Invariant Points Research Articles

Phase Equilibria in the System Na,K∥SO4,CO3,HCO3,F-H2O at 0°C in the Nahcolite (NaHCO3) Crystallization Region

Phase equilibria were determined by the translation method on geometric images of the system Na,K∥SO4,CO3,HCO3, F–H2O at 0°C in the nahcolite (NaHCO3) crystallization region. Nahcolite was found to participate in the formation of 18 divariant fields, 16 monovariant curves, and 5 invariant points. The closed phase diagram (phase complex) of the studied system at 0°C in the nahcolite crystallization region was constructed.

A strategy based on non-extensive statistics to improve frame-matching algorithms under large viewpoint changes

In recent decades, methods to find invariant points in digital images, called fiducial points, have gained great attention, mainly due to the demands of several applications in computer vision and image processing, such as the geometric matching of global structures, objects or specific regions. Among the most well known approaches are algorithms like SIFT, HOG, SURF, and their variations as A-SIFT, PCA-SIFT, surrounded by many others. Though the number of researches demonstrating the efficiency of such methods is undoubtedly enormous, the vast majority compares their performances only on pairs of images with little changes in view perspectives, objects or specific regions of the scenes. Thereby, the study of this type of technique under large viewpoint changes, called here LVC, has received little attention from the researchers. On the other hand, with the aim of filtering points of interest, most techniques have used the traditional extensive statistics. However, methods for image processing based on a new type of statistics, called non-extensive statistics, have shown to be efficient in several applications. In this paper, we present a new method, called q-SIFT, based on the non-extensive Tsallis statistics, to find fiducial points in a sequence of frames of videos under large viewpoints changes. We experimentally show the efficiency of the proposed method in video databases and propose new measurement metrics for this type of algorithm.

Experimental Determination and Thermodynamic Model of Solid–Liquid Equilibria in the Ternary System (LiCl + SrCl2 + H2O) at 273.15 K and Its Application in Industry

Solubilities, densities and refractive indices of the ternary system (LiCl + SrCl2 + H2O) at 273.15 K were determined using the isothermal dissolution equilibrium method. The phase diagram of the system consists of two invariant points, three univariant solubility curves and three crystallization areas corresponding to LiCl·2H2O, SrCl2·2H2O and SrCl2·6H2O. Applying the Pitzer and Harvie–Weare model, the Pitzer mixing parameters and the solubility product constants (Ksp) of solid phases were fitted to the solubility data; then, the solubilities of this ternary system were predicted. A comparison shows that the calculated solubilities agree well with the experimental results. The phase diagram and Pitzer model for the ternary system at 273.15 and 298.15 K were then used to conduct a computer simulation of brine separation. Lithium chloride and strontium chloride salts can be separated completely by six steps in the brine separation process. This result of the simulation of brine separation can be used as a theoretical reference guideline for salt separation and purification of brine systems.

Similarity Solutions for the Complex Burgers’ Hierarchy

Abstract A detailed analysis of the invariant point transformations for the first four partial differential equations which belong to the complex Burgers’ hierarchy is performed. Moreover, a detailed application of the reduction process through the Lie-point symmetries is presented while we construct the similarity solutions. We conclude that the differential equations of our consideration are reduced to first-order equations such as the Abel, Riccati, and to a linearisable second-order differential equation by using similarity transformations.

Phase Equilibrium and Solvation Effect of the Ternary Mixture Solvent System (LiCl + CH3OH + H2O) at 298.15, 308.15 and 318.15 K

The solubility and physicochemical properties, including density and refractive index in the ternary mixture solvent system (LiCl + CH3OH + H2O) at 298.15, 308.15 and 318.15 K were determined using the isothermal equilibrium method. Based on the experimental data, the phase diagrams and physicochemical properties versus composition diagrams at different temperatures were plotted and compared. It was found that the isothermal phase diagram at each temperature consists of one invariant point (LiCl + LiCl·H2O), two univariant curves and two crystalline regions corresponding to LiCl·H2O and LiCl, respectively. The densities and refractive indices in the ternary mixture solvent system at each temperature change regularly with the increase of LiCl concentration. The calculated densities and refractive indices at the three temperatures agree well with the experimental results. The salting-out rate of LiCl solution increases gradually with the increase of CH3OH concentration at each temperature. The results obtained in this work can be used to produce anhydrous LiCl at relatively low temperatures, even room temperature.

