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1.
The variation of a martingale $p_{0}^{k}=p_{0},\ldots,p_{k}$ of probabilities on a finite (or countable) set X is denoted $V(p_{0}^{k})$ and defined by $$ V\bigl(p_0^k\bigr)=E\Biggl(\sum_{t=1}^k\|p_t-p_{t-1}\|_1\Biggr). $$ It is shown that $V(p_{0}^{k})\leq\sqrt{2kH(p_{0})}$ , where H(p) is the entropy function H(p)=?∑ x p(x)logp(x), and log stands for the natural logarithm. Therefore, if d is the number of elements of X, then $V(p_{0}^{k})\leq\sqrt{2k\log d}$ . It is shown that the order of magnitude of the bound $\sqrt{2k\log d}$ is tight for d≤2 k : there is C>0 such that for all k and d≤2 k , there is a martingale $p_{0}^{k}=p_{0},\ldots,p_{k}$ of probabilities on a set X with d elements, and with variation $V(p_{0}^{k})\geq C\sqrt{2k\log d}$ . An application of the first result to game theory is that the difference between v k and lim j v j , where v k is the value of the k-stage repeated game with incomplete information on one side with d states, is bounded by $\|G\|\sqrt{2k^{-1}\log d}$ (where ∥G∥ is the maximal absolute value of a stage payoff). Furthermore, it is shown that the order of magnitude of this game theory bound is tight.  相似文献   

2.
For a symmetric function F, the eigen-operator Δ F acts on the modified Macdonald basis of the ring of symmetric functions by $\Delta_{F} \tilde{H}_{\mu}= F[B_{\mu}] \tilde{H}_{\mu}$ . In a recent paper (Int. Math. Res. Not. 11:525–560, 2004), J. Haglund showed that the expression $\langle\Delta_{h_{J}} E_{n,k}, e_{n}\rangle$ q,t-enumerates the parking functions whose diagonal word is in the shuffle 12?J∪∪J+1?J+n with k of the cars J+1,…,J+n in the main diagonal including car J+n in the cell (1,1) by t area q dinv. In view of some recent conjectures of Haglund–Morse–Zabrocki (Can. J. Math., doi:10.4153/CJM-2011-078-4, 2011), it is natural to conjecture that replacing E n,k by the modified Hall–Littlewood functions $\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots\mathbf{C}_{p_{k}} 1$ would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting the parking function touches the diagonal according to the composition p=(p 1,p 2,…,p k ). We prove this conjecture by deriving a recursion for the polynomial $\langle\Delta_{h_{J}} \mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}} 1 , e_{n}\rangle $ , using this recursion to construct a new $\operatorname{dinv}$ statistic (which we denote $\operatorname{ndinv}$ ), then showing that this polynomial enumerates the latter parking functions by $t^{\operatorname{area}} q^{\operatorname{ndinv}}$ .  相似文献   

3.
Assume that L p,q , $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \{ (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \} $ . We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ , provided 1/p ≠ 1/p 1 + … + 1/p n . In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L p spaces.  相似文献   

4.
It is a result by Lacey and Thiele (Ann. of Math. (2) 146(3):693–724, 1997; ibid. 149(2):475–496, 1999) that the bilinear Hilbert transform maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2}}(\mathbb{R}) $ into $L^{p_{3}}(\mathbb{R})$ whenever (p 1,p 2,p 3) is a Hölder tuple with p 1,p 2>1 and $p_{3}>\frac{2}{3}$ . We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when $p_{3}=\frac{2}{3}$ . We show that the quartile operator maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2}}(\mathbb{R}) $ into $L^{\frac{2}{3},\infty}(\mathbb{R})$ when p 1,p 2>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2},\frac{2}{3}}(\mathbb{R}) $ into $L^{\frac{2}{3},\infty}(\mathbb{R})$ . We also provide weak type estimates and boundedness on Orlicz-Lorentz spaces near p 1=1,p 2=2 which improve, in the Walsh case, the results of Bilyk and Grafakos (J. Geom. Anal. 16 (4):563–584, 2006) and Carro et al. (J. Math. Anal. Appl. 357(2):479–497, 2009). Our main tool is the multi-frequency Calderón-Zygmund decomposition from (Nazarov et al. in Math. Res. Lett. 17(3):529–545, 2010).  相似文献   

5.
We generalize the well-known Lax-Milgram theorem on the Hilbert space to that on the Banach space. Suppose that ${a(\cdot, \cdot)}$ is a continuous bilinear form on the product ${X\times Y}$ of Banach spaces X and Y, where Y is reflexive. If null spaces N X and N Y associated with ${a(\cdot, \cdot)}$ have complements in X and in Y, respectively, and if ${a(\cdot, \cdot)}$ satisfies certain variational inequalities both in X and in Y, then for every ${F \in N_Y^{\perp}}$ , i.e., ${F \in Y^{\ast}}$ with ${F(\phi) = 0}$ for all ${\phi \in N_Y}$ , there exists at least one ${u \in X}$ such that ${a(u, \varphi) = F(\varphi)}$ holds for all ${\varphi \in Y}$ with ${\|u\|_X \le C\|F\|_{Y^{\ast}}}$ . We apply our result to several existence theorems of L r -solutions to the elliptic system of boundary value problems appearing in the fluid mechanics.  相似文献   

6.
Given X,Y two ${\mathbb{Q}}$ -vector spaces, and f : XY, we study under which conditions on the sets ${B_{k} \subseteq X, k=1,\ldots,s}$ , if ${\Delta_{h_1h_2 \cdots h_s}f(x) = 0}$ for all ${x \in X}$ and ${h_k \in B_k, k = 1,2,\ldots,s}$ , then ${\Delta_{h_1h_2\cdots h_{s}}f(x) = 0}$ for all ${(x,h_{1},\ldots,h_{s}) \in X^{s+1}}$ .  相似文献   

7.
8.
This paper shows that if μ 1 , . . . , μ 5 are nonzero real numbers, not all negative, at least one of μ j $ \left( {1\leqslant j\leqslant 5} \right) $ is irrational, and k is a positive integer, then there exist infinitely many primes p 1 , . . . , p 5 , p such that $ \left[ {{\mu_1}p_1^3+\cdots +{\mu_5}p_5^3} \right]=kp $ . In particular, $ \left[ {{\mu_1}p_1^3+\cdots +{\mu_5}p_5^3} \right] $ represents infinitely many primes.  相似文献   

9.
This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups $ \mathfrak{X} $ Wr $ \mathfrak{Y} $ = { X Wr Y | X ∈ $ \mathfrak{X} $ , Y $ \mathfrak{Y} $ } and $ \mathfrak{X} $ wr $ \mathfrak{Y} $ = {X wr Y | X $ \mathfrak{X} $ , Y $ \mathfrak{Y} $ } generate the product variety $ \mathfrak{X} $ var ( $ \mathfrak{Y} $ ).  相似文献   

10.
We consider processes of the form [s,T]?t?u(t,X t ), where (X,P s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if $u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})$ with $\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})$ then there is a quasi-continuous version $\tilde{u}$ of u such that $\tilde{u}(t,X_{t})$ is a P s,x -Dirichlet process for quasi-every (s,x)∈[0,T)×? d with respect to parabolic capacity, and we describe the martingale and the zero-quadratic variation parts of its decomposition. We also give conditions on u ensuring that $\tilde{u}(t,X_{t})$ is a semimartingale.  相似文献   

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