Given a partial action of a topological group G on a space X we determine properties P which can be extended from X to its globalization. We treat the cases when P is any of the following: Hausdorff, regular, metrizable, second countable, and having invariant metric. Further, for a normal subgroup H, we introduce and study a partial action of G/H on the orbit space of X; applications to invariant metrics and inverse limits are presented.

We show that for a partial action η with closed domain of a compact group G on a profinite space X the space of orbits X/~G is profinite, this leads to the fact that when G is profinite the enveloping space XG is also profinite. Moreover, we provide conditions for the induced quotient map πG : X → X / ∼G of η to have a continuous section. Relations between continuous sections of πG and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

The classical concept of a twisted product is examined within the framework of partial actions. Specifically, we demonstrate that the globalization of a partial action results in a twisted product. Additionally, we outline conditions for the metrizability of twisted products and prove certain homotopy and categorical properties. Moreover, sufficient conditions are established for the enveloping space to be an equivariant absolute neighborhood extensor.

Let Λ be an Artin algebra and Kb(proj(Λ)) be the triangulated category of bounded co-chain complexes in proj(Λ). It is well known [1] that two-terms silting complexes in Kb(proj(Λ)) are described by the τ-tilting theory. The aim of this paper is to give a characterization of certain n-term silting complexes in Kb(proj(Λ)) which are induced by Λ-modules. In order to do that, we introduce the notions of τn-rigid, τn-tilting and τn,m-tilting Λ-modules. The latter is both a generalization of τ-tilting and tilting in mod(Λ). It is also stated and proved some variant, for τn-tilting modules, of the well known Bazzoni's characterization for tilting modules [12, Theorem 3.11]. We give some connections between n-terms presilting complexes in Kb(proj(Λ)) and τn-rigid Λ-modules. Moreover, a characterization is given to know when a τn-tilting Λ-module is n-tilting. We also study more deeply the properties of the τn,m-tilting Λ-modules and their connections of being m-tilting in some quotient algebras. We apply the developed τn,m-tilting theory to the finitistic dimension and thus for n=m=1, we get as a particular case [22, Theorem 2.5]. Finally, at the end of the paper we discuss and state some open questions (conjectures) that we consider crucial for the future develop of the τn,m-tilting theory.

We provide a different proof of the equivariant version of the Borsuk-Whitehead-Hanner Theorem in the category of proper G-spaces which are metrizable by a G-invariant metric.

The active blueberry compounds called anthocyanins have poor oxidation stability, but, if encapsulated by protein nanoparticles, they can be protected due to the slowing down of the oxidation process. This work describes the advantages of using a γ-irradiated bovine serum albumin nanoparticle bound to anthocyanins. The interaction was characterized biophysically, mainly by rheology. By computational calculation and simulation based on model nanoparticles, we estimated the number of molecules forming to the albumin nanoparticles, which allowed us to infer a ratio of anthocyanin/nanoparticles. Measurements by UV-VIS spectroscopy, FT-IR spectroscopy, fluorescence spectroscopy, dynamic light scattering (DLS), ζ potential, electron transmission microscopy and rheology at room (25 °C), and physiological (37 °C) temperatures were performed. The spectroscopy measurements allowed identifying additional hydrophobic sites created during the irradiation process of the nanoparticle. Based on the rheological studies, it was observed that for all the temperatures selected, the BSA-NP trend is a Newtonian flow behavior type, and there is a direct correlation between dynamic viscosity and temperature values. Furthermore, when anthocyanins are added, the system increases its resistance to the flow reflected in the morphological changes observed by TEM, confirming the relationship between viscosity values and aggregate formation

Abstract Let G be a group, let A = ⊕ g ∈ G A g {A=\bigoplus_{g\in G}A_{g}} be an epsilon-strongly graded ring over G, let R := A 1 {R:=A_{1}} be the homogeneous component associated with the identity of G, and let 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [ A g ] {[A_{g}]} is an element of 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} for all g ∈ G {g\in G} . Moreover, the association g ↦ [ A g ] {g\mapsto[A_{g}]} determines a partial representation of G on 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} which induces a partial action γ of G on the center Z ⁢ ( R ) {Z(R)} of R. Sufficient conditions for A to be an Azumaya R γ {R^{\gamma}} -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n ⁡ ( A ) {B=\operatorname{M}_{n}(A)} is the ring of matrices with entries in A, or B = END A ( M ) = ⊕ l ∈ G Mor A ( M , M ) l {B=\operatorname{END}_{A}(M)=\bigoplus_{l\in G}\operatorname{Mor}_{A}(M,M)_{l}} is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = END A ⁡ ( M ) {B=\operatorname{END}_{A}(M)} where M = A ⊗ R N {M=A\otimes_{R}N} is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product.

