The emission spectrum of iodine bromide excited in the presence of Argon

IBr vapour was excited in the presence of argon by an uncondensed transformer discharge. Four band systems were obtained in the regions 5425–5360 Å, 4520–4415 Å, 4120–4010 Å and 3915–3540 Å of which the first three are discussed in this paper. The wavelengths and wavenumbers of the band heads in three systems as measured from the plates obtained with a 3-prism Steinheil glass spectrograph are given along with their visually estimated relative intensities. The three band systems, which are new, are analysed and the following vibrational constants expressed in cm.^?1 are obtained:

Band system	v e	w e ″	w e ″x e ″	w e ″y e ″	w e ′	w e ′x e ′
5425-5360 Å	18613	65·5	0·24	?0·01	43·0	0·026
4520-4415 Å	22312	65·5	0·24	?0·01	77·0	0·5
4120-4010 Å	24540	160·6	1·125	..	128·4	0·1

     

Emission spectrum of chlorine excited in the presence of argon

The weak band system in the region 2365-2239 Å is discussed in this paper. The wavelengths and the wavenumbers of the bands photographed with the first order of a 21-ft. grating spectrograph are recorded. The vibrational analysis of the bands and their corresponding intensity distribution are also given. The analysis shows that the lower state of the system is the same as that of the 2600-2390 Å system discussed in the earlier paper and is the 3π (O_u ⁺) state established by Elliott at 17658 cm.^?1 The constants of the upper state arew ₀′ = 261·5 cm.^?1,w ₀′x ₀′ = 0·812 cm.^?1, T₀=61290 cm.^?1.     

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.     

Emission spectrum of bromine excited in the presence of argon

The wavelengths and wavenumbers of the band heads of the system 2660-2590 Å as obtained from the plates taken on the first order 21-feet grating spectrograph are given along with its vibrational analysis. This system is shown as the transition from an upper state at T _e =56776 cm.^?1 withω _e = 108·0 cm.^?1 to the³ Π _u (O _u ⁺) state at T _e =15918 cm.^?1 The lower state is the same as that of the two systems in the regions 2950-2670 Å and 3150-2970 Å reported earlier.     

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.     

F2Δr−A2Πi band system of CN

The spectrum of CN has been studied in condensed electrical discharge through flowing CH₃CN vapour. Ten new bands of F²?,?A²Π_i system in the region 2100–2700 Å are obtained. The vibrational constantsω ₀′ = 1229.7 cm.^?1 and ω₀′χ₀′= 14.0 cm.^?1 are obtained for the first time for the²Δ state of CN.     

Microwave spectrum of pentafluorobenzene

The microwave spectrum of pentafluorobenzene has been measured in the frequency region 12·5 to 18·00 KMHz. The spectrum is analysed on the basis of rigid asymmetric rotor theory. The rotational constants obtained are A=1471·963 MHz, B=1026·278 MHz, C=604·693 MHz and the asymmetry parameterκ=?0·0278. The inertial defect is Δ=I _c -I _a -I _b =?0·014 amu Ų. The bond distances ared _CF=1·328 Å andd _CH=1·089 Å, assumingd _CC=1·395 Å.     

Rotational analysis of some bands of C1 Σ u + -X1 Σ g + system of P2 molecule involving lowν′,ν″ values

Ten ultraviolet bands of the C¹ Σ _u ⁺ -X¹ Σ _g ⁺ system of P₂ involving lowv′ andv″ values have been photographed at dispersion of 0·38 and 0·56 Å/mm. and analysed for their rotational structure. While four of these bands were analysed earlier, six of them,viz., 0–10, 1–12, 2–7, 2–14, 4–8 and 6–9 have been analysed for the first time during the present studies. The rotational constants, B _v S with lowv″ quantum numbers are obtained from which value of B _θ ^″ has been derived. The value of B _θ ^″ is found to be in agreement with the value obtained by Douglas and Rao from their study of A¹ Π _g-X¹ Σ _g ⁺ bands of P₂. Earlier findings on the perturbation ofν′=2 level of the C¹ Σ _u ⁺ state have been confirmed from the analysis of the 2–7, 2–14 and 2–15 bands. Theν ₀₀ values of the bands show large deviations from the expected values.     

