C2 Space Research Articles

On the Analytic Continuation of Appell’s Hypergeometric Function F2 to Some Symmetric Domains in the Space C2

The paper considers the problem of representation and extension of Appell’s hypergeometric functions by a special family of functions—branched continued fractions. Here, we establish new symmetric domains of the analytical continuation of Appell’s hypergeometric function F2 with real and complex parameters, using their branched continued fraction expansions whose elements are polynomials in the space C2. To do this, we used a technique that extends the domain of convergence of the branched continued fraction, which is already known for a small domain, to a larger domain, as well as the PC method to prove that it is also the domain of analytical continuation. A few examples are provided at the end to illustrate this.

Reaction ignition during the oxidative coupling of methane over Mn–Na2WO4/SiO2

The present study investigates the oxidative coupling of methane (OCM) as a viable solution for CH4 upgrading with the benchmark Mn–Na2WO4/SiO2 catalyst. The effects of space velocity on catalytic performance and ignition responses were evaluated. Experimental catalytic tests revealed a complex relationship between space velocity and C2+ product yields and a self-igniting behavior when the reaction temperature reached about 700 °C. The results indicated that the temperature difference (∆T) between the catalyst bed and the flowing gas increased with the feed flow rate and decreased with dilution of the catalyst bed (increasing with catalyst loading), which can allow the reaction ignition at lower temperatures. The present study sheds light on the interplay between reaction conditions and the ignition behavior, providing valuable insights for optimization of OCM processes and catalyst development.

Crystal Structure Determination of Nucleotide-sugar Binding Domain of Human UDP-glucuronosyltransferases 2B10.

UDP-glucuronosyltransferases (UGTs) play a crucial role in maintaining endobiotic homeostasis and metabolizing xenobiotic compounds, particularly clinical drugs. However, the detailed catalytic mechanism of UGTs has not been fully elucidated due to the limited availability of reliable protein structures. Determining the catalytic domain of human UGTs has proven to be a significant challenge, primarily due to the difficulty in purifying and crystallizing the full-length protein. This study focused on the human UGT2B10 C-terminal cofactor binding domain, aiming to provide structural insights into the fundamental catalytic mechanisms. In this study, the C-terminal sugar-donor binding domain of human UGT2B10 was purified and crystallized using the vapor-diffusion method. The resulting UGT2B10 CTD crystals displayed high-quality diffraction patterns, allowing for data collection at an impressive resolution of 1.53 Å using synchrotron radiation. Subsequently, the structure of the UGT2B10 CTD was determined using the molecule replacement method with a homologous structure. The crystals were monoclinic, belonging to the space C2 with unit-cell parameters a = 85.90 Å, b = 58.39 Å, c = 68.87 Å, α = γ = 90°, and β = 98.138°. The Matthews coefficient VM was determined to be 2.24 Å3 Da-1 (solvent content 46.43%) with two molecules in the asymmetric unit. The crystal structure of UGT2B10 CTD was solved at a high resolution of 1.53 Å, revealing a conserved cofactor binding pocket. This is the first study determining the C-terminal cofactor binding domain of human UGT2B10, which plays a key role in additive drug metabolism.

Linear systems, Hankel products and the sinh-Gordon equation

Let (−A,B,C) be a linear system in continuous time t>0 with input and output space C2 and state space H. The scattering (or impulse response) functions ϕ(x)(t)=Ce−(t+2x)AB determines a Hankel integral operator Γϕ(x); if Γϕ(x) is trace class, then the Fredholm determinant τ(x)=det⁡(I+Γϕ(x)) determines the tau function of (−A,B,C). The paper establishes properties of algebras containing Rx=∫x∞e−tABCe−tAdt on H, and obtains solutions of the sinh-Gordon PDE. The tau function for sinh-Gordon satisfies a particular Painlevé III′ nonlinear ODE and describes a random matrix model, with asymptotic distribution found by the Coulomb fluid method to be the solution of an electrostatic variational problem on an interval.