The rheological and viscoelastic properties of hybrid formulations composed of vehicles designed for cutaneous topical application and loaded with ultradeformable liposomes (UDL) were assessed. UDL were selected for their established ability to transport both lipophilic and hydrophilic compounds through the skin, and are applicable in pharmaceuticals and cosmetics. Formulations underwent flow analysis and were fitted to the Herschel–Bulkley model due to their prevalent non-Newtonian behavior in most cases. Linear viscoelastic regions (LVR) were identified, and G′ and G″ moduli were determined via frequency sweep steps, considering the impact of temperature and aging. The formulations exhibited non-Newtonian behavior with pseudoplastic traits in most cases, with UDL incorporation inducing rheological changes. LVR and frequency sweep tests indicated predominantly elastic solid behavior, with G′ higher than G″, at different temperatures and post-production times. Tan δ values also illustrated a predominant solid-like behavior over liquid. This study provides pivotal insights into the rheological and viscoelastic features of topical formulations, emphasizing the crucial role of meticulous vehicle and formulation selection when incorporating UDL or analogous liposomal drug delivery systems.

. The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let G be a group, A = ⊕ g ∈ G A g an epsilon-strongly graded ring and PicS ( R ) the Picard semigroup of R := A 1 . We prove that the isomorphism class [ A g ] is an element of PicS ( R ), for all g ∈ G . Thus, the association g 7→ [ A g ] determines a partial representation of G on PicS ( R ) which induces a partial action γ of G on the center Z ( R ) of R . Sufficient conditions for A to be an Azumaya R γ -algebra are presented in the case that R is commutative. We study when B is a partial crossed product in the following cases: B = M n ( A ) is the ring of matrices with entries in A , or B = END A ( M ) = L l ∈ G Mor A ( M, M ) l is the direct sum of graded endomorphisms of left graded A -module M with degree l , or B = END A ( M ) where M = A ⊗ R N is the induced module of a left R -module N . Finally, assuming that R is semiperfect, we prove that there exists an epsilon-strongly graded subring of A which is graded equivalent to a partial crossed product.

We extend Dade's theorem to the realm of nearly epsilon-strongly graded rings, and present new characterizations of strongly and epsilon-strongly graded rings. Further, we give conditions for an epsilon-strongly graded ring to be written as a direct sum of strongly graded rings and a ring with trivial gradation. Our results are applied to characterize strongly graded Leavitt path algebras endowed with the canonical gradation. Finally we show that for any n is an element of Z(+) there is a Leavitt path algebra which is a sum of n - 1 strongly graded rings and a ring with trivial gradation.

Abstract We present some generalizations of the well-known correspondence, found by Exel, between partial actions of a group G on a set X and semigroup homomorphism of 𝒮⁡(G){\operatorname{\mathcal{S}}(G)} on the semigroup I⁢(X){I(X)} of partial bijections of X, with 𝒮⁡(G){\operatorname{\mathcal{S}}(G)} being an inverse monoid introduced by Exel. We show that any unital premorphism θ:G→S{\theta:G\to S}, where S is an inverse monoid, can be extended to a semigroup homomorphism θ*:T→S{\theta^{*}:T\to S} for any inverse semigroup T with 𝒮⁡(G)⊆T⊆P*⁢(G)×G{\operatorname{\mathcal{S}}(G)\subseteq T\subseteq P^{*}(G)\times G}, with P*⁢(G){P^{*}(G)} being the semigroup of non-empty subsets of G, and such that E⁢(S){E(S)} satisfies some lattice-theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on Γ⁢(X){\Gamma(X)}, the inverse semigroup of partial homeomorphisms between open subsets of a locally compact Hausdorff space X.

Let X be a compact Hausdorff space. In this work we translate partial actions of X to partial actions on some hyperspaces determined by X, this gives an endofunctor 2- in the category of partial actions on compact Hausdorff spaces which generates a monad in this category. Moreover, structural relations between partial actions θ on X and partial determined by 2θ as well as their corresponding globalizations are established.

The Cu-catalyzed reaction of substituted alpha-diazoesters with fluoride gives alpha-fluoroesters with ee values of up to 95 %, provided that chiral indane-derived bis(oxazoline) ligands are used that carry bulky benzyl substituents at the bridge and moderately bulky isopropyl groups on their core. The apparently homogeneous solution of CsF in C6F6/hexafluoroisopropanol (HFIP) is the best reaction medium, but CsF in the biphasic mixture CH2Cl2/HFIP also provides good results. DFT studies suggest that fluoride initially attacks the Cu- rather than the C-atom of the transient donor/acceptor carbene intermediate. This unusual step is followed by 1,2-fluoride shift; for this migratory insertion to occur, the carbene must rotate about the Cu-C bond to ensure orbital overlap. The directionality of this rotatory movement within the C-2-symmetric binding site determines the sense of induction. This model is in excellent accord with the absolute configuration of the resulting product as determined by X-ray diffraction using single crystals of this a priori wax-like material grown by capillary crystallization.

