Fishing Boats For Sale Las Vegas Quart

NCERT Solutions for Class 8 Maths| Updated

All the�. Real and complex number sets. What are the Parts of a Circle So far, we have discussed about the triangle and quadrilateral that have linear boundaries.

Circle is a closed figure that has a curvilinear boundary. When class 10 maths ch 6 ex 6.3 q 10 nv think of circles, the very first thing that comes to our mind is its round shape, for example, bangles, coins, rings, plates, [�].

What are the Inverse Trigonometric Functions? Graphs of inverse trigonometric functions Domain and Range of inverse trigonometric functions Properties of inverse trigonometric functions 5 Principal [�]. What is the half life of a radioactive element? Half-life: Radioactive decay is a random process.

This means that all the unstable nuclei have the same probability to decay at a certain instant. It is not possible to predict when a particular nucleus is going to decay. The unstable nuclei will decay at different times, [�]. For graphing on a number line, see [�]. The organisms that depend upon outside sources for obtaining organic nutritens are called heterotrophs.

Heterotrophic nutrition is of three types � saprophytic, parasitic [�]. Heights And Distances Angle Of Elevation The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.

Angle Of Depression The [�].

Final:

- caulking. For most who do not have the list beheld as well as who do not need to buy a singleBattleships, prosaic. If sea conditions assentas well as dual relating angles have been over lapped as well as glued collectively.

Furloughed kayaks have heavier growth as well as additional areas for rigging storage.

Sky is rolling out its latest update for the Sky Q platform, and it'll see further improvements to c�. Apple only launched its most recent array of iPhones at the end of , but we're already looki�. Intel has finally launched its 11th Generation Intel Core desktop processors, which are also known a�. Home Tech News. Fender Mustang Micro guitar amp looks perfect for lockdown riffs and beyond How we'd have loved the Fender Mustang Micro guitar amp during the long months of lockdown.

Honeywell and Will. When is the next Steam sale? Android Auto is about to get an influx of popular driving apps Compared to most Android platforms, the in-car Android Auto platform seems pretty closed-off.

In the vast majority of cases in mathematics, the set of objects will have at least the structure of an additive group, which means that you can add and, more importantly, subtract. In particular, you can tell right away whether it represents 0. We know that any fraction whose numerator is 0 stands for 0. It seemed to us to be of such astounding generality. No matter what numerical values we would plug in for x and y, we would find that the left side equals the right side.

Let us try to explain why. The reason is that both sides are polynomials in the two variables x, y. Even though they are completely routine, polynomial identities and by clearing denominators, also identities between rational functions can be very important. When the computer terminal logo appears with a darkened screen, the normal mathematical flow will resume, at which point you may either resume reading, or flee to the next terminal logo, again depending, respectively, on your proclivities.

Are the following proofs acceptable? Verify this for the 64 triangles for which! Since the theorem is true in these cases it is always true. The above proofs are completely rigorous. Hence if they agree at 64 points a, b , they are identical. In practice, however, to prove a polynomial identity it is just as easy to expand the! A complete computer-era proof of Theorem 1. Then use your computer to fit a fourth degree polynomial to the data points 0, 0 , 1, 1 , 2, 9 , 3, 36 , and 4, Thanks to modern computer algebra systems, they are also routine in practice.

Here is the Maple code for proving Theorem 1. All quantities are expressed in terms of ta and tb and are easily seen to be rational functions in them.

A and! B, and of! It should be 0,0. In the files hex. So equality of rational expressions in trigonometric functions can be reduced to equality of polynomial expressions in w. This is a routinely verifiable algebraic identity.

Below is the short Maple Code that proves it. If its output is zero then the identity has been proved. Of course, for every fixed n, no matter how big, the above is a routine polynomial identity. Now it follows from the theory of symmetric functions e.

This is also true if we have several sets of variables, ai , bi. Jacobi [Shalosh B. It is possible that they would have prevented the discovery of one of the most beautiful theories in the whole of mathematics: the theory of elliptic functions, which leads naturally to the theory of modular forms, and which, besides being gorgeous for its own sake [Knop93], has been applied all over mathematics e.

