1 one has that

U supp(pi[i:fc_i]^) = U supp(pj...j>) U U supp(pi[i k_i]•) = a.

i[1:k_1]€a jea iv =iM

a

(ii) ^ (iii) : Suppose that any subset a C Im with |a| ^ k — 1 is absorbing. We then obtain that supp(pi[1:fc_1]^) C a for any ii,... ,ifc_i £ a. Particularly, since the subset a° = (j} is absorbing, we have that supp(pj...j^) = (j} for any j £ Im. It follows from Proposition 2.2, (ii) that P(int ra) C int ra for any absorbing subset a C Im. Moreover, if P-i (int ra) \ int ra = 0 then there exists y £ Sm-i with P := supp(y) such that P \ a = 0 and P(y) £ intra. Then it follows from Proposition 2.1, (iv) that P(int r^) C int ra. Since P : Sm-i ^ Sm-i is continuous, we have that P(r/j) = p(int T^) C int ra = ra. Particularly, P(ej) = £ ra (or equivalently supp(pj„j^) C a) for j £ P \ a. However, this contradicts to the fact that the singleton (j} j £ P \ a is an absorbing set (or equivalently supp(pj„j^) = (j}). Therefore, we have that P-i(intra) = intra.

(iii) ^ (i) : Suppose t hat P^^nt ra) = intra for any sub set a C Im wit h |a| ^ k — 1.

(iii)

U supp(pi[1:k_1]^) = a. i[1:1] ea

Particularly, we get that supp(pi[1:fc_1]^) C a for any ii,... ,ifc_i £ a. Let us now fix indices ii,... ,ik_i £ Im (the repetition of indices is allowed). Then, for the set a° = (ii,... ,ik_i} we have that supp(pi^„i£ C (ii,... ,ik_i} = a°. Since the indices ii,... ,ik_i £ Im are arbitrary chosen, the last inclusion means that P : Sm_i ^ Sm_i is the Lotka-Volterra operator. This completes the proof. >

We are now ready to formulate the main result of the paper.

Theorem 3.1. Let P : Sm_i ^ Sm_i be a polynomial operator. Then the following statements are equivalent:

(i) The polynomial op erator P : Sm_i ^ Sm_i is a surjection;

(ii) There exists a permutation n of the set Im such that for any 1 ^ l ^ k — 1 and for any ii,...,il £ Im one has that P^int rei1 ...ei() = int ren(i1)...en( ) where rei1 ...ei( = conv(ei1,..., e^ };

(iii) There exists a permutation matrix n such that n o P is the Lotka-Volterra operator.

Remark 3.2. We always assume that intrei := (ei} for any i £ Im.

< We prove the following imp lieations (i) ^ (ii) ^ (iii) ^ (i).

(i) ^ (ii) : Suppose that the polynomial operator P : Sm ^ Sm-1 is a surjeetion. Let P-1(ej) be a preimage (which is nonempty) of the vertex ej for j G Im. Obviously, if x G P-1(ej) with supp(x) = a then ra C P-1 (ej) (see Proposition 2.1, (iv)). Hence, P-1(ej) is a face or a union of faces of the simplex Sm-1 for any j G Im. Consequently, the set {P-1 (e1), ■ ■ ■ , P-1 (em)} consists of (at 1 east) m mutually disjoint faces of the simplex Sm-1. This is possible if and only if there exists a permutation n of the set Im such that P-1 (ej) = en(j) to any j G Im- Let us now show P^int^...ei() ^tren(il)...en(i|) for any i1,... ,ii G Im by means of mathematical induction with respect to l where 1 ^ l ^ ft — 1. Obviously, if y G P-1 (int reil...ei() with supp(y) = ft then intrg C P-1 (intreil...ei() and rg C P-1 (reil...ei() (see Proposition 2.1, (iv)). Moreover, if ft \ {n(i1),...,n(i)} = 0 (or equivalently n-1(ft) \ {i1, } = 0) then en(j) G P-1 (reil ...ei;) for some j G n (ft) \ {i 1,..., ii}. However, it contradicts to en(j) = P-1(ej). Therefore, we must have that ft C {n(i1),... ,n(ii)}. On the other hand, due to mathematical induction, we also have that {n(i1),..., n(ii)} \ ft = 0. Hence, we get that ft = {n(i1),..., n(ii)}. Since the point y G P-1(int reil...ei;) is arbitrary chosen, we obtain that P-1 (int reil...ei;) C int r^.^...^ ). The inclusion int r^.^...^ ) C P-1(intreil...ei;) follows from Proposition 2.1, (iv). Consequently, P-1(int reil...ei() = int r^.^...^ ) for any i1,... ,ii G Im and 1 ^ l ^ ft — 1.

(ii) ^ (iii) : Suppose that there exists a permutation n such th at P-1(int reil ...ei() = int ren(il)...en(.i) for any i1,...,ii G Im rnd 1 ^ l ^ ft — 1. Particularly, we have that P-1 (ej) = en(j) for any j G We now define a permutation matrix n (associated with the permutation n) as follows n(ej) := enj) for any j G Im. Obviously, we obtain that

n(P(intren ...eij )) ^trefl ...eij , V i1,...,il G Im, V 1 < l < ft — 1.

Due to Lemma 3.1, the polynomial operator n o P is the Lotka-Volterra operator.

(iii) ^ (i) : Suppose that there exists a permutation matrix n such th at Pn := n o P is the Lotka-Volterra operator. Due to Lemmas 2.1 and 3.1, the Lotka-Volterra operator Pn is a surjeetion and so is the polynomial operator P = n-1 o Pn. This completes the proof. >

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Received, February 6, 2011 Saburov Mansur

Department of Computational & Theoretical Sciences, Faculty of Science,

International Islamic University Malaysia, Associate Professor

P.O. Box, Kuantan, Pahang, 25200, Malaysia E-mail: [email protected]; [email protected]

ЗАМЕЧАНИЕ О СЮРЪЕКТИВНЫХ ПОЛИНОМИАЛЬНЫХ ОПЕРАТОРАХ

Сабуров М.

Линейная цепь Маркова является случайным процессом с дискретными состояниями, переходы которого зависят только от текущего состояния процесса. Нелинейная цепь Маркова — случайный процесс с дискретными состояниями, переходы которого могут зависеть как от текущего состояния, так и текущего распределения процесса. Эти процессы естественным образом возникают при изучении предельного поведения большого количества слабо взаимодействующих марковских процессов. Нелинейные марковские процессы были введены Маккином и широко изучались в контексте нелинейных уравнений Чапмана - Колмогорова, а также нелинейных уравнений Фоккера - Планка. Нелинейная цепь Маркова над конечным пространством состояний может быть определена непрерывным отображением (нелинейным оператором Маркова), определяемым на множестве всех вероятностных распределений (являющемся симплексом) конечного пространства состояний семейством матриц перехода, зависящих от распределения вероятностей занятия состояний. В частности, линейный оператор Маркова является линейным оператором, связанным с квадратной стохастической матрицей. Хорошо известно, что линейный оператор Маркова будет сюръекцией симплекса в том и только в том случае, когда он является биекцией. Аналогичная задача для нелинейного оператора Маркова, связанного со стохастической гипер-матрицей, оставалась открытой. Она решена в данной статье, а именно, показано, что нелинейный оператор Марков, связанный со стохастической гиперматрицей, является сюръекцией симплекса, если и только если он является перестановкой оператор Лотки — Вольтерра.

Ключевые слова: стохастическая гипер-матрица, полиномиальный оператор, оператор Лотки — Вольтерра.