A generalization of the spherical ensemble to even-dimensional spheres

In a recent article [1], Alishahi and Zamani discuss the spherical ensemble, a rotationally invariant determinantal point process on S2. In this paper we extend this process in a natural way to the 2d-dimensional sphere S2d. We prove that the expected value of the Riesz s-energy associated to this determinantal point process has a reasonably low value compared to the known asymptotic expansion of the minimal Riesz s-energy.

Equi-affine differential invariants for invariant feature point detection

Image feature points are detected as pixels which locally maximise a detector function, two commonly used examples of which are the (Euclidean) image gradient and the Harris–Stephens corner detector. A major limitation of these feature detectors is that they are only Euclidean-invariant. In this work, we demonstrate the application of a 2D equi-affine-invariant image feature point detector based on differential invariants as derived through the equivariant method of moving frames. The fundamental equi-affine differential invariants for 3D image volumes are also computed.

Phase Equilibria in the Ternary Al–Ti–Pt System. II. Liquidus Surface of the Al–Ti–Pt System in the Compositions Range 0–50 at.% Pt

The liquidus surface of the Al–Ti–Pt system has been constructed for the first time on the composition triangle based on studies of the structure and phase composition and measurement of the liquidus temperatures for the as-cast alloys in the composition range 0–50 at.% Pt produced using a series of techniques. The liquidus surface has 23 primary crystallization fields corresponding to τ1–τ9 and τ11 ternary phases and binary phases formed in the boundary systems and to solid solutions based on components. The extension of these fields and the positions of univariant curves and invariant points have been determined. The liquidus surface exists in the temperature range from 1830°C (corresponding to congruent melting of TiPt compound) to 655°C (temperature of the l ⇄ + eutectic in the binary Al–Pt system).

Principle of Independence of Continuation of Functions Multivalued Logic from Coding

In this paper we consider not everywhere defined functions of multivalued logic. The problem of logical continuation of functions of multivalued logic in the class of disjunctive normal forms is investigated. We prove a theorem on the invariant extension of functions of multivalued logic in the class of disjunctive norms, which does not depend on the accepted coding. An algorithm is constructed for constructing a set of invariant points that are independent of the received encoding.

Investigation of Solid–Liquid Equilibria on the Na+//Cl–, NO3–, SO42––H2O System and the Na+//NO3–, SO42––H2O System at 313.15 K

Equilibria on the Na+//Cl–, NO3–, SO42––H2O system and the Na+//NO3–, SO42––H2O system at 313.15 K are investigated used the isothermal solution saturation method. The composition and physicochemical properties (refractive index nD, viscosity η, density ρ, and pH) of saturated solutions are determined. The phase diagram of the Na+//NO3–, SO42––H2O system includes two invariant points, three invariant curves, and three crystallization fields of single salts, Na2SO4, Na2SO4·NaNO3·H2O, NaNO3. Invariant points are cosaturated with (Na2SO4 + Na2SO4·NaNO3·H2O) salts and (NaNO3 + Na2SO4·NaNO3·H2O) salts, respectively. The Na+//Cl–, NO3–, SO42––H2O system consists of five univariant curves, two invariant points cosaturated with three salts, and four crystallized regions. Double salt Na2SO4·NaNO3·H2O can be verified to exist in the Na+//Cl–, NO3–, SO42––H2O and Na+//NO3–, SO42––H2O systems under equilibrium conditions at 313.15 K. In addition, comparing with reported phase diagrams in different temperatures, crystallized regions of Na2SO4·NaNO3·H2O decrease with increasing temperature and disappear at 373.15 K. The crystallized region of sodium sulfate widens, while both crystallized regions of NaNO3 and NaCl decrease with increasing temperature. All of the physicochemical properties (nD, η, ρ, pH) of equilibrium solutions alter with a composition change of the liquid phase. Phase equilibria of Na+//C1–, SO42–, NO3––H2O at 313.15 K were also investigated by the through and intermediate translation techniques. All data and results from our work are important to design and optimize the crystallization of high-saline wastewater in the coal chemical industry and could be used to solve inorganic salt separation from the system containing the above-mentioned salts.

