On characterizing equilibrium points in two person strictly competitive games

A game is strictly competitive if all possible outcomes are Pareto optimal. It has been known that ifs′ = (s′ ₁,s′ ₂) ands″ = (s″ ₁ s″ ₂) are equilibrium points in a two person strictly competitive game, then payoffs are the same ats′ as ats″ and that (s′ ₁,s″ ₂) and (s″ ₁ s′ ₂) are equilibrium points as well. It is proved in this paper that, for 0 ?k?1,ks′ + (1?k)s″ is also an equilibrium point.     

Regularized traces of higher-order singular differential operators

We consider singular differential operators of order 2m, m ∈ ?, with discrete spectrum in L ₂[0, + ∞). For self-adjoint extensions given by the boundary conditions y(0) = y″(0) = ? = y ^(2m?2)(0) = 0 or y′(0) = y?(0) = ? = y ^(2m?1)(0) = 0, we obtain regularized traces. We present the explicit form of the spectral function, which can be used for calculating regularized traces.     

Rotational analysis of the C2Σ+ — X2Σ+ system of AlO

The 2-0, 1-0, 0-0 and 0-1 bands of the C²Σ⁺-X²Σ⁺ system of AlO, lying at 2391.88 Å, 2438.35 Å, 2487.32 Å and 2548.51 Å respectively were excited in a low pressure arc and photographed on a 6.6-metre concave grating spectrograph in the third order at a dispersion of 0.37 Å/mm. From the rotational analysis of these bands, the following constants have been determined: B_e′ = 0.565₃ cm.^?1 a_e′ = 0.004₈ cm.^?1 T_e =40267.0 cm.^?1 B_e″ = 0.641₃ cm.^?1 a_{″ = 0.005 7} cm.^?1     

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.     

Thermal expansion of ferroelectrics

Using the interference method the thermal expansion of lithium hydrazinium sulphate [Li(N₂H₅)SO₄] has been investigated in the temperature range ?170° C. to +220° C. The principal expansion coefficients along the crystallographic axes have been determined and at room temperature (25° C.) the values ofa _a,a _b anda _c are respectively 17·7, 13·5, and 43·0, ×10^?6 (°C.)^?1. The expansion coefficients are found to vary nonlinearly, and, in particular, along thec-axis the thermal expansion shows an anomalous behaviour between ?160°C. and ?60°C. and also in the neighbourhood of +130°C. These anomalies are explained as due to homomorphous transitions in which there is a reorientation of the ?NH ₊ ³ group about the N-N axis or a re-alignment of the ?NH₂ group.     

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.     

Series inversion of some convolution transforms

We study degeneration for ? → + 0 of the two-point boundary value problems

$\text{τ?}{}^{\mathrm{\pm }}\text{u := ?((au′)′ + bu′ + cu) ± xu′ ? κu = h, u(±1) = A ± B}$

, and convergence of the operators T_?⁺ and T_?^? on $\text{L}$²(?1, 1) connected with them, T_?^±u := τ_?^±u for all

$\text{u?D(T}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{, D(T}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{) := {u ? L}{}^2\text{(?1, 1) ∣ u″ ? L}{}^2\text{(?1, 1) \&; u(?1) = u(1) = O}, T}{}_0{}^{\mathrm{+}}\text{u: = xu′}$

for all

$\text{u?D(T}{}_{\mathrm{O}}{}^{\mathrm{+}}\text{), D(T}{}_{\mathrm{O}}{}^{\mathrm{+}}\text{) := {u ? L}{}^2\text{(?1, 1) ∣ xu′ ? L}{}^2\text{(?1, 1) \&; u(?1) = u(1) = O}}$

. Here ? is a small positive parameter, λ a complex “spectral” parameter; a, b and c are real $\text{b}$^∞-functions, a(x) ? γ > 0 for all x? [?1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T_?⁺ and of T_?^? are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a  1 and b  c  0 and in which we can compute the limits exactly. We show that (T_?⁺ ? λ)^?1 converges for ? → +0 strongly to (T₀⁺ ? λ)^?1 if $\text{R e λ > ?}\text{1}\text{2}$. In an analogous way, we define the operator T_?⁺, n (n ? $\text{N}$ in the Sobolev space H₀^?n(? 1, 1) as a restriction of τ_?⁺ and prove strong convergence of (T⁺_?,n ? λ)^?1 for ? → +0 in this space of distributions if $\text{R e λ > ?n ?}\text{1}\text{2}$. With aid of the maximum principle we infer from this that, if h?$\text{C}$¹, the solution of τ_?⁺u ? λu = h, u(±1) = A ± B converges for ? → +0 uniformly on [?1, ? ?] ∪ [?, 1] to the solution of xu′ ? λu = h, u(±1) = A ± B for each p > 0 and for each λ ? $\text{C}$ if ? ?$\text{N}$.Finally we prove by duality that the solution of τ_?^?u ? λu = h converges to a definite solution of the reduced equation uniformly on each compact subset of (?1, 0) ∪ (0, 1) if h is sufficiently smooth and if 1 ? ?$\text{N}$.     

