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Physical geodesy applies potential theory to study the Earth’s gravitational field in space outside and up to a few km inside the Earth’s mass. Among various tools offered by this theory, boundary-value problems are particularly popular for the transformation or continuation of gravitational field parameters across space. Traditional problems, formulated and solved as early as in the nineteenth century, have been gradually supplemented with new problems, as new observational methods and data are available. In most cases, the emphasis is on formulating a functional relationship involving two functions in 3-D space; the values of one function are searched but unobservable; the values of the other function are observable but with errors. Such mathematical models (observation equations) are referred to as deterministic. Since observed data burdened with observational errors are used for their solutions, the relevant stochastic models must be formulated to provide uncertainties of the estimated parameters against which their quality can be evaluated. This article discusses the boundary-value problems of potential theory formulated for gravitational data currently or in the foreseeable future used by physical geodesy. Their solutions in the form of integral formulas and integral equations are reviewed, practical estimators applicable to numerical solutions of the deterministic models are formulated, and their related stochastic models are introduced. Deterministic and stochastic models represent a complete solution to problems in physical geodesy providing estimates of unknown parameters and their error variances (mean squared errors). On the other hand, analyses of error covariances can reveal problems related to the observed data and/or the design of the mathematical models. Numerical experiments demonstrate the applicability of stochastic models in practice. [ABSTRACT FROM AUTHOR]

Relict vegetation of water-splashed petrifying rocks with calcareous tufa formation dominated by the fern Adiantum capillus-veneris and wetland bryophytes (phytosociological class Adiantetea) was investigated in Georgia, Caucasus biodiversity hotspot. The study brings the first phytosociological data on this scarce community in the Caucasus based on a novel dataset of vegetation plot records. A classification analysis revealed two main vegetation communities. They are represented by the Caucasian community recorded in the Mtkvari River valley inside the Tbilisi city (E Georgia) and the Colchic community developed under the humid, warm-temperate climate of Colchis (W Georgia), one of the key refugia of Tertiary flora across W Eurasia. To compare newly recorded vegetation types and so far described associations of the Adiantetea class, we used a broader dataset of the analogous communities reported from Europe and surroundings. It associated the Caucasian community with the pan-Mediterranean association Eucladio-Adiantetum. It was characterized by subhalophytes (e.g. Samolus valerandi), Eucladium verticillatum as a dominant bryophyte, and the absence of relict or endemic vascular plant species. The Colchic community represented a previously undescribed community. Therefore, we designated a new association Saxifrago cymbalariae-Adiantetum capilli-veneris delineated by the characteristic species of the Colchic-Caucasian territory (e.g. Hedera colchica, Saxifraga cymbalaria) and Palustriella commutata as the dominant bryophyte. Both associations belong to the Mediterranean-Atlantic alliance Adiantion. Described communities require conservation attention for their rarity, refugial character and presence of relict and endemic species. Modifications of the hydrological regime and construction activities are among their most important potential threats. [ABSTRACT FROM AUTHOR]

Physical heights is one of the most important topics in physical geodesy. Their original concept, introduced in the 19-th century, defined physical heights as lengths of plumblines of the Earth's gravity field between the geoid and points of interest. There are orthometric heights of surface points, that have been traditionally estimated by spirit levelling and measured gravity; however, the knowledge of the density distribution of topographic masses (masses between the geoid and Earth's surface) is required that significantly affects their determinability. This was also the main reason why a new type of physical heights was proposed in the mid of the 20-th century. Normal heights approximate orthometric heights in a sense that the Earth's gravity field is replaced by the normal gravity field, an analytic model based on the theory of an equipotential ellipsoid. This height system has been introduced since that time in different countries in Europe and beyond. Contrary to the classical height system based on orthometric heights, its counterpart based on normal heights may have slightly different definitions. Moreover, normal heights are often defined as heights of points above the quasigeoid. This contribution reviews alternative definitions of normal heights and respective height systems. It is argued that both orthometric and normal heights refer to the geoid. In the case physical heights are estimated by satellite positioning, normal heights must be computed through the height anomaly estimated at each point of interest, whether it is below, at or above the Earth's surface. On the contrary, orthometric heights of all points along the same plumbline, be it below, at or above the Earth's surface, are estimated by introducing one value of the geoid height. Normal heights of surface points can be estimated by spirit levelling easier than orthometric heights as no topographic mass density hypothesis is required; however, one has to keep in mind the gravity field approximation used both for their definition and realization. [ABSTRACT FROM AUTHOR]

Composites consisting of iron aluminide and iron silicide phases were studied in this work. Powders of iron aluminide and iron silicide were prepared by mechanical alloying separately. Subsequently, they were blended in three different proportions and sintered by the SPS method under various conditions. After sintering, the composites are composed of FeAl and amounts of other silicides (Fe5Si3 and Fe3Si). Ternary Fe–Al–Si phases were not determined, even though their presence was predicted by DFT calculations. This disagreement was explained by steric factors, i.e., by differences in the space lattice of the present phases. Hardness and tribological properties were measured on composites with various weight ratios of iron aluminide and iron silicide. The results show that sintered silicides with the matrix composed of iron aluminide reach comparable hardness to tool steels. The composites with higher mass ratios of iron aluminide than silicide have higher hardness and better tribological properties. [ABSTRACT FROM AUTHOR]

Niobium is recently considered one of the potential alloying elements for tool steels due to the formation of hard and stable carbides of MC type. Its use is limited by the fact that these carbides tend to coarsen during conventional melting metallurgy processing. This work explores the potential of additive manufacturing for processing Nb-alloyed tool steel with a high content of carbon. Directed energy deposition was used as the processing method. It was found that this method allowed us to obtain a microstructure very similar to that obtained after the use of consolidation via spark plasma sintering when subsequent heat treatment by soft annealing, austenitizing, oil quenching and triple tempering for secondary hardness was applied. Moreover, the soft annealing process could be skipped without affecting the structure and properties when machining would not be required. The hardness of the steel was even higher after additive manufacturing was used (approx. 800–830 HV 30) than after spark plasma sintering (approx. 720–750 HV 30). The wear resistance of the materials processed by both routes was almost comparable, reaching 5–7 × 10−6 mm3N−1m−1 depending on the heat treatment. [ABSTRACT FROM AUTHOR]

