Modeling,Mathematical and Numerical Analysis of Electrorheological Fluids

Many electrorheological fluids are suspensions consisting of solid particles and a carrier oil. If such a suspension is exposed to a strong electric field the effective viscosity increases dramatically. In this paper we first derive a model which captures this behaviour. For the resulting system of equations we then prove local in time existence of strong solutions for large data. For these solutions we finally derive error estimates for a fully implicit time-discretization.     

On Fraenkel’s <Emphasis Type="Italic">N</Emphasis>-Heap Wythoff’s Conjectures

The &lt;i>N&lt;/i>-heap Wythoff&lt;img src="/content/w0an0n4x4a8l8cyt/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">s game is a two-player impartial game with &lt;i>N&lt;/i> piles of tokens of sizes &lt;img src="/content/w0an0n4x4a8l8cyt/26_2004_Article_217_TeX2GIFIEq1.gif" alt=" $$ A^1, \ldots , A^N, A^1 \leq \ldots \leq A^N $$ " align="middle" border="0"> Players take turns removing any number of tokens from a single pile, or removing (&lt;i>a&lt;/i>&lt;sub>1&lt;/sub>,..., &lt;i>a&lt;/i>&lt;sub>&lt;i>N&lt;/i>&lt;/sub>) from all piles - &lt;i>a&lt;/i>&lt;sub>i&lt;/sub> tokens from the &lt;i>i&lt;/i>-th pile, providing that &lt;img src="/content/w0an0n4x4a8l8cyt/26_2004_Article_217_TeX2GIFIEq2.gif" alt=" $$ 0 \leq a_i \leq A^i, \sum^{N}_{i=1} a-i > = \quad \mathrm{and} \quad a_1 \bigotimes \ldots \bigotimes a_N = 0 $$ " align="middle" border="0"> where &lt;img src="/content/w0an0n4x4a8l8cyt/xxlarge8853.gif" alt="oplus" align="MIDDLE" BORDER="0"> is the nim addition. The first player that cannot make a move loses. Denote all the &lt;i>P&lt;/i>-positions (i.e., losing positions) by &lt;img src="/content/w0an0n4x4a8l8cyt/26_2004_Article_217_TeX2GIFIEq3.gif" alt=" $$ (A^1, \ldots , A^{N-2}, A^{N-1}_n , A^{N}_{n}), A^{N-2} \leq A^{N-1}_{n} \leq A^{N}_{n} \quad \mathrm{and} \quad A^{N-1}_{n} &lt; A^{N-1}_{n+1} \quad \mathrm{for all} \quad N \geq 1 $$ " align="middle" border="0"> Two conjectures were proposed on the game by Fraenkel [7]. When &lt;img src="/content/w0an0n4x4a8l8cyt/26_2004_Article_217_TeX2GIFIEq4.gif" alt=" $$ A^1, \ldots, A^{N-2} $$ " align="middle" border="0"> are fixed, i) there exists an integer &lt;i>N&lt;/i>&lt;sub>1&lt;/sub> such that when &lt;img src="/content/w0an0n4x4a8l8cyt/26_2004_Article_217_TeX2GIFIEq5.gif" alt=" $$ n > N_1, A^{N}_{n} = A^{N-1}_{n} + n $$ " align="middle" border="0"> . ii) there exist integers &lt;i>N&lt;/i>&lt;sub>2&lt;/sub> and &lt;img src="/content/w0an0n4x4a8l8cyt/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">_2 such that when &lt;img src="/content/w0an0n4x4a8l8cyt/26_2004_Article_217_TeX2GIFIEq6.gif" alt=" $$ n > N_2, A^{N-1}_{n} = \lfloor n \Phi \rfloor + \epsilon_n + \alpha_2 \quad \mathrm{and} \quad A^{N}_{n} = A^{N-1}_{n} + n, \quad \mathrm{where} \quad -1 \leq \epsilon_n \leq 1 \quad \mathrm{and}\quad \Phi = (1 + \sqrt{5})/2$$ " align="middle" border="0"> , the golden section.In this paper, we provide a sufficient condition for the conjectures to hold, and subsequently prove them for the three-heap Wythoff&lt;img src="/content/w0an0n4x4a8l8cyt/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">s game with the first piles having up to 10 tokens.AMS Subject Classification: 91A46, 68R05.     

α-Particle Loss from Cylindrical Magnetized DT Plasma

&lt;img src="/content/gn8787l43lt35126/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">-particles from fusion reactions in a DT magnetized plasma are confined inside the plasma not only by magnetic fields but also by induced electric fields which are due to the different mobilities of &lt;img src="/content/gn8787l43lt35126/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">-particles and electrons. Criteria are derived for the &lt;img src="/content/gn8787l43lt35126/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">-confinement in a cylindrical plasma, inertially confined by a heavy liner. It is observed that the suprathermal &lt;img src="/content/gn8787l43lt35126/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0"> population in the DT plasma will influence the overall dynamics.     

