Ionospheric total electron content and slab-thickness at low latitudes in Indian zone

Observations of Faraday rotation of beacon signals from low orbiting satellite BE-B recorded at one station near the dip equator (Kodaikanal, dip 3·4° N) and at another station near the peak of the equatorial anomaly (Ahmedabad, dip 34° N) give a complete coverage of the equatorial anomaly belt in Indian zone. Contours of total elctron content (TEC) are obtained on a grid of latitudeversus local time for the different seasons of low (1964–66) and high (1967–69) solar activity epochs in the latitude belt 10° S to 26° N dip latitude. The development of the equatorial anomaly and its dependence on season and solar activity are discussed. Using similar contours of F₂ layer critical frequency, f₀F₂ contours of equivalent slab-thickness, τ are also constructed. The dependence of τ on season and solar activity and its implications on temperature are discussed.     

A study of blanketing sporadic E in the Indian equatorial region

The paper describes the nature of the sporadic E layer of blanketing type observed at Kodaikanal (Lat. 10°·2 N; Long. 77°·5 E; dip: 3°·5 N). It is shown that in the Indian equatorial region, the frequency of occurrence of this type of E _s is abnormally large when compared to similar latitudes in the west and that changes of large magnitudes occur in the F2 layer of the ionosphere as well as in the horizontal force of the earth’s magnetic field simultaneously with the appearance of the blanketing E _s . Lunar effects have also been shown to exist both in the time of appearance of blanketing E _s and in its strength. Some evidence exists to show that blanketing E _s as well as the blanketing frequency,f _b E _s , have a biennial maximum. These characteristics are discussed as part of the Far East Anomaly.     

The disappearance of equatorial E s and the reversal of electrojet current

Based on simultaneous observations of the horizontal geomagnetic field component H, sporadic E (E _s ) and E-W electron drifts at stations close to the dip equator within the equatorial electrojet region, it has been found that on quiet days and sometimes on disturbed days, when there is an abnormal large decrease in H during daytime, there is a simultaneous disappearance of E _s and a reversal of the direction of drift of electrons from westward to eastward. This suggests that the disappearance of equatorial E _s during day-time is due to a temporary reversal of the electrojet current, which is caused by the imposition of an additional electrostatic field opposite in direction to that of normal S _a field.     

The influence of disturbed conditions and increased solar activity on geomagnetic distortion of the equatorial ionospheric F2 region

The principal diurnal anomalies in the maximum electron density of ionospheric F2 layer over Kodaikanal are discussed. From an analysis of data obtained over half a solar cycle, it has been found that the phenomena of rapid increase in the electron densities following sunrise, the diurnal asymmetry and the maintenance of abnormally high ionic densities after sunset undergo systematic changes during stormy conditions. In order to examine whether these changes are accounted for by additional movements during disturbed conditions, magnitudes of movement terms of the continuity equation at three heights, for the quiet as well as the disturbed days, have been computed and discussed. The changes in the pattern of diurnal variation of N _m F2 with solar activity and the mechanism responsible for these changes are also discussed. From an analysis of published F2 layer data obtained at a number of stations with magnetic dip between ± 40°, it has been shown that the ratio of morning to afternoon peak densities yields a fair measure of equatorial F2 layer distortion anomaly. From solar cycle variation in diurnal asymmetry, an attempt has been made to estimate the extent by which the anomalous belt widens during sunspot maximum.     

Equatorial sporadic E ionization and electrojet effect of geomagnetic disturbances

In this paper, the ionospheric data of Kodaikanal (magnetic dip 3·5° N) have been examined to study the relation between the critical frequency of the equatorial sporadic E layer and the horizontal component of the earth’s magnotic field, with special reference to the behaviour of the electrojet both under normal and under disturbed conditions. Some specific SC type magnetic storms have been analysed to study the effect of the storms on the electrojet. It is shown that the blanketing frequency of Es can be taken to be equal to the critical frequency of the normal E-layer and it is suggested that the high value of foEs might be due to the subsidence and conversion of E layer ionization into eddy-clouds forming as a result of strong wind shear in the electrojet. It is also shown that foEs and H are fairly well connected undernormal conditions, but the connection is not simple under disturbed conditions. Further, the effect of a magnetic storm in reducing the electrojet current during the main phase is greater in the morning and evening hours; it appears that a depression of about 100 gammas in H-field is necessary before any significant reduction in the electrojet occurs at midday. Finally, some abnormalities in the variations of foEs and H are pointed out.     

