Generalising a characterisation of Hermitian curves

This article proves a characterisation of the classical unital that is a generalisation of a characterisation proved in 1982 by Lefèvre-Percsy. It is shown that if is a Buekenhout-Metz unital with respect to a line in such that a line of not through meets in a Baer subline, then is classical. An immediate corollary is that if is a unital in PG such that is Buekenhout-Metz with respect to two distinct lines, then is classical. Received 5 August 1999; revised 15 February 2000.     

OPTIMAL INSTAR PARASITIZATION IN A STAGE STRUCTURED HOST-PARASITOID MODEL

A stage structured host-parasitoid model is derived and the equilibria studied. It is shown under what conditions the parasitoid controls an exponentially growing host in the sense that a coexistence equilibrium exists. Furthermore, for host populations whose inherent growth rate is not too large it is proved that in order to minimize the adult host equilibrium level it is necessary that the parasitoids attack only one of the larval stages. It is also proved in this case that the minimum adult host equilibrium level is attained when the parasitoids attack that larval stage which also maximizes the expected number of emerging adult parasitoid per larva at equilibrium. Numerical simulations tentatively indicate that the first conclusion remains in general valid for the model. However, numerical studies also show that it is not true in general that the optimal strategy will maximize the number of emerging adult parasitoid per larva at equilibrium.     

关于区间上的Lorenz映射

本文引进了变换ζ．首先，用比文［４］简捷的方法证明了Ｌｏｒｅｎｚ映射的Ｓａｒｋｏｖｓｋｉ定理并更正了文［４］中的一个错误，还得到了扩张的Ｌｏｒｅｎｚ映射中周期点的存在性与其扩张常数之间的关系．其次，证明了Ｌｏｒｅｎｚ映射与其在变换ζ的像具有相等的捏制行列式．此外，给出了广义的Ｌｏｒｅｎｚ映射的周期点的存在性适合Ｓａｒｋｏｖｓｋｉ序关系的一个充分条件     

Quenching of solutions to a class of semilinear parabolic equations with boundary degeneracy

This paper concerns the quenching phenomenon of solutions to a class of semilinear parabolic equations with boundary degeneracy. In the case that the degeneracy is not strong, it is shown that there exists a critical length, which is positive, such that the solution exists globally in time if the length of the spatial interval is less than it, while quenches in a finite time if the length of the spatial interval is greater than it. Whereas in the case that the degeneracy is strong enough, the solution must be quenching in a finite time no matter how long the spatial interval is. Furthermore, for each quenching solution, the set of quenching points is determined and it is proved that its derivative with respect to the time must blow up at the quenching time.     

Notes on sectionally complemented lattices. IV How far does the Atom Lemma go?

There are two results in the literature that prove that the ideal lattice of a finite, sectionally complemented, chopped lattice is again sectionally complemented. The first is in the 1962 paper of G. Grätzer and E. T. Schmidt, where the ideal lattice is viewed as a closure space to prove that it is sectionally complemented; we call the sectional complement constructed then the 1960 sectional complement. The second is the Atom Lemma from a 1999 paper of the same authors that states that if a finite, sectionally complemented, chopped lattice is made up of two lattices overlapping in an atom and a zero, then the ideal lattice is sectionally complemented. In this paper, we show that the method of proving the Atom Lemma also applies to the 1962 result. In fact, we get a stronger statement, in that we get many sectional complements and they are rather close to the componentwise sectional complement.     

Equilibria in load balancing games

A Nash equilibrium （NE） in a multi-agent game is a strategy profile that is resilient to unilateral deviations. A strong Nash equilibrium （SE） is one that is stable against coordinated deviations of any coalition. We show that, in the load balancing games, NEs approximate SEs in the sense that the benefit of each member of any coalition from coordinated deviations is well limited. Furthermore, we show that an easily recognizable special subset of NEs exhibit even better approximation of SEs.     

Proof validation in real analysis: Inferring and checking warrants

In the study reported here, we investigate the skills needed to validate a proof in real analysis, i.e., to determine whether a proof is valid. We first argue that when one is validating a proof, it is not sufficient to make certain that each statement in the argument is true. One must also check that there is good reason to believe that each statement follows from the preceding statements or from other accepted knowledge, i.e., that there is a valid warrant for making that statement in the context of this argument. We then report an exploratory study in which we investigated the behavior of 13 undergraduates when they were asked to determine whether or not a particular flawed mathematical argument is a valid mathematical proof. The last line of this purported proof was true, but did not follow legitimately from the earlier assertions in the proof. Our findings were that only six of these undergraduates recognized that this argument was invalid and only two did so for legitimate mathematical reasons. On a more positive note, when asked to consider whether the last line of the proof followed from previous assertions, a total of 10 students concluded that the statement did not and rejected the proof as invalid.     