The Order Parameter and Phase Transformations in Compounds with Ionic Stoichiometry

The paper considers instability of a symmetric linear ion chain against a spontaneous distortion that orders ions of opposite charge without a multiple change in the spacing. Transverse deformation in dense chains is observed. The dipolar interaction responsible for 3D ordering of similar structures of Cu—O chains in yttrium perovskites has been assessed. The superclose ion packing in Cu—O chains is explained by the existence of an oriented hole orbital in the electronic structure of bivalent copper. The possible role of this orbital as an additional degree of freedom leading to thermodynamic singularities at intermediate temperatures below the tetra–ortho transition temperature has been investigated.     

Calculation of Families of Stationary Filtration Convection Regimes in a Narrow Container

A plane Darcy filtration convection problem for rectangular containers elongated in the vertical direction is considered. By the spectral&lt;img src="/content/pg34625r94l45381/xxlarge8208.gif" alt="dash" align="MIDDLE" BORDER="0">difference method, which preserved cosymmetry of the initial problem, evolution of families of stationary regimes from the onset of instability on the primary family till the collision of the families is studied.     

On Cesàro Means of Double Trigonometric Fourier Series

Zhizhiashvili studied questions associated with the approximation properties of Cesàro means of trigonometric Fourier series for functions of two variables in the spaces H&lt;sup>&lt;img src="/content/k24615g1l45436l6/xxlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">&lt;/sup>. It is proved here that the corresponding estimate cannot be improved for &lt;i>p&lt;/i> = 1 or &lt;i>p&lt;/i> = &lt;img src="/content/k24615g1l45436l6/xxlarge8734.gif" alt="infin" align="MIDDLE" BORDER="0">.     

A Modal Logic Based on Linearly Ordered f-Spaces

A modal logic associated with the &lt;img src="/content/t672867j5641206l/10469_2004_Article_469356_TeX2GIFIE2.gif" alt=" $$f$$ " align="middle" border="0"> -spaces introduced by Ershov is examined. We construct a modal calculus that is complete w.r.t. the class of all strictly linearly ordered &lt;img src="/content/t672867j5641206l/10469_2004_Article_469356_TeX2GIFIE3.gif" alt=" $${f_0 }$$ " align="middle" border="0"> -frames, and the class of all strictly linearly ordered &lt;img src="/content/t672867j5641206l/10469_2004_Article_469356_TeX2GIFIE4.gif" alt=" $$f$$ " align="middle" border="0"> -frames.     

Chemical Composition of Certain Sedum Species of Kazakhstan

&lt;i>The chemical composition of seven&lt;/i>Sedum&lt;i>species growing in Kazakhstan was studied by chromatography. The contents of seven groups of natural compounds were determined. Comparison with markers and literature data identified 24 compounds including 9 that have not been described previously in&lt;/i>Sedum&lt;i>species&lt;/i>     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space &lt;i>N&lt;/i>, we show that if &lt;img src="/content/l1654304464l5236/xxlarge937.gif" alt="OHgr" align="BASELINE" BORDER="0"> is a finite or &lt;img src="/content/l1654304464l5236/xxlarge963.gif" alt="sgr" align="BASELINE" BORDER="0">-finite measure space and 1&lt;img src="/content/l1654304464l5236/xxlarge8804.gif" alt="le" align="MIDDLE" BORDER="0">&lt;i>p&lt;/i>&lt;img src="/content/l1654304464l5236/xxlarge8804.gif" alt="le" align="MIDDLE" BORDER="0">&lt;img src="/content/l1654304464l5236/xxlarge8734.gif" alt="infin" align="MIDDLE" BORDER="0">, then the functors &lt;i>L&lt;/i> &lt;sub>loc&lt;/sub> &lt;sup> &lt;i>p&lt;/i> &lt;/sup>(&lt;img src="/content/l1654304464l5236/xxlarge937.gif" alt="OHgr" align="BASELINE" BORDER="0">,&lt;i>N&lt;/i>&lt;img src="/content/l1654304464l5236/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0">.) and &lt;i>N&lt;/i>&lt;img src="/content/l1654304464l5236/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0">&lt;i>L&lt;/i> &lt;sup> &lt;i>p&lt;/i> &lt;/sup>(&lt;img src="/content/l1654304464l5236/xxlarge937.gif" alt="OHgr" align="BASELINE" BORDER="0">,.) are isomorphic on the category of b-spaces of L. Waelbroeck.