Extinction, persistence and global stability in models of population growth

First, we systematize earlier results on the global stability of discrete model An+1=λAn+F(An−m) of population growth. Second, we invent the effect of delay m when F is unimodal. New, deep and strong results are discussed in Section 4, although Theorems 3-5 (Section 3) are still freshly new. This paper may be considered as a discrete version of our earlier work on the model [D.V. Giang, Y. Lenbury, Nonlinear delay differential equations involving population growth, Math. Comput. Modelling 40 (2004) 583-590]. We are mainly using ω-limit set of persistent solution, which is discussed in more general by P. Walters [An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982].     

On classical adjoint-commuting mappings between matrix algebras

Let F be a field and let m and n be integers with m,n?3. Let M_n denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:M_n→M_m that satisfy one of the following conditions:

1.

|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈M_n and α∈F with ψ(I_n)≠0.

2.

ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈M_n.

Here, adjA denotes the classical adjoint of the matrix A, and I_n is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(I_n)≠0 in our results.     

On the number of invariant polynomials of the matrix XAX−1+B

Let F be a field. If A is an n×n matrix over F, we denote by i(A) the number of invariant polynomials of A different from 1. We shall prove that if A,B are n×n matrices over F and t ∈ {1,…,n}, then i(A)+i(B)⩽n+t if and only if there exists a nonsingular matrix X over F such that i(XAX⁻¹+B)⩽t, except in a few cases.     

On the asymptotic convergence factor of the total step method in interval computation

For a special class of n×n interval matrices A we derive a necessary and sufficient condition for the asymptotic convergence factor α of the total step method x^(m+1)=Ax^(m)+b to be less than the spectral radius ϱ(|A|) of the absolute value |A| of A.     

Helly-type theorems about sets

Let A be a finite set and $\text{F}$ be a family of its subsets. Two players pick m resp. n points alternately from A. I wins if he picks all the points of some element of $\text{F}$, otherwise II wins. We give a sufficient condition for II to have a winning strategy. Using this we prove the following statement. If in Go-moku I occupies m places, and II occupies n places in each turn, then, in case of m?n, I may have an arbitrarily long row of his places; in case m?n, II may bound the lenght of the rows occupied I.     

Linear transformations which preserve fixed rank

Let M_{m, n}(F) denote the set of all m×n matrices over the algebraically closed field F. Let T denote a linear transformation, T:M_{m, n}(F)→M_{m, n}(F). Theorem: If max(m, n)?2k?2, k?1, and T preserves rank k matrices [i.e.?(A)=k implies ?(T(A))=k], then there exist nonsingular m×m and n×n matrices U and V respectively such that either (i) T:A→UAV for all A?M_{m, n}(F), or (ii) m=n and T:A→UA^tV for all A?M_n(F), where A^t denotes the transpose of A.     

On averaging Frankl's conjecture for large union-closed-sets

Let F be a union-closed family of subsets of an m-element set A. Let n=|F|?2 and for a∈A let s(a) denote the number of sets in F that contain a. Frankl's conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a∈A with n−2s(a)?0. Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280-288] we verify the conjecture for the particular case when m?3 and n?m²−²m/2. Moreover, for these “large” families F we prove an even stronger version via averaging. Namely, the sum of the n−2s(a), for all a∈A, is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices.     

Lie properties of symmetric elements in group rings II

Let F be a field of characteristic different from 2, and G a group with involution ∗. Write (FG)⁺ for the set of elements in the group ring FG that are symmetric with respect to the induced involution. Recently, Giambruno, Polcino Milies and Sehgal showed that if G has no 2-elements, and (FG)⁺ is Lie nilpotent (resp. Lie n-Engel), then FG is Lie nilpotent (resp. Lie m-Engel, for some m). Here, we classify the groups containing 2-elements such that (FG)⁺ is Lie nilpotent or Lie n-Engel.     

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω₁(F)=s, where ω₁(F) is the number of components of order one in F. We denote by σ₃(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ₃ conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ₃(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ₃(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ₃(G)≥n+κ(G)+2, then G is hamiltonian-connected.     

