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In highly repetitive strings, like collections of genomes from the same species, distinct measures of repetition all grow sublinearly in the length of the text, and indexes targeted to such strings typically depend only on one of these measures. We describe two data structures whose size depends on multiple measures of repetition at once, and that provide competitive tradeoffs between the time for counting and reporting all the exact occurrences of a pattern, and the space taken by the structure. The key component of our constructions is the run-length encoded BWT (RLBWT), which takes space proportional to the number of BWT runs: rather than augmenting RLBWT with suffix array samples, we combine it with data structures from LZ77 indexes, which take space proportional to the number of LZ77 factors, and with the compact directed acyclic word graph (CDAWG), which takes space proportional to the number of extensions of maximal repeats. The combination of CDAWG and RLBWT enables also a new representation of the suffix tree, whose size depends again on the number of extensions of maximal repeats, and that is powerful enough to support matching statistics and constant-space traversal.

Concurrent data structures are the data sharing side of parallel programming. Data structures give the means to the program to store data, but also provide operations to the program to access and manipulate these data. These operations are implemented through algorithms that have to be efficient. In the sequential setting, data structures are crucially important for the performance of the respective computation. In the parallel programming setting, their importance becomes more crucial because of the increased use of data and resource sharing for utilizing parallelism. The first and main goal of this chapter is to provide a sufficient background and intuition to help the interested reader to navigate in the complex research area of lock-free data structures. The second goal is to offer the programmer familiarity to the subject that will allow her to use truly concurrent methods.

v, 350 p. : 28 cm

Motivation: Second generation sequencing technology makes it feasible for many researches to obtain enough sequence reads to attempt the de novo assembly of higher eukaryotes (including mammals). De novo assembly not only provides a tool for understanding wide scale biological variation, but within human bio-medicine, it offers a direct way of observing both large scale structural variation and fine scale sequence variation. Unfortunately, improvements in the computational feasibility for de novo assembly have not matched the improvements in the gathering of sequence data. This is for two reasons: the inherent computational complexity of the problem, and the in-practice memory requirements of tools. Results: In this paper we use entropy compressed or succinct data structures to create a practical representation of the de Bruijn assembly graph, which requires at least a factor of 10 less storage than the kinds of structures used by deployed methods. In particular we show that when stored succinctly, the de Bruijn assembly graph for homo sapiens requires only 23 gigabytes of storage. Moreover, because our representation is entropy compressed, in the presence of sequencing errors it has better scaling behaviour asymptotically than conventional approaches.

Tries are among the most versatile and widely used data structures on words. They are pertinent to the (internal) structure of (stored) words and several splitting procedures used in diverse contexts ranging from document taxonomy to IP addresses lookup, from data compression (i.e., Lempel-Ziv'77 scheme) to dynamic hashing, from partial-match queries to speech recognition, from leader election algorithms to distributed hashing tables and graph compression. While the performance of tries under a realistic probabilistic model is of significant importance, its analysis, even for simplest memoryless sources, has proved difficult. Rigorous findings about inherently complex parameters were rarely analyzed (with a few notable exceptions) under more realistic models of string generations. In this paper we meet these challenges: By a novel use of the contraction method combined with analytic techniques we prove a central limit theorem for the external path length of a trie under a general Markov source. In particular, our results apply to the Lempel-Ziv'77 code. We envision that the methods described here will have further applications to other trie parameters and data structures.

The importance of Safety monitoring of old structures e.g., Bridges, Dams etc need not be overemphasized. Monitoring of several significant parameters always essential for prediction of structural behavior of the unit. Need for proper calibration and maintenance of embedded instruments are discussed. Recording of data collected in a regular interval on site from few significant instruments embedded in the structure followed by Statistical analysis of correlated data is an important tool for health monitoring of structure. The issue is discussed with a mention of dam instrumentation and data analysis. It helps in commenting about trend of permanent settlement component of dam deflection of a few geographically supported dam, hence justifies instrumentation in giant structures from safety point of view in turn. B Datta"Role of Instrumentation Data Analysis in Safety Monitoring of Giant Structures" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd5921.pdf Article URL: http://www.ijtsrd.com/engineering/civil-engineering/5921/role-of-instrumentation-data-analysis-in-safety-monitoring-of-giant-structures/b-datta