Suppose that the trigonometric func- tions had not been known before calculus. We can call this the complete circular integral. More generally, suppose that we want to know F z , the arc length of the circle above the interval [0, z], for general z. Furthermore, that genius would have soon realized how to express the sine and cosine functions in terms of the exponential function.

Consider the analogous problem for the arc length of the ellipse. The study of these integrals was at the frontier of mathematical research in the first half of the nineteenth century.

Legendre struggled with them for a long time, and must have been frustrated when Jacobi had the great idea of inverting F z. These are the once famous Jacobi elliptic functions. Jacobi realized that the counterparts of the exponential function are the so-called Jacobi theta functions, and he was able to express his elliptic functions in terms of his theta functions. With the aid of his fa- mous transformation formula see, e. This enabled him or his human computers to compile highly accurate tables of his elliptic functions, and hence, of course, of the incomplete elliptic integral F z.

Much more importantly, it led to a beautiful theory, which is still flourishing. Had they had computer algebra, they would have also realized that all identities between elliptic functions are routine, and that it is not necessary to introduce theta functions. Take for example the addition formula for sn w e. Try to work this out by hand, and see that it would have been a formidable task for any human, even a Jacobi or Legendre.

We will show you how to evaluate and to prove such sums entirely mechanically, i. Everybody knows that computers are fast. The computers do that by themselves, unassisted by hints or nudges from humans. It means also that not only can your PC find such a proof, but you will be able to check the proof easily. That is a very important point. People get unhappy when a computer blinks its lights for a while and then announces a result, if people cannot easily check the truth of the result for themselves.

In this book you will be pleased to note that although the computers will have to blink their lights for quite a long time, when they are finished they will give to us people a short certificate from which it will be easy to check the truth of what they are claiming.

Computers not only find proofs of known identities, they also find completely new identities. Lots of them. Some very pretty. Some not so pretty but very useful. Some neither pretty nor useful, in which case we humans can ignore them. It showed how recurrences for certain polynomial sequences could be found algorithmically.

See Chapter 4. Gosper, Jr. Such a! See Chapter 6. See Chapter 7. A definite! You would like to know whether or not that sum can be expressed in a much simpler way, say as a single term that involves factorials, etc. In this book we will show you how several recently developed computer algorithms can do the job for you. If there is a simple form, the algorithms will find it.

Nobody will ever be able to simplify that expression, within a certain set of conventions about what simplification means, anyway. We will present the underlying mathematical theory of these methods, the princi- pal theorems and their proofs, and we also include a package of computer programs that will do these tasks see Appendix A. The main theme that runs through these methods is that of recurrence. By the way, a recurrence that!

All you have to do is type in the summand nk. Would you like to know how all of that is done? Read on. In that case this book has some powerful tools for you to use. This book contains both mathematics and software, the former being the theoret- ical underpinnings of the latter. For those who have not previously used them, the programs will likely be a revelation. Imagine the convenience of being able to input a sum that one is interested in and having the program print out a simple formula that evaluates it!

Think also of inputting a complicated sum and getting a recurrence formula that it satisfies, automatically. Taken together, they are the story of a sequence of very recent developments that have changed the field on which the game of discrete mathematics is played.

The computer methods tend in certain directions that seem not to come naturally to humans. We illustrate the thought processes by a small example. Example 2. Next, instead of trying to prove that the two sides of the 1 See page Thus F is constant.

Four of these, any one of which would certainly fill the bill, are Macsyma2 , Maple3 , Mathematica4 , or Axiom5. What one needs from such programs are a large number of high level mathematical commands and a built-in programming language. In this book we will for the most part use Maple and Mathematica, and we will also discuss some public domain packages that are available. There are beautiful identities in many branches of mathematics.

It is a fun activity for people to try to prove identities. Here we will not, of course, be able to discuss all kinds of identities. Far from it. The main purpose of this book is to explain how the discoveries and the proofs of hypergeometric identities have been very largely automated.

The book is not primarily about computing; it is the mathematics that underlies the computing that will be the main focus. Automating the discovery and proof of identities is not something that is immediately obvious as soon as you have a large computer. The theoretical devel- opments that have led to the automation make what we believe is a very interesting story, and we would like to tell it to you. The proof theory of these identities has gone through roughly three phases of evolution.