Dirichlet's Theorem, Vojta's Inequality, and Vojta's Conjecture

This paper addresses questions involving the sharpness of Vojta's conjecture and Vojta's inequality for algebraic points on curves over number fields. It is shown that one may choose the approximation term mS(D,-) in such a way that Vojta's inequality is sharp in Theorem 2.3. Partial results are obtained for the more difficult problem of showing that Vojta's conjecture is sharp when the approximation term is not included (that is, when D=0). In Theorem 3.7, it is demonstrated that Vojta's conjecture is best possible with D=0 for quadratic points on hyperelliptic curves. It is also shown, in Theorem 4.8, that Vojta's conjecture is sharp with D=0 on a curve C over a number field when an analogous statement holds for the curve obtained by extending the base field of C to a certain function field.     

Neo-Mooreanism,contextualism, and the evidential basis of scepticism

Two of the main forms of anti-scepticism in the contemporary literature—namely, neo-Mooreanism and attributer contextualism—share a common claim, which is that we are, contra the sceptic, able to know the denials of sceptical hypotheses. This paper begins by surveying the relative merits of these views when it comes to dealing with the standard closure-based formulation of the sceptical problem that is focussed on the possession of knowledge. It is argued, however, that it is not enough to simply deal with this version of the sceptical challenge, since there is a more fundamental sceptical problem underlying the standard closure-based sceptical argument that can be expressed in terms of the evidential basis of our beliefs. Whilst it is argued that neo-Mooreanism has a slight edge over attributer contextualism when it comes to dealing with the closure-based formulation of the sceptical problem, it is claimed that this view is in an ever stronger dialectical position when it comes to the more pressing evidential formulation of the sceptical problem. It is shown that this is so even if one adapts the attributer contextualist thesis along the lines suggested by Michael Williams and Ram Neta so that it is explicitly designed to deal with the evidential variant of the sceptical problem.     

Semiring and Semimodule Issues in MV-Algebras

It is known that an atomic right LCM domain need not be a UFD but is a projectivity-UFD if it is also modular. This paper studies a slightly weaker and easier condition, the RAMP (acronym for the property in the title) , which also ensures that an atomic right LCM domain will be a projectivity-UFD. Among other things it is shown that in an atomic LCM domain, modularity is equivalent to the pair RAMP and LAMP (the left-right analog of RAMP). This result is then used to show that an atomic LCM domain with conjugation is modular. An example is given of an atomic LCM domain that has neither the RAMP nor the LAMP. All rings are not-necessarily commutative integral domains. Recall that an atomic ring is one in which every nonzero nonunit is a product of atoms (i.e. irreducibles) . A ring R is a right LCM domain if for any two elements a and b in R, aR ∩ bR is a principal right ideal. A right LCM domain need not be a left LCM domain [3] . If a ring has both properties it is called an LCM domain. It Is known (see Example 2 below) that, unlike the commutative case, an atomic right LCM domain need not be a UFD (unique factorization domain). In [1] it is shown that if the ring is also modular then it is a projectivity-UFD (definition of the latter recalled below)     

Tyson型集及Borel函数的图的拟对称极小性

本文将欧氏空间R^d中形如[0，1]×Z的集称为Tyson型集，其中d>1，Z⊂R^d-1.已知当Z是R^d-1中的紧集时，Tyson型集是拟对称极小集.本文改进了这个结果，证明了当Z是R^d-1中的Borel集时，Tyson型集仍是拟对称极小集.作为应用，我们证明了Tyson型集三个形变版本的拟对称极小性，其中一个结果是：设Z是R^d-1中的任一Borel集，h：Z→R¹是Borel函数，满足dim_H（{h≠0}∩Z）=dim_H Z，则h的图G（h）是拟对称极小集，其中h的图G（h）定义为G（h）={（z，y）：z∈Z，y∈[0，h（z）]}.     

ASYMPTOTIC NORMALITY OF QUASI MAXIMUM LIKELIHOOD ESTIMATE IN GENERALIZED LINEAR MODELS

For the Generalized Linear Model (GLM), under some conditions including that the specification of the expectation is correct, it is shown that the Quasi Maximum Likelihood Estimate (QMLE) of the parameter-vector is asymptotic normal. It is also shown that the asymptotic covariance matrix of the QMLE reaches its minimum (in the positive-definte sense) in case that the specification of the covariance matrix is correct.     

Autonomous Integral Functionals with Discontinuous Nonconvex Integrands: Lipschitz Regularity of Minimizers,DuBois—Reymond Necessary Conditions,and Hamilton—Jacobi Equations

This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois—Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally Lipschitz and satisfies (in a generalized sense) a Hamilton—Jacobi equation.     

