Hausdorff dimension of some groups acting on the binary tree

Based on the work of Abercrombie [A. G. Abercrombie. Subgroups and subrings of profinite rings. Math. Proc. Cambridge Philos. Soc. 116 (1994), 209-222.], Barnea and Shalev [Y. Barnea and A. Shalev. Hausdorff dimension, pro-p groups, and Kac-Moody algebras. Trans. Amer. Math. Soc. 349 (1997), 5073-5091.] gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree . Abért and Virág [M. Abért and B. Virág. Dimension and randomness in groups acting on rooted trees. J. Amer. Math. Soc. 18 (2005), 157-192.] showed that there exist finitely generated (but not necessarily level-transitive) subgroups of Aut of arbitrary dimension in [0, 1]. In this article we explicitly compute the Hausdorff dimension of the level-transitive spinal groups. We then give examples of 3-generated spinal groups which have transcendental Hausdorff dimension, and construct 2-generated groups whose Hausdorff dimension is

Wreath products of cocommutative Hopf algebras

We define wreath products of cocommutative Hopf algebras, and show that they enjoy a universal property of classifying cleft extensions, analogous to the Kaloujnine-Krasner theorem for groups. We show that the group ring of a wreath product of groups is the wreath product of their group rings, and that (with a natural definition of wreath products of Lie algebras) the universal enveloping algebra of a wreath product of Lie algebras is the wreath product of their enveloping algebras. We recover the aforementioned result that group extensions may be classified as certain subgroups of a wreath product, and that Lie algebra extensions may also be classified as certain subalgebras of a wreath product

Hausdorff dimension of some groups acting on the binary tree

Based on the work of Abercrombie, Barnea and Shalev gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree T. Abert and Virag showed that there exist finitely generated (but not necessarily level-transitive) subgroups of AutT of arbitrary dimension in [0,1]. In this article we explicitly compute the Hausdorff dimension of the level-transitive spinal groups. We then show examples of 3-generated spinal groups which have transcendental Hausdroff dimension, and exhibit a construction of 2-generated groups whose Hausdorff dimension is 1.Comment: 10 pages; full revision; simplified some proof

Augmenting Large Language Models with Audio Generation Capabilities

Chatbots or conversational agent interfaces utilize large language models (LLMs) to provide text responses to user queries. However, such chatbots are not capable of receiving audio input and providing generated audio as a response. This disclosure describes techniques to augment a LLM with an interface to an audio generation model. The LLM is fine-tuned to train it to leverage an API to access the audio generation model when input queries request query response in audio form. The trained LLM performs reasoning tasks and generates prompts for the audio generation model. The user-provided audio input and the LLM-generated prompts are fed to the audio generation model which generates audio. The output audio is analyzed to determine attributes as textual description. The LLM can perform multiple rounds of reasoning, prompt generation, and calling the audio generation model based on previously generated audio and associated textual descriptions. The ultimate audio output as generated by the audio generation model is provided as a response to the user query

Enantiospecific pharmacokinetics of intravenous dexmedetomidine in beagles.

The goal of this study was to investigate the pharmacokinetic (PK) behaviour of dexmedetomidine in dogs administered as a pure enantiomer versus as part of a racemic mixture. Eight unmedicated intact purpose-bread beagles were included. Two intravenous treatments of either medetomidine or dexmedetomidine were administered at 10- to 14-day intervals. Atipamezole or saline solution was administered intramuscularly 45 min later. Venous blood samples were collected into EDTA collection tubes, and the quantification of dexmedetomidine and levomedetomidine was performed by chiral LC-MS/MS. All dogs appeared sedated after each treatment without complication. Plasma concentrations of levomedetomidine were measured only in the racemic group and were 51.4% (51.4%-56.1%) lower than dexmedetomidine. Non-compartmental analysis (NCA) was performed for both drugs, while dexmedetomidine data were further described using a population pharmacokinetic approach. A standard two-compartment mammillary model with linear elimination with combined additive and multiplicative error model for residual unexplained variability was established for dexmedetomidine. An exponential model was finally retained to describe inter-individual variability on parameters of clearance (Cl1 ) and central and peripheral volumes of distribution (V1 , V2 ). No effect of occurrence, levomedetomidine or atipamezole could be observed on dexmedetomidine PK parameters. Dexmedetomidine did not undergo significantly different PK when administered alone or as part of the racemic mixture in otherwise unmedicated dogs

Diskrete und pro-endliche Gruppen, die auf regulären Bäumen mit einem Fixpunkt operieren

Wir entwickeln eine neue Methode um Gruppen zu untersuchen, die auf regulären Bäumen mit einem Fixpunkt operieren. Hierzu betrachten wir die Menge aller stetigen Funktionen von der Automorphismengruppe $\operatorname{Aut}X^*$ eines regulären Baumes in einen endlichen Körper, und ordnen jeder Untergruppe $G\leq\operatorname{Aut}X^*$ das Ideal der Funktionen zu, die auf $G$ verschwinden. Unser Ansatz liefert viele neue Ergebnisse sowohl für diskrete Gruppen, als auch für abgeschlossene Untergruppen von $\operatorname{Aut}X^*$ und einige derer Untergruppen. Wir zeigen die erfolgreiche Anwendung dieses Ansatzes in vielen Bereichen, z.B. in der Berechnung von Zentralreihen, Automorphismentürmen und Hausdorff Dimensionen, ihre Anwendung bei Kongruenzproblemen sowie für eine funktoriale Konstruktion von Lie Algebren durch Gruppenschemata

Effect of Medetomidine, Dexmedetomidine, and Their Reversal with Atipamezole on the Nociceptive Withdrawal Reflex in Beagles

The objectives were: (1) to compare the antinociceptive activity of dexmedetomidine and medetomidine, and (2) to investigate its modulation by atipamezole. This prospective, randomized, blinded experimental trial was carried out on eight beagles. During the first session, dogs received either medetomidine (MED) (0.02 mg kg-1 intravenously (IV)] or dexmedetomidine (DEX) [0.01 mg kg-1 IV), followed by either atipamezole (ATI) (0.1 mg kg-1) or an equivalent volume of saline (SAL) administered intramuscularly 45 min later. The opposite treatments were administered in a second session 10-14 days later. The nociceptive withdrawal reflex (NWR) threshold was determined using a continuous tracking approach. Sedation was scored (0 to 21) every 10 min. Both drugs (MED and DEX) increased the NWR thresholds significantly up to 5.0 (3.7-5.9) and 4.4 (3.9-4.8) times the baseline (p = 0.547), at seven (3-11) and six (4-9) minutes (p = 0.938), respectively. Sedation scores were not different between MED and DEX during the first 45 min (15 (12-17), p = 0.67). Atipamezole antagonized sedation within 25 (15-25) minutes (p = 0.008) and antinociception within five (3-6) minutes (p = 0.008). Following atipamezole, additional analgesics may be needed to maintain pain relief