共查询到20条相似文献,搜索用时 31 毫秒
1.
Hiroaki Minami 《Archive for Mathematical Logic》2010,49(4):501-518
We investigate splitting number and reaping number for the structure (ω)
ω
of infinite partitions of ω. We prove that
\mathfrakrd £ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and
\mathfraksd 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency
\mathfrakrd < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and
\mathfraksd < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants
\mathfrakrpair{\mathfrak{r}_{pair}} and
\mathfrakspair{\mathfrak{s}_{pair}} . We also study the relation between
\mathfrakrpair, \mathfrakspair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that
cov(M),cov(N) £ \mathfrakrpair £ \mathfraksd,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and
\mathfraks £ \mathfrakspair £ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} . 相似文献
2.
De-Jun Feng 《Israel Journal of Mathematics》2003,138(1):353-376
Let (Σ,σ) be a full shift space on an alphabet consisting ofm symbols and letM: Σ→L
+(ℝ
d
, ℝ
d
) be a continuous function taking values in the set ofd×d positive matrices. Denote by λ
M
(x) the upper Lyapunov exponent ofM atx. The set of possible Lyapunov exponents is just an interval. For any possible Lyapunov exponentα, we prove the following variational formula,
, where dim is the Hausdorff dimension or the packing dimension,P
M(q) is the pressure function ofM, μ is aσ-invariant Borel probability measure on Σ,h(μ) is the entropy ofμ, and
.
The author was partially supported by a HK RGC grant in Hong Kong and the Special Funds for Major State Basic Research Projects
in China. 相似文献
3.
Paul Terwilliger 《Designs, Codes and Cryptography》2005,34(2-3):307-332
Let ${\mathbb K} $ denote a field. Let it d denote a nonnegative integer and consider a sequence p=( $\theta_i, \theta^*_i,i=0...d; \varphi_j, \phi_j,j=1...{\it d})$ consisting of scalars taken from ${\mathbb K} $ . We call p a parameter array whenever: (PA1) $\theta_i \not=\theta_j, \; \theta^*_i\not=\theta^*_j$ if $$i\not=j$, $(0 \leq i, j\leq d)$; (PA2) $ \varphi_i\not=0$, $\phi_i\not=0$ $(1 \leq i \leq d)$; (PA3) $\varphi_i = \phi_1 \sum_{h=0}^{i-1} ({\theta_h-\theta_{d-h}})/({\theta_0-\theta_d}) + (\theta^*_i-\theta^*_0)(\theta_{i-1}-\theta_d)$ $(1 \leq i \leq d)$; (PA4) $\phi_i = \varphi_1 \sum_{h=0}^{i-1} ({\theta_h-\theta_{d-h}})/({\theta_0-\theta_d}) + (\theta^*_i-\theta^*_0)(\theta_{d-i+1}-\theta_0)$ $(1 \leq i \leq d)$; (PA5) $(\theta_{i-2}-\theta_{i+1})(\theta_{i-1}-\theta_i)^{-1}$, $(\theta^*_{i-2}-\theta^*_{i+1})(\theta^*_{i-1}-\theta^*_i)^{-1}$$ are equal and independent of i for $2 \leq i \leq d-1$ . In Terwilliger, J. Terwilliger, Linear Algebra Appl., Vol. 330(2001) p. 155 we showed the parameter arrays are in bijection with the isomorphism classes of Leonard systems. Using this bijection we obtain the following two characterizations of parameter arrays. Assume p satisfies PA1 and PA2. Let A, B,A^*, B^* denote the matrices in ${Mat}_{{\it d}+1}$ ( ${\mathbb K} $ ) which have entries A ii =θ i , B ii =θ d-i , A * ii =θ* i , B * ii =θ* i (0 ≤ i ≤ d), A i,i-1=1, B i,i-1=1, A * i-1,i =φ i , B * i-1,i =? i (1 ≤ i ≤ d), and all other entries 0. We show the following are equivalent: (i) p satisfies PA3–PA5; (ii) there exists an invertible G ∈Mat d+1( ${\mathbb K} $ ) such that G ?1 AG=B and G ?1 A * G=B *; (iii) for 0 ≤ i ≤ d the polynomial $$ \sum_{n=0}^i \frac{ (\lambda-\theta_0) (\lambda-\theta_1) \cdots (\lambda-\theta_{n-1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\varphi_1\varphi_2\cdots \varphi_n}$$ is a scalar multiple of the polynomial $$\sum_{n=0}^i \frac{ (\lambda-\theta_d) (\lambda-\theta_{d-1}) \cdots (\lambda-\theta_{d-n+1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\phi_1\phi_2\cdots \phi_n}.$$ We display all the parameter arrays in parametric form. For each array we compute the above polynomials. The resulting polynomials form a class consisting of the q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, Bannai/Ito, and Orphan polynomials. The Bannai/Ito polynomials can be obtained from the q-Racah polynomials by letting q tend to ?1. The Orphan polynomials have maximal degree 3 and exist for ( ${\mathbb K} $ )=2 only. For each of the polynomials listed above we give the orthogonality, 3-term recurrence, and difference equation in terms of the parameter array. 相似文献
4.
The 2-norm distance from a matrix A to the set ${\mathcal{M}}$ of n × n matrices with a zero eigenvalue of multiplicity ≥3 is estimated. If $$Q(\gamma _1 ,\gamma _2 ,\gamma _3 ) = \left( {\begin{array}{*{20}c} A & {\gamma _1 I_n } & {\gamma _3 I_n } \\ 0 & A & {\gamma _2 I_n } \\ 0 & 0 & A \\ \end{array} } \right), n \geqslant 3,$$ then $$\rho _2 (A,{\mathcal{M}}) \geqslant {\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in {\mathbb{C}}}} \sigma _{3n - 2} (Q(\gamma _1 ,\gamma _2 ,\gamma _3 )),$$ where σi(·)is the ith singular value of the corresponding matrix in the decreasing order of singular values. Moreover, if the maximum on the right-hand side is attained at the point $\gamma ^ * = (\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )$ , where $\gamma _1^ * \gamma _2^ * \ne 0$ , then, in fact, one has the exact equality $$\rho _2 (A,{\mathcal{M}}) = \sigma _{3n - 2} (Q(\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )).$$ This result can be regarded as an extension of Malyshev's formula, which gives the 2-norm distance from A to the set of matrices with a multiple zero eigenvalue. 相似文献
5.
For the Dirichlet series F(s) = ?n = 1¥ anexp{ sln } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ
a
=0, we establish conditions for (λ
n
) and (a
n
) under which lnM( s, F ) = TR( 1 + o(1) )exp{ rR