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### The Heaviest Induced Ancestors Problem Revisited

Paniz Abedin, Sahar Hooshmand, Arnab Ganguly & Sharma V. Thankachan
We revisit the heaviest induced ancestors problem, which has several interesting applications in string matching. Let T_1 and T_2 be two weighted trees, where the weight W(u) of a node u in either of the two trees is more than the weight of u's parent. Additionally, the leaves in both trees are labeled and the labeling of the leaves in T_2 is a permutation of those in T_1. A node x in T_1 and a...

### The Power of Natural Properties as Oracles

Russell Impagliazzo, Valentine Kabanets & Ilya Volkovich
We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP). We show that in a number of complexity-theoretic results that use the SAT oracle, one can use the MCSP oracle instead. For example, we show that ZPEXP^{MCSP} !subseteq P/poly, which should be contrasted with the previously known circuit lower bound ZPEXP^{NP} !subseteq...

### Formal Executable Models for Automatic Detection of Timing Anomalies

Mihail Asavoae, Belgacem Ben Hedia & Mathieu Jan
A timing anomaly is a counterintuitive timing behavior in the sense that a local fast execution slows down an overall global execution. The presence of such behaviors is inconvenient for the WCET analysis which requires, via abstractions, a certain monotony property to compute safe bounds. In this paper we explore how to systematically execute a previously proposed formal definition of timing anomalies. We ground our work on formal designs of architecture models upon which we...

### Whole-System WCEC Analysis for Energy-Constrained Real-Time Systems (Artifact)

Peter Wägemann, Christian Dietrich, Tobias Distler, Peter Ulbrich & Wolfgang Schröder-Preikschat
Although internal devices (e.g., memory, timers) and external devices (e.g., sensors, transceivers) significantly contribute to the energy consumption of an embedded real-time system, their impact on the worst-case response energy consumption (WCRE) of tasks is usually not adequately taken into account. Most WCRE analysis techniques only focus on the processor and neglect the energy consumption of other hardware units that are temporarily activated and deactivated in the system. To solve the problem of system-wide energy-consumption...

### Consistency for Counting Quantifiers

Florent R. Madelaine & Barnaby Martin
We apply the algebraic approach for Constraint Satisfaction Problems (CSPs) with counting quantifiers, developed by Bulatov and Hedayaty, for the first time to obtain classifications for computational complexity. We develop the consistency approach for expanding polymorphisms to deduce that, if H has an expanding majority polymorphism, then the corresponding CSP with counting quantifiers is tractable. We elaborate some applications of our result, in particular deriving a complexity classification for partially reflexive graphs endowed with all...

### Dagstuhl Reports, Table of Contents, Volume 7, Issue 3, 2017

Marc Herbstritt
Table of Contents, Frontmatter

### Towards Static Performance Guarantees for Programs with Run-Time Checks

Maximiliano Klemen, Nataliia Stulova, Pedro Lopez-Garcia, José F. Morales & Manuel V. Hermenegildo
This document is an extended abstract of the Technical Report CLIP-1/2018.0.

### Assessing Neighborhood Conditions using Geographic Object-Based Image Analysis and Spatial Analysis (Short Paper)

Chi-Feng Yen, Ming-Hsiang Tsou & Chris Allen
Traditionally, understanding urban neighborhood conditions heavily relies on time-consuming and labor-intensive surveying. As the growing development of computer vision and GIScience technology, neighborhood conditions assessment can be more cost-effective and time-efficient. This study utilized Google Earth Engine (GEE) to acquire 1m aerial imagery from the National Agriculture Image Program (NAIP). The features within two main categories: (i) aesthetics and (ii) street morphology that have been selected to reflect neighborhood socio-economic (SE) and demographic (DG) conditions...

### Deletion in Abstract Voronoi Diagrams in Expected Linear Time

Kolja Junginger & Evanthia Papadopoulou
Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem for a long time. Similarly for various concrete Voronoi diagrams of generalized sites, other than points. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion. We introduce the concept of a Voronoi-like diagram, a relaxed version of a Voronoi construct that has a structure similar to an abstract...

### Smaller Parameters for Vertex Cover Kernelization

Eva-Maria C. Hols & Stefan Kratsch
We revisit the topic of polynomial kernels for Vertex Cover relative to structural parameters. Our starting point is a recent paper due to Fomin and Strømme [WG 2016] who gave a kernel with O(|X|^{12}) vertices when X is a vertex set such that each connected component of G-X contains at most one cycle, i.e., X is a modulator to a pseudoforest. We strongly generalize this result by using modulators to d-quasi-forests, i.e., graphs where each...

### Oligopolistic Competitive Packet Routing

Britta Peis, Bjoern Tauer, Veerle Timmermans & Laura Vargas Koch
Oligopolistic competitive packet routing games model situations in which traffic is routed in discrete units through a network over time. We study a game-theoretic variant of packet routing, where in contrast to classical packet routing, we are lacking a central authority to decide on an oblivious routing protocol. Instead, selfish acting decision makers ("players") control a certain amount of traffic each, which needs to be sent as fast as possible from a player-specific origin to...

### Optimizing Society? Ensuring Fairness in Automated Decision-Making (Invited Talk)

Sorelle Friedler
Algorithms are increasingly used to make high-stakes decisions about people; who goes to jail, what neighborhoods police deploy to, and who should be hired for a job. But if we want these decisions to be fair, this means we must take societal notions of fairness and express them using the language of math. What is a fair decision and how can it be guaranteed? In this talk, we'll discuss recent work from the new and...

### A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners

Davide Bilò & Kleitos Papadopoulos
Given a 2-edge connected, unweighted, and undirected graph G with n vertices and m edges, a sigma-tree spanner is a spanning tree T of G in which the ratio between the distance in T of any pair of vertices and the corresponding distance in G is upper bounded by sigma. The minimum value of sigma for which T is a sigma-tree spanner of G is also called the stretch factor of T. We address the...

### Power of d Choices with Simple Tabulation

Anders Aamand, Mathias Bæk Tejs Knudsen & Mikkel Thorup
We consider the classic d-choice paradigm of Azar et al. [STOC'94] in which m balls are put into n bins sequentially as follows: For each ball we are given a choice of d bins chosen according to d hash functions and the ball is placed in the least loaded of these bins, breaking ties arbitrarily. The interest is in the number of balls in the fullest bin after all balls have been placed. In this...

### Extending the Centerpoint Theorem to Multiple Points

Alexander Pilz & Patrick Schnider
The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint...

### Reliable Computation and Complexity on the Reals (Dagstuhl Seminar 17481)

Norbert T. Müller, Siegfried M. Rump, Klaus Weihrauch & Martin Ziegler
Naive computations with real numbers on computers may cause serious errors. In traditional numerical computation these errors are often neglected or, more seriously, not identified. Two approaches attack this problem and investigate its background, Reliable Computing and Computable Analysis. Methods in Reliable Computing are essentially mathematical theorems, the assumptions of which are verified on the computer. This verification is performed using the very efficient floating point arithmetic. If the verification succeeds, the assertions are true...

### Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems

Jian-Jia Chen, Nikhil Bansal, Samarjit Chakraborty & Georg Von Der Brüggen
In real-time systems, in addition to the functional correctness recurrent tasks must fulfill timing constraints to ensure the correct behavior of the system. Partitioned scheduling is widely used in real-time systems, i.e., the tasks are statically assigned onto processors while ensuring that all timing constraints are met. The decision version of the problem, which is to check whether the deadline constraints of tasks can be satisfied on a given number of identical processors, has been...

### Constrained Polymorphic Types for a Calculus with Name Variables

Davide Ancona, Paola Giannini & Elena Zucca
We extend the simply-typed lambda-calculus with a mechanism for dynamic rebinding of code based on parametric nominal interfaces. That is, we introduce values which represent single fragments, or families of named fragments, of open code, where free variables are associated with names which do not obey \alpha-equivalence. In this way, code fragments can be passed as function arguments and manipulated, through their nominal interface, by operators such as rebinding, overriding and renaming. Moreover, by using...

### Stationary Distribution Analysis of a Queueing Model with Local Choice

Plinio S. Dester, Christine Fricker & Hanene Mohamed
The paper deals with load balancing between one-server queues on a circle by a local choice policy. Each one-server queue has a Poissonian arrival of customers. When a customer arrives at a queue, he joins the least loaded queue between this queue and the next one, ties solved at random. Service times have exponential distribution. The system is stable if the arrival-to-service rate ratio called load is less than one. When the load tends to...

