The ∞‐Besov capacity problem

A theory of ∞‐Besov capacities is developed and several applications are provided. In particular, we solve an open problem in the theory of limits of the ∞‐Besov semi‐norms, we obtain new restriction‐extension inequalities, and we characterize the pointwise multipliers acting on the ∞‐Besov spaces.     

Representability of actions in the category of (Pre)crossed modules in Leibniz algebras

We investigate the representability of actions in the category of (pre)crossed modules in Leibniz algebras. For this, we construct an actor of a (pre)cat¹-Leibniz algebra and then by using the natural equivalence of the categories of (pre)cat¹-Leibniz algebras and that of (pre)crossed modules, we construct the split extension classifier of the corresponding (pre)crossed module.     

Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with $p^4$ elements, (p a prime) are given. Second, the number and structure of γ-constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined.     

Isometries and additive mapping on the unit spheres of normed spaces

In this paper, we investigate isometric extension problem in general normed space. We prove that an isometry between spheres can be extended to a linear isometry between the spaces if and only if the natural positive homogeneous extension is additive on spheres. Moreover, this conclusion still holds provided that the additivity holds on a restricted domain of spheres.     

Isolation of Volatile Compounds with Repellent Properties against Aedes albopictus (Diptera: Culicidae) Using CPC Technology

The present work describes the use of Centrifugal Partition Chromatography (CPC) for the bio-guided isolation of repellent active volatile compounds from essential oils. Five essential oils (EOs) obtained from three Pinus and two Juniperus species were initially analyzed by gas chromatography–mass spectrometry (GC/MS) and evaluated for their repellent properties against Aedes albopictus. The essential oil from needles of P. pinea (PPI) presented the higher activity, showing 82.4% repellency at a dose of 0.2 μL/cm². The above EO, together with the EO from the fruits of J. oxycedrus subsp. deltoides (JOX), were further analyzed by CPC using the biphasic system n-Heptane/ACN/BuOH in ratio 1.6/1.6/0.2 (v/v/v). The analysis of PPI essential oil resulted in the recovery of (−)-limonene, guaiol and simple mixtures of (−)-limonene/β-pheladrene, while the fractionation of JOX EO led to the recovery of β-myrcene, germacrene-D, and mixtures of α-pinene/β-pinene (ratio 70/30) and α-pinene/germacrene D (ratio 65/45). All isolated compounds and recovered mixtures were tested for their repellent activity. From them, (−)-limonene, guaiol, germacrene-D as well the mixtures of (−)-limonene/β-pheladrene presented significant repellent activity (>97% repellency) against Ae. albopictus. The present methodology could be a valuable tool in the effort to develop potent mosquito repellents which are environmentally friendly.     

Chemical Composition and Antioxidant Activity of Essential Oils from Eugenia patrisii Vahl,E. punicifolia (Kunth) DC., and Myrcia tomentosa (Aubl.) DC., Leaf of Family Myrtaceae

Essential oils (EOs) were extracted from Eugenia patrisii, E. punicifolia, and Myrcia tomentosa, specimens A and B, using hydrodistillation. Gas chromatography coupled with mass spectrometry (GC/MS) was used to identify the volatile constituents present, and the antioxidant capacity of EOs was determined using diphenylpicryl-hydrazyl (DPPH) and trolox equivalent antioxidant capacity (TEAC) assays. For E. patrisii, germacrene D (20.03%), bicyclogermacrene (11.82%), and (E)-caryophyllene (11.04%) were identified as the major constituents of the EOs extracted from specimen A, whereas specimen B primarily comprised γ-elemene (25.89%), germacrene B (8.11%), and (E)-caryophyllene (10.76%). The EOs of E. punicifolia specimen A contained β-Elemene (25.12%), (E)-caryophyllene (13.11%), and bicyclogermacrene (9.88%), while specimen B was composed of (E)-caryophyllene (11.47%), bicyclogermacrene (5.86%), β-pinene (5.86%), and γ-muurolene (5.55%). The specimen A of M. tomentosa was characterized by γ-elemene (12.52%), germacrene D (11.45%), and (E)-caryophyllene (10.22%), while specimen B contained spathulenol (40.70%), α-zingiberene (9.58%), and γ-elemene (6.89%). Additionally, the chemical composition of the EOs was qualitatively and quantitatively affected by the collection period. Furthermore, the EOs of the studied specimens, especially specimen A of E. punicifolia, showed a greater antioxidant activity in DPPH rather than TEAC, as represented by a significantly high inhibition percentage (408.0%).     

