Spread-F in the ionosphere over ahmedabad during the years 1954–57

From a study of spread-F or F-scatter at Ahmedabad during the four years 1954–57 of increasing sunspot activity, it was found that the time of its maximum occurrence receded from 03 hr. in low sunspot years to an hour or two before midnight in high sunspot years. This was particularly well seen in the winter and equinoctial months. Also, maximum spread-F activity which was found in summer in sunspot minimum mum years, occurred in equinoxes in maximum sunspot years. The frequency of occurrence of spread-F was found to be a maximum whenhpF₂ was in the range 300–350 km. F-scatter and F₂-stratification were found to be anti-correlated both in their diurnal and seasonal variations. The general trend was towards decreased spread-F with increased sunspot activity. It is concluded that (1) spread-F at Ahmedabad geomagnetic latitude (Φ=13·6° N) undergoes variations similar to those at equatorial stations, more so in high sunspot years, (2) the change-over from low-latitude type to middle-latitude type of variation of spread-F takes place at about geomagnetic latitude 22°, and (3) spread-F at Ahmedabad decreases with increase in magnetic activity, which is the reverse of that observed at high latitudes.     

A high resolution spectral analysis of daily relative sunspot number and 10·7 cm. Solar flux

Auto and cross-spectra of relative sunspot number and slowly varying component of solar radiation at 10·7 cm. have been computed for a 52-month period beginning September 1, 1958. Significant features of the spectra are relatively high variance at periods corresponding to one, three and four solar rotations. Statistically significant spectral peaks have been observed at three other frequencies and are ascribed to amplitude modulation of the 27-day component. The cross-spectral analysis indicates that during the period under investigation the solar 10·7 cm. flux leads the sunspot number for periods in excess of about 27·7 days; for shorter periods the flux lags behind the sunspot number. The coherence between the two time series, after an initial decrease from unity at zero frequency, assumes a maximum value of 0·985 at 27·7 days. The phase and coherence indicate that long-lived radio emission regions and spots appeared to co-rotate during 1958–62 with a period of 27·7 days.     

Rigidity and stability of the Leibniz and the chain rule

We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T ₁, T ₂,A: C ^k (?) → C(?) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ C ^k (?), namely, V (f · g) = (T ₁ f) · g + f · (T ₂ g) for k = 1, V (f · g) = (T ₁ f) · g + f · (T ₂ g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T ₁ f) ○ g · (T ₂ g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T ₁ f) · (Ag) + (Af) · (T ₂ g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T ₁ and T ₂ must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · Tg + B(f, g), and show under suitable conditions on B in the first case that B = 0 and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation.     

A polarographic study of methionine complexes of cadmium in mixed solvents

Methionine complexes of cadmium in 25 and 50 per cent aqueous mixtures of ethyl and methyl alcohol and dioxan have been studied. The half-wave potentials measured in both the alcohols were the same and the reduction was reversible. Three complex species withβ ₁=1·0×10⁴,β ₂=1·1×10⁷ andβ ₃=1·2×10⁹ were found in 25 per cent alcohol while four complexes withβ ₁=3·0×10⁴,β ₂=4·3×10⁷,β ₃=4·0×10⁹ andβ ₄=1·6×10¹¹ were observed in 50 per cent solutions. In the case of dioxan, the reduction was quasi-reversible (k _s=1·0×10^?3 cm sec^?1) in 25 per cent and irreversible (k _s=2·0×10^?4 cm sec^?1) in 50 per cent solutions. The stability constants, evaluated using the formal potentials, wereβ ₁=7·0×10³,β ₂-3·9×10⁵;β ₂=3·9×10⁸ andβ ₄=3·4×10¹⁰ in 25 per cent dioxan andβ ₁=1·5×10⁴,β ₂=3·4×10⁷.β ₃=7·5×10⁹ andβ ₄=9·0×10¹¹ in 50 per cent solutions.     