The aim of this study was to evaluate the effect of the housing system on the behaviour, performance, and health of dairy calves. The two-year study was conducted on 56 Holstein bull calves (from birth to 60 days of age) that were placed in individual hutches immediately after their birth. At the age of 31 days, 28 bull calves were moved to group hutches by four. The calves were weighed at birth, at 30 days, and at 60 days of age. The starter intake and health were recorded once a day. The behavioural activities of calves (lying, standing and cross-sucking, etc.) were analysed in 24-h cycles at 38 days and 53 days of age. The results showed that grouphoused calves spent less time lying and resting (P < 0.01) and more time standing (P < 0.01), receiving the starter (P < 0.05), drinking (P < 0.01), cross-sucking (P < 0.01), and social playing (P < 0.01) compared to individually housed calves. Conversely, calves from individually housing spent more time licking the housing surfaces (P < 0.05) and individual playing (P < 0.05). Bull calves housed from day 31 of age in the group hutches achieved a non-significantly higher body weight at weaning (by +3.0 kg), weight gain (by +2.8 kg), and a significantly higher (P < 0.05) starter intake (by +7.9 kg) compared to the individually housed calves. These results showed that housing in small groups allowed for the natural behaviour of calves and improved their growth without having adverse effects on their health. [ABSTRACT FROM AUTHOR]

The oxidative addition of CF3SO3CH2Si(CH3)3 (NpSiOTf) toward organopnictogen(I) N,C,N-pincer compounds, i.e. [2,6-(DippN＝CH)2C6H3]E (1-E, where E = Sb, Bi; Dipp = 2,6-iPr2C6H3) produced compounds [2,6-(DippN＝CH)2C6H3]E(NpSi)(OTf) (2-E, where E = Sb, Bi). By analogy, the in situ reduction of [2,6-(Me2NCH2)2C6H3]ECl2 (3-E, where E = Sb, Bi) followed by treatment with NpSiOTf or MeI gave compounds [2,6-(Me2NCH2)2C6H3]E(R)(X) (R/X = NpSi/OTf 4-E, where E = Sb, Bi; R/X = Me/I 5-Sb). The reactivity of these compounds toward 1 eq. of K[BEt3H] was examined showing remarkable differences depending both on the ligand backbone and the pnictogen used. Thus in the case of 2-E, the addition of the hydride across the imino-function was achieved thereby yielding azapnicta-heterocyclic compounds [2-(DippNCH2)-6-(DippN＝CH)C6H3]E(NpSi) (6-E, where E = Sb, Bi). The same reaction of 4-Bi produced dibismuthine {[2,6-(Me2NCH2)2C6H3]Bi(NpSi)}2 (7-Bi), but in the case of 4/5-Sb the analogous distibines {[2,6-(Me2NCH2)2C6H3]Sb(R)}2 (R = NpSi7-Sb, Me 8-Sb) were not formed directly and hydrides [2,6-(Me2NCH2)2C6H3]Sb(R)H (R = NpSi9-Sb, Me 10-Sb) could be isolated instead. Nevertheless, heating of both 9-Sb and 10-Sb led to an activation of a labile Sb–H bond and the formation of distibines 7/8-Sb. [ABSTRACT FROM AUTHOR]

The Caucasus Region belongs among the most remarkable biodiversity hotspots globally. Local mountain floodplain forests represent highly endangered ecosystems and possess the status of protective forests. However, they have not been extensively phytosociologically assessed using Braun-Blanquet methods to date. Here, we present a novel dataset of vegetation plots recorded in two remarkable areas of forest plant diversity in this region. They were Enguri (Black Sea watershed) and Aragvi (Caspian Sea watershed) River basins in the Western and Eastern Greater Caucasus, respectively. Based on 49 relevés, sampled on a broad elevation gradient (980-1830 m), we distinguished three vegetation types using unsupervised classification. Main gradients in their species composition reflect biogeographical and climatic differences. Alnus incana dominated two types. The Enguri type (Brunnero macrophyllae-Alnetum incanae) was characterized by numerous species typical of the western part of the Greater Caucasus, including tall forbs and mountain species. The Aragvi type (Veronico filiformis-Alnetum incanae) showed a diagnostic combination of common Caucasian species and relatively thermophilous forest species. The subdivision of this unit included three subtypes according to grazing intensity. The last type (Sedo stoloniferi-Alnetum barbatae) was significant by a putative Tertiary relict Alnus glutinosa subsp. barbata as a canopy dominant. It occupied lower elevations of the Enguri Region, significantly influenced by the nearby Colchic Region. To provide a regional context of the sampled vegetation, we compiled an expanded dataset of alluvial alder forests from the Caucasus and its surroundings. Their joint classification highlighted the uniqueness of the recorded communities with A. incana and clearly distinguished them from the Euxinian alliance Alnion barbatae delimited by relict species typical of the Colchic refugium (e.g. Diospyros lotus, Pterocarya fraxinifolia). A comparison of the Caucasian A. incana forests with the alluvial forests of the boreonemoral alliance Alnion incanae supported the individuality of the Caucasian stands in terms of floristic composition and allowed us to describe them as a new alliance (Veronico filiformis-Alnion incanae all. nova). It is characterized by the dominance of A. incana accompanied by numerous geographically restricted species (e.g. Senecio propinquus, Symphytum asperum) coupled with boreo-montane forest species (e.g. Oxalis acetosella, Polygonatum verticillatum). Caucasian alluvial forests face many ongoing threats, including constructing of new water reservoirs and hydropower plants, overgrazing and illegal cutting. [ABSTRACT FROM AUTHOR]