In this work we study the classes of epsilon and nearly epsilon-strongly graded rings by a group $G$. In particular, we extend Dade's theorem to the realm of nearly epsilon-strongly graded rings. Moreover, we introduce the category SIM$S-$gr of symmetrically graded modules and use it to present a new characterization of strongly graded rings. A functorial approach is used to obtain a characterization of epsilon-strongly graded rings. Finally, we determine conditions for which an epsilon-strongly graded ring can be written as a direct sum of strongly graded rings and a trivially graded ring.

BACKGROUND:Ultradeformable liposomes are promising carriers for cosmeceutical actives as they can be loaded with molecules of different polarities, and they present unique penetration properties. AIMS:While those features have already been tested, we wanted to know whether their special penetration properties could be maintained after incorporation in diverse cosmetic vehicles, including commercial products already in the market. METHODS:Ultradeformable liposomes loaded with a lipophilic and a hydrophilic fluorescent probe were prepared by lipid film resuspension, followed by extrusion and incorporation to different vehicles and commercial products. Penetration was determined in human and pig skin by incubation, with the Saarbrücken penetration model, followed by the recovery of the probes or by fluorescence microscopy. RESULTS:The incorporation of ultradeformable liposomes to cosmetic vehicles did not alter their penetration in most of the cases for human skin explants. Pig skin penetration presented significant differences compared with human explants. CONCLUSIONS:Ultradeformable liposomes could be useful as versatile cosmeceutical carriers in final product formulations.

BackgroundEthanolic extract from blueberry (Vaccinium myrtillus) is rich in anthocyanins and thus exhibits antioxidant activity. On the other hand, ultradeformable liposomes are capable of penetrating to the impermeable barrier of skin. Nanoberries are ultradeformable liposomes carrying blueberry extract. ObjectivesIn this study, their capacity to penetrate the stratum corneum and photodamage prevention were tested, with the aim of developing a topical formulation for skin protection from environmental damage. MethodsNanoberries were prepared by lipid film resuspension with ethanolic extract from blueberry, followed by sonication and incorporation to a gel. Size, zeta potential, deformability, rheology, and viscoelasticity were determined. Toxicity was assessed in vivo in zebrafish model, while in vitro cytotoxicity assay was performed on HaCaT and HEK-293T cell lines. Skin penetration was evaluated with the Saarbrucken penetration model followed by tape stripping, cryosection, or optical sectioning. UV-damage protection and photoprotection were determined by ad hoc methods with UVA, UVB, and UVC radiation on HaCaT cells. Wound assay was performed on HaCaT cells. ResultsNanoberries of about 100 nm, with differential elastic properties, did penetrate the stratum corneum, with low toxicity. When HaCaT cells were exposed to UV radiation in the presence of nanoberries, their viability was maintained. ConclusionsNanoberries could be effective to protect the skin from sun photodamage.

Catalytic ring-opening of bio-sourced non-strained lactones with aromatic amines can offer a straightforward, 100 % atom-economical, and sustainable pathway towards relevant N-aryl amide scaffolds. Herein, the first general, metal-free, and highly efficient N-aryl amide formation is reported from poorly reactive aromatic amines and non-strained lactones under mild operating conditions using an organic bicyclic guanidine catalyst. This protocol has high application potential as exemplified by the formal syntheses of drug-relevant molecules.

A variety of cavitand-based polyphenols was prepared from cheap and accessible aldehyde and resorcinol/pyrogallol reagents to give the respective resorcin[4]- or pyrogallol[4]arenes. The preorganization of the phenolic units allows intra- and intermolecular hydrogen bond (HB) networks that affect both the reactivity and stability of these HB-donor catalysts. Unexpectedly, we found that the resorcin[4]arenes show cooperative catalysis behavior compared to the parent resorcinol in the catalytic coupling of epoxides and CO2 with a significantly higher turnover. At elevated reaction temperatures, the resorcin[4]arene-based catalyst 3d displays the best catalytic performance with very high turnover numbers and frequencies, combining increased reactivity and stability compared to pyrogallol, and an ample substrate scope. This type of polyphenol structure thus illustrates the importance of a new, highly competitive organocatalyst design to devise sustainable CO2 conversion processes.

The first general catalytic and highly stereoselective formation of (Z)-1,4-but-2-ene diols is described from readily available and modular vinyl-substituted cyclic carbonate precursors using water as a nucleophilic reagent. These 1,4-diol scaffolds can be generally prepared in high yields and with ample scope in reaction partners using a simple synthetic method that does not require the presence of any additive or any special precaution unlike the stoichiometric approaches reported to date. Control experiments support the mechanistic view that hyperconjugation within the catalytic intermediate after decarboxylation plays an imperative role to control the stereoselective outcome of these reactions.

An entry from the Cambridge Structural Database, the world’s repository for small molecule crystal structures. The entry contains experimental data from a crystal diffraction study. The deposited dataset for this entry is freely available from the CCDC and typically includes 3D coordinates, cell parameters, space group, experimental conditions and quality measures.