Proof Of Optimality Research Articles

Stability assessment and feature analysis of slope in Nanfen Open Pit Iron Mine

Under the combined influences of special topography and the long term mining in Nanfen Open Pit Iron Mine, many large scale landslide masses appeared in heading side of stope, and tens of millions of iron mine is buried underside, making great economic losses. In order to guide the safety mining and increase the supply quantity, this paper through the use of MSARMA-method and the “MSARMA evaluation and analytical system for slope stability analysis” based on this method, which aimed at the quantitative evaluating of the slope stability before and after the actual mining below the slip mass, and the sensitivity analysis for the main influencing factor, providing scientific proof for the parameter optimization of open mine surface slope, the security and sustainable exploitation.

Experimental proof of optimality of interfacing of chlorosome BChl c and membrane BChl a subantennae in superantenna of photosynthetic green bacteria from the oscillochloridaceae family

This work continues the cycle of studies of the strategy of efficient functioning of natural lighthar� vesting antennae, in particular, inhomogeneous super� antennae of green bacteria, consisting of several homogeneous subantennae (1). In our previous paper (2), mathematical modeling of the dynamics of excita� tion energy transfer in inhomogeneous superantenna of photosynthetic green bacteria Oscillochloris (Osc.) trichoides (using formalism of probability matrices) allowed us to determine the theoretically optimal spectral composition of homogeneous superantennae of Osc. trichoides, which showed the importance of optimal interfacing of their energy levels for efficient and stable functioning of the entire superantenna as a whole. In particular, we theoretically substantiated the necessity of existence in the Osc. trichoides superan� tenna of an intermediate BChl a subantenna with the energy level in the range of ~790-800 nm for optimiz� ing energy transfer from the chlorosome B750 suban� tenna to the membrane B805 subantenna. This paper presents an experimental proof of the validity of the theoretical inferences made in our pre� vious work. The lightharvesting superantenna of Osc. tri� choides is located in separate subcellular structures: the chlorosome containing the oligomeric BChl c sub� antenna B750 and the cytoplasmic membrane con� taining the BChl a subantenna B805-860, which is an analogue of the membrane subantenna B808-866 of Chloroflecsus aurantiacus (3).

Tower-of-sets analysis for the Kise–Ibaraki–Mine algorithm

Kise et al. (Oper. Res. 26:121–126, 1978) give an O(n2) time algorithm to find an optimal schedule for the single-machine number of late jobs problem with agreeable job release dates and due dates. Li et al. (Oper. Res. 58:508–509, 2010a) point out that their proof of optimality for their algorithm is incorrect by giving a counter-example. In this paper, using the concept of “tower-of-sets” from Lawler (Math. Comput. Model. 20:91–106, 1994), we construct the tower-of-sets of the early job set generated by the algorithm. Then we give a correct proof of optimality for the algorithm and show a new result that the early job set by the algorithm obtains not only the maximum number of jobs but also the smallest total processing time among all optimal schedules. The result can be applied to study the problems of the hard real-time systems.

The L1-norm best-fit hyperplane problem

We formalize an algorithm for solving the L(1)-norm best-fit hyperplane problem derived using first principles and geometric insights about L(1) projection and L(1) regression. The procedure follows from a new proof of global optimality and relies on the solution of a small number of linear programs. The procedure is implemented for validation and testing. This analysis of the L(1)-norm best-fit hyperplane problem makes the procedure accessible to applications in areas such as location theory, computer vision, and multivariate statistics.

A Proof of the Optimality of Volatility Weighting Over Time

We provide a proof that volatility weighting over time increases the Sharpe or Information Ratio. The higher the degree of volatility smoothing achieved by volatility weighting, the higher the risk-adjusted performance. Our results apply to risky portfolios managed against a risk free or risky benchmark (so including alpha strategies) and to volatility targeting strategies. We provide an empirical illustration of our results.

