Unspecified Base Research Articles

Stringy Hirzebruch classes of Weierstrass fibrations

A Weierstrass fibration is an elliptic fibration $Y\to B$ whose total space $Y$ may be given by a global Weierstrass equation in a $\mathbb{P}^2$-bundle over $B$. In this note, we compute stringy Hirzebruch classes of singular Weierstrass fibrations associated with constructing non-Abelian gauge theories in $F$-theory. For each Weierstrass fibration $Y\to B$ we then derive a generating function $\chi^{\text{str}}_y(Y;t)$, whose degree-$d$ coefficient encodes the stringy $\chi_y$-genus of $Y\to B$ over an unspecified base of dimension $d$, solely in terms of invariants of the base. To facilitate our computations, we prove a formula for general characteristic classes of blowups along (possibly singular) complete intersections.

Quantal two-center Coulomb problem treated by means of the phase-integral method. I. General theory

The present paper concerns the derivation of phase-integral quantization conditions for the two-center Coulomb problem under the assumption that the two Coulomb centers are fixed. With this restriction we treat the general two-center Coulomb problem according to the phase-integral method, in which one uses an a priori unspecified base function. We consider base functions containing three unspecified parameters C,C̃, and Λ. When the absolute value of the magnetic quantum number m is not too small, it is most appropriate to choose Λ=|m|≠0. When, on the other hand, |m| is sufficiently small, it is most appropriate to choose Λ=0. Arbitrary-order phase-integral quantization conditions are obtained for these choices of Λ. The parameters C and C̃ are determined from the requirement that the results of the first and the third order of the phase-integral approximation coincide, which makes the first-order approximation as good as possible. In order to make the paper to some extent self-contained, a short review of the phase-integral method is given in the Appendix.

Phase-integral formulas for quantal matrix elements

Simple phase-integral formulas, not involving wave functions, are derived for quantal matrix elements associated with bound states of a quantal particle in a smooth single-well potential. In these formulas one uses an arbitrary order of the phase-integral approximation generated from an unspecified base function.

Rotation of a rigid diatomic dipole molecule in a homogeneous electric field: I. Schrödinger equation. Quantization conditions according to phase-integral method

After a brief historical discussion of the energy quantization according to quantum mechanics of the rigid diatomic dipole molecule in a static electric field, the Schrödinger equation for that system is recalled. Previous attempts to obtain the energy levels clearly indicate that there is a need for a reliable method yielding very accurate eigenvalues for all values of the electric field strength. This is accomplished with the aid of new quantization conditions obtained by means of a phase-integral method involving a general phase-integral approximation of arbitrary order generated from an unspecified base function, which is chosen in two different ways such that, when the electric field strength is equal to zero, simple limiting forms of the quantization conditions give the exact values of the energy levels. The two choices of the base function are expected to be appropriate in the cases when the absolute value of the magnetic quantum number m is sufficiently large and sufficiently small respectively. For every value of m at least one of the two base functions should be useful.

Alternative phase-integral approximations

Phase-integral approximations of a new kind are obtained. They are closely related to the phase-integral approximations generated from an unspecified base function Q(z), due to Professor N. Froman and Professor P.O. Froman (1991). In the search for an optimal base function, a sequence of successively improved base functions QN(z) are derived (N=0,1,2, . .) together with a concise recurrence formula for these functions that is suitable for computer calculations. (The function QN(z) is an approximate solution to the nonlinear q-equation, equivalent to the Schrodinger equation.) The proposed approximation, which is immediately obtained from QN(z), is expected to be somewhat more accurate (for N>or=2) than the corresponding phase-integral approximation of order 2N+1, obtained from the function qN(z) given by qN(z)Q(z) Sigma n=0NY2n(z). A detailed comparison is made between QN(z) and qN(z) for N=0, 1, 2, and 3. The proposed approximations have the correct form required to combine them with the F-matrix technique for solving connection problems.

Measuring the stiffness of concrete shear walls during dynamic tests

Results from a series of experiments designed to measure the stiffness of low-aspect-ratio, reinforced-concrete shear walls subjected to simulated seismic inputs on a shake table are reported. The geometry of the test structures allows them to be modeled as single-degree-of-freedom systems. Forces were estimated from accelerometer measurements on masses attached to the structures. Dynamic relative-displacement measurements were obtained from groups of strain gages wired in series to act as one continuous gage. Because this method measures relative displacements, potential sources of error associated with unspecified base motion are avoided. tiffness values determined from the relative displacement measurements were compared with stiffness values determined indirectly from frequency-response functions. Measured, accelerometer data were used to calculate the frequency-response functions. The stiffness values determined from the relative-displacement measurements gave results similar to those given by mechanics-of-materials beam theory that accounts for shear deformation. The stiffness values determined from the frequency-response functions were considerably less than those from the theory.

