Interdiffusion and solid solution strengthening in Ni–Co–Pt and Ni–Co–Fe ternary systems

Diffusion couple experiments are conducted in Co–Ni–Pt system at 1200?°C and in Co–Ni–Fe system at 1150?°C, by coupling binary alloys with the third element. Uphill diffusion is observed for both Co and Ni in Pt rich corner of the Co–Ni–Pt system, whereas in the Co–Ni–Fe system, it is observed for Co. Main and cross interdiffusion coefficients are calculated at the composition of intersection of two independent diffusion profiles. In both the systems, the main interdiffusion coefficients are positive over the whole composition range and the cross interdiffusion coefficients show both positive and negative values at different regions. Hardness measured by performing the nanoindentations on diffusion couples of both the systems shows the higher values at intermediate compositions.     

Microstructure investigations and magnetic after-effect in amorphous and nanocrystalline Fe–Zr–Ti–B–Cu alloy

The microstructure evolution and low field magnetic properties i.e. initial magnetic susceptibility, stabilization field and magnetic after-effect as disaccommodation of the amorphous and nanocrystalline Fe₈₀Zr₄Ti₃B₁₂Cu₁ alloy were investigated. The heat treatment of the as-quenched Fe₈₀Zr₄Ti₃B₁₂Cu₁ alloy at 773 K for 1 h leads to its nanocrystallization. It was stated that initial magnetic susceptibility increases and intensity of disaccommodation decreases with increasing of annealing temperature. The magnetic after-effect of the investigated nanocrystalline samples is connected with relaxation processes that occur in the amorphous matrix.     

Chern–Simons,Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g., in the Chern–Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as \(p= \langle F, F\rangle \) where F is the curvature 2-form and \(\langle \cdot , \cdot \rangle \) is an invariant scalar product on the corresponding Lie algebra \(\mathfrak g\). The descent for p gives rise to an element \(\omega =\omega _3+\omega _2+\omega _1+\omega _0\) of mixed degree. The 3-form part \(\omega _3\) is the Chern–Simons form. The 2-form part \(\omega _2\) is known as the Wess–Zumino action in physics. The 1-form component \(\omega _1\) is related to the canonical central extension of the loop group LG. In this paper, we give a new interpretation of the low degree components \(\omega _1\) and \(\omega _0\). Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara–Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class p. In more detail, we define a 1-cocycle C which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara–Vergne equation F is mapped to \(\omega _1=C(F)\). Furthermore, the component \(\omega _0\) is related to the associator \(\Phi \) corresponding to F. It is surprising that while F and \(\Phi \) satisfy the highly nonlinear twist and pentagon equations, the elements \(\omega _1\) and \(\omega _0\) solve the linear descent equation.     

Fredholm–Lagrangian–Grassmannian and the Maslov index

This is a review article on the topology of the space, so called, Fredholm–Lagrangian–Grassmannian and the quantity “Maslov index” for paths in this space based on the standard theory of functional analysis. Our standing point is to define the Maslov index for arbitrary paths in terms of the fundamental spectral property of the Fredholm operators as an intersection number with the “Maslov cycle”. This argument was first recognized by J. Phillips and was used to define the “Spectral flow” not only for loops but also for arbitrary paths of selfadjoint Fredholm operators. We make the arguments as elementary as possible.     

Nonlinear dynamical symmetries of Smorodinsky–Winternitz and and Fokas–Lagerstorm systems

General solutions of the Smorodinsky–Winternitz system and the Fokas–Lagerstorm system, which are superintegrable in two-dimensional Euclidean space, are obtained using the algebraic method (structure function). Their dynamical symmetries, which are governed by deformed angular momentum algebras, are revealed.     

Surface tension and density of liquid Bi–Pb,Bi–Sn and Bi–Pb–Sn eutectic alloys

Surface tension and density measurement of liquid Bi₅₆Pb₄₄, Bi₄₃Sn₅₇ and Bi₄₆Pb₂₉Sn₂₅ eutectic alloys was carried out by using the large drop method over the temperature range of 380–750 K. The regular solution model has been used in conjunction with Butler's equation to calculate the surface tension of binary and ternary alloys of the Bi–Pb–Sn system, while the surface tension of ternary alloys has also been predicted by using geometric models. The new experimental results were compared with the calculated values of the surface tension as well as with the data available in the literature.     

Superconductivity of Y–Ba–Cu–O and substituted systems

Superconductivity of Y-Ba-Cu-O system is studied in the composition 2:2:3 and 1:2:3 of Y:Ba:Cu. The effect of replacement of Y or Ba by divalent Sr and Ca, trivalent Ce and tetravalent Zr is studied. X-ray diffraction, SEM and TEM techniques are used for materials characterization. Superconducting transition temperatures are measured resistively. Rapid resistance drop observed above 230 K in Y-Ba-Sr-Cu-O and Y-Ba-Ca-Cu-O systems indicate the possible existence of superconductivity above 230 K. Substitution of Ce in place of Y is found to reduce the onset Tc from 95 K to 80 K. For the first time, replacement of Cu by Zr in Y-Ba-Cu-O has yielded the onset Tc of about 105 K.     

Grothendieck–Teichmüller and Batalin–Vilkovisky

It is proven that, for any affine supermanifold M equipped with a constant odd symplectic structure, there is a universal action (up to homotopy) of the Grothendieck–Teichmüller Lie algebra ${\mathfrak{grt}_1}$ on the set of quantum BV structures (i.e. solutions of the quantum master equation) on M.     

Fermi–Pasta–Ulam recurrence and modulation instability

We give a qualitative conceptual explanation of the Fermi–Pasta–Ulam (FPU) like recurrence in the onedimensional focusing nonlinear Schrodinger equation (NLSE). The recurrence can be considered as a result of the nonlinear development of the modulation instability. All known exact localized solitary wave solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its initial region. Based on the integrability of the NLSE, we demonstrate that the FPU recurrence takes place not only for condensate, but also for a more general solution in the form of the cnoidal wave. This solution is periodic in space and can be represented as a solitonic lattice. That lattice reduces to isolated soliton solution in the limit of large distance between solitons. The lattice transforms into the condensate in the opposite limit of dense soliton packing. The cnoidal wave is also modulationally unstable due to soliton overlapping. The recurrence happens at the nonlinear stage of the modulation instability. Due to generic nature of the underlying mathematical model, the proposed concept can be applied across disciplines and nonlinear systems, ranging from optical communications to hydrodynamics.     

Characterization of bismuth–tin–lead and bismuth–tin–lead–cadmium fusible alloys

Microstructure, electrical, mechanical and thermal properties of quenched bismuth–tin eutectic, Rose (Bi₅₀Sn_22.9Pb_27.1) and Wood’s (Bi₅₀Sn_12.5Pb₂₅Cd_12.5) alloys have been investigated using scanning electron microscopy, X-ray diffraction analysis, the double bridge method, the dynamic resonance method, Vickers hardness measurement and thermal analysis. Wood’s alloy (Bi–Pb–Sn–Cd) has low electrical resistivity and melting point but a high elastic modulus and internal friction when compared with the Rose (Bi–Pb–Sn) alloy. The presence of cadmium in Wood’s alloy decreases its melting point and electrical resistivity with an increase in its elastic modulus, which improves the mechanical properties. Wood’s alloy (Bi–Pb–Sn–Cd) has better properties, which make it useful in various applications such as in protection shields for radiotherapy, locking of mechanical devices and welding at low temperature.     

Hardness and microstructural variation of Al–Mg–Mn–Sc–Zr alloy

Variations of Vickers hardness were observed in Al–Mg–Mn alloy and Al–Mg–Mn–Sc–Zr alloy at different ageing times, ranging from a peak value of 81.2 HV at 54 ks down to 67.4 HV at 360 ks, below the initial hardness value, 71.8 HV at 0 ks for the case of Al–Mg–Mn–Sc–Zr alloy. Microstructures of samples at each ageing stage were examined carefully by transmission electron microscopes (TEMs) both in two-dimensions and three-dimensions. The presence of different types, densities, and sizes of particles were observed dispersed spherical Al₃Sc_1−xZr_x and also block-shaped Al₃Sc precipitates growing along <1 0 0>_Al with facets {1 0 0} and {1 1 0} of the precipitates. TEM analysis both in two-dimensions and three-dimensions, performed on various samples, confirmed the direct correlation between the hardness and the density of Al₃Sc.     

Symmetry Reduction and Exact Solutions of the Euler–Lagrange–Born–Infeld,Multidimensional Monge–Ampere and Eikonal Equations

Abstract

Using the subgroup structure of the generalized Poincaré group P (1, 4), ansatzes which reduce the Euler–Lagrange–Born–Infeld, multidimensional Monge–Ampere and eikonal equations to differential equations with fewer independent variables have been constructed. Among these ansatzes there are ones which reduce the considered equations to linear ordinary differential equations. The corresponding symmetry reduction has been done. Using the solutions of the reduced equations, some classes of exact solutions of the investigated equation have been presented.     

The Hintermann–Merlini–Baxter–Wu and the infinite-coupling-limit Ashkin–Teller models

We show how the Hintermann–Merlini–Baxter–Wu model (which is a generalization of the well-known Baxter–Wu model to a general Eulerian triangulation) can be mapped onto a particular infinite-coupling-limit of the Ashkin–Teller model. We work out some mappings among these models, also including the standard and mixed Ashkin–Teller models. Finally, we compute the phase diagram of the infinite-coupling-limit Ashkin–Teller model on the square, triangular, hexagonal, and kagome lattices.     

From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov

We introduce a long wave scaling for the Vlasov–Poisson equation and derive, in the cold ions limit, the Korteweg–de Vries equation (in 1D) and the Zakharov–Kuznetsov equation (in higher dimensions, in the presence of an external magnetic field). The proofs are based on the relative entropy method.     

Precipitation and fracture behaviour of Fe–Mn–Ni–Al alloys

The effects of Al addition on the precipitation and fracture behaviour of Fe–Mn–Ni alloys were investigated. With the increasing of Al concentration, the matrix and grain boundary precipitates changed from L1₀ θ-MnNi to B2 Ni₂MnAl phase, which is coherent and in cube-to-cube orientation relationship with the α′-matrix. Due to the suppression of the θ-MnNi precipitates at prior austenite grain boundaries (PAGBs), the fracture mode changed from intergranular to transgranular cleavage fracture. Further addition of Al resulted in the discontinuous growth of Ni₂MnAl precipitates in the alloy containing 4.2?wt.% Al and fracture occurred by void growth and coalescence, i.e. by ductile dimple rupture. The transition of the fracture behaviour of the Fe–Mn–Ni–Al alloys is discussed in relation to the conversion of the precipitates and their discontinuous precipitation behaviour at PAGBs.