Back-barrier evolution and along-strike variations in infilling of the Kosi Bay lake system, South Africa

This study examines both the inlet dynamics, recorded in the back-barrier stratigraphy of Kosi Bay, and the competing effects of waterbody segmentation on along-strike changes in the character and nature of an incised valley fill at Kosi Bay, South Africa. The Kosi Bay system comprises four interconnected lakes (Lake Makhawulani, Lake Mpungwini, Lake Nhlange and Lake Amanzimnyama) subject to varying degrees of marine influence. Formerly, the system's only connection to the ocean was through the Bhanga Nek palaeo-inlet. The location of this system behind a continuous coastal barrier predisposes it to excellent archives of environmental change related to changing sea-level and sediment supply. In this context, two cores were extracted from the system (KB2 and KB4), together with over 150 km of seismic reflection profiling, as well as AMS radiocarbon analyses and legacy data in order to present a new stratigraphic evolution model for this back-barrier system. A total of 5 seismic units were imaged (Units A-E). A major unconformity surface (SB), characterised by broad u-shaped incised valleys, incises the underlying unconsolidated sandy acoustic basement (Unit A). Unit B forms as thickly developed central basin deposits, partially infilling the incised valleys. Two forms of prograding spits are evident -Unit C1 progrades from the palaeo-highs of the valley interfluves into the nearest available accommodation space and Unit C2 progrades from the margins of the system into the basin. These are separated by an erosional surface (T-RS) formed by migrating tidal channels as sea-level rose and marine waters entered the system during the Holocene. Unit C2 spits are in turn associated with slump deposits formed on the steepest part of the margins (Unit D) that were dated to 2900 ± 165 cal. BP. The unusual depth of Surface T-RS is attributed to (1) limited sediment supply during sea-level rise, (2) the presence of easily erodible fine sediments in which this surface erodes and (3) a larger tidal prism when the Bhanga Nek inlet was still open and system segmentation by Units C2 and D had not yet occurred. Unit E caps the stratigraphy and is interpreted as lacustrine fines with a thin layer of acoustically transparent material (gyttja) on top. The deposition of Unit E signifies the closure of the system to the ocean and the overall shift from an estuarine/lagoon environment to a lacustrine one. Recalibrated AMS dates place the system closure to between 3160 and 2900 cal. BP, coeval with the formation of the prograding marginal spits and waterbody segmentation (Unit C2). Along-strike variation in the timing of the basin infilling, attributed to changes to the inlet functioning coupled to the segmentation of the waterbody, is characteristic of this area. At 1450 cal. BP, a new tidal inlet formed to the north of Lake Makhawulani, where impounded waters were diverted via a low point in the back-barrier coastal dune cordon. The formation of this inlet suggests a prolonged period of back-barrier flooding until a low point in the barrier was breached during the Holocene sea-level highstand of 1.5 m. Such inlet development and along-strike back flooding of incised valleys is an unusual attribute of coastal waterbodies of South Africa.

Biochemical and crystallization analysis of the CENP-SX-DNA complex.

The CENP-SX (MHF) complex is a conserved histone-fold protein complex that is involved in chromosome segregation and DNA repair. It can bind to DNA on its own as well as in complex with other proteins such as CENP-TW and FANCM to recognize specific substrates. CENP-SX binds nonspecifically to dsDNA, similar to other histone-fold proteins. Several low-resolution structures of CENP-SX in complex with DNA are known, but a high-resolution structure is still lacking. The DNA-binding properties of CENP-SX and FANCM-CENP-SX complexes with various lengths of dsDNA were compared and the band-shift patterns and migration positions were found to differ. To confirm the DNA-binding properties in detail, CENP-SX-DNA and FANCM-CENP-SX-DNA complexes were crystallized. Analysis of the crystals revealed that they all contained the CENP-SX-DNA complex, irrespective of the complex that was used in crystallization. Detailed diffraction data analyses revealed that there were two types of crystal with different space groups, P21 and C2, where the volume of the P21 asymmetric unit is twice as large as that of the C2 asymmetric unit. Analysis of the self-rotation function revealed the presence of twofold and fourfold symmetry in both crystals. This suggests that there may be multiple molecules of CENP-SX and DNA within the asymmetric unit with respective symmetry. Structure determination of the present crystals should reveal details of the DNA-binding properties of CENP-SX.