In vivo restriction. Sequence and structure of endonuclease II-dependent cleavage sites in bacteriophage T4 DNA.

Endonuclease II of bacteriophage T4 is required for in vivo restriction of cytosine-containing DNA from its host, Escherichia coli, (as well as from phage mutants lacking cytosine modification), normally the first step in the reutilization of host DNA nucleotides for synthesis of phage DNA in infected cells. The phage cytosine-DNA is fragmented incompletely to yield genetically defined fragments. This restriction is different from that of type I, II, or III restriction enzymes. We have located seven major endonuclease II-dependent restriction sites in the T4 genome, of which three were analyzed in detail; in addition, abundant sites were cleaved in less than or equal to 5% of all molecules. Sites I, II, and III shared the sequence 5'-CCGNNTTGGC-3' and were cleaved in about 25% (I and III) and 65% (II) of all molecules, predominantly staggered around the first or second of the central unspecified base pairs to yield fragments with one 5' base. The less frequently cleaved sites I and III deviated from site II in predicted helical structure when viewed from the consensus strand, and in sequence when viewed from the opposite strand. Thus, interaction with a particular helical structure as well as recognition of the bases in DNA appears important for efficient cleavage.

A Study of the purine/pyrimidine codon occurrence with a reduced centered variable and an evaluation compared to the frequency statistic

With the three-letter alphabet {R,Y,N} (R = purine, Y = pyrimidine, N = R or Y), there are 26 codons (NNN being excluded): RNN,…,NNY (six codons at two unspecified bases N), RRN,…,NYY (12 codons at one unspecified base N), RRR,…,YYY (eight specified codons). A statistical methodology that uses the codon frequency and a reduced centered variable leads to similar results for a codon occurrence study, regardless of gene function and regardless of a particular protein coding gene taxonomic population. Therefore, this variable can be considered a new codon usage index, whose use removes certain nonsignificant results found with the frequency statistic. This methodology identifies the common and rare codons (i.e., the codons having the highest and lowest occurrence) and leads to a model of codon evolution at three successive states: RNN, then RNY, and finally RYY. Some biological relations between this model and the YRY(N) 6YRY preferential occurrence are also presented.

Detailed behavior of the phase-integral approximations at zeros and singularities of the square of the base function

Approximate solutions to the one-dimensional time independent wave equation, called the phase-integral approximations, are analyzed in the vicinity of characteristic points. The approximations are of arbitrary order and are generated from an unspecified base function. The general theory is illustrated by examples involving the power and/or the exponential behavior of the square of the base function. In these cases simple estimates are derived for the integrals which define the accuracy of the phase-integral approximation, and the optimum approximation order is determined.

An Epstein-Barr virus transcript from a latently infected, growth-transformed B-cell line encodes a highly repetitive polypeptide.

By screening a cDNA library prepared from poly(A)+ RNA isolated from an Epstein-Barr virus latently infected, growth-transformed human B-lymphoblastoid cell line, we have recovered a clone corresponding to a highly spliced viral transcript encoded largely by the major internal repeat (IR1). The 5' region contains one copy of a 26-base-pair (bp) exon (W0) [which is 28 bp downstream from a CAATT-(N)34TATAAA sequence (N, unspecified base)] and seven copies of two small exons (W1, 66 bp; W2, 132 bp). In addition, there are three exons from the "unique" region of the BamHI Y fragment of the viral genome. Two other cDNA clones that have been described, corresponding to latent viral transcripts, share homology in their 5' regions with this clone and are clearly divergent at their 3' ends. The cDNA clone described in this paper contains one long open reading frame that extends through the repeat element. In vitro transcription and translation of this open reading frame yielded a 62-kDa polypeptide that could be immunoprecipitated by an Epstein-Barr virus-positive human serum.

Phase‐integral formulas for Bessel functions and their relation to already existing asymptotic formulas

The phase‐integral method devised by Fröman and Fröman [N. Fröman and P. O. Fröman, JWKB Approximation, Contributions to the Theory (North‐Holland, Amsterdam, 1965); Ann. Phys. (NY) 83, 103 (1974); Nuovo Cimento B 20, 121 (1974); N. Fröman, Ark. Fys. 32, 541 (1966); Ann. Phys. (NY) 61, 451 (1970)], involving a general phase‐integral approximation of arbitrary order, which is generated from an unspecified base function, is used for deriving first‐ and higher‐order phase‐integral formulas for Bessel functions. For different choices of the base function one thus obtains in a systematic way different kinds of asymptotic formulas. By series expansion of these formulas one obtains already existing asymptotic formulas presented is standard handbooks. The phase‐integral formulas are seen to have certain advantages that those latter formulas do not possess.