Search

The Hamiltonian approach to black hole entropy, recently proposed in the framework of Poincar\'e gauge theory, is extended by including the scalar matter. The improved approach is used to analyse asymptotic charges and entropy of a typical black hole with scalar hair in two complementary dynamical settings, in GR and in the teleparallel gravity. In both cases, the results confirm the validity of the first law., Comment: LaTeX, 12 pages

Background: There are many studies on the morphology of the liver and its blood vessels in experimental animals, but such studies are lacking in the mole rat (Spalax leucodon). The aim of this paper was a detailed basic study on the topography, morphology, vascular and biliary branching systems of the liver in the mole rat. Materials and methods: Coloured gelatine and mixture of coloured lead oxide and linseed oil were injection contrast masses used to obtain vascular and biliary branching pattern in the liver. It was revealed that the liver of the mole rat had five lobes (left, quadrate, right medial, right lateral and caudate lobes). Results: The left, undivided lobe was the largest lobe of the liver. The quadrate lobe was divided into two components by a deep notch. The gallbladder, of cylindrical shape, was present and attached to the quadrate lobe. The common bile duct was formed by the union of the left and right hepatic ducts. The pancreatic duct joined the common bile duct before it entered the duodenum. In the present study, only the right medial lobe and quadrate lobe always showed a single lobar artery, portal and hepatic veins. The left lobe showed four lobar arteries, portal and hepatic veins. The caudate lobe with its two processes and the right lateral and medial lobes had different arterial and portal blood supply as well as hepatic and biliary drainage of these lobes. The intrahepatic branches of the proper hepatic artery ran parallel to the branches of the common portal vein in the same lobes of the liver. Conclusions: The results of this study are significant for comparative studies among different species of rodents and other experimental animals. Morphology, vasculature and biliary tract of the liver in the mole rat were similar to that of other experimental animals and identified differences may be related to the adaptation to the mode of life and diet of this rodent.

Cell wall content generally is regarded as the most important factor affecting forage utilization because it comprises the major fraction of forage dry matter and is correlated with forage intake and digestibility. The experiment was carried out to evaluate the cell wall constituents of ensiled pea - oat mixtures sown at five different seeding rates. Three effects were studied: application of bacterial inoculant, seeding rate of pea and oat in the mixtures and stages of plant development. The NDF (Neutral Detergent Fiber), ADF (Acid Detergent Fiber), hemicellulose, lignin and DMD (Dry Matter Digestibility) were determined in silages. Pea and oat were grown in binary mixtures at the experimental field of the Institute for forage crops, Krusevac - Serbia, and were tested at five different mixture rates: pure pea and pure oat crop, 25% pea + 75% oat, 50% pea + 50% oat, 75% pea + 25% oat. Application of bacterial inoculant affected lower content of NDF and hemicellulose (P< 0.05). Content of cell wall constituents decreased with increasing pea content in the mixtures, but DMD increased. On the other hand, content of cell wall constituents increased with plant growth and development, but DMD decreased.

The canonical approach to black hole entropy in Poincar\'e gauge theory without matter is extended to include Maxwell field as a matter source. The new formalism is used to calculate asymptotic charges and entropy of Kerr--Newmann--AdS black holes with torsion. The result implies that the first law, with a nontrivial contribution of the Maxwell field, takes the same form as in general relativity., Comment: LaTex, 17 pages

The micellization behavior and foamability of a mixture of anionic surfactant sodium laureth sulfate (SLES) and nonionic polysorbate surfactants commercially available as Tween series were investigated. The mole fraction and the structure of the hydrophobic part of various polysorbates affect the critical micellar concentration (CMC) and foam height and stability of anionic surfactant. Experimental values of CMC were determined by measuring specific conductivity at 25°C and foam height and stability by modified Ross–Miles method at 23°C. The negative deviation of the experimentally determined CMC from theoretical values for ideal mixed micelle indicate non-ideal mixing behavior due to synergism as a consequence of attractive interactions between the individual surfactants in formed micelles. Thus synergism increase with increasing of both the mole fraction and length of hydrophobic tail of examined Tween in SLES/Tween mixtures. Foam height and stability significantly decrease with increasing length of the hydrophobic tail of Tween surfactants and their mole fractions in the binary mixture.