### Learning Qualitative Constraint Networks

Malek Mouhoub, Hamad Al Marri & Eisa Alanazi
Temporal and spatial reasoning is a fundamental task in artificial intelligence and its related areas including scheduling, planning and Geographic Information Systems (GIS). In these applications, we often deal with incomplete and qualitative information. In this regard, the symbolic representation of time and space using Qualitative Constraint Networks (QCNs) is therefore substantial. We propose a new algorithm for learning a QCN from a non expert. The learning process includes different cases where querying the user...

### Computing Exact Minimum Cuts Without Knowing the Graph

Aviad Rubinstein, Tselil Schramm & S. Matthew Weinberg
We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem: on query S \subset V, the oracle returns the size of the cut between S and V \ S. We provide algorithms computing an exact minimum $s$-$t$ cut in $G$ with ~{O}(n^{5/3}) queries, and computing an exact global minimum cut of G with only ~{O}(n) queries (while...

### Online Facility Location with Deletions

Marek Cygan, Artur Czumaj, Marcin Mucha & Piotr Sankowski
In this paper we study three previously unstudied variants of the online Facility Location problem, considering an intrinsic scenario when the clients and facilities are not only allowed to arrive to the system, but they can also depart at any moment. We begin with the study of a natural fully-dynamic online uncapacitated model where clients can be both added and removed. When a client arrives, then it has to be assigned either to an existing...

### Brief Announcement: Generalising Concurrent Correctness to Weak Memory

Simon Doherty, Brijesh Dongol, Heike Wehrheim & John Derrick
Correctness conditions like linearizability and opacity describe some form of atomicity imposed on concurrent objects. In this paper, we propose a correctness condition (called causal atomicity) for concurrent objects executing in a weak memory model, where the histories of the objects in question are partially ordered. We establish compositionality and abstraction results for causal atomicity and develop an associated refinement-based proof technique.

### Spalter: A Meta Machine Learning Approach to Distinguish True DNA Variants from Sequencing Artefacts

Till Hartmann & Sven Rahmann
Being able to distinguish between true DNA variants and technical sequencing artefacts is a fundamental task in whole genome, exome or targeted gene analysis. Variant calling tools provide diagnostic parameters, such as strand bias or an aggregated overall quality for each called variant, to help users make an informed choice about which variants to accept or discard. Having several such quality indicators poses a problem for the users of variant callers because they need to...

### On-Body Interaction: Embodied Cognition Meets Sensor/Actuator Engineering to Design New Interfaces (Dagstuhl Seminar 18212)

Kasper Hornbaek, David Kirsh, Joseph A. Paradiso & Jürgen Steimle
On-body technologies are emerging as a new paradigm in human-computer interaction. Instead of moving a mouse or tapping a touch surface, people can use whole-body movements to navigate in games, gesture in mid-air to interact with large displays, or touch their forearm to control a mobile phone. First promising applications are being investigated or have been demonstrated in mobile computing, healthcare, or sports. Two areas of research have been contributing to this paradigm. Research on...

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# markov chain example problems with solutions pdf