Aims: Oak–hornbeam forests and related vegetation types (phytosociological order Carpinetalia betuli) are widespread in temperate western Eurasia. However, their national classification systems are poorly compatible, and a broad‐scale classification based on numerical analyses is lacking. Therefore, we aimed to establish a unified formalized classification system based on a large data set of vegetation plots covering the entire range of these forests. Location: Europe, Anatolia, Caucasus and northern Iran. Methods: We compiled a data set of 15,817 vegetation plots from the European Vegetation Archive and the Hyrcanian Forest Vegetation Database, using the formal definition of the EUNIS habitat type T1E Carpinus and Quercus mesic deciduous forest. We classified the data set using TWINSPAN. Biogeographically and ecologically similar plot clusters were merged into oak–hornbeam forest types, which were interpreted as alliances. We also developed expert systems for automatically classifying vegetation at the alliance level for both the EuroVegChecklist (EVC) system and the revised classification. In addition, we calculated ordinations to show the major gradients in the species composition of the data set. Results: We present a revised classification system of the order Carpinetalia betuli with nine alliances, including basic descriptions of their species composition, distribution, ecology and syntaxonomy. The analyses largely supported the biogeographic concept of classification, analogous to EVC. Compared to EVC, we recognized an additional alliance Physospermo verticillati‐Quercion cerridis (southern Italy) but found no support for the alliances Astrantio‐Carpinion, Erythronio‐Carpinion and Scillo‐Quercion. The greatest difference in species composition was found between the southern and northern‐northeastern Carpinetalia types. Expert systems for the revised classification system (~89% of plots classified) and the EVC system (~72%) are also included. Conclusions: We provide the first comprehensive overview of alliances of the order Carpinetalia betuli across its whole distribution range. The associated expert systems allow consistent application of the classification of these forests in nature conservation, habitat monitoring, and biodiversity and ecological research.We provide the first formal classification for oak–hornbeam forests and related vegetation types (phytosociological order Carpinetalia betuli) across Europe and adjacent areas. It includes both traditional and revised classifications of this vegetation. Furthermore, we developed expert systems for vegetation‐plot assignment to vegetation types, which can be applied in research and biodiversity conservation, monitoring and management. [ABSTRACT FROM AUTHOR]

Aluminothermic reduction without the separation of individual metals is currently considered as a possible method for processing ferromanganese sea nodules and creating new alloys. In this study, the product of their reduction—a manganese-based polymetallic mixture—was added to pure aluminum, as a mixture of alloying elements in their natural ratios. After extrusion, two new aluminum alloys with a total percentage of metallic additives ranging from 1 to 6 percent were prepared. The possibilities of the precipitation strengthening of these aluminum alloys, especially those containing Mn, Fe, Si, Ni, and Cu, were investigated under a wide range of heat treatment conditions. After each tested combination of annealing and artificial aging temperatures, the phase composition and the microstructure changes were recorded by X-ray diffraction, optical, and scanning electron microscopy with EDS analysis. Under none of the tested heat treatment conditions is a significant hardening effect observed, even though the precipitate phases are observed by TEM. However, the changes in the morphology of the present intermetallic phases caused by the heat treatment are revealed, which highlights the further possible development of these multicomponent alloys. [ABSTRACT FROM AUTHOR]

This work investigated the possibility of the direct preparation of aluminum alloys by aluminothermic reduction of deep-sea nodules with a high excess of aluminum. The process was found to be unable to obtain aluminum alloy, but an aluminum-rich manganese-based alloy was obtained instead, being composed of intermetallics. The alloy was characterized in the as-reduced state, as well as after crushing and sintering in the temperature range of 800–950 °C. The sample sintered at 900 °C was also heat-treated by annealing at 800 °C for 3 h and rapidly cooled. It was observed that with the increasing sintering temperature, the original matrix phase Al11Mn14 was transformed into a duplex matrix with a structure corresponding to Al11Mn14 and Al4Cu9, and this mixture was further transformed to the matrix with the structure corresponding to Al4Cu9. Furthermore, the mechanical properties and wear resistance of the samples were described. The highest microhardness was reached in the sample, which was annealed after sintering. Sintered samples reached a lower wear rate because of the fragmentation of brittle intermetallics during crushing. [ABSTRACT FROM AUTHOR]

Ferromanganese nodules represent an industrially untapped source of valuable metals (Mn, Cu, Ni, Ti, Co) but are still just the object of scientific research. In comparison with already utilized rocks, deep-sea deposits are too hard to reach for their full-scale mining to be profitable. Despite of this, current marine engineering allows us to conduct complete exploration and obtain mineral samples. We can already now develop low-cost methods for subsequent processing in anticipation of improved and safer future mining technology. But in what form would be the final product the most applicable? In our study, we reduced the pyro metallurgical pretreatment of raw nodules to two simple steps: roasting and aluminothermic reduction. Then we added a product, manganese-based alloy, to pure aluminum as a new alloying additive. As a reference for experimental alloys, commercial cast AlMn10 master alloy was chosen. The tested samples were prepared using two advanced operations of powder metallurgy: melt-spinning and spark plasma sintering (SPS). Microstructure was described in detail, tribological and selected mechanical properties were measured, and a comparative analysis was conducted. New additive gradually improved properties of alloys to a level close to 3xxx industrial alloys with its increasing percentage in pure aluminum. Measured values did not surpass the referent AlMn10, but there was not recorded any adverse impact associated with the origin of alloying mixture. • New aluminum alloys with the natural additive – reduced deep-sea nodules – modified by rapid solidification. • Extremely fine-grained microstructure with the content of supersaturated matrix regions. • Improved mechanical and tribological properties comparable to commercial alloys of 3xxx group. • Uncommonly high plasticity. [ABSTRACT FROM AUTHOR]

The Caucasus harbours unique forest vegetation so far only little studied using the Braun-Blanquet approach. This study is mostly based on a dataset (N = 110) of original phytosociological relevés of oak-hornbeam and ravine forests in the Eastern Greater Caucasus, Georgia. Their unsupervised classification produced seven communities. Five belong to oak-hornbeam forests (order Lathyro-Carpinetalia caucasicae). Of the zonal Caucasian alliance Crataego-Carpinion, the association Corno australis-Carpinetum inhabits valleys of the Greater Caucasus, and Clinopodio umbrosi-Carpinetum is confined to the warm Eastern Greater Caucasus promontories. The association Astrantio maximae-Carpinetum of the alliance Astrantio-Carpinion represents distinctive Caucasian montaneoak-hornbeam forests. The other two communities, documented by a few relevés, were described at the community level only. Within ravine forests (order Aceretalia pseudoplatani), we introduce a new Caucasian alliance Pachyphragmo macrophyllae-Tilion begoniifoliae with two associations. Valeriano tiliifoliae-Ulmetum glabrae comprises Caucasian montane ravine forests whereas Hedero pastuchovii-Aceretum velutini inhabits the foothills of the Eastern Greater Caucasian. To provide a broader context of the recognized communities, an expanded dataset (N = 231) of original relevés and previously published relevés of Georgian deciduous forests was analysed. It indicated a major turnover in species composition following biogeographical patterns presumably driven by macroclimate and vegetation history. [ABSTRACT FROM AUTHOR]

