text:
#### Operations Management

Effectiveness

do what is right

Efficiency

do it right

Growth

Innovation

cost

reduction

4 ways to treat a problem

absolution
resolve
solve
dissolve

What is a system?

Two or more parts with the following properties:
each part can affect the whole
all parts are interconnected and interdependent
any groups of parts has the two properties listed above

A system is a whole that cannot be divided into independent parts

t

A system is not a sum of its parts
it is a product of interactions between parts

t

Typical Operations Decisions

What: What resources are needed, and in what amounts?
When: When will each resource be needed? When should the work be scheduled? When should materials and other supplies be ordered?
Where: Where will the work be done?
How: How will he product or service

decision criteria

Output Volume Total Cost (Lifetime Cost) Quality (Reliability) Timely (Availability, Delivery) Flexibility/Customize (Design, Volume, Service) Innovation Value = Quality/Price Sustainability

__________ refers to sourcing from overseas or getting work done in a foreign country.

off shoring

__________ refers to contracting with a third party or an external company to manufacture a good or deliver a service.

Outsourcing

__________ is a sequence of interconnected organizations that help develop, produce, distribute, and sell a product to the final consumer.

Supply Chain

Which of the following refers to the process of managing the system of designing, producing, and delivering goods or services that add value throughout the supply chain and benefit the final consumer?

Operations Management

A phenomenal growth of which sector has expanded the opportunities for operations and supply chain professionals?

Service

The three general types of decisions in operations and supply chain management are __________.

strategic decisions
tactical decisions
operational decisions

The __________ supervises and manages the work environment, vendor selection process, supply chains, real estate, and budgets.

supply chain manager

Which of the following best describes the responsibilities of a logistics manager?

ensures accurate and efficient transportation and storage of goods and materials both for the outbound and inbound areas of the supply chain

Which of the following is true for services?

often labor intensive

__________ is a giant in international trade and global supply chain management.

china

__________ is the world's leading service provider.

India

Which of the following refers to the process of incorporating the interests of the public into a company's core business?

Corporate social responsibility

Which of the following best describes the purpose of an enterprise resource planning system?

to mange to flow of information across all functions not only with an enterprise but also with its external stakeholders including its suppliers and customers

__________ is an extension of lean manufacturing.

Agile manufacturing

__________ is a management philosophy that focuses on continuously improving the quality of a company's products and processes.

Total quality management

Agile Manufacturing

The ability of an organization to respond quickly to market changes with a set of processes, tools, and training available as needed

A management approach that focuses on continuously improving the efficiency of business processes through innovation, flexibility, and the seamless integration of technology

The radical redesign of a firm's existing workflows and resources to reduce operational costs and better meet the needs of customers and support a firm's overall mission

Corporate Social Responsibility (CSR)

The process of incorporating the interests of the public into a company's core business

Customer Management

The process of interfacing with individuals or companies that buy and use finished products

Enterprise Resource Planning (ERP)

An information system that integrates information across all departments of an organization, as well as facilitates and manages the flow of information across all functions within an enterprise and to its external stakeholders

Inbound Portion

Elements of a supply chain that consist of local or foreign suppliers of product ideas, designs, basic raw materials and components, transportation links, and warehouses that end with the internal operations of the company

Lean Manufacturing

A management philosophy that originated in the 1990s based on Toyota's production system that states that any activity or process that does not add value to the product or service is a waste and, therefore, should be eliminated

Lean Production

A management philosophy that originated in the 1990s based on Toyota's production system that states that any activity or process that does not add value to the product or service is a waste and, therefore, should be eliminated

Offshoring

Sourcing from overseas or getting work done in a foreign country

Operations Management (OM)

The process of managing the system of designing, producing, and delivering goods or services that add value throughout the supply chain and benefit the final consumer

Operations Strategies

Strategies that use an organization's resources efficiently to gain a competitive edge in the marketplace by achieving key operations objectives related to product quality, delivery, flexibility, cost, service, and innovation. These strategies should outl

Outbound Portion

Elements of a supply chain that begin when the organization delivers its output to its immediate customers in the supply chain and may consist of globally dispersed wholesalers, retailers, distribution centers, and transportation companies, the consumer,

Outsourcing

Contracting with a third party or an external company to manufacture a good or deliver a service

Service Support

The process of interfacing with suppliers that provide the company with support

Six Sigma

The methodology of improving quality through reducing the number of defects in a given process

Splintering

The practice of breaking traditional supply chains into smaller and more agile supply chains that can better respond to higher levels of business complexity, save money, and improve customer service

Supplier Management

A business process that enables a company to identify and select the best possible suppliers and negotiate the best possible prices for the resources it purchases from them

Supply Chain

A sequence of interconnected organizations that helps develop, produce, distribute, and sell a product to the final consumer

Sustainability

The use of methods, systems, and materials that won't deplete resources or harm natural cycles

Technology Management

...

Total Quality Management (TQM)

A sequence of interconnected organizations that helps develop, produce, distribute, and sell a product to the final consumer

Demand Management

The process of anticipating and managing customer demand for a product or service

The four elements focused by operations strategy are __________.

customers, operational critical success factors, product factors and firms core competencies

__________ can identify the critical processes in a supply chain and, therefore, better understand exactly what to measure when assessing their performance.

Supply chain operations reference model

Incorporating sustainability throughout a company's supply chain is difficult because __________.

time consuming and may require changes in firms structure and infrastructure

The three performance targets that measure sustainability are __________.

economic value, social value and environmental value

A(n) __________, which is formulated by an organization's top managers and board of directors, attempts to address the fundamental question of what industries and markets the organization should enter and compete in.

corporate strategy

The four elements of an operations strategy are __________.

logistics, distribution capabilities, information process capability and marketing channels

__________ is the time it takes to conceptualize a new good or service, produce it, and make it available to customers.

Product development cycle time

__________ is not only a performance measurement system but also a strategic planning and management system that is used extensively by profit, nonprofit, and governmental organizations worldwide.

balanced scorecard

Identifying the target market for a service, also known as its __________, is the first step in developing the strategic plan.

strategic postioning

__________ is the ratio of outputs produced to the inputs used.

productivity

__________ is a process that begins by identifying the best practices used by other firms within the industry and their achievements.

benchmarking

__________ is measured by calculating the number tasks performed or number of customers served in a given time period.

service productivity

Which of the following is NOT a process in the supply chain operations reference model?

innovation

The supply chain operations reference model has more than __________ key indicators.

150

A __________ strategy is a firm's plan to provide value to its customers by using the firm's internal and external resources.

go to market

The four stages of the project life cycle are __________.

conceptualization, planning, execution and termination

__________ works best when you can be reasonably certain of how long the project's activities are likely to take.

Critical path method

The __________ method calculates the value generated by the project as well as the budgeted and actual costs at a certain point in time.

earned value management

__________ is the process of speeding up a project's remaining activities in order to move its completion date forward.

crashing

A __________ is a unique venture with a beginning and end, conducted by people to meet established goals within certain cost, schedule, and quality parameters.

project

The three ways of structuring project teams are __________.

matrix, functional and pure project

The triple constraint of projects are __________.

time, cost, quality

In the __________ stage, the team members decide what they expect to achieve with the project.

conceptualization

Two of the best-known methods for tracking projects upon their execution are __________.

s-curves and value management

__________ is the manufacturing or assembling of materials to create the product or service.

fabrication

__________ is the latest possible date that an activity may begin without delaying the project completion.

late start time

_________ is the latest possible date an activity can be completed without delaying the project's completion.

late finish time

The most common type of activity network employs __________ network representation.

activity on node

A __________ chart links project activities to a schedule, which is usually a calendar.

Gantt

__________ is the process of creating a chart that shows a project's activities and indicates how they should logically be coordinated with one another.

Network Diagramming

Activity Network

A diagram that illustrates how activities should best be sequenced

Activity on Arrow (AOA)

A type of activity network that uses arrows to represent the activities and nodes separating the arrows as events that indicate the completion of one task prior to the beginning of the next

Activity on Node (AON)

A type of activity network that uses nodes to represent the individual project activity and the path arrows to represent the sequencing of tasks from node to node through the network

Activity Slack

The amount of time an activity may be delayed from its early start (ES) without delaying the finish of the project. Activity slack is defined as Activity Slack = LS - ES (or LF - EF)

Backward Pass

The process of labeling the activity nodes in a network to determine the project's critical path and the total slack time for each activity

Crashing

The process of speeding up a project's remaining activities to move its completion date forward

Critical Path

The sequence of activities in a network that defines the overall length of the project (the longest individual path)

Deterministic Estimation

A duration estimation that assumes that sufficient information from historical records or the opinion of experts is available to make this determination with a reasonable degree of accuracy

Early Finish Time (EF)

The earliest possible date on which an activity can be completed. Early finish is defined as es + activity duration = ef

Early Start Time (ES)

The earliest possible date on which an activity can start, based on the beginning and ending times for previous activities in the network

Earned Value Management (EVM)

A method to calculate the value generated by the project as well as the budgeted and actual costs at a certain point in time

Forward Pass

The process of fully labeling the activity nodes in a network to determine the earliest times each activity can begin and the earliest it can be completed

Gantt Chart

A diagram that links project activities to a schedule, usually a calendar

Late Finish Time (LF)

The latest possible date an activity can be completed without delaying the project's completion. Late finish is defined as ls + activity duration = lf

Late Start Time (LS)

The latest possible date that an activity may begin without delaying the project completion

Life Cycle

The stages of a project's development conceptualization, planning, execution, and termination

Precedence Diagramming

Sometimes called network diagramming, the process of creating a chart that shows a project's activities and indicates how they should logically be coordinated with one another

Probabilistic Estimation

A duration estimation that assigns probabilities to a range of time estimates for each activity

Program Evaluation And Review Technique (PERT)

A duration estimation technique that uses probabilities by applying a weighted average of optimistic, pessimistic, and most likely estimates to derive an expected duration for an activity

Project Management

The application of knowledge, skills, tools, and techniques to project activities to meet the project requirements

Project Risk

Any event that can negatively affect the viability of a project based on the formula Project Risk = (Probability of Event) � (Consequences of Event)

Project scheduling

The process of converting a project's goals into a logical method for completing them on time

Project

A unique venture with a beginning and an end, conducted by people to meet established goals within certain cost, schedule, and quality parameters

A modern iteration of triple constraint that considers the goal of satisfying a customer or client. The elements of quadruple constraint are time, cost, quality, and customer satisfaction

Risk Management

The process of anticipating and figuring out how to respond to a project's risks prior to beginning the project

Scope Creep

A situation in which project specifications are continually modified or improved as new ideas emerge or new technical solutions become possible

Scope Management

The process of determining the best way in which the project's goals can be accomplished given its constraints

S-Curve

a curve that depicts logistic growth; shape of an "S." The leveling off of a J-Curve exponential growth.