A posteriori joint detection of a recurring tuple of reference fragments in a quasi-periodic sequence

The problem of joint detection of a recurring tuple of reference fragments in a noisy numerical quasi-periodic sequence is solved in the framework of the a posteriori (off-line) approach. It is assumed that (i) the total number of fragments in the sequence is known, (ii) the index of the sequence member corresponding to the beginning of a fragment is a deterministic (not random) value, and (iii) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is shown that the problem consists of testing a set of simple hypotheses about the mean of a random Gaussian vector. A specific feature of the problem is that the cardinality of the set grows exponentially as the vector dimension (i.e., the length of the observed sequence) and the number of fragments in the sequence increase. It is established that the search for a maximum-likelihood hypothesis is equivalent to the search for arguments that maximize a special auxiliary objective function with linear inequality constraints. It is shown that this function is maximized by solving the basic extremum problem. It is proved that this problem is solvable in polynomial time. An exact algorithm for its solution is substantiated that underlies an algorithm guaranteeing optimal (maximum-likelihood) detection of a recurring tuple of reference fragments. The results of numerical simulation demonstrate the noise stability of the detection algorithm.     

On optimal polling policies

In a single-server polling system, the server visits the queues according to a routing policy and while at a queue, serves some or all of the customers there according to a service policy. A polling (or scheduling) policy is a sequence of decisions on whether to serve a customer, idle the server, or switch the server to another queue. The goal of this paper is to find polling policies that stochastically minimize the unfinished work and the number of customers in the system at all times. This optimization problem is decomposed into three subproblems: determine the optimal action (i.e., serve, switch, idle) when the server is at a nonempty queue; determine the optimal action (i.e., switch, idle) when the server empties a queue; determine the optimal routing (i.e., choice of the queue) when the server decides to switch. Under fairly general assumptions, we show for the first subproblem that optimal policies are greedy and exhaustive, i.e., the server should neither idle nor switch when it is at a nonempty queue. For the second subproblem, we prove that in symmetric polling systems patient policies are optimal, i.e., the server should stay idling at the last visited queue whenever the system is empty. When the system is slotted, we further prove that non-idling and impatient policies are optimal. For the third subproblem, we establish that in symmetric polling systems optimal policies belong to the class of Stochastically Largest Queue (SLQ) policies. An SLQ policy is one that never routes the server to a queue known to have a queue length that is stochastically smaller than that of another queue. This result implies, in particular, that the policy that routes the server to the queue with the largest queue length is optimal when all queue lengths are known and that the cyclic routing policy is optimal in the case that the only information available is the previous decisions.This work was supported in part by NSF under Contract ASC-8802764.     

Pricing a Nontradeable Asset and Its Derivatives

This paper extends the Black-Scholes methodology to payoffs that are functions of a stochastically varying variable that can be observed but not traded. The stochastic price process proposed in this paper satisfies a partial differential equation that is an extension of the Black-Scholes equation. The resulting price process is based on projection onto the marketed space, and it is universal in the sense that all risk-averse investors will find that, when priced according to the process, the asset cannot improve portfolio performance relative to other assets in the market. The development of the equation and its properties is facilitated by the introduction of an operational calculus for pricing. The results can be put in risk-neutral form. Perfect replication is not generally possible for these derivatives, but the approximation of minimum expected squared error is determined by another partial differential equation.     

Sufficiency theorems for target capture

A dynamical system is assumed to be governed by a set of ordinary differential equations subject to control. The set of points in state space from which there exist permissible controls that can transfer these points to a prescribed target set in a finite time interval is called a capture set. The task of determining the capture set is studied in two contexts. first, in the case of the system subject to a single control vector; and second, in the case of the system subject to two control vectors each operated independently. In the latter case, it is assumed that one controller's aim is to cause the system to attain the target, and the other's is to prevent that from occurring.Sufficient conditions are developed that, when satisfied everywhere on the interior of some subset of the state space, ensure that this subset is truly a capture set. A candidate capture set is assumed to have already been predetermined by independent methods. The sufficient conditions developed herein require the use of an auxiliary scalar function of the state, similar to a Lyapunov function.To ensure capture, five conditions must be satisfied. Four of these constrain the auxiliary state function. Basically, these four conditions require that the boundary of the controllable set be an envelope of the auxiliary state function and that that function be positive inside the capture set, approaching zero value as the target set is approached. The final condition tests the inner product of the gradient of the auxiliary state function with the system state velocity vector. If the sign of that inner product can be made negative everywhere within the test subset, then that subset is a capture set.Dedicated to Professor A. BusemannThe authors are indebted to Professors G. Leitmann and J. M. Skowronskii for their useful comments and discussion.     

拓扑空间的内部运算

讨论了拓扑空间上内部运算的性质,并对满足内部运算前三条性质的一类运算,引入一种等价关系,将这些运算进行等价分类,证明了每个等价类中的所有运算导出同一个拓扑,并进一步证明了每个等价类中仅有一个是内部运算且是等价类中的最小元.     

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Let be a correlated random walk in random environment. For the sub-linear regime, that is, almost surely but , we show that there is ??Let be a correlated random walk in random environment. For the sub-linear regime, that is, almost surely but , we show that there is $0s. This result characterizes the slowdown property of the walk.