Lunar tidal variations inf 0F2 in the american zone during the years of low solar activity at huancayo,panama and buenos aires—A geomagnetic anomaly effect

Chapman’s Phase Rule\(L = \sum\limits_{n = 1}^4 { c_n } sin \left\{ {n_\tau + (n - 2)\nu + a_n } \right\}\) shown by him to be true for geomagnetic data is shown to be valid for the lunar oscillations inf ₀F₂ also at the tropical latitude stations Panama and Buenos Aires. Lunar semi-monthly (M₂) oscillations inf ₀F₂ at Panama are significantly larger during the day hours with a maximum amplitude near noon, while at Buenos Aires the tidal amplitudes are significant for all hours of the day and night and the maximum amplitude occurs in the afternoon around 15 hours. Regarding seasonal variation, the amplitude of lunar semi-diurnal or semi-monthly tides is largest during December solstices and least during June solstices.     

Acidity function in dipolar aprotic media

The H₀ acidity function has been determined for hydrochloric and trichloroacetic acids in dimethyl sulphoxide (DMSO) and acetonitrile using three amine class of indicators. The variation of H₀ with solvent composition at a fixed concentration of 0.1 M HCI in DMSO-water, acetonitrile-water and DMSO acetonitrile mixtures was also studied. Approximate medium effects on the proton in all these solvents were computed from the H₀ data and the results are discussed.     

A simple proof of Ramanujan’s summation of the1ψ1

A simple proof by functional equations is given for Ramanujan’s₁ ψ ₁ sum. Ramanujan’s sum is a useful extension of Jacobi's triple product formula, and has recently become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series.     

Computing the ℤ2-Cocharacter of 3 × 3 Matrices of Odd Degree

Let F be a field of characteristic 0 and A = M _{2, 1}(F) the algebra of 3 × 3 matrices over F endowed with the only non trivial ?₂-grading. Aver'yanov in [1 Aver'yanov , I. V. ( 2009 ). Basis of graded identities of the superalgebra M _{1, 2}(F) . Mathematical Notes 85 ( 4 ): 467 – 483 .[Crossref] , [Google Scholar]] determined a set of generators for the T ₂-ideal of graded identities of A. Here we study the identities in variables of homogeneous degree 1 via the representation theory of the symmetric group, and we determine the decomposition of the corresponding character into irreducibles.     

Vinyl polymerization

1:1 Poly (styrene-co-ethylmethacrylate) (PSEM) was prepared by bulk polymerization and fractionated using methylethyl ketone/petroleum ether system. IR and NMR analyses of the copolymer substantiated the composition kinetically expected. As the composition heterogeneity of the compolymer was narrow, the characterization of the copolymer was effected in a single solvent by light scattering as well as viscosity measurements at 35° C. The molecular weight dependence of [η], the limiting viscosity number;\(\left( {\bar r^2 } \right)w^{_2^1 } \), therms end to end distance and A₂ the second virial coefficient were established. The solvent ethylacetate was found to be thermodynamically more ideal for PSEM, than for polyethylmethacrylate. In an attempt to evaluate K, the Flory’s constant, application of various methods appeared to give almost similar values. In order to explain the low value ofΦ, the Flory’s universal parameter, rough estimates of\(\left( {\bar r_0 ^2 } \right)^{_2^1 } \), the unperturbedrms end to end distance;b, the statistical chain element;q, the persistence length and\({{\bar r_0 ^2 } \mathord{\left/ {\vphantom {{\bar r_0 ^2 } {l\bar r_{\max .} }}} \right. \kern-\nulldelimiterspace} {l\bar r_{\max .} }}\), the semiflexibility parameter were also made.     