Rotational analysis of some of the bands of the orange system of FeO

The orange bands of FeO are excited in a low pressure arc in oxygen and photographed at dispersions of 0·8 and 0·5 Å/mm. respectively. Rotational analysis of five of the bands shows that they involve a¹ Σ-¹ Σ transition. The vibrational and rotational constants (in cm.^?1) of the upper and lowerΣ states are found to be

	Lower state	Upper state
B e	0·3491	0·3063
α e	0·0029	0·0032
w e	870·7	(ΔG½=770·0)
w e x e	4·39	..

     

Quantization causes waves: Smooth finitely computable functions are affine

Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are F _p = {0, 1,..., p - 1} produces a 1-Lipschitz map f _A from the space Z _p of p-adic integers to Z _p . The map fA can naturally be plotted in a unit real square I² ? R²: To an m-letter non-empty word v = γ _m-1γ _m-2... γ0 there corresponds a number 0.v ∈ R with base-p expansion 0.γ _m-1γ _m-2... γ0; so to every m-letter input word w = α _m-1α _m-2 ··· α0 of A and to the respective m-letter output word a(w) = β _m-1β _m-2 ··· β0 of A there corresponds a point (0.w; 0.a(w)) ∈ R². Denote P(A) a closure of the point set (0.w; 0.a(w)) where w ranges over all non-empty words.We prove that once some points of P(A) constitute a C ²-smooth curve in R², the curve is a segment of a straight line with a rational slope. Moreover, when identifying P(A) with a subset of a 2-dimensional torus T² ∈ R³, the smooth curves from P(A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ(x, t) = e ^i(Ax-2πBt) (x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter.     

On real solutions of systems of equations

Systems of equations f ₁ = ··· = f _n?1 = 0 in ? ⁿ = {x} having the solution x = 0 are considered under the assumption that the quasi-homogeneous truncations of the smooth functions f ₁,..., f _n?1 are independent at x ≠ 0. It is shown that, for n ≠ 2 and n ≠ 4, such a system has a smooth solution which passes through x = 0 and has nonzero Maclaurin series.     

Characteristics ofπ −-nucleon collisions at a primary energy of 4·4 GeV

Eight hundred and sixteen nuclear interactions produced by 4·4 GeVπ ^?-mesons in nuclear emulsion have been obtained by “along the track” scanning procedure. Favourable secondary particles from a selected 290π ^?-N (pion-nucleon) collisions have been identified by blob-density and multiple scattering measurements. It is found that the pion often persists in theseπ ^?-N collisions, the average persistence is estimated to be 0·24 per collision. It is estimated thatπ-N andπ-π collisions account for 60% and 28% respectively of the secondary particles. The average number of charged shower particles is 〈n _s〉=2·09±0·12, the average number of created charged particles is 〈n _e±〉=1·94±0·12, the average number of protons with energy greater than 300 MeV is 〈n _p〉=0·15±0·05 and the average number of charged kaons is found to be 〈n _k±〉=0·11±0·06. The integral energy spectra of pions in C-system as well as in L-system are well represented by exponential forms. The average inelasticity of the proton in C-system is found to be 0·52±0·10. The charge retention probability for protons inπ ^?-p collisions is 0·45±0·07.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Isotope shift studies of the ultra-violet and visible bands of P16O and P18O

The spectra of P¹⁶O and P¹⁸O were excited in sealed discharge tubes containing neon (2–3 mm. pressure), oxygen gas enriched to 65 per cent. of¹⁸O and trace amounts of phosphorus vapour and photographed on a 3 m. grating spectrograph at a dispersion of 2·5 Å/mm. Isotope shift studies in theβ-bands confirmed the earlier vibrational scheme of Curryet al. and showed conclusively that the red as well as the violet degraded bands belonged to the sameβ-system. The present studies of isotope shifts also confirmed the vibrational assignments of the extensive ultraviolet bands involving the² Σ ^??X² Π transition and theγ-bands (A² Σ ⁺?X² Π). In the case of the visible bands, they provided evidence for the first time that the bands at 5585 Å, 5962 Å and 6385 Å belonged to one system and involved 0–0, 0–1 and 0–2 transitions respectively.     