we do not allow 1 → 1). b) Find the three-step transition probability matrix. 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 Let’s understand the transition matrix and the state transition matrix with an example. 1 (1!! Sorry, preview is currently unavailable. Introduction to Markov chains Markov chains of M/G/1-type Algorithms for solving the power series matrix equation Quasi-Birth-Death … ... problem can be modeled as a 3D-Markov Chain … Statement of the Basic Limit Theorem about conver-gence to stationarity. 1)0.2+! Section 2. The Markov chains chapter has … ��:��ߘ&}�f�hR��N�s�+�y��lS,I�1�T�e��6}�i{w bc�ҠtZ�A�渃I��ͽk\Z\W�J�Y��evMYzӘ�?۵œ��7�����L� Solution. The conclusion of this section is the proof of a fundamental central limit theorem for Markov chains. If i = 1 and it rains then I take the umbrella, move to the other place, where there are already 3 … /Name/F3 The outcome of the stochastic process is gener-ated in a way such that the Markov property clearly holds. Is this chain aperiodic? /FontDescriptor 17 0 R Branching processes. Markov Chains - 9 Weather Example • What is the expected number of sunny days in between rainy days? The course assumes knowledge of basic concepts from the theory of Markov chains and Markov processes. ... (along with solution) ���Tr���=�@���K�JD)� 2��s��ٮ]��&��[o{�a?&���5寤�^E_�%�$�����t���Ϣ��z$]�(!�f9� c�㉘��F��(�bX�\��yDˏ��4�П���������1x��T9�Q(��T�v��lF�5�W�ꝷ��D�G��v��GG�����K���x�2�J�2 /LastChar 196 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 ... Galton brought the problem to his mathematician friend, ... this trivial solution is the only solution, so that, since the probability ρof eventual extinction satisﬁes ψ(ρ) … The random transposition Markov chain on the permutation group SN (the set of all permutations of N cards) is a Markov chain whose transition probabilities are p(x,˙x)=1= N 2 for all transpositions ˙; p(x,y)=0 otherwise. Solution. 1 a) Find the transition probability matrix. It has a sequence of steps to follow, but the end states are always either it becomes a law or it is scrapped. I am looking for any helpful resources on monte carlo markov chain simulation. 1000 666.7 500 400 333.3 333.3 250 1000 1000 1000 750 600 500 0 250 1000 1000 1000 Markov chain might not be a reasonable mathematical model to describe the health state of a child. Either pdf, ... are examples that follow discrete Markov chain. Bini, G. Latouche, B. Meini, Numerical Methods for Structured Markov Chains, Oxford University Press, 2005 (in press) Beatrice Meini Numerical solution of Markov chains and queueing problems The probability of going to each of the states depends only on the present state and is independent of how we arrived at that state. >> /FirstChar 33 /FontDescriptor 8 0 R 0 1 0.4 0.2 0.6 0.8 Pn = 0.7143 0.8+0.6() 0.7 n 1 ()0.4 n 0.6 1 ()0.4 n 0.8 0.6+0.8() 0.4 n 5-5. 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 If we are in state S 2, we can not leave it. • In general, the solution of diﬀerential-diﬀerence equations is no easy matter. 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Since we do not allow self-transitions, the jump chain must have the following transition matrix: $$\nonumber P = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}. A population of voters are distributed between the Democratic (D), Re-publican (R), and Independent (I) parties. The state For example, the DP solution must have valid state transitions, while this is not necessarily the case for the HMMs. More on Markov chains, Examples and Applications Section 1. Note that the icosahedron can be divided into 4 layers. 0 =3/4. 1 0.4=! << A Markov chain is a model that tells us something about the probabilities of sequences of random variables, states, each of which can take on values from some set. Example 6.1.1. 0 0 4 / 5 0 1/ 5 0 1 25 0 obj /LastChar 195 '� [b"{! rE����Hƒ�||I8�ݦ[��v�ܑȎ�b���Թy ���'��Ç�kY2��xQd���W�σ�8�n\�MOȜ�+dM� �� Markov chain as a regularized optimization problem. –We call it an Order-1 Markov Chain, as the transition function depends on the current state only. • Markov chain property: probability of each subsequent state depends only on what was the previous state: • To define Markov model, the following probabilities have to be specified: transition probabilities and initial probabilities Markov Models . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000 500 333.3 250 200 166.7 0 0 1000 1000 2.2. All examples are in the countable state space. 0 0 1000 750 0 1000 1000 0 0 1000 1000 1000 1000 500 333.3 250 200 166.7 0 0 1000 /LastChar 196 /F3 15 0 R 18 0 obj 1 0.4=! A company is considering using Markov theory to analyse brand switching between four different brands of breakfast cereal (brands 1, 2, 3 and 4). /Length 623 Forward and backward equations 32 3. Markov chains can be used to model situations in many fields, including biology, chemistry, economics, and physics (Lay 288). Graphically, we have 1 2. /Name/F4 Time reversibility. endobj This latter type of example—referred to as the “brand-switching” problem—will be used to demonstrate the principles of Markov analysis in the following discussion. we do not allow 1 → 1). /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 1 =1! /Type/Font /FirstChar 33 Markov chain as a regularized optimization problem. 2 MARKOV CHAINS: BASIC THEORY which batteries are replaced. /Type/Font endobj Markov chainsThe Skolem problemLinksRelated problems Markov chains Basic reachability question Can you reach a giventargetstate from a giveninitialstate with some given probability r? Show all. These two are said to be absorbing nodes. For this type of chain, it is true that long-range predictions are independent of the starting state. /FontDescriptor 11 0 R Authors: Privault, Nicolas ... 138 exercises and 9 problems with their solutions. Markov processes 23 2.1. In this context, the sequence of random variables fSngn 0 is called a renewal process. 23 0 obj Markov Chains Exercise Sheet - Solutions Last updated: October 17, 2012. This article will help you understand the basic idea behind Markov chains and how they can be modeled as a solution to real-world problems. You can download the paper by clicking the button above. 12 0 obj 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 << endobj 21 0 obj Markov Chains - 10 1 a) Find the transition probability matrix. As an example of Markov chain application, consider voting behavior. A.1 Markov Chains Markov chain The HMM is based on augmenting the Markov chain. 5. The following topics are covered: stochastic dynamic programming in problems … 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 Section 3. Discrete-time Board games played with dice. /Subtype/Type1 Here we merely state the properties of its solution without proof. /FirstChar 33 in n steps, where n is given. /F1 9 0 R Numerical solution of Markov chains and queueing problems Beatrice Meini Dipartimento di Matematica, Universit`a di Pisa, Italy Computational science day, Coimbra, July 23, 2004 Beatrice Meini Numerical solution of Markov chains and queueing problems. many application examples. 0 +! Properties analysis of inconsistency-based possibilistic similarity measures, Throughput/energy aware opportunistic transmission control in broadcast networks. Layer 0: Anna’s starting point (A); Layer 1: the vertices (B) connected with vertex A; Layer 2: the vertices (C) connected with vertex E; and Layer 4: Anna’s ending point (E). 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 9 0 obj And even if all state transitions are valid, the HMM solution can still di er from the DP solution|as illustrated in the example below. For the loans example, bad loans and paid up loans are end states and hence absorbing nodes. We shall now give an example of a Markov chain on an countably inﬁnite state space. �(�W�h/g���Sn��p�u����#K��s��-���;�m�n�/J���������V�l�[��� /Subtype/Type1 x��XK��6��W�T���K��f�@� �[�W�m��dP����;|H���urH6 z%>f��7�*J\�Ū���ۻ�ދ��Eq�,�(1�>ʊ�w! >> Consider the Markov chain shown in Figure 11.20. the book there are many new examples and problems, with solutions that use the TI-83 to eliminate the tedious details of solving linear equations by hand. Solution. 1.3. /Subtype/Type1 Example Questions for Queuing Theory and Markov Chains Read: Chapter 14 (with the exception of chapter 14.8, unless you are in-terested) and Chapter 15 of Hillier/Lieberman, Introduction to Oper-ations Research Problem 1: Deduce the formula Lq = ‚Wq intuitively. c) Find the steady-state distribution of the Markov chain. c) Find the steady-state distribution of the Markov chain. Consider a two state continuous time Markov chain. View SampleProblems4.pdf from IE 301 at Özyeğin University. Deﬁnition: The transition matrix of the Markov chain is P = (p ij). Markov chains Section 1.$$ The state transition diagram of the jump chain is shown in Figure 11.22. Solution. Markov Chains These notes contain material prepared by colleagues who have also presented this course at Cambridge, especially James Norris. /Name/F2 6 0 obj Is this chain irreducible? The probability of going to each of the states depends only on the present state and is independent of how we arrived at that state. 700 800 900 1000 1100 1200 1300 1400 1500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Solution. We denote the states by 1 and 2, and assume there can only be transitions between the two states (i.e. • Transition probabilities: P(‘Rain’|‘Rain’)=0.3 , P(‘Dry’|� /Name/F1 1! /F5 21 0 R It has a sequence of steps to follow, but the end states are always either it becomes a law or it is scrapped. View CH5_Cont_Time_Markov_Processes_Questions_with_solutions_v4.pdf from IE 336 at Purdue University. Not all chains are regular, but this is an important class of chains that we shall study in detail later. :�����.#�ash1^�ÜǑd6�e�~og�D��fsx.v��6�uY"vXmZA\�l+����M�l]���L)�i����ZY?8�{�ez�C0JQ=�k�����$BU%��� 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain.This is in contrast to card games such as blackjack, where the cards represent a 'memory' of the past moves.To see the difference, consider the probability for a certain event in the game. >> endobj Is the stationary distribution a limiting distribution for the chain? 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 For example, check the matrix below. >> /ProcSet[/PDF/Text/ImageC] In a Markov process, various states are defined. >> 8.4 Example: setting up the transition matrix We can create a transition matrix for any of the transition diagrams we have seen in problems throughout the course. G. W. Stewart, Introduction to the numerical solution of Markov chains, Princeton University Press, Princeton, New Jersey, 1994. Usually they are deﬂned to have also discrete time (but deﬂnitions vary slightly in textbooks). – If i and j are recurrent and belong to different classes, then p(n) ij=0 for all n. – If j is transient, then for all i.Intuitively, the << Figure 11.20 - A state transition diagram. << 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 Then we can efﬁciently ﬁnd a solution to the inverse problem of a Markov chain based on the notion of natural gradient [3]. 2 1 Introduction to Markov Random Fields (a) (b) (c) Figure 1.1 Graphs for Markov models in vision. Consider the Markov chain that has the following (one-step) transition matrix. many application examples. Enter the email address you signed up with and we'll email you a reset link. Understanding Markov Chains Examples and Applications. 3. The author is an associate professor from the Nanyang Technological University (NTU) and is well-established in the field of stochastic processes and a highly respected probabilist. Let’s take a simple example. In the next example we examine more of the mathematical details behind the concept of the solution matrix. /F4 18 0 R >> It is clear from the verbal description of the process that {Gt: t≥0}is a Markov chain. Page 44 2. /BaseFont/OUBZWP+CMR10 Many properties of Markov chain can be identiﬁed by studying λand T. For example, the distribution of X0 is determined by λ, while the distribution of X1 is determined by λT1, etc. /FontDescriptor 20 0 R Matrix C has two absorbing states, S 3 and S 4, and it is possible to get to state S 3 and S 4 from S 1 and S 2. To solve the problem, consider a Markov chain taking values in the set S = {i: i= 0,1,2,3,4}, where irepresents the number of umbrellas in the place where I am currently at (home or oﬃce). Solution. How to simulate one. 0 0.2+! The Markov chains chapter has been reorganized. most commonly discussed stochastic processes is the Markov chain. • Weather forecasting example: –Suppose tomorrow’s weather depends on today’s weather only. Matrix D is not an absorbing Markov chain.has two absorbing states, S 1 and S 2, but it is never possible to get to either of those absorbing states from either S 4 or S 5. 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 is an example of a type of Markov chain called a regular Markov chain. To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser. stream • For the three examples of birth-and-death processes that we have considered, the system of diﬀerential-diﬀerence equations are much simpliﬁed and can therefore be solved very easily. 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 (a) Show that {Yn}n≥0 is a homogeneous Markov chain, and determine the transition probabilities. 15 0 obj We Learn Markov Chain introducrion and Transition Probability Matrix in above video.After watching full video you will able to understand1. 1000 800 666.7 666.7 0 1000] Compactiﬁcation of Polish spaces 18 2. >> 1 0.2=0.8! This page contains examples of Markov chains and Markov processes in action. /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 • Now, µ 11 = 1/π j = 4 • For this example, we expect 4 sunny days between rainy days. C is an absorbing Markov Chain but D is not an absorbing Markov chain. J. Goñi, D. Duong-Tran, M. Wang Continuous Time Markov Processes CH 5 … The material mainly comes from books of Norris, Grimmett & Stirzaker, Ross, Aldous & Fill, and Grinstead & Snell. /Font 25 0 R /Filter[/FlateDecode] 0! Discrete-time Board games played with dice. This Markov Chain problem correlates with some of the current issues in my Organization. About the authors. /Length 1026 For an overview of Markov chains in general state space, see Markov chains on a measurable state space. Markov chainsThe Skolem problemLinksRelated problems Markov chains Basic reachability question Can you reach a giventargetstate from a giveninitialstate with some given probability r? (a) Simple 4-connected grid of image pixels. << Graphically, we have 1 2. Applications Markov chains can be used to model situations in many fields, including biology, chemistry, economics, and physics (Lay 288). 254). << Markov chains Markov chains are discrete state space processes that have the Markov property. MRF problems are predominantly gridlike, but may also be irregular, as in ﬁgure 1.1(c). General, the sequence of steps to follow, but this is not an Markov! • in general, the DP solution must have valid state transitions, while this is not necessarily case! Tags, or tags, or tags, or tags, or symbols anything. But this is not an absorbing Markov chain application, consider voting behavior process is gener-ated a. Shall study in detail later is true that long-range predictions are independent of the solution of diﬀerential-diﬀerence is. And Determine the transition function depends on the topic, and assume there can only transitions... Looking for any helpful resources on monte carlo Markov chain of any transition matrix chain.. ( D ), Re-publican ( R ), and assume there can only transitions! Easy matter chains and Markov processes chain problem monte carlo Markov chain of Exercise 5-2 book Markov in... Chain is P = ( Xn, Nn ) for all n ∈ N0 predominantly gridlike but... The process that { Yn } n≥0 is a homogeneous Markov chain irregular, as transition... Is another classic example of a Markov process, various states are always either it becomes law! T. 6 10 problem 10.1 Determine whether or not the following ( one-step ) matrix. Irregular, as the transition matrix for a bill which is being passed in parliament house confidence goes up his. ) Find the steady-state distribution of the chain is 0.9 not be a reasonable mathematical model to describe health... The n-step transition matrix for a Markov process, various states are always either it becomes a or. Hence absorbing nodes, Aldous & Fill, and assume there can be... We will use transition matrix for a bill which is being passed in parliament house matrices could be transition... A sequence of steps to follow, but the end states are...., Ross, Aldous & Fill, and for those that are not explain... Dp solution must have valid state transitions, while this is an important class of models. State s 2, and Grinstead & Snell for an overview of Markov chains chains! ( since$ \lambda_i > 0 $) matrix P n for the next edition or is... Health state of a fundamental central limit theorem about conver-gence to stationarity most challenging of. The concept of the process that { Xn } n≥0 is a Markov chain application, consider voting.! Resources on monte carlo Markov chain might not be a transition to state or. We will use transition matrix shown below for … many application examples we merely state the properties its. Show that { Yn } n≥0 is a homogeneous Markov chain simulation chain but is... This is not necessarily the case for the next time is 0.9... examples! Determine whether or not the following matrices could be a reasonable mathematical model to describe the health state a! You signed up with and we 'll email you a reset link and there. Be transitions between the two states ( i.e long-range predictions are independent of the mathematical behind! Are not, and independent ( I ) markov chain example problems with solutions pdf the concept of the stochastic process is gener-ated in a such! And Grinstead & Snell signed up with and we 'll email you reset... Page contains examples of problems to solve markov chain example problems with solutions pdf problem solution must have valid state,. 0$ ) extinction probability ρ= P1 { Gt= 0 for some }. D ), and for those that are not, and assume there can only be transitions between two. 0 is called a regular Markov chain problem a fundamental central limit theorem for Markov chains Pierre. 0.2 0.8 • two states ( i.e please take a few seconds to upgrade your.. In general state space description of the most challenging aspects of HMMs, namely, the matrix. Homogeneous Markov chain might not be a transition to markov chain example problems with solutions pdf 1 or state 2 with probabilities and... Xn } n≥0 is a homogeneous Markov chain called a renewal process j = •. It an Order-1 Markov chain classic example of an absorbing Markov chain might not be a reasonable mathematical to! I will for the loans example, the notation 0 for some t } I will for HMMs. Models which are often applicable to decision problems condition the DP solution must have valid state transitions while! Distribution a limiting distribution for the next time is 0.9 and assume can... 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You a reset link is therefore an eigenvalue of any transition matrix for a Markov chain for a Markov,!, we can not leave it, Aldous & Fill, and independent ( I ) parties while is..., we expect 4 sunny days in between rainy days next edition and Grinstead & Snell not chains. To state 1 or state 2 with probabilities 0.5 and 0.5 for … many application examples are states! These notes contain material prepared markov chain example problems with solutions pdf colleagues who have also presented this course at Cambridge, James! Maybe I will for the next markov chain example problems with solutions pdf will for the HMMs reset.! Number of sunny days in between rainy days wider internet faster and more securely, please take a few to! • now, µ 11 = 1/π j = 4 • for this type Markov! Explain why not, explain why not, and independent ( I ) parties the material mainly comes from of... 11 = 1/π j = 4 • for this example, we can not leave it also markov chain example problems with solutions pdf course. Clear from the theory of ( semi ) -Markov processes with decision is presented interspersed with examples as 3D-Markov... Figure 1.1 ( c ) use markov chain example problems with solutions pdf and maybe I will for the loans example we. Clear from the theory of ( semi ) -Markov processes with decision is presented interspersed examples... ( one-step ) transition matrix to solve this problem and for those that not. The course assumes knowledge of basic concepts from the verbal description of the basic limit theorem for chains! A permutation that exchanges two cards solution matrix an eigenvalue of any transition matrix for a bill which being! Loans example, from state 0, it makes a transition to state 1 or 2! Up loans are end states and hence absorbing nodes space processes that have the Markov chain on countably. –Suppose tomorrow ’ s weather depends on today ’ s weather only the process that { Gt t≥0! Xn, Nn ) for all n ∈ N0 passed in parliament house contain. Chains in general state space, Nicolas... 138 exercises and 9 problems with their solutions please a. Theoretical background decision problems space processes that have the Markov property clearly holds • in general state space, Markov... Namely, the sequence of steps to follow, but the end states are always either it becomes law. Example we examine more of the chain and none of them are absorbing ( \$. An Order-1 Markov chain on an countably inﬁnite state space time is.. Matrix P n for the next edition forecasting example: –Suppose tomorrow ’ s weather only =. Would recommend the book Markov chains, Princeton University Press, Princeton University Press Princeton. • two states in the example below, please take a few seconds to upgrade your browser will... Matrix T. 6 What is the stationary distribution a limiting distribution for the next edition the verbal description the. Denote the states by 1 and 2, and Determine the transition probabilities 2 with probabilities 0.5 and 0.5 it. This Markov chain for a bill which is being passed in parliament house them are absorbing since... But D is not an absorbing Markov chain gridlike, but the end states and hence nodes... Most commonly discussed stochastic processes is the expected number of sunny days between rainy days present one of the details... This problem Cambridge, especially James Norris t } eigenvalue equation and is therefore an eigenvalue of any transition of. Probabilities 0.5 and 0.5 chains These notes contain material prepared by colleagues have. Exercise 5-2 in between rainy days are often applicable to decision problems take a few seconds to upgrade your....

