Optimal primitive sets with restricted primes

A set of natural numbers is primitive if no element of the set divides another. Erd\H{o}s conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in \P} 1/(p log p), where \P denotes the set of primes. In this paper, we make progress towards this conjecture by restricting the setting to smaller sets of primes. Let P denote any subset of \P, and let N(P) denote the set of natural numbers all of whose prime factors are in P. We say that P is Erd\H{o}s-best among primitive subsets of N(P) if the inequality \sum_{n\in S} 1/(n log n) \le \sum_{n\in P} 1/(p log p) holds for every primitive set S contained in N(P). We show that if the sum of the reciprocals of the elements of P is small enough, then P is Erd\H{o}s-best among primitive subsets of N(P). As an application, we prove that the set of twin primes exceeding 3 is Erd\H{o}s-best among the corresponding primitive sets. This problem turns out to be related to a similar problem involving multiplicative weights. For any real number t>1, we say that P is t-best among primitive subsets of N(P) if the inequality \sum_{n\in S} n^{-t} \le \sum_{n\in P} p^{-t} holds for every primitive set S contained in N(P). We show that if the sum on the right-hand side of this inequality is small enough, then P is t-best among primitive subsets of N(P).Comment: 10 page

Long Duration Exposure Facility (LDEF) space environments overview

The Long Duration Exposure Facility (LDEF) was retrieved from Earth orbit in January 1990 after spending almost six years in space. It had flown in a near-circular orbit with an inclination of 28.5 degrees. Initially, the orbit altitude was approximately 257 nautical miles; however, when the LDEF was retrieved the orbit altitude had decayed to approximately 179 nautical miles. The LDEF was passively stabilized about three axes while in free flight, making it an ideal platform for exposing experiments which were measuring the environments of near-Earth space and investigating the long-term effects of these environments on spacecraft. A brief overview of the encountered environments that were of most interest to the LDEF investigators is presented

A Stochastic Infinite-Horizon Economy with Secured Lending, or Unsecured Lending and Bankruptcy

Modeling problems for a monetary economy are discussed and some examples are presented in the context of an infinite-horizon economy with one or two types of traders, who use fiat money to buy a single perishable consumption good. Three instances are considered, all with transactions in fiat money. The first model has no borrowing or lending. The second model permits both borrowing and lending, but all loans are secured. The third model has borrowing and unsecured lending, and takes into account the presence of debtors who are unable to honor their debts and go bankrupt. Borrowing and depositing take place through an outside bank, although in some circumstances a money market could be used instead. Conditions for different forms of lending are discussed. This is a survey of three technical papers, where the mathematical models are developed in detail and the proofs are supplied.

A Strategic Market Game with Secured Lending

We study stationary Markov equilibria for strategic, competitive games, in a market-economy model with one non-durable commodity, fiat money, borrowing/lending through a central bank or a money market, and a continuum of agents. These use fiat money in order to offset random fluctuations in their endowments of the commodity, are not allowed to borrow more than they can pay back (secured lending), and maximize expected discounted utility from consumption of the commodity. Their aggregate optimal actions determine dynamically prices and/or interest rates for borrowing and lending, in each period of play. In equilibrium, random fluctuations in endowment- and wealth-levels offset each other, and prices and interest rates remain constant. As in our related recent work, KSS (1994), we study in detail the individual agents' dynamic optimization problems, and the invariance measures for the associated, optimally controlled Markov chains. By appropriate aggregation, these individual problems lead to the construction of stationary Markov competitive equilibrium for the economy as a whole. Several examples are studied in detail, fairly general existence theorems are established, and open questions are indicated for further research.

Information and the Existence of Stationary Markovian Equilibrium

We describe conditions for the existence of a stationary Markovian equilibrium when total production or total endowment is a random variable. Apart from regularity assumptions, there are two crucial conditions: (i) low information -- agents are ignorant of both total endowment and their own endowments when they make decisions in a given period, and (ii) proportional endowments -- the endowment of each agent is in proportion, possibly a random proportion, to the total endowment. When these conditions hold, there is a stationary equilibrium. When they do not hold, such equilibrium need not exist.Information, stochastic process, money, and disequilibrium

Financial Control of a Competitive Economy without Randomness

The monetary and fiscal control of a simple economy without outside randomness is studied here from the micro-economic basis of a strategic market game. The government's bureaucracy is treated as a public good that provides services at a cost. A conventional public good is also considered.Dynamic programming, Public goods, Bureaucracy, Taxation

A Stochastic Overlapping Generations Economy with Inheritance

An overlapping generations model of an exchange economy is considered, with individuals having a finite expected life-span. Conditions concerning birth, death, inheritance and bequests are fully specified. Under such conditions, the existence of stationary Markov equilibrium is established in some generality, and several explicitly solvable examples are treated in detail.Overlapping generations, inheritance, stochastic process, life span

Trivalent scandium, yttrium and lanthanide complexes with thia-oxa and selena-oxa macrocycles and crown ether coordination

Complexes of the oxa-thia macrocycles [18]aneO4S2, [15]aneO3S2 and the oxa-selena macrocycle [18]aneO4Se2 (L) of types [MCl2(L)]FeCl4 (M = Sc or Y) were prepared from [ScCl3(thf)3] or [YCl2(THF)5][YCl4(THF)2] and the ligand in anhydrous MeCN, using FeCl3 as a chloride abstractor. The [MI2(L)]I, [LaI3(L)] and [LuI2(L)]I have been prepared from the ligands and the appropriate anhydrous metal triiodide in MeCN. Complexes of type [LaI3(crown)] and [LuI2(crown)]I (crown = 18-crown-6, 15-crown-5) were made for comparison. Use of the metal iodide results in complexes with high solubility compared to the corresponding chlorides, although also with increased sensitivity to moisture. All complexes were characterised by microanalysis, IR, (1)H, (45)Sc and (77)Se NMR spectroscopy as appropriate. X-ray crystal structures are reported for [ScCl2([18]aneO4S2)][FeCl4], [ScI2([18]aneO4S2)]I, [YCl2(18-crown-6)]3[Y2Cl9], [YCl2([18]aneO4S2)][FeCl4], [LaI3(15-crown-5)], [LaI2(18-crown-6)(MeCN)]I, [LuI(18-crown-6)(MeCN)2]I2, [Lu(15-crown-5)(MeCN)2(OH2)]I3, [LaI3([18]aneO4S2)], [LaI([18]aneO4S2)(OH2)]I2, [LaI3([18]aneO4Se2)] and [LuI2([18]aneO4Se2)]I. In each complex all the neutral donor atoms of the macrocycles are coordinated to the metal centre, showing very rare examples of these oxophilic metal centres coordinated to thioether groups, and the first examples of coordinated selenoether donors. In some cases MeCN or adventitious water displaces halide ligands, but not the S/Se donors from La or Lu complexes. A complex of the oxa-tellura macrocycle [18]aneO4Te2, [ScCl2([18]aneO4Te2)][FeCl4] was isolated, but is unstable in MeCN solution, depositing elemental Te. YCl3 and 18-crown-6 produced [YCl2(18-crown-6)]3[Y2Cl9], the asymmetric unit of which contains two cations with a trans-YCl2 arrangement and a third with a cis-YCl2 group