Functions In Cn Research Articles

Theoretical study of the dipole moment function of the X2Sigma(+) state of CN

The X2Sigma(+) state dipole moment function of CN is determined from accurate ab initio calculations. The calculated Einstein coefficient of 13.0 /s for the fundamental 1-0 vibrational band is in excellent agreement with the value measured by Treffers (1975) using a King furnace. The theoretical vibrational band strengths should be valuable in interpreting the fluorescence spectrum of CN in comets.

Crystal chemistry of complex indium(III) and other M(III) halides, with a discussion of M—Cl bond lengths in complex M(III) chlorides and of the structures of and hydrogen bonding in (NH4)2[InCl5(H2O)], K3InCl6•nH2O, (MeNH3)4[InCl6]Cl, and (Me2NH2)4[InCl6]Cl

The crystal chemistry of complex In(III) halides is discussed and compared with that of other Group III elements and Fe(III), with critical comments and analysis of the available information. The cation/anion packing types occurring among A+[M3+X4] halides are classified according to volume per formula unit and the relative ionic sizes of M, X, and A, and the structural trends in this class of compounds are identified. The effect of systematic factors responsible for variation in M(III)—Cl bond lengths is examined in detail. It is shown that the mean M(III)—Cl bond lengths in [MClnLm]ε(M = Al, Ga, Fe, In, Tl; L = Cl or a neutral ligand) complexes can be approximated, to within about 0.02 Ă, by a linear function of the coordination number CN = n + m and the charge ε over the range 3 ≤ CN ≤ 6 and.−3 ≤ ε ≤2. This analytical expression provides a norm for comparing M(III)—Cl bond lengths and demonstrates that the CN is always a more important factor than ε in determining the variation in the mean bond length and that it becomes dominant when M is small. The crystal structure of (Me2NH2)4[InCl6]Cl (P21212, a = 10.156(4) Å, b = 13.007(4) Å, c = 8.751(3) Å, Z = 2) has been determined and those of (NH4)2InCl5(H2O)] (Pnma, a = 13.953(4) Å, b = 10.086(5) Å, c = 7.152(3) Å, Z = 4), K3InCl6•nH2O (I4mm, a = 15.659(8) Å, c = 18.106(5) Å, Z = 14) and (MeNH3)4[InCl6]Cl (P2/n, a = 16.113(3) Å, b = 7.446(2) Å, c = 16.163(4) Å, β = 103.75(2)°, Z = 4) redetermined. The water in K3InCl6•nH2O appears to be in part zeolitic; the hydrate examined contained more water than the monohydrate reported previously. Hydrogen bonding in these and related structures is described and discussed in detail; descriptions of the hydrogen-bonding schemes in (NH4)2[InCl5(H2O)] and (MeNH3)4[InCl6]Cl constitute the main improvement over earlier structure determinations.

Exit times for elliptic diffusions and BMO

In 1948 P. Lévy formulated the following theorem: If U is an open subset of the complex plane and f:U → ℂ is a nonconstant analytic function, then f maps a 2-dimensional Brownian motion Bt (up to the exit time from U) into a time changed 2-dimensional Brownian motion. A rigorous proof of this result first appeared in McKean [22]. This theorem has been used by many authors to solve problems about analytic functions by reducing them to problems about Brownian motion where the arguments are often more transparent. The survey paper [8] is a good reference for some of these applications. Lévy's theorem has been generalized, first by Bernard, Campbell, and Davie [5], and subsequently by Csink and Øksendal [7]. In Section 1 of this note we use these generalizations of Lévy's theorem to extend some results about BMO functions in the unit disc to harmonic morphisms in ℝn to holomorphic functions in ℂn and to analytic functions on Riemann surfaces. In Section 2, we characterize the domains in ℝn which have the property that the expected exit time of elliptic diffusions is uniformly bounded as a function of the starting point. This extends a result of Hayman and Pommerenke [15], and Stegenga [24] about BMO domains in the complex plane.

Determination of low levels of cyanide with a silver/silver sulphide wire electrode

A silver/silver sulphide electrode is prepred quickly by holding a cleaned silver wire in vapours from molten sulphur. In 1000–10 mg l −1 cyanide solutions, the electrode exhibits a linear E/log C CN function which becomes slightly sinusoidal for 10–0.1 mg l –1 cyanide. The average slope is slightl super-Nerstian (10 mV/decade concentration). The applicability of the electrode is demonstrated for the determinations of microgram quantities of water-soluble cyanide from the Prussian blue pigments which are constituents of externally applied cosmetics. The home-made electrode provides results agreing with those obtained with commercially available electrodes.

