Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.     

A remark on compact matrix quantum groups

A simpler set of axioms of the theory of compact matrix quantum groups (pseudogroups) is found.Supported by Japan Society for Promoting Science. On leave from the Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoa 74, 00-682 Warsaw, Poland.     

Cocycles d'Euler et de Maslov

Conclusion Nous espérons avoir convaincu le lecteur qu'il peut être utile de considérer la classe de Maslov comme une classe bornée. Dans [Gh], nous avons montré que la classe d'Euler bornée pour un groupe d'homéomorphismes directs du cercle rend compte de la dynamique topologique de ce groupe. Existe-t-il un résultat analogue pour Sp(2n,)? En d'autres termes, soit un groupe discret et ₁, ₂ deux représentations de dans Sp(2n,). On suppose que les cocycles ₁ ^* et ₂ ^* définissent la même classe bornée. Que peut-on en conclure sur ₁ et ₂?Par ailleurs, l'article [At l] traite aussi d'invariants sur SL(2,) différents de ceux que nous avons considérés, comme par exemple les fonctionsL de Shimizu. Est-il possible de les faire rentrer naturellement dans notre cadre?

---

     

The exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket

The authors rigorously prove that the exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket is

Designs as maximum codes in polynomial metric spaces

Finite and infinite metric spaces % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] that are polynomial with respect to a monotone substitution of variable t(d) are considered. A finite subset (code) W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] is characterized by the minimal distance d(W) between its distinct elements, by the number l(W) of distances between its distinct elements and by the maximal strength (W) of the design generated by the code W. A code W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] is called a maximum one if it has the greatest cardinality among subsets of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with minimal distance at least d(W), and diametrical if the diameter of W is equal to the diameter of the whole space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\]. Delsarte codes are codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with (W)2l(W)–1 or (W)=2l(W)–2>0 and W is a diametrical code. It is shown that all parameters of Delsarte codes W) % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] are uniquely determined by their cardinality |W| or minimal distance d(W) and that the minimal polynomials of the Delsarte codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] are expansible with positive coefficients in an orthogonal system of polynomials {Q _i(t)} corresponding to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\]. The main results of the present paper consist in a proof of maximality of all Delsarte codes provided that the system {Q _i)} satisfies some condition and of a new proof confirming in this case the validity of all the results on the upper bounds for the maximum cardinality of codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \]% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with a given minimal distance, announced by the author in 1978. Moreover, it appeared that this condition is satisfied for all infinite polynomial metric spaces as well as for distance-regular graphs, decomposable in a sense defined below. It is also proved that with one exception all classical distance-regular graphs are decomposable. In addition for decomposable distance-regular graphs an improvement of the absolute Delsarte bound for diametrical codes is obtained. For the Hamming and Johnson spaces, Euclidean sphere, real and complex projective spaces, tables containing parameters of known Delsarte codes are presented. Moreover, for each of the above-mentioned infinite spaces infinite sequences (of maximum) Delsarte codes not belonging to tight designs are indicated.     

Some new domains with complete Kähler metrics of negative curvature

We add to the known examples of complete Kähler manifolds with negative sectional curvature by showing that the following three classes of domains in euclidean spaces also belong: perturbations of ellipsoidal domains in ?ⁿ, intersections of complex-ellipsoidal domains in ?², and intersections of fractional linear transforms of the unit ball in ?². In the process, we prove the following theorem in differential geometry: in the intersection of two complex-ellipsoidal domains in ?², the sum of the Bergman metrics is a Kähler metric with negative curvature operator.     

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

On the homogeneity of additive mappings

LetK be a ring with an identity 1 0 andM, L two unitaryK-modules. Then, for any additive mappingf:M L, the setH _f :={ K f(x)=f(x) for allx M} forms a subring ofK, the homogeneity ring off. It is shown that, forM {0},L {0} and any subringS ofK for whichM is a freeS-module, there exists an additive mappingf:ML such thatH _f =S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce onM a multiplication ·:M × M M makingM into a ring but not into aK-algebra.     

Water confined in two-dimensions: Fundamentals and applications

The behavior of water in close proximity to other materials under ambient conditions is of great significance due to its importance in a broad range of daily applications and scientific research. The structure and dynamics of water at an interface or in a nanopore are often significantly different from those of its bulk counterpart. Until recently, experimental access to these interfacial water structures was difficult to realize. The advent of two-dimensional materials, especially graphene, and the availability of various scanning probe microscopies were instrumental to visualize, characterize and provide fundamental knowledge of confined water. This review article summarizes the recent experimental and theoretical progress in a better understanding of water confined between layered Van der Waals materials. These results reveal that the structure and stability of the hydrogen bonded networks are determined by the elegant balance between water-surface and water-water interactions. The water-surface interactions often lead to structures that differ significantly from the conventional bilayer model of natural ice. Here, we review the current knowledge of water adsorption in different environments and intercalation within various confinements. In addition, we extend this review to cover the influence of interfacial water on the two-dimensional material cover and summarize the use of these systems in potential novel applications. Finally, we discuss emerged issues and identify some flaws in the present understanding.     

Central limit theorems for realized volatility under hitting times of an irregular grid

We consider a continuous semi-martingale sampled at hitting times of an irregular grid. The goal of this work is to analyze the asymptotic behavior of the realized volatility under this rather natural observation scheme. This framework strongly differs from the well understood situations when the sampling times are deterministic or when the grid is regular. Indeed, neither Gaussian approximations nor symmetry properties can be used. In this setting, as the distance between two consecutive barriers tends to zero, we establish central limit theorems for the normalized error of the realized volatility. In particular, we show that there is no bias in the limiting process.