Infrared absorption spectra of some lanthanide acetylacetonate complexes

The infrared absorption spectra of 12 lanthanide acetylacetonate complexes were measured in the region 400–2000 cm^?1 and discussed. Assignments of the bands especially those due to metal-oxygen (M—O), C=O and C=C stretching vibration is given. It is found that the band at 530 ± 5 cm^?1 is due mainly to Ln—O stretching vibration. In the carbonyl region, it is confirmed that the band at lower frequency is a C=C stretching vibration while that at higher frequency is due to C=O stretching vibration. Integral intensities for the M—O, C=O and C=C stretching vibrations were calculated in KB₁ and CHCl₃ solutions. The variation of the intensities of the M—O bands witn complexed cation were discussed in terms of crystal field stabilization energy (CFSE).     

Finite Groups with Given Quantitative Non-Nilpotent Subgroups II

As an extension of Shi and Zhang's 2011 article [4 Shi , J. , Zhang , C. ( 2011 ). Finite groups with given quantitative non-nilpotent subgroups . Comm. Algebra 39 : 3346 – 3355 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we prove that any finite group having at most 23 non-normal non-nilpotent proper subgroups is solvable except for G ? A ₅ or SL(2, 5), and any finite group having at most three conjugacy classes of non-normal non-nilpotent proper subgroups is solvable except for G ? A ₅ or SL(2, 5).     

A unified approach to singular problems arising in the membrane theory

We consider the singular boundary value problem \(({t^n}u't))' + {t^n}f(t,u(t)) = 0,{\rm{ }}\mathop {\lim }\limits_{t \to 0 + } {t^n}u'(t) = 0,{\rm{ }}{a_0}u(1) + {a_1}u'(1 - ) = A,\) where f(t, x) is a given continuous function defined on the set (0, 1]×(0,∞) which can have a time singularity at t = 0 and a space singularity at x = 0. Moreover, n ∈ ?, n ? >2, and a ₀, a ₁, A are real constants such that a ₀ ∈ (0,1), whereas a ₁,A ∈ [0,∞). The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in characterizing positive solutions. We illustrate the analytical findings by numerical simulations based on polynomial collocation.     

There exist no Ramanujan congruences mod 6912

Let τ(n) be Ramanujan's function, $$x\prod _{m = 1}^\infty (1 - x^m )^{24} = \sum\nolimits_{n = 1}^\infty {\tau (n)x^n .} $$ In this paper it is shown that the Ramanujan congruence τ(n)=σ_d/nd¹¹ mod 691 cannot be improved mod 691². The following result is proved: for arbitrary r, s mod 691 the set of primes such that p ≡ r mod 691,τ (p) ≡ p¹¹+1+691 · s mod 691² has positive density.     

Unique ergodicity for horocycle foliations

The notion of unique ergodicity is extended to multidimensional foliations, and it is shown that ifg is the strong stable or strong unstable foliation of a topologically mixing basic set Ω_κ for an Axiom A diffeomorphism or flow theng is uniquely ergodic.     

On the deficiency index of the vector-valued Sturm–Liouville operator

Let R₊:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R₊ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R₊ and the elements of the matrix functions P^?1, Q, and R are measurable on R₊ and summable on each of its closed finite subintervals. We study the operators generated in the space L_n²(R₊) by formal expressions of the form l[f] = ?(P(f' ? Rf))' ? R*P(f' ? Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = ?(P₀f')' + i((Q₀f)' + Q₀f') + P'₁f, where everywhere the derivatives are understood in the sense of distributions and P₀, Q₀, and P₁ are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P₀^?1 exists and ∥P0∥, ∥P₀^?1∥, ∥P₀^?1∥∥P₁∥², ∥P₀^?1∥∥Q₀∥² ∈ L_loc¹(R₊). Themain goal in this paper is to study of the deficiency index of the minimal operator L₀ generated by expression l[f] in L_n²(R₊) in terms of the matrix functions P, Q, and R (P₀, Q₀, and P₁). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where x_k, k = 1, 2,..., is an increasing sequence of positive numbers, with lim_k→+∞x_k = +∞, H_k is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.     

