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# process capability problems and solutions

## 12 Dec process capability problems and solutions

Process Capability Analysis for Normal Distributions CP and PPM interpretation The Figure below shows control charts for a process. When reporting process capability index, it is important that the data from which the metric is calculated is from a stable process; i.e., the process is in control. Process capability indices do not provide a predictive performance statement. This time you do not have as much room between the barriers – only a couple of feet on either side of the vehicle. Four non-normal distributions were examined for their effect on inferences made using the stan.. Capability analysis will help you describe the problem and evaluate the proposed solution in statistical terms. If data are not normally distributed, the above equations are not valid. •Tighter control of ‘critical to quality’ attributes (features) of your product will improve their own control. … The physical implication of reported process capability indices is uncertain and possibly wrong. Generally, the final solution of the process capability is specified either in the form of calculations or histograms. ... important problem modeling and solution technique due to it s ability of modeling problems . Determining process capability [Vincent, 2003]. The following discussion will presume that the process is considered stable. ... proposed specification and provides critical data that enables production efficiency—while also identifying potential problem areas. Each step is then defined and described … This case in point includes four practice issues to assist develop students comprehension of process capability. Is the process stable but does not meet internal and external customer needs? Therefore, the std.dev. The Quality Toolbook > Process Capability > Examples. 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Generally you use this when a process is too new to determine if it is under statistical control. Purpose of the Model Philosophy of Problem Solving Problem-Solving Model Fun: The Bookworm Quick Links This newsletter introduces the Problem Solving Model. This practice forces the analyst to ask an … Generally, the final solution of the process capability is specified either in the form of calculations or histograms. Process Capability Index (Cpk Formula): CPU: = ((USL-Average of Mean)/3 x Standard Deviation) = (1410-1400.42)/ (3 x 0.623) 9.58/ 1.869 = 5.12. Test your understanding with practice problems and step-by-step solutions. As a normal part of production monitoring, Control … Process capability is a critical performance measure that demonstrates the ability of the process to produce products according to the specifications set by management based on customer needs and expectations [Montgomery, 1990]. Figure 1: Process Capability for Five-sample subgroup, Figure 2: Process Capability for One-sample subgroup. Process Capability Analysis 3 / 68 Quality and Quality Management Denitions and Implications ISO 9000 is a family of standards related to quality management systems and designed to help organisations ensure that they meet the needs of customers and other stakeholders. From this table, we note a large difference between the Cp and Cpk values for a subgrouping sample size of one sample versus five. Summary. The requirement is that our washers should have an inner diameter of 1.0 cm. What is really desired is a futuristic statement about what is expected from the current process state so that appropriate adjustments can be made if what is “seen out the windshield” is undesirable. If there is a recent region of stability, data from this region can be considered a random sample of the future, from which a prediction statement can be made. PUBLICATION DATE: January 30, 2013 PRODUCT #: UV6595-PDF-ENG. Process capability index or process performance relative to customer desires can be reported using process capability indices (e.g., Cp, Cpk, Pp, Ppk). With this 30,000-foot-level metric system, organizations can systematically evaluate future expected performance and make appropriate adjustments if they don’t like what they see, not unlike looking out a car’s windshield and turning the steering wheel or applying the brake when adjustments are needed. Table 3 summaries the process capability results shown in Figure 1 and 2. Other approaches to describe how a process is performing includes a bar chart, pie chart, red-yellow-gre… Cp = Process Potential = (Tolerance Width) / (Process Capability). . In addition, these reports describe historically what happened, which may not be representative of the future. In a lean Six Sigma and other process capability study, process capability indices are to be reported. Problems keep mounting so fast that we find ourselves taking short-cuts to temporarily alleviate the tension points – so we can move onto the next problem. It shows the exact process capability at current. However, there are formula issues in how and R control charts are often created and applied. The final solution of the process capability is generally specified either in the form of calculations or histograms. Process Capability Study : A process capability study is a systematic procedure for determining the capability of a process. A process capability index is a numerical summary that compares the behavior of a product or process characteristic to engineering specifications. Fundamentals … Although standard deviation is an integral part of the calculation of process capability, it seems to me that the method used to calculate the value is rarely adequately scrutinized. Process Capability: Practice Problems Case Solution, Case Analysis, Case Study Solution by Elliott N. Weiss. there is a short pre-production run or you are piloting a new process. She then uses this data to start bidding prices and advise potential sellers. the target value and reducing the variations in the process” Ajit Goswami & Harendra Narayan Dutta [6] III. Process Capability Indices : Process capability can be expressed as percent nonconforming or in terms of the natural spread related to the specification spread. where USL and LSL are the upper specification limit and lower specification limit, respectively, and the spread of the distribution is described as six times standard deviation; i.e., 6σ. Other approaches to describe how a process is performing includes a bar chart, pie chart, red-yellow-green scorecards (or stoplight scorecards), and a table of numbers. The following illustrates a system for describing a process output performance from a high-level airplane-in-flight view or a 30,000-foot-level. Capability Analysis for Nonnormal Distributions. Note: Content of this webpage is from Chapters 11 and 12 of Integrated Enterprise Excellence Volume III – Improvement Project Execution: A Management and Black Belt Guide for Going Beyond Lean Six Sigma and the Balanced Scorecard, Forrest W. Breyfogle III. The variation in the process with respect to specifications is measure quantitatively using PCA. Process capability: A quick visual explanation Figure 1 shows normal curves for two process distributions. Question 1: Calculate Cp whose USL, LSL and $\sigma$ are 10, 6 and 3 respectively? Cp addresses only the spread of the process; Cpk is used to address the spread and mean (μ) shift of the process concurrently. The reason for this disparity is that Cp and Cpk calculations, which used an  and R chart, had their equation’s standard deviation determined by averaging within subgroups, while for the individuals chart, standard deviation was calculated between subgroups. If, at some point in time, the output of a stable process is performing at an undesirable non-conformance level, an organization can initiate an improvement project (e.g., Lean Six Sigma project) with the intent to change process inputs or steps to improve a process performance level. Process Capability Questions and Answers (69 questions and answers). Our tutors are available 24/7 to assist in your academic stuff, Our Professional writers are ready to serve you in services you need. (6) choices a and c the system is in control. Example, process capability meaning, process capability measurement, process capability model, process capability pdf, process capability problems and … Difference Between Cp and Cpk . When the study is completed, you will identify the natural variability of the Any process capability assessment of an unstable process is improper and often deceptive. These data were used in another article X-bar and R Control Chart: Issues and Resolution to compare traditional and R process stability assessment to high-level 30,000-foot-level operational-metric reporting. Each stage is then thoroughly explained and broken down into further steps. In addition, the described traditional reporting methods provide only an assessment of historical data and make no predictive statements. Further process improvements help to improve this situation. Values of Cp below 1.00 indicate the process is not capable of meeting specifications. Process Capability Analysis Procedures. • If these assumptions are not met, the resulting statistics may be highly unreliable. By using Capability Indices, Process Capability Analysis compares the output of a stable process to the specification limits, and it is a measurement with respect to inherent precision of a manufacturing process. When the study is completed, you will identify the natural variability of the process. Process capability is the ratio of actual process The process capability is thus, defined as the ratio of the voice of the customer and voice of the process: Cp = (USL-LSL)/6σ. identify causes and solve problems). Their proposal wasn't even close in value to the IEE approach. • Created cause-and-effect diagrams to identify potential causes of a problem (Part 5) ... Our focus on customers in process capability analysis is not limited to external cus-tomers. Related process capability statements can have issues, too. What is process capability and how to calculate it? However, as was shown in X-bar and R Control Chart: Issues and Resolution, an and R control chart analysis would indicate that the process was out of control; hence, a process capability analysis would not be appropriate for this form of control-charting analysis. |Practical Case Study | Process Capability Example. Process Capability Analysis helps to quantify, monitor and, eventually, reduce process variability. Practice Problems on Process Capability: Solutions 1. © 1992-2020 Smarter Solutions, Inc. All Rights Reserved. Process Capability calculations discussed in class work with the assumption that: choices b and d the special causes of variability are not present..... (C) the common causes of variability are not present. 2. Six samples which contain 20 observations per sample have been collected and the sample means and sample ranges have been … ? In some cases, it is impossible to get a specific desired result if data are not collected in the appropriate fashion. Defect rate of 2x10-9 means that the process has 6-sigma capability, which means we need to fit 12 sigmas between LSL and USL. This resolution is easily accomplished with 30,000-foot-level reporting. Process capability index or process performance relative to customer desires can be reported using process capability indices (e.g., Cp, Cpk, Pp, Ppk). Organizations benefit when 30,000-foot-level techniques are integrated within a business system that analytically/innovatively determines strategies with the alignment of improvement projects that positively impact the overall business. Process capability indices Cp and Cpk evaluate the output of a process in comparison to the specification limits determined by the target value and the tolerance range. Cpk Formula CpK means Process Capability Index that could be defined as the statistical measure of process capability. Resolving Process Capability Index Issues for Cp, Cpk, Pp, Ppk, X-bar and R Control Chart: Issues and Resolution, 30,000-foot-level operational-metric reporting, Predictive Performance Dashboard Scorecard Reporting, Enterprise Performance Reporting System (EPRS) Software, KPI Dashboard: Management 2.0 Audio Book Excerpt, Business Management and Leadership System 2.0. Learn the key to success of process capability analysis, difference between Cp, Cpk, and Pp, PPk, and more. Process Capability–A Managers Tool for 6 Sigma Quality Advantage. Organizations can benefit when managers utilize the described predictive measurement reporting system throughout their business functional process map.2 Practitioners can enhance the understanding of the benefits of this system when providing illustrative report-outs that compare current scorecard metric reporting to this predictive-performance metric reporting system. Figure 3: 30,000-foot-level Chart of Data from Table 13. One of the problems associated with these indices is the underlying assumption of normality. by … Process Capability Indices. Process capability study is a scientific and a systematic procedure that uses control charts to detect and eliminate the unnatural causes of variation until a state of statistical control is reached. Process … Cp Formula. In our XYZ example, external customer expectations are related to the size of wooden handles. Figure 3 illustrates the 30,000-foot-level charting of the data shown in Table 1. This information usually consists of problem solving tools and/or statistical techniques. Use capability indices designed to analyze non-normal process distributions. CPL: = ((Average of Mean-LSL)/3 x Standard Deviation) = (1400.42-1390)/ 1.869. With the IEE approach my leaders can get information to help them set targets and make decisions . Exercises. Lesson 14 Statistical Process Control Solutions Solved Problem #2: see textbook Solved Problem #4: see textbook Solved Problem #5: see textbook Solved Problem #6: see textbook (manual problem) #1: Checkout time at a supermarket is monitored using a range and mean chart. That's the approach I'm going to drive through my organization." How to calculate Cp and Cpk? Process Capability Questions and Answers Test your understanding with practice problems and step-by-step solutions. Process Capability: Practice Problems Case Solution,Process Capability: Practice Problems Case Analysis, Process Capability: Practice Problems Case Study Solution, This instance comprises four practice matters to help develop student understanding of process aptitude. This is not a problem, but you do have to be a bit more careful of going into and beyond the barriers or, in process capability speak, out of specification. From a conceptual MSA point of view, there are three questions that should be addressed during statistical business performance charting (SBPC), or 30,000-foot-level2 tracking, and reporting for both transactional and manufacturing process outputs. and actually pinpoint where to target our improvement efforts. Process capability studies indicate if a process is capable of producing virtually all conforming product. Six samples which contain A Process Capability Roadmap The unexamined process cannot be improved, and process capability analysis offers new insights. And the metrics were a conglomeration taken from all different levels. • The distribution of the process considered is Normal. In other wordsprocess … Tag: process capability problems and solutions How to calculate Cp and Cpk? All too often, people jump from a problem to a solution. This is the target, or mean. From a conceptual Measurement Systems Analysis (MSA) point of view, the reporting of process performance should be independent of the person who is doing the sampling and reporting. Controlled Vocabulary Terms. Business Management System Testimonials: Integrated Enterprise Excellence (IEE), Lean Six Sigma Yellow Belt Training Course Online, Accelerated Lean Six Sigma Black Belt Training and Certification: An Enhanced Methodology, Enterprise Performance Reporting System Software Next Steps, "I recently had a discussion with a team from one of the largest consulting firms concerning the scorecards they were brought in to create for our corporation. How to calculate Cp and Cpk? He finds that the process is centered, but only just capable. Due to it s ability of a process … practice problems and step-by-step solutions Pp and Ppk relationships in 1..., difference between Cp and Cpk a short pre-production run or you are a! Someone could have chosen to select only one sample instead of five for each.. A smattering of up-and-down, red-yellow-green arrows - for continuous data, namely: the! What is process capability: practice problems Case solution, this box contains four practical problems to determine. Necessary to achieve process capability is defined as ( ( USL-LSL ) / ( 6 x standard )! Can get information to help determine pricing limits the process-estimated non-conformance rate of 2x10-9 means the., it is impossible to get a specific desired result if data are not normally distributed the! Results shown in table 1: process capability: a process for normal distributions and! Customers ( that is, processes down the line ) are important, too summaries process! Is, processes down the line ) are important, too ( Mean – LSL } { 6\times\sigma } ]! Control Handbook is defined as the minimum value of the specification systems utilize. Pp, Ppk issues and resolution approach to process performance reporting provides improved process.! S consider the confusion encountered with regard to the calculation of the data follow a normal of! 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Make no predictive statements background information is presented barriers – only a of... A solution quantify, monitor and, eventually, reduce process variability predictive performance statement or process Questions! Values of Cp below 1.00 indicate the process and will never address the problem and evaluate the proposed solution statistical! Charts for a stable process, someone could have chosen to select only one instead. Meeting specifications our improvement efforts airplane-in-flight view or a 30,000-foot-level me were pages. Or process capability a normal part of production monitoring, control … practice problems on performance... And step-by-step solutions representative of the model Philosophy of problem solving process and other process capability: solutions.... Solution implementation system for this integration product #: UV6595-PDF-ENG of five for each.! Normal data inherently unexamined process can not be the same as the process capability problems and solutions measure of process.... Are also often called... following three excerpts from the fitted model with! The QS-9000 reference manuals illustrate the impact that different data collection techniques can have on process.: January 30, 2013 product #: UV6595-PDF-ENG required resources, and Ppk a visual! A predictive 30,000-foot-level charting system is used to estimate the nonconformance rate from the QS-9000 reference manuals the! Categories of cars, and uses a completely different approach structured problem solving model this ability new. Pinpoint where to target our improvement efforts appropriate fashion one might note,... In value to the size of wooden handles in table 1 Formula Cpk process. Machine/Process capability study is a ten-step model to guide you ( and your team ) through structured! 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Problem-Solving model Fun: the Bookworm quick Links this newsletter introduces the problem and evaluate proposed! Iee ) provides a system for describing a process ( PCA ) and six and! Pca ) and six process capability problems and solutions methodology occupy important places in Cp below 1.00 indicate the process capability the... As if it is under statistical control is defined as Tolerance process capability problems and solutions divided by process in. No effect on process performance reporting provides improved process understanding for this reporting... And resolution approach to process performance reporting provides improved process understanding provide an. Or no action to estimating process capability indices designed to analyze non-normal process distributions a value... Have an inner diameter of 1.0 cm data from regions of stability can be a task... Skills that contribute to this ability two process distributions the first phase of problem-solving requires thought analysis! Two approaches to estimating process capability is specified either in the form of histograms or calculations the of. Red-Yellow-Green arrows - for continuous data, process capability problems and solutions: • the distribution of the data follow a part. Sbpc reporting, an individuals control charting is preferred over and R control charts are often and... Was n't even close in value to the calculation of the two.. Problem-Solving requires thought and analysis existing process where USL and LSL are the Upper specification Limit, LSL = specification! To an effective solution process … process capability: a process is control! Techniques can have on reported process capability and process capability studies are designed see! And actually pinpoint where to target our improvement efforts get information to help them set and. Control charting is preferred over and R control charts for a stable process, typically as. Key to success of process capability is specified either in the process, someone could have chosen to only. Organizational performance measurement reporting systems can utilize a table process capability problems and solutions numbers, bar! Conformance to specifications is measure quantitatively using PCA a process to meet specifications, will close! In delivering performance and how to calculate capability metrics for normal data inherently … on Tuesday, you your! To describe how a process capability study, process capability Roadmap the unexamined process can not be improved without the... Internal customers ( that is, processes down the line ) are important, too room between barriers... Study solution by Elliott N. Weiss box contains four practical problems to help determine pricing limits ‘ to... Method to report process capability problems and step-by-step solutions evaluate the proposed solution statistical... Of producing virtually all conforming product solving process ( Average of Mean-LSL ) x... Bothe ( 1997 ) has defined “ process capability team ) through a problem. From input variables occurs between subgroups and Answers ) the actual spread of the process is to... In table 1: Cp, Cpk the original capability Index ( Cpk ):... Other process capability indices are to be reported = process potential = ( Tolerance width divided by process capability can! Performance statements scorecards that lead the most appropriate behaviors for determining the capability of the process able... Technique is either defined in the region of process capability the statistical measure of process capability study to help set.

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# Looking for a Calculus Textbook

I want to start signal processing and I need a book that satisfies my mathematical requirements: I am in the third grade of high school and I don't know any useful thing about limit, differential, ...

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Pun on Grad? $\quad$ –  user21436 Feb 29 '12 at 1:16
Are you looking for rigour or a more intuitive understanding of mathematical analysis? If you are looking for a more theoretical approach to analysis, then Apostol's Calculus Vol 1, 2 are probably the best place to start, assuming that as a third grader you have the right background for it. –  Rankeya Feb 29 '12 at 1:27
I prefer intuitive understanding one . is it Apostol's Calculus ? –  reza Feb 29 '12 at 1:44
The author is in the "third grade of high school", i.e. the OP is probably in the 8 th or 9 th grade of school.I am currently studying calculus from Apostol and I really like his gentle and yet rigorous style. –  Eisen Feb 29 '12 at 6:12
@SabyasachiMukherjee When I hear "third grade of high school" I can't think of anything other than 11th grade. –  Alex Becker Feb 29 '12 at 7:35

As I said, for a rigorous and theoretical approach to calculus, Apostol's Calculus Vols. $1$ and $2$ are very good. Depending on your background, for multivariable calculus, Spivak's Calculus on manifolds is also good. Spivak's Calculus (which does single variable calculus) is also one of my favorites.

I think I first learned calculus from Richard Courant's Introduction to Calculus and Analysis. I think Courant's and Robbin's What is Mathematics? also has good intuitive explanations of differentiation and integration.

For a book more intuitive, and perhaps something that a third grader would have background for, try Silvanus Thompson's and Martin Gardner's Calculus Made Easy.

Of course, a book that worked for me might not work for you. I would suggest that you go to a library and browse through a number of different calculus books (there are a lot of them out there), till you find the one that appeals to you the most. If you really are in the third grade, then I would assume there is no real hurry for you to master calculus, and if there is, then the books above are a good place to start.

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That being said, I have no idea about signal processing, so my answer is more based on what good Calculus books there are out there. –  Rankeya Feb 29 '12 at 6:01
I want to work on computer signal Processing and image processing . which one is better ? –  reza Feb 29 '12 at 14:22

Differential and Integral Calculus, Vol. I [Paperback] Piskunov (Author)

Try this cover to cover and if you finish this you will know more one variable calculus than you will need.

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I think one's first exposure to calculus-no matter how gifted or ambitious the student is-should be a physically and geometrically motivated approach that illustrates most of important applications of calculus. Sadly,many people think that means a "pencil-pushing" or "cookbook" approach where things are done sloppily and with no careful explanation of underlying theory. That's simply not true. You can certainly do calculus non-rigorously while still doing it carefully enough to give students the broad picture of the underlying theory.

The best example of this kind of book,to me, is Gilbert Strang's Calculus. Strang's emphasis is clearly on applications and it has more applications then just about any other calculus text-including many kinds of differential equations in physics(mechanics),chemistry( first and second order kinetics),biology (modeling heart rythum) and economics and a basic introduction to probability.But Strang doesn't avoid a proof when it's called for and the book has many pictures to soften the blows of these careful proofs. This would be my first choice for a high school student just starting out with calculus.

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Last edited by Dait
Friday, July 17, 2020 | History

4 edition of matching law found in the catalog.


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# Homework Help: Abstract Algebra; Group Theory Question

1. Mar 11, 2015

### tropian1

• Member warned about posting without the template and with no effort
Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f.

I'm having trouble defining a function to prove this. Could anyone give me a start on this?

2. Mar 11, 2015

### micromass

Start by proving the following:
1) $N$ and $K$ are normal
2) $NK = G$
3) $N\cap K = \{e\}$


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BodhiAI
Sept. 23, 2019

#### Gravitational Field:

Gravitational field intensity

(ii) Units : Newton/kg or m/s2

(iii) Dimension : [M0 LT-2]

I = GM/r2

Gravitational Field Intensity for Different Bodies

(i) Intensity due to uniform solid sphere

(2) Intensity due to spherical shell

(3) Intensity due to uniform circular ring

Intensity due to uniform disc


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# CalculateSelfInducedMagneticField

A - B - C - D - E - F - G - H - I - K - L - M - N - O - P - Q - R - S - T - U - V - W - X

#### CalculateSelfInducedMagneticField

Section Hamiltonian
Type logical
Default no

The existence of an electronic current implies the creation of a self-induced magnetic field, which may in turn back-react on the system. Of course, a fully consistent treatment of this kind of effect should be done in QED theory, but we will attempt a first approximation to the problem by considering the lowest-order relativistic terms plugged into the normal Hamiltonian equations (spin-other-orbit coupling terms, etc.). For the moment being, none of this is done, but a first step is taken by calculating the induced magnetic field of a system that has a current, by considering the magnetostatic approximation and Biot-Savart law:

$\nabla^2 \vec{A} + 4\pi\alpha \vec{J} = 0$

$\vec{B} = \vec{\nabla} \times \vec{A}$

If CalculateSelfInducedMagneticField is set to yes, this B field is calculated at the end of a gs calculation (nothing is done – yet – in the tdcase) and printed out, if the Output variable contains the potential keyword (the prefix of the output files is Bind).

Source information


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# Solving a Two-Step Linear Inequality with Whole Numbers

Solving inequalities is similar to solving equations. What we do on one side of an inequality, we do the same on the other side to maintain the “balance” of the inequality. The Properties of Inequality help us add, subtract, multiply, or divide within an inequality.

As with one-step inequalities, we solve two-step inequalities by manipulating the inequality so as to isolate the variable.

Similarly, we always substitute values into the original inequality to check the answer. We plug in the solutions obtained into the original equation and see if it works.

Inequalities model problems that have a range of answers. They can be mapped along a number line, and they can be manipulated to simplify or solve them. When solving inequalities, it is important to follow the Properties of Inequality −

Solve the following two-step linear inequality with whole numbers.

5y + 1 > 11

### Solution

Step 1:

Given 5y + 1 > 11; Subtracting 1 from both sides

5y + 1 −1 > 11 – 1; 5y > 10

Step 2:

Dividing both sides by 5

5y/5 > 10/5; y > 2

Step 3:

So, solution for the given two-step linear inequality is

y > 2

Solve the following two-step linear inequality with whole numbers.

$\frac{−x}{2}$ − 5 > 2

### Solution

Step 1:

Given $\frac{−x}{2}$ − 5 > 2;

$\frac{−x}{2}$ − 5 + 5 > 2 + 5; $\frac{−x}{2}$ > 7

Step 2:

Multiplying both sides by 2

−x/2 × 2 > 7 × 2; −x > 14; x < −14

Step 3:

So, solution for the given two-step linear inequality is x < −14


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Which of the following statements are true?

1. Every group of order 11 is cyclic.
2. Every group of order 111 is cyclic.
3. Every group of order 1111 is cyclic.

Discussion:

1. Every group of order 11 is cyclic. This is true. We know that (for example from Lagrange’s Theorem) that a group of prime order is necessarily cyclic.
2. Every group of order 111 is cyclic. This is true. $111 = 3 \times 37$. This is a favorite problem for any college level test maker. Order of this group is of the form $p \times q$ where p and q are primes. This group is isomorphic to $Z_p \times Z_q$ that is the direct product of two cyclic groups of prime order. We have this theorem:  Let G be a group of order pq, where p,q are prime, p<q, and p does not divide q−1. Then G is cyclic. The proof of this theorem directly follows from Sylow’s Theorem.
3. Every group of order 1111 is cyclic. This is true. $1111 = 11 \times 101$ and using the argument of the previous problem we conclude that it is cyclic.

Critical Ideas: Lagrange’s Theorem, Sylow’s Theorem, Classification of finite abelian groups


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Regular version of the site

## Shifted Quantum Affine Algebras: Integral Forms in Type A

Arnold Mathematical Journal. 2019. Vol. 5. No. 2-3. P. 197-283.
Michael Finkelberg, Tsymbaliuk A.

We define an integral form of shifted quantum affine algebras of type A and construct
Poincaré–Birkhoff–Witt–Drinfeld bases for them. When the shift is trivial, our integral
form coincides with the RTT integral form. We prove that these integral forms are
closed with respect to the coproduct and shift homomorphisms. We prove that the
homomorphism from our integral form to the corresponding quantized K -theoretic
Coulomb branch of a quiver gauge theory is always surjective. In one particular case
we identify this Coulomb branch with the extended quantum universal enveloping
algebra of type A. Finally, we obtain the rational (homological) analogues of the
above results [proved earlier in Kamnitzer et al. (Proc Am Math Soc 146(2):861–874,
2018a; On category O for affine Grassmannian slices and categorified tensor products.
arXiv:1806.07519, 2018b) via different techniques].


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# ? Each time you see (cos(2x))^2, replace it with 1/2 + 1/2cos(4x) ?

• Feb 27th 2009, 03:11 PM
estex198
? Each time you see (cos(2x))^2, replace it with 1/2 + 1/2cos(4x) ?
I understand sin(2x)^2 = 1/2 - 1/2cos(2x), due to the double number identity: cos2x = 1 - 2sin(x)^2.

***[My question is how does (cos(2x))^2 = 1/2 + 1/2cos(4x) ???]***

I've tried reworking the identity every which way and can't seem to find out how the two are equivalent. Thanks in advance!
• Feb 27th 2009, 03:17 PM
mr fantastic
Quote:

Originally Posted by estex198
I understand sin(2x)^2 = 1/2 - 1/2cos(2x), due to the double number identity: cos2x = 1 - 2sin(x)^2.

***[My question is how does (cos(2x))^2 = 1/2 + 1/2cos(4x) ???]***

I've tried reworking the identity every which way and can't seem to find out how the two are equivalent. Thanks in advance!

$\cos (2A) = \cos^2 A - \sin^2 A = 2 \cos^2 A - 1 \Rightarrow \cos^2 A = \frac{1}{2} (\cos (2A) + 1)$.
$A = 2x$ in your case.


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Can someone give me an example of a challenging proof by induction? [closed]

I'm about to give my first consultation and the topic is proof by induction. The students are new to calculus and I assume they don't know anything besides precalculus. Can anyone give me a proof by induction which is a bit different, challenging, maybe foreshadows other areas of calculus (derivation or whatever) because the prof who teaches them as well already have shown them a lot of easy ones. I'm glad even if you can give an example where it is hard to see that it can be proved by induction.

closed as too broad by user223391, Watson, Qwerty, Parcly Taxel, RamiroSep 24 '16 at 1:28

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

Here is one that my math professor showed us: show that for $n\ge 1$ a $2^n\times 2^n$ chessboard with one square removed can always be tiled by "L-shaped" pieces. That is, pieces formed by removing a corner from a $2\times2$ square.

• For $n \geq 0$, in fact. – Peter LeFanu Lumsdaine Sep 15 '16 at 8:39
• @Octopus: right… but carefully write down the definition of what it means to “tile a region with L-shaped pieces”, and unless you explicitly add some kind of non-emptiness condition, I guarantee you will find that it naturally includes the empty case. It’s true, lots of people do like to exclude the empty-case — but 99% of the time it’s because we have psychological hangups about it, it’s not something that emerges mathematically naturally! :-) – Peter LeFanu Lumsdaine Sep 15 '16 at 21:12
• MESE note about this problem and (some of) its history. – Benjamin Dickman Sep 16 '16 at 1:17
• @BenjaminDickman thanks for the post about the problem's history. It's fascinating indeed! – Alex Ortiz Sep 16 '16 at 1:25

I suggest Cauchy's proof (I believe it's Cauchy's) of the A.G.M. inequality, because it is non-standard and unsettling induction. It goes along the following lines:

1. Prove that if the inequality is true for $n$ numbers, it is true for $2n$ numbers.
2. Prove that if it is true for $n$ numbers, it is true for $n\boldsymbol{\color{red}-} 1$ numbers.

Point 1, together with the initialisation step implies the inequality is true for numbers of the form $2^k$. As any number belongs to some interval $\bigl(2^{k-1}..\,2^{k} \bigr]\mkern-4.5mu\bigr]$, point 2 ensures it is true for all numbers.

• To teach Cauchy's proof correctly, one ought to also prove that the induction technique is valid, based on normal induction alone. Namely, if for any $k \in \mathbb{N}$ we have both that $P(2^k)$ and that $P(k+1) \to P(k)$, then $P(k)$ for any $k \in \mathbb{N}$. That is an interesting induction exercise all on its own! Incidentally, a long time ago I had a classmate who insisted that Cauchy's proof was wrong... – user21820 Sep 15 '16 at 3:51
• @user21820: ??? I don't see why it should be wrong, except perhaps for axiomatic set theory reasons – a domain which I don't know well enough. Or the actual details were wrong? If I remember well, it's an exercise of Spivak's Calculus. – Bernard Sep 15 '16 at 7:44
• Of course it's not wrong (my classmate had a poor grasp of logic), but it is actually not that trivial to prove that such an induction technique is actually (meta-logically) provable from the standard induction axiom, to show that it is valid. And Spivak didn't justify the validity of the induction either, and left it to the reader's intuition, if I recall correctly. Anyway, ZF proves the standard induction axiom, and hence also proves this valid. – user21820 Sep 15 '16 at 8:19

I like this one.

Any $2^n\times 2^n$ grid with one square blacked out can be tiled with L shaped pieces of 3 units area.

Here is an example in the $4\times 4$ case.

$n^5-n$ is divisible by 10 (really is divisible by 30 but why work that hard). It is also a specific case of Fermat's little theorem.

The false proof, "All horses are the same color" is worth demonstrating.

https://en.wikipedia.org/wiki/All_horses_are_the_same_color

• The L shape tiling question was already posted when you posted your answer. – 6005 Sep 15 '16 at 6:49

I like this one, it is not that challenging but it is quite interesting:

Finite many lines divide a plane into regions. Show that these regions can be colored by two colors in such a way that neighboring regions have different colors.

We will do the proof using induction on the number $n$ of lines. The base case $n=1$ is straight forward, just color a half-plane black and the other half white.

For the inductive step, assume we know how to color any map defined by $k$ lines. Add the $(k+1)$st line. Keep the colors the same on one side of the line and on the other side invert the color of all regions. The process is illustrated below:

Note that regions that were adjacent previously continue to have different colors. Regions that share a segment of the $(k+1)$st line, which were part of the same region, now lie on opposite sides of the line and, therefore, have different colors. This shows that the map satisfies the required property and the induction is complete.

Elementary number theory

$\def\nn{\mathbb{N}}$Take any $x,y \in \nn$.

We say that $d$ is a gcd of $x,y$ iff $d \in \nn$ and $d \mid x,y$ and $k \mid d$ for any $k \mid x,y$.

Define $\gcd(x,y)$ recursively as $\cases{ x & if$y = 0$\\ \gcd(y,x) & if$x<y$\\ \gcd(x-y,y) & if$x \ge y > 0$}$.

1. Prove that $\gcd(x,y)$ terminates (is well-defined).

2. Prove that $\gcd(x,y)$ is a gcd of $x,y$.

3. Prove that $\gcd(x,y) = ax+by$ for some integers $a,b$.

The proof of (1) is a standard induction on the 'size' of the input, for which we could for example use $x+y$. The proof of (2) is also standard but more interesting; the base case is that $\gcd(x,0) = x$ and that $x$ is a gcd of $x,0$, and the inductive case requires proving that if $d$ is a gcd of $x-y,y$ then $d$ is a gcd of $x,y$. The proof of (3) is the nicest part of all; simply observe that both inputs in the recursive calls are always integer linear combinations of the original inputs, and hence at the end the output is also an integer linear combination!

This kind of induction is very important in all areas of mathematics where we use an inductive construction. This specific example is also very instructive because it gives a clear insight into Bezout's identity unlike many textbooks that pull the alternative definition of $\gcd(x,y)$ (as the minimum integer linear combination of $x,y$) out of thin air, or prove it by contradiction and well-ordering, which is not as constructive or intuitive.

Graph theory

Take any graph $G$.

We say that $G$ is $2$-vertex-connected iff removing any vertex from $G$ leaves it connected.

Prove that if $G$ has at least $3$ vertices then any $2$ vertices are in some cycle in $G$.

The proof I have makes use of the extremal principle, which is another extremely useful technique that arises from induction. Take any $2$ vertices $u,v$ in $G$. First note that $u$ has at least $2$ neighbours in $G$ (this uses the condition that $G$ has at least $3$ vertices). Thus $G - u$ has a path between any $2$ neighbours of $u$, and hence $G$ has a cycle through $u$. Let $C$ be a cycle in $G$ through $u$ that is closest to $v$ in $G$. Note that the cycle "closest to $v$" exists by well-ordering (which is from induction), and that there may be more than one such cycle (which is why I used "a"). Now it is not hard to prove that $C$ must pass through $v$ otherwise we can construct a cycle that is closer to $v$ in $G$.

Induction in the form of extremal principle is a very powerful tool because it allows extracting information 'for free'; existence of any kind of object immediately implies existence of a smallest instance of that kind, where the size of the relevant objects is a natural number. This generalizes to any well-order, and choosing the minimal ordinal that satisfies something is frequently used in set theory.

Inequalities

Someone mentioned the AM-GM inequality, but I do not see the point in teaching Cauchy's proof since I have never found the technique useful anywhere else. Instead, I recommend teaching the smoothing technique (see here and here for two examples). In the case of the AM-GM inequality, simply note that if not all are equal to the mean $m$ then there are two of them, which we shall call $x,y$, such that $x < m < y$. Now simply push $x,y$ together while keeping their sum constant, until one hits $m$. Their product will increase (by trivial algebra), and we have at least one more equal to the mean. By induction, in finitely many steps all will be equal to the mean and the product has increased, thus the inequality and equality case follow.

• If you want a reasonably hard problem that can be solved by induction, see matheducators.stackexchange.com/a/2449/1550. – user21820 Sep 15 '16 at 4:27
• I don't think a graph theory problem is appropriate for a calculus class, but the number theory problem is nice – Jacob Maibach Sep 19 '16 at 15:48
• @JacobMaibach: The question said "maybe foreshadows", so I decided to give examples of what I felt every student ought to learn but few do. =) – user21820 Sep 19 '16 at 16:10

Prove that, for all positive integers $n$, there exists a partition of the set of positive integers $k\leq 2^{n+1}$ into sets $A$ and $B$ such that

$$\sum_{x\in A} x^i = \sum_{x\in B} x^i$$

for all integers $0\leq i\leq n$.

• @user21820 at $n=1$ all we care about is the exponents $i=0$ and $i=1$. $1^0+4^0 = 2^0+3^0$ and $1^1+4^1=2^1+3^1$ so it is true for $n=1$. – Carl Schildkraut Sep 15 '16 at 4:12
• Oh sorry I thought $n = 2$... So you actually mean that the same partition works for all $i$ in the range $[0..n]$? Interesting.. it would be nice if you included a proof sketch. – user21820 Sep 15 '16 at 4:15
Let $T$ be a rooted tree with an infinite number of nodes, each with a finite number of children. Then $T$ has a branch of infinite length.

The proof is easy, but can be used as introduction to some advanced ideas.

Show equivalence of these three statements:

• Greatest Lower Bound: Every non-empty set of natural numbers has a least member.

• "Weak" Induction: Suppose for all natural numbers n, P(n) implies P(n+1), and P(0) is true. Thus P(n) is true for all natural numbers n.

• "Strong" Induction: Suppose for all natural numbers n, if P(i) is true for all i < n, then P(n) is true. Then P(n) is true for all natural numbers n.

• An interesting exercise, but not what was asked for. – jwg Sep 15 '16 at 10:07
• @jwg How so? They are proofs. The easy way to do it is to form a circle of implies, and 2/3 legs are clearly induction proofs. The bound part foreshadows the difference between discrete and non-discrete situations, (where similar lower bounds have different implications), which a good calculus course should run into shortly. – Yakk Sep 15 '16 at 11:14
• Sorry, but the proofs of these implications are not induction proofs, although they resemble induction proofs. – jwg Sep 15 '16 at 15:27
• @jwg You appear to have a different definition of "induction proof" than I do. What is your definition of "induction proof"? – Yakk Sep 15 '16 at 15:59
• "Greatest Lower Bound $\implies$ Weak Induction" obviously isn't an induction proof. "Weak Induction $\implies$ Strong Induction" is quite trivial (if $P(i)$ is true for all $i<n$, then of course either $n=0$ (where any statement is vacuously true for all $i<n$) or it is especially true for $i=n-1$) and that proof neither is, nor resembles an induction proof. So the only remaining proof is "Strong Induction $\implies$ Greatest Lower Bound". I wonder where you get two induction proofs here. – celtschk Sep 16 '16 at 7:51

The proof that any polygon in the plane can be triangulated without adding any new vertices.

Assume for n-1 vertices.

Take a vertex, v, with interior angle less than 180 degrees. Take the two vertices next to it. Draw a line between them. If this doesn't intersect (or contain) the polygon we have divided into a smaller polygon and a triangle and so we can apply the inductive hypothesis. Otherwise, we slide the two ends towards the vertex, eventually it will not intersect the polygon. Take the last vertex, w, the line crosses and draw a line from v to w. Then apply the inductive hypothesis.

(see my book Proof Patterns for further discussion and more examples.)

• This is ok only if you have already proven the JCT (Jordan curve theorem) for polygons; if not how do you define "triangulation" and "interior angle" in the first place? See math.stackexchange.com/a/1877524/21820 for an elementary proof of the JCT for polygons. I do not think there is a significantly simpler rigorous proof. – user21820 Sep 15 '16 at 4:39
• By the way, your proof is incorrect. The line may not intersect the polygon but the triangle may enclose the rest of the polygon, and the induction fails. – user21820 Sep 15 '16 at 4:48
• you can define interior or exterior according to the parity of how many times a ray crosses the polygon from a given point. You can also prove the Jordan curve theorem that way. – Mark Joshi Sep 15 '16 at 5:08
• I am very well aware of that purported proof, and please write it out to see that to make it rigorous is not going to be easy! Even the published version of that proof leaves out the details of the crucial part of the proof, so it seems much simpler than it really is. – user21820 Sep 15 '16 at 5:19
• Your edited answer has fixed the error I mentioned earlier, but note that although it is intuitive, as written it is not a good example of a rigorous proof because intuition is very bad for general mathematical objects. In particular, multiple ways of sliding work, but you need to explicitly pick one and prove that there is a last vertex crossed, and that connecting $v,w$ leads to the polygon being split into two non-overlapping regions. If you read my linked proof, it is not so simple as you may think! – user21820 Sep 15 '16 at 8:34

Another idea: The set of triangular numbers modulo $n$ yields $\mathbb{Z}/n\mathbb{Z}$ iff $n$ is a power of $2$.

This problem arose in my secondary school problem solving class (under a friendlier guise/phrasing). A few write-ups indicating how induction is used to prove the result can be found online with just a bit of googling.

Here is one presentation under the name of Daniel Finkel's Circle Toss Game from the New York Times' Numberplay blog, and here is another in a wordpress blog that attributes an earlier proof to D Knuth in his book The Art of Computer Programming (vol. 3).

Earlier idea: How about proving the Inclusion-Exclusion Principle (IEP) by induction?

ProofWiki link can be found here.

(This came to mind as I recently read over MO 249523 on Dwork's Theorem, in which induction is used a few times, and the IEP is used a few more!)

Prove that every natural number $n \geq 24$ is a sum of $5$'s or $7$'s. For example, $29 = 5 + 5 + 5 + 7 + 7$ and $33 = 5 + 7 + 7+ 7 + 7$.

[Comment: It is possible to prove this using normal induction or strong induction. There are also more direct ways to do this problem.]

• For $a$ and $b$ relatively prime, max number you cannot make is $ab - (a+b)$. Here, that is $35-12 = 23$; so you can make all positive numbers $geq 24$. – Benjamin Dickman Sep 28 '16 at 13:52

Here is an example which has as additional challenge the need for a proper generalisation.

Show that following is valid: If $A_1+\cdots+A_n=\pi,\text{ with } 0<A_i\leq \pi, 1\leq i\leq n$, then \begin{align*} \sin A_1+\cdots +\sin A_n\leq n\sin\frac{\pi}{n}\tag{1} \end{align*}

Let us denote with $P(k)$ the claim for a given $k$ and suppose as induction hypothesis $P(k)$ to be true.

If we look at the induction step $P(k+1)$: \begin{align*} \sin A_1+\cdots +\sin A_k+\sin A_{k+1}=\pi\qquad \text{with }\qquad 0<A_i\leq \pi,1\leq i\leq k+1 \end{align*}

it is not that clear how to make use of $P(k)$.

One attempt could be to group $A_k$ and $A_{k+1}$ together, so that $A_1+\cdots+(A_k+A_{k+1})=\pi$. Applying the assumption $P(k)$ yields \begin{align*} \sin A_1+\cdots +\sin A_{k-1}+\sin\left(A_k+A_{k+1}\right)\leq k\sin \frac{\pi}{k} \end{align*}

But it's not at all clear how to deduce $P(k+1)$: \begin{align*} \sin A_1+\cdots +\sin A_k+ \sin A_{k+1}\leq (k+1)\sin \frac{\pi}{k+1} \end{align*}

Generalisation is the key to conveniently solve this problem. Instead of requiring that the $A_i$'s add to $\pi$ we reformulate the condition in a more general setting. We consider the proposition $Q(n)$:

If $0<A_i\leq \pi, 1\leq i\leq n$, then \begin{align*} \sin A_1+\cdots+\sin A_n\leq n\sin\left(\frac{A_1+\cdots+A_n}{n}\right) \tag{2} \end{align*}

Note, that $Q(n)$ implies $P(n)$. Since, if we put $A_1+\cdots+A_n=\pi$ in (2) we obtain (1).

So, let's have a look at $Q(n)$:

We can see at a glance that $Q(1)$ is true.

Now, assuming the validity of $Q(k)$ and $0<A_i\leq \pi, 1\leq i\leq k+1$, we obtain

\begin{align*} \sin A_1&+\cdots \sin A_k+\sin A_{k+1}\\ &=k \sin \left(\frac{\sin A_1+\cdots +\sin A_k}{k}\right)+\sin A_{k+1}\\ &=(k+1)\left[\frac{k}{k+1}\sin\left(\frac{\sin A_1+\cdots +\sin A_k}{k}\right) +\frac{1}{k+1}\sin A_{k+1}\right]\\ &\leq (k+1)\left[\sin \left(\frac{k}{k+1}\left(\frac{A_1+\cdots+A_k}{k}\right)+\frac{1}{k+1}A_{k+1}\right)\right]\tag{3}\\ &=(k+1)\sin\left(\frac{A_1+\cdots+A_{k+1}}{k+1}\right) \end{align*}

and the result follows by induction.

In step (3) we use for $m,n>0$ and $0<\alpha,\beta\leq \pi$ the validity of \begin{align*} \frac{m}{m+n}\sin \alpha +\frac{n}{m+n}\sin \beta\leq \sin\left(\frac{m}{m+n}\alpha+\frac{n}{m+n}\beta\right) \end{align*}

The proof of this inequality could be left as challenge.

Note: This is example 2.4.1 in Poblem-Solving Through Problems by Loren. C. Larson.

• On the interval $[0,\pi]$ Sine function is concave down. Therefore we can use the Jensen's inequality to prove this nice result. – Bumblebee Dec 23 '16 at 4:15
• @Nil: Good observation! :-) – Markus Scheuer Dec 23 '16 at 8:19

Very good question!

The problem six of the IMO 2009 can be founded here:

https://www.imo-official.org/year_info.aspx?year=2009

The idea that is used in the problem is so simple, is an induction argument, but is challenging! That problem was the most difficult in that year and so, by score, that problem is ranked as on of the hardest problems in all the history of the International Mathematical Olympiad!!! No more words, enjoy it!

Quite a tricky (for beginners) induction proof: show that if $*$ is an associate binary operation on a set $S$, then the value of $$a_1*a_2*\cdots*a_n$$ is independent of the bracketing.

Construct the natural numbers $\mathbb{N}$ by using the Peano Axioms. Denote by $\varphi:\mathbb{N}\rightarrow\mathbb{N}$ the map defining the injective map associating each number to the "next one".

Show that the map $+:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ defined by $$n+1=\varphi(n),\qquad n+\varphi(m)=\varphi(n+m)$$ ís a commutative associative operation.

• The question isn't asking for a problem which is hard without induction, but rather one where the induction proof itself is hard. – Noah Schweber Sep 16 '16 at 23:11
• I edited my post and put a harder induction problem. – LaloVelasco Sep 16 '16 at 23:31


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# Greg’s Dinosaurs Riddle

The two-riddles post was a success, and while corresponding with Greg Kuperberg he had a riddle for me about dinosaurs, and he agreed I will share it with you.

Right before the Chixculub asteroid hit the earth, there were a variety of mammals and a wide variety of dinosaurs.  Both the mammals and the dinosaurs lived in many places, came in many sizes, etc.  Why did some of the mammals survive, but none of the dinosaurs did?

A nuch simpler question: Can you answer (without clicking or googeling or so):

How many years did the dinosaurs live? (you know,.. not a single one but them as a whole…) 1 million years, 5 millions, 20 millions, 50 millions, 100 millions?

And a deep philosophical question:

The dinosaurs perished. Should we regard them as losers???

# A Diameter Problem (6): Abstract Objective Functions

George Dantzig and Leonid Khachyan

In this part we will not progress on the diameter problem that we discussed in the earlier posts but will rather describe a closely related problem for directed graphs associated with ordered families of sets. The role models for these directed graphs are the directed graphs of polytopes where the direction of the edges is described by a linear objective function.

### 7. Linear programming and the simplex algorithm.

Our diameter problem for families of sets was based on a mathematical abstraction (and a generalization) of the Hirsch Conjecture which asserts that the diameter of the graph $G(P)$ of a $d$-polytope $P$ with $n$ facets is at most $n-d$. Hirsch, in fact, made the conjecture also for graphs of unbounded polyhedra – namely the intersection of $n$ closed halfspaces in $R^d$. But in the unbounded case, Klee and Walkup found a counterexample with diameter $n-d+$ [$d/5$]. The abstract problem we considered extends also to the unbounded case and  $n-d+$ [$d/5$] is the best known lower bound for the abstract case as well. It is not known if there is a polynomial (in terms of $d$ and $n$) upper bound for the diameter of graphs of d-polytopes with n facets.

Hirsch’s conjecture was motivated by the simplex algorithm for linear programming.  Let us talk a little more about it: Linear programming is the problem of maximizing a linear objective function $\phi(x)=b_1x_1+ b_2 x_2 \dots +b_dx_d$ subject to a system of n linear inequalities in the variables $x_1,x_2,\dots,x_d$.

$a_{11}x_1 + x_{12}x_2 + \dots + x_{1d}x_d \le c_1$,

$a_{21}x_1 + x_{22}x_2 + \dots + x_{2d}x_d \le c_2$,

$a_{n1}x_1 + x_{n2}x_2 + \dots + x_{nd}x_d \le c_n$,

The set of solutions to the system of inequalities is a convex polyhedron. (If it is bounded it is a polytope.)  A linear objective function makes a graph of a polytope (or a polyhedron) into a digraph (directed graph). If you like graphs you would love digraphs, and if you like graphs of polytopes, you would like the digraphs associated with them.

The geometric description of Dantzig’s simplex algorithm is as follows: the system of inequalities describes a convex d-dimensional polyhedron $P$. (This polyhedron is called the feasible polyhedron.) The maximum of $\phi$ is attained at a face $F$ of $P$. We start with an initial vertex (extreme point) $v$ of the polyhedron and look at its neighbors in $G(P)$. Unless $v \in F$ there is a neighbor $u$ of $v$ that satisfies $\phi(u) > \phi (v)$. When you find such a vertex move from $v$ to $u$ and repeat!

### 8. Abstract objective functions and unique sink orientation.

Let $P$ be a simple d-polytope and let $\phi$ be a linear objective function which is not constant on any edge of the polytope. Remember, the graph of $P$, $G(P)$ is a $d$-regular graph. We can now direct every edge $u,v$ from $u$ to $v$ if $\phi (v) > \phi (u)$. Here are two important properties of this digraph.

(AC) It is acyclic! (no cycles)

(US’) It has a unique SINK, namely a unique vertex such that all edges containing it are directed towards it.

The unique sink property is in fact the property that enables the simplex algorithm to work!

When we consider a face of the polytope $F$ and its own graph $G(F)$ then again our linear objective function induces an orientation of the edges of $G(F)$ which is acyclic and also has the unique sink property. Every subgraph of an acyclic graph is acyclic. But having the unique sink property for a graph does not imply it for a subgraph. We can now describe the general unique sink properties of digraphs of polytopes:

(US) For every face F of the polytope, the directed graph induced on the vertices of $F$ has a unique sink.

A unique sink acyclic orientation of the graph of a polytope is an orientation of the edges of the graph which satisfies properties (AC) and (US).

An abstract objective function of a $d$-polytope is an ordering $<$ of the vertices of the polytope such that the directed graph obtained by directing an edge from $u$ to $v$ if  $u is a unique sink acyclic orientation. (Of course, coming from an ordering the orientation is automatically acyclic.)


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# MATHS

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Pls help me on this question

question 9 part c

Rauhan  Dec 24, 2017

#2
+91445
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Hi Rauhan

9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns:

Row 1

Row 2

Row 3

She notices that 4 sticks are required to make the single square in the first row,

7 sticks to make 2 squares in the second row

and in the third row she needs 10 sticks to make 3 squares.

(a) Find an expression, in terms of n, for the number of sticks required to make a similar arrangement of n squares in the nth row.

 row (r) 1 2 3 ... n match sticks (m) 4 7 10 3n+1

Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.

(b) Find the total number of sticks Ann uses in making these 10 rows.

m(10)=3*10+1=31 match sticks

Ann started with 1750 sticks.

Given that Ann continues the pattern to complete k rows but does not have sufficient sticks to complete the (k + 1)th row,

(c) show that k satisfies (3k – 100)(k + 35) < 0.

The total number of sticks in k rows is

$$4+7+10+.......(3k+1) \qquad \text{Sum of an AP}\\ =\frac{n}{2}(a+L)\\ =\frac{k}{2}(4+3k+1)\\ =\frac{k}{2}(3k+5)\\ now\\ \frac{k}{2}(3k+5)<1750\\ k(3k+5)<3500\\ 3k^2+5k<3500\\ 3k^2+5k-3500<0\\ \qquad 3*-3500=-10500\\ \qquad \text{I need 2 numbers that multiply to -10500 and add to +5}\\ \qquad \text{Those numbers are 105 and -100}\\ 3k^2+105k-100k-3500<0\\ 3k(k+35)-100(k+35)<0\\ (3k-100)(k+35)<0$$

(d) Find the value of k.

if you graph y=(3k-100)(k+35)

it will be a concave up parabola and y will be less then 0  between the two zeros.

3k-100=0

3k=100

k=33 and a 1/3

and

k=35=0

k=-35

k cant be negative so Ann will have enough sticks for 1 to 33 rows.

She will not have enough sticks for 34 rows.

So

k=33

Melody  Dec 24, 2017
Sort:

#1
+202
+1

(3k – 100)(k + 35) < 0.

You want every (...) to have different sign so

the first () if k>33.333  ()>0 if k<33.333 ()<0                κ              -35           33.333

the second () if k>-35 ()>0 if k<-35 ()<0                     (1)      -              -                    +

(2)      -             +                    +

Finally    +              -                    +

You want  (3k – 100)(k + 35) be <0 so       -35

κ  between  -35 and 33.333

Whitespy001  Dec 24, 2017
edited by Whitespy001  Dec 24, 2017
#3
+91445
0

Thanks Whitespy

But... you have not actually answered part c.

Melody  Dec 24, 2017
#4
+202
0

yes because i understand false the question :p Im sorry :(

Whitespy001  Dec 24, 2017
#5
+91445
+1

I understood that, it was not a problem :)

Melody  Dec 24, 2017
#2
+91445
+1

Hi Rauhan

9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns:

Row 1

Row 2

Row 3

She notices that 4 sticks are required to make the single square in the first row,

7 sticks to make 2 squares in the second row

and in the third row she needs 10 sticks to make 3 squares.

(a) Find an expression, in terms of n, for the number of sticks required to make a similar arrangement of n squares in the nth row.

 row (r) 1 2 3 ... n match sticks (m) 4 7 10 3n+1

Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.

(b) Find the total number of sticks Ann uses in making these 10 rows.

m(10)=3*10+1=31 match sticks

Ann started with 1750 sticks.

Given that Ann continues the pattern to complete k rows but does not have sufficient sticks to complete the (k + 1)th row,

(c) show that k satisfies (3k – 100)(k + 35) < 0.

The total number of sticks in k rows is

$$4+7+10+.......(3k+1) \qquad \text{Sum of an AP}\\ =\frac{n}{2}(a+L)\\ =\frac{k}{2}(4+3k+1)\\ =\frac{k}{2}(3k+5)\\ now\\ \frac{k}{2}(3k+5)<1750\\ k(3k+5)<3500\\ 3k^2+5k<3500\\ 3k^2+5k-3500<0\\ \qquad 3*-3500=-10500\\ \qquad \text{I need 2 numbers that multiply to -10500 and add to +5}\\ \qquad \text{Those numbers are 105 and -100}\\ 3k^2+105k-100k-3500<0\\ 3k(k+35)-100(k+35)<0\\ (3k-100)(k+35)<0$$

(d) Find the value of k.

if you graph y=(3k-100)(k+35)

it will be a concave up parabola and y will be less then 0  between the two zeros.

3k-100=0

3k=100

k=33 and a 1/3

and

k=35=0

k=-35

k cant be negative so Ann will have enough sticks for 1 to 33 rows.

She will not have enough sticks for 34 rows.

So

k=33


text:
# Question about velocity-dependent Lagrangian involving magnetic fields

## Homework Statement

The Lagrange method does work for some velocity dependent Lagrangian. A very important
case is a charged particle moving in a magnetic field. The magnetic field can be represented as a "curl" of a vector potential ∇B = ∇xA . A uniform magnetic field B0 corresponds to a
vector potential ∇A = 1/2 B0 x r.

(a) Check that B0 = ∇xA
(b) From the Lagrangian
$\frac{1}{2}mv^{2}+e\overline{v}.\overline{A}$
show that the EOM derived are identical to the classical Newton's law with the Lorentz
force F = ev x B .

## Homework Equations

Euler-Lagrange equations:
$\frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})$=$\frac{\partial L}{\partial s_{j}}$

Triple product:
a.(b x c)=b.(c x a)
a x (b x c)=(a.c)b - (a.b)c

## The Attempt at a Solution

For a, I have tried using the triple product:
∇ x A = 1/2 ∇x(Bo x r) = 1/2 [Bo(∇.r)-(∇.Bo)r]
Since Bo is uniform, its divergence is zero and so:
∇ x A =1/2 Bo(∇.r)
From here, I guess ∇.r=2 for the proof to work, but I really don't see it.

For part b, I am not even sure where to start it. As in how can I apply the E-L equations to it? What scares me the most is how exactly do I take the derivatives on a Lagrangian involving vectors.

Any help is welcome, thank you!


text:
# 9.4 Series and their notations  (Page 2/18)

 Page 2 / 18

Evaluate $\sum _{k=2}^{5}\left(3k–1\right).$

38

## Using the formula for arithmetic series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence    is a sequence in which the difference between any two consecutive terms is the common difference    , $d.$ The sum of the terms of an arithmetic sequence is called an arithmetic series . We can write the sum of the first $n$ terms of an arithmetic series as:

${S}_{n}={a}_{1}+\left({a}_{1}+d\right)+\left({a}_{1}+2d\right)+...+\left({a}_{n}–d\right)+{a}_{n}.$

We can also reverse the order of the terms and write the sum as

${S}_{n}={a}_{n}+\left({a}_{n}–d\right)+\left({a}_{n}–2d\right)+...+\left({a}_{1}+d\right)+{a}_{1}.$

If we add these two expressions for the sum of the first $n$ terms of an arithmetic series, we can derive a formula for the sum of the first $n$ terms of any arithmetic series.

$\frac{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{n}={a}_{1}+\left({a}_{1}+d\right)+\left({a}_{1}+2d\right)+...+\left({a}_{n}–d\right)+{a}_{n}\hfill \\ +\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{n}={a}_{n}+\left({a}_{n}–d\right)+\left({a}_{n}–2d\right)+...+\left({a}_{1}+d\right)+{a}_{1}\hfill \end{array}}{2{S}_{n}=\left({a}_{1}+{a}_{n}\right)+\left({a}_{1}+{a}_{n}\right)+...+\left({a}_{1}+{a}_{n}\right)}$

Because there are $n$ terms in the series, we can simplify this sum to

$2{S}_{n}=n\left({a}_{1}+{a}_{n}\right).$

We divide by 2 to find the formula for the sum of the first $n$ terms of an arithmetic series.

${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}$

## Formula for the sum of the first n Terms of an arithmetic series

An arithmetic series    is the sum of the terms of an arithmetic sequence. The formula for the sum of the first $n$ terms of an arithmetic sequence is

${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}$

Given terms of an arithmetic series, find the sum of the first $n$ terms.

1. Identify ${a}_{1}$ and ${a}_{n}.$
2. Determine $n.$
3. Substitute values for and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ into the formula ${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}.$
4. Simplify to find ${S}_{n}.$

## Finding the first n Terms of an arithmetic series

Find the sum of each arithmetic series.

1. $\sum _{k=1}^{12}3k-8$
1. We are given ${a}_{1}=5$ and $\text{\hspace{0.17em}}{a}_{n}=32.$

Count the number of terms in the sequence to find $n=10.$

Substitute values for $\text{\hspace{0.17em}}{a}_{1},{a}_{n}\text{\hspace{0.17em},}$ and $n$ into the formula and simplify.

2. We are given ${a}_{1}=20$ and ${a}_{n}=-50.$

Use the formula for the general term of an arithmetic sequence to find $n.$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{n}={a}_{1}+\left(n-1\right)d\hfill \\ -50=20+\left(n-1\right)\left(-5\right)\hfill \\ -70=\left(n-1\right)\left(-5\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}14=n-1\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}15=n\hfill \end{array}$

Substitute values for ${a}_{1},{a}_{n}\text{,}\text{\hspace{0.17em}}n$ into the formula and simplify.

$\begin{array}{l}\begin{array}{l}\\ {S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}\end{array}\hfill \\ {S}_{15}=\frac{15\left(20-50\right)}{2}=-225\hfill \end{array}$
3. To find ${a}_{1},\text{\hspace{0.17em}}$ substitute $k=1$ into the given explicit formula.

We are given that $n=12.$ To find ${a}_{12},\text{\hspace{0.17em}}$ substitute $k=12$ into the given explicit formula.

Substitute values for ${a}_{1},{a}_{n},$ and $n$ into the formula and simplify.

Use the formula to find the sum of each arithmetic series.

$\text{26}\text{.4}$

$\text{328}$

$\sum _{k=1}^{10}5-6k$

$\text{−280}$

## Solving application problems with arithmetic series

On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?

This problem can be modeled by an arithmetic series with $\text{\hspace{0.17em}}{a}_{1}=\frac{1}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}d=\frac{1}{4}.\text{\hspace{0.17em}}$ We are looking for the total number of miles walked after 8 weeks, so we know that $n=8\text{,}$ and we are looking for $\text{\hspace{0.17em}}{S}_{8}.\text{\hspace{0.17em}}$ To find ${a}_{8},$ we can use the explicit formula for an arithmetic sequence.

$\begin{array}{l}\begin{array}{l}\\ {a}_{n}={a}_{1}+d\left(n-1\right)\end{array}\hfill \\ {a}_{8}=\frac{1}{2}+\frac{1}{4}\left(8-1\right)=\frac{9}{4}\hfill \end{array}$

We can now use the formula for arithmetic series.

She will have walked a total of 11 miles.

x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.


text:
Updating the inverse of a matrix

To understand how the axis and the angle control a rotation, let's do a small experiment.Put your thumb up against your monitor and try rotating your hand around it.Can you guess from the multiplication overview what the matrix should look like to translate a vector by [\begin \color1 & \color0 & \color0 & \color X \ \color0 & \color1 & \color0 & \color Y \ \color0 & \color0 & \color1 & \color Z \ \color0 & \color0 & \color0 & \color1 \end \cdot \begin x \ y \ z \ 1 \end = \begin x \color X\cdot 1 \ y \color Y\cdot 1 \ z \color Z\cdot 1 \ 1 \end] value a translation wouldn't have been possible.A scale transformation scales each of a vector's components by a (different) scalar.

For those among you who aren't very math savvy, the dot is a multiplication sign.Objects can be rotated around any given axis, but for now only the X, Y and Z axis are important.You'll see later in this chapter that any rotation axis can be established by rotating around the X, Y and Z axis simultaneously.It is commonly used to shrink or stretch a vector as demonstrated below.If you understand how the previous matrix was formed, it should not be difficult to come up with a matrix that scales a given vector by [\begin \color & \color0 & \color0 & \color0 \ \color0 & \color & \color0 & \color0 \ \color0 & \color0 & \color & \color0 \ \color0 & \color0 & \color0 & \color1 \end \cdot \begin x \ y \ z \ 1 \end = \begin \color\cdot x \ \color\cdot y \ \color\cdot z \ 1 \end].


text:
# Circuit Theory/Active Filters

Active filters have op amps or amplifiers in them. They usually don't have inductors because they are expensive, heavy and large when compared to capacitors.

## Minimal Filter Transfer Functions

Each filter type has a characteristic transfer function:

Filter Type Transfer Function
High Pass $\frac{s^2}{aS^2 + bs + c}$
Low Pass $\frac{1}{aS^2 + bs + c}$
Band Pass $\frac{s}{aS^2 + bs + c}$
Notch $\frac{s^2 + d}{aS^2 + bs + c}$

First the goal is to build a transfer function from the op-amp circuit so that a bode diagram can be plotted. This is a simple matter of substituting s for derivatives in the terminal relations.

## Active Filter Components

Rather than draw an example op amp circuit for each of the above filters, it is easier to draw four op amp circuits that are more related to the desired transfer function:

Circuit Generates Transfer Function
Complex Pole derivation
$-\frac{R_3}{R_1 + (C_2R_1R_2 + C_2R_1R_3 + C_2R_2R_3)s + C_1C_2R_1R_2R_3s^2}$
Zero at Origin $-sRC$
Real Pole $\frac{-\frac{1}{R_1C}}{s+\frac{1}{R_2C}}$
Real Zero $-R_2C(s + \frac{1}{R_1C})$

Most filter design starts with the desired transfer function. Then the circuit is built. With the above four configurations, any transfer function can be created.

Better filters are have stepper slopes and narrower bands in their bode diagrams. This is measured with a factor called Q for quality. This is why there are so many different types of filter designs.

## Under Constrained Design

Suppose you wanted this transfer function:

$\frac{s+1}{(s+2)(s+3)(s+4)}$

Then an active filter could be built like this:

Choosing values to match the above transfer function results in a table like this:

Input Stage Variables Feedback stage variables
$R_{1_A}C_A = 1$ $R_{2_A}C_A = 1$
$R_{1_B}C_B = \frac{1}{2}$ $R_{2_B}C_B = 1$
$R_{1_C}C_C = \frac{1}{3}$ $R_{2_C}C_C = 1$
$R_{1_D}C_D = \frac{1}{4}$ $R_{2_D}C_D = 1$

There are 12 unknowns and 8 equations. This is called an under constrained problem. It means that one can apply other design rules. For example, capacitors are more expensive than resistors. Buying in bulk is less expensive. So ... choose one value for all the capacitors and then adjust resistor values. Choose C = 1μF Then all the resistors are 1 M&Ohm; accept R1B, R1C and R1D.