A geometric characterization of Powell-Sabin triangulations allowing the construction of C2 quartic splines

The paper deals with the characterization of Powell-Sabin triangulations allowing the construction of bivariate quartic splines of class C2. The result is established by relating the triangle and edge split points provided by the refinement of each triangle. For a triangulation fulfilling the characterization obtained, a normalized representation of the splines in the C2 space is given.

About the calculation of hypergeometric function F4 (1,2;2,2; z1, z2 ) by the branched continued fraction of a special kind

In the paper, using some recurrent relations, the expansion of the hypergeometric Appel function F4 (1,2;2,2; z1, z2 ) into a branched continued fraction of special form is constructed. Explicit formulas for the coefficients of constructed development are obtained. The structure of the obtained branched continued fraction is investigated. The values of the suitable fractions and the corresponding partial sums of the hypergeometric series at different points of the two-dimensional complex space are calculated. A comparative analysis of the obtained values is carried out, the results of which confirm the efficiency of using branched continued fractions to calculate the values of the hypergeometric function F4 (1,2;2,2; z1, z2 ) in space C2.

Algebras of Symmetric *-Polynomials in the Space ℂ2

We study symmetric *-polynomials in the space ℂ2. In particular, we construct a system of generators of the algebra of symmetric *-polynomials on ℂ2 and show that elements of this system are algebraically dependent. We also study the algebras obtained as completions of the algebra of symmetric * -polynomials in the space ℂ2. The spectrum and the *-spectrum of the Frechet algebra of absolutely convergent series of symmetric *-polynomials are described.

Infinite transitivity on the Calogero-Moser space C2

We prove a particular case of the conjecture of Berest--Eshmatov--Eshmatov by showing that the group of unimodular automorphisms of C[x,y] acts in an infinitely-transitive way on the Calogero-Moser space C2.

Americium pertraction across supported liquid membranes containing multiple diglycolamide ligands: Role of alkyl substitution and spacer length in carrier ligands

Liquid–liquid extraction and subsequent supported liquid membrane (SLM) transport studies involving Am3+ ion were carried out from nitric acid feed solutions using three N-pivot tripodal diglycolamide (DGA) ligands with varying substituent at the inner carboxamide N atom and spacer length. The ligand with an isopropyl substituent and with C2 spacer is termed as iPr3-TREN-DGA, while the one without alkyl substituent is termed as TREN-DGA. The third ligand with no substituent but a C3 spacer is termed as TRPN-DGA. The trend of metal ion extraction was iPr3-TREN-DGA > TRPN-DGA > TREN-DGA when solutions of ca. 1.0×10−3 M ligand in 95% n-dodecane+5% isodecanol were used. Stripping studies were carried out using 0.01M HNO3, 0.01M EDTA (at pH 3.0) and 1M α-HIBA (α-hydroxyisobutyric acid) The extraction and stripping kinetics studies suggested attainment of equilibrium in about 30minutes. The SLM studies were carried out in a two-compartment cell with 20mL feed as well as receiver phase volumes and PTFE (polytetrafluoroethylene) flat sheets as the solid support. The transport rates followed the same trend as that of the metal ion extraction study, suggesting comparable diffusion coefficients of the complexes (obtained by the lag-time method). Membrane stability studies indicated the best stability for the TRPN-DGA ligand.

Multiplicative Convolution on the Algebra of Block-Symmetric Analytic Functions

We introduce and study a multiplicative convolution operator on the spectrum of the algebra of block-symmetric analytic functions of bounded type on an infinite l1 -sum of copies of the Banach space ℂS. The case of the algebra of block-symmetric functions on the space ℂ2 and the action of multiplicative convolution on its spectrum are studied separately.

