Optimal control of a stochastic hybrid system with discounted cost

We address the optimal control problem of a very general stochastic hybrid system with both autonomous and impulsive jumps. The planning horizon is infinite and we use the discounted-cost criterion for performance evaluation. Under certain assumptions, we show the existence of an optimal control. We then derive the quasivariational inequalities satisfied by the value function and establish well-posedness. Finally, we prove the usual verification theorem of dynamic programming.

Solutions of cubic equations in quadratic fields

Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r + s + t = rst = 1 in O-K. This Diophantine equation gives an elliptic curve defined over Q with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of Q(i) and Q(root 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

The complexity of certain incremental code generation problems

This paper looks at the complexity of four different incremental problems. The following are the problems considered: (1) Interval partitioning of a flow graph (2) Breadth first search (BFS) of a directed graph (3) Lexicographic depth first search (DFS) of a directed graph (4) Constructing the postorder listing of the nodes of a binary tree. The last problem arises out of the need for incrementally computing the Sethi-Ullman (SU) ordering [1] of the subtrees of a tree after it has undergone changes of a given type. These problems are among those that claimed our attention in the process of our designing algorithmic techniques for incremental code generation. BFS and DFS have certainly numerous other applications, but as far as our work is concerned, incremental code generation is the common thread linking these problems. The study of the complexity of these problems is done from two different perspectives. In [2] is given the theory of incremental relative lower bounds (IRLB). We use this theory to derive the IRLBs of the first three problems. Then we use the notion of a bounded incremental algorithm [4] to prove the unboundedness of the fourth problem with respect to the locally persistent model of computation. Possibly, the lower bound result for lexicographic DFS is the most interesting. In [5] the author considers lexicographic DFS to be a problem for which the incremental version may require the recomputation of the entire solution from scratch. In that sense, our IRLB result provides further evidence for this possibility with the proviso that the incremental DFS algorithms considered be ones that do not require too much of preprocessing.

The Herzog-Vasconcelos conjecture for affine semigroup rings

Let S be a simplicial affine semigroup such that its semigroup ring A = k[S] is Buchsbaum. We prove for such A the Herzog-Vasconcelos conjecture: If the A-module Der(k)A of k-linear derivations of A has finite projective dimension then it is free and hence A is a polynomial ring by the well known graded case of the Zariski-Lipman conjecture.

Advantage of the ether linkage between the positive charge and the cholesteryl skeleton in cholesterol-based amphiphiles as vectors for gene delivery

Twelve novel cationic cholesterol derivatives with different linkage types between the cationic headgroup and the cholesteryl backbone have been developed. These have been tested for their efficacies as gene transfer agents as mixtures with dioleoyl phosphatidylethanolamine (DOPE). A pronounced improvement in transfection efficiency was observed when the cationic center was linked to the steroid backbone using an ether type bond. Among these, cholest-5-en-3b-oxyethane-N, N,N-trimethylammonium bromide (2a) and cholest-5-en-3b-oxyethane-N, N-dimethyl-N-2-hydroxyethylammonium bromide (3d) showed transfection efficiencies considerably greater than commercially available reagents such as Lipofectin or Lipofectamine. To achieve transfection, 3d did not require DOPE. Increasing hydration at the headgroup level for both ester- and ether-linked amphiphiles resulted in progressive loss of transfection efficiency. Transfection efficiency was also greatly reduced when a 'disorder'-inducing chain like an oleyl (cis-9-octadecenyl) segment was added to these cholesteryl amphiphiles. Importantly, the transfection ability of 2a with DOPE in the presence of serum was significantly greater than for a commercially available reagent, Lipofectamine. This suggests that these novel cholesterol-based amphiphiles might prove promising in applications involving liposome-mediated gene transfection. This investigation demonstrates the importance of structural features at the molecular level for the design of cholesterol-based gene delivery reagents that would aid the development of newer, more efficient formulations based on this class of molecules.

Providing quality of service via opportunistic splitting using stochastic approximation

We consider the problem of wireless channel allocation to multiple users. A slot is given to a user with a highest metric (e.g., channel gain) in that slot. The scheduler may not know the channel states of all the users at the beginning of each slot. In this scenario opportunistic splitting is an attractive solution. However this algorithm requires that the metrics of different users form independent, identically distributed (iid) sequences with same distribution and that their distribution and number be known to the scheduler. This limits the usefulness of opportunistic splitting. In this paper we develop a parametric version of this algorithm. The optimal parameters of the algorithm are learnt online through a stochastic approximation scheme. Our algorithm does not require the metrics of different users to have the same distribution. The statistics of these metrics and the number of users can be unknown and also vary with time. Each metric sequence can be Markov. We prove the convergence of the algorithm and show its utility by scheduling the channel to maximize its throughput while satisfying some fairness and/or quality of service constraints.

On optimal timer-based distributed selection for rate-adaptive multi-user diversity systems

We develop an optimal, distributed, and low feedback timer-based selection scheme to enable next generation rate-adaptive wireless systems to exploit multi-user diversity. In our scheme, each user sets a timer depending on its signal to noise ratio (SNR) and transmits a small packet to identify itself when its timer expires. When the SNR-to-timer mapping is monotone non-decreasing, timers of users with better SNRs expire earlier. Thus, the base station (BS) simply selects the first user whose timer expiry it can detect, and transmits data to it at as high a rate as reliably possible. However, timers that expire too close to one another cannot be detected by the BS due to collisions. We characterize in detail the structure of the SNR-to-timer mapping that optimally handles these collisions to maximize the average data rate. We prove that the optimal timer values take only a discrete set of values, and that the rate adaptation policy strongly influences the optimal scheme's structure. The optimal average rate is very close to that of ideal selection in which the BS always selects highest rate user, and is much higher than that of the popular, but ad hoc, timer schemes considered in the literature.

