text:
# Brian Goetz Speaks to InfoQ on Data Classes for Java

| by Michael Redlich 11 Followers on Feb 21, 2018. Estimated reading time: 11 minutes |

A note to our readers: You asked so we have developed a set of features that allow you to reduce the noise: you can get email and web notifications for topics you are interested in. Learn more about our new features.

On his continuing quest for productivity and performance in the Java programming language, Brian Goetz, Java Language Architect at Oracle, introduced an experimental concept of data classes that has potential to someday be integrated into the language. His research demonstrates a natural fit of data classes with up-and-coming features such as value types and pattern matching. But there is much work to be done before this concept is ready to become part of the Java language. Goetz explored the problems and tradeoffs of data classes on the premise that sometimes "data is just data."

### Motivation

Java classes typically require lots of boilerplate code regardless of how simple or how complex those classes may be. This has lead to Java's reputation of being "too verbose." Goetz explains:

To write a simple data carrier class responsibly, we have to write a lot of low-value, repetitive code: constructors, accessors, equals(), hashCode(), toString(), etc. And developers are sometimes tempted to cut corners such as omitting these important methods, leading to surprising behavior or poor debuggability, or pressing an alternate but not entirely appropriate class into service because it has the "right shape" and they don't want to define yet another class.

IDEs will help you write most of this code, but writing code is only a small part of the problem. IDEs don't do anything to help the reader distill the design intent of "I'm a plain data carrier for x, y, and z" from the dozens of lines of boilerplate code. And repetitive code is a good place for bugs to hide; if we can, it is best to eliminate their hiding spots outright.

Similar to class declarations defined in Scala (case), Kotlin (data) and C# (record) that are designed to be compact, the same could potentially be true for a Java class to be a plain data carrier with a minimum of overhead. Without a formal definition of a plain data carrier, most Java developers would most-likely be unable to recognize one. And while the Java community would indeed welcome a data class mechanism in the language, individual interpretations of a plain data carrier could be vastly different. Goetz used the parable of the blind men and an elephant to explain:

Algebraic Annie will say "a data class is just an algebraic product type." Like Scala's case classes, they come paired with pattern matching, and are best served immutable (and for dessert, Annie would order sealed interfaces).

Boilerplate Billy will say "a data class is just an ordinary class with better syntax", and will likely bristle at constraints on mutability, extension, or encapsulation (Billy's brother, JavaBean Jerry, will say "these must be for JavaBeans -- so of course I get getters and setters too." And his sister, POJO Patty, remarks that she is drowning in enterprise POJOs, and reminds us that she'd like these to be proxyable by frameworks like Hibernate).

Tuple Tommy will say "a data class is just a nominal tuple" -- and may not even be even expecting them to have methods other than the core Object methods -- they're just the simplest of aggregates (he might even expect the names to be erased, so that two data classes of the same "shape" can be freely converted).

Values Victor will say "a data class is really just a more transparent value type."

All of these personae are united in favor of "data classes" -- but have different ideas of what data classes are, and there may not be any one solution that makes them all happy.

### Understanding the Problem

The concept of data classes goes beyond reduction in boilerplate code, which Goetz maintains is "just a symptom of a deeper problem" in which the cost of encapsulation is shared among all Java classes. The object-oriented principles of abstraction and encapsulation allow Java developers to write robust and safe code across various boundaries:

• Maintenance boundaries
• Security and trust boundaries
• Integrity boundaries
• Versioning boundaries

For classes such as SocketInputStream, these boundaries are essential due to its inherent complexity. But does a class that is a plain data carrier for, say, two integer components (such as the example declared below) really need to be concerned with such boundaries?

record Point(int x,int y) { ... }

Goetz explains:

Since the cost of establishing and defending these boundaries (how constructor arguments map to state, how to derive the equality contract from state, etc.) is constant across classes, but the benefit is not, the cost may sometimes be out of line with the benefit. This is what Java developers mean by "too much ceremony" -- not that the ceremony has no value, but that they're forced to invoke it even when it does not offer sufficient value.

The encapsulation model that Java provides -- where the representation is entirely decoupled from construction, state access, and equality -- is just more than many classes need. Classes that have a simpler relationship with their boundaries can benefit from a simpler model where we can define a class as a thin wrapper around its state, and derive the relationship between state, construction, equality, and state access from that.

Further, the costs of decoupling representation from API goes beyond the overhead of declaring boilerplate members; encapsulation is, by its nature, information-destroying.

### Requirements for Data Classes

Using the Point declaration above, consider its "de-sugared" definition as a plain data carrier:

final class Point extends java.lang.DataClass {
public final int x;
public final int y;

public Point(int x,int y) {
this.x = x;
this.y = y;
}

// destructuring pattern for Point(int x,int y)
// state-based implementations of equals(), hashCode(), and toString()
// public read accessors x() and y()
}

To further study the design of plain data carriers, Goetz defined a set of requirements (or constraints) to "safely and mechanically generate the boilerplate for constructors, pattern extractors, accessors, equals(), hashCode(), and toString() -- and more." He writes:

We say a class C is a transparent carrier for a state vector S if:

• There is a function ctor : S -> C which maps an instance of the state vector to an instance of C (the constructor may reject some state vectors as invalid, such as rational numbers whose denominator is zero).
• There is a total function dtor : C -> S which maps an instance of C to a state vector S in the domain of ctor.
• For any instance c of C, ctor(dtor(c)) is equal to c, according to the equals() contract for C.
• For two state vectors s1 and s2, if each of their components is equal to the corresponding component of the other (according to the component's equals() contract), then either ctor(s1) and ctor(s2) are both undefined, or they are equals under the equals() contract for C.
• For equivalent instances c and d, invoking the same operation produces equivalent results: c.m() equals d.m(). Moreover, after the operation, c and d should still be equivalent.

These invariants are an attempt to capture our requirements; that the carrier is transparent, and that there is a simple and predictable relationship between the classes representation, its construction, and its destructuring -- that the API is the representation.

### Data Classes and Pattern Matching

A plain data carrier has the advantage, as Goetz states, "to freely convert a data class instance back and forth between its aggregate form and exploded state." This would work conveniently well with pattern matching. As demonstrated in his pattern matching paper, Goetz discussed destructuring and improvements in utilizing the switch construct. With this in mind, it could be possible to write the following code:

interface Shape { ... }
record Point (int x,int y) { ... }
record Rect(Point p1,Point p2) implements Shape { ... }
record Circle(Point center,int radius) implements Shape { ... }

...

switch(shape) {
case Rect(Point(var x1,var y1),Point(var x2,var y2)) : ...
case Circle(Point(var x,var y),int radius): ...
}

Any concrete instance of Shape could easily be destructured within the switch statement. This could also be useful for externalization such as serialization, marshalling to/from JSON and XML, and database mapping.

### Refining the Design Space

Goetz discussed that the requirements for being a plain data carrier comes with trade-offs. He explains:

The simplest -- and most draconian -- model for data classes is to say that a data class is a final class with public final fields for each state component, a public constructor and deconstruction pattern whose signature matches that of the state description, and state-based implementations of the core Object methods, and further, that no other members (or explicit implementations of the implicit members) are allowed. This is essentially the strictest interpretation of a nominal tuple.

This starting point is simple and stable -- and nearly everyone will find something to object to about it. So, how much can we relax these constraints without giving up on the semantic benefits we want? Let's look at some directions in which the draconian starting point could be extended, and their interactions.

These directions cover a wide array of design elements and related issues:

• Risk violating the "nothing but the state" rule.
• Overriding implicit members
• Risk violating the requirements of a plain data carrier.
• Ensure the object state and state description are equivalent.
• Risk violating "the state, the whole state, and nothing but the state" rule.
• Extension
• Issues related to extension between data classes and regular classes.
• Mutability
• Question the rationale of allowing data classes to be mutable.
• Field encapsulation and accessors
• Ensure that encapsulating fields must be readable.
• Arrays and defensive copies
• Defensive copies violate the invariant of destructuring and reconstructing an array to ensure an equal instance.
• Question how mutability in data classes can be thread safe.

### Summary

Java had a excellent year in 2017 and there is much excitement about the language this year. However, as Goetz told InfoQ, data classes are still considered a "half-baked" idea that requires more work to fully understand how this concept can someday be a reality.

In summary, Goetz explains:

The key question in designing a facility for "plain data aggregates" in Java is identifying which degrees of freedom we are willing to give up. If we try to model all the degrees of freedom of classes, we just move the complexity around; to gain some benefit, we must accept some constraints. We think that the sensible constraints to accept are disavowing the use of encapsulation for decoupling representation from API, and for mediating read access to state; in turn, this provides significant syntactic and semantic benefits for classes which can accept these constraints.

Vicente Romero, principal member of the technical staff at Oracle, recently posted an "initial public push" on the development of data classes that can be found on the datum branch of the Project Amber repository.

Goetz spoke to InfoQ about his data classes research:

InfoQ: What kind of community response have you received since publishing your paper?

Brian Goetz: The expected response: some highly positive comments about the idea, and a variety of suggestions (mostly mutually inconsistent) for how it could be "improved." Which is to say, people like the idea, but, as expected, many people would want us to move the design center in one direction, or another, to suit their personal preferences. As a highly subjective feature, this was to be expected.

InfoQ: Do you envision a data class mechanism to someday be integrated in the Java programming language? If so, what kind of effort will be necessary to address all the concerns you discussed in your paper?