Triple Constraint

Elements that characterize a project and are used to evaluate its success time (adherence to schedule), cost, and quality

Work Breakdown Structure (WBS)

process of breaking down a project's overall mission into step-by-step tasks

Work Packages

Activities that comprise the deliverables of a project

Which of the following is a primary reason for the accelerated pace of global interaction?

By improving the quality of goods and services produced, the _______ value of products and services is enhanced.

Functional

Which of the following is a strategic supply chain and operations decision?

quality management

______ manufacturing refers to the ability of an organization to respond quickly to market changes.

agile

Which of the following is an external interface with the operations function?

customer

The transformation process of a hospital is ______.

physiological

Which of the following can be on both the inbound and outbound portion of a supply chain?

transportation

Firms that provide components or finished goods and services to the primary firm are called ______.

first tier suppliers

Which of the following is a significant operation interface that distinguishes a service supply chain from a product supply chain?

service performance management

Which of the following is the type of operations interface that interacts with suppliers to provide the company with equipment maintenance and repair services?

vertical interfaces

L. L. Bean contracts with shirt manufacturers to produce a wide assortment of shirts for men and women. These shirt manufacturers represent ______ suppliers for L. L. Bean.

First Tier

______ will be effective for reducing variations in a company's manufacturing processes.

six sigma

There is a clear separation between tactical and operational supply chain and operations decisions.

false

An apparel manufacturer restructures its traditional supply chains by outsourcing its T-shirts manufacturing division to China while keeping its designer clothing division in the United States. Such practice is called splintering.

true

Budget airlines such as JetBlue are leveraging their ability to offer low-cost flights to price-conscious customers as their core competency.

true

Flexibility is one of the core competencies that do not apply to service operations.

False

Which of the following capabilities enables companies to successfully integrate their operations and supply chain globally?

financial- engineering

Steinway's high-quality, high-cost vertical piano failed because it did not meet the order-______ criterion.

qualifying

The receptionist at the information desk of a hospital represents the ______ to the service operation system.

front stage activity

Companies that manufacture products with a very ______ product life cycle have ______ as their core competency.

short, innovation

Which of the following is an example of a socially responsible company as described in Chapter 2 of the textbook?

Patagonia

Companies increasingly strive to achieve the ______ performance when formulating their corporate strategy.

triple bottom line

Which of the following is a supply chain's delivery performance indicator?

the total number of products delivered correctly and on time

Shell is able to compete on ______ by producing gasoline at a ______ than its competitors.

price, lowercost

A firm's strategy should be aligned to form a top-to-bottom hierarchy, linking ______ levels.

The critical path of a project is always the one with the most activities.

false

Estimating the duration of a project's activities involves both subjective and objective analysis.

true

Which of the following are the stages of a project life cycle?

conceptualization, planning, execution, termination

A critical skill for project managers is ______.

team building

A project manager is evaluating local suppliers to be potential back-ups for main suppliers in Japan. The project manager is doing ______ to manage a project supply chain.

an environmental analysis

A project is considered successful if it meets budget and schedule requirements.

false

The earned value is the ______ of the ______.

estimate of budgeted cost; work actually performed to date

The manager of a crew that installs carpeting has tracked the crew's output over the past several weeks, obtaining these figures. What is the week with the highest labor productivity?

yards installed/ workers

A company offers ID theft protection using leads obtained from client banks. Four employees work 40 hour a week on the lead, at a pay rate of $35 per hour per employee. Each employee identifies an average of 3500 potential leads a week from a list of 6000 Do give positive rating if you think this answer helped. MFP = (Potential leads)(No. of workers)(Fee)(Conversion percentage) /Labor cost + Material cost + Overhead cost MFP = 3,500 (4) ($60) (0.03) / ( 4(40)($35) +$1,500 + \$8,000) = 25200 / 15100 = 1.67

Multifactor Productivity Equation

output/ (labor cost+material cost +overhead cost)

project

A temporary endeavor undertaken to create a unique product, service, or result.

Why projects?

Shortened product life cycles
Narrow product-launch windows
Complex products and services
Integrated global markets/suppliers Fast, Complex, Global

Time, Cost, quality and customer client satisfaction

key decisions

Project selection
Project Conceptualization/Definition (Scope)
Selecting Project Manager and Team
Planning and designing project: Conceptualize, Plan, Risk
Managing and controlling project resources: Scheduling, Managing the Supply Chain
Executing, Evalua

project risk

A risk related to management and control of the (test) project, e.g. lack of staffing, strict deadlines, changing requirements, etc.

project risk

probability of event * consequences of event

Risk Management

is the process of anticipating and figuring out how to respond to a project's risks prior to beginning the project.

Risk Management Techniques

The risk management process involves three steps:
Identifying risks: Financial, Technical, Commercial, Legal/Contractual
Analyzing the probability of risks occurring and their consequences.
Formulating risk mitigation strategies: Accept, Minimize, Share,

Key Decisions

Project selection
Project Conceptualization/Definition (Scope)
Selecting Project Manager and Team
Planning and designing project: Conceptualize, Plan, Risk
Managing and controlling project resources: Scheduling, Managing the Supply Chain
Executing, Evalua

History of PM

Modern project management began with the Manhattan Project, which the U.S. military led to develop the atomic bomb In 1917 Henry Gantt developed the Gantt chart as a tool for scheduling work in job shops
In 1958, the Navy developed PERT charts
In the 197

Professional PM

Program Management Professional (PgMPSM) Credential
Professionals who manage multiple related projects that are aligned with an organization's strategy.
Project Management Professional (PMP�) Credential
One of the most valued credentials in project manage

Tools and Methods

WBS: Work Breakdown Structure
Gantt Chart
Network Diagrams
CPM: Critical Path Method
PERT: Program Evaluation and Review Technique
Crashing Schedules
Earned Value Analysis

Crashing (the schedule)

Crash = shorten the activity duration, and thereby reduce the project duration
So which activities do we need to crash to impact the project duration? (heuristic or rule-of-thumb)
You should be able to crash the activity (ie the work involved can be exped

critical path

all activies have zero slack

slack

early start - late start

Early Start/Early Finish

the earliest times that the activity can start and finish without delaying the project

Early Start/Early Finish

Early start is the Early finish of the letter in front
EF = ES + # given
LS = LF - TIME
LF: TAKE THE EF OF LAST ONE AND GO BACK

crashing the schedule

- Reducing the duration of the critical path resulting in overall shorter project duration
- Additional cost and schedule risks
- Accelerates project completion

Crash activities

activities should be on the critical path
select the least cost option
cheapest avaible on critical path


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# Abstracts

### Monday, April 10th, 2017

9:30 am10:00 am
Green and Tao ([GT04]) used the existence of a dense subset indistinguishable from the primes under certain tests from a certain class to prove the existence of arbitrarily long prime arithmetic progressions. Tao and Ziegler ([TZ06]) showed some general conditions under which such a model exists. In [RTTV08], a quantitatively improved characterization was obtained using an argument based on Nisan’s proof of the Impagliazzo hard-core set theorem ([I95]) from computational complexity. Gowers ([Gow08]) independently obtained a similar improvement.

We show that the existence of dense models can be reduced directly to the improved hardcore distri-bution results of Holenstein ([H05]). Using Holenstein’s uniform proof of an optimal density hard-core set theorem, we show that the dense models that one derives have a canonical form, with models being (sampleable from) functions defined in terms of tests from the original class. We give several applications, including generalizations of weak regularity lemmas ([FK99, K97, COCF]). For example, we show that any graph G with ∆n2 edges has a γ-cut-approximator of rank 2poly(1/γ,1/ log(1/∆), whereas direct application of [FK99] gives rank 2O(1/γ2∆2)
10:00 am10:30 am
We consider random (or sufficiently pseudorandom) instances of a constraint satisfaction problem with n variables, m = m(n) constraints, and constraint predicate F.  Given a certain running time n^r for the "SOS algorithm", how strong an upper-bound 1-delta can be placed on the maximum fraction of simultaneously satisfiable constraints?

We establish the optimal 3-way tradeoff between m, r, and delta, in terms of F, generalizing all previously known lower bounds in the area.

Perhaps the most interesting contribution is a new and improved definition for the "closure" of a set of constraints in a CSP, a graph-theoretic concept heavily used in previous work on proof complexity.

joint work with Pravesh Kothari, Ryuhei Mori, and David Witmer
11:00 am12:00 pm

What structure/randomness dichotomies can be found across families of infinite structures? How precisely can we pinpoint the interactions of randomness and freedom vs structure and control inside a structure? Model theory has long investigated these questions. It turns out that model theoretic theorems about infinite objects can have strong consequences for finite objects. The talk will expand on these questions and some motivating consequences from the speaker's work (joint with S. Shelah), including characterization of the existence of irregular pairs in Szemeredi's regularity lemma and proofs of instances of Erdos-Hajnal.