Coupling between equatorial and auroral ionospheres during polar substorms

The intense substorm activity following the geomagnetic sudden commencement on 8 August 1972 at 2354 UT and on 9 August 1972 at 0037 UT had reversed the eastward electric field at the equatorial station Jicamarca between 0700 and 1200 UT, and this was preceded by similar changes in the westward electric field in the auroral region. This abnormal change at the equatorial latitudes in the American zone (in the night-time) was closely associated with the westward electric field in the Indian zone (in the day-time) causing the counter-electrojet and the sudden disappearance of theq type of equatorialEs. The reversal of the electric field at Jicamarca had caused the generation of theF region irregularities associated with range spreadF. It is suggested that on some occasions the counter equatorial electrojet currents are caused due to the reversal of the equatorial electric fields by the spreading of the auroral electrojet currents to the low latitudes following geomagnetic storms. This suggests a possible link between the equatorial ionosphere and the magnetosphere through the auroral latitudes.     

Lacunary subsets of orthonormal sets

В РАБОтЕ РАссМАтРИВА УтсьS _Р-пОДсИстЕМы О. Н.с. В ЧАстНОстИ, ДОкАжыВА Етсь слЕДУУЩАь тЕОРЕ МА, кОтОРАь НЕУсИльЕМА. тЕОРЕМА.пУсть Р>2 —ЧЕ тНОЕ ЧИслО, δ — пРОИжВО льНОЕ ЧИслО, 0<δ≦p?2,Φ= {Φ _n(x)} _n=1 ^N —O.H.C.,x?[0,1],пРИЧЕМ ∥ Φ _n∥_p≦M, n=1,2,...,N, гДЕР=Р+δ, 0М<∞. тОгДА Иж сИстЕМы Ф МОж НО ВыБРАть пОДсИстЕМ У \(\Phi ' = \left\{ {\varphi _{n_k } } \right\}_{k = 1}^{N'} ,N' \geqq N^{\alpha (\delta )} ,\alpha (\delta ) = \frac{{2\delta }}{{p(p - 2 + \delta )}}\) , тАкУУ, ЧтО Дль лУБОгО п ОлИНОМА \(P(x) = \sum\limits_{k = 1}^{N'} {a_k \varphi _{n_k } (x)} \) ИМЕЕ т МЕстО ОцЕНкА $$(\mathop \sum \limits_{k = 1}^{{\rm N}'} a_k^2 )^{1/2} \leqq \left\| P \right\|_p \leqq c_{p,M,\delta } (\mathop \sum \limits_{k = 1}^{{\rm N}'} a_k^2 )^{1/2} $$ (c _p, m, δ — пОстОьННАь, жАВИ сьЩАь тОлькО Отp, M, δ, НО НЕ От N ИлИ кОЁФФИцИЕНтОВ пО лИ-НОМА). пРИВОДьтсь И ДРУгИЕ РЕжУльтАты А НАлОгИЧНОгО хАРАктЕ РА.     

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.     

Some remarks on the equatorial sporadicE layer at kodaikanal

It is shown that the sporadicE layer at an equatorial station is not always of the commonly known equatorial (q) type Es. Broadly there occur two types of Es at a station near the magnetic equator,viz., (i) diffuseq type and (ii) multiple or blanketing (b) type. The latter may be of diffuse type or of thin layer type. The Es-b is due to highly dense cloud of ions which are localised, whereas Es-q occurs over an extended area at any time. The effect of counter-electrojet is seen on both the types of Es layers. The disappearance of sporadicE during counter-electrojet is valid for only theq type of Es, but sometimes Es-b may occur during periods of very weak or reversed electrojet. It is suggested that published equatorial Es data should be used with caution unless the type of Es is also indicated. The plausible causes of the sudden disappearance of Es-q on some occasions and the appearance of Es-b during others are suggested.     

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.     

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.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-boundedness of a boundary integral operator on a contour with a peak

Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ? _{p, β + 1} (Γ) continuously onto \(\mathcal{N}_{p,\beta } (\Gamma )\). The norm of a function in ? _{p, β} (Γ) is defined as

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, provided that the peak is at the origin. In this case, the norms on the spaces \(\mathcal{N}_{p,\beta }^ \mp (\Gamma )\) are defined by

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, where q _± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ? _{p, β + 1} (Γ) onto ? _{p, β}(Γ), where ? _{p, β}(Γ) is the direct sum of \(\mathcal{N}_{p,\beta }^ + (\Gamma )\) (Γ) and the space

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(Γ) of functions on Γ of the form p(z) = Σ _{k = 0} ^m t ^(k)Rez ^k with the parameter m = [μ ? β ? p ^?1]. The operator I ? 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I ? 2W continuously maps the space ? _{p, β + 1} × ? _{p, β + 1}(Γ) to the space \(\mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma )\).

On a contour with an inward peak, the obtained representation of the operator I ? 2W and theorems on the boundedness of auxiliary integral operators imply that the images of vector-valued functions from ? _{p, β + 1} × ? _{p, β + 1}(Γ) have components representable as sums of functions from the spaces \(\mathcal{N}_{p,\beta }^ - (\Gamma )\)(Γ) and ? _{p, β}(Γ).     