Opportunistic splitting for scheduling via stochastic approximation

We consider the problem of scheduling a wireless channel among multiple users. A slot is given to a user with a highest metric (e.g., channel gain) in that slot. The scheduler may not know the channel states of all the users at the beginning of each slot. In this scenario opportunistic splitting is an attractive solution. However this algorithm requires that the metrics of different users form independent, identically distributed (iid) sequences with same distribution and that their distribution and number be known to the scheduler. This limits the usefulness of opportunistic splitting. In this paper we develop a parametric version of this algorithm. The optimal parameters of the algorithm are learnt online through a stochastic approximation scheme. Our algorithm does not require the metrics of different users to have the same distribution. The statistics of these metrics and the number of users can be unknown and also vary with time. We prove the convergence of the algorithm and show its utility by scheduling the channel to maximize its throughput while satisfying some fairness and/or quality of service constraints.

Partial instantiation methods for inference in first-order logic

Satisfiability algorithms for propositional logic have improved enormously in recently years. This improvement increases the attractiveness of satisfiability methods for first-order logic that reduce the problem to a series of ground-level satisfiability problems. R. Jeroslow introduced a partial instantiation method of this kind that differs radically from the standard resolution-based methods. This paper lays the theoretical groundwork for an extension of his method that is general enough and efficient enough for general logic programming with indefinite clauses. In particular we improve Jeroslow's approach by (1) extending it to logic with functions, (2) accelerating it through the use of satisfiers, as introduced by Gallo and Rago, and (3) simplifying it to obtain further speedup. We provide a similar development for a "dual" partial instantiation approach defined by Hooker and suggest a primal-dual strategy. We prove correctness of the primal and dual algorithms for full first-order logic with functions, as well as termination on unsatisfiable formulas. We also report some preliminary computational results.

Matching sender and receiver: poikilothermy and frequency tuning in a tree cricket

Animals communicate in non-ideal and noisy conditions. The primary method they use to improve communication efficiency is sender-receiver matching: the receiver's sensory mechanism filters the impinging signal based on the expected signal. In the context of acoustic communication in crickets, such a match is made in the frequency domain. The males broadcast a mate attraction signal, the calling song, in a narrow frequency band centred on the carrier frequency (CF), and the females are most sensitive to sound close to this frequency. In tree crickets, however, the CF changes with temperature. The mechanisms used by female tree crickets to accommodate this change in CF were investigated at the behavioural and biomechanical level. At the behavioural level, female tree crickets were broadly tuned and responded equally to CFs produced within the naturally occurring range of temperatures (18 to 27 degrees C). To allow such a broad response, however, the transduction mechanisms that convert sound into mechanical and then neural signals must also have a broad response. The tympana of the female tree crickets exhibited a frequency response that was even broader than suggested by the behaviour. Their tympana vibrate with equal amplitude to frequencies spanning nearly an order of magnitude. Such a flat frequency response is unusual in biological systems and cannot be modelled as a simple mechanical system. This feature of the tree cricket auditory system not only has interesting implications for mate choice and species isolation but may also prove exciting for bio-mimetic applications such as the design of miniature low frequency microphones.

A lower bound for the hitting set size for combinatorial rectangles and an application

We prove a lower bound of Omega(1/epsilon (m + log(d - a)) where a = [log(m) (1/4epsilon)] for the hitting set size for combinatorial rectangles of volume at least epsilon in [m](d) space, for epsilon is an element of [m(-(d-2)), 2/9] and d > 2. (C) 2002 Elsevier Science B.V. All rights reserved.

Almost quarter-pinched Kähler metrics and Chern numbers

Given n is an element of Z(+) and epsilon > 0, we prove that there exists delta = delta(epsilon, n) > 0 such that the following holds: If (M(n),g) is a compact Kahler n-manifold whose sectional curvatures K satisfy -1 -delta <= K <= -1/4 and c(I)(M), c(J)(M) are any two Chern numbers of M, then |c(I)(M)/c(J)(M) - c(I)(0)/c(J)(0)| < epsilon, where c(I)(0), c(J)(0) are the corresponding characteristic numbers of a complex hyperbolic space form. It follows that the Mostow-Siu surfaces and the threefolds of Deraux do not admit Kahler metrics with pinching close to 1/4.

Variants of Miyachi�s Theorem for Nilpotent lie Groups

We formulate and prove two versions of Miyachi�s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi�s theorem for the group Fourier transform.

Variants Of Miyachi’S Theorem For Nilpotent Lie Groups

We formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

Boxicity of Line Graphs

The boxicity of a graph H, denoted by View the MathML source, is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in View the MathML source. In this paper we show that for a line graph G of a multigraph, View the MathML source, where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)−1), where χ(G) denotes the chromatic number of G, and therefore, View the MathML source. For the d-dimensional hypercube Qd, we prove that View the MathML source. The question of finding a nontrivial lower bound for View the MathML source was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800]. The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once).