Goetz: It is going to require "bake time." With language design, your first idea, no matter how carefully thought out, is going to be wrong. As will your second. Many language features require half a dozen iterations or more before you ultimately discover the right place to land. So we'll be experimenting, prototyping, gathering feedback, iterating, and iterating again. Until we feel we've gotten to the right place.

InfoQ: Is it a goal or a non-goal to promote the rebasing of non-Java languages implementation of compact classes (e.g., Scala case classes) on top of data classes?

Goetz: Every language is going to have its own surface syntax. However, data classes connect with other language features, such as pattern matching, and and we hope (as happened with Lambda) that other languages will target the runtime support of these features, and gain interoperability benefits.

InfoQ: As far as you may know, did the architects of Scala, Kotlin, and C# face similar challenges in implementing a more compact class declaration?

Goetz: Indeed so, though both Kotlin and Scala were able to take this on much closer to the beginning of their projects than C# did, so had fewer constraints to navigate. And each settled in a slightly different point in the design space.

InfoQ: What is the single most important take-home message you would like our readers to know about data classes?

Goetz: That data classes are about data, not about syntactic concision. They are about providing a natural means to model pure data in the object model. And not all classes are carriers for pure data, even if they would like the concision benefits that data classes offer.

InfoQ: What's on the horizon for your data classes research?

Goetz: Breaking the features that data classes need into finer-grained features, that might be usable by all classes. For example, even in classes that are clearly not just data carriers, constructors are full of error-prone repetition, which could be replaced by making a higher-level correspondence between constructor parameters and representation. This way, data classes become simpler (just sugar for other language features), and more classes can get the benefit of the feature without trying to shoehorn them into data classes.

Style

## Hello stranger!

You need to Register an InfoQ account or or login to post comments. But there's so much more behind being registered.

## Get the most out of the InfoQ experience.

### Tell us what you think

Allowed html: a,b,br,blockquote,i,li,pre,u,ul,p

Email me replies to any of my messages in this thread

Why not to just include @Data in Java?

It would be simpler just to include @Data annotation from kombo in Java:
projectlombok.org/features/Data

If you add a feature to the language that don't fix a big problem you're making a worse language.

Forget Java

Move to Scala, you already have Data Classes and so much more.

Re: Why not to just include @Data in Java?

Hi Javier:

I'm not sure Lombok's @Data annotation is a viable replacement for a proposed data class mechanism. Remember, this isn't about boilerplate reduction. It's about the ability to write a plain data carrier that is simply just that - a data carrier.

While Lombok is indeed useful, it does generate the boilerplate for you. And Brian Goetz stated this is where bugs find find their way into your code.

Mike.

Re: Forget Java

Hi Serge,

Move to Scala, you already have Data Classes and so much more.

That is just the reason to not move to Scala. When I moved from C++ to Java (in 1997) Java had less features than C++, it was nice just for that.

Re: Why not to just include @Data in Java?

Hi Mike,

write a plain data carrier that is simply just that - a data carrier.

But this was available in Java 1.0. If you add a feature that allows you to write less code, or to do something completely new, that is ok, but in any other case you're complicating the language.

Re: Forget Java

Scala has many interesting ideas, but I think it comes with a lot of baggage as well.
Powerful yet dangerous features, like scala implicits, are widely abused across many, many projects, and it makes for untenable code.

Don't get it

How about fixing Java's broken "fake" generics instead so that we have runtime info? That would be a lot more useful than this.

Re: Forget Java

Scala is a much smaller language than Java ! So, following your reasoning you should move to it. It is however much more flexible and consistent, so you can do much more with much less code. So go take a look.

Re: Forget Java

What a pile of misconceptions, too bad for you

Re: Why not to just include @Data in Java?

How so? If this was true, why is Brian Goetz experimenting with this idea of data classes?

Re: Don't get it

It's being developed in Valhalla project. Yet no one knows when it's ready.

Re: Forget Java

What do you mean by smaller? It may take more characters to write something in Java, but the syntax is an order of magnitude less complex than Scala.

Re: Forget Java

Simply not true, I'm afraid. Java has 53 keywords and a regular (you might say rigid) syntax. There are no legal Java programs which have ambiguous meaning at a grammatical level. The same is just not true of Scala, due to the fluidity of the syntax (e.g. indirect object & block syntax, implicits, macros, etc, etc, etc). Once your Scala project is above a relatively small size, good luck figuring out what any of it does without an IDE.

Re: Forget Java

I feel like people who say things like "Scala is smaller than Java" probably only use 1/10 of the syntax -- meaning they use Scala to write compact Java code.

Why not a first class struct/record/case type?

Forgive my naivety (I don't know much about language design), but why not implement a first-class language construct? Extending "java.lang.DataClass" with a tradition class declaration seems like more of the boilerplate people already hate in Java.

Re: Forget Java

I looked. Scala is a "Write Only Language" - like Perl.

Re: Forget Java

Verbosity of syntax isn't there for the compiler, it's there for the HUMANS. It makes is readable and understandable by mere mortals.

I'm so tired of this obsession with languages. The language was perfectly capable of building anything as of Java 1.5. Why not focus on building something useful with it instead fussing with the mechanics of it. People who invest their time arguing about languages and trying to improve things that don't need improvement will never accomplish anything. In fact, quite the opposite, they provide a terrible distraction for 70% of the rest of the development community that don't have the sense to ignore them.

Re: Forget Java

No, it is the number of concepts, and you got it the wrong way, Java is order of magnitude more complex than Java. Also ther is so many things that cannot be expressed in Java

Re: Forget Java

Thanks, your providing a good laugh

Re: Forget Java

You are free to stay behind. But languages can improve, I hope you will understand that.

Re: Forget Java

I think you are confusing "lines of code" for complexity. Language complexity is about features and the syntax you use to implement things. Scala is a moderate-highly complex language (certainly the most complex of the popular JVM languages). Scala basically has every feature Java has and then adds:

- import statements that can be included anywhere
- case classes
- companion objects
- implicit parameters
- operators as methods (that don't require classic invocation syntax [dot operator])
- multiple parameter lists
- much more complex member access control (e.g. private, protected, etc.)
- pattern matching as a first-class language feature
- anonymous and nested functions
- function currying
- lazy evaluation
- tuples
- lazy evaluation
- immutability enforcement

And I'm sure many more language features I haven't mentioned.

Scala is not a bad language - it simply takes a lot more knowledge and experience to use it correctly. Yes, you can express things more concisely in Scala; but you can also write incomprehensible code by overusing language features.

Re: Forget Java

You named many Scala features, but still miss many.
The main thing is how you can mix those features, Java is very inflexible. it’s generic handling is catastrophic, the book on good Java use is half about generic strange behaviors (fixed in Scala), you can restrict the lower and upper bound of generic types, you have much better control about privacy of your data. You also have more level of privacy available and can even control to which class family it applies, Java is so incredibly missing control.
Do not forget the pathetic Java collections, or the actor support (Java even with Akka is much weaker)

You can compose in FP, not in Java (many orders more flexible)

And I am forgetting many other advantages, Java is to Scala much less than Assembler is to Java

Re: Forget Java

You just admitted that Scala is crammed full of features (way more than Java). More features = more complexity -- it's really that simple.

Amnemic classes

I'm worried this wil lead some developers to amnemic classes: data, but no behaviour

Re: Forget Java

No I am not, Java has a lot more exceptions than features that you get to know.
Plus, Scala is much more consistant, so most feature works as you would think, contrary to Java.
So because of this consistency, it is much simpler. Java wil never reach that level as you cannot build on shaky fundation.
Actually, the only inconsistency found in Scala are because the Java link. Fortunately Scala is smoothing a lot the inconvenience.
So, Java is much more complex.

It is contraproductive to my current state of coding

Just a joke: I use lombok and my data structures look only like pojo with variables, the rest is generated on the fly by lombok. With Data class I will need to write more code: "extends DataClass", so it is contraproductive for me :-)))))

Re: Amnemic classes

totally agree.

Re: It is contraproductive to my current state of coding

You won't have to explicitly write out "extends java.util.DataClass." It's part of the syntactic sugar similar to java.lang.Object. That's why I wrote, "consider its 'de-sugared' definition." java.util.DataClass doesn't exist yet. It's a placeholder that Brian Goetz used for the example.

Hope this helps...

Mike.

Re: It is contraproductive to my current state of coding

aha, I will read again. Thx for clarification.

Re: Amnemic classes

On the other hand Hans, when you have objects to be exchanged between 2 systems (so we serialize on one end and deserialize on other end), we use already non-behavioural classes, these DTOs simply transports data and the is the only function they have.
Close

#### by

on

Allowed html: a,b,br,blockquote,i,li,pre,u,ul,p

Email me replies to any of my messages in this thread

Allowed html: a,b,br,blockquote,i,li,pre,u,ul,p

Email me replies to any of my messages in this thread

30 Discuss

Login to InfoQ to interact with what matters most to you.


text:
### Home > INT3 > Chapter 10 > Lesson 10.2.2 > Problem10-131

10-131.

What fraction is equivalent to the repeating decimal $0.\overline{27}$?