Optimal cloning of arbitrary mirror-symmetric distributions on the Bloch sphere: a proposal for practical photonic realization

We study state-dependent quantum cloning that can outperform universal cloning (UC). This is possible by using some a priori information on a given quantum state to be cloned. Specifically, we propose a generalization and optical implementation of quantum optimal mirror phase-covariant cloning, which refers to optimal cloning of sets of qubits of known modulus of the expectation value of Pauli's Z operator. Our results can be applied to cloning of an arbitrary mirror-symmetric distribution of qubits on the Bloch sphere including in special cases UC and phase-covariant cloning. We show that the cloning is optimal by adapting our former optimality proof for axisymmetric cloning (Bartkiewicz and Miranowicz 2010 Phys. Rev. A 82 042330). Moreover, we propose an optical realization of the optimal mirror phase-covariant 1→2 cloning of a qubit, for which the mean probability of successful cloning varies from 1/6 to 1/3 depending on prior information on the set of qubits to be cloned. The qubits are represented by polarization states of photons generated by the type-I spontaneous parametric down-conversion. The scheme is based on the interference of two photons on an unbalanced polarization-dependent beam splitter with different splitting ratios for vertical and horizontal polarization components and the additional application of feedforward by means of Pockels cells. The experimental feasibility of the proposed setup is carefully studied including various kinds of imperfections and losses. Moreover, we briefly describe two possible cryptographic applications of the optimal mirror phase-covariant cloning corresponding to state discrimination (or estimation) and secure quantum teleportation.

Regular design equations for the discrete reduced-order Kalman filter

Regular design equations for the discrete reduced-order Kalman filter In the presence of white Gaussian noises at the input and the output of a system Kalman filters provide a minimum-variance state estimate. When part of the measurements can be regarded as noise-free, the order of the filter is reduced. The filter design can be carried out both in the time domain and in the frequency domain. In the case of full-order filters all measurements are corrupted by noise and therefore the design equations are regular. In the presence of noise-free measurements, however, they are not regular so that standard software cannot readily be applied in a time-domain design. In the frequency domain the spectral factorization of the non-regular polynomial matrix equation causes no problems. However, the known proof of optimality of the factorization result requires a regular measurement covariance matrix. This paper presents regular (reduced-order) design equations for the reduced-order discrete-time Kalman filter in the time and in the frequency domains so that standard software is applicable. They also allow to formulate the conditions for the stability of the filter and to prove the optimality of the existing solutions.

Re‐visiting structural optimization of the Ziegler pendulum: singularities and exact optimal solutions

AbstractStructural optimization of non‐conservative systems with respect to stability criteria is a research area with important applications in fluid‐structure interactions, friction‐induced instabilities, and civil engineering. In contrast to optimization of conservative systems where rigorously proven optimal solutions in buckling problems have been found, for non‐conservative optimization problems only numerically optimized designs were reported. The proof of optimality in the non‐conservative optimization problems is a mathematical challenge related to multiple eigenvalues, singularities on the stability domain, and non‐convexity of the merit functional. We present a study of the optimal mass distribution in a classical Ziegler's pendulum where local and global extrema can be found explicitly. In particular, for the undamped case, the two maxima of the critical flutter load correspond to a vanishing mass either in a joint or at the free end of the pendulum; in the minimum, the ratio of the masses is equal to the ratio of the stiffness coefficients. The role of the singularities on the stability boundary in the optimization is highlighted and extension to the damped case as well as to the case of higher degrees of freedom is discussed. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Quasi-distinct parsing and optimal compression methods

In this paper, the optimality proof of Ziv–Lempel coding is re-studied, and a more general compression optimality theorem is derived. In particular, the property of quasi-distinct parsing is defined. This property allows infinitely many repetitions of phrases in the parsing as long as the total number of repetitions is o(n/logn), where n is length of the parsed string. The quasi-distinct parsing property is weaker than distinct parsing used in the original proof which does not allow repetitions of phrases in the parsing. Yet we show that the theorem holds with this weaker property as well. This provides a better understanding of the optimality proof of Ziv–Lempel coding, together with a new tool for proving optimality of other compression schemes which is applicable for a much wider family of codes. To demonstrate the possible use of this generalization, a new coding method–the Arithmetic Progression Tree coding (APT)–is presented. This new coding method is based on a principle that is very different from Ziv–Lempel’s coding. Nevertheless, the APT coding is analyzed in this paper and using the generalized theorem shown to be asymptotically optimal up to a constant factor,11Of course, a constant factor for a compression method may indicate that the method is practically useless. In the APT proof presented in this paper the constant is less than 10, however, this constant may be only a byproduct of an inefficiency of the proof. The true constant may be much less. if the APT quasi-distinctness hypothesis holds. An empirical evidence that this hypothesis holds is also given.