The classical 1966 theorem of Tverberg with its numerous variations was and still is a motivating force behind many important developments in convex and computational geometry as well as the testing ground for methods from equivariant algebraic topology. In 2018, B\'ar\'any and Sober\'on presented a new variation, the "Tverberg plus minus theorem." In this paper, we give a new proof of the Tverberg plus minus theorem, by using a projective transformation. The same tool allows us to derive plus minus analogues of all known affine Tverberg type results. In particular, we prove a plus minus analogue of the optimal colored Tverberg theorem., Comment: 4 + \eps pages, 1 figure

In 1960 Grünbaum asked whether for any finite mass in R d \mathbb {R}^d there are d d hyperplanes that cut it into 2 d 2^d equal parts. This was proved by Hadwiger (1966) for d ≤ 3 d\le 3 , but disproved by Avis (1984) for d ≥ 5 d\ge 5 , while the case d = 4 d=4 remained open. More generally, Ramos (1996) asked for the smallest dimension Δ ( j , k ) \Delta (j,k) in which for any j j masses there are k k affine hyperplanes that simultaneously cut each of the masses into 2 k 2^k equal parts. At present the best lower bounds on Δ ( j , k ) \Delta (j,k) are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska, Vrećica and Živaljević (2006). The problem has been an active testing ground for advanced machinery from equivariant topology. We give a critical review of the work on the Grünbaum–Hadwiger–Ramos problem, which includes the documentation of essential gaps in the proofs for some previous claims. Furthermore, we establish that Δ ( j , 2 ) = 1 2 ( 3 j + 1 ) \Delta (j,2)= \frac 12(3j+1) in the cases when j − 1 j-1 is a power of 2 2 , j ≥ 5 j\ge 5 .

Planiranjem razvoja se obezbeđuje da sistem odbrane, odnosno vojska u užem smislu, poseduje potrebne sposobnosti za zastitu nacionalnih interesa od aktuelnih i buducih bezbednosnih izazova, rizika i pretnji. Presudan uticaj na rezultate planiranja razvoja vojske ima usklađivanje motiva i ciljeva bitnih subjekata. Subjekti koji najvise uticu na planiranje razvoja su nacija, odnosno građani, reprezentativna politicka partija i subjekti unutar vojske (vojni rukovodioci i zaposleni). Dugogodisnja neusklađenost motiva i ciljeva navedenih subjekata, kao i primena loseg modela usklađivanja, dovela je do kontinuiranog smanjenja budžeta odbrane. Smanjenje budžeta odbrane je uticalo na sistem motivacije zaposlenih, a to je rezultiralo time da veliki broj pripadnika Ministarstva odbrane i Vojske Srbije prekine radni odnos. Krajnji efekti ovakvog usklađivanja se ogledaju u padu operativnih sposobnosti Vojske Srbije.

The experiment was carried out to evaluate the possibility of ensiling common vetch - oat mixtures sown at five different seeding rates. Two effects were studied: seeding rate of common vetch and oat in the mixtures and application of bacterial inoculant. The pH, DM (Dry Matter) content, ammonia nitrogen, soluble nitrogen, lactic, acetic and butyric acids were determined on silages. DLG method was utilized for classification the silage quality. Common vetch and oat were grown in binary mixtures at the experimental field of the Institute for forage crops, Krusevac - Serbia, and were tested at five different mixture rates: pure vetch, 25% vetch + 75% oat, 50% vetch + 50% oat, 75% vetch + 25% oat and pure oat. Application of bacterial inoculant affected higher content of ammonia nitrogen and acetic acid (P< 0.05), but lower content of soluble nitrogen (P< 0.05). Depending on the seeding rates of common vetch and oat, 75 : 25 common vetch - oat silage had the highest content of lactic acid and the lowest content of butyric acid. Contents of DM, pH and ammonia nitrogen were similar in all silages ranged from 307.2 to 318.5 g kg-1, from 4.27 to 4.54 and from 16.1 to 19.1% ?N, respectively. According to the DLG method for silage quality evaluation, similar quality grades were founded.