Monge-Ampère measures associated to extremal plurisubharmonic functions in 𝐶ⁿ

We consider the extremal plurisubharmonic functions L E ∗ L_E^\ast and U E ∗ U_E^\ast associated to a nonpluripolar compact subset E E of the unit ball B ⊂ C n B \subset {{\mathbf {C}}^n} and show that the corresponding Monge-Ampère measures ( d d c L E ∗ ) n {(d{d^c}L_E^\ast )^n} and ( d d c U E ∗ ) n {(d{d^c}U_E^\ast )^n} are mutually absolutely continuous. We then discuss the polynomial growth condition ( L ∗ ) ({L^\ast }) , a generalization of Leja’s polynomial condition in the plane, and study the relationship between the asymptotic behavior of the orthogonal polynomials associated to a measure on E E and the ( L ∗ ) ({L^\ast }) condition.

The structure and function of carboxypeptidase CN.

Carboxypeptidase CN from Citrus natsudaidai HAYATA is an acidic protease (M, 93000). The enzyme was inactivated by the incorporation of 1.0 mol of 32P-labeled diisopropylphosphofluoridate, 1.3 mol of 203HgCl2, or 2.2 mol of 110mAgNO3 per mol of enzyme at pH 5.5. Four or five radioactive peptides were isolated from a partial acid hydrolysate of the 32P-labeled enzyme. These peptides provided evidence indicating the amino acid sequence Glx-Gly-Asx-Ser-Gly-Gly-Glu-Leu-Val around the reactive serine residue. The enzyme reacted with 0.4 mol of 203Hg-labeled p-chloromercuribenzoate without appreciable loss of enzymatic activity. On the other hand, 1 mol of 203Hg-labeled p-chloromercuribenzoate was incorporated into the enzyme which had been denatured with 6 M guanidine hydrochloride and 8 M urea at pH 7.0 in the absence of dithiothreitol. Thus, the enzyme has one sulfhydryl group which possesses only poor reactivity. Photooxidation using methylene blue as a sensitizer caused a loss of enzymatic activity and specific destruction of approximately 1 mol of histidine residue per mol of enzyme. The photooxidized, p-bromophenacyl bromide-treated and HgCl2-treated enzyme failed to react with 32P-labeled diisopropylphosphofluoridate. From these findings, we inferred that one serine, one histidine, and the carboxyl groups are essential for catalytic activity.

Cnoidal Waves as Exact Sums of Repeated Solitary Waves: New Series for Elliptic Functions

Via Poisson summation of the corresponding Fourier series, it is shown that the cnoidal wave of the Korteweg–de Vries equation can be written as an infinite sum of periodically repeated solitary waves, a result first proved by Toda through a different method. Similar series of hyperbolic secant or tangent functions are also derived for the elliptic dn, sn, and cn functions which make it possible to express the cnoidal wave of almost any evolution equation in terms of a series of the corresponding solitary waves. As a by-product, new series are also obtained for the complete elliptic integral K and modulus k.

Certain contour-solid properties of holomorphic functions in Cn

Certain contour-solid properties of holomorphic functions in Cn

A high-order cnoidal wave theory

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

Orientational anisotropy of the CN - ions in f.c.c. NaCN: an optical spectroscopy measurement

We present a new method for measuring the anisotropy of the orientational probability function of CN - ions in a f.c.c. NaCN crystal. The results are discussed and compared with previous neutron diffraction measurements.

Cycles separating zeros of analytic functions in Cn

Cycles separating zeros of analytic functions in Cn

Inner functions in Cn

In this paper we prove that if f is a holomorphic function in a strictly pseudoconvex region D in Cn, n > 1, with radial limit equal to 1 in modulus at each point of some nonempty open subset S of the boundary of D, then f ≡ const in D.

Plurisubharmonic functions and convexity properties for general function algebras

A “natural system” consists of a Hausdorff space Σ \Sigma plus an algebra A \mathfrak {A} of complex-valued continuous functions on Σ \Sigma (which contains the constants and determines the topology in Σ \Sigma ) such that every continuous homomorphism of A \mathfrak {A} onto C {\mathbf {C}} is given by an evaluation at a point of Σ \Sigma (compact-open topology in A \mathfrak {A} ). The prototype of a natural system is [ C n , P ] [{{\mathbf {C}}^n},\mathfrak {P}] , where P \mathfrak {P} is the algebra of polynomials on C n {{\mathbf {C}}^n} . In earlier papers (Pacific J. Math. 18 and Canad. J. Math. 20), the author studied A \mathfrak {A} -holomorphic functions, which are generalizations of ordinary holomorphic functions in C n {{\mathbf {C}}^n} , and associated concepts of A \mathfrak {A} -analytic variety and A \mathfrak {A} -holomorphic convexity in Σ \Sigma . In the present paper, a class of extended real-valued functions, called A \mathfrak {A} -subharmonic functions, is introduced which generalizes the ordinary plurisubharmonic functions in C n {{\mathbf {C}}^n} . These functions enjoy many of the properties associated with plurisubharmonic functions. Furthermore, in terms of the A \mathfrak {A} -subharmonic functions, a number of convexity properties of C n {{\mathbf {C}}^n} associated with plurisubharmonic functions can be generalized. For example, if G G is an open A \mathfrak {A} -holomorphically convex subset of Σ \Sigma and K K is a compact subset of G G , then the convex hull of K K with respect to the continuous A \mathfrak {A} -subharmonic functions on G G is equal to its hull with respect to the A \mathfrak {A} -holomorphic functions on G G .