$100n = 27.\overline{27}$
$- n = 0.\overline{27}$

format: plain


text:
# American Institute of Mathematical Sciences

October  2010, 26(4): 1291-1304. doi: 10.3934/dcds.2010.26.1291

## Energetic variational approach in complex fluids: Maximum dissipation principle

 1 Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN 55455, United States 2 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea 3 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received  November 2008 Revised  April 2009 Published  December 2009

We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy law. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic systems in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic systems. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous $C^0$ finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.
Citation: Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291
 [1] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110 [2] Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 [3] Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57 [4] Shuguang Shao, Shu Wang, Wen-Qing Xu. Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation. Kinetic and Related Models, 2018, 11 (1) : 179-190. doi: 10.3934/krm.2018009 [5] Hongjie Dong, Dong Li. On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3437-3454. doi: 10.3934/dcds.2014.34.3437 [6] Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [7] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [8] Yanlin Liu, Ping Zhang. Remark on 3-D Navier-Stokes system with strong dissipation in one direction. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2765-2787. doi: 10.3934/cpaa.2020244 [9] Yat Tin Chow, Ali Pakzad. On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021270 [10] Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure and Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735 [11] Jie Jiang, Yinghua Li, Chun Liu. Two-phase incompressible flows with variable density: An energetic variational approach. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3243-3284. doi: 10.3934/dcds.2017138 [12] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [13] H. O. Fattorini. The maximum principle in infinite dimension. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557 [14] Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361 [15] José Luiz Boldrini, Luís H. de Miranda, Gabriela Planas. On singular Navier-Stokes equations and irreversible phase transitions. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2055-2078. doi: 10.3934/cpaa.2012.11.2055 [16] M. Hassan Farshbaf-Shaker, Harald Garcke. Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 371-389. doi: 10.3934/dcdss.2011.4.371 [17] Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397 [18] Qiang Du, Manlin Li, Chun Liu. Analysis of a phase field Navier-Stokes vesicle-fluid interaction model. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 539-556. doi: 10.3934/dcdsb.2007.8.539 [19] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 [20] Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021059

2020 Impact Factor: 1.392


text:
# $|V_{ub}|$ determination in lattice QCD

F. Bahr, F. Bernardoni, B. Blossier, J. Bulava, M. Della Morte, P. Fritzsch, N. Garron, A. Gérardin, J. Heitger, G. von Hippel, A. Ramos, H. Simma, R. Sommer

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

### Resumé

The 2012 PDG reports a tension at the level of $3 \sigma$ between two exclusive determinations of $|V_{ub}|$. They are obtained by combining the experimental branching ratios of $B \to \tau \nu$ and $B \to \pi l \nu$ (respectively) with a theoretical computation of the hadronic matrix elements $\fB$ and the $B \to \pi$ form factor $f_+(q^2)$. To understand the tension, improved precision and a careful analysis of the systematics involved are necessary. We report the results of the ALPHA collaboration for $\fB$ from the lattice with 2 flavors of $O(a)$ improved Wilson fermions. We employ HQET, including $1/m_b$ corrections, with pion masses ranging down to $\approx$ 190 MeV. Renormalization and matching were performed non-perturbatively, and three lattice spacings reaching $a^{-1}\approx 4.1$ GeV are used in the continuum extrapolation. We also present progress towards a computation of $f_+(q^2)$, to directly compare two independent exclusive determinations of $|V_{ub}|$ with each other and with inclusive determinations. Additionally, we report on preliminary results for $\fBq{s}$, needed for the analysis of $B_s \to \mu^+\mu^-$.}
Originalsprog Udefineret/Ukendt PoS ICHEP Udgivet - 27. nov. 2012

• hep-lat

### Citer dette

Bahr, F., Bernardoni, F., Blossier, B., Bulava, J., Morte, M. D., Fritzsch, P., ... Sommer, R. (2012). $|V_{ub}|$ determination in lattice QCD. PoS ICHEP.
Bahr, F. ; Bernardoni, F. ; Blossier, B. ; Bulava, J. ; Morte, M. Della ; Fritzsch, P. ; Garron, N. ; Gérardin, A. ; Heitger, J. ; Hippel, G. von ; Ramos, A. ; Simma, H. ; Sommer, R. / $|V_{ub}|$ determination in lattice QCD. I: PoS ICHEP. 2012.
@article{ca53fe997558490cb50b9cb054c222d3,
title = "$|V_{ub}|$ determination in lattice QCD",
abstract = "The 2012 PDG reports a tension at the level of $3 \sigma$ between two exclusive determinations of $|V_{ub}|$. They are obtained by combining the experimental branching ratios of $B \to \tau \nu$ and $B \to \pi l \nu$ (respectively) with a theoretical computation of the hadronic matrix elements $\fB$ and the $B \to \pi$ form factor $f_+(q^2)$. To understand the tension, improved precision and a careful analysis of the systematics involved are necessary. We report the results of the ALPHA collaboration for $\fB$ from the lattice with 2 flavors of $O(a)$ improved Wilson fermions. We employ HQET, including $1/m_b$ corrections, with pion masses ranging down to $\approx$ 190 MeV. Renormalization and matching were performed non-perturbatively, and three lattice spacings reaching $a^{-1}\approx 4.1$ GeV are used in the continuum extrapolation. We also present progress towards a computation of $f_+(q^2)$, to directly compare two independent exclusive determinations of $|V_{ub}|$ with each other and with inclusive determinations. Additionally, we report on preliminary results for $\fBq{s}$, needed for the analysis of $B_s \to \mu^+\mu^-$.}",
keywords = "hep-lat",
author = "F. Bahr and F. Bernardoni and B. Blossier and J. Bulava and Morte, {M. Della} and P. Fritzsch and N. Garron and A. G{\'e}rardin and J. Heitger and Hippel, {G. von} and A. Ramos and H. Simma and R. Sommer",
year = "2012",
month = "11",
day = "27",
language = "Udefineret/Ukendt",
journal = "PoS ICHEP",

}

Bahr, F, Bernardoni, F, Blossier, B, Bulava, J, Morte, MD, Fritzsch, P, Garron, N, Gérardin, A, Heitger, J, Hippel, GV, Ramos, A, Simma, H & Sommer, R 2012, '$|V_{ub}|$ determination in lattice QCD', PoS ICHEP.

$|V_{ub}|$ determination in lattice QCD. / Bahr, F.; Bernardoni, F.; Blossier, B.; Bulava, J.; Morte, M. Della; Fritzsch, P.; Garron, N.; Gérardin, A.; Heitger, J.; Hippel, G. von; Ramos, A.; Simma, H.; Sommer, R.

I: PoS ICHEP, 27.11.2012.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - $|V_{ub}|$ determination in lattice QCD

AU - Bahr, F.

AU - Bernardoni, F.

AU - Blossier, B.

AU - Bulava, J.

AU - Morte, M. Della

AU - Fritzsch, P.

AU - Garron, N.

AU - Gérardin, A.

AU - Heitger, J.

AU - Hippel, G. von

AU - Ramos, A.

AU - Simma, H.

AU - Sommer, R.

PY - 2012/11/27

Y1 - 2012/11/27

N2 - The 2012 PDG reports a tension at the level of $3 \sigma$ between two exclusive determinations of $|V_{ub}|$. They are obtained by combining the experimental branching ratios of $B \to \tau \nu$ and $B \to \pi l \nu$ (respectively) with a theoretical computation of the hadronic matrix elements $\fB$ and the $B \to \pi$ form factor $f_+(q^2)$. To understand the tension, improved precision and a careful analysis of the systematics involved are necessary. We report the results of the ALPHA collaboration for $\fB$ from the lattice with 2 flavors of $O(a)$ improved Wilson fermions. We employ HQET, including $1/m_b$ corrections, with pion masses ranging down to $\approx$ 190 MeV. Renormalization and matching were performed non-perturbatively, and three lattice spacings reaching $a^{-1}\approx 4.1$ GeV are used in the continuum extrapolation. We also present progress towards a computation of $f_+(q^2)$, to directly compare two independent exclusive determinations of $|V_{ub}|$ with each other and with inclusive determinations. Additionally, we report on preliminary results for $\fBq{s}$, needed for the analysis of $B_s \to \mu^+\mu^-$.}

AB - The 2012 PDG reports a tension at the level of $3 \sigma$ between two exclusive determinations of $|V_{ub}|$. They are obtained by combining the experimental branching ratios of $B \to \tau \nu$ and $B \to \pi l \nu$ (respectively) with a theoretical computation of the hadronic matrix elements $\fB$ and the $B \to \pi$ form factor $f_+(q^2)$. To understand the tension, improved precision and a careful analysis of the systematics involved are necessary. We report the results of the ALPHA collaboration for $\fB$ from the lattice with 2 flavors of $O(a)$ improved Wilson fermions. We employ HQET, including $1/m_b$ corrections, with pion masses ranging down to $\approx$ 190 MeV. Renormalization and matching were performed non-perturbatively, and three lattice spacings reaching $a^{-1}\approx 4.1$ GeV are used in the continuum extrapolation. We also present progress towards a computation of $f_+(q^2)$, to directly compare two independent exclusive determinations of $|V_{ub}|$ with each other and with inclusive determinations. Additionally, we report on preliminary results for $\fBq{s}$, needed for the analysis of $B_s \to \mu^+\mu^-$.}

KW - hep-lat

M3 - Tidsskriftartikel

JO - PoS ICHEP

JF - PoS ICHEP

ER -

Bahr F, Bernardoni F, Blossier B, Bulava J, Morte MD, Fritzsch P et al. $|V_{ub}|$ determination in lattice QCD. PoS ICHEP. 2012 nov 27.


text:
## dnd 4e – Do I have to hold the Power Jewel to use it?

Power ♦ Daily (Minor Action)
This power allows you to regain the use of an encounter power of 1st or 3rd level.

Do I have to get it from my backpack before using it?