The physicochemical properties of initial formulation, that is anionic/amphoteric surfactants mixture SLES/AOS/CAB (sodium lauryl ether sulfate (SLES), alpha-olefin sulfonates (AOS) and cocamidopropyl betaine (CAB) at ratio 80 : 15 : 5) with nonionic surfactant of amine oxide type (lauramine oxide (AO)) in various concentration (1-5%) were studied. To characterize the surfactants mixture, the critical micelle concentration (CMC), surface tension (gamma), foam volume, biodegradability and irritability were determined. This study showed that adding of AO in those mixtures lowered both gamma and CMC as well as enhanced SLES/AOS/CAB foaming properties, but did not significantly affect biodegradability and irritability of initial formulation. Moreover, an increase in AO concentration has a meaningful synergistic effect on the initial formulation properties. All those results indicates that a nonionic surfactant of amine oxide type significantly improves the performance of anionic/amphoteric mixed micelle systems, and because of that anionic/amphoteric/nonionic mixture can be used in considerably lower concentrations as a cleaning formulation.

The amplituhedra arise as images of the totally nonnegative Grassmannians by projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed \& Trnka (Journal of High Energy Physics, 2014) as model spaces that should provide a better understanding of the scattering amplitudes of quantum field theories. The topology of the amplituhedra has been known only in a few special cases, where they turned out to be homeomorphic to balls. The amplituhedra are special cases of Grassmann polytopes introduced by Lam (Current Developments in Mathematics 2014, Int.\ Press). In this paper we show that that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible., 7 pages, to appear in Selecta Mathematica

We describe how a powerful new “constraint method” yields many different extensions of the topological version of Tverberg’s 1966 Theorem in the prime power case— and how the same method also was instrumental in the recent spectacular construction of counterexamples for the general case. © 2016. All rights reserved.

The production and utilization of intercropped legume- cereal bi-crops as ruminant feeds is beneficial for many reasons – legumes are rich in terms of protein concentration, whereas cereals are characterized by higher carbohydrate contents. To improve utilization of common vetch-oat mixtures as feed in Serbia, the objective of this investigation was to evaluate the crude protein (CP) and carbohydrate (CHO) fractions in these feeds using the Cornell Net Carbohydrate and Protein System (CNCPS) depending on the stage of plant development and seeding rate of common vetch and oat in the mixtures. The experiment was conducted at the experimental field of Institute for forage crops, Kruševac – Serbia, in 2015. The field trial was arranged in a randomized block design with three replications. In this research, pure stands of common vetch and oat, and their seed mixtures (common vetch:oat – 1:1.5; 1:1 and 1:0.5) were studied harvested at three different cutting stages: beginning of vetch flowering – 10% of flowering; forming the first pods on 2/3 vetch plants and forming green seeds in 2/3 pods. The samples were analyzed for protein and carbohydrate fractions according to CNCPS. Stage of growth affected significantly all carbohydrate and protein fractions, except unavailable PC fraction of crude protein which did not change with plant growth and development. The highest protein fraction was intermediately degraded PB2 fraction of crude protein. The highest carbohydrate fraction was available cell wall – CB3 fraction of carbohydrate. According to the results obtained in these investigations, it was concluded that common vetch-oat mixtures varied significantly with respect to their carbohydrate and protein fraction. Due to lower CC fraction, oat was superior feed in terms of total carbohydrates supply to ruminants. Among all investigated mixtures and pure stand of oat and common vetch, it could be conclude that mixture with 1:1 common vetch-oat rate was superior because of higher CA and CB3 fractions of total carbohydrate and the highest PB2 fraction of crude protein.