At first each identity was treated on its own merits. The theory of hypergeometric functions was initiated by C. Gauss early in the nineteenth century, and in the course of developing that theory some very general identities were found.

It was not until , however, that the recognition mentioned above occurred. A similar, but much shorter, time lag took place before the third phase of the proof theory flowered. In the s, the main ideas for the automated discovery of recurrence relations for hypergeometric sums were discovered by Sister Mary Celine Fasenmyer see Chapter 4.

It was not Class 10 Maths Ch 6 Ex 6.3 Zen until that it was recognized, by Doron Zeilberger [Zeil82], that these ideas also provided tools for the automated proofs of hypergeometric identities. With that realization the idea of finding recurrence relations that sums satisfy was elevated to the first priority task in the analysis of identities.

As the many facets of that realization have been developed, the emergence of powerful high level computer algebra programs for personal computers and workstations has brought the whole chain of ideas to your own desktop.

Anyone who has access to such equipment can use the programs of this book, or others that are available, to prove and discover many kinds of identities.

There are k ways to choose k " n letters from among the letters 1, 2,. But every one of the 2n n ways of choosing n letters from the 2n letters 1, 2,.

So as we go through this book whose main theme is that computers can prove all of these identities, please note that we will never7 claim that computerized proofs are better than human ones, in any sense.

When an elegant proof exists, as in the above example, the computer will be hard put to top it. To continue, the pre-computer proof of 2. It found the combinatorial interpretations of both sides of the identity, and showed that they both count the same thing. We preface it with some remarks about standardized proofs and certificates. Then it would clearly be nice to have a rather standardized proof outline, one that would work on all of the thousands of examples.

That small detail will be called the certificate. Since the rest of the proof is standard, and not dependent on the particular example, we will be able to describe the complete proof for a given example just by describing the proof certificate.

The only thing that changes in the proof, as we go from one 7 Well, hardly ever. Otherwise, all of the proofs are the same. The rest of the proof is standardized. The rational function R n, k certifies the proof. Here is the standardized WZ proof algorithm:! Divide through by the right hand side, so the identity that you wish to prove!

Sum that equation over all integers k, and note that the right side telescopes to 0. The lock is the proof outlined above. If you want to prove an identity, and you have the key, then just put it into the lock and watch the proof come out. In Step 3 we use the key. That phrase means roughly that your pet chimpanzee could check out the equation.

More precisely it means this. Finally, cancel out all of the factorials by suitable divisions, leaving only a polynomial identity that involves n and k. In an article in the American Mathematical Monthly , p. Well, it turns out that if you would really like to simplify ratios of factorials then the thing to do is to read in the package RSolve, because in that package there lives a command FactorialSimplify, which does the simplification that you would like to see.

So it needs to be coaxed. The best approach seems to be to divide the whole expression by f n, k , then expand it to get rid of all of the factorials , and then simplify it, in order to collect terms. Human beings might have a great deal of trouble in finding one of these proofs, but the verification procedure, as we have seen, is perfectly civilized, and involves only a medium amount of human labor.

In Chapter 3 we will meet the hypergeometric database. We will see that this gives us access to a database of theorems and identities, and we will learn how to interrogate that database. We will also see some of its limitations.

Following this are five consecutive chapters that deal with the fundamental algo- rithms of the subject of computer proofs of identities. Chapter 4 describes the original algorithm of Sister Mary Celine Fasenmyer. She developed, in her doctoral dissertation of , the first computerizable method for finding recurrence relations that are satisfied by sums. We also prove the validity of her algorithm here, since that fact underlies the later developments.

Chapter 5 is about the fundamental algorithm of Gosper, which is to summation as finding antiderivatives is to integration. This algorithm allows us to do indefinite hypergeometric sums in simple closed form, or it furnishes a proof of impossibility if, in a given case, that cannot be done. Beyond its obvious use in doing indefinite sums, it has several nonobvious uses in executing the WZ method, in finding recurrences for definite sums, and even for finding the right hand side of a definite sum whose evaluation we are seeking.

Again, this is an algorithm that finds recurrence relations that are satisfied by sums. It is in most cases much faster than the method of Sister Celine, and it has made possible a whole generation of computerized proofs of identities that were formerly inaccessible to these ideas. It is the cornerstone of the methods that we present for finding out if a given combinatorial sum can be simplified, and it is guaranteed to work every time.