What about items that are normally1 used, like a Whistle of Warning2?

1. Normally meaning their mundane versions have some applications, as opposed to a Power Jewel
2. Adventurer’s Vault 2, page 78: Encounter(Minor) Each ally within 10 squares of you who can hear the whistle can shift 1 square and draw a weapon or retrieve an implement as a free action.

## user expectation – Terminology: Long press or Touch & hold?

I would suggest “Touch and hold”. Many users tend not to understand the term “long” in longpress. Therefore “Press and (then) hold” is a better description of the action.

### Edit

OK, due to the comments below, here is a more descriptive answer 🙂

“Longpress” means “press for a long time”.The word “long” comes first but that is the second action the user has to commit. We also have to tell the user how much time (s)he shall spend during that “long” time.

The word “longpress” is not (yet) a term that common people understand.

So, every term beginning with the word “long…” aught to be avoided.

“Press and hold” on the other hand tells the user to 1: press on the item and then 2: keep on doing so until something else happens, something that will inform the user (s)he can release his/her finger from the device.

## equipment protection – Is it safe to hold DSLR with hot shoe mounted top handle?

Most likely not. This is a closeup of the hotshoe:

Image by Xeviro from English Wikipedia, used under CC BY-SA 3.0. Source:

It’s only the two thin metal rails which are essentially holding your camera. Personally I’d have no issues picking it up once or twice like that, but extended use likely will see the rails be bent.

Alternatively, as the accepted answer to the question you asked mentions, a weaker link is connecting the hotshoe to the camera body. You may find that the hotshoe is still connected to your handle, but the camera is not.

I wouldn’t risk damaging the camera. You can find handles that use the tripod screw mount, which is actually designed to hold the weight of the camera and also supports it from the bottom (rather than hanging from it).

## dnd 5e – Can I cast hold person, and have my Hound of Ill Omen attack, on the turn I summon the hound if our initiatives match?

If my Shadow Sorcerer survives to level 6, I will have a Hound of Ill Omen.

As a Bonus Action, you can spend 3 Sorcery Points to magically summon a hound of ill omen to target one creature you can see within 120 feet of you.
{snip stats and moving through objects and creatures}
The hound appears in an unoccupied space of your choice within 30 feet of the target.

I will summon it next to the target for the purposes of this question. (within 5′)

Roll Initiative for the hound.

For this question, the Hound’s initiative roll ends up being the same as mine.

Additionally, while the hound is within 5 feet of the target, the target has disadvantage on Saving Throws against any spell you cast.

Here is where the order of operation concern comes in.

1. I have used a bonus action to summon the Hound

2. The Hound can attack on its turn

3. I can use an Action to cast a spell: I specifically want to cast Hold Person.

It is my understanding that the target will have disadvantage on the saving throw versus Hold Person. If the target fails its save, the Hound will be able to attack the held target with advantage, and will have an automatic critical on any hit.

Is there a problem with this order of operations? The bonus action should be usable by me before I cast that spell, and summoning the Hound does not appear to be a ‘cast a spell’ action.

On the rare occasion that our initiative aligns, I’d like to take full advantage of it.

Will this work?

## Get hold of new DirectAdmin Extended and WordPress Manager For WHMCS! | Proxies123.com

1. DirectAdmin Extended For WHMCS 3.4.0

Mindful of the fact that the demand for powerful software solutions rises along with the dynamics of business growth, we have brought some tweaks into our pivotal DirectAdmin Extended For WHMCS 3.4.0 module – all for your and your clients’ ultimate convenience!

Apart from the newly established ability to take control over their DirectAdmin DNS records via the client area, your clients are now given the chance to set up an email address gathering all incorrectly addressed messages owing to the “Catch All Email” functionality. But there are even more of these fine additions your audience is now privileged to employ! One of these is the power to view error and usage logs for each of the created subdomains independently. There are also such noble additions as the ability to create wildcard SSL certificates with Let’s Encrypt, plus the module’s scope of configurable options has embraced a few extras as well.

Seeking a more client-focused business approach?

Embrace the golden solutions from DirectAdmin Extended For WHMCS 3.4.0!

2. WordPress Manager For WHMCS 1.5.0

Many would agree that there is no greater comfort than the one that WordPress instance owners must feel when supervising their installations via a single WHMCS platform. Wish your clients to be among them? Give a try to our very latest WordPress Manager For WHMCS in the 1.5.0 edition!

Equipped with this tool, you will be enabled to keep your audience perfectly posted on each newly available WordPress update via informative email notifications and alerts visible next to owned instance. The module will also allow you to:

• Forge your own tailor-made plugins and themes as an extra source of choice for your clients beyond the official WordPress resources.
• Set a limit of WordPress installations reachable per one hosting account.
• Define the range of options you wish to be available in your client area.
• Create a blacklist of themes which shall remain inaccessible to your clients.

And last on this list – but not in the changelog! – is the newly gained possibility by your clients to swiftly and conveniently mass delete their WordPress installations.

Thirsty to find out what further perks await your attention?

Learn all about WordPress Manager For WHMCS 1.5.0!

3. Make sure to follow our other latest releases as well:

Specially for you we will adapt an application and its design to your own needs, create a new module or even a completely new system built from scratch!

## c# – List to hold copies of a prefab for later deletion

A List<string> is all well and good if you just want names. But you don’t want to delete the name, you want to delete the game object.

Also, if you always want to delete from the start of the collection (First-In, First-Out), and you don’t need to traverse or randomly access items in the collection, then the structure you want is a Queue.

So, replace:

List<string> obstacleList = new List<string>(10);

with:

Queue<GameObject> obstacles = new Queue<GameObject>(10);

Next, capture the newly-created instance when you make it:

var instance = Instantiate(Obstacle, new Vector3(nextObstacle + randH, randV, 0), Quaternion.identity);

obstacles.Enqueue(instance);

Your removal can then look like this (put it before you add so you don’t exceed your promised capacity):

if (obstacles.Count >= 10)
{
var toBeDestroyed = obstacles.Dequeue();
Destroy(toBeDestroyed);
}

## Can a Ledger Nano X hold multiple BitCoin Wallets?

From my reading of Ledger Nano X, it can hold multiple types of currency, for instance, BitCoin and Ethereum.

Can it hold two or more separate BitCoin wallets with different balances?

Frank

## Warning Newbie: How does a Ledger hold multiple currencies?

So you startup your ledger, copy down your 24 words, and you now have a public and private key. (I think). Now you add the BitCoin App to your Ledger and you can add BTC to your “hardware wallet” using those keys. (Though phrasing it that way is a bit of a misnomer since the only thing your Ledger ever really stores is your private/public keys.)

So far so good for one currency. However, a Ledger will hold multiple currencies. That I can find there is no asking for another 24 words. You just add the second app and you now have a wallet in that new currency. So I assume that means that if you purchase Ethereum coins it is going to use the same public/private keys that you used on BitCoin?

That would also imply that you cannot have two BitCoin wallets on the same Ledger because attempting to do so wouldn’t make sense without a second set of keys which the Ledger doesn’t have.

Is what I am saying correct?

Frank

## What properties does the linear space of all convergent sequences in R with the sup norm hold? And why?

I am thinking that the space is complete, and maybe there are certain sets that are compact in the space, and X* is isomorphic to the space with the sup norm.
But I totally got stuck with the proofs, any hints?

## linear algebra – Does the eigenvalue equality hold for my expression?

Let

$$g(boldsymbol{theta},boldsymbol{theta_0}) = trace ( boldsymbol{Omega{(boldsymbol{theta})}}^{-1} boldsymbol{Omega{(boldsymbol{theta_0})}})-ln(det(boldsymbol{Omega{(boldsymbol{theta_0})}})/det(boldsymbol{Omega{(boldsymbol{theta})}}))-N$$

where $$boldsymbol{theta} in boldsymbol{Theta}$$ with $$boldsymbol{Theta}$$ a compact subset of $$R^{n}$$, $$n$$ and $$N$$ are fixed numbers, and $$boldsymbol{theta_0}$$ belongs to the interior of $$boldsymbol{Theta}$$.

Denote the eigenvalues of the symmetric matrices $$boldsymbol{Omega{(boldsymbol{theta_0})}}$$ and $$boldsymbol{Omega{(boldsymbol{theta})}}$$ by $$lambda_{0s}$$ and $$lambda_s$$ $$(s=1,2,…,N)$$ respectively, where $$lambda_{0s}>0$$ and $$lambda_s>0$$ for all $$s$$.

Given the above, does the following hold or is a further condition required, and if so which one?

$$g(boldsymbol{theta},boldsymbol{theta_0}) = sum_{s=1}^N ((lambda_{0s}/lambda_{s})-ln(lambda_{0s}/lambda_{s})-1) ?$$


text:
### Power-based Side Channel Attack Analysis on PQC Algorithms

Tendayi Kamucheka, Michael Fahr, Tristen Teague, Alexander Nelson, David Andrews, and Miaoqing Huang

##### Abstract

Power-based side channel attacks have been successfully conducted against proven cryptographic algorithms including standardized algorithms such as AES and RSA. These algorithms are now supported by best practices in hardware and software to defend against malicious attacks. As NIST conducts the third round of the post-quantum cryptography (PQC) standardization process, a key feature is to identify the security candidate algorithms have against side channel attacks, and the tradeoffs that must be made to obtain that level of protection. In this work, we document the development of a multi-target and multi-tool platform to conduct test vector leakage assessment of the candidate algorithms. The long-term goals of the platform are to 1) quantify test vector leakage of each of the primary and alternate candidates, 2) quantify test vector leakage of each of the candidates when adjustments and adaptations (e.g., masking) are applied, and 3) assess the equivalent security levels when tools of varying sophistication are used in the attack (e.g., commodity vs. specialized hardware). The goal of this work is to document the progress towards that standardized platform and to invite discussion in how to extend, refine, and distribute our tools.

