THREE NOTES ON NICE EXTENSION DOMAINS

The first note shows that the integral closure L′ of certain localities L over a local domain R are unmixed and analytically unramified, even when it is not assumed that R has these properties. The second note considers a separably generated extension domain B of a regular domain A, and a sufficient condition is given for a prime ideal p in A to be unramified with respect to B (that is, p B is an intersection of prime ideals and B/P is separably generated over A/p for all P ∈ Ass (B/p B)). Then, assuming that p satisfies this condition, a sufficient condition is given in order that all but finitely many q ∈ S = {q ∈ Spec(A), p ? q and height(q/p) = 1} are unramified with respect to B, and a form of the converse is also considered. The third note shows that if R′ is the integral closure of a semi-local domain R, then I(R) = ∩{R′ _p′ ;p′ ∈ Spec(R′) and altitude(R′/p′) = altitude(R′) ? 1} is a quasi-semi-local Krull domain such that: (a) height(N ^*) = altitude(R) for each maximal ideal N ^* in I(R); and, (b) I(R) is an H-domain (that is, altitude(I(R)/p ^*) = altitude(I(R)) ? 1 for all height one p ^* ∈ Spec(I(R))). Also, K = ∩{R _p ; p ∈ Spec(R) and altitude(R/p) = altitude(R) ? 1} is a quasi-semi-local H-domain such that height (N) = altitude(R) for all maximal ideals N in K.     

On pointwise and analytic similarity of matrices

LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ _p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈_a B(ε) ifB(ε)=T(ε)A(ε)T ⁻¹(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ _a B(ε) provided thatA(ε)≈ _p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024     

On the a-Browder and a-Weyl spectra of tensor products

Given Banach space operators A ∈ B( ) and B ∈ B( ), let A?B ∈ B( ? ) denote the tensor product of A and B. Let σ _a , σ _aw and σ _ab denote the approximate point spectrum, the Weyl approximate point spectrum and the Browder approximate point spectrum, respectively. Then σ _aw (A?B) ? σ _a (A)σ _aw (B) ? σ _aw (A)σ _a (B) ? σ _a (A)σ _ab (B) ? σ _ab (A)σ _a (B) = σ _ab (A?B), and a sufficient condition for the (a-Weyl spectrum) identity σ _aw (A?B) = σ _a (A)σ _aw (B) ? σ _aw (A)σ _a (B) to hold is that σ _aw (A?B) = σ _ab (A?B). Equivalent conditions are proved in Theorem 1, and the problem of the transference of a-Weyl’s theorem for a-isoloid operators A and B to their tensor product A?B is considered in Theorem 2. Necessary and sufficient conditions for the (plain) Weyl spectrum identity are revisited in Theorem 3.     

Infinite dimensional linear groups with many G — invariant subspaces

Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.     

Normal series and character values in p-standard table algebras

We investigate the character values and structures of p-standard table algebras (A,B) with o(B) = p^N. If N≤3, then B has a complete normal series. If for every χ∈Irr(B), χ has at most p distinct classes of character values, and if either B has a complete normal series or p = 2, then B is an elementary abelian p-group.     

Equivalent measures of dependence

The maximal correlation between a pair of σ-fields $\text{A}$ and $\text{B}$ becomes arbitrarily small as sup{|P(A ? B) ? P(A) P(B)|/[P(A) P(B)]^1/2, A ∈ $\text{A}$, B ∈ $\text{B}$, P(A) > 0, P(B) > 0} becomes sufficiently small.     

Bilinear isometries on subspaces of continuous functions

Let A and B be strongly separating linear subspaces of C₀(X) and C₀(Y), respectively, and assume that ?A ≠ ?? (?A stands for the set of generalized peak points for A) and ?B ≠ ??. Let T: A × B → C₀(Z) be a bilinear isometry. Then there exist a nonempty subset Z₀ of Z, a surjective continuous mapping h: Z₀ → ?A × ?B and a norm‐one continuous function a: Z₀ → K such that T (f, g)(z) = a (z)f (π_x (h (z))g (π_y (h (z)) for all z ∈ Z₀ and every pair (f, g) ∈ A × B. These results can be applied, for example, to non‐unital function algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equations in finite fields with restricted solution sets. I (Character sums)

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z _p and for h(x) ∈ Z _p [x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with a ∈ A, b ∈ B, x ∈ Z _p , and for large subsets A, B, C, D of Z _p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.     