The importance of Safety monitoring of old structures e.g., Bridges, Dams etc need not be overemphasized. Monitoring of several significant parameters always essential for prediction of structural behavior of the unit. Need for proper calibration and maintenance of embedded instruments are discussed. Recording of data collected in a regular interval on site from few significant instruments embedded in the structure followed by Statistical analysis of correlated data is an important tool for health monitoring of structure. The issue is discussed with a mention of dam instrumentation and data analysis. It helps in commenting about trend of permanent settlement component of dam deflection of a few geographically supported dam, hence justifies instrumentation in giant structures from safety point of view in turn. B Datta"Role of Instrumentation Data Analysis in Safety Monitoring of Giant Structures" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd5921.pdf Article URL: http://www.ijtsrd.com/engineering/civil-engineering/5921/role-of-instrumentation-data-analysis-in-safety-monitoring-of-giant-structures/b-datta

The importance of Safety monitoring of old structures e.g., Bridges, Dams etc need not be overemphasized. Monitoring of several significant parameters always essential for prediction of structural behavior of the unit. Need for proper calibration and maintenance of embedded instruments are discussed. Recording of data collected in a regular interval on site from few significant instruments embedded in the structure followed by Statistical analysis of correlated data is an important tool for health monitoring of structure. The issue is discussed with a mention of dam instrumentation and data analysis. It helps in commenting about trend of permanent settlement component of dam deflection of a few geographically supported dam, hence justifies instrumentation in giant structures from safety point of view in turn. B Datta"Role of Instrumentation Data Analysis in Safety Monitoring of Giant Structures" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd5921.pdf Article URL: http://www.ijtsrd.com/engineering/civil-engineering/5921/role-of-instrumentation-data-analysis-in-safety-monitoring-of-giant-structures/b-datta

The importance of Safety monitoring of old structures e.g., Bridges, Dams etc need not be overemphasized. Monitoring of several significant parameters always essential for prediction of structural behavior of the unit. Need for proper calibration and maintenance of embedded instruments are discussed. Recording of data collected in a regular interval on site from few significant instruments embedded in the structure followed by Statistical analysis of correlated data is an important tool for health monitoring of structure. The issue is discussed with a mention of dam instrumentation and data analysis. It helps in commenting about trend of permanent settlement component of dam deflection of a few geographically supported dam, hence justifies instrumentation in giant structures from safety point of view in turn. B Datta"Role of Instrumentation Data Analysis in Safety Monitoring of Giant Structures" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd5921.pdf  http://www.ijtsrd.com/engineering/civil-engineering/5921/role-of-instrumentation-data-analysis-in-safety-monitoring-of-giant-structures/b-datta

The importance of Safety monitoring of old structures e.g., Bridges, Dams etc need not be overemphasized. Monitoring of several significant parameters always essential for prediction of structural behavior of the unit. Need for proper calibration and maintenance of embedded instruments are discussed. Recording of data collected in a regular interval on site from few significant instruments embedded in the structure followed by Statistical analysis of correlated data is an important tool for health monitoring of structure. The issue is discussed with a mention of dam instrumentation and data analysis. It helps in commenting about trend of permanent settlement component of dam deflection of a few geographically supported dam, hence justifies instrumentation in giant structures from safety point of view in turn. B Datta"Role of Instrumentation Data Analysis in Safety Monitoring of Giant Structures" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd5921.pdf  http://www.ijtsrd.com/engineering/civil-engineering/5921/role-of-instrumentation-data-analysis-in-safety-monitoring-of-giant-structures/b-datta

A Time History Data Base (THDB) is to be developed for the Photovoltaic Array and Space Power Experiment - Plus Diagnostics (PASP+) sensor package flown aboard the APEX satellite. A THDB system has been developed for the generation and display of the THDB data.