When it does exist, however, which is very often, it gives us the unique opportunity to find new identities, as well as to prove old ones. We will see here how to do that, and give some examples of the treasures that can be found in this way. In all of the examples earlier in the book, the computer analysis of an identity will produce just such a recurrence that the sum satisfies.

If we want to prove that a certain right hand side is the correct one, then we just check that the claimed right hand side satisfies the same recurrence and we check a few initial conditions. Then we need to know how to recognize when a higher order recurrence has simple solutions of a certain form, and when it does not.

Use a computer algebra program to check the following pairs. For each of the four parts of Problem 2 above, write out the complete proof of the identity, using the full text of the standardized WZ proof together with the appropriate rational function certificate.

For each of the parts of Problem 2 above, say exactly what the standardized summand F n, k is, and in each case evaluate lim F n, k and lim F n, k.

Write a procedure, in your favorite programming language, whose input will be the summand t n, k , and the right hand side rhs n , of a claimed identity! Test your program on the examples in Problem 2 above.

Be sure to check the initial conditions as well as the WZ equation. Chapter 3 The Hypergeometric Database 3. Roughly see the formal definition below , a hypergeometric sum is one in which the summand involves only factorials, polynomials, and exponential functions of the summation variable.

Whenever this happens we have an identity. In Chapter 2 we saw a few examples of such identities. There is such a wealth of information available now that it is important to have systematic ways of searching the literature for information that may help us to deal with a particular sum.

So our main task in this chapter will be to show how a given sum is described by using standardized hypergeometric notation. We must emphasize that the main thrust of this book is away from this approach, to look instead at an alternative to such database lookups. We will develop com- puterized methods of such generality and scope that instead of attempting to look up a sum in such a database, which is a process that is far from algorithmic, and which has no theorem that guarantees success under general conditions, it will often be preferable to ask the computer to prove the identity directly or to find out if a simple evaluation of it exists.

Nevertheless, hypergeometric function theory is the context in which this activity resides, and the language of that theory, and its main theorems, are important in all of these applications. The k th term of a geometric series is of the form cxk where c and x are constants, i. In this case we will call the terms hypergeometric terms. Hypergeometric series are very important in mathematics. Many of the familiar functions of analysis are hypergeometric.

These include the exponential, logarithmic, trigonometric, binomial, and Bessel functions, along with the classical orthogonal polynomial sequences of Legendre, Chebyshev, Laguerre, Hermite, etc.

If we have a hypergeometric series that interests us for some reason, we might wonder what is known about it. Is it possible to sum the series in simple form?

Is it possible to transform the series into another form that is easier to work with? Is some result that we have just discovered about this series really new or is it well known? These questions can often be answered by consulting the extensive literature on hypergeometric series. Example 3. Given a series k tk.

If this cannot be done, the series is not hypergeometric. Whatever numerical factors are needed to achieve this are absorbed into the factor x. You have now identified the input series. A first step might be to identify it as a hypergeometric series. Hence by 3. In terms of the rising factorial function, here is what the general hypergeometric series looks like: 4 5 a1 a2.

The series is well defined as long as the lower parameters b1 , b2 ,. The series terminates automatically if any of the upper parameters a1 , a2 ,. If the series is well defined and nonterminating, then questions of convergence or divergence become relevant. In this book we will be concerned for the most part with terminating series.

Computers can help even with this humble task. First, Mathematica has a limited capability for transforming sums that are given in customary summation form into standard hypergeometric form. But Mathematica is too smart for us. It knows how to evaluate the sum in simple form, and so it proudly replies 2n! We can force Mathematica to identify our sum as a p Fq only when it does not know how to express it in simple form. Finally we illustrate the use of the package Hyp.

This package, whose purpose is to facilitate the manipulation of hypergeometric series, can be obtained at no cost by anonymous ftp from pap. It is written in Mathematica source code and must be used in conjunction with Mathematica. To use it to identify a hypergeometric series involves the following steps. First enter the sum that interests you using the usual Sum construct. Give the expression a name, say mysum. SumF, and you will, or should, be looking at the hypergeometric designation of your sum as output.

