Variations on seven points: An introduction to the scope and methods of coding theory and finite geometries

We use a simple example (the projective plane on seven points) to give an introductory survey on the problems and methods in finite geometries — an area of mathematics related to geometry, combinatorial theory, algebra, group theory and number theory as well as to applied mathematics (e.g., coding theory, information theory, statistical design of experiments, tomography, cryptography, etc.). As this list already indicates, finite geometries is — both from the point of view of pure mathematics and from that of applications related to computer science and communication engineering — one of the most interesting and active fields of mathematics. It is the aim of this paper to introduce the nonspecialist to some of these aspects.To Professor Günter Pickert on the occasion of his 65th birthday     

Formulation of Zeeman modulation as a signal filter

The Bloch equation containing a Zeeman modulation field is solved analytically by treating the Zeeman modulation frequency as a perturbation. The absorption and dispersion signals at both 0 degrees and 90 degrees modulation phase are obtained. The solutions are valid to first order in the modulation frequency, but are otherwise valid for any value of modulation amplitude or microwave amplitude. A first order treatment of modulation frequency is shown to be a valid approximation over a wide range of typical experimental EPR conditions. The solutions derived from the Bloch equation suggest that the effect of over-modulation on first and second harmonic EPR spectra can be formulated as a mathematical filter that smoothes and broadens the under-modulated signal. The only adjustable filter parameter is a width that is equivalent to the applied peak-to-peak modulation amplitude. The true spin-spin and spin-lattice relaxation rates are completely determined from the under-modulated spectrum. The filters derived from the analytic solutions of the Bloch equation in the linear limit of modulation frequency are tested against numerical solutions of the Bloch equation that are valid for any modulation frequency to show their applicability. The filters are further tested using experimental EPR spectra. Experimental under-modulated spectra are mathematically filtered and compared with the experimental over-modulated spectra. The application of modulation filters to STEPR spectra is explored and limitations are discussed.     

The rank of the AG(2,n) incidence matrix over any prime field

The dimension of the F-row spaces of the incidence matrices of all affine planes AG(2,n) is determined. The proof presented here gives a unified method for the cases when charF is or is not a divisor of n.     

Co-transport of H+/Cl- by a synthetic prodigiosin mimic

An amidopyrrole with appended imidazole group can bind and co-transport H+/Cl- across vesicle membranes much more effectively than an analogue with an appended pyridyl group.     

The effect of CpG-ODN on antigen presenting cells of the foal

Background  

Cytosine-phosphate-guanosine oligodeoxynucleotide (CpG-ODN) has been used successfully to induce immune responses against viral and intracellular organisms in mammals. The main objective of this study was to test the effect of CpG-ODN on antigen presenting cells of young foals.     

Thermodynamic properties of liquids,including solutions

Summary A new theoretical treatment of the dependence of surface pressure on the concentration of linear polymer molecules in a monolayer is presented. The development follows closely that in the author's new theory of the thermodynamic properties of three-dimensional solutions.

---

Zusammenfassung Es wird eine neuartige theoretische Untersuchung vorgelegt über die Abhängigkeit des Oberflächendruckes von der Konzentration an linearen Makromolekülen in einer monomolekularen Schicht. Die Entwicklung schließt sich eng an des Autors neue Theorie der dynamischen Eigenschaften von dreidimensionalen Lösungen an.