In geophysical and geodetic studies, gravity inversion is typically performed such that observed gravity values are first continued downward onto a regular (planar, spherical or spheroidal) surface by solving an inverse integral transform, which originates from a classical solution to the first boundary-value problem in potential theory. A typical example is continuing gravity observed at the topographic surface down to the mean sea level (geoid). Nowadays, gravity-dedicated satellite missions and aerial gravimetry provide gravity data above the topographic surface in addition to classical terrestrial gravity observations. For specific purposes (such as gravity data combination and validation, or quasigeoid determination), satellite and aerial gravity observations have to be continued to the irregular topographic surface. In this study, we address this issue by formulating a functional model for a spectral downward continuation of selected gravitational field quantities to an irregular topographic surface. Moreover, we generalize this functional model to allow for transformation between different types of gravitational field quantities. In particular, we derive spectral weights for estimation of the disturbing potential or disturbing/anomalous gravity at the Earth's surface by combining the first-, second- and third-order radial gradients of the disturbing potential (disturbing gradients). The correctness of the developed combined spectral estimator is verified in a closed-loop test based on synthetic satellite disturbing gradients. The combined spectral estimator is applied to simulated satellite disturbing gradients polluted by a realistic Gaussian noise. Results of the numerical experiments show that the combined spectral estimator puts the highest importance on the least polluted disturbing gradient, while the contribution of the least accurate disturbing gradient is negligible. An important advantage of this spectral combination method is that no matrix inversion with numerical instabilities requiring regularization is needed. [ABSTRACT FROM AUTHOR]

The Heat Treatment of Metallic Materials in Modern Industry is a Special Issue of the journal I Materials i , which aims to publish original full-length articles and review papers on basic and applied research centered around the given topic, and thereby make the understanding of the metallurgical background of the contemporary state of heat treatment techniques used in the industrial branches in the 21st century. Finally, the papers on laser- or electron-beam surface treatments as well as the physical vapor deposition of thin films are of interest to the Editors of this Special Issue. The papers which explore the research and development works on bulk heat and cryogenic treatment of steel, aluminium, titanium, magnesium, or copper alloys are particularly welcome. [Extracted from the article]

Aim: The aim of the paper is to describe species composition, ecology and syntaxonomy of unique mesophilous forests in Georgian Colchis in the context of the Euxinian forests. Location: SW Georgia, western part of the Caucasus Region, slopes of the Lesser Caucasus (80-990 m). Methods: Forest vegetation was sampled following the Braun-Blanquet approach and basic environmental characteristics were recorded for each relevé (n = 53). The dataset was clustered using modified TWINSPAN and diagnostic species for each cluster were defined by phi fidelity index. Links between the species composition of vegetation and environmental factors were analysed by detrended correspondence analysis. Numerical comparison of the distinguished clusters with previously published associations from northern Turkey was carried out (expanded dataset, n = 173). Results: Among the new relevés, we distinguished three vegetation types subsequently described as new associations: (i) Digitali schischkinii-Carpinetum betuli, chestnut-hornbeam forests of slightly dry sites, with low cover of evergreen shrubs, (ii) Rusco colchici-Castaneetum sativae, chestnut-hornbeam forests with a dense evergreen shrub layer, and (iii) Polysticho woronowii-Ulmetum glabrae, ravine forests with noble hardwood trees and ravine forest specialists. These associations were classified to the alliances of Euxinian forests, the first two to Castaneo-Carpinion and the third to Alnion barbatae. Major turnover in species composition within the dataset followed gradients of rockiness, slope inclination, altitude and mean annual temperature. The analysis of the expanded dataset revealed a decreasing occurrence of Mediterranean species and an increasing number of fern species along an easterly gradient. This change in species composition is likely to be driven by macroclimatic gradients. Conclusions: This paper presents the first numerical comparison of chestnut-hornbeam and ravine forests across central and eastern parts of the Euxinian Province. Our results have highlighted the uniqueness of Georgian Colchic forests, described as new associations, in the context of Euxinian forest vegetation. [ABSTRACT FROM AUTHOR]

Suitably protected (R)‐ and (S)‐HPEP {9‐[3‐hydroxy‐2‐(phosphonoethoxy)propyl]} derivatives containing adenine and thymine nucleobases were prepared as monomers for the introduction of acyclic nucleotide units into oligonucleotides. The solid phase synthesis of a series of ribo and 2′‐deoxyribo nonamers containing (R)‐HPEP and (S)‐HPEP units was successfully performed in both 5′ → 3′ and 3′ → 5′ directions. The measurement of thermal characteristics of the complexes of modified nonamers with complementary DNA and RNA strands revealed a general destabilizing effect of the introduced acyclic units on the thermal stability, but in several cases [modif.RNA*RNA and modif.DNA*RNA with (S)‐HPEPA units], the duplexes were characterized in terms of Tm values. A solid‐phase synthesis afforded a series of ribo and 2′‐deoxyribo nonamers containing acyclic nucleotide units. A general destabilizing effect of the introduced acyclic units on the thermal stability of complexes of modified nonamers with the complementary DNA and RNA strands was observed, but in cases with (S)‐HPEPA units, the duplexes were characterized in terms of Tm values. [ABSTRACT FROM AUTHOR]

Galium divaricatum Pourr. ex Lam. is an annual plant species occurring mainly in dry grasslands. The centre of its distribution range is situated in the Mediterranean region, while in Central Europe it has been known only from a few isolated sites in Slovakia and Hungary. In 2016 we discovered this species in open dry grassland of the class Sedo-Scleranthetea on the upper edge of a basaltic andesite quarry near the village of Siltse, Irshavskyi District, Zakarpatska Region, as a new species for the Ukrainian flora. The site is situated in the transitional region between the Pannonian Basin and the Eastern Carpathians. The new locality represents the northernmost recent occurrence of this species. [ABSTRACT FROM AUTHOR]

The article focuses on the conversion of the names Galio-Carpinetum Oberdorfer 1957, Lithospermo-Carpinetum Oberdorfer 1957, and Stellario-Carpinetum Oberdorfer 1957. It mentions about mesophytic oak-hornbeam forests of the western and central parts of Central Europe, and thermophytic oak-hornbeam forests of southwestern Central Europe and the adjacent parts of France.