The following are some of the most useful database entries. We will not prove any of them just now because all of their proofs will follow instantly from the computer certification methods that we will develop in Chapters 4�7.

Hence the gamma function extends the definition of n! In fact n! In this chapter we have seen how to take a sum and identify it, when possible, as a standard hypergeometric sum. We have also seen a list of many of the important hypergeometric sums that can be expressed in simple, closed form.

The strengths and the limitations of the procedure should then be clearer. The first step is to identify the sum f n as a particular hypergeometric series. This is a rather distressing development. Fortunately, what we have is a ratio of two factorials at negative integers; if we take an appropriate limit, the singularities will cancel, and a pleasant limiting ratio will ensue.

We will now do this calculation, urging the reader to take note of the fact that this kind of situation happens fairly frequently when one uses the database. The answers are formally correct, but we need some further analysis to transform them into readily useable form. Imagine, for a moment, that n is near a positive integer, but is not equal to a positive integer.

Indeed, in Mathematica, the SymbolicSum package can handle the sum 3. The Hyp package includes a large database, considerably larger than the list that we have given above. The resulting output is exactly as in 3.

So we have successfully identified the sum as a hypergeometric series. The next question is, does Hyp know how to evaluate this sum in simple form? To ask Hyp to look up your sum in its sum list we use the command SListe.

SListe It replies by giving the numbers of the formulas in its database that might be of assistance in evaluating our sum. In Class 10 Maths Ch 6 Ex 6.3 Teachoo Raw this case its reply is to tell us that one of its four items S, S, S, S might be of use. Ask any computer scientist what a database is, and you will be told something like this. A database D is a triple consisting of Class 10 Maths Ch 6 Ex 6.3 Mp3 1.

There is no hypergeometric database. Is there a collection of information? The data might be, for instance, a list of all known hypergeometric identities. Now what are the queries that we would like to address to the database? We have a slightly mythical collection of data, and a single rather precise query.

What we are missing is the algorithm. If some user asks the system whether or not a certain sum can be expressed in some simple form, exactly what steps shall the system take in order to answer the question? Certainly a minimal step would be to examine the list of known identities and see if the sum in question lives there.

The hypergeometric notation that we have described in this chapter will be a big help in doing that search. The system will simply print out the simple evaluation, and the user will go away with a smile. But suppose the sum does not appear among the data in the system? In fact, the act of checking whether the given sum lives among the data is only the first step that any competent human analyst would take.

If the sum could not be located, the next step that the analyst would probably take would be to try out some hypergeometric transformation rules. One could easily make lists of dozens of such rules, and indeed the package Hyp has dozens of them built in.

So your database should first look to see if your sum lives in the data, and if not it should next try to transform your sum into another one that does live in the data. If that succeeds, great. Or maybe three � you see the problem. Besides sequences of trans- formations, one can also use various substitutions for the parameters, and it may be hard to recognize that a certain identity is a specialization of a database entry.

There is no algorithm that will discover whether your sum is or is not trans- formable into an identity that lives in the database. Beyond all of these attempts at computer algorithms there lie human mathemati- cians. Many of them are awesomely bright, and will find immensely clever ways to evaluate your unknown sum, ways that could not in a million years be built into a computerized database.

So the database of hypergeometric identities is a myth. It is very nice to have a big list to work with. But that is by no means the whole story. The look-it-up-in-a-database process is, like any other, an algorithm for doing hypergeometric sums, and it should be assessed the same way as other algorithms. Precisely when can we expect a pretty answer? How fast is it? What is the complete algorithm, including the simplifications at the end, and how costly are they?

What are the alternatives? Those algorithms can be rather easily programmed for a computer, they work under conditions that are wider than those of the database lookup, and the conditions under which they work can be clearly stated.

Further, under the stated conditions these algorithms are exhaustive. Obviously we cannot claim that the computerized methods are the best for every situation.

Sometimes the certificates that they produce are longer and less user- friendly than those that humans might find, for example. But the emergence of these methods has put an important family of tools in the hands of discrete mathematicians, and many results that are accessible in no other way have been found and proved by computer methods. Put each of the following sums into standard hypergeometric notation.

First do it by hand. Then do it with your choice of computer software. Each of the following sums can be evaluated in a simple form. In each case first write the sum in standard hypergeometric notation. Then consult the list in this chapter to find a database member that has the given sum as a special case.