2:00 pm3:00 pm
We determine the asymptotics of the log-probability that the number of k-term arithmetic progressions in a random subset of integers exceeds its expectation by a constant factor. This is the arithmetic analog of subgraph counts in a random graph. The solution uses nonlinear large deviation principles developed by Chatterjee and Dembo, and, more recently, by Eldan.

I will highlight some open problems in additive combinatorics that we encountered in our work, namely concerning the "complexity" of the dual functions of AP-counts. These problems seem to extend beyond the current reach of higher-order Fourier analysis.
3:30 pm4:00 pm
We consider the standard two-party communication model. The central problem studied in this talk is how much one can save in information complexity by allowing an error of epsilon.

We answer a question of Braverman by establishing a gap between the two different notions of the prior-free information cost. This implies that amortized randomized communication complexity is not necessarily equal to the amortized distributional communication complexity with respect to the hardest distribution.

This is based on a joint work with Yuval Dagan, Yuval Filmus, and Yaqiao Li.
4:00 pm4:30 pm
We address the question when randomly perturbed graphs are sparse (bounded expansion and nowhere dense).
This relates to special coloring problems. Joint work with P. Ossona de Mendez.

### Tuesday, April 11th, 2017

9:30 am10:30 am
Speaker: Terry Tao, UCLA

We discuss a variant of the density and energy increment arguments that we call an "entropy decrement method", which can be used to locate a scale in which two relevant random variables share very little mutual information, and thus behave somewhat like independent random variables.  We were able to use this method to obtain a new correlation estimate for multiplicative functions, which in turn was used to establish the Erdos discrepancy conjecture that any sequence taking values in {-1,+1} had unbounded sums on homogeneous arithmetic progressions.

11:00 am12:00 pm
Vinogradov showed in 1937 that every large enough odd integer can be represented as a sum of three primes. One may ask what if these primes are restricted to some subset of the primes. In general, if the set is badly distributed in congruence classes or Bohr sets, the result does not necessarily hold. Examples of such badly distributed sets include the set of primes that are 1 mod 5, and the set of primes $p$ for which $|| \sqrt{2} p || < 1/10$, where $||x||$ denotes the distance from $x$ to the nearest integer.

In the talk I discuss joint works with X. Shao and J. Maynard and X. Shao on Vinogradov's three primes theorem for subsets of primes. In particular I will describe two transference type'' results aimed to show the obstructions describe above are only sort of obstructions. As applications of the transference principles we get that Vinogradov's three primes theorem holds for Chen's primes (i.e. primes $p$ for which $p+2$ has at most two prime factors) and for almost equal primes (i.e., for any $\theta > 0.55$, every large enough odd integer $N$ can be represented as $N = p_1 + p_2 + p_3$, where $|p_i - N/3| < N^\theta$ for each $i$).
2:00 pm3:00 pm
The collection of large Fourier coefficients of a function, whether they be called 'major arcs' or the 'large spectrum', are one way of representing the linearly structured component of a function, and as such plays an important role in many problems in additive combinatorics, analytic number theory, theoretical computer science, and beyond. In this talk I will discuss some results concerning what kind of additive structure such sets can have, and how this structure can be exploited to improve density increment arguments.
3:30 pm4:00 pm
A graph G has a threshold rank k if the normalized adjacency matrix of G has at most k large’’ eigenvalues. In this talk we discuss structural properties of graphs that have small threshold rank. We see that these graphs can be seen as a union of at most k expander graphs. We also discuss a natural extension of the weak regularity lemma to (sparse) low threshold rank graphs.

Based on joint works with Luca Trevisan.

### Wednesday, April 12th, 2017

9:30 am10:30 am
The symmetry of a circle is easily destroyed: just individualize'' two non-opposite points -- color one of them red, the other blue -- and all the symmetry is gone.  In fact, the resulting structure is completely irregular: every point is uniquely identified by a pair of numerical invariants, namely, its pair of distances to the two individualized points. We shall say that the circle has a high degree of hidden irregularity.

In contrast, Johnson graphs are objects with robust symmetry: individualizing a small number of vertices of a Johnson graph hardly makes a dent in its symmetry.

It turns out that Johnson graphs are unique in this regard: Every finite relational structure of small arity either has a measurable (say 10%) hidden irregularity (exposed by individualizing a small number of elements) or has a high degree of hidden structure, manifested in a canonically embedded Johnson graph on more than 90% of the underlying set.

This dichotomy provides the combinatorial Divide-and-Conquer tool to recent progress on the complexity of the Graph Isomorphism problem.
11:00 am12:00 pm
The symmetry of a circle is easily destroyed: just individualize'' two non-opposite points -- color one of them red, the other blue -- and all the symmetry is gone.  In fact, the resulting structure is completely irregular: every point is uniquely identified by a pair of numerical invariants, namely, its pair of distances to the two individualized points. We shall say that the circle has a high degree of hidden irregularity.

In contrast, Johnson graphs are objects with robust symmetry: individualizing a small number of vertices of a Johnson graph hardly makes a dent in its symmetry.

It turns out that Johnson graphs are unique in this regard: Every finite relational structure of small arity either has a measurable (say 10%) hidden irregularity (exposed by individualizing a small number of elements) or has a high degree of hidden structure, manifested in a canonically embedded Johnson graph on more than 90% of the underlying set.

This dichotomy provides the combinatorial Divide-and-Conquer tool to recent progress on the complexity of the Graph Isomorphism problem.
2:00 pm3:00 pm

This is part of our ongoing efforts to understand high-dimensional combinatorial objects. These include simplicial complexes (a 1-dimensional simpicial complex is a graph), high-dimensional permutations (e.g., a 2-dimensional permutation is synonymous with a Latin square), hypertrees and more. For every type of combinatorial object that we wish to study it is crucial to understand their typical behavior. Namely, to understand the properties of random objects of the type that we study. In this talk I describe recent work with my student Maya Dotan which allows us to efficiently generate random one-factorizations. I will also mention ongoing work with my student Michael Simkin on random generation of Latin squares. In addition I will speak about joint work with Zur Luria on low-discrepancy high-dimensional permutations.

### Thursday, April 13th, 2017

9:30 am10:30 am

Positional games are perfect information games between two players with no chance moves. As such, they do not seem to allow much room for randomness and random considerations. Yet, as convincingly shown by intensive research in the last decades, randomness is omnipresent in positional games and manifests itself in a variety of ways, including:
- probabilistic (looking) winning criteria;
- guessing the threshold bias of biased games; - winning using random play;
- playing on random boards;
- introducing randomness into game rules.

In this survey talk I will discuss and illustrate the rich interplay between positional games and randomness, concentrating mostly on Maker-Breaker games. Essentially no prior experience with positional games will be assumed.

11:00 am12:00 pm
I will describe a journey that starts at error correcting codes for distributed storage, and leads to graph labeling, the study of the Birkhoff polytope graph and the representation theory of the symmetric group.

On the technical side, we prove tight bounds for the independence number of the Birkhoff polytope graph, and use it to characterize the alphabet size needed for maximally recoverable codes in grid topologies.

Joint work with Daniel Kane (UCSD) and Sankeerth Rao (UCSD).
2:00 pm2:30 pm

Let f:F_2^n --> {0,1} be a function and suppose F_2^n x F_2^n is partitioned into k sets A_i x B_i such that f is constant on each sumset A_i + B_i.  We show this implies a partitioning of F_2^n to quasipoly(k) affine subspaces such that f is constant on each. In other words, up to polynomial factors, deterministic communication complexity and parity decision tree complexity of f are equal. This relies on a novel technique of entropy decrement combined with Sanders' Bogolyubov-Ruzsa lemma.

Joint work with Hamed Hatami and Shachar Lovett.

2:30 pm3:00 pm

A random subset with density alpha of an n-dimensional vector space over a finite field on p elements likely has about an alpha^3 fraction of the three-term arithmetic progressions with each nonzero common difference. Note that, for a subset of density alpha, the density of three-term arithmetic progressions can be much smaller than alpha^3. However, Green proved that for each epsilon>0 there is n_p(epsilon) such that if the dimension n of the vector space is at least n_p(epsilon), then there always exists a nonzero common difference d with three-term arithmetic progression density at least alpha^3-epsilon, what we expect from the random bound. The proof uses the arithmetic regularity lemma and gives a bound on n_p(epsilon) which is an exponential tower with height polynomial in 1/epsilon.

We provide a construction which shows that n_p(epsilon) is at least a tower with height logarithmic in 1/epsilon when epsilon is small, and prove a matching upper bound up to constant factor in the tower height. This gives the first natural application of a regularity lemma where the tower type bound is really necessary. Tight bounds are also obtained for other ranges of epsilon (in terms of the density alpha).

This is joint work with Jacob Fox and Yufei Zhao.

3:30 pm4:00 pm

Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. In this work, we give a new characterization of LDCs using distributions over smooth Boolean functions whose expectation is hard to approximate (in~$L_\infty$~norm) with a small number of samples. We give several candidates for such distributions  coming from finite field incidence geometry and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters.

4:00 pm4:30 pm
A basic combinatorial interpretation of Shannon's entropy function is via the 20 questions'' game.
This cooperative game is played by two players, Alice and Bob:
Alice picks a distribution $\pi$ over the numbers $\{1,\ldots,n\}$,
and announces it to Bob.
She then chooses a number $x$ according to $\pi$,
and Bob attempts to identify $x$ using as few
Yes/No queries as possible, on average.

An optimal strategy for the 20 questions'' game is given by a Huffman code for $\pi$:
Bob's questions reveal the codeword for $x$ bit by bit.
This strategy finds $x$ using fewer than $H(\pi)+1$ questions on average.
However, the questions asked by Bob could be arbitrary.
In this paper, we investigate the following question:
\emph{Are there restricted sets of questions that match the performance of Huffman codes, either exactly or approximately?}

Our first main result shows that for every distribution $\pi$,
Bob has a strategy that uses only questions of the form $x < c$?'' and $x = c$?'’,
and uncovers $x$ using at most $H(\pi)+1$ questions on average,
matching the performance of Huffman codes in this sense.
We also give a natural set of $O(rn^{1/r})$ questions that achieve a performance of at most $H(\pi)+r$,
and show that $\Omega(rn^{1/r})$ questions are required to achieve such a guarantee.