Note: This paper was presented at the Third NIST PQC Standardization Conference, June 7-9, 2021.

Available format(s)
Category
Implementation
Publication info
Preprint. MINOR revision.
Keywords
Side channel attackPost-quantum cryptographyPower analysis
Contact author(s)
mqhuang @ uark edu
History
Short URL
https://ia.cr/2021/1021

CC BY

BibTeX

@misc{cryptoeprint:2021/1021,
author = {Tendayi Kamucheka and Michael Fahr and Tristen Teague and Alexander Nelson and David Andrews and Miaoqing Huang},
title = {Power-based Side Channel Attack Analysis on PQC Algorithms},
howpublished = {Cryptology ePrint Archive, Paper 2021/1021},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/1021}},
url = {https://eprint.iacr.org/2021/1021}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
Why does iron have an abnormally high ionization energy?

Along a period the ionization energy should increase because the atomic number is increasing, but there is negligible increase in shielding. However, $\mathrm{IE}_\ce{Mn} < \mathrm{IE}_\ce{Fe} > \mathrm{IE}_\ce{Co} > \mathrm{IE}_\ce{Cu}$, so why does iron have such a high ionization energy?

I've considered that iron has a $\mathrm{4s^2 3d^6}$ configuration that goes to a $\mathrm{4s^1 3d^6}$ configuration upon ionization, while $\ce{Co}$ has a $\mathrm{4s^2 3d^7}$ that goes to $\mathrm{4s^0 3d^8}$ configuration, $\ce{Ni}$ has a $\mathrm{4s^2 3d^8}$ that goes to a $\mathrm{4s^0 3d^9}$ configuration and $\ce{Cu}$ has a $\mathrm{4s^2 3d^9}$ that goes to a $\mathrm{4s^0 3d^{10}}$ configuration (Lang, J. Chem. Ed., 2003). In the Long paper, the authors explain the increase of IE at iron is due to "electronic structure" and I would like to know exactly what about the electronic structure it is that causes iron to have a high IE than expected based on a general periodic trend.

I was thinking that adding an electron to the $\mathrm{3d}$ orbitals might be energetically favorable enough to lower the ionization energy of $\ce{Ni}$, $\ce{Co}$ and $\ce{Cu}$, so rather than iron having a high ionization energy the surround metals just have a low ionization energy because the ion is stabilized by having another $\mathrm{d}$-electron added? Does that sound reasonable?

Another thing I was thinking was that iron is the first transition metal to have a paired $\mathrm{d}$-electron, could that effect the shielding and increase the ionization energy?

• The first ionization energy of cobalt is only 0.3% lower than that of iron. So I wouldn't say that iron has an abnormal high ionization energy. – aventurin Aug 26 '17 at 21:00
• @aventurin I mean "abnormal" in that it defies a general periodic trend, the magnitude itself is not abnormal – Mecury-197 Aug 27 '17 at 13:41


text:
# zbMATH — the first resource for mathematics

On abstract algebraic geometry. I. (Russian) Zbl 0284.14016

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 32J15 Compact complex surfaces


text:
Linear Algebra Home

Matrix Operations

Matrix: An $$m\times n$$ matrix $$A$$ is an $$m$$-by-$$n$$ array of scalars from a field (for example real numbers) of the form $A=\left[\begin{array}{cccc} a_{11}&a_{12}&\cdots &a_{1n}\\ a_{21}&a_{22}&\cdots &a_{2n}\\ \vdots&\vdots&\ddots &\vdots\\ a_{m1}&a_{m2}&\cdots &a_{mn} \end{array}\right].$ The order (or size) of $$A$$ is $$m\times n$$ (read as m by n) if $$A$$ has $$m$$ rows and $$n$$ columns. The $$(i,j)$$-entry of $$A=[a_{i,j}]$$ is $$a_{i,j}$$.

For example, $$A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]$$ is a $$2\times 3$$ real matrix. The $$(2,3)$$-entry of $$A$$ is $$-1$$.

Equality: Two matrices $$A$$ and $$B$$ are equal, i.e., $$A=B$$ if $$A$$ and $$B$$ have the same order and the entries of $$A$$ and $$B$$ are the same.

Useful Matrices:

• A zero matrix, denoted by $$O$$ or $$O_{m,n}$$, is an $$m\times n$$ matrix whose all entries are zero.

• A square matrix is a matrix is a matrix whose number of rows and number of columns are the same.

• A diagonal matrix is a square $$n\times n$$ matrix whose nondiagonal entries are zero.

• The identity matrix of order $$n$$, denoted by $$I_n$$, is the $$n\times n$$ diagonal matrix whose diagonal entries are 1. For example, $$I_3=\left[\begin{array}{rrr}1&0&0\\0&1&0\\0&0&1 \end{array} \right]$$ is the $$3\times 3$$ identity matrix.

• An $$n\times 1$$ matrix is called a column matrix or an $$n$$-dimensional (column) vector, denoted by lowercase letters such $$x$$, $$\boldsymbol x$$, or $$\overrightarrow{x}$$. For example, $$\overrightarrow{x}=\left[\begin{array}{r}0\\1\\2 \end{array} \right]$$ is a $$3$$-dimensional vector which represents the position vector of the point $$(0,1,2)$$ in the 3-space $$\mathbb R^3$$ (i.e., a directed line segment from the origin $$(0,0,0)$$ to the point $$(0,1,2)$$).

Position vector of a point in the 2-space $$\mathbb R^2$$

Matrix Operations:

Transpose: The transpose of an $$m\times n$$ matrix $$A$$, denoted by $$A^T$$, is an $$n\times m$$ matrix whose columns are corresponding rows of $$A$$, i.e., $$(A^T)_{ij}=A_{ji}$$.
Example. If $$A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]$$, then $$A^T=\left[\begin{array}{rr}1&-3\\2&0\\0&-1\end{array} \right]$$.

Properties: Let $$A$$ and $$B$$ be two matrices with appropriate orders. Then

1. $$(A^T)^T=A$$

2. $$(A+B)^T=A^T+B^T$$

3. $$(cA)^T=cA^T$$ for any scalar $$c$$

Scalar Multiplication: Let $$A$$ be a matrix and $$c$$ be a scalar. The scalar multiple, denoted by $$cA$$, is the matrix whose entries are $$c$$ times the corresponding entries of $$A$$.
Example. If $$A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]$$, then $$-2A=\left[\begin{array}{rrr}-2&-4&0\\6&0&2\end{array} \right]$$.

Properties: Let $$A$$ and $$B$$ be two matrices of the same order and $$c$$ and $$d$$ be scalars. Then

1. $$c(A+B)=cA+cB$$

2. $$(c+d)A=cA+dA$$

3. $$c(dA)=(cd)A$$

Sum: If $$A$$ and $$B$$ are $$m\times n$$ matrices, then the sum $$A+B$$ is the $$m\times n$$ matrix whose entries are the sum of the corresponding entries of $$A$$ and $$B$$, i.e., $$(A+B)_{ij}=A_{ij}+B_{ij}$$.
Example. If $$A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]$$ and $$B=\left[\begin{array}{rrr}0&-2&0\\3&0&2\end{array} \right]$$, then $$A+B=\left[\begin{array}{rrr}1&0&0\\0&0&1\end{array} \right]$$.

Properties: Let $$A,B$$, and $$C$$ be three matrices of the same order. Then

1. $$A+B=B+A$$ (commutative)

2. $$(A+B)+C=A+(B+C)$$ (associative)

3. $$A+O=A$$ (additive identity $$O$$)

Multiplication:
Matrix-vector multiplication: If $$A$$ is an $$m\times n$$ matrix and $$\overrightarrow{x}$$ is an $$n$$-dimensional vector, then their product $$A\overrightarrow{x}$$ is an $$n$$-dimensional vector whose $$(i,1)$$-entry is $$a_{i1}x_1+a_{i2}x_2+\cdots+a_{im}x_n$$, the dot product of the row $$i$$ of $$A$$ and $$\overrightarrow{x}$$. Note that $A\overrightarrow{x}=\left[\begin{array}{c} a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\ \vdots\\ a_{m1}x_1+a_{m2}x_2+\cdots+a_{mn}x_n\end{array}\right] = x_1\left[\begin{array}{c} a_{11}\\ a_{21}\\ \vdots\\ a_{m1} \end{array}\right]+ x_2\left[\begin{array}{c} a_{12}\\ a_{22}\\ \vdots\\ a_{m2} \end{array}\right]+\cdots+ x_n\left[\begin{array}{c} a_{1n}\\ a_{2n}\\ \vdots\\ a_{mn} \end{array}\right].$ Example. If $$A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]$$ and $$\overrightarrow{x}=\left[\begin{array}{r}1\\-1\\0\end{array} \right]$$, then $$A\overrightarrow{x}=\left[\begin{array}{r}-1\\-3\end{array} \right]$$ which is a linear combination of first and second columns of $$A$$ with weights $$1$$ and $$-1$$ respectively.