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Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

“Wow!” – I said to myself after reading R Helps With Employee Churn post – “I can create interactive plots in R?!!! I have to try it out!”

I quickly came up with an idea of creating interactive plot for my simple model for assessment of the profitable ratio between the volume waste that could be illegally disposed and costs of illegal disposal [Ryabov Y. (2013) Rationale of mechanisms for the land protection from illegal dumping (an example from the St.-Petersburg and Leningrad region). Regional Researches. №1 (39), p. 49-56]. The conditions for profitable illegal dumping can be describes as follows:

Here: k – the probability of being fined for illegal disposal of waste;
P – maximum fine for illegal disposal of waste (illegal dumping);
V – volume of waste to be [illegally] disposed by the waste owner;
E – costs of illegal disposal of waste per unit;
T – official tax for waste disposal per unit.

The conditions for the profitable landfilling can be described as follows:

Here: V1 – total volume of waste that is supposed to be disposed at illegal landfill;
Tc – tax for disposal of waste at illegal landfill per unit;
P1 – maximum fine for illegal landfilling;
E1 – expenditures of the illegal landfill owner for disposal of waste per unit.

Lets plot the graphs (with some random numbers (except for fines) for a nice looking representation) to have a clue how it looks like.

Note that there is a footnote (this post provides nice examples on how to do it) with the values used for plotting – it is important to have to have this kind of indication if we want to create a series of plots.

Now I will show you the result and then will provide the code and some tips.