Growth of Cu2Cl2 and Cu2Br2 on single crystal faces and polycrystalline copper during dissolution in CuSO4 + H2SO4

Cu₂Cl₂ and Cu₂Br₂ precipitate and crystallise in the form of triangular pyramids and dendrites when Cu is immersed in aerated, acid CuSO₄ solution containing HC1 or HBr. The critical concentration of HC1 or HBr for this precipitation depends on the surface of the copper: Poly-crystalline < (110) < (100) < (111). In the deaerated solution there is no precipitation of Cu₂Cl or Cu₂Br₂ even at a high concentration of HC1 or HBr when only preferential etching occurs.     

On some new nonlocal solvability theorems for various classes of nonlinear differential equations

We study nonlinear boundary value problems of the form $$ [\Psi u']' + F(x;u',u) = g, u(0) = u(1) = 0 $$ , where Φ is a coercive continuous operator from L _p to L _q , and $$ F(x;u'',u',u) = g, u(0) = u(1) = 0 $$ ; first- and second-order partial differential equations $$ \Phi (x_1 ,x_2 ;u'_1 ,u'_2 ,u) = 0, \sum\limits_{i = 1}^\infty {[\Psi _i (u'_{x_i } )]'_{x_i } + F(x; \ldots ,u'_{x_i } , \ldots ,u) = g_i } $$ ; and general equations F(x; ..., u″ _ii , ...., ...., u′ _i , ...; u) = g(x) of elliptic type. We consider the corresponding boundary value problems of parabolic and hyperbolic type. The proof is based on various a priori estimates obtained in the paper and a nonlocal implicit function theorem.     

Solutions to m-point boundary value problems of third order ordinary differential equations at resonance

In this paper, we study the third order ordinary differential equation: $$x'''(t) = f(t,x(t),x'(t),x''(t)), t \in (0,1)$$ subject to the boundary value conditions: $$x'(0) = x'(\xi ), x'(1) = \sum\limits_{i - 1}^{m - 3} {\beta _i x'(\eta _i )} , x''(1) = 0.$$ Hereβ _i ∈R, $\sum\limits_{i = 1}^{m - 3} {\beta _i = 1, 0< \eta _1< \eta _2< \ldots< \eta _{m - 3}< 1, 0< \xi< 1} $ . This is the case dimKerL=2. When theβ _i have different signs, we prove some existence results for the m-point boundary value problem at resonance by use of the coincidence degree theory of Mawhin [12, 13]. Since all the existence results obtained in previous papers are for the case dimKerL=1, our work is new.     

Electron spin resonance of Cr5+ in CaWO4

Electron Spin Resonance of CaWO₄ with 0·1% of Cr has been investigated at liquid nitrogen and liquid helium temperatures. The observed ESR spectrum is attributed to Cr⁵⁺ ion in the substitutional site of W which has a compressed tetrahedral surroundings. A simple point charge calculation based on this geometry explains the observedg anisotropy and hyperfine anisotropy and places the magnetic electron in a predominantly \(3d_{z^2 } \) orbital. A comparison of these results with those obtained on other isoelectronic systems in similar and different co-ordinations justifies our assignment.     

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K _s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K _s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K _s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K _s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5 Cui , J. , Chen , J. ( 2011 ). When is a 2 × 2 matrix ring over a commutative local ring quasipolar? Comm. Alg. 39 : 3212 – 3221 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] are improved or extended.     

Faraday rotation tensor-extraction of components

This paper reports methods of obtaining the components of Faraday rotation tenser in anisotropic crystals, using longitudinal and transverse Faraday rotation along the optic axis. It is found that in crystals belonging to triclinic, monoclinic and orthorhombic symmetry, one can get 6 out of 9, 3 or 4 out of 5, and 2 out of 3 components respectively. In the case of uniaxial crystals only 1 component can be obtained. By measuring Faraday rotation in a randomly oriented polycrystal one more component in all the above classes can be obtained.