https://arxiv.org/abs/1802.05471

Longitudinal data are important in exposure and risk assessments, especially for pollutants with long half-lives in the human body and where chronic exposures to current levels in the environment raise concerns for human health effects. It is usually difficult and expensive to obtain large longitudinal data sets for human exposure studies. This article reports a new simulation method to generate longitudinal data with flexible numbers of subjects and days. Mixed models are used to describe the variance-covariance structures of input longitudinal data. Based on estimated model parameters, simulation data are generated with similar statistical characteristics compared to the input data. Three criteria are used to determine similarity: the overall mean and standard deviation, the variance components percentages, and the average autocorrelation coefficients. Upon the discussion of mixed models, a simulation procedure is produced and numerical results are shown through one human exposure study. Simulations of three sets of exposure data successfully meet above criteria. In particular, simulations can always retain correct weights of inter- and intrasubject variances as in the input data. Autocorrelations are also well followed. Compared with other simulation algorithms, this new method stores more information about the input overall distribution so as to satisfy the above multiple criteria for statistical targets. In addition, it generates values from numerous data sources and simulates continuous observed variables better than current data methods. This new method also provides flexible options in both modeling and simulation procedures according to various user requirements.

88 p. 28 cm

In this paper we will investigate transformations that serve as tools in the design of new data structures. Specifically, we study general methods for converting static structures (in which all elements are known before any searches are performed) to dynamic structures (in which insertions of new elements can be mixed with searches). We will exhibit three classes of such transformations, each based on a different counting scheme for representing the integers, and then use a combinatorial model to show the optimality of many of the transformations. Issues such as online data structures and deletion of elements are also examined. To demonstrate the applicability of these tools, we will study six new data structures that have been developed by applying the transformations. (Author)

Data from five series of movable bed laboratory experiments are presented in this report. These experiments were conducted in the Large-scale Sediment Transport Facility at the U.S. Army Engineer Research and Development Center, Vicksburg, MS. The data collected from these experiments is being used to improve longshore sand transport relationships under the combined influence of waves and currents and the enhancement of predictive numerical models of beach morphology evolution, in particular, with respect to modeling of tombolo development at detached breakwaters and T-groins. These data were instrumental in the development and validation of GENESIS-T, an enhanced version of GENESIS that allows for continued simulation of shoreline evolution after tombolo formation at detached breakwaters.

Data from five series of movable bed laboratory experiments are presented in this report. These experiments were conducted in the Large-scale Sediment Transport Facility at the U.S. Army Engineer Research and Development Center, Vicksburg, MS. The data collected from these experiments is being used to improve longshore sand transport relationships under the combined influence of waves and currents and the enhancement of predictive numerical models of beach morphology evolution, in particular, with respect to modeling of tombolo development at detached breakwaters and T-groins. These data were instrumental in the development and validation of GENESIS-T, an enhanced version of GENESIS that allows for continued simulation of shoreline evolution after tombolo formation at detached breakwaters.

Includes index

This article is from Nucleic Acids Research , volume 42 . Abstract Chromosomes are not positioned randomly within a nucleus, but instead, they adopt preferred spatial conformations to facilitate necessary long-range gene–gene interactions and regulations. Thus, obtaining the 3D shape of chromosomes of a genome is critical for understanding how the genome folds, functions and how its genes interact and are regulated. Here, we describe a method to reconstruct preferred 3D structures of individual chromosomes of the human genome from chromosomal contact data generated by the Hi-C chromosome conformation capturing technique. A novel parameterized objective function was designed for modeling chromosome structures, which was optimized by a gradient descent method to generate chromosomal structural models that could satisfy as many intra-chromosomal contacts as possible. We applied the objective function and the corresponding optimization method to two Hi-C chromosomal data sets of both a healthy and a cancerous human B-cell to construct 3D models of individual chromosomes at resolutions of 1 MB and 200 KB, respectively. The parameters used with the method were calibrated according to an independent fluorescence in situ hybridization experimental data. The structural models generated by our method could satisfy a high percentage of contacts (pairs of loci in interaction) and non-contacts (pairs of loci not in interaction) and were compatible with the known two-compartment organization of human chromatin structures. Furthermore, structural models generated at different resolutions and from randomly permuted data sets were consistent.

