共查询到20条相似文献,搜索用时 523 毫秒
1.
Letγ(n, R) andγ * (n, R) stand for the semigroups of all stochastic and all invertible stochasticn × n real matrices respectively, and letK be a commutative field. The following theorem is proved in the paper. Every functionf :γ(n, R) → K orf :γ * (n, R) → K satisfying the multiplicative functional equationf(X)f(Y) = f(XY) is of the formf(X) = ψ(detX), whereψ: 〈?1, 1〉 → K orψ: 〈?1, 0) ∪ (0, 1〉 → →K, respectively, is an arbitrary multiplicative function. The concept of pseudostochastic matrix \(\left( {\sum\limits_{j = 1}^n {a_{ij} = 1} } \right)\) and the canonical form of such a matrix derived in [3] are used. 相似文献
2.
We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F *, thenT is an additive injective operator preserving rank-additivity onS n(F) if and only if there exists an invertible matrixU∈M n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ?UT, ?X=(xij)∈Sn(F) wherec∈F *,X ?=(?(x ij)). As applications, we determine the additive operators preserving minus-order onS n(F) over the fieldF. 相似文献
3.
We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of ? n of dimension greater than two? We call an n-immersion f(x) in ? m isothermic k if the normal bundle of f is flat and x is a line of curvature coordinate system such that its induced metric is of the form $\sum_{i=1}^{n} g_{ii}\,\mathrm{d} x_{i}^{2}$ with $\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0$ , where ?? i =1 for 1??i??n?k and ?? i =?1 for n?k<i??n. A smooth map (f 1,??,f n ) from an open subset ${\mathcal{O}}$ of ? n to the space of m×n matrices is called an n-tuple of isothermic k n-submanifolds in ? m if each f i is an isothermic k immersion, $(f_{i})_{x_{j}}$ is parallel to $(f_{1})_{x_{j}}$ for all 1??i,j??n, and there exists an orthonormal frame (e 1,??,e n ) and a GL(n)-valued map (a ij ) such that $\mathrm{d}f_{i}= \sum_{j=1}^{n} a_{ij} e_{j}\,\mathrm {d} x_{j}$ for 1??i??n. Isothermic1 surfaces in ?3 are the classical isothermic surfaces in ?3. Isothermic k submanifolds in ? m are invariant under conformal transformations. We show that the equation for n-tuples of isothermic k n-submanifolds in ? m is the $\frac{O(m+n-k,k)}{O(m)\times O(n-k,k)}$ -system, which is an integrable system. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries. 相似文献
4.
Prof. Dr. Ulrich Krengel 《Monatshefte für Mathematik》1978,86(1):3-6
Given any ergodic invertible measure preserving transformation τ of [0,1] and any null-sequence (α N ) of positive reals, there exists a continuousf such that $$\lim \sup \alpha _{\rm N}^{ - 1} \left| {N^{ - 1} \sum\limits_{k = 0}^{N - 1} {f \circ \tau ^k - \smallint f} } \right| = \infty a. e.,$$ i.e. there is no “speed of convergence” in the ergodic theorem for any τ. The analogous result holds also for norm-convergence. 相似文献
5.
V. Maier 《Analysis Mathematica》1978,4(4):289-295
Оператор Канторович а дляf∈L p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W p 2 (I), т.е.f абсол ютно непрерывна наI иf″∈L p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L p(I),p>1 и ∥P n f-f∥р=О(n ?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ 0 x h(t)dt, гдеh∈L p(I) и ∝ 0 1 h(t)dt=0}. Если жеf∈Lp(I),p>1, то из условия ∥P n(f)?f∥pL=o(n?1) вытекает, чтоf постоянна почти всюду. 相似文献
6.
For functionsf(x) ε KH(α) [satisfying the Lipschitz condition of order α (0 < α < 1) with constant K on [?1, 1], the existence is proved of a sequence Pn (f; x) of algebraic polynomials of degree n = 1, 2,..., such that $$|f(x) - P_{n - 1} (f;x)| \leqslant \mathop {\sup }\limits_{f \in KH^{(\alpha )} } E_n (f)[(1 - x^2 )^{a/2} + o(1)],$$ when n → ∞, uniformly for x ε [?1, 1], where En(f) is the best approximation off(x) by polynomials of degree not higher than n. 相似文献
7.