Relative widths of function classes of L 2(T) defined by a linear differential operator in L q (T)

In this paper, the smallest number M which makes the equality $$ K_n (W_2^{L_r } (T),MW_2^{L_r } (T),L_2 (T)) = d_n (W_2^{L_r } (T),L_2 (T)) $$ valid, is established and the asymptotic order of $$ K_n (W_2^{L_r } (T),W_2^{L_r } (T),L_q (T)),1 \leqslant q \leqslant \infty $$ , is obtained, where $ W_2^{L_r } $ (T) is a periodic smooth function class which is determined by a linear differential operator, K _n (·, ·, ·) and d _n (·, ·) are the relative width and the width in the sense of Kolmogorov, respectively.     

Measure preserving composition operators

We consider composition operators T induced on functional Hilbert spaces H = L₂(S, ∑, μ) byTf(·) = f(h(·)) where h: S → S is a nonsingular transformation. For these mappings T: H → H we give conditions under which they accept invariant Borel probability measures, and we relate the two structures of T, i.e., that of a bounded linear operator to that of a measure preserving transformation.     

A polarographic study of methionine complexes of cadmium and indium

The stability constants of methionine complexes of cadmium were determined polarographically by ths method of DeFord and Hume as β₁ =6·5 × 10³, β₂ = 1·7 × 10⁶ and β₃ = 2·1 · 10⁸. The indium complexes were studied by the modified method of Momoki and Ogawa and two complexes, with β₁ = 1·7 × 10⁸ and β₂ = 8·4 × 10¹³, were identified in the concentration range studied. The haf-wave potential of uncomplexed indium ion which cannot be measured directly owing to the irreversible nature of the reduction was calculated as — 0·503 Vvs. SCE.     

A class of nonlinear elliptic problems

We obtain a strict coercivity estimate, (generalizing that of T. I. Seidman [J. Differential Equations 19 (1975), 242–257] in considering spatial variation) for second order elliptic operators A: u ? ?▽ · γ(·, ▽u) with γ “radial in the gradient” ?γ(·, ξ) = a(·, |ξ|)ξ for ξ ? $\text{R}$^m. The estimate is then applied to obtain existence of solutions of boundary value problems: $\text{?▽ ·}\text{a}\text{?}\text{(·, u, |▽u|) ▽u = f(·, u, ▽u)}$ with Dirichlet conditions.     

Emission spectrum of bromine excited in the presence of argon

The wavelengths and wavenumbers of the band heads of the system 3150–2970 Å as obtained from the plates taken on the first order 21′ grating spectrograph are given along with the vibrational analysis. This system is shown to be due to a transition from an upper electronic state at T_e = 48516 cm.^-1 with ω′ _e = 162·0 cm.^?1 and ω′ _e χ′ _e = 0·29 cm.^?1 to the well-known³ Π _u (O _u ⁺) state at T_e = 15918 cm.^-1 This lower state is common with that of the system 2950–2670 Å.     

On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)?B(t)y(t) = Δ(t) f(t) on an interval ${\mathcal{I}=[a,b) }$ with the regular endpoint a. It is assumed that the deficiency indices n _±(T _min) of the minimal relation T _min associated with this system in ${L^2_\Delta(\mathcal{I})}$ satisfy ${n_-(T_{\rm min})\leq n_+(T_{\rm min})}$ . We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size ${N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}$ . Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform ${V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}$ where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension ${\tilde{T}}$ of T _min induced by the boundary problem is unitarily equivalent to the multiplication operator in L ²(Σ). Hence the multiplicity of spectrum of ${\tilde{T}}$ does not exceed N _Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.     

Trivectors yielding spreads in PG(5, 2)

Given an alternating trilinear form ${T\in {\rm Alt}(\times^{3}V_{6})}$ on V ₆ = V(6, 2) let ${\mathcal{L}_{T}}$ denote the set of those lines ${\langle a, b \rangle}$ in ${{\rm PG}(5,2)=\mathbb{P}V_{6}}$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all ${x\in {\rm PG}(5, 2).}$ If ${\mathcal{L}_{21}}$ is a Desarguesian line-spread in PG(5, 2) it is shown that ${\mathcal{L}_{T}=\mathcal{L}_{21}}$ for precisely three choices T ₁,T ₂,T ₃ of T, which moreover satisfy T ₁ + T ₂ + T ₃ = 0. For ${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$ the ${\mathcal{G}_{T}}$ -orbits of flats in PG(5, 2) are determined, where ${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each ${\mathcal{G}_{T}}$ -orbit, the T-associate U ^# is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .     

