On the Turán number for the hexagon

A long-standing conjecture of Erd?s and Simonovits is that ex(n,C_2k), the maximum number of edges in an n-vertex graph without a 2k-gon is asymptotically as n tends to infinity. This was known almost 40 years ago in the case of quadrilaterals. In this paper, we construct a counterexample to the conjecture in the case of hexagons. For infinitely many n, we prove that

     

Quadratic forms on graphs

We introduce a new graph parameter, called the Grothendieck constant of a graph G=(V,E), which is defined as the least constant K such that for every A:E→ℝ,

Nonembeddability theorems via Fourier analysis

Various new nonembeddability results (mainly into L₁) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0,1}^d has L₁ distortion We also give new lower bounds on the L₁ distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.     

On topological spaces of singular density and minimal weight

We introduce a weakening of the generalized continuum hypothesis, which we will refer to as the prevalent singular cardinals hypothesis, and show it implies that every topological space of density and weight ℵω₁ is not hereditarily Lindelöf.The assumption PSH is very weak, and in fact holds in all currently known models of ZFC.     

Characterization of regular Diophantine quadruples

---

Characterization of regular Diophantine quadruples

---

     

Minami–Webb splittings and some exotic <Emphasis Type="Italic">p</Emphasis>-local finite groups

We give a new proof of the Minami–Webb formula for classifying spaces of finite groups by exploiting Symonds’s resolution of Webb’s conjecture. The methods are applicable to obtain a stable decomposition of Minami’s type for the classifying spaces of the three exotic p-local finite groups which were introduced by Ruiz and Viruel at the prime 7. We obtain a similar decomposition for the classifying spaces of a family of exotic p-local finite groups which were constructed by Broto, Levi and Oliver. The author was supported by the Nuffield Foundation Grant NAL/00735/G.     

Dynamic Routing and Admission Control in High-Volume Service Systems: Asymptotic Analysis via Multi-Scale Fluid Limits

Motivated by applications in telephone call centers, we consider a service system model with m customer classes and r server pools. The model is one with doubly stochastic arrivals, which means that the m-vector λ of instantaneous arrival rates is allowed to vary both temporally and stochastically. Two levels of dynamic control are considered: customers may be either blocked or accepted at the time of their arrival, and then accepted customers of each class must be routed, either immediately upon acceptance or after some period of waiting, to a server pool that is qualified to handle that class. Customers who are made to wait before commencement of their service are liable to defect. The objective is to minimize the expected sum of blocking costs, waiting costs and defection costs over a fixed and finite planning horizon. We consider an asymptotic parameter regime in which (i) the arrival rates, service rates and defection rates are uniformly accelerated by a large factor κ, then (ii) arrival rates are increased by an additional factor g(κ), and the number of servers in each pool is increased by g(κ) as well. This produces a separation of time scales, justifying a pointwise stationary stochastic fluid approximation for our original system model. In the stochastic fluid approximation, optimal admission control and routing decisions are determined by a simple linear program that uses the current arrival rate vector λ as data. We explain how to implement the fluid model's optimal control policy in our original service system context, and prove that the proposed implementation is asymptotically optimal in the first-order sense. AMS subject classification: 60K30, 90B15, 90B36     

Anodic polarization of Al-Sn alloy in H3PO4 solutions

The anodic polarization behaviour of Al-Sn alloy (5.6% Sn) was studied in aerated 1, 1.5 and 2 M H₃PO₄ acid solutions using potentiodynamic and potentiostatic techniques. Anodic behaviour of Al and Sn metals was investigated for comparison. The results show that the alloy exhibits active-passive transition behaviour. The passivation of the examined alloy is due to the formation of oxide film for both Al and Sn incorporated with their phosphates. In general, at constant concentration of PO₄ ³⁻ ion, the passive current density (I _pass) is decreased with increase of pH in the range 2.5–5. Further, the influence of Cl⁻ ions on the passivity of the given alloy was studied. It was found that the aggressive effect of Cl⁻ ions on the passive film is inhibited with increase in phosphate concentration and pH. However, the addition of organic compounds (2- and 3-picoline and 2-aminopyridine) shows that only 2-aminopyridine inhibited the attack of Cl⁻ ions. Received: 24 October 1997 / Accepted: 5 February 1998     

Factors controlling charge recombination under dark and light conditions in dye sensitised solar cells

A simple and powerful approach for assessing the recombination losses in dye sensitised solar cells (DSSCs) across the current voltage curve (j-V) as a function of TiO(2) electron concentration (n) is demonstrated. The total flux of electrons recombining with iodine species in the electrolyte and oxidised dye molecules can be thought of as a recombination current density, defined as j(rec) = j(inj)-j where j(inj) is the current of electrons injected from optically excited dye states and j is the current density collected at cell voltage (V). The electron concentration at any given operating conditions is determined by charge extraction. This allows comparison of factors influencing electron recombination rates at matched n. We show that j(rec) is typically 2-3 times higher under 1 sun equivalent illumination (j(inj) > 0) relative to dark (j(inj) = 0) conditions. This difference was increased by increasing light intensity, electrolyte iodine concentration and electrolyte solvent viscosity. The difference was reduced by increasing the electrolyte iodide concentration and increasing the temperature. These results allowed us to verify a numerical model of complete operational cells (Barnes et al., Phys. Chem. Chem. Phys., DOI: 10.1039/c0cp01554g) and to relate the differences in j(rec) to physical processes in the devices. The difference between j(rec) in the light and dark can be explained by two factors: (1) an increase in the concentration of electron acceptor species (I(3)(-) and/or I(2)) when current is flowing under illumination relative to dark conditions where the current is flowing in the opposite direction, and (2) a non-trivial contribution from electron recombination to oxidised dye molecules under light conditions. More generally, the technique helps to assign the observed relationship between the components, processing and performance of DSSCs to more fundamental physical processes.