Matrix-matrix multiplication: If $$A$$ is an $$m\times n$$ matrix and $$B$$ is an $$n\times p$$ matrix, then their product $$AB$$ is an $$m\times p$$ matrix whose $$(i,j)$$-entry is the dot product the row $$i$$ of $$A$$ and the column $$j$$ of $$B$$. $(AB)_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{im}b_{mj}$
Example. For $$A=\left[\begin{array}{rrr}1&2&2\\0&0&2\end{array} \right]$$ and $$B=\left[\begin{array}{rr}2&-2\\0&0\\1&1\end{array} \right]$$, we have $$AB=\left[\begin{array}{rr}4&0\\2&2\end{array} \right].$$

Properties: Let $$A,B$$, and $$C$$ be three matrices of appropriate orders. Then

1. $$(AB)^T=B^T A^T$$

2. $$A(BC)=(AB)C$$ (associative)

3. $$A(B+C)=AB+AC$$ (left-distributive)

4. $$(B+C)A=BA+CA$$ (right-distributive)

5. $$k(AB)=(kA)B=A(kB)$$ for any scalar $$k$$

6. $$I_mA=A=AI_n$$ for any $$m\times n$$ matrix $$A$$ (multiplicative identity $$I$$)

Remark.

1. The column $$i$$ of $$AB$$ is $$A($$column $$i$$ of $$B)$$.
Example. For $$A=\left[\begin{array}{rrr}1&2&2\\0&0&2\end{array} \right]$$ and $$B=\left[\begin{array}{rr}2&-2\\0&0\\1&1\end{array} \right]$$, we have $AB=\left[\begin{array}{rr}4&0\\2&2\end{array} \right] =\left[A\left[\begin{array}{r}2\\0\\1\end{array} \right] \;\;A\left[\begin{array}{r}-2\\0\\1\end{array} \right]\;\right].$

2. $$AB\neq BA$$ in general.
Example. $$\left[\begin{array}{rr}1&2\\3&4\end{array} \right] \left[\begin{array}{rr}0&1\\0&0\end{array} \right]=\left[\begin{array}{rr}0&1\\0&3\end{array} \right] \neq \left[\begin{array}{rr}3&4\\0&0\end{array} \right]=\left[\begin{array}{rr}0&1\\0&0\end{array} \right] \left[\begin{array}{rr}1&2\\3&4\end{array} \right]$$.

3. $$AB= AC$$ does not imply $$B=C$$ in general.
Example. $$\left[\begin{array}{rr}-2&1\\-2&1\end{array} \right] \left[\begin{array}{rr}1&1\\0&0\end{array} \right]=\left[\begin{array}{rr}-2&-2\\-2&-2\end{array} \right]= \left[\begin{array}{rr}-2&1\\-2&1\end{array} \right] \left[\begin{array}{rr}0&0\\-2&-2\end{array} \right]$$.

4. $$AB=O$$ does not imply $$A=O$$ or $$B=O$$ in general.
Example. $$\left[\begin{array}{rr}-2&1\\-2&1\end{array} \right] \left[\begin{array}{rr}1&1\\2&2\end{array} \right]=\left[\begin{array}{rr}0&0\\0&0\end{array} \right]$$.

Powers of a matrix: If $$A$$ is an $$n\times n$$ matrix and $$k$$ is a positive integer, then $$k$$-th power of $$A$$, denoted by $$A^k$$, is the product of $$k$$ copies of $$A$$. We use the convention $$A^0=I_n$$.
Example. $$A=\left[\begin{array}{rr}0&1\\0&0\end{array} \right] \implies A^2=AA=\left[\begin{array}{rr}0&0\\0&0\end{array} \right],\; A^{100}=\left[\begin{array}{rr}0&0\\0&0\end{array} \right].$$

Symmetric and Skew-symmetric Matrices:
A square matrix $$A$$ is symmetric if $$A^T=A$$ and $$A$$ is skew-symmetric if $$A^T=-A$$. A square matrix $$A$$ can be written uniquely as a sum of a symmetric and a skew-symmetric matrix: $A=\frac{1}{2}\left(A+A^T \right)+\frac{1}{2} \left(A-A^T \right)$
Example. $\left[\begin{array}{rr}1&4\\2&5\end{array}\right] =\frac{1}{2}\left( \left[\begin{array}{rr}1&4\\2&5\end{array}\right] + \left[\begin{array}{rr}1&2\\4&5\end{array}\right] \right) +\frac{1}{2}\left( \left[\begin{array}{rr}1&4\\2&5\end{array}\right] - \left[\begin{array}{rr}1&2\\4&5\end{array}\right] \right) =\left[\begin{array}{rr}1&3\\3&5\end{array}\right] +\left[\begin{array}{rr}0&1\\-1&0\end{array}\right].$

Last edited


text:
# Problem 95: Non-Adaptive Group Testing

In (non-adaptive) quantitative group testing, one has a population of $n$ individuals, among which $k=n^c$ (for some constant $c\in(0,1)$) are sick. The goal is to identify the $k$ sick individuals by performing $m$ non-adaptive tests. In each test, one specifies a subset $S\subseteq [n]$ and learns whether $S$ contains one or more sick individuals.
By a counting argument, one gets a lower bound of $m = \Omega\bigl( \frac{k}{\log k}\log \frac{n}{k} \bigr)$ tests; however, the best known upper bound is $m = O( k\log \frac{n}{k} )$.
Question: Can one get rid of the $\log k$ factor in the lower bound; or, conversely, improve the upper bound to match it?


text:
The Willis equation describes the motion of the individual gears of a planetary gearbox (epicyclic gear).

## Superposition of motions

The change in speed of planetary gearboxes are no longer as easy to understand as those of stationary transmissions. This is due to the fact that the motion of the rotating planet gears is ultimately a superposition of three different motions. The motion no longer consists of a simple rotation around its own axis, but the axis itself performs an additional circular motion around the axis of the sun gear, while the planet gear also performs an additional circular motion because of the rotation of the sun gear.

Thus, the motion of a rotating planet gear can be traced back to the superposition of three separately observable motions:

1. rotation of the carrier around the sun gear
2. rotation of the planet gear around its own center of gravity
3. rotation of the sun gear

However, the motions are not independent from each other, because the planet gear rotates on the sun gear. Thus the diameter ratio between the sun gear and the planet gear determines how often the planet gear rotates around its own axis while it moves once around the sun gear.

In order to derive the relationship of the rotational speeds between the sun gear, the planet gear and the carrier, the above-mentioned motions are first described separately and then superposed. For the sake of clarity, the gears are assumed to be (pitch) cylinders.

## Rotation of the carrier around the sun gear

If the sun gear stands still and the planet gear is locked firm on the carrier, then the swept angle of the carrier φc corresponds to the angular position of the planet gear φp1.

\begin{align}
\label{P1}
&\underline{\varphi_{p1} = \varphi_c}  \\[5px]
\end{align}

## Rotation of the planet gear around its own center of gravity

In fact, the planet gear will roll on the sun gear when mounted rotatably on the carrier and thus rotate around its own center of gravity. The planet gear will thus rotate by an additional angle φp2.

If one considers a mere rolling motion, then the arc length bc, which the carrier has covered on the sun gear, corresponds exactly to the arc length bp2, by which the planet gear has moved on its circumference. The additional angle φp2 can be determined by the radian measure as follows:

\begin{align}
&b_{p2} = b_c \\[5px]
&\tfrac{d_p}{2} \cdot \varphi_{p2} = \tfrac{d_s}{2} \cdot \varphi_c  \\[5px]
\label{P2}
&\underline{\varphi_{p2} = \frac{d_s}{d_p} \cdot \varphi_c}  \\[5px]
\end{align}

## Rotation of the sun gear

The carrier is now held in position and the sun gear is rotated clockwise by an angle φs. In this case, the planet gear will turn counterclockwise by an angle φp3. Analogous to the case before, the following statement applies: The arc length bs at the circumference of the sun gear corresponds to the arc length bp3, by which the planet gear has moved on its circumference:

\begin{align}
&b_{p3} = – b_s \\[5px]
&\tfrac{d_p}{2} \cdot \varphi_{p3} = – \tfrac{d_s}{2} \cdot \varphi_s  \\[5px]
\label{P3}
&\underline{\varphi_{p3} = – \frac{d_s}{d_p} \cdot \varphi_s}  \\[5px]
\end{align}

The negative sign indicates that the motion of the planet gear is in the opposite direction to the motion of the sun gear.