Ranks of matrices with few distinct entries

It is consistent that P(ω ₁) is the union of less than \({2^{{\aleph _1}}}\) parts such that if A ₀,..., A _n?1, B ₀,..., B _m?1 are distinct elements of the same part, then |A ₀ ∩ · · · ∩ A _n?1 ∩ (ω ₁ ? B ₀) ∩ · · ·∩ (ω ₁ ? B _m?1)| = N₁.     

Word maps and word maps with constants of simple algebraic groups

In the present paper, we consider word maps w: G ^m → G and word maps with constants w _Σ: G ^m → G of a simple algebraic group G, where w is a nontrivial word in the free group F _m of rank m, w _Σ = w ₁ σ ₁ w ₂ ··· w _r σ _r w _{r + 1}, w ₁, …, w _{r + 1} ∈ F _m , w ₂, …, w _r ≠ 1, Σ = {σ ₁, …, σ _r | σ _i ∈ G Z(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γ_w, G) of the group Γ_w = F _m /<w>.     

Vertical soundings at Ahmedabad during partial solar eclipse on 14th December 1955

The results of vertical ionospheric soundings at Ahmedabad during the partial (magnitude 47%) solar eclipse on 14th December 1955 are discussed. The critical frequency of the E layer on control days varied according to the lawf _oE=kcos ⁿ X, k andn having the mean values 4·06 Mc./s. and 0·37 respectively. The variation off _oE during eclipse period was roughly consistent with that of Chapman Layer with a recombination coefficient of about 0·8 × 10^?8 cm.³ sec.^?1 The critical frequencies of E₁, E₂ and F₁ layers decreased during eclipse, though the variations were irregular. The percentage deviation off ₀F₂ on the eclipse day from the control day values showed a minimum at about the middle of the eclipse period. There were indications of the formation off _1·5 during the latter part of the eclipse. Semi-thickness of the F₂ layer had increased during the eclipse period. Sporadic E layer was not affected by the eclipse.     

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.     

An extended class of stabilizable uncertain systems

The system of equations \(\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u\), where A(·) ∈ ?^{n × n}, B(·) ∈ ?^{n × m}, S(·) ∈ R^{n × m}, is considered. The elements of the matrices A(·), B(·), S(·) are uniformly bounded and are functionals of an arbitrary nature. It is assumed that there exist k elements \({\alpha _{{i_i}{j_l}}}\left( \cdot \right)\left( {l \in \overline {1,k} } \right)\) of fixed sign above the main diagonal of the matrix A(·), and each of them is the only significant element in its row and column. The other elements above the main diagonal are sufficiently small. It is assumed that m = n ?k, and the elements β_ij(·) of the matrix B(·) possess the property \(\left| {{\beta _{{i_s}s}}\left( \cdot \right)} \right| = {\beta _0} > 0\;at\;{i_s}\; \in \;\overline {1,n} \backslash \left\{ {{i_1}, \ldots ,{i_k}} \right\}\). The other elements of the matrix B(·) are zero. The positive definite matrix H = {h_ij} of the following form is constructed. The main diagonal is occupied by the positive numbers h_ii = h_i, \({h_{{i_l}}}_{{j_l}}\, = \,{h_{{j_l}{i_l}}}\, = \, - 0.5\sqrt {{h_{{i_l}}}_{{j_l}}} \,\operatorname{sgn} \,{\alpha _{{i_l}}}_{{j_l}}\left( \cdot \right)\). The other elements of the matrix H are zero. The analysis of the derivative of the Lyapunov function V(x) = x*H^–1x yields h_i\(\left( {i \in \overline {1,n} } \right)\) and λ_i ≤ 0 \(\left( {i \in \overline {1,n} } \right)\) such that for S(·) = H^?1ΛB(·), Λ = diag(λ₁, ..., λ_n), the system of the considered equations becomes globally exponentially stable. The control is robust with respect to the elements of the matrix A(·).     

The limit absorption principle and homogenization procedure for periodic elliptic operators

For a periodic matrix elliptic operator \(A_\varepsilon \) with (x ?-dependent) rapidly oscillating coefficients, a certain analog of the limit absorption principle is proved. It is shown that the bordered resolvent 〈x〉^?1/2?· (\(A_\varepsilon \) ? (η ± i? ^σ )I)^?1〈x〉^?1/2?· has a limit in the operator norm in L ₂ as ? → 0 provided that η > 0, · > 0, and 0 < σ < 1/2.