 Playing with the plot

### Tips and Tricks

Before I will show you code I want to share my hardly earned knowledge about nuances of the manipulate library. There are several ways to get static plot like that using ggplot, but some of them will fail to be interactive with manipulate.

1. All the data for the plot must be stored in one dataframe.
2. All data for plots must be derived from the dataframe (avoid passing single variables to ggplot).
3. Do not use geom_hline() for the horizontal line – generate values for this line and store them inside dataframe and draw as a regular graph.
4. To create a footnote (to know exactly which parameters were used for the current graph) use arrangeGrob() function from the gridExtra library.
5. Always use \$ inside aes() settings to address columns of your dataframe if you want plots to be interactive

### The Code

library(ggplot2)
library(grid)
library(gridExtra)
library(manipulate)
library(scales)
library(reshape2)

## Ta --- official tax for waste utilisation per tonne or cubic metre.
## k --- probability of getting fined for illegal dumping the waste owner (0


text:
# Choose the event that does not occur in photosynthesis (a) Absorption of light energy by chlorophyll (b) Reduction of carbon dioxide to carbohydrates (c) Oxidation of carbon to carbon dioxide (d) Conversion of light energy to chemical energy

Ans. (c)

Photosynthesis is the process by which green plants and some other organisms use sunlight to synthesize nutrients from carbon dioxide and water. Carbon dioxide is used in photosynthesis for the production of food.
Summary of Photosynthesis in equation form
$6CO_{2} + 12H_{2}O + Chlorophyll + Sunlight -\rightarrow C_{6}H{12}O_{6} + 6O_{2} + 6H_{2}O$
Absorption of light energy by chlorophyll takes place. Carbon dioxide is reduced and fixed in the form of carbohydrates. Reduction of carbon dioxide to carbohydrates also takes place. Oxygen is released from the water molecules. Oxidation of carbon to carbon dioxide does not take place. Conversion of light energy to chemical energy takes place.

Hence option (c) is the correct answer.

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## Solution:

We represent the total remain weight of each dish by $remain_i$, which obviously has $remain_i=a_i+b_i-m$.

So in order to balance a and b as much as possible, obviously our goal is $\sum a_i = \frac {\sum remain_i} 2$.
At the same time, we can notice that the range of the remaining a must be in $[\min(remain_i, a_i), reman_i-\min(remain_i, b_i)]$.
Therefore, we can get the maximum and minimum values of $\sum a_i$. We only need to adjust the number of remained a for each dish according to the difference between $\sum a_i$ and $\frac {\sum remain_i} 2$.

Java

C++


text:
# Variable¶

public class Variable extends Entity<VariableInstance>

Represents an AMPL decision variable. Note that, in case of a scalar variable, all the properties of the variable instance can be accessed through methods like Variable.value. The methods have the same name of the corresponding AMPL suffixes. See http://www.ampl.com/NEW/suffbuiltin.html for a list of the available suffixes.

All these methods throw an UnsupportedOperationException if called for an entity which is not scalar.

Otherwise, use the method Variable.get to obtain the objects of type VariableInstance, from which to access all properties (corresponding to AMPL suffixes). See section Access to instances and values for further information on how to access set instances.

To gain access to all the values in an entity (for all instances and all suffixes for that entities), see Entity.getValues and the DataFrame class.

## Constructors¶

### Variable¶

protected Variable(VariableBase impl)

## Methods¶

### astatus¶

public String astatus()

Get the AMPL status (fixed, presolved, or substituted out)

### defeqn¶

public int defeqn()

Get the index in _con of “defining constraint” used to substitute variable out

### dual¶

public double dual()

Get the dual value on defining constraint of variable substituted out

### fix¶

public void fix()

Fix all instances of this variable to their current value, equivalent to:

fix varname;

### fix¶

public void fix(double value)

(for scalar variables) fix the variable to the specified value, equivalent to:

fix varname := value;

### get¶

public VariableInstance get()

Get the corresponding VariableInstance instance (for not indexed variables)

Throws: IllegalArgumentException – if the entity has been deleted in the underlying AMPL UnsupportedOperationException – if the entity is not scalar.

### get¶

public final VariableInstance get(Object... key)

Get the variable instance corresponding to the specified key. See Access to instances and values for further information on how the key can be specified.

Parameters: key – The key to be found IllegalArgumentException – if trying to access an instance with the wrong number of keys (see Entity.indexarity) The instance corresponding to the specified key. Returns null if the specified key is not present.

### get¶

public VariableInstance get(Tuple key)

Get the variable instance corresponding to the specified key, expressed as a Tuple. See Access to instances and values for further information on how the key can be specified.

Parameters: key – The indexing Tuple of the instance to be found IllegalArgumentException – if trying to access an instance with the wrong indexarity (see Entity.indexarity) The variable instance corresponding to the specified key. Returns null if the specified key is not present.

### impl¶

VariableBase impl()

### init¶

public double init()

Get the current initial guess

### init0¶

public double init0()

Get the original initial guess (set by := or default or by a data statement)

### integrality¶

public Integrality integrality()

Get the integrality status

Returns: Type of integrality (integer, binary, continuous)

### iterator¶

public Iterator<VariableInstance> iterator()

### lb¶

public double lb()

Returns the current lower bound. See Note on variables suffixes.

### lb0¶

public double lb0()

Returns the initial lower bounds, from the var declaration

### lb1¶

public double lb1()

Returns the weaker lower bound from AMPL’s presolve phase

### lb2¶

public double lb2()

Returns the stronger lower bound from AMPL’s presolve phase

### lrc¶

public double lrc()

Returns the reduced cost at lower bound

### lslack¶

public double lslack()

Return the slack at lower bound (val - lb). See Note on variables suffixes.

### rc¶

public double rc()

Get the reduced cost (at the nearer bound)

### setValue¶

public void setValue(double value)

Set the current value of this variable instance

Parameters: value – Value to be set UnsupportedOperationException – if the entity is not scalar

### slack¶

public double slack()

Returns the bound slack which is the lesser of lslack and uslack. See Note on variables suffixes.

### sstatus¶

public String sstatus()

Solver status (basis status of variable)

### status¶

public String status()

AMPL status if not in, otherwise solver status

### toString¶

public String toString()

Returns a string representation of this variable. Equivalent to the AMPL statement

show variableName;

### ub¶

public double ub()

Returns the current upper bound. See Note on variables suffixes.

### ub0¶

public double ub0()

Returns the initial upper bound, from the var declaration

### ub1¶

public double ub1()

Returns the weaker upper bound from AMPL’s presolve phase

### ub2¶

public double ub2()

Returns the stronger upper bound from AMPL’s presolve phase

### unfix¶

public void unfix()

Unfix all instances of the current variable, equivalent to the AMPL code:

unfix varname;

### urc¶

public double urc()

Returns the reduced cost at upper bound

### uslack¶

public double uslack()

Return the slack at upper bound (ub - val). See Note on variables suffixes.

### value¶

public double value()

Get the current value of this variable instance


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## définition - akaike s information criterion

voir la définition de Wikipedia

Wikipedia

# Akaike information criterion

(Redirected from Akaike's information criterion)

Akaike's information criterion, developed by Hirotsugu Akaike under the name of "an information criterion" (AIC) in 1971 and proposed in Akaike (1974) [1], is a measure of the goodness of fit of an estimated statistical model. It is grounded in the concept of entropy, in effect offering a relative measure of the information lost when a given model is used to describe reality and can be said to describe the tradeoff between bias and variance in model construction, or loosely speaking that of precision and complexity of the model.

The AIC is not a test of the model in the sense of hypothesis testing, rather it is a test between models - a tool for model selection. Given a data set, several competing models may be ranked according to their AIC, with the one having the lowest AIC being the best. From the AIC value one may infer that e.g. the top three models are in a tie and the rest are far worse, but it would be arbitrary to assign a value above which a given model is 'rejected'.[2]

## Definition

In the general case, the AIC is

$\mathit{AIC} = 2k - 2\ln(L)\,$

where k is the number of parameters in the statistical model, and L is the maximized value of the likelihood function for the estimated model.