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.     

Ramanujan — Fourier series and a theorem of Ingham

Given two arithmetical functions f, g, we derive, under suitable conditions, asymptotic formulas for the convolution sums ∑ _n≤N f (n) _g (n + h) for a fixed number h. To this end, we develop the theory of Ramanujan expansions for arithmetical functions. Our results give new proofs of some old results of Ingham proved by him in 1927 using different techniques.     

Raman spectrum of dimethyl sulfoxide (DMSO) and the influence of solvents

The Raman spectrum of DMSO is recorded with a Hilger two-prism spectrograph andλ 4358 Å excitation. In addition to all the Raman lines reported earlier, six new lines at 898, 925, 1223, 1309, 2811 and 2871 cm.^?1 are observed and tentative assignments are given. The influence of solvents (CCl₄, CHCl₃, CH₃COOH) on the S=O bond is also studied. A shift from the liquid phase value,i.e., 1043 cm.^?1 to 1054, 1052 and 1009 cm.^?1 in the respective solvents is observed. The possibilities of association effects and hydrogen bonding are discussed.     

NMR studies of solvent-solute interactions of OH,SH and CH protons in dimethyl formamide and dimethyl acetamide

The H-bonding of CH₃OH, C₆H₅SH and CHCl₃ with N:N Dimethyl formamide and N:N Dimethyl acetamide has been studied using proton magnetic resonance. It has been observed that the strength of interaction is greater for dimethyl acetamide than for dimethyl formamide for the same solvent. For the same solute, however, the strength increases in the order chloroform, thiol and alcohol. The equilibrium constant for 1:1 complex of these solutes in the chloroform solvent has been estimated. In the case of benzene thiol, complex formation between the pi-electrons of the phenyl ring and the amides has been detected. It has also been observed that the exchange rate of OH proton of CH₃OH in the dimethyl formamide decreases considerably.     

An SDP randomized approximation algorithm for max hypergraph cut with limited unbalance

We consider the design of semidefinite programming(SDP) based approximation algorithm for the problem Max Hypergraph Cut with Limited Unbalance(MHC-LU): Find a partition of the vertices of a weighted hypergraph H =(V, E) into two subsets V1, V2 with ‖V2|- |V1‖ u for some given u and maximizing the total weight of the edges meeting both V1 and V2. The problem MHC-LU generalizes several other combinatorial optimization problems including Max Cut, Max Cut with Limited Unbalance(MC-LU), Max Set Splitting,Max Ek-Set Splitting and Max Hypergraph Bisection. By generalizing several earlier ideas, we present an SDP randomized approximation algorithm for MHC-LU with guaranteed worst-case performance ratios for various unbalance parameters τ = u/|V|. We also give the worst-case performance ratio of the SDP-algorithm for approximating MHC-LU regardless of the value of τ. Our strengthened SDP relaxation and rounding method improve a result of Ageev and Sviridenko(2000) on Max Hypergraph Bisection(MHC-LU with u = 0), and results of Andersson and Engebretsen(1999), Gaur and Krishnamurti(2001) and Zhang et al.(2004) on Max Set Splitting(MHC-LU with u = |V|). Furthermore, our new formula for the performance ratio by a tighter analysis compared with that in Galbiati and Maffioli(2007) is responsible for the improvement of a result of Galbiati and Maffioli(2007) on MC-LU for some range of τ.     

Bergman type operators on a class of weakly pseudoconvex domains

A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK _x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω _a . As an appli cation, it is proved thatf?L _H ^p (ω _a ,dv _λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α₁,?,α _n and ß = (ß₁, —ß). An interesting question is whether the converse holds.     

Asymptotic integration of symmetric second-order quasidifferential equations

This paper presents conditions on the coefficients of the equations $$\begin{array}{*{20}c} { - (p(f' - rf))' - \bar rp(f' - rf) + qf = 0,} \\ { - (P(f' - Rf))' - \bar RP(f' - Rf) + Qf = 0,} \\ \end{array}$$ where 1/p, 1/P, q, Q, r, R ∈ ? _loc ¹ (?₊), p, P, q, and Q are real-valued functions, while r and R are complex-valued functions, as well as on the fundamental system of solutions of the second equation, which ensure the asymptotic proximity of the solutions of these equations. The results obtained are applied to the study of the spectral properties of the differential operator generated by the expression $$ - y'' + \sum\limits_{k = 0}^{ + \infty } {h_k \delta (x - x_k )y, x_k \in \mathbb{R}_ + ,} h_k \in R,$$ , in the space ?²(?₊). In particular, we obtain conditions on h _k , x _k under which the limit-disk case is realized for this operator.     

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u _ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V ₀(y) w = w ^p , w > 0 in Ω₀, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V ₀(x) and Ω₀ depend on the local behaviors of V(x).