Our second main result gives a set $\Q$ of $1.25^{n+o(n)}$ questions such that
for every distribution $\pi$, Bob can implement an \emph{optimal} strategy for $\pi$ using only questions from $\cQ$.
We also show that $1.25^{n-o(n)}$ questions are needed, for infinitely many $n$.
Here, the set $\Q$ is derived via a random construction.
If time will allow, we will discuss the challenge of derandomizing
this construction, and a connection with hitting maximal antichains.

Based on joint work with Yuval Dagan, Yuval Filmus, and Ariel Gabizon

### Friday, April 14th, 2017

9:30 am10:30 am
A word w is said to contain the pattern P if there is a way to substitute a nonempty word for each letter in P so that the resulting word is a subword of w. Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised the patterns P which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains P. Zimin's characterisation says that a pattern is unavoidable if and only if it is contained in a Zimin word, where the Zimin words are defined by Z_1 = x_1 and Z_n=Z_{n-1} x_n Z_{n-1}.  We study the quantitative aspects of this theorem, obtaining essentially tight tower-type bounds for the function f(n,q), the least integer such that any word of length f(n, q) over an alphabet of size q contains Z_n.

Joint work with Jacob Fox and Benny Sudakov.
11:00 am12:00 pm
Suppose we have a set of integers S, and we wish to count solutions to a certain system of linear equations in S; in particular, we want to know when we get the expected "random" answer.  A by-now standard approach to this is to show that the count is controlled by a certain uniformity norm for some s, and use tools from higher order Fourier analysis to understand when S is pseudorandom by that measure.

One question is: what is the smallest s you can get away with?  In other wards, how high up in the higher Fourier hierarchy do you have to go to get the right answer?  When can we get away with looking just for linear discrepancies?

This quantity s is called the "true complexity" of the system, and these question were addressed in work of Gowers--Wolf and Green--Tao.  A striking feature, though, is the significant gap between what elementary methods can show (the "Cauchy--Schwarz complexity") and the truth, which is currently only accessible using very heavy machinery and which gives ineffective bounds.

We will explore some questions in this area.

format: plain


text:
# Simplicity of group $C^\ast$-algebra implies fullness of group-von Neumann algebra?

Let $$\Gamma$$ be a discrete group whose reduced group $$C^\ast$$-algebra is simple. Can we conclude that the corresponding group-von Neumann algebra $$\mathcal{L}(G)$$ is a full $$\text{II}_1$$-factor, meaning that every uniformly bounded net $$(x_i)_{i \in I}$$ that satisfies $$\lim_i \Vert x_ia-ax_i\Vert_2 =0$$ for all $$a \in \mathcal L(\Gamma)$$ we must have $$\lim_i \Vert x_i -\tau (x_i)1\Vert=0$$ where $$\tau$$ is the canonical tracial state of $$\mathcal{L}(\Gamma)$$?

No, whenever $$\Gamma$$ is an infinite direct product of C$$^*$$-simple groups, we obtain a counterexample. For instance, taking $$\Gamma = \mathbb{F}_2^{(\mathbb{N})}$$ to be the direct sum of infinitely many copies of the free group $$\mathbb{F}_2$$, we get that $$C^*_r(\Gamma)$$ is simple (because this is true for every finite direct product). On the other hand, taking a sequence of group elements $$g_n \in \Gamma$$ in the $$n$$-th factor of the direct product, the corresponding sequence of unitaries in $$L(\Gamma)$$ is a nontrivial central sequence, so that $$L(\Gamma)$$ is not full.
• I meant to say: "because $C^*_r(\Gamma_n)$ is simple for every direct product of $n$ copies of $\mathbb{F}_2$". Apr 8 '21 at 12:39
• @YCor: yes, indeed, and the argument goes as follows. For a C$^*$-algebra, simplicity amounts to saying that every representation on a Hilbert space is isometric. By the minimality of the minimal tensor product, the minimal tensor product of finitely many simple C$^*$-algebras is simple. So a representation of the infinite minimal tensor product will be isometric on every finite tensor product, and hence isometric as a whole. Apr 8 '21 at 12:58
• @worldreporter14: there are also finitely generated counterexamples. You could take the wreath product $\mathbb{F_2} \wr \mathbb{Z}$, i.e. the semidirect product of an infinite direct sum of copies of $\mathbb{F_2}$ with $\mathbb{Z}$ acting by the shift. Both the proof of C$^*$-simplicity and fullness would require a bit more space than this comment section. Apr 8 '21 at 13:00
• For a finitely generated example that is even finitely presented (and even type $F_\infty$), I believe the Lodha--Moore group $G_0$ works. It is $C^*$-simple (see Theorem 1.10 of arxiv.org/pdf/1605.01651.pdf), and I strongly suspect $L(G_0)$ is a McDuff factor (and hence has a non-trivial central sequence) for reasons identical to those in Jolissaint's proof that this is the case for Thompson's group $F$ (see numdam.org/item/AIF_1998__48_4_1093_0). (Also, if $F$ itself turned out to be non-amenable, then it would be $C^*$-simple and provide yet another example.) Apr 8 '21 at 14:32


text:
# Equation of State¶

Consider a composite polytrope composed of $$\nreg$$ regions extending from the origin out to the stellar surface. In the $$i$$’th region ($$1 \leq i \leq \nreg$$), the pressure $$P$$ and density $$\rho$$ are related by the polytropic equation-of-state

(11)$\frac{P_{i}}{P_{i,0}} = \left( \frac{\rho_{i}}{\rho_{i,0}} \right)^{(n_{i} + 1)/n_{i}}$

where the normalizing pressure $$P_{i,0}$$ and density $$\rho_{i,0}$$, together with the polytropic index $$n_{i}$$, are constant across the region but may change from one region to the next. At the $$\nreg-1$$ boundaries between adjacent regions, the pressure and interior mass $$M_{r}$$ are required to be continuous, but the density may jump.


text:
# Algorithm for approximation with sum or difference of 2 unit fractions

Let $U =\{\frac{1}{n}: n\in \mathbb{N}\} \cup \{-\frac{1}{n}: n\in \mathbb{N}\}$ be the set of positive or negative unit fractions.

Given positive integers $m<n \in\mathbb{N}$, how do you find $u_1, u_2 \in U$ such that $|\frac{m}{n} - (u_1+u_2)|$ becomes minimal?

A weaker problem would be the following: Given positive integers $m<n \in\mathbb{N}$ and a positive integer $k\in\mathbb{N}$ devise an algorithm that outputs "Yes" if there exist $u_1, u_2\in U$ such that $|\frac{m}{n} - (u_1+u_2)| < \frac{1}{k}$, and "No" otherwise.

• What have you tried? What research have you done? I expect you to do a significant amount of research before asking, and to show us what you've tried and what you know in the question. Are you familiar with continued fractions? (This might affect the level of answer.) Also, what's the context in which you ran across this question? – D.W. Aug 27 '14 at 16:16
• I'm familiar with continued fractions and have tried a similar (but less efficient) approach comparable to Pseudonym's approach below. My research on this question didn't take me a long way - I couldn't even find out whether there always is an optimal answer. (I.e. I didn't find out whether there are $m<n\in \mathbb{N}$ such that for all $u,v\in U$ there are $u',v'\in U$ such that $|\frac{m}{n} - (u'+v')| < |\frac{m}{n} - (u'+v')|$.) There seems to be little research on approximations by sums of positive or negative unit fractions. – Dominic van der Zypen Aug 28 '14 at 6:42

Here's a method that gets close. Do a continued fraction expansion of $\frac{m}{n}$:

$$\frac{m}{n} = \frac{1}{a + \frac{1}{b + \epsilon}} \approx \frac{1}{a + \frac{1}{b}} = \frac{1}{a} - \frac{1}{a(1 + ba)}$$

If $\epsilon$ is small, then this is probably very close to optimal if not optimal in most cases.

The method fails when the truncated continued fraction approximation isn't very good. Take $\frac{15403}{26685}$ (a rational approximation to the Euler-Mascheroni constant) as an example. The continued fraction approximation is:

$$\frac{15403}{26685} = \frac{1}{1 + \frac{1}{1 + \epsilon}}$$

Which suggests:

$$\frac{15403}{26685} \approx \frac{1}{1} - \frac{1}{2}$$

But clearly $\frac{1}{2} + \frac{1}{14}$ is closer.