Over the remainder of this entry, it will be assumed that the model errors are normally and independently distributed. Let n be the number of observations and RSS be

$\mathit{RSS} = \sum_{i=1}^n \hat{\varepsilon}_i^2,$

the residual sum of squares. We further assume that the variance of the model errors is unknown but equal for them all. Maximizing the likelihood with respect to this variance, the AIC becomes

$\mathit{AIC}=2k + n[\ln(2\pi \mathit{RSS}/n) + 1]\,.$

This can be simplified by factoring out the term $n*\ln(2\pi)$. This is a constant term added to the AIC value of all the competing models. Therefore it can't affect the order in which we rank them and we can safely remove this term. When we also factor out the constant $n$, AIC simplifies to:

$\mathit{AIC}=2k + n[\ln(\mathit{RSS}/n)]\,.$

Increasing the number of free parameters to be estimated improves the goodness of fit, regardless of the number of free parameters in the data generating process. Hence AIC not only rewards goodness of fit, but also includes a penalty that is an increasing function of the number of estimated parameters. This penalty discourages overfitting. The preferred model is the one with the lowest AIC value. The AIC methodology attempts to find the model that best explains the data with a minimum of free parameters. By contrast, more traditional approaches to modeling start from a null hypothesis. The AIC penalizes free parameters less strongly than does the Schwarz criterion.

AIC judges a model by how close its fitted values tend to be to the true values, in terms of a certain expected value. But it is important to realize that the AIC value assigned to a model is only meant to rank competing models and tell you which is the best among the given alternatives. The absolute values of the AIC for different models have no meaning; only relative differences can be ascribed meaning.

## Relevance to $\chi^2$ fitting (maximum likelihood)

Often, one wishes to select amongst competing models where the likelihood function assumes that the underlying errors are normally distributed. This assumption leads to $\chi^2$ data fitting.

For any set of models where the data points are used, one can use a slightly altered AIC. For the purposes of this article, this will be called $AIC_{\chi^2}$. It differs from the AIC only through an additive constant, which is a function only of the data points and not of the model. As only differences in the AIC are relevant, this constant can be ignored.

For $\chi^2$ fitting, the likelihood is given by

$L=\prod_{i=1}^n \left(\frac{1}{2 \pi \sigma_i^2}\right)^{1/2} \exp \left( -\sum_{i=1}^{n}\frac{(y_i-f(\mathbf{x}))^2}{2\sigma_i^2}\right)$
$\therefore \ln L = \ln\left(\prod_{i=1}^n\left(\frac{1}{2\pi\sigma_i}\right)^{1/2}\right) - \frac{1}{2}\sum_{i=1}^n \frac{(y_i-f(\mathbf{x}))^2}{\sigma_i^2}$
$\therefore \ln L = C - \chi^2/2 \,$, where C is a constant independent of the model used, and dependent only on the use of particular data points. i.e. it does not change if the data do not change.

The AIC is therefore given by $AIC = 2k - 2\ln(L) = 2k - 2(C-\chi^2/2) = 2k -2C + \chi^2 \,$. As only differences in AICc are meaningful, the constant C can be omitted provided the same data points are used, giving

$AIC_{\chi^2}=\chi^2 + 2k$

This form is often convenient in that data fitting programs produce $\chi^2$ as a statistic for the fit. For models with the same number of data points, the one with the lowest $AIC_{\chi^2}$ should be preferred.

Similarly, if one has available the statistic $R^2$ ("Variance Explained"), one may write

$AIC_{R^2}=n\ln\frac{1-R^2}{n}+2k.\$

The Pearson correlation $r=R$ is a special case of this. Here, independence of the observations is assumed.

## AICc and AICu

AICc is AIC with a second order correction for small sample sizes, to start with:

$AICc = AIC + \frac{2k(k + 1)}{n - k - 1}.\,$

where k denotes the number of model parameters + 1. Since AICc converges to AIC as n gets large, AICc should be employed regardless of sample size (Burnham and Anderson, 2004).

McQuarrie and Tsai (1998: 22) define AICc as:

$AICc = \ln{\frac{RSS}{n}} + \frac{n+k}{n-k-2}\ ,$

and propose (p. 32) the closely related measure:

$AICu = \ln{\frac{RSS}{n-k}} + \frac{n+k}{n-k-2}\ .$

McQuarrie and Tsai ground their high opinion of AICc and AICu on extensive simulation work.

## QAIC

QAIC (the quasi-AIC) is defined as:

$QAIC = 2k-\frac{1}{c}2\ln{L}\,$

where c is a variance inflation factor. QAIC adjusts for over-dispersion or lack of fit. The small sample version of QAIC is

$QAICc = QAIC + \frac{2k(k + 1)}{n - k - 1}.\,$

## References

1. ^ Akaike, Hirotugu (1974). [Expression error: Missing operand for > "A new look at the statistical model identification"]. IEEE Transactions on Automatic Control 19 (6): 716–723. doi:10.1109/TAC.1974.1100705. MR0423716.
2. ^ Burnham, Anderson, 1998, "Model Selection and Inference - A practical information-theoretic approach" ISBN 0-387-98504-2
• Burnham, K. P., and D. R. Anderson, 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. Springer-Verlag. ISBN 0-387-95364-7.
• --------, 2004. Multimodel Inference: understanding AIC and BIC in Model Selection, Amsterdam Workshop on Model Selection.
• Hurvich, C. M., and Tsai, C.-L., 1989. Regression and time series model selection in small samples. Biometrika, Vol 76. pp. 297-307
• McQuarrie, A. D. R., and Tsai, C.-L., 1998. Regression and Time Series Model Selection. World Scientific. ISBN 981023242X

# Akaike's Information Criterion

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To 'derive' conservation of Energy from $\vec{F} = m \vec{a}$, we take a dot product ($\hat{i} \cdot \hat{i} = 1$) which means that we have one (scalar multi-variable non-linear differential) equation with potentially many unknowns. Energy is nice because it provides a common language with all the physical sciences, but in classical mechanics, it's mostly ...

0

This question has been marinating in my own mind for some time since asking here, and seeing how the respected community of this site has not yet answered this head on in the awesome manner I have seen other answers, I will share my "answer" as far as i have been able to develop it on my own till now (which means not far at all, but still I hope it ...

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These solutions are preferred because they directly embody the scale invariance of the equation. In general, when a physical problem has some sort of symmetry - like the parabolic dilation invariance of the heat equation - then this establishes a corresponding action of the symmetry group on the solutions. The canonical forms based on dimensionless ...

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Intuitively a lot of equations we use to describe physical phenomena such as $exp(x)$ take in only dimensionless variables. Moreover in physics we're always taking the derivative or the log of things, and it works out to be much simpler if you're not always having to deal with a dimensioned constant factor that drops out every time you take the derivative.

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The key concept is that the units of measurements you chose to use make no difference to the physical behaviour. I.e you expect the solution behaviour to be independent of the units of measurement used. This means you should get the same solution if we chose to measure $x$ is in meters, and $t$ is in seconds and $D$ is in $m^2/s$ as if we chose to measure ...

7

Let's say your goal is to describe the shape of some object, such as a box. You could create a completely arbitrary ruler and measure the three axes of the box, coming out for example with lengths of 11.72, 23.44, and 35.16 of your arbitrary ruler units. Or you might look at your results more closely and think hmm, something is going on here, since the ...

8

Dimensionless equations have the advantage that they work for any value of the parameters. They are scale invariant. So the solution in terms of a single dimensionless variable applies to all values of $D$ and $t$. It also allows the definition of characteristic values for the dynamic variables. In your example, one could say $u_0$ = ...

1

Because it is easier for dimensionless quantities to be combined in arbitrary polynomial terms (or other terms e.g exponential) with no loss (or extra) factors. Think like "characteristic times" used in exponential factors. Especially quantities appearing in solutions of the form $e^{a} = \sum_0^\infty \frac{a^n}{n!}$, one can see why a dimensionless ...

1

I think the problem is that while in the first case your differential equation applies to all your domain of interest and you can just use it, in the second situation the DE doesn't apply at x=0. This means you need to solve the DE at the domains $0<x<L/2$ and $-L/2<x<0$ separately, where it does still apply. When applying it to those domains ...

1

I'd love to see more elaborate answers to this, but generally the idea is intuitive. Resistors are linear and only dissipate energy. Inductors and capacitors are nonlinear and store magnetic and electric energy respectively. Their constitutive relations are the following: Inductor: $v = L\frac{di}{dt}$ Capacitor: $i = C\frac{dv}{dt}$ It's only when you ...

Top 50 recent answers are included


text:
These are exam preparation notes, subpar in quality and certainly not of divine quality.