ix, 407 pages ; 26 cm

ix, 407 pages ; 26 cm

ix, 407 pages ; 26 cm

ix, 407 pages ; 26 cm

We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph $G = (V,E)$, with $|V| = n$ and $|E| =m$, in $o(\sqrt{m}\,)$ time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a $(2+\eps)$ approximation in $O(\log n/\eps^2)$ amortized time per update. For maximum matching, we show how to maintain a $(3+\eps)$ approximation in $O(\min(\sqrt{n}/\epsilon, m^{1/3}/\eps^2))$ {\em amortized} time per update, and a $(4+\eps)$ approximation in $O(m^{1/3}/\eps^2)$ {\em worst-case} time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld from STOC' 2010.

This paper shows the weighted matching problem on general graphs can be solved in time $O(n(m + n\log n))$ for $n$ and $m$ the number of vertices and edges, respectively. This was previously known only for bipartite graphs. The crux is a data structure for blossom creation. It uses a dynamic nearest-common-ancestor algorithm to simplify blossom steps, so they involve only back edges rather than arbitrary nontree edges. The rest of the paper presents direct extensions of Edmonds' blossom algorithm to weighted $b$-matching and $f$-factors. Again the time bound is the one previously known for bipartite graphs: for $b$-matching the time is $O(\min\{b(V),n\log n\}(m + n\log n))$ and for $f$-factors the time is $O(\min\{f(V),m\log n\}( m + n\log n) )$, where $b(V)$ and $f(V)$ denote the sum of all degree constraints. Several immediate applications of the $f$-factor algorithm are given: The generalized shortest path structure of \cite{GS13}, i.e., the analog of the shortest path tree for conservative undirected graphs, is shown to be a version of the blossom structure for $f$-factors. This structure is found in time $O(|N|(m+n\log n))$ for $N$ the set of negative edges ($0

Talk by Joshua Vogelstein of Johns Hopkins University.  Given to the Redwood Center for Theoretical Neuroscience at UC Berkeley. Abstract Determining whether certain properties are related to other properties is fundamental to scientific discovery. As data collection rates accelerate, it is becoming increasingly difficult yet ever more important to determine whether one property of data (e.g., cloud density) is related to another (e.g., grass wetness). Only if two properties are related are further investigations into the geometry of the relationship warranted. While existing approaches can test whether two properties are related, they may require unfeasibly large sample sizes in real data scenarios, and do not provide insight into the geometry underlying the structure of the relationship. We juxtapose hypothesis testing, manifold learning, and harmonic analysis to obtain Multiscale Generalized Correlation (MGC). Our key insight is that one can adaptively restrict the analysis to the "jointly local" observations - that is, one can estimate the scale with the most informative neighbors for determining the existence and geometry of a relationship. We prove that to achieve a given true positive rate, MGC typically requires far fewer samples than existing methods for all investigated dependence structures and dimensionalities, while maintaining computational efficiency. Moreover, MGC uniquely provides a simple and elegant characterization of the potentially complex latent geometry underlying the relationship. We used MGC to detect the presence and reveal the geometry of the relationships between mental and brain properties, to perform a proteomics screening, and to develop an imaging biomarker for disease, while avoiding the false positive inflation problems that have plagued conventional parametric approaches. Our open source implementation of MGC is easy to use, parameter-free, and applicable to previously vexing statistical questions that are ubiquitous in science, government, finance, and other disciplines.

PURL domain: /linked-data-structures/

The subroutine package TREEPAK is designed to interface a PL/I program with a graphic display unit, allowing display, creation and manipulation of tree structures, while providing the user with a graphic representation of the tree. The package is implemented under IBM System/360 using the IBM 2250-1 graphic display unit as the primary input/output device. Facilities include creation of trees from the alphanumeric keyboard and cards, adding and deleting subtrees in existing trees, and saving and retrieving trees. Trees may be passed between TREEPAK and the user written main program. An implementation of the Wang Algorithm is given as an example application program.