A nanocomposite of magnetic hydroxyapatite was synthesized and tested as an adsorbent for the removal of copper (Cu (II)) and nickel (Ni(II)) from aqueous solution. The adsorbent was investigated using Transmission Electron Microscopy (TEM), Scanning Electron Microscopy equipped with an Energy Dispersive Spectrometer (SEM/EDS), X-ray powder diffraction (XRD) and the Brunauer–Elmet–Teller nitrogen adsorption technique (BET-N 2 adsorption). Batch experiments were carried out to determine and compare the adsorption parameters of Fe 3 O 4 and its composite with hydroxyapatite. It was found that the adsorbent is nanostructured and has a specific surface area of 101.2 m 2  g −1 . The Langmuir adsorption isotherm was found to be an appropriate model to describe the adsorption processes, showing the adsorption capacities of Cu(II) and Ni(II) of 48.78 mg g −1 and 29.07 mg g −1 , respectively. In addition to the high adsorption capacity, the fully-adsorbed material could be easily separated from aqueous media using an external magnetic field. These results suggested that the utilization of new hydroxyapatite – Fe 3 O 4 nanocomposite for the removal of Cu(II) and Ni(II) is a promising method in water technology. [ABSTRACT FROM AUTHOR]

The static Earth’s gravitational field has traditionally been described in geodesy and geophysics by the gravitational potential (geopotential for short), a scalar function of 3-D position. Although not directly observable, geopotential functionals such as its first- and second-order gradients are routinely measured by ground, airborne and/or satellite sensors. In geodesy, these observables are often used for recovery of the static geopotential at some simple reference surface approximating the actual Earth’s surface. A generalized mathematical model is represented by a surface integral equation which originates in solving Dirichlet’s boundary-value problem of the potential theory defined for the harmonic geopotential, spheroidal boundary and globally distributed gradient data. The mathematical model can be used for combining various geopotential gradients without necessity of their re-sampling or prior continuation in space. The model extends the apparatus of integral equations which results from solving boundary-value problems of the potential theory to all geopotential gradients observed by current ground, airborne and satellite sensors. Differences between spherical and spheroidal formulations of integral kernel functions of Green’s kind are investigated. Estimated differences reach relative values at the level of 3% which demonstrates the significance of spheroidal approximation for flattened bodies such as the Earth. The observation model can be used for combined inversion of currently available geopotential gradients while exploring their spectral and stochastic characteristics. The model would be even more relevant to gravitational field modelling of other bodies in space with more pronounced spheroidal geometry than that of the Earth. [ABSTRACT FROM AUTHOR]

The paper provides information about a finding of a remote population of the Pteris cretica fern which occurs mainly in subtropical regions of the World. The species was discovered in 2016 in an old well of the Letovice Castle (W Moravia, Czech Republic). Since this fern has never been recorded in the Czech Republic before, it is the first record for the national flora. Only a single specimen was discovered growing in the gap in the lining of the well, at a depth ca 3 m. The plant probably belongs to the var. albolineata which is often cultivated as a houseplant; this is a probable source of spores, and an explanation for this new occurrence. Since this taxon was recorded in similar habitats in Germany and Belgium, details related to this new occurrence can be compared. Several other fern species were observed in the well including the critically endangered Asplenium scolopendrium. [ABSTRACT FROM AUTHOR]

In this work, the structure and mechanical properties of innovative Al-Cu-Fe based alloys were studied. We focused on preparation and characterization of rapidly solidified and hot extruded Al-Cu-Fe, Al-Cu-Fe-Ni and Al-Cu-Fe-Cr alloys. The content of transition metals affects mechanical properties and structure. For this reason, microstructure, phase composition, hardness and thermal stability have been investigated in this study. The results showed exceptional thermal stability of these alloys and very good values of mechanical properties. Alloying by chromium ensured the highest thermal stability, while nickel addition refined the structure of the consolidated alloy. High thermal stability of all tested alloys was described in context with the transformation of the quasicrystalline phases to other types of intermetallics. [ABSTRACT FROM AUTHOR]

New materials with appropriate mechanical properties and an antibacterial effect are constantly being sought for orthopedic and dental applications. The aim of this study was to investigate newly developed TiSi alloys coated with titania sol-gel containing silver. Titanium alloys with 5 or 10 wt% of silicon were prepared by vacuum arc remelting and dip-coated with titania sol containing either AgNO 3 or Ag 3 PO 4 in two concentrations. The size and distribution of the particles in the layer were evaluated, as well as layer compactness (SEM). The antibacterial effect (against E. coli and S. epidermidis ) and cytotoxicity (towards L929 and U-2 OS cell lines) of these materials were then tested. Despite cracking of the coatings after firing, the coatings demonstrated very good antibacterial effects against both E. coli and S. epidermidis after 24 h of interaction. None of the tested materials were toxic to both cell lines. Collectively, our results suggest that these materials are promising candidates for orthopedic applications. [ABSTRACT FROM AUTHOR]

The downward continuation (DWC) of the gravity anomalies from the Earth's surface to the geoid is still probably the most problematic step in the precise geoid determination. It is this step that motivates the quasi-geoid users to opt for Molodenskij's rather than Stokes's theory. The reason for this is that the DWC is perceived as suffering from two major flaws: first, a physically meaningful DWC technique requires the knowledge of the irregular topographical density; second, the Poisson DWC, which is the only physically meaningful technique we know, presents itself mathematically in the form of Fredholm integral equation of the 1st kind. As Fredholm integral equations are often numerically ill-conditioned, this makes some people believe that the DWC problem is physically ill-posed. According to a revered French mathematician Hadamard, the DWC problem is physically well-posed and as such gives always a finite and unique solution. The necessity of knowing the topographical density is, of course, a real problem but one that is being solved with an ever increasing accuracy; so sooner or later it will allow us to determine the geoid with the centimetre accuracy. [ABSTRACT FROM AUTHOR]

Ti–Al–Si alloys are prospective material for high-temperature applications. Due to low density, good mechanical properties, and oxidation resistance, these intermetallic alloys can be used in the aerospace and automobile industries. Ti–Al–Si alloys were prepared by powder metallurgy using reactive sintering, milling, and spark plasma sintering. One of the novel SPS techniques is high-pressure spark plasma sintering (HP SPS), which was tested in this work and applied to a Ti–10Al–20Si intermetallic alloy using a pressure of 6 GPa and temperatures ranging from 1318 K (1045 °C) to 1597 K (1324 °C). The low-porosity consolidated samples consist of Ti5Si3 silicides in an aluminide (TiAl) matrix. The hardness varied between 720 and 892 HV 5. [ABSTRACT FROM AUTHOR]