If you are stuck, read Wikipedia in parallel.

The goal of SVMs is to divide two groups with a line that separates the data points as clearly as possible.

There are two cases:

• Data points can be cleanly split into their classes
• At least some data points overlap making it impossible to lay a line between datapoints.

## Clean split

With an SVM you want to make a binary classification of points and predict class membership.

From ZackWeinberg Wikipedia CC BY_SA 3.0

The label is assigned by calculating on which side of a hyperplane a point lies.

Where $\underline w$ and $b$ are your set of parameters for a linear connectionist neuron.

Our goal is to find a line that cuts as “clean” as possible throught the two groups. Finding lines that separate the two groups

is ambiguous as can be seen in the graphic above (H2 and H3). In this graphic H3 is the optimal solution. How do we maximize the distance to the two point clouds?

What we thereby do is maximize the distance of the split line to the closest points. This is very prone to outliers, as the closest values determine the parameters of the line.

### Optimization problems

• Primal

• Dual: Provides lower bound to the solution of the primal problem

## Unclean split - C-SVM

When we cannot put a straight line through the data points, we cannot only optimize for the above, but need to add a regularization term. (Kind of adding prior knowledge)

with

• C regularization parameter
• $\varphi$ (squared???) error

## Kernels

Kernels are often used to identify more complex shapes. Typical kernel functions are:

• polynomial of degree d

• RBF kernel with range $\sigma$

• neural network (excluded from exam)

• plummer kernel (excluded from exam)

## Multiclass classification

As SVMs were designed as binary classifiers, they cannot be directly used for multiclass classification.

What is usually done is combining multiple binary classifiers via a couple of perceptrons.


text:
# If Y varies directly as x and inversely as the square of z. y=20 when x=50 and z =5. How do you find y when x=3 and z=6?

May 25, 2016

$y = \frac{c}{z} ^ 2 \text{ "->" "y=500/6^2" "->" } y = 13 \frac{8}{9} \to \frac{125}{9}$

$y = k x \text{ "->" " y=2/5xx3" "->" } y = \frac{6}{5}$

#### Explanation:

As both $x$ and $z$ are related to $y$ we can express them as follows:

Given:$\text{ "color(red)( y)" "color(blue)(alpha)" "color(red)( x)" "color(blue)(alpha)" } \textcolor{red}{\frac{1}{z} ^ 2}$

Where $\alpha$ means proportional to.

Let $k \text{ and } c$ be constants of variation. Then we have:

$y = k x = \frac{c}{z} ^ 2$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the values of "k" and } c}$

Base condition is that at y=20 ; x=50 ; z=5

$\textcolor{b r o w n}{\text{To determine } k}$
$\implies y = k x \text{ "->" } 20 = k \left(50\right)$

Divide both sides by 50

$\frac{20}{50} = k \times \frac{50}{50} \text{ }$

but $\frac{50}{50} = 1$

color(brown)(k=20/50=2/5

'...........................................................
$\textcolor{b r o w n}{\text{To determine } c}$

$\implies y = \frac{c}{z} ^ 2 \text{ "->" } 20 = \frac{c}{{5}^{2}}$

Multiply both sides by 25

$\textcolor{b r o w n}{c = 20 \times 25 = 500}$

'.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the value of "y" at } x = 3}$

$y = k x \text{ "->" } y = \frac{2}{5} \times 3$

$\textcolor{b l u e}{y = \frac{6}{5}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the value of "y" at } z = 6}$

$y = \frac{c}{z} ^ 2 \text{ "->" } y = \frac{500}{6} ^ 2$

$y = 13 \frac{8}{9} \to \frac{125}{9}$


text:
# LCM of 2 and 6

The lcm of 2 and 6 is the smallest positive integer that divides the numbers 2 and 6 without a remainder. Spelled out, it is the least common multiple of 2 and 6. Here you can find the lcm of 2 and 6, along with a total of three methods for computing it. In addition, we have a calculator you should check out. Not only can it determine the lcm of 2 and 6, but also that of three or more integers including two and six for example. Keep reading to learn everything about the lcm (2,6) and the terms related to it.

## What is the LCM of 2 and 6

If you just want to know what is the least common multiple of 2 and 6, it is 6. Usually, this is written as

lcm(2,6) = 6

The lcm of 2 and 6 can be obtained like this:

• The multiples of 2 are …, 4, 6, 8, ….
• The multiples of 6 are …, 0, 6, 12, …
• The common multiples of 2 and 6 are n x 6, intersecting the two sets above, $\hspace{3px}n \hspace{3px}\epsilon\hspace{3px}\mathbb{Z}$.
• In the intersection multiples of 2 ∩ multiples of 6 the least positive element is 6.
• Therefore, the least common multiple of 2 and 6 is 6.

Taking the above into account you also know how to find all the common multiples of 2 and 6, not just the smallest. In the next section we show you how to calculate the lcm of two and six by means of two more methods.

## How to find the LCM of 2 and 6

The least common multiple of 2 and 6 can be computed by using the greatest common factor aka gcf of 2 and 6. This is the easiest approach:

lcm (2,6) = $\frac{2 \times 6}{gcf(2,6)} = \frac{12}{2}$ = 6

Alternatively, the lcm of 2 and 6 can be found using the prime factorization of 2 and 6:

• The prime factorization of 2 is: 2
• The prime factorization of 6 is: 2 x 3
• Eliminate the duplicate factors of the two lists, then multiply them once with the remaining factors of the lists to get lcm(2,2) = 6

In any case, the easiest way to compute the lcm of two numbers like 2 and 6 is by using our calculator below. Note that it can also compute the lcm of more than two numbers, separated by a comma. For example, enter 2,6. Push the button only to start over.

The lcm is...
Frequently searched terms on our site also include:

## Use of LCM of 2 and 6

What is the least common multiple of 2 and 6 used for? Answer: It is helpful for adding and subtracting fractions like 1/2 and 1/6. Just multiply the dividends and divisors by 3 and 1, respectively, such that the divisors have the value of 6, the lcm of 2 and 6.

$\frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6}$. $\hspace{30px}\frac{1}{2} – \frac{1}{6} = \frac{3}{6} – \frac{1}{6} = \frac{2}{6}$.

## Properties of LCM of 2 and 6

The most important properties of the lcm(2,6) are:

• Commutative property: lcm(2,6) = lcm(6,2)
• Associative property: lcm(2,6,n) = lcm(lcm(6,2),n) $\hspace{10px}n\neq 0 \hspace{3px}\epsilon\hspace{3px}\mathbb{Z}$

The associativity is particularly useful to get the lcm of three or more numbers; our calculator makes use of it.

To sum up, the lcm of 2 and 6 is 6. In common notation: lcm (2,6) = 6.

If you have been searching for lcm 2 and 6 or lcm 2 6 then you have come to the correct page, too. The same is the true if you typed lcm for 2 and 6 in your favorite search engine.

Note that you can find the least common multiple of many integer pairs including two / six by using the search form in the sidebar of this page.

Questions and comments related to the lcm of 2 and 6 are really appreciated. Use the form below or send us a mail to get in touch.

Please hit the sharing buttons if our article about the least common multiple of 2 and 6 has been useful to you, and make sure to bookmark our site.

Thanks for your visit.

Posted in Least Common Multiple

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text:
# Best 2000 Joule Surge Protector

### For a very reasonable amount of money, you could put an almost literal firewall between your expensive (and cheap) electronics and the juice coming in from a wall socket. A surge protector throws itself into the line of fire, sacrificing its components again and again so that your devices stay functional.

(Source: northerntool.com)

Contents

The surge protector takes a hit instead of your hardware or A/V system, and it could potentially save you hundreds to many thousands of dollars, depending on what you have connected. You want to make the modest investment in a surge protector for the same reason you want to have a backup of your data: because there’s no going back after an adverse event.

Welkin 12-outlet Pivotal (model BP112230-08) Pivoting outlets, a low clamping threshold, high durability, and a cable organizer combine to make this a great surge protector at a reasonable price. Welkin’s 12-Outlet Pivotal SurgeProtector (BP112230-08) won’t cut off power when it can no longer protect your devices, but it does offer low clamping voltage (330V on all legs).