A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on $R^d$ is bisected by the arrangement of affine hyperplanes $H$ if the measure on the "non-negative side" of the arrangement $\{x\in R^d : p_{H}(x)\ge 0\}$ is the same as the measure on the "non-positive" side $\{x\in R^d : p_{H}(x)\le 0\}$. In 2017 Barba, Pilz \& Schnider considered special cases of the following measure partition hypothesis: For a given collection of $j$ finite Borel measures on $R^d$ there exists a $k$-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when $d=k=2$ and $j=4$. They conjectured that every collection of $j$ measures on $R^d$ can be simultaneously bisected with a $k$-element affine hyperplane arrangement provided that $d\ge \lceil j/k \rceil$. The conjecture was confirmed in the case when $d\ge j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojevi��, Frick, Haase \& Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Gr��nbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+\ell$ measures on $R^{2^a+\ell}$, where $1\leq \ell\leq 2^a-1$, there exists a $(2h+1)$-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb in 2020., 2 figures

The system KSCN-H2O2-CuSO4-NaOH, also known as the OrbA n-Epstein oscillatory reaction, is exposed to external perturbations by several phenolic compounds: catechol, resorcinol, hydroquinone, 2,5-dihydroxybenzoic acid, 2,6-dihydroxybenzoic acid, 2,3-dihydroxybenzoic acid, 2,4-dihydroxybenzoic acid, 3,4-dihydroxybenzoic acid, 3,5-dihydroxybenzoic acid, caffeic acid, and ferulic acid. As a result of the performed perturbation in most examined cases, oscillations have been inhibited for some characteristic time, and resumed afterwards. The evaluated inhibition time is typical for each substance and strongly dependent on its concentration. The chemical mechanism of the interaction between the OrbA n-Epstein system and phenolic compounds is briefly discussed. Numerical simulations are performed using the original OrbA n-Epstein model with 30 reactions, extended by three reactions describing the interaction with inhibitory substances. The rate constants of three added reactions are adjusted to fit experimental inhibition times, and compared among used compounds. The observed effects are discussed further in relation with the bond dissociation enthalpy theory. Unlike other tested compounds, 2,5-dihydroxybenzoic acid remains off from predicted order of activity.

In this paper we use the strength of the constraint method in combination with a generalized Borsuk–Ulam type theorem and a cohomological intersection lemma to show how one can obtain many new topological transversal theorems of Tverberg type. In particular, we derive a topological generalized transversal Van Kampen–Flores theorem and a topological transversal weak colored Tverberg theorem.

We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dimensional Finsler space with possibly irreversible Finsler metric. An example of such a system is a billiard in a sufficiently weak magnetic field. The $r$-periodic Finsler billiard trajectories correspond to $r$-gons inscribed in $M$ and having extremal Finsler length. The cyclic group ${\mathbb Z}_r$ acts on these extremal polygons, and one counts the ${\mathbb Z}_r$-orbits. Using Morse and Lusternik-Schnirelmann theories, we prove that if $r\ge 3$ is prime, then the number of $r$-periodic Finsler billiard trajectories is not less than $(r-1)(d-2)+1$. We also give stronger lower bounds when $M$ is in general position. The problem of estimating the number of periodic billiard trajectories from below goes back to Birkhoff. Our work extends to the Finsler setting the results previously obtained for Euclidean billiards by Babenko, Farber, Tabachnikov, and Karasev.

In this study grape pomace was ensilaged without and with the addition of NPN substances (Benural) at the dose of 0.5, 1.0 and 1.5% of the amount of husk and with the addition of inoculant based on homo and hetero fermentative lactic acid bacteria. The greatest effect on the nutritional value had application of Benural, especially in increasing the CP content from 126.9 to 178.3 g kg-1DM, an increase of over 40%. Application of Benural increased the ammonia and soluble nitrogen several times, but even with the maximum addition of Benural percentage of ammonia nitrogen in the total nitrogen reached only slightly above 5% NH3 -N/Σ N (5.38%), while the percentage of soluble nitrogen in total nitrogen was 28.29%. The application of inoculants generally had no significant impact, both on the chemical composition and the fermentation process. U istraživaniima komina grožđa je silirana bez i uz dodatak NPN supstanci (Benural) u količini od 0,5; 1,0 i 1,5% od količine komina i sa dodatkom inokulantana na bazi homo i heterofementativnih mlečno-kiselinskih bakterija. Najveći efekat na hranljivu vrednost imala je primena NPN, posebno u povećanju sadržaja SP od 126,9 na 178,3 gkg-1SM. Primena NPN supstanci je uticala na povećanje amonijačnog i rastvorljivog azota nekoliko puta, ali i pri najvećoj količini dodatog NPN supstanci udeo amonijačnog azota u ukupnom azotu je neznatno prešao vrednost od 5% (%NH3-N/ΣN 5,38) dok je udeo rastvorljivog azota u ukupnom azotu iznosio 28,29%. Primena inokulanata uglavnom nije imala značajnog uticaja, kako na hemijski sastav, tako i na proces fermentacije.

Vetch (Vicia sativa L.) is annual plant from the legume family (Fabaceae) and originates from the temperate zone of Europe and Asia. It has a special place in the provision of ani­mal feed (fodder) in the zone of moderate climate. It belongs to high-quality protein fodder plants. There has not been a systematic research of vetch mycoflora in Serbia. This research aims to present the results of preliminary research of mycopopulation of 15 different genotypes of vetch. Total of 600 plant parts has been examined and 9 genera of fungi were isola­ted: Fusarium, Phythophthora, Rhizoctonia, Phoma, Verticillium, Alternaria, Sclerotinia, Botrytis and Ascochyta. On the plants from which the fungi were isolated, there were ma­croscopically clearly visible symptoms of infection. Grahorica (Vicia sativa L.) je jednogodišnja biljka iz porodice mahunarki (Fabaceae). Potiče iz umerenog klimatskog pojasa Evrope i Azije. Ima posebno mesto u obezbeđenju kabaste stočne hrane (krme) u zoni umerenog klimata. Pripada visokokvalitetnim proteinskim krmnim biljkama. Sistematskih istraživanja mikoflore grahorice u Srbiji do sada nije bilo. U ovome radu iznosimo rezultate preliminarnih istraživanja mikopopulacije 15 različitih genotipova grahorice. Ukupno je pregledano 600 biljnih delova sa kojih je izolovano 9 rodova gljiva: Fusarium, Phythophthora, Rhizoctonia, Phoma, Verticillium, Alternaria, Sclerotinia, Botrytis i Ascochyta. Na biljkama iz kojih su izolovane gljive bili su jasno vidljivi makroskopski simptomi zaraze.

Ground squirrel is the only representative of its genus in our country. As experimental animal is used in microbiology, parasitology, immunology and pharmacology. The aim of this study was to examine a part of ground squirrel cardiovascular system and thus help better understanding of anatomy of the body of this specific animal as well as to contribute to comparative anatomy. The studies were perfomed on six ground squirrels, both sexes, weight between 200- 300 g. In order to obtain the arterial vascularization of the kidney, contrast mass gelatin stained with painting tempera was injected into the abdominal aorta after bleeding out. After the injection, blood vessels were prepared and photographed. Corosive preparations of the vein blood vessels of the kidneys were obtained by injection of Byocril into the right azygos vein after bleeding out. After injection, the preparations were placed into 5% NaOH for 96 hours or 10% NaOH for 48 hours. After that the preparations were rinsed with hot water and photographed. A. renalis dextra arises from the lateral wall of the abdominal aorta, 3-4 mm caudal to A. mesenterica cranialis. In most cases, this vessel divides into two or three branches before entering the hilus of the right kidney. A. renalis sinistra arises from the lateral wall of the abdominal aorta, 7-9 mm caudal to the right renal artery. Often, instead of one left renal artery, there are two, rarely three. Based on the results of our study, we concluded that in ground squirrel there is one A. renalis dextra and often two, rarely three Aa. renales sinistrae. In renal venous vascularization, both right and left renal vein are involved. Before entering the kidney, both of them divide into cranial and caudal branch, undergoing renal hilus, enter the renal sinus and continues to branch out into smaller branches. U oblasti eksperimentalne, humane i veterinarske medicine sve se više pridaje značaj eksperimentalnim životinjama. Na njima se mogu pratiti funkcije određenih sistema i njihove farmakodinamske manifestacije, koje nastaju zavisno od vrste, načina unošenja i metaboličkih puteva pojedinih supstanci ili bioloških agenasa u organizam ove životinje. Sa stepenom razvoja nauke, poznavanje građe tela tekunice (Citellus citellus) u cilju eksperimentalnih ispitivanja podrazumeva poznavanje makroskopske i mikroskopske građe kao i odnosa pojedinih organa i organskih sistema životinje za određene vidove istraživačke delatnosti. Za pomenuta istraživanja, pored ostalog, značajno je i poznavanje vaskularizacije bubrega. To je bio jedan od glavnih razloga da obradimo deo kardiovaskularnog sistema kod tekunice i boljim poznavanjem njene građe damo doprinos komparativnoj anatomiji.