Burden of Proof

The burden of proof is a central feature of all systems of adjudication, yet one that has been subject to little normative analysis. This Article examines how strong evidence should have to be in order to assign liability when the objective is to maximize social welfare. In basic settings, there is a tradeoff between deterrence benefits and chilling costs, and the optimal proof requirement is determined by factors that are almost entirely distinct from those underlying the preponderance of the evidence rule and other traditional standards. As a consequence, these familiar burden of proof rules have some surprising properties, as do alternative criteria that have been advanced. The Article also considers how setting the proof burden interacts with other features of legal system design: the determination of enforcement effort, the level of sanctions, and the degree of accuracy of adjudication. It compares and contrasts a variety of legal environments and methods of enforcement, explaining how the appropriate proof requirements differ qualitatively across contexts. Most of the questions raised and answers presented differ in kind — as well as in result — from those in prior literature.

Numerical Simulation of Rock Fragmentation Process Induced by Two Disc Cutters

Rock fragmentation processes induced by two cutters were simulated by ABAQUS software with different cutter spacing. A series of numerical experiments reproduce the progressive process of rock fragmentation in indentation. The influence of different cutter spacing on the penetration process can also be clearly researched. Rock fragmentation process induced by two TBM disc cutters was performed using the finite element method. The simulation not only provided a realistic description of the rock fragmentation mechanism, but also supported the sufficient proof for the cutter spacing optimization between adjacent disc cutters for a given penetration depth. When the penetration is 6 mm, the cutter spacing optimization is 48 mm. The result shows that the numerical simulations will contribute to an improved knowledge of rock fragmentation in indentation and will be useful for performance assessment of TBM, which will enhance the efficiency of rock breaking.

An optimal algorithm for solving partial target coverage problem in wireless sensor networks

This paper deals with the partial target coverage problem in wireless sensor networks under a novel coverage model. The most commonly used method in previous literature on the target coverage problem is to divide continuous time into discrete slots of different lengths, each of which is dominated by a subset of sensors while setting all the other sensors into the sleep state to save energy. This method, however, suffers from shortcomings such as high computational complexity and no performance bound. We showed that the partial target coverage problem can be optimally solved in polynomial time. First, we built a linear programming formulation, which considers the total time that a sensor spends on covering targets, in order to obtain a lifetime upper bound. Based on the information derived in previous formulation, we developed a sensor assignment algorithm to seek an optimal schedule meeting the lifetime upper bound. A formal proof of optimality was provided. We compared the proposed algorithm with the well-known column generation algorithm and showed that the proposed algorithm significantly improves performance in terms of computational time. Experiments were conducted to study the impact of different network parameters on the network lifetime, and their results led us to several interesting insights. Copyright © 2011 John Wiley & Sons, Ltd.

On Optimal Heuristic Randomized Semidecision Procedures, with Applications to Proof Complexity and Cryptography

The existence of an optimal propositional proof system is a major open question in proof complexity; many people conjecture that such systems do not exist. Krajíček and Pudlák (J. Symbol. Logic 54(3):1063, 1989) show that this question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe (Theor. Comput. Sci. 412(4–5):478, 2011) recently presented a conjecture implying that such an algorithm does not exist.We show that if one allows errors, then such optimal algorithms do exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow an algorithm, called a heuristic acceptor, to claim a small number of false “theorems” and err with bounded probability on other inputs. The amount of false “theorems” is measured according to a polynomial-time samplable distribution on non-tautologies. Our result remains valid for all recursively enumerable languages and can also be viewed as the existence of an optimal weakly automatizable heuristic proof system. The notion of a heuristic acceptor extends the notion of a classical acceptor; in particular, an optimal heuristic acceptor for any distribution simulates every classical acceptor for the same language.We also note that the existence of a co-NP-language L with a polynomial-time samplable distribution on \(\overline{L}\) that has no polynomial-time heuristic acceptors is equivalent to the existence of an infinitely-often one-way function.

Optimal signal states for quantum detectors

Quantum detectors provide information about the microscopic properties of quantum systems by establishing correlations between those properties and a set of macroscopically distinct events that we observe. The question of how much information a quantum detector can extract from a system is therefore of fundamental significance. In this paper, we address this question within a precise framework: given a measurement apparatus implementing a specific POVM measurement, what is the optimal performance achievable with it for a specific information readout task and what is the optimal way to encode information in the quantum system in order to achieve this performance? We consider some of the most common information transmission tasks—the Bayes cost problem, unambiguous message discrimination and the maximal mutual information. We provide general solutions to the Bayesian and unambiguous discrimination problems. We also show that the maximal mutual information is equal to the classical capacity of the quantum-to-classical channel describing the measurement, and study its properties in certain special cases. For a group covariant measurement, we show that the problem is equivalent to the problem of accessible information of a group covariant ensemble of states. We give analytical proofs of optimality in some relevant cases. The framework presented here provides a natural way to characterize generalized quantum measurements in terms of their information readout capabilities.

