The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Weak Compactness and Variational Characterization of the Convexity

We prove that if X is a Banach space and ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a proper function such that f ? ? attains its minimum for every ? ε X ^*, then the sublevels of f are all relatively weakly compact in X. As a consequence we show that a Banach space X where there exists a function ${f : X \rightarrow \mathbb{R}}$ such that f ? ? attains its minimum for every ? ε X ^* is reflexive. We also prove that if ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a weakly lower semicontinuous function on the Banach space X and if for every continuous linear functional ? on X the set where the function f ? ? attains its minimum is convex and non-empty then f is convex.     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

Non-negative Spectral Measures and Representations of C*-Algebras

Regular normalized W-valued spectral measures on a compact Hausdorff space X are in one-to-one correspondence with unital *-representations ${\rho : C(X, \mathbb{C}) \to W}$ , where W stands for a von Neumann algebra. In this paper we show that for every compact Hausdorff space X and every von Neumann algebras W ₁, W ₂ there is a one-to-one correspondence between unital *-representations ${\rho : C(X, W_1) \to W_2}$ and special B(W ₁, W ₂)-valued measures on X that we call non-negative spectral measures. Such measures are special cases of non-negative measures that we introduced in our previous paper (Cimpri? and Zalar, J Math Anal Appl 401:307–316, 2013) in connection with moment problems for operator polynomials.     

Cubic symmetric polynomials yielding variations of the Erdős–Ginzburg–Ziv theorem

Let p be a prime and let $\varphi\in\mathbb{Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ be a symmetric polynomial, where  $\mathbb {Z}_{p}$ is the field of p elements. A sequence T in  $\mathbb {Z}_{p}$ of length p is called a φ-zero sequence if φ(T)=0; a sequence in $\mathbb {Z}_{p}$ is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Motivated by the EGZ theorem for the prime p, we consider symmetric polynomials $\varphi\in \mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ , which satisfy the following two conditions: (i) every sequence in  $\mathbb {Z}_{p}$ of length 2p?1 contains a φ-zero subsequence, and (ii) the φ-zero free sequences in  $\mathbb {Z}_{p}$ of maximal length are all those containing exactly two distinct elements, where each element appears p?1 times. In this paper, we determine all symmetric polynomials in $\mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ of degree not exceeding 3 satisfying the conditions above.     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let ${\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)}$ with the non-negative potential V belonging to reverse H?lder class with respect to the measure ??(x)dx, where ??(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying ${\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. }$ We obtain some estimates for ${V^{\alpha}\mathcal{L}^{-\alpha}}$ on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator ${[b, V^{\alpha} \mathcal{L}^{-\alpha}]}$ when ${b\in BMO_\omega}$ and 0?<??? ?? 1.     

Numerically Hypercyclic Operators

An operator T acting on a normed space E is numerically hypercyclic if, for some \({(x, x^*)\in \Pi(E)}\), the numerical orbit \({\{x^*(T^n(x)):n\geq 0\}}\) is dense in \({\mathbb{C}}\). We prove that finite dimensional Banach spaces with dimension at least two support numerically hypercyclic operators. We also characterize the numerically hypercyclic weighted shifts on classical sequence spaces.     

Tameness of holomorphic closure dimension in a semialgebraic set

Given a semianalytic set S in ${\mathbb{C}^n}$ and a point ${p \in \bar{S}}$ , there is a unique smallest complex-analytic germ X _p which contains S _p , called the holomorphic closure of S _p . We show that if S is semialgebraic then X _p is a Nash germ, for every p, and S admits a semialgebraic filtration by the holomorphic closure dimension. As a consequence, every semialgebraic subset of a complex vector space admits a semialgebraic stratification into CR manifolds.     

Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems

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.     

