Graph minors. I. Excluding a forest

The path-width of a graph is the minimum value ofk such that the graph can be obtained from a sequence of graphsG₁,…,G_r each of which has at mostk + 1 vertices, by identifying some vertices ofG_i pairwise with some ofG_i+1 (1 ≤ i < r). For every forestH it is proved that there is a numberk such that every graph with no minor isomorphic toH has path-width?k. This, together with results of other papers, yields a “good” algorithm to test for the presence of any fixed forest as a minor, and implies that ifP is any property of graphs such that some forest does not have propertyP, then the set of minor-minimal graphs without propertyP is finite.     

Cohomology of certain moduli spaces of vector bundles

LetX be a smooth irreducible projective curve of genusg over the field of complex numbers. LetM ₀ be the moduli space of semi-stable vector bundles onX of rank two and trivial determinant. A canonical desingularizationN _o ofM _o has been constructed by Seshadri [17]. In this paper we compute the third and fourth cohomology groups ofN _o. In particular we give a different proof of the theorem due to Nitsure [12], that the third cohomology group ofN _o is torsion-free.     

The algebraA p((0, ∞)) and its multipliers

LetI =xεR: 0≤x≤∞ be the locally compact semigroup with addition as binary operation and the usual interval topology. The purpose of this note is to study the algebraA _p(I) of elements inL ₁(I) whose Gelfand transforms belong toL _p(Î), whereÎ denotes maximal ideal space ofL ₁(I). The multipliers ofA _p(I) have also been identified.     

The nuclear magnetic resonance spectra of four spin systems

The ring proton resonance spectra ofα, β andγ picolines have been studied at 40 Mc./sec. The spectra which belong to the classes ABCX, ABXY and A₂X₂ respectively have been completely analyzed to determine all the involved parameters.     

The ionization constants of somep-alkylthio- andp-alkylsulphonyl-anilines

The ionization constants of somep-alkylthio- andp-alkylsulphonyl anilines have been determined by the spectrophotometric method at 30°. The results have been interpreted by invoking an electron pair acceptor type conjugation of the sulphur atom with thep-amino group. The pK_a values of thep-alkylthioanilines are of the order, methyl > ethyl > isopropyl >t-butyl. In the case ofp-alkylsulphonylanilines, the order is, methyl > ethyl < isopropyl. The higher strength ofp-isopropyl-sulphonylaniline compared to its ethyl analogue, is attributed to the steric effect.     

Force constants and mean amplitudes for some anions of MX6 type (X=F,Cl, Br)

Force constants of the Urey Bradley type for twelve MX₆ ^n? ions ofOh symmetry have been reported. Generalized mean square amplitudes and mean amplitudes of vibration for bonded and non-bonded distances have also been calculated. Thermodynamic quantities for these molecules have been calculated at various temperatures from 50° K to 1600° K using a rigid rotator, harmonic oscillator approximation.     

On the day airglow

The altitude variation of the intensity ofλ 6300 line of OI atom and that ofλ 3914 (0–0 band of the first negative system of N ₂ ⁺ ) in the day airglow have been calculated. It has been found that photodissociation of O₂ by the Schumann-Range continuum is mainly responsible for the emission ofλ 6300 line (intensity 65 KR). Emission produced by dissociative recombination of O ₂ ⁺ and NO⁺ ions with electrons is of less importance (intensity 22 KR). The low intensity ofλ 6300 line during a solar eclipse, when the photo dissociation process is absent, is in conformity with the intensity (<6 KR) obtained by Tanabe and Tohmatsu (1959). Assuming that the resonance fluorescence is responsible for the emission ofλ 3914 band, the intensity of this band is calculated. The calculated intensity (7 KR) is within the limits of the values (>1 KR and <70 KR) obtained from theoretical considerations by Chamberlain and Sagan (1960).     

On right congruences associated with the input semigroupS of automata

LetA=(M, S, δ) be an automaton without outputs whereM is a nonemptyset andS is a nonempty semigroup. Then the right congruences μM and μ _m associated withS have been expressed in many different ways (μ _M is called the Myhill-Nerode congruence onS). Also, their algebraic properties have been investigated. We have introduced the right congruences μ _S and μ_α onM and we have obtained necessary and sufficient conditions thatS/μ andM/w have nontrivialS-homomorphisms where μ andw are any right congruences onS andM respectively. The faithfulness ofS has been introduced.     

The cut cone III: On the role of triangle facets

The cut polytopeP _n is the convex hull of the incidence vectors of the cuts (i.e. complete bipartite subgraphs) of the complete graph onn nodes. A well known class of facets ofP _n arises from the triangle inequalities:x _ij +x _ik +x _jk ≤ 2 andx _ij -x _ik -x _jk ≤ 0 for 1 ≤i,j, k ≤n. Hence, the metric polytope M_n, defined as the solution set of the triangle inequalities, is a relaxation ofP _n . We consider several properties of geometric type for P_n, in particular, concerning its position withinM _n . Strengthening the known fact ([3]) thatP _n has diameter 1, we show that any set ofk cuts,k ≤ log₂ n, satisfying some additional assumption, determines a simplicial face ofM _n and thus, also, ofP _n . In particular, the collection of low dimension faces ofP _n is contained in that ofM _n . Among a large subclass of the facets ofP _n , the triangle facets are the closest ones to the barycentrum of P_n and we conjecture that this result holds in general. The lattice generated by all even cuts (corresponding to bipartitions of the nodes into sets of even cardinality) is characterized and some additional questions on the links between general facets ofP _n and its triangle facets are mentioned.     