Stable Phase Equilibria in Quaternary LiCl–NaCl–MgCl<sub>2</sub>–H<sub>2</sub>O and NaCl–Na<sub>2</sub>B<sub>4</sub>O<sub>7</sub>–MgCl<sub>2</sub>–MgB<sub>4</sub>O<sub>7</sub>–H<sub>2</sub>O Systems at 273 K

Phase equilibria of quaternary LiCl–NaCl–MgCl2–H2O and NaCl–Na2B4O7–MgCl2–MgB4O7–H2O systems at 273 K were studied by isothermal dissolution. The isothermal solubility of each salt was measured. Based on the experimental data and the corresponding equilibrium solid phases, equilibrium phase diagrams and water content diagrams were drawn. The results show that the quaternary LiCl–NaCl–MgCl2–H2O system forms a double salt LiCl·MgCl2·7H2O at 273 K. The phase diagram contains two invariant points, five univariant curves and four crystalline regions. The crystalline regions are LiCl·2H2O, NaCl, MgCl2·6H2O and LiCl·MgCl2·7H2O, respectively. There are two invariant points, five univariant curves and four crystallizing fields in the quaternary NaCl–Na2B4O7–MgCl2–MgB4O7–H2O system at 273 K, and the corresponding crystallizing fields are NaCl, Na2B4O7·10H2O, MgCl2·6H2O and MgB4O7·9H2O.

Solid-liquid equilibrium for the ternary systems (Na2CO3 + NaCl + H2O) and (Na2CO3 + Na2SO4 + H2O) at 313.15 K and atmospheric pressure

The stable (solid + liquid) phase equilibrium and physico-chemical properties of the ternary systems (Na2CO3 + NaCl + H2O) and (Na2CO3 + Na2SO4 + H2O) at T = 313.15 K have determined by the method of isothermal solution saturation. The solid phases formed in the systems studied were determined by the Schreinemaker wet residue method. According to the experimental data, the phase diagrams and the diagrams of physico-chemical properties versus composition were plotted. It was found that there is one invariant point, two univariant curves and three fields of crystallization in the ternary systems (Na2CO3 + NaCl + H2O) at T = 313.15 K. The three crystallization regions correspond to NaCl (I), Na2CO3·H2O (II), the mixed of (NaCl + Na2CO3·H2O) (III). Field EFUC (IV) corresponds to unsaturated solution. Based on the phase diagrams obtained, we analyzed each invariant point. When Na2CO3 and NaCl were analyzed as (15.80, 17.04) wt%, respectively, the ternary system (Na2CO3 + NaCl + H2O) reached saturation at T = 313.15 K. The phase diagram of the ternary system at 313.15 K includes two invariant points, three invariant curves, and three fields of crystallization of single salts corresponding to Na2CO3, Na2CO3·2Na2SO4, Na2SO4. The two invariant points are cosaturated with two salts, that is (Na2CO3·H2O + Na2CO3·2Na2SO4) and (Na2SO4 + Na2CO3·2Na2SO4), respectively. In the phase diagram of the two ternary systems at 313.15 K, the crystallization region of Na2SO4 was significantly larger than the crystallization region of Na2CO3·H2O and NaCl. The quaternary phase equilibrium in Na+//C1−, CO32−, SO42−-H2O system at 313.15 K was also researched by means the through and intermediate translation techniques together. Thus, knowledge of (solid + liquid) equilibrium, especially the crystalline fields at T = 313.15 K, would be extremely valuable to design and optimize solvent extraction process of Na2SO4 and other crystallization processes involving the two ternary systems in the industrial production.

Stable Phase Diagram of Quaternary Water–Salt System Li+, Na+, Cs+//SO42––H2O at T = 298.2 K

The stable phase diagram of quaternary water–salt system Li+, Na+, Cs+//SO42––H2O was studied at T = 298.2 K by the isothermal dissolution equilibrium method. The solubility and physicochemical properties (density and refractive index) of the equilibrium solution were measured. Moreover, the composition of the solid phase was identified using the X-ray diffraction method. There are seven solids formed in this system at research temperature, including four single salts and three double salts, which are lithium sulfate monohydrate (Li2SO4·H2O), sodium sulfate (Na2SO4), sodium sulfate decahydrate (Na2SO4·10H2O), cesium sulfate (Cs2SO4), and double salts Li2SO4·Cs2SO4, 3Li2SO4·Cs2SO4·2H2O, and 3Na2SO4·Li2SO4·12H2O, respectively. Thus, the stable phase diagram of the quaternary water–salt system contains 5 invariant points, 11 univariant curves, and 7 crystallization regions. The size order of the salts’ crystalline phase region is Li2SO4·Cs2SO4 > 3Na2SO4·Li2SO4·12H2O > 3Li2SO4·Cs2SO4·2H2O > Na2SO4·10H2O > Li2SO4·H2O > Na2SO4 > Cs2SO4. Obviously, the double salts have larger crystallization fields, which means that in the sulfate system coexisting with lithium, sodium, and cesium, double salts are relatively easier to form than a single salt of various components. As a result, lithium and cesium are scarcely possible separated as Li2SO4 and Cs2SO4 from this system at 298.2 K.