On the evaluation of the Eigenvalues of a banded toeplitz block matrix

Let A = (a_ij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, a_ij = a_j−i, i, J = 1,… ,n, a_i = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if a_i are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m³k(log² k + log n) + m²k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B.     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.      

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.     

Characterization of Lie Derivations on Prime Rings

Let ? be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent P. It is shown that, under some mild conditions, an additive map L on ? satisfies L([A, B]) = [L(A), B] + [A, L(B)] whenever AB = 0 (resp., AB = P) if and only if it has the form L(A) = ?(A) + h(A) for all A ∈ ?, where ? is an additive derivation on ? and h is an additive map into its center.     

A Problem in Linear Matrix Approximation

Let A be a normal operator in ??(H), H a complex Hilbert space, and let ? _A = ? {AX - XA:X ∈ ??(H)} be the commutator subspace of ??(H) associated with A. If B in ??(H) commutes with A, then B is orthogonal to ?_A with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in ? _A. This was proved by J. Anderson in 1973 and extended by P. J. Maher with respect to the Schatten p-norm recently. We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in R_A and prove that the metric projection of H onto ?_A is continuous.     

Interpolating sequences in the ball ofC n

LetB be the unit ball ofC ⁿ , I give necessary conditions on sequenceS of points inB to beH ^∞(B) interpolating in term of aC ⁿ valued holomorphic function zero onS (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequenceS is interpolating for ∩ _p>1 (B) and is also interpolating forH ^p (B) for 1≤p<∞.     

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A ^m such that (x ₁, …, x _m ) ≡ (y ₁, …, y _m ) if there exists a permutation σ on {1, …, m} such that y _σ(i) = x _i for all i. Let A ^(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A ^(m) are said to be adjacent if (x ₁, …, x _m?1, a) ∈ X and (x ₁, …, x _m?1, b) ∈ Y for some x ₁, …, x _m?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A ^(m) to B ⁽ⁿ⁾ that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

Polynomials in a matrix

Let A be a matrixp(x) a polynomial. Put B=p(A). It is shown that necessary and sufficient conditions for A to be a polynomial in B are (i) if λ is any eigenvalue of A, and if some elementary divisor of A corresponding to λ is nonlinear, thenp ^′(λ)≠0;and (ii) if λ,μ are distinct eigenvalues of A, then p(λ)p(μ) are also distinct. Here all computations are over some algebraically closed field.     

Rigidity of commutators and elementary operators on Calkin algebras

LetA=(A ₁,...,A _n ),B=(B ₁,...,B _n )εL(ℓ ^p ) ⁿ be arbitraryn-tuples of bounded linear operators on (ℓ ^p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε _a,b on the Calkin algebraC(ℓ ^p )≡L(ℓ ^p )/K(ℓ ^p ); , where quotient elements are denoted bys=S+K(ℓ ^p ) forSεL(ℓ ^p ). It is shown among other results that the kernel Ker(ε _a,b ) is a non-separable subspace ofC(ℓ ^p ) whenever ε _a,b fails to be one-one, while the quotient is non-separable whenever ε _a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A ₁,...,A _n }, {B ₁,...,B _n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces. Supported by the Academy of Finland, Project 32837.     

Completely strong path-connected tournaments

Let T = (V, A) be a tournament with p vertices. T is called completely strong path-connected if for each arc (a, b) ∈ A and k (k = 2, 3,…, p), there is a path from b to a of length k (denoted by P_k(a, b)) and a path from a to b of length k (denoted by P′_k(a, b)). In this paper, we prove that T is completely strong path-connected if and only if for each arc (a, b) ∈ A, there exist P₂(a, b), P′₂(a, b) in T, and T satisfies one of the following conditions: (a) T/T₀-type graph, (b) T is 2-connected, (c) for each arc (a, b) ∈ A, there exists a P′_p?1(a, b) in T.