The conjugacy problem in extensions of Thompson’s group F

We solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut₊(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Gonçalves in [4] showing that F has property R_∞, and which can be extended to show that Thompson’s group T also has property R_∞.     

Relaxation results for functions depending on polynomials changing sign on rank-one matrices

In this paper, we are interested in computing the different convex envelopes of functions depending on polynomials, especially those having it is main part change sign on rank-one matrices. Our main result applies to functions of the type W(F)=φ(P(F)), W(F)=φ(P(F))+f(detF) or W(F)=φ(P(F))+g(adjnF) defined on the space of matrices, where φ, f:R→R and g:R³→R are three continuous functions, and P=P₀+P₁+?+Pd is a polynomial such that Pd has the property of changing sign on rank-one matrices. Then the polyconvex, quasi-convex and rank-one convex envelopes of W are equal.     

On the matrices of given rank in a large subspace

Let V be a linear subspace of M_n,p(K) with codimension lesser than n, where K is an arbitrary field and n?p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K?F₂. Here, we give a sufficient condition on codim V for V to be spanned by its rank r matrices for a given r∈?1,p-1?. This involves a generalization of the Gerstenhaber theorem on linear subspaces of nilpotent matrices.     

On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I～J if and only ifV _R (I)=V _R(J), whereV _R() is the centralizer inR. LetI _R=F/～. Then we can see thatn(I _R), the cardinality ofI _R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st?ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I _R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I _R)=m for any given natural numberm.     

Points entiers de certains schémas de matrices

Let R be an Euclidean domain, M _n (R) the R-module of matrices with n rows and n columns, V an intersection of n?1 affine hyperplanes of M _n (R). Assume V is not empty. Then, for all a in R, there exists a matrix in V with determinant equal to a.     

The variety of the asymptotic values of a real polynomial etale map

A polynomial map F: R² → R² is said to satisfy the Jacobian condition if ∀(X, Y)ϵ R², J(F)(X, Y) ≠ 0. The real Jacobian conjecture was the assertion that such a map is a global diffeomorphism. Recently the conjecture was shown to be false by S. Pinchuk. According to a theorem of J. Hadamard any counterexample to the conjecture must have asymptotic values. We give the structure of the variety of all the asymptotic values of a polynomial map F: R² → R² that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps that have only X- or Y-finite asymptotic values. We prove that a Y-finite asymptotic value can be realized by F along a rational curve of the type (X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N), where X → 0, Y is fixed and K, N > 0 are integers. More precisely we prove that the coordinate polynomials P(U, V) of F(U, V) satisfy finitely many asymptotic identities, namely, identities of the following type, P(X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N) = A(X, Y)ϵ R[X, Y], which ‘capture’ the whole set of asymptotic values of F.     

Fractional weak discrepancy and split semiorders

The fractional weak discrepancywd_F(P) of a poset P=(V,?) was introduced in Shuchat et al. (2007) [6] as the minimum nonnegative k for which there exists a function f:V→R satisfying (i) if a?b then f(a)+1≤f(b) and (ii) if a∥b then |f(a)−f(b)|≤k. In this paper we generalize results in Shuchat et al. (2006, 2009) [5] and [7] on the range of wd_F for semiorders to the larger class of split semiorders. In particular, we prove that for such posets the range is the set of rationals that can be represented as r/s for which 0≤s−1≤r<2s.     

Linear transformations on matrices: the invariance of sets of ranks

Let R_E denote the set of all m × n matrices over an algebraically closed field F whose ranks lie in the set E, where E is a subset of {1,2,…,m}. Let T be a linear transformation which maps R_E into itself. Under some restrictions on E, or when T is nonsingular, there are nonsingular matrices U and V such that T(A) = UAV for every m × n matrix A.     

Some results on V-ary asymmetric tries

Tries (radix search tries) find many applications in computer science and telecommunications. It is assumed that a trie is built over an alphabet U = {σ₁,…,σ_v} (V-ary trie), and n (possible infinite) strings of elements from U (i.e., keys) are stored in external nodes of the trie. The occurrence of an element σ_i in a key is represented by a probability p_i (asymmetric trie). Our main interest is to compute all moments of the depth of a leaf (external node) in a random family of tries. By solving a system of recurrences we find an exact formula for all factorial moments of the depth, and—using the Mellin transform technique—we derive asymptotic approximations for them. We prove that the m th factorial moment of the depth of a leaf in a trie with n keys is equal to αln^mn + βln^m−1n + O(ln^m−2n), where the constants α and β are functions of the probabilities, p_i, i = 1,…,V. In particular, we show that for symmetric tries the variance of the depth is O(1), while for asymmetric tries it is αlnn + O(1), and we determine explicitly the constant O(1). These results extend previous analyses by Knuth [12], Flajolet and Sedgewick [6], Jacquet and Regnier [10], and Kirschenhofer and Prodinger [11].     