I would wager that the worst case for any method is the conjugate golden ratio $\phi' = \frac{\sqrt{5} - 1}{2}$. This isn't rational (in fact it's the "most" irrational number in a technical sense; find its continued fraction expansion if you're curious), but you can get arbitrarily close by choosing $m$ and $n$ to be consecutive Fibonacci numbers, e.g. $\frac{4181}{6765}$. The best approximation I could find is $\frac{1}{2} + \frac{1}{9}$.


text:
Still have questions? Join our Discord server and get real time help.
1

# How do I print the contents of a player?

I was trying to make a simple script to print the contents of a player's character:

local chr1 = game.Players.LocalPlayer.Character
local chr2 = chr1:GetChildren()

print(chr2)

0
You can use "unpack(array Table)" Ziffixture 5707 — 8d
0
for i = 1, #chr2 do print(chr2[i]) end TheBeastMare 2 — 7d

0
Glacitron 147
8 days ago

You can use this script, basically you want to loop through all the contents and print each one individually you can't just print a table,

local plr = game.Players.LocalPlayer
local Char = plr.Character or plr.CharacterAdded:Wait()

for i , v in pairs(Char:GetChildren()) do --Char:GetDescendants() if you want to get all of them
print(v.Name)
end

0
I'd like to advise you to use ipairs instead, instances serve as number-type references to tables so we can perfectly use ipairs here. In the current version of lua roblox uses ipairs is generally faster then pairs. Its just good practice as pairs usually shows that the object your traversing through has non-numeric indices. 4thAxis 0 — 8d
0
Yeah my bad Glacitron 147 — 7d


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• 1985-1989  (7)
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Publisher
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Year
• 1
Electronic Resource
Hoboken, NJ : Wiley-Blackwell
Advances in Polymer Technology 6 (1986), S. 407-407
ISSN: 0730-6679
Keywords: Chemistry ; Polymer and Materials Science
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Chemistry and Pharmacology , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
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• 2
Electronic Resource
Bognor Regis [u.a.] : Wiley-Blackwell
ISSN: 0887-6266
Keywords: Chemistry ; Polymer and Materials Science
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Chemistry and Pharmacology , Physics
Notes: Linear low-density polyethylenes (LLDPES) and polypropylene (PP) have been recovered from solutions of varying initial polymer concentration. Melts of these polymers show significant reductions in viscosity and elasticity, and the effects are attributed to changes in the entanglement density of the polymer. Measurements of entanglement densities have been attempted from experimental values of the apparent zero-shear melt viscosity. These indicate that solution treatments in trichlorobenzene at 135°C reduce the entanglement density more effectively in PP than in LLDPE. In all cases the observed effects are reversible by annealing at elevated temperatures. Analytic data point to entanglement changes as the true origin of changes in viscoelastic properties, since solution treatments produce no changes in molecular weights and weight distributions, and the samples tested are free of solvent residues.
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• 3
Electronic Resource
Springer
Rheologica acta 25 (1986), S. 95-101
ISSN: 1435-1528
Keywords: End effect ; elastic-constant viscosity fluid ; Boger fluid ; primary normalstress difference ; shear-thinning properties
Source: Springer Online Journal Archives 1860-2000
Topics: Chemistry and Pharmacology , Physics
Notes: Abstract In the light of a new interpretation, we have studied the end effects for highly elastic-constant viscosity fluids commonly called Boger fluids. In terms of entrance effect only, the presence of primary normal-stress differences in absence of shear-thinning properties results in a decrease of the entrance correction below the Couette (Newtonian) value, whereas the total end correction can be substantially increased by an amount which is strongly dependent on the Weissenberg number or recoverable shear.
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• 4
Electronic Resource
Springer
Rheologica acta 27 (1988), S. 241-254
ISSN: 1435-1528
Keywords: Rheological equation ; dilute polymersolution ; polyelectrolyte ; shear thickening ; polyacrylamide solution
Source: Springer Online Journal Archives 1860-2000
Topics: Chemistry and Pharmacology , Physics
Notes: Abstract A mathematical model based on the diffusion-convection equations is used to describe the rheological properties of dilute polymer solutions. The model uses a second-order conformation tensor as a measure of the internal strain; this avoids the mathematical complexity resulting from the use of a more detailed description of the macromolecules and also avoids the necessity of introducing additional ad-hoc assumptions (closure approximations) commonly used in other molecular theories. The rheological equation is obtained in terms of the rate-of-deformation tensor $$\dot \gamma$$ and a scalar functionf(σ) relating the extra stress tensorσ to the internal strain tensorc. The functionf(σ) depends on the physical insight introduced in the Helmholtz free energyA(c) of the solvent-polymer system. This approach is illustrated for an intra-molecular potential of a “FENE-charged” type. The concept of an isotropic, but conformation-dependent, friction coefficient, is also introduced to account for the “coil-stretch” transformation of macromolecules in solution. Viscosity and first normal-stress data, of partially hydrolyzed polyacrylamide solutions, (polyelectrolytes) are analyzed and compared to the model predictions in steady shear and elongational flows.
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• 5
Electronic Resource
Springer
Rheologica acta 25 (1986), S. 214-224
ISSN: 1435-1528
Keywords: Constitutive equation ; viscoelastic fluid ; kernel function ; simple shear flow ; uniaxial extension
Source: Springer Online Journal Archives 1860-2000
Topics: Chemistry and Pharmacology , Physics
Notes: Abstract A selection of kernel functions is given to be used in a new integral constitutive equation proposed by Piau whereby the deviatoric stress is calculated from the integral of the history of the past intrinsic rate of rotation and rate of deformation tensors through a representation theorem. Piau has demonstrated the objectivity of a frame moving with a given particle whose axis are directed along the eigenvectors of the rate of deformation tensor. The use of such a framework provides a new approach in the attempt to reduce the computational difficulties associated with conventional constitutive equations written in co-deformational or co-rotational reference frames. The shear and primary normal-stress material functions and the extensional (elongational) stress growth function are defined for the proposed integral constitutive equation. These material functions are used to calculate the kernel functions using steady state, stress relaxation and stress growth data of Attané in simple shear flow for monodisperse polystyrene solutions. The shear and extensional stress growth data of Meissner for a polyethylene melt are also used to show the flexibility of the rheological model. The material functions are first written in terms of five monotonically decreasing functions of the time lag between the past and the present time. Then kernel functions are chosen such that when substituted in the new integral constitutive equation they yield the functions used to describe the data. A further condition imposed on the normalized kernel functions is that they be decreasing functions of time lag.
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• 6
Electronic Resource
Stamford, Conn. [u.a.] : Wiley-Blackwell
Polymer Engineering and Science 27 (1987), S. 1627-1633
ISSN: 0032-3888
Keywords: Chemistry ; Chemical Engineering
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Chemistry and Pharmacology , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
Notes: This second paper of a series continues the examination of the tensile properties of two series of linear low density polyethylene/polypropylene, (LLDPE/PP) blends. The blends were prepared using a twin-screw extruder and cover the whole concentration range, An Instron Universal Tensile Tester was used to measure the tensile properties of the blends between 10 and 70°C, and the temperature and composition dependences of the modulus were examined. A comparison is established between the solid state and melt properties to underline the behavior in the PP rich region. Results of dynamic mechanical experiments and differential scanning calorimetry on the same materials are also given, and the mechanical behavior is discussed in terms of the variation of the system's crystallinity.
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• 7
Electronic Resource
Stamford, Conn. [u.a.] : Wiley-Blackwell
Polymer Engineering and Science 29 (1989), S. 1265-1272
ISSN: 0032-3888
Keywords: Chemistry ; Chemical Engineering
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Chemistry and Pharmacology , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
Notes: Primary normal stress difference data of some 32 different polymer solutions have been examined and correlated to the shear stress by a simple power-law model. The normal stress at constant shear stress is shown to increase with the molecular weight of the polymer, but decreases with the polymer concentration. The method proposed by Abdel-Khalik, et al., for predicting the primary normal stress difference from viscosity data for these polymer solutions is inadequate. The FENE-P dumbbell model is used to obtain master curves combining normal stress and viscosity data as a function of a dimensionless shear rate. It is found that this presents several drawbacks. It predicts shear-thinning effects with a power-law slope of -2/3, which is found to be too restrictive. The model fails to predict shear-thinning effects for dimensionless shear rates λEγ lower than approximately 0.5. It also fails to predict satisfactorily the low shear rate value of the dimensionless combination of viscosity and first normal stress coefficient Ψ1/λE(η - ηs). An empirically modified correlation is proposed. The fit for a series of polyacrylamide (Dow Separan AP-30) solutions is very good. In the case of a series of monodisperse polystyrene solutions, the fit is not as good, as the data show a more pronounced effect of polymer concentration at high shear rates.


text:
How is Area Approximated Using Sums?

Problem 1 Approximate the area under $displaystyle f(x)=5x+2$ with 4 rectangles using a) right hand rectangles, b) left hand rectangles, and c) midpoint rectangles.

a) Right hand rectangles

b) Left hand rectangles
c) Midpoint Rectangles

Notice that each of these parts does essentially the same thing…multiply an f(x) value for the height of a rectangle by a width. Notationally, we can write this as

$displaystyle f({{x}_{i}})cdot Delta x$

where $displaystyle {{x}_{i}}$ is the x value where the rectangle touches the curve and $displaystyle Delta x$ is the width of the rectangle. So we can use summation notation for the sum,

$displaystyle sumlimits_{i=1}^{n}{f({{x}_{i}})cdot Delta x}$

where there are n rectangles. The sigma indicates that we are adding up terms with a particular format…in this case, function values time a width.


text:
Drake
RigidBodyLoop< T > Class Template Reference

Defines a "loop joint" that models a kinematic loop formed by a chain of rigid bodies and their regular joints. More...

#include <drake/attic/multibody/rigid_body_loop.h>

## Public Member Functions

RigidBodyLoop (std::shared_ptr< RigidBodyFrame< T >> frameA, std::shared_ptr< RigidBodyFrame< T >> frameB, const Eigen::Vector3d &axis)
Constructs a kinematic loop by fully constraining the origins and partially constraining the orientations of two frames. More...

## Public Attributes

const std::shared_ptr< RigidBodyFrame< T > > frameA_

const std::shared_ptr< RigidBodyFrame< T > > frameB_

const Eigen::Vector3d axis_

## Friends

std::ostream & operator<< (std::ostream &os, const RigidBodyLoop< double > &obj)

## Detailed Description

### template<typename T> class RigidBodyLoop< T >

Defines a "loop joint" that models a kinematic loop formed by a chain of rigid bodies and their regular joints.

The loop joint is specified by two RigidBodyFrame objects that must be attached to two different RigidBody objects. The coordinate frames defined by the two RigidBodyFrame objects are constrained to have the same origin. The orientations of the two frames are partially constrained based on the axis of rotation (i.e., the two frames are only allowed to rotate relative to each other along the axis of rotation).

