Optimality Results for Coupon Collection

2016 ◽  
Vol 53 (3) ◽  
pp. 930-937
Author(s):  
Mark Brown ◽  
Sheldon M. Ross
Keyword(s):  
Expected Time ◽  
Coupon Collection

AbstractWe consider the coupon collection problem, where each coupon is one of the types 1,…,s with probabilities given by a vector 𝒑. For specified numbers r1,…,rs, we are interested in finding 𝒑 that minimizes the expected time to obtain at least ri type-i coupons for all i=1,…,s. For example, for s=2, r1=1, and r2=r, we show that p1=(logr−log(logr))∕r is close to optimal.

scholarly journals Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

2021 ◽  
Vol 174 (1) ◽  
Author(s):  
Amirlan Seksenbayev
Keyword(s):  
Dynamic Programming ◽  
Asymptotic Expansions ◽  
Dual Problem ◽  
Infinite Length ◽  
Expected Time ◽  
Selection Strategies ◽  
Dynamic Programming Equation ◽  
Optimal Values ◽  
Design Selection ◽  
Selection Of

AbstractWe study two closely related problems in the online selection of increasing subsequence. In the first problem, introduced by Samuels and Steele (Ann. Probab. 9(6):937–947, 1981), the objective is to maximise the length of a subsequence selected by a nonanticipating strategy from a random sample of given size $n$ n . In the dual problem, recently studied by Arlotto et al. (Random Struct. Algorithms 49:235–252, 2016), the objective is to minimise the expected time needed to choose an increasing subsequence of given length $k$ k from a sequence of infinite length. Developing a method based on the monotonicity of the dynamic programming equation, we derive the two-term asymptotic expansions for the optimal values, with $O(1)$ O ( 1 ) remainder in the first problem and $O(k)$ O ( k ) in the second. Settling a conjecture in Arlotto et al. (Random Struct. Algorithms 52:41–53, 2018), we also design selection strategies to achieve optimality within these bounds, that are, in a sense, best possible.

scholarly journals The Rate of Compensatory Evolution

Genetics ◽  
1996 ◽  
Vol 144 (1) ◽  
pp. 419-426 ◽  
Author(s):  
Wolfgang Stephan
Keyword(s):  
Diffusion Theory ◽  
Secondary Structures ◽  
Compensatory Mutation ◽  
Weak Selection ◽  
Mutation Pressure ◽  
Theory Of Evolution ◽  
Compensatory Evolution ◽  
Expected Time ◽  
Individual Fitness ◽  
Shifting Balance

Abstract A two-locus model is presented to analyze the evolution of compensatory mutations occurring in stems of RNA secondary structures. Single mutations are assumed to be deleterious but harmless (neutral) in appropriate combinations. In proceeding under mutation pressure, natural selection and genetic drift from one fitness peak to another one, a population must therefore pass through a valley of intermediate deleterious states of individual fitness. The expected time for this transition is calculated using diffusion theory. The rate of compensatory evolution, kc, is then defined as the inverse of the expected transition time. When selection against deleterious single mutations is strong, kc, depends on the recombination fraction r between the two loci. Recombination generally reduces the rate of compensatory evolution because it breaks up favorable combinations of double mutants. For complete linkage, kc, is given by the rate at which favorable combinations of double mutantS are produced by compensatory mutation. For r > 0, kc, decreases exponentially with r. In contrast, kc, becomes independent of r for weak selection. We discuss the dynamics of evolutionary substitutions of compensatory mutants in relation to Wright'S shifting balance theory of evolution and use our results to analyze the substitution process in helices of mRNA secondary structures.

Ovarian and endocrine responses to prostaglandin F2α (PGF2α) given at the expected time of the endogenous FSH peak at mid-cycle in ewes

2006 ◽  
Vol 66 (4) ◽  
pp. 811-821 ◽  
Author(s):  
Xinyu Liu ◽  
Qiao Dai ◽  
Ezra J. Hart ◽  
Rajesha Duggavathi ◽  
David M.W. Barrett ◽  
...  
Keyword(s):  
Prostaglandin F2α ◽  
Expected Time

scholarly journals An Efficient Motion Strategy to Compute Expected-Time Locally Optimal Continuous Search Paths in Known Environments

2009 ◽  
Vol 23 (12-13) ◽  
pp. 1533-1560 ◽  
Author(s):  
Alejandro Sarmiento ◽  
Rafael Murrieta-Cid ◽  
Seth Hutchinson
Keyword(s):  
Expected Time ◽  
Motion Strategy

scholarly journals Superconductor qubits hamiltonian approximations effect on quantum state evolution and control

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Javad Sharifi
Keyword(s):  
Quantum State ◽  
Charge Qubit ◽  
Gate Operation ◽  
Goal State ◽  
Expected Time ◽  
State Evolution ◽  
Angle Of Deviation ◽  
Hadamard Gate ◽  
Phase Qubit ◽  
And Control

AbstractMicrowave IQ-mixer controllers are designed for the three approximated Hamiltonians of charge, phase and flux qubits and the controllers are exerted both on approximate and precise quantum system models. The controlled qubits are for the implementation of the two quantum-gates with these three fundamental types of qubits, Quantum NOT-gate and Hadamard-gate. In the charge-qubit, for implementation of both gates, in the approximated and precise model, we observed different controlled trajectories. But fortunately, applying the controller designed for the approximated system over the precise system leads to the passing of the quantum state from the desired state sooner that the expected time. Phase-qubit and flux qubit have similar behaviour under the control system action. In both of them, the implementation of NOT-gate operation led to same trajectories which arrive at final goal state at different times. But in both of those two qubits for implementation of Hadamard-gate, desired trajectory and precise trajectory have some angle of deviation, then by exerting the approximated design controller to precise system, it caused the quantum state to approach the goal state for Hadamard gate implementation, and since the quantum state does not completely reach the goal state, we can not obtain very high gate fidelity.

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