The Double Centralizer Theorem for Semiprime Algebras

In the paper we prove the double centralizer theorem for semiprime algebras. To be precise, let R be a closed semiprime algebra over its extended centroid F, and let A be a closed semiprime subalgebra of R, which is a finitely generated module over F. Then C _R (A) is also a closed semiprime algebra and C _R (C _R (A))?=?A. In addition, if C _R (A) satisfies a polynomial identity, then so does the whole ring R. Here, for a subset T of R, we write C _R (T):?=?{x?∈?R|xt?=?tx???t?∈?T}, the centralizer of T in R.     

Rate of decay for solutions of stochastic differential equations

A theorem is given which demonstrates that solutions of a stochastic differential equation decay to zero at least as fast as a function ψ(t) provided $\text{L[}\text{ln}\text{V] ?}\text{(d}\text{dt)}\text{(}\text{ln}\text{ψ)}$, where V is a Liapunov-like function and L is the operator associated with the stochastic equation. The result is an extension of a recent result of A. Friedman and M. Pinsky. A further generalization is discussed.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

Optimality conditions for fractional minmax programming

Let V ?H be real separable Hilbert spaces. The abstract wave equation u′' + A(t)u = g(u), where u(t) ?V, A(t) maps V to its dual $\text{V}{}^1$, and g is a nonlinear map from the ball S(R₀) = {u?V: ∥u∥ < R₀} into H, is considered. It is assumed that g is locally Lipschitz in S(R₀) and possibly singular at the boundary. Local existence and continuation theorems are established for the Cauchy problem u(0) = u₀?S(R₀), u′(0) = u₁?H. Global existence is shown for g(u) = εφ(u), where φ has a potential and ε is small. Global nonexistence is shown for g(u) = εφ(u), where φ satisfies an abstract convexity property and ε is large.     

Idempotent elements determined matrix algebras

Let M_n(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from M_n(R)×M_n(R) to V satisfies the condition that {u,u}={e,u} whenever u²=u, then there exists a linear map f from M_n(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on M_n(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on M_n(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.     

The centralizer of a semi-linear transformation

Let V be a finite-dimensional vector space over a division ring D, where D is finite-dimensional over its center F. Suppose T is a semi-linear transformation on V with associated automorphism σ of D. The centralizer of T is the ring C(T) of all linear transformations on V which commute with T. If σ^r is the identity on D for some r ? 1 and no smaller positive power of σ is an inner automorphism, then the center of C(T) is computed to be polynomials in T^r with coefficients from F₀, where F₀ is the subfield of F left elementwise fixed by σ. A matrix version of this theorem is also given.     

Standard identities for skew-symmetric matrices

Let n be a positive, even integer and let K_n(F) denote the subspace of skew-symmetric matrices of Mn(F), the full matrix algebra with coefficients in a field F. A theorem of Kostant states that K_n(F) satisfies the (2n-2)-fold standard identity s_2n-2. In this paper we refine this result by showing that s_2n-2 may be written non-trivially as the sum of two polynomial identities of K_n(F).     

Some common fixed point theorems for Banach operator pairs with applications in best approximation

The ordered pair (T,I) of two self-maps of a metric space (X,d) is called a Banach operator pair if the set F(I) of fixed points of I is T-invariant i.e. T(F(I))⊆F(I). Some common fixed point theorems for a Banach operator pair and the existence of common fixed points of best approximation are presented in this paper. The results prove, generalize and extend some results of Al-Thagafi [M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318-323], Carbone [A. Carbone, Applications of fixed point theorems, Jnanabha 19 (1989) 149-155], Chen and Li [J. Chen, Z. Li, Common fixed points for Banach operator pairs in best approximations, J. Math. Anal. Appl. 336 (2007) 1466-1475], Habiniak [L. Habiniak, Fixed point theorems and invariant approximation, J. Approx. Theory 56 (1989) 241-244], Jungck and Sessa [G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japon. 42 (1995) 249-252], Sahab, Khan and Sessa [S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349-351], Shahzad [N. Shahzad, Invariant approximations and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39-45] and of few others.     

An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L _p,k, and the finite field F(p ^k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ₁ of the variety V(L _p,k) generated by L _p,k into the variety V(F(p ^k)) generated by F(p ^k) and an interpretation Φ₂ of V(F(p ^k)) into V(L _p,k) such that Φ₂Φ₁(B) = B for every B ε V(L _p,k) and Φ₁Φ₂(R) = R for every R ε V(F(p ^k)).     

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.