Do there exist complete sets for promise classes?

In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language L? Q2: Do there exist complete problems for a given promise class ? For concrete languages L (such as TAUT or SAT) and concrete promise classes (such as , , , disjoint -pairs etc.), these questions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new characterizations for Q1 and Q2 that apply to almost all promise classes and languages L, thus creating a unifying framework for the study of these practically relevant questions. While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount of advice is available in the underlying machine model. For promise classes with promise condition in , the advice can be replaced by a tally -oracle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

An Optimal Algorithm for Relay Node Assignment in Cooperative Ad Hoc Networks

Recently, cooperative communications, in the form of having each node equipped with a single antenna and exploit spatial diversity via some relay node's antenna, is shown to be a promising approach to increase data rates in wireless networks. Under this communication paradigm, the choice of a relay node (among a set of available relay nodes) is critical in the overall network performance. In this paper, we study the relay node assignment problem in a cooperative ad hoc network environment, where multiple source-destination pairs compete for the same pool of relay nodes in the network. Our objective is to assign the available relay nodes to different source-destination pairs so as to maximize the minimum data rate among all pairs. The main contribution of this paper is the development of an optimal polynomial time algorithm, called ORA, that achieves this objective. A novel idea in this algorithm is a “linear marking” mechanism, which maintains linear complexity of each iteration. We give a formal proof of optimality for ORA and use numerical results to demonstrate its capability.

Optimal Proof Burdens, Deterrence, and the Chilling of Desirable Behavior

The optimal stringency of the burden of proof is characterized in a model in which relaxing the proof burden enhances deterrence but also chills desirable behavior. The result are strikingly different from those in prior work that uses a simpler model in which individuals only choose whether to commit a harmful act (so only deterrence is at stake). Moreover, the qualitative differences between the optimal rule and the familiar preponderance of the evidence rule—and related rules that look to Bayesian posteriors—are great, much more so than revealed by prior work.

Combinatorial theorems about embedding trees on the real line

We consider the combinatorial problem of embedding the metric defined by an unweighted graph into the real line, so as to minimize the distortion of the embedding. This problem is inspired by connections to Banach space theory and to computer science. After establishing a framework in which to study line embeddings, we focus on metrics defined by three specific families of trees: complete binary trees, fans, and combs. We construct asymptotically optimal (i.e., distortion-minimizing) line embeddings for these metrics and prove their optimality via suitable lower bound arguments. It appears that even such specialized metrics require nontrivial constructions and proofs of optimality require sophisticated combinatorial arguments. The combinatorial techniques from our work might prove useful in further algorithmic research on low distortion metric embeddings. © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 153–168, 2011 © 2011 Wiley Periodicals, Inc.

Optimality proofs of quantum weight decision algorithms

Recently two quantum algorithms have been proposed for the symmetric and asymmetric weight decision problems. Although they show the quadratic speedup over the classical approaches, their optimality has not been proven formally. In this article, we prove their optimality in a formal fashion.

Allocating reforestation areas for sediment flow minimization: an integer programming formulation and a heuristic solution method

Policy and decision makers dealing with environmental conservation and land use planning often require identifying potential sites for contributing to minimize sediment flow reaching riverbeds. This is the case of reforestation initiatives, which can have sediment flow minimization among their objectives. This paper proposes an Integer Programming (IP) formulation and a Heuristic solution method for selecting a predefined number of locations to be reforested in order to minimize sediment load at a given outlet in a watershed. Although the core structure of both methods can be applied for different sorts of flow, the formulations are targeted to minimization of sediment delivery. The proposed approaches make use of a Single Flow Direction (SFD) raster map covering the watershed in order to construct a tree structure so that the outlet cell corresponds to the root node in the tree. The results obtained with both approaches are in agreement with expert assessments of erosion levels, slopes and distances to the riverbeds, which in turn allows concluding that this approach is suitable for minimizing sediment flow. Since the results obtained with the IP formulation are the same as the ones obtained with the Heuristic approach, an optimality proof is included in the present work. Taking into consideration that the heuristic requires much less computation time, this solution method is more suitable to be applied in large sized problems.