Self-Affine Sets and Graph-Directed Systems

Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ? Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.     

UCB revisited: Improved regret bounds for the stochastic multi-armed bandit problem

In the stochastic multi-armed bandit problem we consider a modification of the UCB algorithm of Auer et al. [4]. For this modified algorithm we give an improved bound on the regret with respect to the optimal reward. While for the original UCB algorithm the regret in K-armed bandits after T trials is bounded by const · $ \frac{{K\log (T)}} {\Delta } $ , where Δ measures the distance between a suboptimal arm and the optimal arm, for the modified UCB algorithm we show an upper bound on the regret of const · $ \frac{{K\log (T\Delta ^2 )}} {\Delta } $ .     

Molecular lines and continuum from W51A (I) —CO and NH3 spectra and 3 mm continuum

As for the 5′ × 4′(～llpc × 9pc) region centered at W51 lRSl the observations of the 3.4 mm continuum, CO (J = 1-0) line and simultaneous NH₃(1,1), (2,2), (3,3), (4,4) inverse lines were made for studying the massive star formation region located in the main spiral arms of the Galaxy. In the directions of W51 IRS1, IRS2 and el/e2 in 3.4 mm continuum, analyses of the line profiles show that the absorption lines of ammonia, which arise from the gas in front of the HII region, are red-shifted with respect to the emission lines, which arise from the surrounding cloud. Furthermore, a radiation transfer and statistical equilibrium calculation of ammonia molecules show that the densities increase by 3–10 times from the eastern border to the center. These points hint that the collapse is happening in the molecular cloud core obscured in optical wavelengths. The effects of the radiation fields from radio, infrared and UCHII sources is non-negligible on the excitation of various molecules (e.g. NH₃) within the circle of radius 40″ centered at IRS1. The profiles of the COJ = 1–0 line in the circle change from double peaks ( ～ 60, ～ 68 km. s^-1) to triple peaks, i.e. the component ～53 km·s^?1, which associates with UCHII, also appears in the spectra. There are indications that the circle of radius 40″ centered at IRSI is a region of massive star forming activity     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

Wermer Examples and Currents

We give the first examples of positive closed currents T in ${{\mathbb C}^2}We give the first examples of positive closed currents T in \mathbb C²{{\mathbb C}^2} with continuous potentials, T ùT = 0{T \wedge T = 0}, and whose supports do not contain any holomorphic disk. This gives in particular an affirmative answer to a question of Forn?ss and Levenberg. We actually construct examples with potential of class C ^1,α for all α < 1. This regularity is expected to be essentially optimal.     

Curvature inequalities for operators of the Cowen–Douglas class

Let T be an operator tuple in the Cowen–Douglas class B _n (Ω) for Ω ? C ^m . The kernels Ker(T ? w) ^l , for w ∈ Ω, l = 1, 2, ···, define Hermitian vector bundles E _T ^l over Ω. We prove certain negativity of the curvature of E _T ^l . We also study the relation between certain curvature inequality and the contractive property of T when Ω is a planar domain.     

Microwave spectrum of pentafluorobenzonitrile

The absorption spectrum of pentafluorobenzonitrile has been investigated in the frequency range of 18–26·5 GHz using a 100 KHz stark modulated microwave spectrometer. The analysis of the spectrum is based on the rigid asymmetric rotor theory. The rotational constants obtained are A=1026·82±0·3 MHz, B=776·34±0·1 MHz, C=442·06±0·1 MHz and the asymmetry parameterχ=+0·1433. The inertial defect is I _o ?I _a ?I _b =0·081 amu Ų. The bond distances ared _CF=1·328 Å andd _CN=1·157 Å. The results are in good agreement with the assumed planarity of the molecule and the normal values of bond distances.     

ON THE MINIMUM DISTANCE DETERMINED BY n（≤7）POINTS IN AN ISOSCELE RIGHT TRIANGLE

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