Convergence of random walks on double transitive group generated by its permutational character

Let $P$ be a probability on a finite group $G$, $U(g)=\frac{1}{|G|}$ the uniform (trivial) probability on the group $G$, $P^{(n)}=P *\ldots*P$ an $n$-fold convolution of $P$. A lot of estimates of the rate of the convergence $P^{(n)}\rightarrow U$ are found in different norms. It is well known conditions under which $P^{(n)}\rightarrow U$ if $n\rightarrow\infty$. Many papers are devoted to estimating the rate of this convergence for different norms. We consider finite groups that have a double transitive representation by substitutions and the probability that naturally arises in this image. This probability on each element of the group is proportional to the number of fixed (or stationary) points of this element, which is considered as a substitution. In other words, this probability is a character of the substitution representation of the group. A probability is called class if it takes the same values on each class of conjugate elements of a group, that is, it is a function of the class. The considered probability is class because any character of a group takes on the same values on conjugate elements. Any probability (and, in general, functions with values in an arbitrary ring) on a group can be associated with an element of the group algebra of this group over this ring. The class probability corresponds to an element of the center of this group algebra; that is why the class probability is also called central. On an abelian group, any probability is class (central). In the paper convergence with respect to the norm $\|F\|=\sum\limits_{g\in G} |F(g)|$, where $F(g)$ is a function on group $G$, is considered. For the norm an exact formula not estimate only, as usual for rate of convergence of convolution $P^{(n)}\rightarrow U$ is given. It turns out that the norm of the difference $\|P^{(n)}-U\|$ is determined by the order of the group, degree the group as a substitution group, and the number of regular substitutions in the group. A substitution is called regular if it has no fixed points. Special cases are considered the symmetric group, the alternating group, the Zassenhaus group, and the Frobenius group of order $p(p-1)$ with the Frobenius core of order $p$ ($p$ is a prime number). A Zassenhaus group is a double transitive substitution group of a finite set in which only a trivial substitution leaves more than two elements of this set fixed.

Substituent-tuned structure and luminescence sensitizing towards Al3+ based on phenoxy bridged dinuclear EuIII complexes

We present herein a novel luminescent enhanced EuIII complex-supported chemosensor by fine tuning the substitution group on ligands.

The Safety of Substitution of Antiretroviral Regimen in Non-Clinical Trial Settings in Asian Countries

Background Although substitutions of antiretroviral regimen are generally safe, most data on substitutions are based on results from clinical trials. The objective of this study was to evaluate the safety of substituting antiretroviral regimen in virologically suppressed HIV-infected patients in non-clinical trial settings in Asian countries. Methods HIV-infected patients enrolled in the TREAT Asia HIV Observational Database (TAHOD) were included in this analysis if they started combination antiretroviral therapy (cART) after 2002, were being treated at a center that documented a median rate of viral load (VL) monitoring ≥ 1 tests/patient/year, and experienced a minor or major treatment substitution while on virally suppressive cART (VL < 200 copies/mL). Minor regimen substitutions were defined as within-class changes and major regimen substitutions were defined as changes to a drug class. Virologic failure was defined as having had two viral load measurements > 400 copies/mL. The patterns of substitutions and rate of virologic failure after substitutions were analyzed. Results Of 3,994 adults who started ART after 2002, 3,119 (78.1%) had at least one period of virological suppression. Among these, 1,170 (37.5%) underwent a minor regimen substitution, and 296 (9.5%) underwent a major regimen substitution during suppression. The rates of virological failure were 1.48/100person years (95% CI 1.14–1.91) in the minor substitution group and 2.85/100person years (95% CI 1.88–4.33) in the major substitution group, and 2.53/100person years (95% CI 2.20–2.92) among patients that did not undergo a treatment substitution. Conclusion The rate of virological failure was relatively low in both major and minor substitution groups, showing that regimen substitution is generally safe in non-clinical trial settings in Asian countries. Disclosures All authors: No reported disclosures.