Path-fan Ramsey numbers

For two given graphs F and H, the Ramsey number R(F,H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers R(P_n,F_m), where P_n is a path on n vertices and F_m is the graph obtained from m disjoint triangles by identifying precisely one vertex of every triangle (F_m is the join of K₁ and mK₂). We determine the exact values of R(P_n,F_m) for the following values of n and m: 1?n?5 and m?2; n?6 and 2?m?(n+1)/2; 6?n?7 and m?n-1; n?8 and n-1?m?n or ((q·n-2q+1)/2?m?(q·n-q+2)/2 with 3?q?n-5) or m?(n-3)²/2; odd n?9 and ((q·n-3q+1)/2?m?(q·n-2q)/2 with 3?q?(n-3)/2) or ((q·n-q-n+4)/2?m?(q·n-2q)/2 with (n-1)/2?q?n-5). Moreover, we give nontrivial lower bounds and upper bounds for R(P_n,F_m) for the other values of m and n.     

On the sum of powers of terms of a linear recurrence sequence

Let (F _n ) _n??0 be the Fibonacci sequence given by F _n+2 = F _n+1 + F _n , for n ?? 0, where F ₀ = 0 and F ₁ = 1. There are several interesting identities involving this sequence such as F _n ² + F _n+1 ² = F _2n+1, for all n ?? 0. In a very recent paper, Marques and Togbé proved that if F _n ^s + F _n+1 ^s is a Fibonacci number for all sufficiently large n, then s = 1 or 2. In this paper, we will prove, in particular, that if (G _m ) _m is a linear recurrence sequence (under weak assumptions) and G _n ^s + ... + G _n+k ^s ?? (G _m ) _m , for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on k and the parameters of G _m .     

Generalized inverses of matrices over a finite field

For a given m × n matrix A of rank r over a finite field F, the number of generalized inverses, of reflexive generalized inverses, of normalized generalized inverses, and of pseudoinverses of A are determined by elementary methods. The more difficult problem of determining which m × n matrices A of rank r over F have normalized generalized inverses and which have pseudoinverses is solved. Moreover, the number of such matrices which possess normalized generalized inverses and the number which possess pseudoinverses are found.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

Products of reflections in the general linear group over a division ring

Let F be a division ring and A?GL_n(F). We determine the smallest integer k such that A admits a factorization A=R₁R₂?R_k?1B, where R₁,…,R_k?1 are reflections and B is such that rank(B?I_n)=1. We find that, apart from two very special exceptional cases, k=rank(A?I_n). In the exceptional cases k is one larger than this rank. The first exceptional case is the matrices A of the form I_m⊕αI_n?m where n?m?2, α≠?1, and α belongs to the center of F. The second exceptional case is the matrices A satisfying (A?I_n)²=0, rank(A?I_n)?2 in the case when char F≠2 only. This result is used to determine, in the case when F is commutative, the length of a matrix A?GL_n(F) with detA=±1 with respect to the set of all reflections in GL_n(F).     

A Grobman-Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators Am, we establish a version of the Grobman-Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(n), determined by a sequence ρ(n). For all sufficiently small Lipschitz perturbations Am+fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators Am. We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ(n)=n, but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ(n)=logn.     

On the multiplicity and the Cohen-Macaulayness of fiber cones of graded algebras

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J. Let S=?_n≥0S_n be a finitely generated standard graded algebra over A. Set S⁺=?_n>0S_n. Denote by F_J(S)=?_n≥0→(S_n/JS_n) the fiber cone of S with respect to J. The paper characterizes the multiplicity and the Cohen-Macaulayness of F_J(S) in terms of minimal reductions of S⁺.     

Jordan derivations of full matrix algebras

Let A be a unital associative ring and M be a 2-torsion free A-bimodule. Using an elementary and constructive method we show that every Jordan derivation from M_n(A) into M_n(M) is a derivation.     

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(? ⁿ ) is a Banach space as well as an algebra, while B(? ⁿ , ? ^m ) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σ _r be the set of all operators of finite rank r in B(E, F) (or B(? ⁿ , ? ^m )). In fact, we have that 1) suppose Σ _r ∈ B(? ⁿ , ? ^m ), and then Σ _r is a smooth and path-connected submanifold of B(? ⁿ , ? ^m ) and dimΣ _r = (n + m)r ? r ², for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σ _r and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σ _r ∈ B(E, F), and dimF = ∞, and then Σ _r is a smooth and path-connected submanifold of B(E, F) with the tangent space T _A Σ _r = {B ∈ B(E, F): BN(A) ? R(A)} at each A ∈ Σ _r for 0 ? r ? ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(? ⁿ ), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.