TRI 2 2015/2016

Tri 2 2017/2018

Tri 2 – 2015/2016

Tri 2 – 2016/2017

Tri 2 – 2016/2017

Tri 2 – 2016/2017

A book on JavaScript data structures and algorithms that provides both an introduction and a more in-depth look into the topic.

A new technique, based on virtual backpointers, for local concurrent error detection and correction in linked data structures is presented. Two new data structures, the Virtual Double Linked List, and the B-tree with Virtual Backpointers, are described. For these structures, double errors can be detected in 0(1) time and errors detected during forward moves can be corrected in 0(1) time. The application of a concurrent auditor process to data structure error detection and correction is analyzed, and an implementation is described, to determine the effect on mean time to failure of a multi-user shared database system. The implementation utilizes a Sequent shared memory multiprocessor system operating on a shared databased of Virtual Double Linked Lists.

All of the publicly available online resources for the book Data Structures and Algorithms in Python by John Canning. (All sourced from https://datastructures.live )

All of the publicly available online resources for the book Data Structures and Algorithms in Python by John Canning. (All sourced from https://datastructures.live )

The volume is the second of two and presents additional statistical analyses of data discussed in the first, which presented 1960 and 1961 census data from 53 countries in an attempt to identify and quantify factors which determined the occupational and educational structure of the labor force. The second volume consists of eight chapters: (1) a discussion on classification and aggregation problems; (2) supplementary analyses of the data in multiple regression models; (3) analyses of occupational and educational coefficients; (4) analyses of production function equations; (5) a short exposition of information theory as it relates to the aggregation problems in cross-classifications; (6) basic data used for the occupational analysis; (7) basic data used for the occupational/educational analysis; and (8) basic data used for the sectoral/educational analyses. (Author/MDW)

A mode of a multiset $S$ is an element $a \in S$ of maximum multiplicity; that is, $a$ occurs at least as frequently as any other element in $S$. Given a list $A[1:n]$ of $n$ items, we consider the problem of constructing a data structure that efficiently answers range mode queries on $A$. Each query consists of an input pair of indices $(i, j)$ for which a mode of $A[i:j]$ must be returned. We present an $O(n^{2-2\epsilon})$-space static data structure that supports range mode queries in $O(n^\epsilon)$ time in the worst case, for any fixed $\epsilon \in [0,1/2]$. When $\epsilon = 1/2$, this corresponds to the first linear-space data structure to guarantee $O(\sqrt{n})$ query time. We then describe three additional linear-space data structures that provide $O(k)$, $O(m)$, and $O(|j-i|)$ query time, respectively, where $k$ denotes the number of distinct elements in $A$ and $m$ denotes the frequency of the mode of $A$. Finally, we examine generalizing our data structures to higher dimensions.

A mode of a multiset $S$ is an element $a \in S$ of maximum multiplicity; that is, $a$ occurs at least as frequently as any other element in $S$. Given a list $A[1:n]$ of $n$ items, we consider the problem of constructing a data structure that efficiently answers range mode queries on $A$. Each query consists of an input pair of indices $(i, j)$ for which a mode of $A[i:j]$ must be returned. We present an $O(n^{2-2\epsilon})$-space static data structure that supports range mode queries in $O(n^\epsilon)$ time in the worst case, for any fixed $\epsilon \in [0,1/2]$. When $\epsilon = 1/2$, this corresponds to the first linear-space data structure to guarantee $O(\sqrt{n})$ query time. We then describe three additional linear-space data structures that provide $O(k)$, $O(m)$, and $O(|j-i|)$ query time, respectively, where $k$ denotes the number of distinct elements in $A$ and $m$ denotes the frequency of the mode of $A$. Finally, we examine generalizing our data structures to higher dimensions.

