Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces

We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝ n + 1 {\mathbb{R}^{n+1}} and every p > n {p>n} , the L p {L^{p}} -norm of the trace-free part of the anisotropic second fundamental form controls from above the W 2 , p {W^{2,p}} -closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p ≤ n {p\leq n} , the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.

VASCULAR ANATOMY OF GREATER OMENTUM IN MIDDLE AGE ADULTS

The variability of artery and vein topography of greater omenta from 20 corpses of middle age adults was investigated by applying the anatomical and morphometric methods. The right and left gastroepiploic arteries supply the greater omentum with blood. In 80% of the cases studied these arteries form the superior arterial arch, in 15% of the cases they are not connected, and in 5% of the cases they form anastomosis through the right and left lateral branches, thus, forming the middle arterial arch of the greater omentum. In 10% of the cases, the left gastroepiploic artery is located in the thick of the posterior plate of the greater omentum. The right gastroepiploic artery branches into central, right medial, intermediate and lateral arteries and in 10% left additional artery. The right gastroepiploic artery and vein are longer in men (16.95 ± 5.5 cm) that in women (15.77 ± 2.9 cm). The outer diameter of these vessels is larger in women: 0.24 ± 0.03 cm of arteries and 0.27 ± 0.03 cm of veins. The central artery is the longest and widest of all omental branches. The area of the central artery is more preferred for flaps in quadrangular and triangular shapes of the greater omentum. The branches of the right gastroepiploic artery supply blood to the right half of the greater omentum and reach the lower edge of its free part. The left half of the greater omentum is supplied by the branches of the left gastroepiploic artery, namely by the left medial, intermediate and lateral arteries, which do not reach the lower edge of the free part of the omentum. In 10% of the cases, the superior left part of the omentum is additionally supplied with blood by the splenic artery. On such case of blood vessels presence, it can serve as a source for obtaining a flap. One vein accompanies the artery of the same name. All the veins of the omentum have a larger diameter than the arteries. In 15% of the cases greater omentum is quadrangular, in 25% of the cases is triangular and in 60% it is of irregular shape with two or more parts. In the one-part omentum, the left and right parts have the same blood supply. At the two-part structure, the right half is in more favourable condition in terms of the blood supply that enables to recommend its usage in transplantation surgery.

Staging of Plantar Fasciitis According to Ultrasonography

Objective. Analysis of sonographic examination of plantar aponeurosis in norm and in plantar fasciitis, determination of the staging of plantar fasciitis depending on the duration of the pain syndrome and sonographic changes in plantar aponeurosis. Materials and Methods. 193 patients (272 cases) with plantar fasciitis (131 females and 62 males). The average age of patients was 47.69±0.97 years (18-81 years). Unilateral pathology was noted in 114 patients (114 cases) and bilateral – in 79 patients (158 cases). The average body mass index was 26.68±0.24. The average duration of pain syndrome was 101.12±5.83 days (7-390 days). The pain syndrome lasted up to 1 month in 26 patients (37 cases), from 1 to 6 months – in 126 patients (181 cases), and more than 6 months – in 41 patients (54 cases). The control group included 20 healthy volunteers (40 feet). Clinical radiological, sonographic, and statistical methods were used. Results. The average thickness of plantar aponeurosis in the study group was 6.14±1.49 mm (2.3-7.7 mm) and in control group – 3.5±0.1 mm (2.5-4.4 mm). There were three stages of the plantar fasciitis. Stage I was characterized by the absence of thickening of the plantar aponeurosis and structural changes, and by signs of local perifocal edema in the subcutaneous fat of the area adjacent to the aponeurosis, which are manifested by a zone of heterogeneous echogenicity. Stage II was characterized by a spindle-shaped form of aponeurosis, a thickening of enthesis more than 4 mm, significant reduction in echogenicity, erased or missing fibrous pattern, fuzzy contour of the aponeurosis, and modified contour of the calcaneus. At stage III, inequality of the cortical calcaneal contour and small focal hyperechogenicity points in enthesis were detected. Distal free part of the aponeurosis also changed. Loci of vascularization were detected using power Doppler sonography. Conclusions. A quantitative sonographic sign of plantar fasciitis is the thickening of the enthesis of the plantar aponeurosis of more than 4 mm. Qualitative signs were changes in echogenicity and structure, changes in the contours of the cortical layer of enthesis and plantar aponeurosis, and distribution of echographic changes distally to the free part of the aponeurosis. The described signs, when compared with the duration of the pain syndrome, determine the stage of plantar fasciitis.

$$L^p$$-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions

For $$n\ge 3$$ n ≥ 3 and $$1<p<\infty $$ 1 < p < ∞ , we prove an $$L^p$$ L p -version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields $$P:\Omega \rightarrow \mathbb {R}^{n\times n}$$ P : Ω → R n × n having p-integrable generalized $${\text {Curl}}_{n}$$ Curl n and generalized vanishing tangential trace $$P\,\tau _l=0$$ P τ l = 0 on $$\partial \Omega $$ ∂ Ω , denoting by $$\{\tau _l\}_{l=1,\ldots , n-1}$$ { τ l } l = 1 , … , n - 1 a moving tangent frame on $$\partial \Omega $$ ∂ Ω . More precisely, there exists a constant $$c=c(n,p,\Omega )$$ c = c ( n , p , Ω ) such that $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\left( \Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p\left( \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}}\right) }\right) , \end{aligned}$$ ‖ P ‖ L p ( Ω , R n × n ) ≤ c ‖ dev n sym P ‖ L p ( Ω , R n × n ) + ‖ Curl n P ‖ L p Ω , R n × n ( n - 1 ) 2 , where the generalized $${\text {Curl}}_{n}$$ Curl n is given by $$({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$$ ( Curl n P ) ijk : = ∂ i P kj - ∂ j P ki and "Equation missing" denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.

Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

Let X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

Two new Nearctic genera in the tribe Odynerini s. str. revealed on the bionomics and morphology, with a comment on the cocoons of the eumenine wasps (Hymenoptera: Vespidae: Eumeninae)

Two new genera are described for three species in the “red” group of the North American eumenine wasps from the genus Odynerus Latreille, 1802: Bohartodynerus Fateryga, gen. n. for O. margaretellus Rohwer, 1915 (type species) and O. cinnabarinus Bohart, 1939, and Parkerodynerus Fateryga, gen. n. (type species O. erythrogaster Bohart, 1939). Bohartodynerus cinnabarinus comb. n. is newly recorded from Nevada and New Mexico and B. margaretellus comb. n. from California, New Mexico, Utah, and Wyoming. Bionomics of the species included to these genera are summarized. Nectar robbing is reported for wasps in the genus Bohartodynerus at flowers of Astragalus spp. (Fabaceae), Penstemon sp. (Plantaginaceae), and Calylophus sp. (Onagraceae). It is speculated that nectar is used by them to bond the gravel during the construction of the closing plug of the nest. Parkerodynerus erythrogaster comb. n. apparently does not use liquid for the nest construction. Bionomics of other genera in the Odynerus group of the tribe Odynerini s. str. are discussed. Neither Bohartodynerus nor Parkerodynerus can be treated as members of the genus Odynerus according to their ethology, and the differences between these two newly described taxa are also enough to recognize them as separate genera. A cocoon of P. erythrogaster is described in detail; it has a sandwich-like structure with two layers of silk strands and a layer of sand in-between. Similar structure is found in the cocoons of other genera in the Odynerus group but only at the free part of the cocoon instead over its whole surface; that was overlooked during the previous studies of those genera.

The Small Imitation Form in the Sacred Chorale Concertos by Samuel Scheidt

Известный немецкий теоретик музыки XVII века Вольфганг Каспар Принц в труде „Historische Beschreibung der edelen Sing- und Kling-Kunst“ («Историческое описание благородного искусства вокальной и инструментальной музыки», 1690) назвал Генриха Шютца, Иоганна Германа Шейна и Самуэля Шейдта тремя лучшими композиторами их времени в Германии. Композиторское письмо двух из упомянутых Принцем персон, Шейна и Шейдта, остается малоизученным в отечественном музыкознании. В статье рассматривается малая имитационная форма в хоральных концертах Шейдта из собрания „Geistliche Concerte“ («Духовные концерты»), четыре части которого были опубликованы в 1631, 1634, 1635 и 1640 годах. Тридцатилетняя война (1618–1648), непосредственно коснувшаяся Шейдта, стала причиной обращения композитора к разновидности вокального духовного концерта для небольшого состава, включающего несколько вокальных голосов (в концертах Шейдта это, как правило, три партии) и бассо континуо. Исследуются тексто-музыкальные единицы, образующиеся на основе строки или сегмента строки хорала и становящиеся материалом для имитации, раскрываются особенности имитационной техники, выявляется связь свободной части малой имитационной формы с типом обработки — Cantionalsatz, типологизируются структуры, различающиеся порядком поступления материала строки в малую имитационную форму. В конце статьи прослеживаются изменения, которые претерпевает малая имитационная форма в хоральных концертах по сравнению с хоральными мотетами из первого собрания вокальной музыки Шейдта “Cantiones sacrae” («Священные песнопения», 1620). The famous German music theorist of the 17th century, Wolfgang Caspar Printz, in his work „Historische Beschreibung der edelen Sing- und Kling-Kunst“ (“Historical Description of the Noble Art of Vocal and Instrumental Music”, 1690) named Heinrich Schütz, Johann Hermann Schein and Samuel Scheidt as the three best composers of their time in Germany, the compositional technique of two of them, Schein and Scheidt, remaining poorly studied in Russian musicology. The article examines the small imitation form in Scheidt’s chorale concertos from the „Geistliche Concerte“ (“Sacred Concertos”) collection, four parts of which were published in 1631, 1634, 1635 and 1640. The Thirty Years’ War (1618–1648), which directly affected Scheidt, caused the composer to turn to a kind of vocal sacred concerto for a small cast, including few voices (in Scheidt’s concertos, these are, as a rule, three parts) and basso continuo in the organ. The textual-and-musical units that are formed on the basis of a chorale line or a segment of a line and become the material for imitation are investigated; the features of the imitation technique are revealed; a connection of the free part with the type of arrangement — Cantionalsatz, is established, mapping the structures that differ in the order in which the line material arrives in the small imitation form. At the end of the article, the changes are traced that the small imitation form has undergone in chorale concertos in comparison with chorale motets from Scheidt’s first collection of vocal music “Cantiones sacrae” (“Sacred chants”, 1620).

The Arnon bases in the Steenrod algebra

Let {\mathcal{A}=\mathcal{A}_{p}} be the {\mathrm{mod}\,p} Steenrod algebra, where p is a fixed prime and let {\mathcal{A}^{\prime}} denote the Bockstein-free part of {\mathcal{A}} at odd primes. Being a connected graded Hopf algebra, {\mathcal{A}} has the canonical conjugation χ. Using this map, we introduce a relationship between the X- and Z-bases of {\mathcal{A}^{\prime}}. We show that these bases restrict to give bases to the well-known sub-Hopf algebras {\mathcal{A}(n-1)}, {n\geq 1}, of {\mathcal{A}^{\prime}}.