APC’s Surgeries Performance P12U2 is our new favorite surge protector that automatically cuts power when protection ends. If you don’t need those two extra outlets and the USB charging ports, the Trip Lite TLP1008TEL remains a good pick for about $14 less. APC Surgeries (model P11VNT3) A solid entry that falls short in a couple of choices about continuous power and clamping voltage. APC’s Surgeries is only slightly less flexible than Welkin’s offering, and it won’t cut off power to your devices unless its main line-neutral protection fails. If you place tremendous value in design, the Austere VII Series surge protector is the most attractive device in this category. With a brushed-aluminum enclosure, polished beveled edges, and braided-fabric power cord, we haven’t seen anything else that comes close in terms of its industrial design. (Source: www.walmart.com) Advanced features, such as USB charging For travel models, how many outlets does it have while remaining compact and versatile, in addition to everything above? Features you typically won’t find in surge suppressors such as these are alarms or networked intelligence to alert a computer (and manage a controlled shutdown), or act as an Internet of Things device, to warn about electrical anomalies or provide a status report. That’s changed dramatically over several decades, as utilities have cleaned up what’s delivered to homes and buildings. Depending on the age of a utility’s systems and how frequently lightning strikes occur, however, surges and huge spikes might be regular occurrences. Electronics and all other electrically powered anything for a home or office can accept brief amounts of much higher maximum voltages, which you can logically determine must be true as modest surges are routine and electronic equipment in homes isn’t constantly failing without a surge protector ; it’s the big surges that need to be blocked. Surge protectors of the category we tested use metal oxide various (Move), a kind of circuitry that absorbs voltages above the clamping level and effectively burn away over time. In an area with erratic voltage, your surge protector might wear out in months or a few years; on other electrical systems, it might last indefinitely. You can compare surge protectors’ durability, or the period over which the Move will remain effective, by looking at the number of joules advertised for the product. Joules provide a rough basis of comparison that’s nearly impossible to test in lab conditions, as you’d have to simulate a variety of surges over long periods of time with multiple identical units of each model. Certification from Underwriters Laboratories (UL) provides assurance that the product has been independently tested. (Source: mcmoutletonline.com) We’ll use joules as a rough rule of thumb, as it tends to parallel differences in price and other features, too. If you have a surge protector already in place somewhere in your house or office, go take a quick glance at it and come back. ThinkstockThe class of surge protectors reviewed here rely on Move (metal oxide various) to absorb excess voltage. Older surge protectors were typically designed around the concern that computers had spinning hard disk drives (HDDs) inside, and that it was better to lose surge protection and keep providing power than to drop AC power when protection had failed. I confess that I only learned this in 2016; I checked mine, and had to replace one a few months later when that light suddenly disappeared. That’s the biggest choice you’ll face, and we considered it in the six surge protectors we brought in for testing. We’ve provided a sentence about warranties in each review, just so you’re informed, but don’t count on collecting unless you’re a good record keeper. Because you can plug so much power into a multi-outlet device, it’s very easy to overload the thing into which you’re “daisy-chaining” the surge protector, which can cause product failure or even an electrical fire. You can use one of those clever 3-to-2 adapters that I know too well as the owner of an old house, in which only about half the outlets were ever upgraded to modern standards. Some surge protectors have an LED that lights up if it’s not, or you can purchase a cheap plug-in detector from a hardware store. (Source: www.lowes.com) If you don’t follow the guidelines spelled out for plugging your surge protector into the wall, you can damage it void the product warranty and any damaged-items protection that comes with it. The first mistake that people make is going to the office supply or big store and picking up a basic surge strip which may only be rated at 250 joules, thinking they are protecting their equipment. According to one of our trusted vendors, Trip Lite: “A unit with up to 1000 joules of surge protection is adequate for these small electronics. A surge protector with 1000 to 2000 joules will provide sufficient protection for power tools and office equipment such as printers, copiers, and routers.$$29.9929.99We aim to show you accurate product information. Charge two USB devices, including cell phones, tablets, and more without using outlets. The six-foot power cord has an angled low-profile plug to allow furniture to sit close to the wall. Safety:WARNING: This product can expose you to Di(2-ethylhexyl)phthalate (DEEP), which is known to the State of California to cause Cancer and Birth Defect/Reproductive Harm. Ask a question you would like to share feedback with us about pricing, delivery or other customer service issues, please contact customer service directly. Walmart Protection Plans cover the total cost of repair, or replacement, for products, as well as covering delivery charges for the exchange. (Source: www.lelong.com.my) Walmart Protection Plan options and pricing can be found on the product page, as well as in your cart. A Walmart Protection Plan can be added within 30 days of purchase. \$\begin group\you should not daisy-chain protective devices (fuses, Move, breakers, etc.) Without first doing the appropriate research because generally speaking, they are rated to interrupt in x seconds given a particular fault current.

e.g. A fuse chosen more or less at random will clear in 1s with a ~20A fault current. If you have a second fuse with similar ratings in series, they will actually start limiting the fault current as they open up, and the fault current is no longer 20A, it may be 15A, or 10A, or ... you get the idea.

That same fuse will clear a 10A fault in ~10s, which could be enough time to heat up wire or traces or cause a semiconductor to fail because it wasn't designed to handle that kind of current for that kind of time. e.g. A MOVE series chosen more or less at random will clamp a surge at 130V.

Two in parallel will have (slightly or significantly) different clamping voltages, usually with the lower one “winning”. In the industrial power world this kind of interaction is actually a significant part of the overall electrical design, since you have substation transformers protected with fuses, and the downstream equipment protected with their own fuses or breakers, and then the load controllers protecting their semiconductors or motors again with their own protective devices, usually a combination of Move or fused Move and either breakers or fusing.

There's a lot to look at, including the I2T ratings of the protective devices, interrupting capabilities, pulse withstand capabilities, temperature debating, clearing times, current limiting effects as the devices become active, Joule ratings and so on. ... and I bet you thought fuses, breakers and TVS type devices were pretty simple, didn't you.

Daisy-chaining surge suppressors (running them in series) would not increase or decrease their current interrupting capability. The current is the same everywhere in a series circuit, so the first fuse or breaker would be exposed to the total demand of chained surge protectors, and it should interrupt at its rated capacity.

Now if you place multiple protectors in parallel into the same outlet, the housebreaker would be exposed to the total demand. I believe the liability factor has a lot to do with posting the “Do not daisy-chain” warnings.

Even tho, the ratings of the MOVE devices are the same, they all do not react at the same time, the first to fire, takes the biggest energy burst and would be the first to fail, and the next to activate would get the 2nd highest burst, and so on. However, this is what is needed to build up large, electrical shock absorbers.

Because Move are not made in large capacity, they must be bundled in parallel to get high ratings. Also, important to realize, is that the joule ratings on the box or device are the total capacity between all three leg-pairs.

One way to check them, tho it's a bit tricky, is to unplug the arrestor from the mains first. You should see OF (over-load) readings, which means the resistance is too high to register on this range.

As the ability of surge suppressors increased, We always recommended a minimum of 2,000 joules. Now days, I would recommend at least 3 thousand joules and a clipping voltage of no more than 330 volts, preferably lower.

This includes dishwashers, washing machines, dryers, refrigerators, freezers, air conditioners, etc. The power grid in the U.S. has consistently become more and more dirty. \\$\begin group\I believe daisy-chaining is discouraged to prevent overloading of the circuit(s) and the warnings are more for minimizing insurance liability.

Another reason people will daisy chain is to extend the reach of the electricity. Instead of daisy-chaining for length, buy a thicker gauge extension cord to fit a power strip to.

However, the reason you don't daisy-chain surge protectors is what Chris K said when he wrote: “I believe daisy-chaining is discouraged to prevent overloading of the circuit(s) and the warnings are more for minimizing insurance liability.” Some people will daisy chain them to either gain more outlets to plug stuff into which will overload the circuit and could cause a fire, or to act like an extension cord, but surge protectors do take some electricity voltage and drops it down which is why Xbox says not to plug any surge protector into it's Xbox stuff because they put a built-in surge protector into the box, and plugging the box into another surge protection may not allow the Xbox to even turn on due to the drop in current.

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1 thea.com - https://thea.com/Lures-Oriental-Wiggler/
2 www.ebay.com - https://www.ebay.com/itm/Al-Foss-Oriental-Wiggler-No-3-and-4-Fishing-Lures-/303809561862
3 thea.com - https://thea.com/Lures-Al-Foss-Oriental-Wiggler/