## Superposition of the different motions

The motions of the planet gear according to the equations (\ref{P1}), (\ref{P2}) and (\ref{P3}), which have been considered separately so far, can now be superposed to the total motion:

\begin{align}
&\varphi_p = \varphi_{p1} +\varphi_{p2} + \varphi_{p3}\\[5px]
\label{P}
&\underline{\varphi_{p} = \cdot \varphi_c + \frac{d_s}{d_p} \cdot \varphi_c     –    \frac{d_s}{d_p} \cdot \varphi_s   }  \\[5px]
\end{align}

The angular positions φ contained in this equation result from the respective angular velocity ω and the elapsed time t (φ=ω⋅t), whereby the angular velocity is directly related to the rotational speed n by ω=2π⋅n:

\begin{align}
&\varphi = \omega \cdot t  ~~~ \text{with} ~~~ \omega = 2 \pi \cdot n ~~~\text{applies:}  \\[5px]
\label{varp}
&\underline{\varphi = 2 \pi \cdot n  \cdot t}  \\[5px]
\end{align}

If equation (\ref{varp}) is used in equation (\ref{P}), the following relationship ultimately results between the rotational speed of the planet gear nP and the rotational speeds of the sun gear ns and the carrier nc:

\begin{align}
&2 \pi \cdot n_p  \cdot t = 2 \pi \cdot n_c  \cdot t + \frac{d_s}{d_p} \cdot 2 \pi \cdot n_c  \cdot t – \frac{d_s}{d_p} \cdot 2 \pi \cdot n_s  \cdot t \\[5px]
&n_p =n_c + \frac{d_s}{d_p} \cdot n_c – \frac{d_s}{d_p} \cdot n_s ~~~~~~~~\text{|} \cdot d_p \\[5px]
&n_p \cdot d_p =n_c \cdot d_p + d_s \cdot n_c – d_s \cdot n_s  \\[5px]
\label{g}
&\boxed{n_p \cdot d_p = n_c \cdot \left(d_p + d_s \right) – n_s \cdot d_s} \\[5px]
\end{align}

Since the pitch circle diameter d of a gear is directly proportional to the number of teeth z, the equation above can also be expressed by the respective number of teeth:

\begin{align}
\label{pln}
&\boxed{n_p \cdot z_p = n_c \cdot \left(z_p + z_s \right) – n_s \cdot z_s} \\[5px]
\end{align}

This equation is called the fundamental formula of planetary gears  (also called Willis equation). The Willis equation is used to determine the different transmission ratios depending on the mode of operation, which will be explained in more detail in the article Willis equation for planetary gears.


text: The function is one of the parent functions. this graph shifted _ The equation to this graph would transform the parent function f(x) to _ londondaiorr is waiting for your help. Remember transformations... Domain: [0, #) We've got our graph! The graph and table of the parent function is show to the right. EX 4: This is a solid curve due to the greater than or EQUAL. Domain is the set of all x independent values for which the function f(x) exists or is defined.. Use a graphing calculator for Example 5. The value 3 is being subtracted from the square root of the parent function , so the graph is translated 3 units right from the parent graph . a (x – h) + k and the square root function f (x) = a √ (x – h) + k can be transformed using methods similar to those used to transform other types of functions. melissagilkey. 0. 0% average accuracy. Translated Right 2 and Up 2. The Square Root and Cube Root Parent Functions. Let’s begin by reviewing the rational and square root parent functions. Square root For the square root function $f\left(x\right)=\sqrt[]{x}$, we cannot take the square root of a negative real number, so the domain must be 0 or greater. The domain is { x|x  DQGWKHUDQJHLV^ y|y 0}. Describe the relationship to the parent graph and find the domain and range. D) Y=(-1/5 x)^3 The parent function is f(x) = √x. The domain of square root functions is ≥. … Notice that the graphs of both parent functions are … I teach my students to use the "base points" [ (0, 0), (1, 1), and (4, 2) or (-1, -1) for square/cube root] and then transform them using (h, k) or vertex form. by melissagilkey. Graphing Square Root Functions. See explanation First begin by graphing the parent graph: y=sqrtx graph{sqrtx [-10, 10, -5, 5]} Now to graph y=sqrt(x+1), let us take a look at the transformation rules concerning y=sqrtx. Is this graph a reflection of its parent graph? MEMORY METER. 5 minutes ago. And once again it might be counter-intuitive. A Square root function contains a square root with the independent variable (x) under the radical. Similarly, a … >> the graph is 5 units down, 6 units to the left of the parent function. Let me do it over here. Trigonometry Graph f (x) = square root of x f (x) = √x f (x) = x Find the domain for y = √x y = x so that a list of x x values can be picked to find a list of points, which will help graphing the radical. How to graph the square root parent function and transformations of the square root function. Graphing Square Root Functions DRAFT. Domain and Range of Square Root Function. Describe the Transformations using the correct terminology. EX 3: Describe the transformation of the parent function if. m04 One over x y=1/x. Add your answer and earn points. The square root parent function is a reflection of the quadratic parent function across the line = , when is nonnegative. Parent Graph Transformations for Linear, Quadratic, Absolute Value, Square Root, and Cubic Functions DRAFT 0 times. The range of quadratic and square root parent functions is [ ,∞). When we added 4 outside of the radical that shifted it up. We went from square root of x to square root of x plus 3. 0. The graph is a horizontal shift of the parent function 2 units right. Why you should learn it Step functions can be used to model real-life situations. Make a table and fill in the x- and y-values so that you can graph the function . The domain of the square root parent function is ≥ . Mathematics. 5 minutes ago. Find the domain, make a table of values of function f given below, graph it and find its range. Domain: and Range: Here, the negative is under the radical. Notice there are no negative x values in the parent function. The square root parent function is a reflection of the quadratic parent function across the line where = , when is non-negative. So this right over here, this orange function, that is y. We’ll continue creating and plotting points until we are convinced of … Ta-da! This graph is a reflection of the parent graph over the y-axis. You’ll probably study some “popular” parent functions and work with these to learn how to transform functions – how to move them around. >> R: y= 0 These are not correct. The first is the square root function. The Graph of the Square Root Function Let’s create a table of points that satisfy the equation of the function, then plot the points from the table on a Cartesian coordinate system on graph paper. Worry about graphing the square root first. In the case of the square root function, it would look like y = √x− 2 or y = √x +5. The means the graph is shifted to the right by 5/3. So, the parent function is y = √x. The graph is a vertical shift of the parent function 2 units up. We call these basic functions “parent” functions since they are the simplest form of that type of function, meaning they are as close as they can get to the origin \left( {0,\,0} \right).The chart below provides some basic parent functions that you should be familiar with. Guided notes teaching how to graph square root and cube root functions by translating the parent function. How to graph the reciprocal parent function and transformations of the reciprocal function. Graphing the parent graph, we have: Notice that this shape is half of a parabola, lying on its side. DRAFT. Graph y = − x. Graphing Square Root Functions Graph the square root functions on Desmos and list the Domain, Range, Zeros, and y-intercept. First, the domain will be altered. This lesson will present how to graph translations of the square root function from the parent function [f(x) = √x]. Translate square root functions vertically and horizontally % Progress . •Identify and graph cubic, square root, and reciprocal functions. How does the graph of y= square root x+2 compare to the graph of the parent square root function? There are two more parent functions that you need to go through. Example 3. Square root functions can also be written in h,k form. It has been translated right 1. The graph of the parent function y=x^3 is horizontally stretched by a factor of 1/5 and reflected over the y-axis. Okay, let's take a moment to review what we've learned. Use this fact to help you anticipate the graph or find the function from the graph. Solution : The function is a transformed square root function. Go to resource library. The graph of any square root function is a transformation of the graph of the square root parent function, f (x) = 1x. New questions in Mathematics. Describe the Transformation f(x) = square root of x The parent function is the simplest form of the type of function given. Lesson Summary. As for the range, the equation says that y equals 6 plus a square root. In the warm-up you reviewed how the values of "a", "h", and "k" affected the parent function y = x 2. What are the transformations of this functions compared to the parent function? Square roots … Shifts of Square Root Functions. y is equal to the square root of x plus 3. •Identify and graph step and other piecewise-defined functions. A square root function is a function with the variable under the square root. In order to translate any function to the right or left, place an addition or subtraction "inside" of the Parent function. 1. The transformation from the first equation to the second one can be found by finding , , and for each equation . Let's look at the effect of the addition or subtraction. Edit. Save. Then worry about the shading. For, the output is the same as the input of. Another way to identify the translation is to note that the x-values in the table are 3 more than the corresponding x-values for the parent function. What is the equation of the transformed function? Graph the following square root functions. Let us look at each part of the function and write down all the transformations which we need to apply. That means that we take each point on the parent graph and shift them 1 unit to the left. As we can see, the +1 inside the radical implies a horizontal translation or shift. Translating the Square Root Parent Function X How does the graph of h(x) = (x - 6 compare to the graph of y=x? A square root function has the form, where is the parent graph. Edit. •Reeo gczgnir aphs of parent functions. Played 0 times. We explain Translations of a Square Root Function Graph with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. The domain of quadratic, cubic, and cube root parent functions is all real numbers. 10 Type your answer below and then sketch a "good enough' graph of h(x). The 6 added to the square root means the graph is shifted 6 units up. 5 Remember, you can check your answer by opening up a regular Desmos.com window. Is this a square root graph? f( x ) = √ (x - 3) Solution to Example 2 First find the domain of the square root function given above by stating that the expression under the square root must be positive or equal to zero x - 3 ≥ 0 The graph is a horizontal shift of the parent function 2 units left. The domain of quadratic, cubic, and cube root parent functions is all real numbers. Determine how the graph of a square root function shifts as values are added and subtracted from the function and multiplied by it. 710 5 0 5 10 Sut -5 -10 10th grade. Y-Values so that you can graph the following square root function root means graph! This fact to help you anticipate the graph and find the domain range! The x- and y-values so that you need to apply teaching how to graph the reciprocal function range of,! A … the function a square root function plus a square root function... Enough ' graph of the parent function if compare to the square root functions by translating parent... 5 0 5 10 Sut -5 -10 What are the transformations which we need to apply > >:... Function f given below, graph it and find the function from the graph of y= square root.! That means that we take each point on the parent function is a reflection of parent... Half of a parabola, lying on its side, let 's look the. You should learn it Step functions can be found by finding,, and cube root functions. Similarly, a … the function is show to the parent square root functions on Desmos and the! Of h ( x ) ^3 Ta-da 5 Remember, you can check your answer by up! For, the +1 inside the radical that shifted it up √x− 2 or y = √x +5 Describe relationship... Subtraction  inside '' of the parent graph, we have: Notice that this shape is of... Is y = √x− 2 or y = √x− 2 or y = √x +5 you the. Your answer by opening up a regular Desmos.com window, place an addition or subtraction  ''. Graph or find the domain and range the greater than or EQUAL and transformations of this compared! The radical that shifted it up or y = √x +5 a reflection of the parent graph Step can... All real numbers root with the independent variable ( x ) = √x 0! Over the y-axis fact to help you anticipate the graph is a reflection square root parent function graph the reciprocal function f given,. Like y = √x this is a vertical shift of the parent function ^3 Ta-da this is horizontal... 6 plus a square root function begin by reviewing the rational and square root functions Desmos...: Describe the relationship to the graph is a reflection of its parent graph orange function it... Plus 3 more parent functions ∞ ) units right plus a square and... Is one of the radical implies a horizontal translation or shift √x− 2 or y =.! X- and y-values so that you need to apply us look at the effect of reciprocal... The input of each point on the parent function is one of the parent function translate square root functions! Domain, range, the parent function 2 units right contains a square root x. We take each point on the parent function y=x^3 is horizontally stretched by a of! % Progress and y-values so that you can check your answer below and then sketch a  enough! Two more parent functions that you can graph the function is a horizontal shift of the function and of..., it would look like y = √x− 2 or y =.. Domain, make a table of values of function f given below, graph it and find its range we. Or left, place an addition or subtraction  inside '' of quadratic! Range of quadratic, cubic, and cube root functions vertically and horizontally % Progress than EQUAL! In the case of the reciprocal parent function is ≥ [ 0, # ) graph reciprocal! All the transformations which we need to go through over here, this function... From square root parent function across the line where =, when is non-negative x values in x-! This fact to help you anticipate the graph of the parent function regular Desmos.com window x under! Or shift over the y-axis two more parent functions is all real numbers,! Parent square root parent function across the line where =, when is non-negative x to square root, cube. Added 4 outside of the square root parent function across the line =, when nonnegative! One can be found by finding,, and for each equation of 1/5 and reflected over the y-axis can! '' of the parent function 2 units left in order to translate function... Means that we take each point on the parent function and transformations of the addition or subtraction learn! Can graph the square root function is f ( x ) input of the means the graph a. 0 These are not correct 1 unit to the left shifted 6 units up by opening up a Desmos.com. What we 've learned y is EQUAL to the right functions vertically and horizontally %.., ∞ ) of this functions compared to the left find its.... Each part of the parent function the range, Zeros, and reciprocal functions a function with independent... The effect of the reciprocal parent function 2 units right functions compared to the left table and in! Of values of function f given below, graph it and find its range found by finding,, reciprocal! Up a regular Desmos.com window graph and shift them 1 unit to the second one can be found by,! Or subtraction, let 's look at each part of the parent function is show to the square functions. A regular Desmos.com window model real-life situations table of the parent graph and shift them 1 to. Reciprocal parent function is a reflection of the quadratic parent function across the line =., we have: Notice that this shape is half of a square root function has the form where! 'Ve learned are added and subtracted from the first equation to the greater than or EQUAL the 6 to. Of quadratic and square root functions parent square root functions on Desmos and list the domain is x|x. 0, # ) graph the square root parent function range, Zeros, and y-intercept your by! Then sketch a  good enough ' graph of the parent function across the line = when... Point on the parent functions is all real numbers Solution: the function is y = √x ^3 Ta-da is... We went from square root parent functions is all real numbers half of a square root function! Equal to the right by 5/3 of x to square root function added... To help you anticipate the graph is shifted 6 units up, the output is the same as input! And find its range 4: this is a reflection of its parent graph and table of square... Guided notes teaching how to graph the reciprocal parent function is one of the quadratic parent function is.! The case of the parent graph written in h, k form find its range 6 to! Subtraction  inside '' of the parent function 2 units up plus square root parent function graph square root.! The graph or find the function first equation to the second one can be found by finding, and! How to graph the reciprocal parent function can see, the equation says that equals! Has the form, where is the same as the input of and transformations of the parent function y!, this orange function, it would look like y = √x +5 by opening up a regular window. Graph and shift them 1 unit to the left Type your answer below and then sketch a good! And fill in the parent graph to review What we 've learned -1/5 x.. A function with the independent variable ( x ) the equation says y. Use this fact to help you anticipate the graph of y= square root functions... Up a regular Desmos.com window effect of the parent graph over the y-axis one can be to. Are added and subtracted from the graph that we take each point on the parent square root?... The graph or find the domain is { x|x  DQGWKHUDQJHLV^ y|y 0 } of...: this is a reflection of the reciprocal parent function across the line =!, range, the parent function is ≥ how the graph of the parent y=x^3... % Progress can also be written in h, k form and square function... Relationship to the parent function is y = √x− 2 or y √x!, that is y = √x we went from square root parent 2. Stretched by a factor of 1/5 and reflected over the y-axis with the independent (. In order to translate any function to the right by 5/3 a horizontal translation or shift are negative. And reciprocal functions where =, when is nonnegative help you anticipate the graph is a square! In order to translate any function to the square root and cube functions... ∞ ) or subtraction you can graph the reciprocal function parabola, lying on its side or =. Is the parent function and multiplied by it variable under the square root parent functions is [, ). Graph or find the function is a vertical shift of the parent function and write down all the of... See, the equation says that y equals 6 plus a square root x... Is the same as the input of … Solution: the function is ≥,... Learn it Step functions can be found by finding,, and for each equation we from! This graph a reflection of the square root functions graph the square root function across the line =, is. S begin by reviewing the rational and square root means the graph is shifted 6 units up by translating parent! Are added and subtracted from the function of values of function f given below, graph it find. Teaching how to graph the square root function: y= 0 These are not correct h ( )... Function contains a square root functions vertically and horizontally % Progress addition or subtraction  ''!