---

Glossary A Area of the monolayer surface - B ₁,B ₂ Functions in the expansion of in powers ofm/A. See [38] - E Intermolecular energy in the monolayer. See [2]. It is negative for attraction energy - G Gibbs energy. See [1] - H Enthalpy. See [1] - K Equilibrium constant, defined by [3]. Equal to 1 for perfect randomness of segment contacts - K A constant, related toK by [8] - M ₁,M ₂ Molar masses (molecular weights) of substrate and polymer - N _a Avogadros number=6.0225×10²³ mol^–1 - R Molar gas constant=8.3143 JK^–1 mol^–1 - S Entropy. See [1] - S _cc Correction to the combinatorial entropy to allow for imperfect randomness - S _or Contribution to the entropy of mixing to allow for concentration dependence of entropy of orientation, vibration and rotation of molecules and segments - S _rm Combinatorial entropy of mixing, assuming perfect randomness - T Temperature in kelvins - a ₀ ,a ₀ Areas in monolayer occupied by one mole (Avogadros number) of solvent and polymer segments, respectively. See [10] - g _K A function ofK and the concentration, defined by [7] - m Mass of polymer in the monolayer - m/A Mass/area ratio. Related to area fractions by [14] - n Average number of segments per polymer molecule - r _a =a ₀ /a ₀ Segment area ratio. See [16] - r = ₀ / ₀ Contacting segment outline ratio. See [16] - r _a/ =r _a /r See [16] - r _/a =r /r _a See [16] - x ₁,x ₂ Mole fractions of solvent and polymer in the monolayer - z ,z Contacting segment outline fractions, defined by [9] - Surface pressure required to keep monolayer in areaA - , Designating solvent and polymer segments, respectively - , , Energy per unit length of contact between pairs of segments of the type indicated. See [6] - Energy change when two unit lengths of contacts between like segments are replaced by two unit lengths of contacts between unlike segments. Defined by [6] - Energy parameter, defined by [5] - ₁, ₂ Densities of the pure components (mass/volume) - _m Surface density of polymer in a completely filled monolayer (mass/area) - , Sum of the contacting outlines, for segments of the type indicated, in the monolayer - ⁰ , ⁰ Average contacting outline per segment, of the type indicated - , , Sum of the contact lengths, for contacts of the type indicated, in the monolayer - ₁, ₂ Area fractions, measuring the fractions of the monolayer area occupied by solvent and polymer molecules, respectively.     

Density functional approach to quantum lattice systems

For quantum lattice systems, we consider the problem of characterizing the set of single-particle densities,, which come from the ground-state eigenspace of someN-particle Hamiltonian of the form whereH ₀ is a fixed, bounded operator representing the kinetic and interaction energies. We show that the conditions on are that it be strictly positive, properly normalized, and consistent with the Pauli principle. Our results are valid for both finite and infinite lattices and for either bosons or fermions. The Coulomb interaction may be included inH ₀ if the lattice dimension is 2. We also characterize those single-particle densities which come from the Gibbs states of such Hamiltonians at finite temperature. In addition to the conditions stated above, must satisfy a finite entropy condition.Research supported by the National Science Foundation under grant No. PHY-82-03669.Research supported by Office of Naval Research under grant No. 0014-80-G-0084.On leave from Department of Mathematics, University of Lowell, Massachusetts 01854.     

Theory of polymer fractionation efficiency

The theory of the equilibrium distribution of a mixture of different size species of a polymer between two liquid phases is reviewed and used as a basis for the calculation of the dependence of fractionation efficiency on overall concentration, interaction coefficient (hence, choice of solvent and temperature), average molecular weight, and the spread of the molecular weight distribution in the polymer. The special case of a single type of polymer in a single solvent, for which the polymer–solvent interaction coefficient is independent of concentration, is dealt with quantitatively. The ultimate aim is to make it possible to determine systematically the optimum solvent (or mixture of solvents), temperature, and overall concentration for fractionation of a given type of polymer, having a given average molecular weight and molecular weight distribution, considering practical limitations, e. g., the total volume which can conveniently be handled and the time required to achieve a sufficiently close approach to equilibrium.     

Glaucolide-A and -B,new germacranolide-type sesquiterpene lactones from Vernonia (compositae)

The structures of two new germacranolides, glaucolide-A (1a) and -B (1b), isolated from more than 25 species of Vernonia from the Western Hemisphere, have been established by a series of chemical transformations, and spectral and X-ray crystallographic studies.     

Design machines: Algebraically well described interconnection networks

We present a method constructing efficient multiprocessor networks based on combinatorial designs and group theory. The principal idea is to combine 1-factorizations of complete graphs on each block of certain 2-designs, to obtain efficient networks with a short maximum distance between any pair of processors. Using a smart multiplexing technique derived from the 1-factorization of complete graphs we can circumvent the restrictive Moore bound for (, d)-graphs.