Equilibrium of Surfaces in a Vertical Force Field

In this paper, we study $$\varphi $$ φ -minimal surfaces in $$\mathbb {R}^3$$ R 3 when the function $$\varphi $$ φ is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in $$\mathbb {R}^2$$ R 2 . We describe a full classification of complete flat-embedded $$\varphi $$ φ -minimal surfaces if $$\varphi $$ φ is strictly monotone and characterize rotational $$\varphi $$ φ -minimal surfaces by its behavior at infinity when $$\varphi $$ φ has a quadratic growth.

Subcategory of land as an important element of the conceptual apparatus of land law science

This article is devoted to the insufficiently studied in the theory of land law the concept of «subcategory of lands». The author established the fact that despite the widespread use of the phrase «subcategory of lands» by domestic and foreign scientists, there are no studies aimed at revealing this concept and outlining the same subcategories of lands. In the course of the research it was established that the current land legislation has about 119 subcategories of land, which are within different categories of land. Using the methods of theoretical and legal science, an attempt was made to reveal the concept of «subcategory of lands» and give it an appropriate definition. It is stated that subcategories of land play an important role in the proper functioning of current land legislation of our state, the lack of allocation and proper legal regulation of subcategories of land can lead to mass violations (intentional or negligent) in the use of land for its intended purpose. To effectively address the issue of proper legal regulation of land subcategories, the author analyzed the successful experience of the United States on this issue and proposed the development of a legal act that could perfectly regulate the relevant land categories, establish a full classification of major categories and subcategories. Lands, as well as provide them with a detailed description. In the final stage of the study, the author emphasizes that there is an urgent need to pay due attention to this issue by lawmakers, scholars and practitioners. Keywords: subcategory of lands, category of lands, division of lands into subcategories, purpose of lands, land legislation

Triharmonic Curves in 3-Dimensional Homogeneous Spaces

We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi–Cartan–Vranceanu spaces.

Duality for Outer $$L^p_\mu (\ell ^r)$$ Spaces and Relation to Tent Spaces

We study the outer $$L^p$$ L p spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for $$1< p \leqslant \infty , 1 \leqslant r < \infty $$ 1 < p ⩽ ∞ , 1 ⩽ r < ∞ or $$p=r \in \{ 1, \infty \}$$ p = r ∈ { 1 , ∞ } , the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces uniformly in the cardinality of the set. Moreover, for $$p=1, 1 < r \leqslant \infty $$ p = 1 , 1 < r ⩽ ∞ , we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range $$1 \leqslant p,r \leqslant \infty $$ 1 ⩽ p , r ⩽ ∞ . These results are obtained via greedy decompositions of functions in the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces. As a consequence, we establish the equivalence between the classical tent spaces $$T^p_r$$ T r p and the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $$\mathbb {R}^d$$ R d .

Classification of Vertex-Transitive Zonotopes

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $$\Gamma $$ Γ -permutahedron for some finite reflection group $$\Gamma \subset {{\,\mathrm{O}\,}}(\mathbb {R}^d)$$ Γ ⊂ O ( R d ) . The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length. The classification of these then follows from the classification of finite reflection groups. We prove that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set to be a root system.

Two-step solvable SKT shears

We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting a left-invariant SKT structure. We reduce this to a specification of SKT shear data on Abelian Lie algebras, and which then is studied more deeply in different cases. We obtain classifications and structure results for $$\mathfrak {g}$$ g almost Abelian, for derived algebra $$\mathfrak {g}'$$ g ′ of codimension 2 and not J-invariant, for $$\mathfrak {g}'$$ g ′ totally real, and for $$\mathfrak {g}'$$ g ′ of dimension at most 2. This leads to a large part of the full classification for two-step solvable SKT algebras of dimension six.