Valuation Extensions of Algebras Defined by Monic Gr?bner Bases

Let K be a field, $\mathcal {O}_v$ a valuation ring of K associated to a valuation v: K → Γ?∪?{?∞?}, and m _v the unique maximal ideal of $\mathcal {O}_v$ . Consider an ideal $\mathcal {I}$ of the free K-algebra $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ on X ₁,...,X _n . If ${\cal I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$ which is a monic Gr?bner basis of ${\cal I}$ in $K\langle X\rangle$ , where $\mathcal {O}_v\langle X\rangle =\mathcal{O}_v\langle X_1,...,X_n\rangle$ is the free $\mathcal{O}_v$ -algebra on X ₁,...,X _n , then the valuation v induces naturally an exhaustive and separated Γ-filtration F ^v A for the K-algebra $A=K\langle X\rangle /\mathcal {I}$ , and moreover $\mathcal{I}\cap\mathcal{O}_v\langle X\rangle =\langle\mathcal{G}\rangle$ holds in $\mathcal{O}_v\langle X\rangle$ ; it follows that, if furthermore $\mathcal{G}\not\subset {\bf m}_v{O}_v\langle X\rangle$ and $k\langle X\rangle /\langle\overline{\mathcal G}\rangle$ is a domain, where $k=\mathcal{O}_v/{\bf m}_v$ is the residue field of $\mathcal{O}_v$ , $k\langle X\rangle =k\langle X_1,...,X_n\rangle$ is the free k-algebra on X ₁,...,X _n , and $\overline{\mathcal G}$ is the image of $\mathcal{G}$ under the canonical epimorphism $\mathcal{O}_v\langle X\rangle\rightarrow k\langle X\rangle$ , then F ^v A determines a valuation function A → Γ?∪?{?∞?}, and thereby v extends naturally to a valuation function on the (skew-)field Δ of fractions of A provided Δ exists.     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

A certain generalization of the law of the iterated logarithm

Let X₁ ,..., X_n be independent random variables and let \(S_n = \sum\limits_{i = 1}^n {X_i }\) . For the sequence of random variables $$T_n = \sum\limits_1^p {(S_{t_j } - S_{t_{j - 1} } )} ,$$ where t₀=01<...p=n, p?1, under certain conditions on t_i, \(i = \overline {1,n}\) , one proves a series of general theorems of the type of the iterated logarithm laws.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Let ψ and φ be analytic functions on the open unit disk $\mathbb{D}$ with φ( $\mathbb{D}$ ) ? $\mathbb{D}$ . We give new characterizations of the bounded and compact weighted composition operators W _ψ,? from the Hardy spaces H ^p , 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A _α ^p , α > ? 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal{D}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W _ψ,? f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H ¹ as well as of compactness for H ^p , 1 ≤ p < ∞, and $\mathcal{D}$ purely in terms of the symbols ψ and φ.     

The Degree of Weakly Discretely Generated Spaces

A space X is discretely generated at a point \({x \in X}\) if for any \({A \subseteqq X}\) with \({x \in \textsf{cl}(A)}\) , there exists a discrete set \({D \subseteqq A}\) such that \({x \in \textsf{cl}(D)}\) . The space X is discretely generated if it is discretely generated at every point \({x \in X}\) . We say that X is weakly discretely generated if for any non-closed set \({A \subseteqq X}\) , there exists a discrete set \({D \subseteqq A}\) such that \({\textsf{cl}(D) \setminus A \neq \emptyset}\) . New results about these properties in the classes of pseudocompact and ?ech-complete spaces are obtained and a theorem of Ivanov and Osipov concerning the ordinal function idc is generalized to the class of ?ech-complete spaces.     

Multiplication Operators Defined by a Class of Polynomials on {L_a^2(\mathbb{D}^2)}

In this paper, we consider those multiplication operators M _p on \({L_a^2(\mathbb{D}^2)}\) defined by a class of polynomials p. Also, this paper consider the reducing subspaces of M _p , the von Neumann algebra \({ \mathcal{W}^*(p)}\) generated by M _p , and its commutant \({\mathcal{V}^*(p) = \mathcal{W}^*(p)'.}\) The structure of \({\mathcal{V}^*(p)}\) is completely determined, along with those reducing subspaces of M _p .     

On the weights of simple paths in weighted complete graphs

Consider a weighted simple graph G on the vertex set {1,..., n}. For any path p in G, we call ω _G (p) the sum of the weights of the edges of the path, and, for any {i, j} ? {1,..., n}, we define the multiset $$D_{\{ i,j\} } (G) = \left\{ {\left. {w_G \left( p \right)} \right|p a simple path between i and j} \right\}.$$ We establish a criterion for a multisubset of ? to be of the form $\mathcal{D}_{\{ i,j\} } (G)$ for some weighted complete graph G and for some i, j vertices of G. Besides we establish a criterion for a family of multi-subsets of ? to be of the form for some weighted complete graph G with vertex set {1,..., n}.     

The Maximal Variation of Martingales of Probabilities and Repeated Games with Incomplete Information

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.