A mode of a multiset $S$ is an element $a \in S$ of maximum multiplicity; that is, $a$ occurs at least as frequently as any other element in $S$. Given a list $A[1:n]$ of $n$ items, we consider the problem of constructing a data structure that efficiently answers range mode queries on $A$. Each query consists of an input pair of indices $(i, j)$ for which a mode of $A[i:j]$ must be returned. We present an $O(n^{2-2\epsilon})$-space static data structure that supports range mode queries in $O(n^\epsilon)$ time in the worst case, for any fixed $\epsilon \in [0,1/2]$. When $\epsilon = 1/2$, this corresponds to the first linear-space data structure to guarantee $O(\sqrt{n})$ query time. We then describe three additional linear-space data structures that provide $O(k)$, $O(m)$, and $O(|j-i|)$ query time, respectively, where $k$ denotes the number of distinct elements in $A$ and $m$ denotes the frequency of the mode of $A$. Finally, we examine generalizing our data structures to higher dimensions.

The current study has several aims. First, we will evaluate to what extent a fine-tuned large language model (LLM) can produce synthetic inter-item correlations (based solely on the items’ text) that approximate the items’ observed correlations calculated from self-reported ratings of the same items in a large sample of individuals. Numerous items had been developed as part of the development of the Hierarchical Taxonomy of Psychopathology (HiTOP) self-report measure (HiTOP-SR) but had never been made public, meaning that the LLM could not have trained on the items–these would be testing data. Second, we will conduct separate factor analyses of the synthetic and observed correlations to test the similarity of the two approaches’ implied latent factor structures of psychopathology. We will thus explore the potential of using synthetic correlations generated by LLMs, which are based solely on trained lexical associations and thus require no human survey data collection, for measure development in both bivariate and multivariate contexts.

The current study has several aims. First, we will evaluate to what extent a fine-tuned large language model (LLM) can produce synthetic inter-item correlations (based solely on the items’ text) that approximate the items’ observed correlations calculated from self-reported ratings of the same items in a large sample of individuals. Numerous items had been developed as part of the development of the Hierarchical Taxonomy of Psychopathology (HiTOP) self-report measure (HiTOP-SR) but had never been made public, meaning that the LLM could not have trained on the items–these would be testing data. Second, we will conduct separate factor analyses of the synthetic and observed correlations to test the similarity of the two approaches’ implied latent factor structures of psychopathology. We will thus explore the potential of using synthetic correlations generated by LLMs, which are based solely on trained lexical associations and thus require no human survey data collection, for measure development in both bivariate and multivariate contexts.

Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang Baxter equation. For any non trivial Belavin-Drinfeld data of minimal size for $SL_{n}$, we give an algorithm for constructing an initial seed $\Sigma$ in $\mathcal{O}(SL_{n})$. The cluster structure $\mathcal{C}=\mathcal{C}(\Sigma)$ is then proved to be compatible with the Poisson bracket associated with that Belavin-Drinfeld data, and the seed $\Sigma$ is locally regular. This is the first of two papers, and the second one proves the rest of the conjecture: the upper cluster algebra $\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{C})$ is naturally isomorphic to $\mathcal{O}(SL_{n})$, and the correspondence of Belavin-Drinfeld classes and cluster structures is one to one.

Full Course Of Alogrithms

The Complete Python Bootcamp From Zero to Hero in Python (20 - Advanced Python Objects and Data Structures)

The analysis and design of robust controllers for multivariable discrete-time feedback systems in the frequency domain are studied. The robustness stability conditions for discrete-time systems with additive alteration, and with multiplicative alteration are developed. The robust LQG/LTR method has been extended from the continuous-time system to the discrete-time system. It has been proven that the LQG/LTR method is also valid for the LQG control of the discrete-time system with the filtering observer. As an application of the LQG/LTR technique for discrete-time systems, the design of reduced order optimal digital LQG controllers for the orbiting flexible shallow spherical shell system is considered. The comparison between the digital optimal LQG controller with the filtering observer and with the predicting observer for the orbiting flexible shallow spherical shell has been made.... Digital control, Large space structures, LQG/LTR method for discrete-time system, Robust controller design, Design of reduced order LQG controller.