On an optimal quadrature formula for approximation of Fourier integrals in the space [formula omitted

This paper deals with the construction of an optimal quadrature formula for approximation of Fourier integrals in the Sobolev space L2(1)[a,b] of non-periodic, complex valued functions which are square integrable with first order derivative. Here the quadrature sum consists of linear combination of the given function values in a uniform grid. The difference between the integral and the quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. Analytic formulas for optimal coefficients can also be obtained using discrete analogue of the differential operator d2∕dx2. In addition, the convergence order of the optimal quadrature formula is studied. It is proved that the obtained formula is exact for all linear polynomials. Thus, it is shown that the convergence order of the optimal quadrature formula for functions of the space C2[a,b] is O(h2). Moreover, several numerical results are presented and the obtained optimal quadrature formula is applied to reconstruct the X-ray Computed Tomography image by approximating Fourier transforms.

Immobilization of laccase from Trametes versicolor on Lifetechtm supports for applications in degradation of industrial dyes

In this study, immobilization of laccase from Trametes versicolor on eight Lifetech? supports, with different characteristics (pore size, length of the spacer arm and functional groups), was studied and optimized for intended use in bioremediation for decolorization of industrial wastewaters. Out of six tested amino-functionalized supports, the most promising carrier was proved to be porous Lifetech? ECR8309F with primary amino groups and a C2 spacer arm. Onto this support, laccase is attached by forming electrostatic interactions so that the most active preparation has shown the activity of 66876 U/g support. On the other hand, during immobilization of laccase on epoxy-functionalized Lifetech? ECR8285F, via hydrophobic interactions and covalent bonding confirmed by a desorption assay, immobilization yield of 60 % and the activity of 118929 U/g were accomplished. Furthermore, immobilized enzyme on this support showed high capacity for decolorization of dyes (Lanaset? Violet B, Lanaset? Blue 2R, bromothymol blue and bromocresol green), by combination of both adsorption and enzyme degradation. Decolorization was in the range of 88 to 96 % after 4 h, with more than 80 % achieved after only 45 min. Also, this preparation demonstrated high operational stability during seven consecutive reuses in all examined dye reaction systems.

Pertraction of Np(IV) and Pu(IV) across a flat sheet supported liquid membrane containing two N-pivoted tripodal diglycolamides

Two tripodal diglycolamide (DGA) ligands with a central N atom, dissolved in a mixture of 5% isodecanol and 95% n-dodecane, were evaluated for the liquid-liquid extraction and the pertraction across a flat sheet PTFE membrane of tetravalent Np(IV) and Pu(IV) from dilute nitric acid medium. The ligand with an isopropyl group substituted at the inner amidic N atom of the tripodal DGA ligand (termed as i-Pr3-TREN-DGA) showed higher extraction for the tetravalent actinides than the other ligand with no substitution but with a C3 instead of a C2 spacer (termed as TRPN-DGA). Although the extraction studies indicated higher extraction of Pu(IV) than Np(IV) for both the ligands, an unusual reversal in the pertraction efficiency across the supported liquid membrane was observed in nitric acid feeds. Transport parameters such as the diffusion coefficients (Do) and the permeability coefficients (P) were determined for the metal ion transport with both the ligands. The liquid membranes membrane showed poor stability, limiting their long term use.

On the Parametrization of an Algebraic Curve

At present, a plane algebraic curve can be parametrized in the following two cases: if its genus is equal to 0 or 1 and if it has a large group of birational automorphisms. Here we propose a new polyhedron method (involving a polyhedron called a Hadamard polyhedron by the author), which allows us to divide the space ℝ2 or ℂ2 into pieces in each of which the polynomial specifying the curve is sufficiently well approximated by its truncated polynomial, which often defines the parametrized curve. This approximate parametrization in a piece can be refined by means of the Newton method. Thus, an arbitrarily exact piecewise parametrization of the original curve can be obtained.