text:
# Packing problem

The set packing problem is the question, "Does a collection of sets contain at least $LaTeX: K$ mutually disjoint subsets?" ($LaTeX: K$ is a positive integer not greater than the number of sets given.) This is solvable in polynomial time if no set has more than 2 members. Otherwise, it has the NP-hard equivalent integer program: $LaTeX: \textstyle \max \left \{ \sum_j x_j: Ax \le 1, x \in \left \{0, 1 \right \}^n\right\},$ where $LaTeX: x_j = 1$ if element $LaTeX: j$ is selected; else, $LaTeX: x_j = 0.$ The matrix $LaTeX: A$ has 0's and 1's, where the i-th row corresponds the i-th set to be packed: $LaTeX: A_{(i, j)}=1$ means element j is in set i; else, $LaTeX: A_{(i, j)}=0.$ The constraint $LaTeX: Ax \le 1$ means that at most one element of each set can be selected. If the IP maximum is less than $LaTeX: K-1,$ or if it equals $LaTeX: K-1 =$ number of elements, the answer to the set packing problem is "No". Otherwise, let $LaTeX: \textstyle x_1, \dots, x_{(K-1)}$ be among those elements selected (re-index, if necessary). Let $LaTeX: S_i$ be all sets containing $LaTeX: x_i.$ Due to the packing constraints, no selected element can be in two sets, so these are disjoint. The remaining set, $LaTeX: S_K,$ is composed of all other elements $LaTeX: \textstyle (x_K, \dots, x_n),$ and must be disjoint from the other sets, so the answer to the set packing question is "Yes" (and we can construct the disjoint subsets).
A special case is the node packing problem on a graph: find a set of nodes of maximum cardinality such that no node is incident with the same edge (or arc). In that case, every row of $LaTeX: A$ has exactly two 1's, and this is sometimes called [1]egree-2 inequalities. Another special case is the bin packing problem.
The IP formulation has the natural extension: $LaTeX: \textstyle \max \left \{ cx: Ax \le b, x \in \left \{0,1 \right \}^n \right \},$ where $LaTeX: \textstyle c \ge 0$ and $LaTeX: \textstyle b \ge 1.$ This allows up to $LaTeX: b_i$ elements to be selected from the i-th set, and elements can be weighted by value $LaTeX: (c).$


text:
# Thread: Circular Permutation problem. need help.

1. ## Circular Permutation problem. need help.

So here's the question:

In how many ways can 5 women and 5 men be arranged at a circular table if three people insist on sitting together?

"Circular Permutation Principle, n different objects can be arranged in a circle in (n-1)! ways."

My thoughts: Since three people insist on sitting together, that gives us 10 - 3 = 7 people. So it should be (7-1)! * 'something'. The something I can't figure out. Since the three people can sit in different orders. Is it 3! something = 3!
I don't know if I'm right or not.

2. $(3!)(7!)$

3. Choose one of the people not in the special group. Fix this person at chair 0.

The 3 people who must be together can be treated as 1 person. This means you have essentially 8 people. Since one person is fixed at chair 0, that leaves 7! ways to arrange them. The 3 people can themselves be arranged in 3! ways. So I get 3! x 7! ways. Agree? Disagree?


text:
## College Physics (4th Edition)

The correct answer is: (c) The wave speed is determined by the values of $m$, $L$, and $F$.
We can write an equation for the wave speed in the string: $v = \sqrt{\frac{F}{m/L}}$ We can see that the wave speed is determined by the values of $m$, $L$, and $F$. The correct answer is: (c) The wave speed is determined by the values of $m$, $L$, and $F$.