Electrodeposition of copper on copper single crystal (100) face in presence of chloride ions

Observations of copper electrodeposits on to the (100) plane of copper was made from highly purified solutions of copper sulphate containing known concentration of hydrochloric acid from 10^?10 to 10^?1 m/L. In pure solutions at current densities of 5 and 10 mA/cm.² layers and pyramids were noticed. In the presence of hydrochloric acid of concentration 10^?9 to 10^?5 m/L there is a gradual decrease of distance between successive steps. At 10^?4 m/L of HCl there was the breaking of layers giving rise to ridge type of growth. With the increase of concentration to 3·5×10^?3 m/L pyramids appear again. On increasing the concentration of HCl to 10^?2 m/L there was the formation of triangular pyramids of cuprous chloride and on still increasing the concentration, polycrystalline type of deposit was noticed. The transition from layer to ridge, ridge to pyramids and to polycrystalline deposit occurs at all c.d. studied but the critical concentration of HCl needed for the transition depends upon the current density.     

Analysis of some allowed beta-transitions

Sakai’s method of analysing logft values has been extended to higher excited states in even-even nuclei. Within the context of the rather limited data an enhancement in the matrix element is observed for transitions to 2″⁺, 2?⁺ and 4′⁺ and 4″⁺ levels.     

Optical absorption of Co2+ doped NH4Cl: Phase transformation studies

The absorption spectrum of Co²⁺ doped NH₄Cl has been studied from the room temperature to the liquid nitrogen temperature. A sudden change in the spectrum is observed between 243° K and 233° K which is attributed to the phase transition in the crystal. From the observed spectrum it is suggested that Co²⁺ goes in interstitially as well as substitutionally. Both the types of centers exist at room temperature, but with decrease in temperature substitutional ions migrate to interstitial sites, the process being stimulated at the phase transformation point so that the 77° K spectrum seems to be mostly due to the interstitial centers. The 77° K spectrum is analyzed in the approximation of octahedral symmetry for interstitial ions and the band positions are fitted fairly well with B = 870 cm.^?1 Dq = 850 cm.^?1 and C = 4·4 B. A blue shift of about 100 cm.^?1 is observed for⁴T₁ (P) band at the phase transition which is attributed to the increase in Dq value with the anomalous lattice contraction at the phase transition. The decrease in the lattice parameter calculated from this blue shift is around 0·4% which is in good agreement with the results of X-ray measurements. Two possible models for the interstitial complex are examined and the one with fourfold chlorine coordination associated with two neutral water molecules at the first neighbour (NH₄)⁺ site lying along < 100> direction is suggested to be more probable.     

Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices

The main concern of this paper is linear matrix equations with block-companion matrix coefficients. It is shown that general matrix equations AX ? XB = C and X ? AXB = C can be transformed to equations whose coefficients are block companion matrices: $\text{C}\text{?}{}_{\mathrm{L}}\text{X?XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$ and $\text{X?}\text{C}\text{?}{}_{\mathrm{L}}\text{XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$, respectively, where ?_L and C_M stand for the first and second block-companion matrices of some monic r × r matrix polynomials L(λ) = λ^sI + Σ^s?1_j=0λ^jL_j and M(λ) = λ^tI + Σ^t7minus;1_j=0λ^jM_j. The solution of the equat with block companion coefficients is reduced to solving vector equations Sx = ?, where the matrix S is r²l × r²l[l = max(s, t)] and enjoys some symmetry properties.     

A new2 Π-X2 Π band system of NS

Vibrational and rotational analysis of some bands forming a new band system of NS is given. It is also shown that the system involves the ground X² Π _reg. state of the molecule, and is due to the transition² Π _reg.→X² Π _reg. The bands form a singlev″=0 progression withv′=7, 8, 9 and 10. The assignment of these quantum numbersv′, v″ is supported by (1) Δ₂F″ (J) values which are identical with those for thev″=0 bands of theβ andγ systems and (2) the isotopic shift data from¹⁵NS bands, respectively. The derived vibrational and rotational constants for the new² Π _reg. state are as follows (cm.^?1 units):

	T e	ω e	ω e x e	B e	D e
2 Π 3/2..	30364·8	803·3	3·82	0·6030	2·0×10?6
2 Π ½..	30292·3	797·0	3·63	0·5898	2·0×10?6

     

Nuclear disintegrations produced in nuclear emulsions by α-particles of great energy