Wronskian indices and rational conformal field theories

The classification scheme for rational conformal field theories, given by the Mathur-Mukhi-Sen (MMS) program, identifies a rational conformal field theory by two numbers: (n, l). n is the number of characters of the rational conformal field theory. The characters form linearly independent solutions to a modular linear differential equation (which is also labelled by (n, l)); the Wronskian index l is a non-negative integer associated to the structure of zeroes of the Wronskian.In this paper, we compute the (n, l) values for three classes of well-known CFTs viz. the WZW CFTs, the Virasoro minimal models and the $$ \mathcal{N} $$ N = 1 super-Virasoro minimal models. For the latter two, we obtain exact formulae for the Wronskian indices. For WZW CFTs, we get exact formulae for small ranks (upto 2) and all levels and for all ranks and small levels (upto 2) and for the rest we compute using a computer program. We find that any WZW CFT at level 1 has a vanishing Wronskian index as does the $$ {\hat{\mathbf{A}}}_{\mathbf{1}} $$ A ̂ 1 CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with $$ {\hat{\mathbf{A}}}_{\mathbf{2}} $$ A ̂ 2 and $$ {\hat{\mathbf{G}}}_{\mathbf{2}} $$ G ̂ 2 have the same (n, l) values, (ii) for the same level CFTs with $$ {\hat{\mathbf{B}}}_{\mathbf{r}} $$ B ̂ r and $$ {\hat{\mathbf{D}}}_{\mathbf{r}} $$ D ̂ r have the same (n, l) values for all r ≥ 5.Classifying all rational conformal field theories for a given (n, l) is one of the aims of the MMS program. We can use our computations to provide partial classifications. For the famous (2, 0) case, our partial classification turns out to be the full classification (achieved by MMS three decades ago). For the (3, 0) case, our partial classification includes two infinite series of CFTs as well as fifteen “discrete” CFTs; except three all others have Kac-Moody symmetry.

Organic Petrographical Features of Fly Ashes Originating from Coal and Coal-SRF Co-Combustion

In this study, the features of fly ashes originating from industrial-scale high volatile bituminous coal combustion and co-combustion of coal with 10% admixture of alternative fuel SRF (solid recovered fuel) are presented, with emphasis on the organic petrographical characteristics. The organic petrographical and mineralogical data are co-evaluated with geochemical data, with the aim to provide a full classification of the studied fly ashes, as well as base information toward any potential application of this waste material, according to the recycling economy principles. By applying organic petrographical methods, the assignment of the carbon-rich residuals to the respective feed fuel, either coal or SRF, can be achieved. The obtained quantitative evaluation provides useful information regarding the combustion conditions in the stoker boiler. The analyzed fly ashes contain significant C-residuals, mostly in the form of fused, dense, and anisotropic particles, while the enrichment in sooty particles is caused due to the addition of SRF fuel. In conjunction with the moderate-low content of potential hazardous elements, the features of the contained C-residual phases suggest that these fly ashes could possibly be the subject of further studies for their applicability as soil improvements.

On the symmetries of three-dimensional generalized symmetric spaces

In this work we consider the three-dimensional generalized symmetric space, equipped with the left-invariant pseudo-Riemannian metric. We determine Killing vector fields and affine vectors fields. Also we obtain a full classification of Ricci, curvature and matter collineations

TECHNIQUES FOR FIXED DENTAL RESTORATIONS REMOVAL - CLASSIFICATION, DECISION ON THE CORRECT APPROACH, ADVANTAGES AND DISADVANTAGES

Introduction: Fixed dental restorations possess a predefined period of use. Most often they are removed by means of sectioning which renders them unusable. Reasons exist when practitioners shall preserve the restoration, applying conservative approaches for removing. In the literature, apart from the classic destructive technique with sectioning, conservative, semi-conservative and combined techniques for removal are described. The aim of this article is to present a complete classification and description of different approaches to remove permanently and temporarily fixed prosthetic constructions. Materials and methods: A literature study was conducted athe beginning of 2019. Information was gathered using dental textbooks on the topic specified and online scientific databases such as PubMed, ResearchGate, etc. Conclusion Through this article, a conclusion was drawn that a full classification or description of removal techniques for fixed dentures has not been done in the Bulgarian literature so far. It is reported that information for patients about advantages, disadvantages and dangers of removing a permanently cemented restoration is of great importance. Except for the destructive ones, all methods described here may be used when removing temporarily fixated constructions. The selection of the ideal system or a combination depends on the clinical situation. Safest and most atraumatic for underlying structures when removing permanently cemented restorations is the destructive approach. Practitioners must be precisely familiar with the techniques so as to be able to preserve the construction, to avoid any danger connected with the clinical case. According to data gathered, no approach is universally applicable when removing fixed prosthetic restorations.