Some generating functions of modified Gegenbauer polynomials

L Weisner's group theoretic method has been introduced in the study of special function. In this paper we obtain two differential operators, one of which simultaneously raises the index and lowers the parameter of modified Gegenbauer polynomialsC _n ^v+n (x) by unity and the other acts onC _n ^v+n(x) in the reversed way by suitable interpretation to the indexn and the parameterv ofC _n ^v+n(x) . We have also found out the extended form of the groups generated by the operatorsA _ij(i,j=1,2). We have also derived some novel generating functions ofC _n ^v+n (x) from which several special generating functions can be easily derived.     

Lindelöf models of the reals: Solution to a problem of Sikorski

We show that it is consistent with ZFC that there is a modelM of ZF + DC such that the integers ofM areω ₁-like, the reals ofM have cardinalityω ₂, and the unit interval [0, 1] ^M is Lindelöf (i.e. every open cover has a countable subcover). This answers an old question of Sikorski.     

Generalized prolate spheroidal wave functions

The following equation $$(1 - x^2 )d^2 y/dx^2 + [(\beta - \alpha - (\alpha + \beta + 2)x]dy/dx + (\chi (c) - c^2 x^2 )y = 0$$ has been solved wherex(c) a separation constant is the characteristic value and is a function ofc. This solution is a generalization of spheroidal wave function into the series form ofP _n ^α;β (x),α andβ both separately are greater than ?1. The finite transform and its properties have been defined and a boundary value problem has been solved applying these tools.     

A note on reflexivity in Banach spaces

For any normed spaceX, the unit ball ofX is weak *-dense in the unit ball ofX ^**. This says that for any ε>0, for anyF in the unit ball ofX ^**, and for anyf ₁,…,f _n inX ^*, the system of inequalities |f _i(x)?F(f _i)|≤ε can be solved for somex in the unit ball ofX. The author shows that the requirement that ε be strictly positive can be dropped only ifX is reflexive.     

A complete parametrization of cyclic field extensions of 2-power degree

Letq be a power of 2 at least equal to 8 and ζ be a primitiveq-th root of unity, and letK be any field of characteristic zero. We define the group of special projective conormsS _K as a quotient of the group of elements ofK(ζ) of norm 1:S _K is obviously trival if the groul Gal (K(ζ)/K) is cyclic. We prove that for some fieldsK, the groupS _K is finite, and it is even trivial for certain fields such as ? or ?(X ₁,...,X _m). We then prove that the groupS _K completely paramatrizes the cycle extensions ofK of degreeq. We exhibit an explicit polynomial defined over ?(T ₀,...,T _q/2) which parametrizes all cyclic extensions ofK of degreeq associated to the trivial element ofS _K. In particular, this polynomial parametrizes all cyclic extensions ofK of degreeq whenever the groupS _K is trivial.     

Structure and analysis of some bands of theβ-system of PO molecule

Rotational analysis of (4–4), (7–6), (7–8) bands of the² Σ?² II _1/2 sub-system and (3–3) band of² Σ–² II _3/2 sub-system of theβ-system of PO have been made. Vibrational assignments of fifteen new bands of this system have also been reported.     

Polynomial approximation on the boundary and strictly inside

We investigate the possibility of approximating a function on a compact setK of the complex plane in such a way that the rate of approximation is almost optimal onK, and the rate inside the interior ofK is faster than on the whole ofK. We show that ifK has an external angle smaller than π at some point z_o∈δK, then geometric convergence insideK is possible only for functions that are analytic at z_o. We also consider the possibility of approximation rates of the form exp(?cn ^β), for approximation insideK, where β is related to the largest external angle ofK. It is also shown that no matter how slowly the sequence {γ _n } tends to zero, there is aK and a Lip β, β<1, functionf such that approximation insideK cannot have order {γ _n }.     

Constrained best approximation in Hilbert space III. Applications ton-convex functions

This paper continues the study of best approximation in a Hilbert spaceX from a subsetK which is the intersection of a closed convex coneC and a closed linear variety, with special emphasis on application to then-convex functions. A subtle separation theorem is utilized to significantly extend the results in [4] and to obtain new results even for the “classical” cone of nonnegative functions. It was shown in [4] that finding best approximations inK to anyf inX can be reduced to the (generally much simpler) problem of finding best approximations to a certain perturbation off from either the coneC or a certain subconeC _F. We will show how to determine this subconeC _F, give the precise condition characterizing whenC _F=C, and apply and strengthen these general results in the practically important case whenC is the cone ofn-convex functions inL ₂ (a,b),     

Three examples ofΛBe7 hyperfragments

In a systematic study of hyperfragments (HFs), three uniquely identified examples of_ΛBe⁷ hyperfragments have been observed; one of them decayed byπ ^?1-mesic mode and the other two by non-mesic modes. The data from the presentπ ^?mesic decay of_ΛBe⁷ when combined with that from the other two similar events reported in the literature, yielded a mean binding energy of 4·99±0·36 MeV for Λ^o relative to the lowest level of Be⁶.     

The extraordinary spectral properties of radially periodic Schrödinger operators

Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operatorst = - (d²/dr²) +q(r) does not survive a spherically symmetric extension to Schrödinger operatorsT =- Δ+ V with V(x) =q(¦x¦) for x ∈ ?^d,d ∈ ? 1, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [μ₀, ∞[ ofT with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm-Liouville operatorst _c = t +(c/r ²). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues ofT more closely. An eigenvalue was discovered below the essential spectrum in the cased = 2, and it turned out that there are in fact infinitely many, accumulating at μ₀. Moreover, a method based on oscillation theory made it possible to count eigenvalues oft _c contributing to an interval of dense point spectrum ofT. We gained evidence that an asymptotic formula, valid forc → ∞, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained.