Topological properties of ordinary nematics in 3-space

This paper describes the topologically possible global defect behavior of ordinary nematics in 3-space. It is written for physicists interested in defects of ordered media as well as for topologists, but instead of using an intermediate way of presentation, which might appeal to no one, we first state the result for physicists and then, discussing the proof, turn to mathematicians and physicists who are inclined to read a mathematical paper.     

Uniqueness in the Freedericksz transition with weak anchoring

In this paper we consider a boundary value problem for a quasi-linear pendulum equation with non-linear boundary conditions that arises in a classical liquid crystals setup, the Freedericksz transition, which is the simplest opto-electronic switch, the result of competition between reorienting effects of an applied electric field and the anchoring to the bounding surfaces. A change of variables transforms the problem into the equation xττ=−f(x) for τ∈(−T,T), with boundary conditions at τ=?T, for a convex non-linearity f. By analysing an associated inviscid Burgers' equation, we prove uniqueness of monotone solutions in the original non-linear boundary value problem.This result has been for many years conjectured in the liquid crystals literature, e.g. in [E.G. Virga, Variational Theories for Liquid Crystals, Appl. Math. Math. Comput., vol. 8, Chapman & Hall, London, 1994] and in [I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis, London, 2003].     

Soliton interaction in a dispersion-decreasing fiber with effective gain and effective phase modulation

We have considered soliton pulse compression in a hyperbolic dispersion-decreasing fiber with fiber loss and phase modulation. We have shown that there is exact balancing between the effective gain and the effective phase modulation as it provides fairly good compression even after compensating for the fiber loss. We have also considered the interaction scenario in detail.     

A planning and control algorithm for metal melting

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 33–38, 1988.     

A numerical method to compute solidification and melting processes

This paper deals with thermodynamically consistent numerical predictions of solidification and melting processes of pure materials using moving grids. Till date, enthalpy-porosity-based formulations of numerical codes have been generally the popular choice, although because of an artificial numerical smearing of the interface, it is virtually impossible to reproduce a sharp melting/solidification front that is supposed to exist for phase changes of pure substances. Numerical techniques based on moving grid methods have been relatively less used as they rely on complex and time-consuming adaptive grid generations. Using the moving grid approach, the authors present a method to solve solidification and melting problems. A simple linear interpolation is used to slide grid nodes along the interface to handle the otherwise obtained grid skewness near the interface. The numerical approach employed is validated with standard test cases available in the literature. The capability of capturing very complex flow field structures and the superiority of the present approach over enthalpy-porosity-based formulations is discussed. The authors also demonstrate the ability of the set-up computer code to solve complex thermofluid processes such as occur during crystal growth in Czochralski reactors.     

Fluctuations,effective learnability and metastability in analysis

This paper discusses what kind of quantitative information one can extract under which circumstances from proofs of convergence statements in analysis. We show that from proofs using only a limited amount of the law-of-excluded-middle, one can extract functionals (B,L)$(B,L)$, where L   is a learning procedure for a rate of convergence which succeeds after at most B(a)$B(a)$-many mind changes. This (B,L)$(B,L)$-learnability provides quantitative information strictly in between a full rate of convergence (obtainable in general only from semi-constructive proofs) and a rate of metastability in the sense of Tao (extractable also from classical proofs). In fact, it corresponds to rates of metastability of a particular simple form. Moreover, if a certain gap condition is satisfied, then B and L yield a bound on the number of possible fluctuations. We explain recent applications of proof mining to ergodic theory in terms of these results.     

Software Engineering and complexity in effective Algebraic Geometry

One may represent polynomials not only by their coefficients but also by arithmetic circuits which evaluate them. This idea allowed in the past fifteen years considerable complexity progress in effective polynomial equation solving. We present a circuit based computation model which captures all known symbolic elimination algorithms in effective Algebraic Geometry and exhibit a class of simple elimination problems which require exponential size circuits to be solved in this model. This implies that the known, circuit based elimination algorithms are already optimal.     

Recursion theorems and effective domains

Every acceptable numbering of an effective domain is complete. Hence every effective domain admits the 2nd recursion theorem of Eršov[1]. On the other hand for every effective domain, the 1st recursion theorem holds. In this note, we establish that for effective domains, the 2nd recursion theorem is strictly more general than the 1st recursion theorem, a generalization of an important result in recursive function theory.     

Numerical study of melting inside concentric and eccentric horizontal annulus

This paper presents numerical investigations on melting of phase change material using N-eicosane inside a cylindrical container. Numerical simulations are performed for symmetric melting of phase change material between two cylinders in concentric and eccentric arrays using the FLUENT software which is sub-cooled initially to 1 °C. Inner cylindrical tube is considered hot wall while outer tube is insulated. Predicted result shows that melting rate is the same approximately for concentric and eccentric array before time of 15 min. After this time, melting rate decreases in concentric array. It is due to the pure conduction between hot tube and cold solid phase change material.     

Cold flow and melting of plastics under severe friction conditions

Dry friction of a series of crystalline polymers (polypropylene, nylon, PTFE) against steel over a wide range of sliding velocities (4.4–4.4×10^–4 cm/sec) and loads (P=7.5–360 kg) with almost complete mutual overlap of the friction surfaces is studied. It is shown that friction of polymer materials against steel may be accompanied by melting of the crystalline polymer (at high velocities) or cold flow (at low velocities) in the surface layers associated with an orientation effect and the appearance of anisotropy.Mekhanika Polimerov, Vol. 1, No. 5, pp. 95–100, 1965     

Experimental and numerical study of the cold crucible melting process

The cold crucible, or induction skull melting process as is otherwise known, has the potential to produce high purity melts of a range of difficult to melt materials, including Ti–Al and Ti6Al4V alloys for Aerospace, Ti–Ta and other biocompatible materials for surgical implants, silicon for photovoltaic and electronic applications, etc. A water cooled AC coil surrounds the crucible causing induction currents to melt the alloy and partially suspend it against gravity away from water-cooled surfaces.     

An effective computational attempt in DDEA

Dynamic DEA (DDEA) is a mathematical programming-based technique which assesses the performance of decision making units (DMUs) in the presence of time factor. This paper provides a new technique for reducing the computational complexity of some recently introduced DDEA models.     

A simple and effective mesh adaptation

Simulation accuracy is greatly influenced by grid quality. Here, mesh quality indicates orthogonality of grid lines at the boundaries and quasi-orthogonality within critical regions, smoothness, solution adaptive behaviour and bounded aspect ratios. A simple, effective and computationally efficient approach for adapting quadrilateral grids to a given adaptive functional is presented. Several numerical examples are explored for supporting our claim.     

Filtered colimits in the effective topos

It is shown that the “constant sheaves” functor ∇:Sets→Eff does not preserve ω₁-filtered colimits, and that as a consequence of this, the full subcategory of Eff on the countable projective objects is not dense.     

Formal spaces and their effective presentations

The theory of formal spaces is developed in terms of presentations, in order to study effectivity.The present paper is extracted from the author's doctoral dissertation [16]     

Modular forms and effective Diophantine approximation

After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Frey?s height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Frey?s height conjecture which applies to all elliptic curves over $\mathbb{Q}$, not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of ${\mathbb{P}}^1?\{0,1,\infty \}$), with a completely explicit and effective bound, without using any variant of Baker?s theory or the Thue–Bombieri method.