New spherical integral formulas between components of the second- and third-order gravitational tensors are formulated in this article. First, we review the nomenclature and basic properties of the second- and third-order gravitational tensors. Initial points of mathematical derivations, i.e., the second- and third-order differential operators defined in the spherical local North-oriented reference frame and the analytical solutions of the gradiometric boundary-value problem, are also summarized. Secondly, we apply the third-order differential operators to the analytical solutions of the gradiometric boundary-value problem which gives 30 new integral formulas transforming (1) vertical-vertical, (2) vertical-horizontal and (3) horizontal-horizontal second-order gravitational tensor components onto their third-order counterparts. Using spherical polar coordinates related sub-integral kernels can efficiently be decomposed into azimuthal and isotropic parts. Both spectral and closed forms of the isotropic kernels are provided and their limits are investigated. Thirdly, numerical experiments are performed to test the consistency of the new integral transforms and to investigate properties of the sub-integral kernels. The new mathematical apparatus is valid for any harmonic potential field and may be exploited, e.g., when gravitational/magnetic second- and third-order tensor components become available in the future. The new integral formulas also extend the well-known Meissl diagram and enrich the theoretical apparatus of geodesy. [ABSTRACT FROM AUTHOR]

Although the Štramberk Karst is one of the most important botanical sites of the northeastern Moravia, the local forest vegetation has been poorly studied so far. For this reason, we conducted field sampling focusing on mesophilous forest vegetation, especially on the scree- and rock-outcrop forests. We recorded 16 phytosociological relevés and basic environmental variables of the plots. We divided our relevés into three main vegetation types using numerical analysis: beech forests, rock-outcrop forests and ravine forests. Among them, we distinguished six associations, four of them belonging to the alliance Tilio platyphylli-Acerion (1 to 4) and two of them to the alliance Fagion sylvaticae (5 and 6): (1) Aceri-Tilietum, (2) Arunco dioici-Aceretum, (3) Mercuriali perennis-Fraxinetum, (4) Seslerio albicantis-Tilietum, (5) Galio odorati-Fagetum and (6) Mercuriali perennis-Fagetum. Occurrence of the rare rock-outcrop forest association Seslerio albicantis-Tilietum is remarkable, since several endangered and relict species occur here (e.g. Laserpitium latifolium, Saxifraga paniculata, Scabiosa lucida subsp. calcicola). This is the first record of this association in the north-eastern Moravia. [ABSTRACT FROM AUTHOR]

In this article boundary-value problems of potential theory, that are required for transforming and continuing selected parameters of a potential field, are discussed. By the potential field parameters we understand the potential and its gradients (potential gradients) up to the third order. In particular, integral equations defined for a spherical boundary transforming potential gradients of different orders and continuing potential gradients of the same order through the 3-D space are reviewed and classified. This mathematical apparatus can be used for any harmonic potential, such as electric, magnetic or gravitational, under the assumption of its conservativeness, i.e., neglecting possible temporal variations. Integral transforms are discussed in context of geoscience applications, namely in terms of the Earth's gravitational field; however, the article can serve as a general reference for integral transforms of potential field parameters in any scientific or engineering area where potential fields are used. [ABSTRACT FROM AUTHOR]

Due to the lack of mineral resources, unconventional raw materials have to be considered. In this work, the use of reduced deep-sea nodules from Clarion-Clippertion Fracture Zone of Pacific Ocean was tested as the natural mixture of alloying elements for aluminum alloys. The nodules were reduced by aluminothermic process without the extraction of individual elements. As the technology for the manufacture of the alloys, the powder metallurgy comprising mechanical alloying and spark plasma sintering was applied. The produced Al-Mn based alloys are characterized by uncommon spherical morphology of Al-Mn intermetallics, which leads to a significant ductility even in the case of high total amount of additives (up to 20 wt%). The wear resistance of the alloys strongly increases with growing amount of the reduced deep-sea nodules. • New aluminum alloys with reduced deep-sea nodules prepared by powder metallurgy. • Unique rounded morphology of intermetallics was obtained. • Improved strength compared with common Al-Mn alloys. • Reduced deep sea nodules improve tribological properties of aluminum. [ABSTRACT FROM AUTHOR]

One of the current problems concerning the re-use of aluminium alloys is the recycling of aluminium alloys with increased content of iron. One of the recent trends is the processing by powder metallurgy, producing completely new grades of aluminium alloys. The aim of the present work was to test the properties of Al–Si–Fe alloy, which can be manufactured from the high-irony scrap, as a possible material for elevated temperature applications. Since chromium and nickel accompany the iron in the scrap when stainless steels are admixed, the effect of chromium and nickel on thermal stability and mechanical properties at room and elevated temperature was also studied. Al–Si–Fe–X alloys were rapidly solidified by using melt spinning technique. The rapidly solidified ribbons were pulverized by planetary ball mill and the obtained powders were used for compaction by means of Spark Plasma Sintering. It was found that nickel improves the thermal stability as well as the mechanical properties at the elevated temperature. On the other hand, the effect of chromium on the mechanical properties at elevated temperature was almost detrimental. [ABSTRACT FROM AUTHOR]

Our previous works on the synthesis of intermetallics by Self-propagating High-temperature Synthesis revealed strong dependence of microstructure of the products on heating rate. In this paper, the application of various heating regimes and sources was tested in preparation of Ni Ti shape memory alloy. It was found that high heating rate (approx. over 100 °C min −1 ) was required to obtain the material with maximized amount of NiTi shape memory phase and no unreacted metals. Heating in electric resistance furnace preheated to the process temperature or induction heating furnace seemed to be promising for this purpose. On the other hand, Spark Plasma Sintering was found to be inapplicable, because the strongest increase of the temperature occurred on the surface of the particles, producing layers of intermetallics that further acted as diffusion barriers. [ABSTRACT FROM AUTHOR]

Values of scalar, vector and second-order tensor parameters of the Earth's gravitational field have been collected by various sensors in geodesy and geophysics. Such observables have been widely exploited in different parametrization methods for the gravitational field modelling. Moreover, theoretical aspects of these quantities have extensively been studied and well understood. On the other hand, new sensors for observing gravitational curvatures, i.e., components of the third-order gravitational tensor, are currently under development. As the gravitational curvatures represent new types of observables, their exploitation for modelling of the Earth's gravitational field is a subject of this study. Firstly, the gravitational curvature tensor is decomposed into six parts which are expanded in terms of third-order tensor spherical harmonics. Secondly, gravitational curvature boundary-value problems defined for four combinations of the gravitational curvatures are formulated and solved in spectral and spatial domains. Thirdly, properties of the corresponding sub-integral kernels are investigated. The presented mathematical formulations reveal some important properties of the gravitational curvatures and extend the so-called Meissl scheme, i.e., an important theoretical framework that relates various parameters of the Earth's gravitational field. [ABSTRACT FROM AUTHOR]