N-Valine-2-(3,5-dimethy-1,1-dioxido-2H-1,2,6-thiadiazin-4-yl)Benzamide: Synthesis, X-ray Structure and Hirshfeld Surface Analysis

Here we report the synthesis and crystal structure of the title compound N-valine-2-(3,5-dimethy-1,1-dioxido-2H-1,2,6-thiadiazin-4-yl)benzamide, C18H23N3O5S. The structure was determined by the single crystal X-ray diffraction method. The compound crystallized in a monoclinic Sohncke space C2 space group with cell values of a = 18.3096 (11) A, b = 7.4858 (5) A, c = 14.1452 (10) A, β = 90.176 (2)° and with Z = 4. Crystal structure analysis showed that the molecule is not planar and the thiadiazine ring is essentially orthogonal to the plane passing through the aromatic ring. The dihedral angle between the thiadiazine ring and the aromatic ring is 93.72°. The carbonyl of the amide bond deviates from co-planarity of the aromatic ring by 60.03° which is perhaps due to the steric hindrance of the carbonyl group towards the thiadiazine ring. The sulfur atom deviates from the mean plane of the ring by 0.102 A. The synthesis of the title compound, N-valine-2-(3,5-dimethy-1,1-dioxido-2H-1,2,6-thiadiazin-4-yl)benzamide, and its structural characterization by means of single crystal X-ray diffraction and Hirshfeld surface analysis are reported.

A surface in W2,p is a locally Lipschitz-continuous function of its fundamental forms in W1,p and Lp, p > 2

The fundamental theorem of surface theory asserts that a surface in the three-dimensional Euclidean space E3 can be reconstructed from the knowledge of its two fundamental forms under the assumptions that their components are smooth enough—classically in the space C2(ω) for the first one and in the space C1(ω) for the second one—and satisfy the Gauss and Codazzi–Mainardi equations over a simply-connected open subset ω of R2; the surface is then uniquely determined up to proper isometries of E3. Then S. Mardare showed in 2005 that this result still holds under the much weaker assumptions that the components of the first form are only in the space Wloc1,p(ω) and those of the second form only in the space Llocp(ω), the components of the immersion defining the reconstructed surface being then in the space Wloc2,p(ω), p>2.The purpose of this paper is to complement this last result as follows. First, under the additional assumption that ω is bounded and has a Lipschitz-continuous boundary, we show that a similar existence and uniqueness theorem holds with the spaces Wm,p(ω) instead of Wlocm,p(ω). Second, we establish a nonlinear Korn inequality on a surface asserting that the distance in the W2,p(ω)-norm, p>2, between two given surfaces is bounded, at least locally, by the distance in the W1,p(ω)-norm between their first fundamental forms and the distance in the Lp(ω)-norm between their second fundamental forms. Third, we show that the mapping that uniquely defines in this fashion a surface up to proper isometries of E3 in terms of its two fundamental forms is locally Lipschitz-continuous.

Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inverse-square potential with zero energy E. The geodesic flow on the full configuration space ℂ3 (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space ℂ2 and shape space ℝ3 (as well as 𝕊3 and the shape sphere 𝕊2 for the inverse-square potential when E = 0). The corresponding Riemannian submersions are described explicitly in “Hopf” coordinates which are particularly adapted to the isometries. For equal masses subject to inverse-square potentials, Montgomery shows that the zero-energy “pair of pants” JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on ℂ2, ℝ3, and 𝕊3 with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits, and observe that the geodesic reformulation “regularizes” pairwise and triple collisions on ℂ2 and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and zero energy, we find that the scalar curvature on ℂ2 is strictly negative though it could have either sign on ℝ3. However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.

Recognizing projections of algebraic curves

Given two irreducible, algebraic space curves C1 and C2, where C2 is contained in some plane Π, we provide algorithms to check whether or not there exist perspective or parallel projections mapping C1 onto C2, i.e. to recognize C2 as the projection of C1. In the affirmative case, the algorithms provide the eye point(s) of the perspective transformation(s), or the direction(s) of the parallel projection(s). Although the problem is mainly discussed for rational curves, an algorithm for implicit curves is also given. The algorithms presented are mostly symbolic; nevertheless, we include an approximate algorithm for rational curves too.