Template Parameters
 T The type being integrated. Must be a valid Eigen scalar.

Instantiated templates for the following kinds of T's are provided:

• double
• AutoDiffXd

They are already available to link against in the containing library. No other values for T are currently supported.

## ◆ RigidBodyLoop()

 RigidBodyLoop ( std::shared_ptr< RigidBodyFrame< T >> frameA, std::shared_ptr< RigidBodyFrame< T >> frameB, const Eigen::Vector3d & axis )

Constructs a kinematic loop by fully constraining the origins and partially constraining the orientations of two frames.

The two frames are defined relative to two rigid bodies that should be connected via a loop joint.

Parameters
 [in] frameA A frame defined relative to the loop joint's "parent" rigid body. [in] frameB A frame defined relative to the loop joint's "child" rigid body. [in] axis The loop joint's axis of rotation expressed in the coordinate frame of frameA.

## ◆ operator<<

 std::ostream& operator<< ( std::ostream & os, const RigidBodyLoop< double > & obj )
friend

## ◆ axis_

 const Eigen::Vector3d axis_

## ◆ frameA_

 const std::shared_ptr > frameA_

## ◆ frameB_

 const std::shared_ptr > frameB_

The documentation for this class was generated from the following files:


text:
## Reading 6 – The Time Value of Money

The Time Value of Money is an important concept for the level 1 exam. You will need to be comfortable discounting the Present Value and Future Value of cash flows for individual and ongoing payments in the exam. Your calculator has functions built in to make these calculations easier. Here are the calculations that you need to know:
$$FV=PV{ \left( 1+\frac { r }{ n } \right) }^{ n\ast t }$$
$$PV=\frac { FV }{ { \left( 1+\frac { r }{ n } \right) }^{ n\ast t } }$$
$$FV=Future\quad value$$
$$PV=Present\quad Value$$
$$r = discount \quad rate$$
$$n = number \quad or\quad discounting\quad periods\quad per\quad year$$
$$t = number\quad of\quad years$$

A simple example of this application is if you needed to determine how much you would pay today to receive $1 in one year, assuming your required rate of return was 5%. The formula would look like this: $$PV=\frac { 1 }{ \left( 1+.05 \right) } =0.95$$ Using this discount rate,$1 one year from now is worth \$0.95 today.

You won’t often have to do the formulas written out by hand, but be familiar with them because the logic here applies to many other applications through this topic area. A similar issue that you’ll have to deal with involves ongoing cash flows such as annuities and perpetuities. Both involve finding the current value of a series of cash flows, just that an annuity continues for a certain period of time and a perpetuity never ends. The formulas for these are as follows:

$$PV\quad Annuity=\frac { \left( 1-{ V }_{ n } \right) }{ r }$$
$$r=rate\quad of\quad return$$
$$n=term \quad of \quad annuity$$
$$V={ \left( 1+r \right) }^{ -1 }$$
$$PV\quad Annuity\quad Due=\frac { \left( 1-{ V }_{ n } \right) }{ d }$$
$$d=\frac { r }{ \left( 1+r \right) }$$
$$PV\quad Perpetuity=\frac { C }{ r }$$
$$C\quad is\quad amount\quad of\quad each\quad payment$$

Your calculator has built-in functionality for PV and FV equations using the row of buttons labeled “N’, “I/Y”, “PV”, “PMT”, and “FV”. To use these for a problem, you input the value of all the parameters that you know and then have the calculator compute the missing value. To input a value, you type in the number for the value and then press the corresponding value’s button (i.e., to input 10,000 as the PV, you type in 10000, then press the “PV” button). Once you have input all figures you have, you press the CPT button and then the parameter button you want to calculate. Don’t forget to input the interest amount in decimals, so that 5% is entered as 0.05.

## Reading 7 – Discounted Cash Flow Applications

The Net Present Value (NPV) and Internal Rate of Return (IRR) methods are both ways of comparing projects with different payoffs to determine where a company should invest its resources. Both methods attempt to make project returns comparable so that a company can tell which one is the best investment.

$$NPV=\sum _{ }^{ }{ { C }_{ t }{ \left( 1+r \right) }^{ t } }$$
$${ C }_{ t }\quad is\quad the\quad cash\quad flow\quad at\quad time\quad t$$
$$r\quad is\quad the\quad discount\quad rate$$
The NPV of a project is the sum of all cash inflows, minus all cash outflows, discounted using a required rate of return so that the final figure is in today’s dollars. A positive NPV means a project will make money for the company, a negative NPV means a company will lose money, and a project with a higher NPV than another project is a more profitable investment.

The IRR is very similar to the NPV. It is the rate of return for a project that gives it an NPV of zero.

$$NPV=0\quad when\quad r=IRR$$

Expect to see problems where you are calculating these figures for a given project and problems where you are choosing a project based on these figures. For both NPV and IRR, higher values are preferred over lower ones. If you are asked to pick the best project and the two don’t give the same answer, go with the highest NPV. Small projects can have high IRRs but not provide as much value to the company, so the NPV is a better comparison of the overall scale of the return.

The calculator function for IRR and NPV works the same as the rest of the PV and FV. You input the values that you do have using the same value buttons as before and then use the CPT and NPV or IRR buttons to calculate those values.

There are several important types of returns you need to know: Holding Period Return (HPR), Money-Weighted Rate of Return (MWRR), and Time-Weighted Rate of Return (TWRR). The HRP is the simple return over the life of an investment, the MWRR is the IRR of a portfolio, and the TWRR is the geometric mean for a period of HPRs (ie for each year of an investment).

The HPR is the sum of the net gains of an investment divided by the beginning value:

$$HPR=(Ending\quad value–Beginning\quad value+Income)/Beginning\quad value$$

The MWRR is the discount rate at which: PV of cash outflows = PV of cash inflows

You solve MWRR the same way that you solve other IRR problems using your calculator. Since this method involves finding the PV of each cash flow, it is sensitive to when funds are invested or removed from the portfolio.

$$The\quad TWRR\quad formula\quad is:\left\{ { \left( 1+{ HPR }_{ 1 } \right) }^{ \ast }{ \left( 1+{ HPR }_{ 2 } \right) { \dots }^{ \ast } }\left( 1+{ HPR }_{ n } \right) \right\} -1$$

By combining each holding period return into a geometric mean, this method gives a more robust return figure that is not influenced by the timing of cash flows.

There are several ways to calculate a yield on a fixed income security that you need to know. One is the Bank Discount Yield, which is based on the trading price of the bond in comparison to its face value. The formula for this yield is:

$${ R }_{ BD }=\frac { D }{ F } \ast \frac { 360 }{ t }$$
$$D=face\quad value-price\left( also\quad known\quad as\quad the\quad Discount \right)$$
$$F=face \quad value$$
$$t=time\quad to\quad maturity\quad in\quad days$$
$$360\quad is\quad the\quad conventional\quad bank\quad days\quad in\quad a\quad year$$

Another method is the Holding Period Yield, which is the same as the Holding Period Return calculation from earlier in this section:

$$HPY=(Ending\quad value–Beginning\quad value+Income)/Beginning\quad value$$

The third method is the Effective Annual Yield (EAY). This is an annualized version of the HPY:

$$EAY={ \left( 1+HPY \right) }^{ \frac { 365 }{ t } }-1$$

## Reading 8 – Statistical Concepts and Market Returns

There are several methods of calculating the average value for a set of numbers as laid out in the curriculum. The formulas you need to know are:

$$Arithmetic\quad Mean=\frac { \Sigma { X }_{ i } }{ N }$$
$$Weighted\quad Mean=\sum _{ }^{ }{ { X }_{ i }{ W }_{ i } }$$
$$Geometric\quad Mean={ \left( { X }_{ 1 }\ast { X }_{ 2 }\ast \dots \ast { X }_{ n } \right) }^{ \frac { 1 }{ n } }$$
$$Harmonic\quad Mean=\frac { N }{ \sum _{ }^{ }{ \frac { 1 }{ { X }_{ i } } } }$$
$$N=number\quad of\quad observations$$
$$W=percentage\quad weight$$

Examples:

The Arithmetic Mean of the numbers $${ \left\{ 1,2,3,4 \right\} }$$ is: $$\frac { 1+2+3+4 }{ 4 } =2.5$$

The Weighted Mean of a portfolio with the following assets is:

$$\begin{array} {} & Stocks & Bonds \\ Portfolio\quad Weight & 60\% & 40\% \\ Return & 10\% & 6\% \\ \end{array}$$
$$\left( 0.6\ast 0.1 \right) +\left( 0.4\ast 0.6 \right) =0.8$$

The Geometric Mean of the numbers $$\left\{ { 1,2,3,4 } \right\}$$ is:
$${ \left( 1\ast 2\ast 3\ast 4 \right) }^{ \frac { 1 }{ 4 } }=2.21$$

The Harmonic Mean of the numbers $$\left\{ { 1,2,3,4 } \right\}$$ is:
$$\frac { 4 }{ \frac { 1 }{ 1 } +\frac { 1 }{ 2 } +\frac { 1 }{ 3 } +\frac { 1 }{ 4 } } =1.92$$

For the same set of numbers, it is always the case that:

Arithmetic Mean > Geometric Mean > Harmonic Mean

In addition to measures of average, there are several measurements of the distribution for datasets that you will need to know.

The Variance is the average of the squared deviations from the mean, and the formula to calculate this for a sample of data looks like this:

$${ \sigma }^{ 2 }=\frac { \Sigma { \left( { X }_{ i }-\bar { X } \right) }^{ 2 } }{ n-1 }$$
$$\bar { X }=Mean$$

The Standard Deviation is simply the square root of the variance.

The Mean Absolute Deviation is a measure of the average of the absolute values of deviations from the mean in a data set. We must use the absolute values because a sum of the deviations from the mean in a data set is always 0. This formula is as follows:

$$MAD=\frac { \Sigma |{ X }_{ i }-\bar { X } | }{ n-1 }$$

Now we’ll calculate these values for the following returns data:

$$\begin{array} \\ Year\quad 1 & Year\quad 2 & Year\quad 3 & Mean \\ 10\% & 7\% & 5\% & 7.3\% \\ \end{array}$$
$$Variance={ \sigma }^{ 2 }=\frac { { \left( 0.1-0.073 \right) }^{ 2 }+{ \left( 0.07-0.073 \right) }^{ 2 }+{ \left( 0.05-0.073 \right) }^{ 2 } }{ 2 } =0.00063$$
$$Standard\quad Deviation={ \sigma }=\sqrt { 0.00063 } =0.025$$
$$MAD=\frac { |0.1-0.073|+|0.07-0.073|+|0.05-0.073| }{ 2 } =0.027$$
Another important formula related to the previous metrics is the Sharpe Ratio. It measures the risk-adjusted returns of a portfolio. The higher the Sharpe Ratio of a portfolio, the more excess return you are getting for each additional unit of risk. This formula shows up repeatedly in all levels of the CFA curriculum, so it’s important to know it well.

$$Sharpe\quad Ratio=\frac { \left( { r }_{ p }-{ r }_{ f } \right) }{ { \sigma }_{ p } }$$
$${ r }_{ p }=portfolio \quad return$$
$${ r }_{ f }=risk-free\quad rate\quad of\quad return$$
$${ \sigma }_{ p }=standard\quad deviation\quad of\quad portfolio\quad returns$$

When looking at data sets, there are different ways to categorize the distribution of the data. Two of these methods that you need to know are Kurtosis and Skewness. Kurtosis refers to the degree to which a data set is distributed, relative to the Normal Distribution. There are three categories of kurtosis, as illustrated in the graph below:

A data set would be described as Mesokurtic if it closely resembled a normal distribution, Leptokurtic if it was more tightly clustered around the mean, and Platykurtic if it is more widely distributed. The kurtosis of a normal distribution has a value of 3, so the excess Kurtosis is calculated as:

$$Excess\quad Kurtosis=\left( \frac { \Sigma { \left( { X }_{ i }-X \right) }^{ 4 } }{ { s }^{ 4 } } \ast \frac { 1 }{ n } \right) -3$$
$$s=standard\quad deviation$$
$$n=number\quad of\quad observations$$

Skewness is another way to describe the distribution of a data set. It refers to how symmetrical a data’s distribution is relative to a completely symmetrical normal distribution. A data set with negative skew has more values to the extreme left side of the distribution, and positive skew indicates the same occurring to the right side of the scale. Either of these conditions would be referred to as the distribution having “fat” tails.

$$Skewness=\frac { \Sigma { \left( { X }_{ i }-X \right) }^{ 3 } }{ { s }^{ 3 } } \ast \frac { 1 }{ n }$$

## Reading 9 – Probability Concepts

There are several concepts involving calculating probabilities you need to know. Most of these involve determining the probability of multiple events occurring, given the probability of each individual event.

The Multiplication Rule applies to the joint probability of multiple events occurring. The formula is as follows:

$$P\left( AB \right) =P\left( A|B \right) P\left( B \right)$$

The joint probability of both events happening (P(AB)), is the conditional probability of A, assuming B occurs (P(A|B)), times the probability of B (P(B)). Since the probability of both events happening is less than the odds of either one individually, the joint probability will always be lower than either input.

The Addition Rule applies to the probability of multiple events occurring, i.e., A and/or B occurs. That formula is:

$$P\left( AorB \right) =P\left( A \right) +P\left( B \right) -P\left( AB \right)$$

Since both P(A) and P(B) include the possibility of both events occurring, we subtract the probability of both so that it’s not double-counted.

The Total Probability rule is similar to the addition rule but deals with the combination of conditional probabilities to create the entire probability of an event occurring. The sum of all conditional probabilities for an event adds up to the non-conditional probability of that event, if the conditions are mutually exclusive.

Two important concepts related to statistics in portfolio management are Correlation and Covariance. Correlation refers to the ratio of covariance between two variables and the product of their standard deviations. Covariance is the degree to which two variables move in sync with one another.

The formula to calculate the covariance of two assets (a and b) in a portfolio appears as follows.

$${ \sigma }_{ { R }_{ a },{ R }_{ b } }=\sum _{ }^{ }{ P\left( { R }_{ a } \right) \left[ { R }_{ a }-E\left( { R }_{ a } \right) \right] } \left[ { R }_{ b }-E\left( { R }_{ b } \right) \right]$$

It is important to understand the nature of covariance, but you should not have to calculate this formula by hand on the exam. You should expect, however, to be calculating correlations, which use the following formula:

$$Correlation\left( { R }_{ a },{ R }_{ b } \right) =\frac { Covariance\left( { R }_{ a },{ R }_{ b } \right) }{ StdDev\left( { R }_{ a } \right) \ast StdDev\left( { R }_{ b } \right) }$$

Be careful in exam questions involving these equations, because sometimes you will be given the portfolio risk expressed as the Standard Deviation or as the Variance. Don’t forget that the Standard Deviation is the square root of the Variance.

The Correlation value ranges between -1 and 1.

• 1 means that the two variables move in sync
• 0 means that they have no relationship
• -1 means that they move exactly opposite

Bayes’ Formula is a method of calculating an updated probability of an event occurring, given a set of prior probabilities.

$$P\left( { E }_{ i }|A \right) =\frac { P\left( { E }_{ i } \right) P{ \left( A|{ E } \right) }_{ i } }{ \Sigma P\left( { E }_{ j } \right) P\left( { A|E }_{ j } \right) }$$
$$P\left( { E }_{ j } \right)=prior\quad probabilities$$
$$A=an\quad event\quad known\quad to\quad have\quad occurred$$
$$P\left( { E }_{ i }|A \right)=posterior\quad probability$$

## Reading 10 – Common Probability Distributions

There are several types of distributions you will need to understand, and this includes understanding a few common kinds of randomly distributed variables.

A Discrete Uniform Random Variable is one where the probabilities of all outcomes are equal, like the roll of a die. A Bernoulli Random Variable is one that can have only two outcomes. Similarly, a Binomial Random Variable is the number of specific outcomes in a Bernoulli situation that are independent of each other, such as the results of flipping a coin.

One way to calculate potential outcomes using a Bernoulli approach is known as a Binomial Tree:

In this case, each final outcome is seen as successive iterations of Bernoulli events. At each point in time, the stock goes up or down. The tree lists all possible outcomes for this stock, and the probability of each final outcome is the multiplied product of the probability at each step along the way. A good way to catch potential errors when calculating the values of a binomial tree is that each column (a specific point in time) should add up to a probability of 100%.

The curriculum specifies several methods of tracking and managing risk exposures using quantitative means. An important one is known as Roy’s Safety-First Ratio. This is based on the principle that an optimal portfolio is one that minimizes the probability of a returning less than a given threshold level. Achieving a return less than a specified required return is known as Shortfall Risk. The Safety First Ratio is expressed as the excess return of the portfolio (expected returns – threshold returns) divided by portfolio risk (standard deviation):

$$SF\quad Ratio=\frac { E\left( { R }_{ p } \right) -{ R }_{ L } }{ { \sigma }_{ p } }$$

Computing technology has introduced new methods of performance forecasting, including Monte Carlo Simulation. This method involves building a model of the portfolio and potential outcomes, then running a simulation of what could happen over time (factoring in all available probabilities and covariances of related variables) thousands of times. The output of this method is a distribution of the potential outcomes far more detailed than what a simple linear model could project. This is compared to the other common technique of Historical Simulation. In that approach, a projected portfolio is run through actual historical situations to see how it would have performed. This approach gives a much more limited set of outcomes than the Monte Carlo, but has the benefit of being based on observed market events rather than projected simulations.

## Reading 11 – Sampling and Estimation

A core tenet of statistics is using samples of data to find information about the entire population of a data set. It’s often impossible to gather all of the data of a population, so we rely on representative samples and try to make sure they properly represent the population from which they are drawn (like polls given to small groups of people to represent a whole state or country). One major assumption that underpins the validity of this approach is the Central Limit Theorem. This posits that a data sample will have a mean and variance that approach the mean and variance of the population it represents as the sample size becomes sufficiently large. It is widely accepted that a sample of size of at least 30 is usually enough to make the sample representative, but that can vary depending on the skewness of the sample distribution.

A common way (that will definitely appear multiple times on the exam) of analyzing the probability distribution of a data sample is to use the Student’s T-Distribution. This distribution is appropriate for small samples when the population variance is unknown. The formula for calculating a t-statistic is:

$$t=\frac { x-\mu }{ \frac { S }{ \sqrt { n } } }$$
$$x = sample \quad mean$$
$$\mu =population\quad mean$$
$$S=sample \quad standard \quad deviation$$
$$n= sample \quad size$$

The t-distribution has an important parameter known as the Degrees of Freedom. The degrees of freedom is calculated as n-1. The t-distribution is symmetrical about its mean and has thicker tails than the normal distribution. The shape of the distribution is dependent on the number of degrees of freedom. As the number of degrees of freedom increases, the distribution becomes more closely gathered around the mean.

A big part of the statistics part of the curriculum deals with the use of Confidence Intervals. A confidence interval is the range of values in which a statistician believes that a certain population parameter lies. They incorporate a trade-off of value range and confidence, such that the smaller the area in which the value is believed to exist, the lower the confidence that it is actually there. There are three primary scenarios for calculating the confidence intervals in the CFA curriculum.