In this study we assume that a gravitational curvature tensor, i.e. a tensor of third-order directional derivatives of the Earth's gravitational potential, is observable at satellite altitudes. Such a tensor is composed of ten different components, i.e. gravitational curvatures, which may be combined into vertical-vertical-vertical, vertical-vertical-horizontal, vertical-horizontal-horizontal and horizontal-horizontal-horizontal gravitational curvatures. Firstly, we study spectral properties of the gravitational curvatures. Secondly, we derive new quadrature formulas for the spherical harmonic analysis of the four gravitational curvatures and provide their corresponding analytical error models. Thirdly, requirements for an instrument that would eventually observe gravitational curvatures by differential accelerometry are investigated. The results reveal that measuring third-order directional derivatives of the gravitational potential imposes very high requirements on the accuracy of deployed accelerometers which are beyond the limits of currently available sensors. For example, for orbital parameters and performance similar to those of the GOCE mission, observing third-order directional derivatives requires accelerometers with the noise level of $${\sim}10^{-17}\,\hbox {m}\,\hbox {s}^{-2}$$ Hz $$^{-1/2}$$ . [ABSTRACT FROM AUTHOR]

The valley of the Třebůvka River between the municipalities Městečko Trnávka and Loštice (Central Moravia, Czech Republic) has been poorly known in a phytosociological sense. Therefore, a field survey was led and vegetation of the site Babí skála near the village Jeřmaň is described in this study. The site is a steep rocky slope above the Třebůvka River built up by acidic Paleozoic shales; it is covered mainly by well-preserved deciduous forests. Dominant forest vegetation types are acidophilous oak forests, mesophilous (association Luzulo-Quercetum), semithermophilous (Viscario-Quercetum) and thermophilous (Sorbo torminalis-Quercetum) ones. On the slope basis, scree forests dominated by noble deciduous trees (Aceri-Tilietum) are growing. Primary shrub vegetation is represented by scrubs of Cotoneaster integerrimus that can be found on the tops of some rock outcrops (Junipero-Cotoneasteretum integerrimi). Among non-forest vegetation an acidophilous rock vegetation from the alliance Asplenion septentrionalis is the most important. The most remarkable species in the flora of the locality are light-demanding (Arabidopsis arenosa, Carex rhizina and Cotoneaster integerrimus) and thermophilous (e. g. Polygonatum odoratum and Vincetoxicum hirundinaria) ones. [ABSTRACT FROM AUTHOR]

This work deals with a problem of paraxial analysis of optical systems for transformation of Gaussian beams. General equations describing the paraxial parameters of a two-element zoom system for transformation of a Gaussian beam are derived. It is shown that such a zoom lens has different kinematics with respect to classical zoom system designed for homocentric beams, which depends on the parameters of the transformed Gaussian beam. [ABSTRACT FROM AUTHOR]

In this work, the formation of intermetallics in Ni–Ti system by reactive sintering at 500 and 650 °C was studied. Mechanism and kinetics of the reactions leading to Ni–Ti phases were determined by DTA and the application of an experimental model consisted of nickel-plated titanium. It was found that Ti 2 Ni phase forms already at 500 °C. At 650 °C this phase undergoes a reaction with nickel to NiTi and Ni 3 Ti. The reactions were determined to be diffusion-controlled. The formation of Ni 3 Ti contributes to the porosity evolution during reactive sintering production of NiTi alloy. [ABSTRACT FROM AUTHOR]

A new mathematical model for evaluation of the third-order (disturbing) gravitational tensor is formulated in this article. Firstly, we construct corresponding differential operators for the components of the third-order (disturbing) gravitational tensor in a spherical local north-oriented frame. We show that the differential operators may efficiently be decomposed into an azimuthal and an isotropic part. The differential operators are even more simplified for a certain class of isotropic kernels. Secondly, the differential operators are applied to the well-known integrals of Newton, Abel-Poisson, Pizzetti and Hotine. In this way, 40 new integral formulas are derived. The new integral formulas allow for evaluation of the components of the third-order (disturbing) gravitational tensor from density distribution, disturbing gravitational potential, gravity anomalies and gravity disturbances. Thirdly, we investigate the behaviour of the corresponding integral kernels in the spatial domain. The new mathematical formulas extend the theoretical apparatus of geodesy, i.e. the well-known Meissl scheme, and reveal important properties of the third-order gravitational tensor. They may be exploited in geophysical studies, continuation of gravitational field quantities and analysing the gradiometric-geodynamic boundary value problem. [ABSTRACT FROM AUTHOR]

Abstract: This work is devoted to the preparation of ultrafine material based on Al–Cu–Fe quasicrystalline phase by powder metallurgy using mechanical alloying and Spark Plasma Sintering. The dependence of microstructure and phase composition of powders on the conditions of mechanical alloying was described. It was found that the Al60Cu30Fe10 quasicrystalline phase forms directly already after 2 h of milling under optimized conditions. The stability of this quasicrystalline phase was studied at various temperatures of Spark Plasma Sintering compaction process. [Copyright &y& Elsevier]

Abstract: Our work is focused on the problem of theoretical evaluation of the surface shape of a variable focus membrane lens. In order to be able to determine the imaging properties of such a lens one has to know the shape of the membrane for a given pressure of the liquid inside the lens chamber with high accuracy. In our work a generalized nonlinear differential equation describing this problem is derived and the solution of this equation using expansion into series and transformation of the problem into constraint optimization problem is proposed. The proposed method enables us to calculate the shape of a membrane lens surface with satisfactory accuracy for further calculations such as e.g. calculation of aberrations of such a lens, imaging properties etc. [Copyright &y& Elsevier]

With the advent of geodetic satellite missions mapping almost globally the Earth's gravitational field, new methods and theoretical approaches have been developed and investigated to fully exploit the potential of their new observables. Besides estimating values of numerical coefficients in harmonic series of the gravitational potential, new applications emerged such as data validation and combination. In this contribution, new integral transformations are presented which transform principal components of the terrestrial deflection of the vertical onto disturbing satellite-to-satellite tracking and gradiometric data at altitude. Using spherical approximation, necessary integral kernel functions are derived in both spectral and closed forms. The behaviour of isotropic kernel functions is studied and the new integral transformations are tested in a closed-loop simulation using synthetic terrestrial and satellite data synthesized from a global gravitational model. New integral transformations can be used for data validation and combination purposes. [ABSTRACT FROM AUTHOR]