We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of $m$ measures in $R^d$ among $r$ thieves using roughly $mr/d$ convex pieces, even in the cases when $m$ is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of $R^d$ into convex parts, and the computation of the Fadell-Husseini ideal valued index of the resulting spaces., 14 pages, 9 figures

Starting from the generalized pp waves that are exact vacuum solutions of general relativity with a cosmological constant, we construct a new family of exact vacuum solutions of Poincar\'e gauge theory, the generalized pp waves with torsion. The ansatz for torsion is chosen in accordance with the wave nature of the solutions. For a subfamily of these solutions, the metric is dynamically determined by the torsion., Comment: LATEX 18 pages, 4 figures

Barany’s “topological Tverberg conjecture” from 1976 states that any continuous map of an N-simplex \(\Delta _{N}\) to \(\mathbb{R}^{d}\), for N ≥ (d + 1)(r − 1), maps points from r disjoint faces in \(\Delta _{N}\) to the same point in \(\mathbb{R}^{d}\). The proof of this result for the case when r is a prime, as well as some colored version of the same result, using the results of Borsuk–Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matousek’s 2003 book “Using the Borsuk–Ulam theorem.”

We show that any $d$-colored set of points in general position in $\mathbb{R}^d$ can be partitioned into $n$ subsets with disjoint convex hulls such that the set of points and all color classes are partitioned as evenly as possible. This extends results by Holmsen, Kyn\v{c}l & Valculescu (2017) and establishes a special case of their general conjecture. Our proof utilizes a result obtained independently by Sober\'on and by Karasev in 2010, on simultaneous equipartitions of $d$ continuous measures in $\mathbb{R}^d$ by $n$ convex regions. This gives a convex partition of $\mathbb{R}^d$ with the desired properties, except that points may lie on the boundaries of the regions. In order to resolve the ambiguous assignment of these points, we set up a network flow problem. The equipartition of the continuous measures gives a fractional flow. The existence of an integer flow then yields the desired partition of the point set., Comment: 8 pages, 1 figure

B\'ar\'any, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex to d-dimensional Euclidean space, if the matroid has sufficiently many disjoint bases. They make a conjecture on the connectivity of k-fold deleted joins of a matroid with many disjoint bases, which would yield a much tighter result - but we provide a counterexample already for the case of k=2, where a tight Tverberg-type theorem would be a topological Radon theorem for matroids. Nevertheless, we prove the topological Radon theorem for the counterexample family of matroids by an index calculation, despite the failure of the connectivity-based approach., Comment: 16 pages, 4 figures

We describe a regular cell complex model for the configuration space F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter., Comment: Revised and improved version with extra explanations, 20 pages, 7 figures, to appear in Israel J. Math

In this paper we compute: the Schwarz genus of the Stiefel manifold V k ( R n ) with respect to the action of the Weyl group W k : = ( Z / 2 ) k ⋊ S k , and the Lusternik–Schnirelmann category of the quotient space V k ( R n ) / W k . Furthermore, these results are used in estimating the number of critically outscribed parallelotopes around a strictly convex body, and Birkhoff–James orthogonal bases of a normed finite dimensional vector space.

In this paper we give the complete answer to a question posed by A. Arhangelʼskii and prove that the sphere S n is diagonal resolvable if and only if S n is an H-space if and only if n ∈ { 0 , 1 , 3 , 7 } . Moreover, we prove that any upper half even dimensional Q -sphere cannot be diagonal resolvable.