1. A normal distribution with a known variance:

$$CI=X\pm { z }_{ \frac { \alpha }{ 2 } }\ast \frac { \sigma }{ \sqrt { n } }$$

2. A normal distribution with unknown variance:

$$CI=X\pm { t }_{ \frac { \alpha }{ 2 } }\ast \frac { S }{ \sqrt { n } }$$

3. A normal distribution with unknown variance and a sample size is large enough:

$$CI=X\pm { z }_{ \frac { \alpha }{ 2 } }\ast \frac { S }{ \sqrt { n } }$$

When the population variance is known, we can use the more accurate z-score to determine the confidence interval instead of the student’s t-score, but the Central Limit Theorem allows us to also use the z-score when the sample size is large enough.

## Reading 12 – Hypothesis Testing

Hypothesis testing is a method of determining the statistical significance of a given data point in a distribution. There are One-Tailed Tests that test the possibility of a change in one direction, and Two-Tailed Tests regarding changes in both positive and negative directions. This method uses calculated test statistics to determine whether a given hypothesis should be accepted or rejected.

The test statistic for the hypothesis testis is calculated as follows:

$$Test\quad statistic=\frac { Sample\quad statistic-Hypothesized\quad value }{ Standard\quad error\quad of\quad the\quad sample\quad statistic }$$

There are two types of errors that could occur when running a hypothesis test. Type 1 error occurs when we reject a true null hypothesis and Type 2 error occurs when we fail to reject a false null hypothesis. The level of significance (α) represents the probability of making a type 1 error.

Another important component of hypothesis testing is the P-Value. If the calculated P-value is lower than the level of significance, then we can reject the null hypothesis.

The correct hypothesis test for a normal distribution of the random variable is the Z-test, which is calculated as:

$$z-statistic=\frac { X-{ \mu }_{ 0 } }{ \frac { \sigma }{ \sqrt { n } } }$$
$$X=sample\quad mean$$
$${ \sigma }_{ 0 }=Hypothesized\quad mean\quad of\quad the\quad population$$
$${ \mu }=standard\quad deviation\quad of\quad the\quad population$$
$$n= sample \quad size$$

For hypothesis testing a population mean when the population variance is unknown and the sample size is large, a T-test is more appropriate:

$${ t }_{ n-1 }=\frac { X-{ \mu }_{ 0 } }{ \frac { s }{ \sqrt { n } } }$$
$$X=sample\quad mean$$
$${ \sigma }_{ 0 }=Hypothesized\quad mean\quad of\quad the\quad population$$
$${ s }=standard\quad deviation\quad of\quad the\quad sample$$
$$n= sample \quad size$$

Your testing materials will include tables of significant values for t- and z-scores, so that you can compare to them in determining whether a calculated value exceeds the level necessary to reject a null hypothesis.

There are two classifications of statistical tests that you will need to know, Parametric and Non-Parametric. Parametric tests involve any testing where the given parameter follows a specific distribution. Non-parametric testing does not involve making any assumptions about the distribution of the parameter being studied. An instance where this could be necessary is when there are significant outliers that could cause the mean value of a dataset to be unrepresentative of the data. For this reason, if you wanted to run a test focused on the median of a dataset instead of the mean you would run a non-parametric test.

## Reading 13 – Technical Analysis

Technical analysis is the practice of using price and volume data to value stocks. There are a number of concepts you’ll be expected to understand related to technical analysis on the exam. One of these is Trend, which is a when a security consistently keeps moving upward or downward over time. This is the market’s way of reflecting a difference between supply and demand that can persist for a while.

Another important concept is that is Support and Resistance levels. These are price points where market activity tends to keep a security from going below (Support) or above (Resistance). These levels can act as floors or ceilings to the trading level of a security.

Related to the previous concept is that of the Change in Polarity. This is the principle that when a Support or Resistance level is breached, it then becomes its opposite. This can occur when a Resistance level is exceeded and that price point becomes a new Support for trading activity at the new higher level.

In the field of technical analysis, there are a number of patterns that appear on stock charts reflecting price movements that can be used when trying to forecast future movements in stock price. These chart patterns fall into a few categories. Reversal Patterns indicate that a trend is likely to reverse from the direction it has been going. A common example is the “Head and Shoulders” pattern that usually signals the end of an uptrend.

Another category of chart pattern is the Continuation Pattern. These patterns are used to identify trends that will continue after a pause. Many common continuation patterns are based on triangles that highlight trading movement that is converging to a given point in the future.

In addition to chart patterns, there are also indicators used in technical analysis that are used to predict future changes in a security’s price. The four categories of indicators are:

### 1. Price-Based Indicators

These indicators use information about the security’s past and current price to try and predict future prices. A simple example of this is the moving average, which is the average price of a security over a number of previous time periods. This average removes the oldest observation every time a new observation is added. Bollinger Bands are another common indicator based on price. These are lines that indicate the standard deviations of prices compared to a moving average of the stock’s price.

### 2. Momentum Oscillators

These are values calculated by finding the difference between the most recent price of a security and a previous price on a specified day in the past. Watching for the calculated value to switch from positive to negative is one indicator of a potential trend reversal

$$M=\left( V-{ V }_{ x } \right) \ast 100$$
$$V=most\quad recent\quad closing\quad price$$
$${ V }_{ x }=closing\quad price\quad at\quad specified\quad date\quad in\quad past$$

### 3. Sentiment Indicators

These are various polls that are conducted to gauge the sentiment of individual investors or investment professionals about the state of the equity market.

### 4. Flow-of-Funds Indicators

These measure the movement of money into and out of rising and declining stocks to identify upward or downward trending behavior. One of the most useful here is the Arms Index:

$$Arms\quad Index=\frac { \frac { Number\quad of\quad advancing\quad issues }{ Number\quad of\quad declinging\quad issues } }{ \frac { Volume\quad of\quad advancing\quad issues }{ Volume\quad of\quad declinging\quad issues } }$$

Values above 1 indicate more volume in declining stocks, while values below 1 indicate more activity in rising stocks.

Another category in the field of technical analysis used to forecast market price movements is Cycles. These are patterns of market movement that reoccur over various time frames. There are cycles that can happen multiple times throughout a trading day or take many years to complete one occurrence.

Related to the analysis of cycles is the Elliott Wave Theory, developed in the late 1930s. This theory posits that there are repetitive and predictable cycles that can be observed in stock price movements. The theory dictates that there is a pattern of cycles consisting of activity waves. A cycle will contain 5 waves moving in the direction of the main market trend and then 3 waves moving in the opposite direction. The size of each wave is related to the one preceding it based on the Fibonacci number sequence.

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text:
Probability of conditional inequality for a random variable

If a positive integer $n$ is picked at random from the positive integers less than or equal to $10$, what is the probability that $5n+3\leq14$?

-

$5n + 3 \leq 14$
Subtract 3 from both sides.
$5n \leq 11$
Divide both sides by $5$.
$n \leq 11/5$
$n \leq 2.2$
Since $n$ has to be a positive integer less than or equal to $10$, there are 10 possible number numbers which n could be. ($n \in \{1,2,3,4,5,6,7,8,9,10\}$).
From the inequality, we can see that $n$ can only be $1$ or $2$. Since only $2$ out of $10$ numbers can fulfill this inequality, the probability that an appropriate $n$ is chosen is $\frac{2}{10} = \frac{1}{5} = .20 = 20\%$

-
 Thank you very much for helping – cary Sep 12 '12 at 21:33 This of course assumes select at random means n has a uniform distribution on the integers from 1 to n. – Michael Chernick Sep 13 '12 at 2:38

Since $n\in\{1,2,3,4,5,6,7,8,9,10\}\implies (5n+3)\in \{8,13,18,23,28,33,38,43,48,53\}$

Now Since $5n+3$ takes only two values out of ten which are less than or equal to $14$

$\implies P(5n+3\leq 14)=\frac{2}{10}=0.2$ (Assuming each number from $1$ to $10$ is equally likely to be chosen)

-


text:
## anonymous 5 years ago A metal beam with Y as the young's modulus is supported at the two ends and loaded at the center. The depression is directly proportional to: a-Y b-Y^2 c-1/Y

1. JamesJ

If the mass were on the end of a wire of this metal, what would say the answer is, out of a,b,c?

2. anonymous

c? I am lost at this question

3. JamesJ

What's the definition of Young's Modulus?

4. JamesJ

Go back and understand what it means. Then answer my question about the wire. Then think again about the beam. ==== This lecture might be helpful, if a bit advanced: http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/lecture-26/

5. anonymous

I studied the topic again, I never get this for some reason. But I got the answer for your question If the mass were on the end of a wire of this metal, what would say the answer is, out of a,b,c? it would then be Y

6. JamesJ

$\frac{F}{A} = Y \frac{\Delta \ell}{\ell}$ So for a given force and length, $$Y \Delta \ell = constant$$. Hence $$\Delta \ell$$ is inversely proportional to $$Y$$. Therefore with the wire alone, the answer will be c) 1/Y. The same principle applies with the beam. Here the cross sectional area is a bit more complicated and related to the length of the beam, but the basic physical principles are these same.

7. JamesJ

Intuitively, this also makes sense. Y is a measure of 'stiffness'. Y of iron is higher than that of copper say. You'd expect the copper beam to bend more, consistent with the fact that 1/Y for copper > 1/Y for iron as $Y_{iron} > Y_{copper}$

8. anonymous

9. JamesJ

I don't know what that is. What is this formula and what are all the variables?

10. anonymous

e=depression M=mass of load g=gravitational acceleration l=length of beam b is the breadth of the beam d is the thickness of the beam Y=modulus

11. JamesJ

Oh, you have a formula for it. Why didn't you tell me?

12. JamesJ

Anyway, you can see immediately that E is proportional to 1/Y. So the answer remains the same.

13. anonymous

yeah, but how do we get this formula?

14. anonymous

anyways, I like you answer better, sweet and simple.

15. JamesJ

I'm not sure, but it'll come from manipulation of the basic formula for Young's modulus one way or another.

16. anonymous

or experiment, I found this: http://amrita.vlab.co.in/?sub=1&brch=74&sim=550&cnt=1

17. JamesJ

ok