Integral transformations of gravitational gradients onto a Gravity Recovery And Climate Experiment (GRACE) type of observable are derived in this article. The gravitational gradients represent components of the gravitational tensor in the local north-oriented frame. The GRACE type of observable corresponds to a difference between two gravitational vectors as projected onto the line of sight between the two GRACE satellites. In total, three integral transformations relating vertical-vertical, vertical-horizontal and horizontal-horizontal gravitational gradients with the GRACE type of observable are provided. Spectral and closed forms of corresponding isotropic kernels are derived for each transformation. Special cases show that the integral transformations are general and relate gravitational gradients to many other quantities of the gravitational field, such as the gravitational vector, and its radial and tangential components. Correctness of the mathematical derivations is validated in a closed-loop simulation using synthetic data. [ABSTRACT FROM AUTHOR]

Abstract: This work deals with a problem of an effective and robust evaluation of Cartesian partial derivatives of Zernike polynomials. Calculations based on explicit relations suffer from high computational costs and are numerically instable for higher orders. Thus, fast and numerically stable methods for calculation of derivatives of Zernike polynomials are needed. Such methods employing recursive techniques based on the relation of Zernike radial polynomials to Jacobi polynomials are proposed. The performance of proposed algorithms is compared to the standard calculation using explicit relations. [Copyright &y& Elsevier]

This work aims to optimize the parameters of the SHS process for the preparation of NiTi shape memory alloy to obtain a high-purity, low-porosity material. For this purpose, high heating rate (over 300 °C min−1), temperature of 1100 °C and process duration of 20 min were determined as the optimal parameters. To prevent nickel depletion of the product and to minimize the porosity, the use of coarse titanium particles (200–600 μm) with fresh surface can be recommended. SHS products were composed of NiTi with the admixture of Ti2Ni phase. The formation of this phase cannot be fully avoided in SHS preparation route. [ABSTRACT FROM AUTHOR]

We investigate the gravity gradient components corrected for major known anomalous density structures within the Earth's crust. Heterogeneous mantle density structures are disregarded. The gravimetric forward modeling technique is utilized to compute the gravity gradients based on methods for a spherical harmonic analysis and synthesis of a gravity field. The Earth's gravity gradient components are generated using the global geopotential model GOCO-03s. The topographic and stripping gravity corrections due to the density contrasts of the ocean and ice are computed from the global topographic/bathymetric model DTM2006.0 (which also includes the ice-thickness dataset). The discrete data of sediments and crust layers taken from the CRUST2.0 global crustal model are then used to apply the additional stripping corrections for sediments and remaining anomalous crustal density structures. All computations are realized globally on a one arc-deg geographical grid at a mean satellite elevation of 255 km. The global map of the consolidated crust-stripped gravity gradients reveals distinctive features which are attributed to global tectonics, lithospheric plate configuration, lithosphere structure and mantle dynamics (e.g., glacial isostatic adjustment, mantle convection). The Moho signature, which is the most pronounced signal in these refined gravity gradients, is superimposed over a weaker gravity signal of the lithospheric mantle. An interpretational quality of the computed (refined) gravity gradient components is mainly limited by a low accuracy and resolution of the CRUST2.0 sediment and crustal layer data and unmodeled mantle structures. [ABSTRACT FROM AUTHOR]

Due to the ESA's satellite mission GOCE launched in March 2009, gravitational gradients sampled along the orbital trajectory approximately 250 km above the Earth's surface have become available. Since 2010, gravitational gradients have routinely been applied in geodesy for the derivation of global Earth's gravitational models provided in terms of fully normalized coefficients in a spherical harmonic series representation of the Earth's gravitational potential. However, in geophysics, gravitational gradients observed by spaceborne instruments have still been applied relatively seldom. This contribution describes their possible geophysical applications in structural studies where gravitational gradients observed at satellite altitudes are compared with those derived by a spectral forward modeling technique using available global models of selected Earth's mass components as input data. In particular, GOCE gravitational gradients are interpreted in terms of a superposition principle of gravitation as combined gravitational effects generated by a homogeneous reference ellipsoid of revolution, mean topographic and ice mass density distributions, depth-dependent mass density contrasts within bathymetry and lateral mass density anomalies with sediments and crustal layers. Respective gravitational effects are one by one removed from gravitational gradients observed at approximately 250 km elevation above ground. Removing respective gravitational gradients from observed gravitational gradients gradually reveals problematic geographic areas with model deficiencies. For the full interpretation of observed gravitational gradients, deficiencies of CRUST2.0 must be corrected and effects of deeper laying mass anomalies not included in the study considered. These findings are confirmed by parameters describing spectral properties of the gravitational gradients. The methodology can be applied for validating Earth's gravitational models and for constraining crustal models in the development phase. [ABSTRACT FROM AUTHOR]

Abstract: This work aims to describe the effect of alloying by transition metals (Co, Cr, Cu, Fe, Ni, Mo) on the structure and properties of TiAl15Si15 alloy prepared by SHS technique. Results show that alloying elements do not form their own phases in detectable amounts, being dissolved in titanium silicide or aluminide. Co, Cu and Ni were determined mainly in aluminide, while iron was found predominantly in the silicide phase. Chromium and molybdenum dissolve in both aluminide and silicide in almost comparable amounts. All applied alloying elements increase the wear resistance and the oxidation resistance at 1000 °C, but reduce the room-temperature mechanical properties. Molybdenum-alloyed material exhibits the best oxidation resistance, followed by chromium-containing alloy. These alloys are also characterized by exceptional thermal stability. [Copyright &y& Elsevier]

Abstract: The work deals with the problem of aspherical surfaces in optics. General relations for the calculation of wave aberration, ray aberrations and the influence of change of radius of the reference sphere on the wave aberration of a normal to aspherical surface are derived. Furthermore, general formulas are derived for the calculation of the direction of the normal to the aspherical surface in case that the directions of the incident and the reflected (or refracted) rays are known. These equations are of a high importance for the design and manufacturing process of aspherical surfaces, design and adjustment of compensation optics in interferometric measurements of aspherical surfaces, measurement of aspherical surfaces by deflectometry, etc. [Copyright &y& Elsevier]