Chemical Profiling,Toxicity and Anti-Inflammatory Activities of Essential Oils from Three Grapefruit Cultivars from KwaZulu-Natal in South Africa

The medicinal potential and volatile composition of different parts of three cultivars of grapefruit (Citrus paradisi) were evaluated for their toxicity and anti-inflammatory activities. Fresh leaf and fruit peel were separately isolated by hydrodistillation for 4 h. The essential oils were subjected to GC/GC-MS analysis for chemical profile. Toxicity of the essential oils in mice were evaluated using Lorke’s method, while an anti-inflammatory assay was performed in a rat model using egg albumin-induced oedema. The oils obtained were light yellow in colour, and odour varied from strong citrus smell to mild. Percentage yield of fresh peel oil (0.34–0.57%) was greater than the fresh leaf oil yield (0.21–0.34%). D-limonene (86.70–89.90%) was the major compound identified in the leaf oil, while β-phellandrene (90.00–91.01%) dominated the peel oil. At a dosage level of 5000 mg/kg, none of the oils showed mortality in mice. An anti-inflammatory bioassay revealed that all the oils caused a significant (p < 0.05–0.01) reduction in oedema size when compared to the negative control group throughout the 5 h post induction assessment period. The study reveals that the oils are non-toxic and demonstrate significant anti-inflammatory activity. Our findings suggest that the leaf and peel oils obtained from waste parts of grapefruit plants can be useful as flavouring agents, as well as anti-inflammatory agents.     

The Chemical Compositions,and Antibacterial and Antioxidant Activities of Four Types of Citrus Essential Oils

Nanfeng mandarins (Citrus reticulata Blanco cv. Kinokuni), Xunwu mandarins (Citrus reticulata Blanco), Yangshuo kumquats (Citrus japonica Thunb) and physiologically dropped navel oranges (Citrus sinensis Osbeck cv. Newhall) were used as materials to extract peel essential oils (EOs) via hydrodistillation. The chemical composition, and antibacterial and antioxidant activities of the EOs were investigated. GC-MS analysis showed that monoterpene hydrocarbons were the major components and limonene was the predominate compound for all citrus EOs. The antibacterial testing of EOs against five different bacteria (Bacillus subtilis, Staphylococcus aureus, Escherichia coli, Pseudomonas aeruginosa and Salmonella typhimurium) was carried out using the filter paper method and the broth microdilution method. Kumquat EO had the best inhibitory effect on B. subtilis, E. coli and S. typhimurium with MIC (minimum inhibitory concentration) values of 1.56, 1.56 and 6.25 µL/mL, respectively. All citrus EOs showed the antioxidant activity of scavenging DPPH and ABTS free radicals in a dose-dependent manner. Nanfeng mandarin EO presented the best antioxidant activity, with IC₅₀ values of 15.20 mg/mL for the DPPH assay and 0.80 mg/mL for the ABTS assay. The results also showed that the antibacterial activities of EOs might not be related to their antioxidant activities.     

The essential spectrum of periodically stationary pulses in lumped models of short-pulse fiber lasers

In modern short-pulse fiber lasers, there is significant pulse breathing over each round trip of the laser loop. Consequently, averaged models cannot be used for quantitative modeling and design. Instead, lumped models, which are obtained by concatenating models for the various components of the laser, are required. As the pulses in lumped models are periodic rather than stationary, their linear stability is evaluated with the aid of the monodromy operator obtained by linearizing the round-trip operator about the periodic pulse. Conditions are given on the smoothness and decay of the periodic pulse that ensure that the monodromy operator exists on an appropriate Lebesgue function space. A formula for the essential spectrum of the monodromy operator is given, which can be used to quantify the growth rate of continuous wave perturbations. This formula is established by showing that the essential spectrum of the monodromy operator equals that of an associated asymptotic operator. Since the asymptotic monodromy operator acts as a multiplication operator in the Fourier domain, it is possible to derive a formula for its spectrum. Although the main results are stated for a particular experimental stretched pulse laser, the analysis shows that they can be readily adapted to a wide range of lumped laser models.     

On graphs with exactly one anti-adjacency eigenvalue and beyond

The anti-adjacency matrix of a graph is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping in each row and each column only the distances equal to 1. The (anti-)adjacency eigenvalues of a graph are those of its (anti-)adjacency matrix. Employing a novel technique introduced by Haemers (2019) [9], we characterize all connected graphs with exactly one positive anti-adjacency eigenvalue, which is an analog of Smith's classical result that a connected graph has exactly one positive adjacency eigenvalue iff it is a complete multipartite graph. On this basis, we identify the connected graphs with all but at most two anti-adjacency eigenvalues equal to ?2 and 0. Moreover, for the anti-adjacency matrix we determine the HL-index of graphs with exactly one positive anti-adjacency eigenvalue, where the HL-index measures how large in absolute value may be the median eigenvalues of a graph. We finally propose some problems for further study.