E. G. Goluzina 《Journal of Mathematical Sciences》1987,38(4):2061-2064
LetP κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF α(z)∈P κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g (v?1)(z?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N ?, on the classP κ,1(λ,β). On the classesP κ,n (λ,β),M κ,n (λ,β,α) one finds the ranges of Cv, v?n, am, n?m?2n-2, and ofg(?),F ?(?), 0<¦ξ¦<1, ξ is fixed. 相似文献
8.
Y. Kaya 《Siberian Mathematical Journal》2006,47(3):452-458
Let f: M m → ? m+1 be an immersion of an orientable m-dimensional connected smooth manifold M without boundary and assume that ξ is a unit normal field for f. For a real number t the map f tξ: M m → ? m+1 is defined as f tξ(p) = f(p) + tξ(p). It is known that if f tξ is an immersion, then for each p ∈ M the number of the focal points on the line segment joining f(p) to f tξ(p) is a constant integer. This constant integer is called the index of the parallel immersion f tξ and clearly the index lies between 0 and m. In case f: $\mathbb{S}^m \to \mathbb{R}^{m + 1} $ is an immersion, we study the presence of a component of index μ in the push-out space Ω(f). If there exists a component with index μ = m in Ω(f) then f is known to be a strictly convex embedding of $\mathbb{S}^m $ . We reveal the structure of Ω(f) when $f(\mathbb{S}^m )$ is convex and nonconvex. We also show that the presence of a component of index μ in Ω(f) enables us to construct a continuous field of tangent planes of dimension μ on $\mathbb{S}^m $ and so we see that for certain values of μ there does not exist a component of index μ in Ω(f). 相似文献
9.
Kirill A. Kopotun 《Constructive Approximation》1994,10(2):153-178
Let Δ q be the set of functionsf for which theqth difference, is nonnegative on the interval [? 1,1],P n is the set of algebraic polynomials of degree not exceedingn, τ k (f, δ) p is the averaged Sendov-Popov modulus of smoothness in theL p [?1,1] metric for 1≦p≦∞, ω k (f, δ) and $\omega _\phi ^k (f,\delta ),\phi (x): = \sqrt {1 - x^2 } ,$ , are the usual modulus and the Ditzian-Totik modulus of smoothness in the uniform metric, respectively. For a functionf∈C[?1,1]?Δ2 we construct a polynomialp n ∈P n ?Δ2 such that $$\begin{gathered} \left| {f(x) - p_n (x)} \right| \leqslant C\omega _3 (f,n^{ - 1} \sqrt {1 - x^2 } + n^{ - 2} ),x \in [ - 1,1]; \hfill \\ \left\| {f - p_n } \right\|_\infty \leqslant C\omega _\phi ^3 (f,n^{ - 1} ); \hfill \\ \left\| {f - p_n } \right\|_p \leqslant C\tau _3 (f,n^{ - 1} )_p . \hfill \\ \end{gathered}$$ As a consequence, for a functionf∈C 2[?1,1]?Δ3 a polynomialp n * ∈P n ?Δ3 exists such that $$\left\| {f - p_n^* } \right\|_\infty \leqslant Cn^{ - 1} \omega _2 (f\prime ,n^{ - 1} ),$$ wheren≥2 andC is an absolute constant. 相似文献
10.
V. I. Nechaev 《Mathematical Notes》1975,17(6):504-511
Supposef is a polynomial of degree n≥3 with integral coefficientsa 0,a 1,...,a n; q is a natural number; (a 1,...,a n, q)=1,f(0) = 0. It is proved that $$\left| {\sum\nolimits_{x = 1}^q {e^{2\pi if(x)/q} } } \right|< e^{5n^2 /\ln n} q^{1 - 1/n} $$ . 相似文献
11.
J. Hu 《Transformation Groups》2010,15(2):333-370
Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let
\mathfrakBn(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra
\mathfrakBn( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e
1
e
3⋯
e
2f-1 where 0 ≤ f ≤ [n/2]. Let HTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V
⊗n
. In this paper we prove that dimHTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim
\textEn\textdK\textSp(V)( V ?n \mathord