Preparation, electronic structure and piezooptical properties of solid solutions Tl3PbBr5–I

Abstact Phase diagram of the Tl3PbBr5–Tl3PbI5 section of the reciprocal system 2TlI + PbBr2⇔2TlBr + PbI2 was investigated by differential thermal and X-ray phase analysis methods on 18 alloys. The section is quasi-binary, of eutectic type, with the invariant point coordinates 76 mol% Tl3PbI5 and 355 °C. The addition of Tl3PbI5 results in stabilization of the high-temperature (HT) modification of Tl3PbBr5 forming a substitution solid solution, which at room temperature occupies the concentration range of ∼9–71 mol% Tl3PbI5. The solid solution range of low-temperature (LT) Tl3PbBr5 is minor (0–∼6 mol% Tl3PbI5), and that of Tl3PbI5 is in the range of ∼71–100 mol% Tl3PbI5. Tl3PbBr2.5I2.5 crystals were grown by Bridgman-Stockbarger method; they belong to wide-gap semiconductors with Eg decreasing from 2.45 eV at 100 K to 2.36 eV at 300 K. Based on X-ray photoelectron (XP) core-level spectra, one can conclude that the Tl3PbBr2.5I2.5 crystal reveals the XP fine-structure peculiarities well ascribed to thallium, lead, bromine and iodine, constituent atoms of the quaternary halide under consideration. Treatment of the Tl3PbBr2.5I2.5 crystal with 3 keV Ar+ ions reveals that the Pb–I bonds are somewhat weaker in this compound, as compared to the Pb–Br, Tl–Br and Tl–I bonds. Comparison with the similar XP measurements of the related ternary bromide and iodide (Tl3PbBr5 and Tl3PbI5) indicates that the positive effective charge of lead atoms gradually decreases when going from Tl3PbBr5 to Tl3PbBr2.5I2.5 and, further, to Tl3PbI5, while the charge state of thallium atoms does not change in the above sequence of compounds. In the sequence Tl3PbBr5 → Tl3PbBr2.5I2.5 → Tl3PbI5 the relative peak intensity of the XP valence-band spectrum monotonously decreases. The laser stimulated piezooptics at 1150 nm excited by 1540 nm Er:glass nanosecond laser and its second harmonic generation are reported.

The Phase and Physicochemical Properties Diagrams of Systems Rb+(Mg2+)//Cl– and Borate–H2O at 323 K

The phase and physicochemical properties diagrams at 323 K of systems Rb+//Cl–, borate–H2O, and Mg2+//Cl–, borate–H2O were plotted using the solubilities, densities, and refractive indices of the systems. The Schreinemakers’ wet residue method and the X-ray diffraction method were used for the determination of the compositions of solid phase. Results show that these two systems belong to the hydrate I type, without solid solution or double salt formation. The phase diagrams of both systems consist of one invariant point, two monovariant curves, and two crystallization zones. The borates formed in the studied systems are RbB5O6(OH)4 · 2H2O, and MgB4O5(OH)4 · 7H2O. The comparison between the stable phase diagrams of the system Rb+//Cl–, borate–H2O at 323 and 348 K shows that no matter at which temperature point, the crystallization form of salts are not changed and the crystallization zones have slightly changed.

The conformal bootstrap: Theory, numerical techniques, and applications

Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory. For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible both due to significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. These developments have led to a number of groundbreaking results, including world record determinations of critical exponents and correlation function coefficients in the Ising and $O(N)$ models in three dimensions. This article will review these exciting developments for newcomers to the bootstrap, giving an introduction to conformal field theories and the theory of conformal blocks, describing numerical techniques for the bootstrap based on convex optimization, and summarizing in detail their applications to fixed points in three and four dimensions with no or minimal supersymmetry.