Saturation of Scintillation Magnitude in Near-Earth Optical Propagation*

Both a pulsed laser and a helium–neon laser have been used to examine the magnitude of scintillation as a function of range and turbulence strength over a near-earth, horizontal path. A photo-optical technique was utilized to record directly a 61-cm cross section of a received laser beam. Photographs of beam cross sections made at ranges of from 200 to 1500 m were used to compute log-irradiance variances. Simultaneous with the optical data, measurements of the index-structure coefficient Cn were made by use of a high-speed thermal technique. The data are used to test the spherical-wave equation that gives the log-amplitude variance as a function of range and Cn. The measurements indicate that the variance increases for ranges up to about 700 m, at which distance saturation occurs, i.e., no further growth of the variance is observed. The data are also compared with the saturation equations of Tatarski and deWolf. In addition, the effect of the transmitter-beam divergence on the magnitude of scintillation is examined. Finally, a modification of the inertial subrange of the Kolmogorov turbulence model is suggested to explain the occurrence of particular optical effects observed during temperature-inversion conditions.

Spectrophotometric studies of hexacyanoferrate(III) ion and its reaction with iron(III) in water and ethanol

The reduction of Fe(CN) 6 3− ion in water has been studied as a function of pH and excess CN − ion. A mechanism is proposed which explains the series of coloured species observed for this reduction and for its reaction with Fe 3+. In ethanol, FeFe(CN) 6 is immediately reduced upon excess addition of CN − ion whereas in water, the reduction is concerned with the Fe(CN) 6 3− ion from the dissociated complex.

On the closure of certain Banach spaces of functions of several variables

1. Statement of the main results. Our primary goal in this note is to establish the following proposition. THEOREM 1. Consider the Banach space 6» of continuous real-valued functions ƒ : En—>R, En standing for the unit cube in n-dimensional Euclidean space. If 0 and \p are any fixed functions of Cn with connected level-sets intersecting pairwise in connected sets, then the subspace of superpositions a o +b 0$ is closed in Cw under the uniform norm. Mark by (£» the indicated space of superpositions. To prove the stated theorem, it suffices to verify THEOREM 2. Every function of (5n has a best uniform approximation in (Bn. Distinguish one of the fixed functions, say, \p; denote its level sets by l+(t), kit) = {pGEni^p) = / } ;

Amplitude changes of auditory potentials evoked at cochlear nucleus during acoustic habituation

1. 1. Evoked auditory potentials at the cochlear nucleus were recorded from sixteen chronically implanted electrodes in nine cats during acoustic habituation at each of two repetition rates — 1/sec and 1/10 sec. Seven animals were given 6 h habituation runs and two were given 5 day runs. 2. 2. Three simultaneous recordings were obtained from each bipolar electrode. Two of these were monopolar tracings, comparing each tip of the electrode to ground, and the third was a bipolar recording between the two electrode tips. 3. 3. Reduction of evoked CN potentials during acoustic habituation was not confirmed. Statistically significant changes in amplitude occurred but these were small and inconsistent in direction. Increases as well as decreases were observed. 4. 4. Amplitude changes were inconsistent between right and left CN as well as between two adjacent points within one CN. 5. 5. No correlation was found between electrode placement and direction of amplitude changes over time. Even from one electrode tip, amplitude changes on one run could be reversed on a second run. 6. 6. No consistent effect on peak to peak amplitudes was observed as a function of rate of stimulus presentation or amount of movement of the animal. 7. 7. The cumulative mean amplitude of potentials in the alert samples was larger than in the sleep samples. It is suggested that the acoustic effects secondary to position differences between the alerted and the sleeping samples would tend to contribute to the amplitude differences observed. 8. 8. The implications of these results for CN function during acoustic habituation are discussed.

Functions which have harmonic support

1. Introduction. This paper presents some properties of the functions in a certain subclass of the class of subharmonic functions. Consider real valued functions v(z) defined in a domain D of the complex plane. DEFINITION 1. The function v is in (hs), the class of functions which have harmonic support in D, if and only if v is uniformly bounded on each compact subset of D and for each zoC=Dah harmonic and single valued in D3h(zo) =v(zo) and h<v in D. Caratheodory [5] used such functions in constructing a general result on solutions of Dirichlet's problem which are based on convergent sequences of continuous operators. The functions of the class (hs) are continuous subharmonic functions, but Brelot [4, footnote 16] showed the existence of subharmonic functions in Cn which are not in (hs). A simple example of a continuous subharmonic function not in (hs) is the function v = z a, 0 <a <1, in zI <1. For suppose v C (hs) and let hn denote the support functions at Zn= l/n, n= 2, 3, * - - . The sequence { hn } constitutes a bounded family of harmonic functions which is thus normal. Writing z = x+iy, the family