A study of nuclear disintegrations caused by α-particles of primary cosmic radiation with energies > 5 BeV per nucleon, has been carried out. In a systematic survey in nuclear emulsions using ‘along the track’ scanning method, 479 α-particles with a total track length of 40·84 metres and 242 interactions were obtained. From the angular distribution of shower particles associated with these interactions, a procedure has been found for distinguishing protons, which originally formed part of the incident α-particle and which have not taken part in the interaction, from other charged particles. The mean free path for nuclear interaction in G-5 emulsion is found to be 17·5±1·1 cm. (68·9±4·3 gm./cm.²). Assigning both to the incident α-particle and to the target nuclei a radius R=r _oA^1/2, one obtains an effective nuclear radiusr _o=1·13±0·04 ×10^?13 cm. Using the number of protons emerging from disintegrations of heavy nuclei (Silver and Bromine) without having participated in the interaction (as can be deduced from the angular distribution) and assuming spherical nuclei of uniform density, the mean free path of nucleons in nuclear matter is calculated to be less than 3·2×10^?13 cm.     

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     

Star-Configurations in ℙ2 Having Generic Hilbert Function and the Weak Lefschetz Property

We prove that a star-configuration 𝕏 in ?² is defined by general forms of degrees ≤2 if and only if 𝕏 has generic Hilbert function. We also show that if 𝕏 and 𝕐 are star-configurations in ?² defined by general forms of degrees ≤2 and σ(𝕏) ≠ σ(𝕐), then the ring R/(I _𝕏 + I _𝕐) has the Weak Lefschetz property. These two results generalize results of Ahn and Shin [3 Ahn , J. , Shin , Y. S. ( 2012 ). The minimal free resolution of a star-configuration in ? ⁿ and the weak lefschetz property . J. Korean Math. Soc. 49 ( 2 ): 405 – 417 . [Google Scholar]]. Furthermore, we find the Lefschetz element of the graded Artinian ring R/(I _𝕏 + I _?) precisely when 𝕏 and ? are two star-configurations in ?² defined by general forms F ₁,…, F _s , and G ₁,…, G _s , L, respectively, with deg F _i  = deg G _i  = 2 for every i ≥ 1, and deg L = 1 with s ≥ 3.     

Viscosity analysis on the Boltzmann equation

This paper is devoted to investigating the asymptotic properties of the renormalized solution to the viscosity equation tfε + v ·▽xfε = Q (fε,fε ) + εΔvfε as ε→ 0+ . We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L1 ((0 , T ) × RN × RN ). The proof is based on compactness analysis and velocity averaging theory.     

Numerical solutions of AXB = C for centrosymmetric matrix X under a specified submatrix constraint

We say that X = [x_ij] is centrosymmetric if x_ij = x_{n ? j + 1, n ? i + 1}, 1?i, j?n. In this paper, we present an efficient algorithm for minimizing ∥AXB ? C∥ where ∥·∥ is the Frobenius norm, A∈?^{m × n}, B∈?^{n × s}, C∈?^{m × s} and X∈?^{n × n} is centrosymmetric with a specified central submatrix [x_ij]_{p?i, j?n ? p}. Our algorithm produces a suitable X such that AXB = C in finitely many steps, if such an X exists. We show that the algorithm is stable in any case, and we give results of numerical experiments that support this claim. Copyright © 2011 John Wiley & Sons, Ltd.     

Sixth-order superstable two-step methods for second-order initial-value problems

For the numerical integration of general second-order initial-value problems y″ = f(x, y, y′), y(x₀) = y₀, y′(x₀) = y′₀, we report a family of two-step sixth-order methods which are superstable for the test equation y″ + 2αy′ + β²y = 0, α, β ⩾ 0, α + β\s>0, in the sense of Chawla [1].     

The effect of a penalty term involving higher order derivatives on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing 0<p<1, h∈?, σ>0, among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable characteristic functions χ:Ω→?. Here ?⁺_h, ?^?, denote quadratic potentials defined on the space of all symmetric d×d matrices, h is the minimum energy of ?⁺_h and ε(u) denotes the symmetric gradient of the displacement field. An equilibrium state û, χ?, of I [·,·,h, σ] is termed one‐phase if χ?≡0 or χ?≡1, two‐phase otherwise. We investigate the way in which the distribution of phases is affected by the choice of the parameters h and σ. Copyright 2002 John Wiley & Sons, Ltd.     

On the several differences between primes

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,4(5 × 10¹⁰) = 118905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computedB ₄(5×10¹⁰)=1.197054473029 andB ₄=1.197054±7×10^?6. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,6(5 × 10¹⁰) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computedB ₆(5×10¹⁰)=0.93087506039231 andB ₆=1.135835±1.2×10^?6.