Phase Equilibria and Phase Diagrams for the Aqueous Ternary System Containing Sodium, Chloride, and Metaborate Ions at 288.15 and 308.15 K and 0.1 MPa

Salt-water phase diagram of the ternary system (NaCl + NaBO2 + H2O) is significant in separate boron resources from brine. Solubilities and physicochemical properties bifurcated to density and refractive index for the ternary system at 288.15 and 308.15 K and 0.1 MPa were investigated experimentally by means of isothermal dissolution. According to the experimental data, the phase diagrams for this system at two temperatures were constructed and their physicochemical properties were drawn. There both contain two invariant points, three univariant isotherm dissolution curves, and three crystallization regions corresponding to sodium metaborate tetrahydrate (NaBO2·4H2O, Smt), double salt teepleite (NaCl·NaBO2·2H2O, Te), and halite (NaCl, Ha) for (NaCl + NaBO2 + H2O). At the invariant points of this ternary system, the coexisting solid phases are both for (Smt + Te) and (Te + Ha), respectively. It was found that the double salt of teepleite was incongruent at 288.15 K and then transformed into congruent at 308.15 K. Comparing the phase diagrams at 288.15 and 308.15 K, it reveals that the area of the crystallization region of NaBO2·4H2O decreased with the increase of temperature obviously, while the area of NaCl·NaBO2·2H2O increased sharply and that of NaCl increased lightly. The refractive indices and densities of the system at two temperatures are similarly increased firstly and then decreased with the increase of NaCl concentration in the solution and attain extreme points at both invariant points. Besides, the calculated values of densities and refractive indices utilizing the empirical equations are in accordance with the experimental data at 288.15 and 308.15 K.

The Stable Phase Equilibria of the Ternary Systems Na2SO4 + Rb2SO4 (Cs2SO4) + H2O at 298.2 K

To understand the crystallographic forms of salts in sulfate systems containing sodium and rubidium (cesium), the stable phase equilibria of ternary systems Na2SO4 + Rb2SO4 + H2O and Na2SO4 + Cs2SO4 + H2O at 298.2 K were studied with the isothermal dissolution method. The composition and crystallographic forms of the equilibrated solid phase was verified by Schreinemaker’s wet residue method combined with X-ray diffraction. Experimental results show that there is no double salt or solid solution formed. The stable phase diagram of Na2SO4 + Rb2SO4 + H2O at 298.2 K is characterized by one invariant point (H), two univariant curves, and two crystallographic forms (corresponding to Na2SO4·10H2O and Rb2SO4). The ternary system Na2SO4 + Cs2SO4 + H2O is of the hydrate type II, there are two invariant points (E1, E2), three univariant curves, and three crystallographic forms (corresponding to salts Na2SO4·10H2O, Na2SO4 and Cs2SO4). Affected by the unique structure of Cs+, both salt Na2SO4·10H2O and Na2SO4 have be...

Phase equilibria in the Na,Ca//SO4,CO3,HCO3-H2O system at 0 °C

Phase equilibria in the Na,Ca//SO4,CO3,HCO3-H2O system have been studied at 0 °C. The studied system at 0 °C involves 4 invariant points, 13 monovariant curves and 15 divariant fields. The obtained results served as a basis for the construction of phase diagram (phase complex) of the studied system at 0 °C.

The reaction routes comparison with respect to slow invariant manifold and equilibrium points

Mathematically, complex chemical reactions can be simplified by “model reduction,” that is, the rigorous way of approximating and representing a complex model in simplified form. Furthermore, to reduce the dimension of the reaction mechanism there are different available model reduction techniques (MRT). Two MRT Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM) are applied here. Both techniques are good approximation techniques but not completely analytical. While for a case study, the complex behavior of the two-route reaction mechanism allied graphically. The characteristic properties of nodes and trees of reaction mechanism are addressed with the graph theory. The invariant solution curves and their comparison for two-route reaction mechanism have never been reported so far. Therefore, the aim of this article is to study such facts. The possible approaches to their solution and key problems are based on the Horiuti’s rules and their slow invariants. Both model-reduction techniques are applied to solve the high dimensional complex problem and comparison between invariant curves of both routes are presented graphically. The efficiency of ILDM with respect to time is compared with the SQEM graphically and in tabulated form by using MATLAB.