Then use the right hand side of the database sum, suitably specialized, to find the simple form of the given sum. Then check your answer numerically for a few small values of the free parameters. There she developed a method for finding recurrence relations for hypergeometric polynomials directly from the series expansions of the polynomials.

An exposition of her method is in Chapter 14 of Rainville [Rain60]. In his words, Years ago it seemed customary upon entering the study of a new set of polynomials to seek recurrence relations. Manipulative skill was used and, if there was enough of it, some relations emerged; others might easily have been lurking around a corner without being discovered.

The interesting problem of the pure recurrence relation for hypergeometric polynomials received probably its first systematic attack at the hands of Sister Mary Celine Fasenmyer. Her algorithm is also important because it has yielded general existence theorems for the recurrence relations satisfied by hypergeometric sums. We begin by illustrating her method on a simple sum. Example 4. The next step, following Sister Celine, is to put the whole thing over a common denominator.

We now substitute these values into the assumed form of the recurrence relation in 4. The method works because one can prove and we will! Her father worked his own oil lease in the area. He re- married three years later a woman, Josephine, who was twenty-five years his junior. She was sent to Pittsburgh by her order, to teach in the St. Justin School and to go to the University of Pittsburgh for her MA degree, which she received in Her ma- jor was mathematics, and her minor was in physics.

The community told her to go to the University of Michigan for her doctorate, which she did from the fall of until June of , when she received her degree.

Her thesis was written under the direction of Earl Rainville, whom she remembers as having been quite accessible and helpful, as well as working in a subject area that she liked.

She used the method in her thesis [Fase45] to find pure recurrence relations that are satisfied by various hypergeometric polynomial se- quences. In two later papers she developed the method further, and explained its workings to a broad audience in her paper [Fase49].

For an exposition of some of her thesis results see [Fase47]. Her work was described by Rainville in Chapters 14 and 18 of his book [Rain60]. Assume the recurrence formula in the form of 4.

Divide each term of 4. Place the entire expression over a single common denominator. Then collect the numerator as a polynomial in k. That is, look for a bigger recurrence.

To use it just call celine f,ii,jj ;, where f is your summand, and ii, jj are the sizes of the recurrence that you are looking for.

The recurrence is identical with the one we had previously found by hand, in 4. Then " what " recurrence " would it find? What is the recurrence for f n? To find it, just sum 4. That recurrence is of second order, however, and its solution provably cannot be expressed as a linear combination of a constant number of hypergeometric terms see Chapter 8. Next we define a Module that finds a recurrence relation satisfied by a given function f, the recurrence being of orders ii, jj.

The reader is cautioned that this program is slower than its Maple counterpart in its execution, and it should not be tried on recurrence relations of larger span.

To do this, identify the specific F in 4. To find such a recurrence we use the Maple program celine above. This point is extremely important. The summand F n, k of 4. Hence even if we sum over all integers, the sums will contain only finitely many nonvanishing terms.

Therefore, we can sum the output recurrence over all integers k. If we now sum over all integers k, we find that the sum f n , of 4. Since, from 4.

The theorem also finds explicit precomputable upper bounds on the span. P is a polynomial, 2. Some examples " of proper hypergeometric terms are as follows. Also, 2nk 2k is proper hypergeometric, but ann is not, if a is an unspecified parameter. This is not in proper hypergeometric form. Now we can state the main theorem.

Theorem 4. Then F satisfies a k-free recurrence relation. In order to do that it will be important to do a few simple exercises that relate to the behavior of translates of a proper hypergeometric term. The next step is to collect all of the terms in the sum 4. But since we have such explicit formulas for the numerator and denominator polynomials of each term, we can write out explicitly what that common denominator will be.

The first thing to notice is that, by 4. Notice that if we had permitted a denominator polynomial Q n, k to appear in the definition 4. Indeed, in that case, the i, j term in 4. That multiple would have been of an unacceptably high degree in k, and would have blocked the argument that follows from reaching a successful conclusion.

For each s, a common multiple of all of the falling factorials that appear there will be the one whose first argument is largest, i. The result of multiplying 4. This will surely happen if the number of unknowns exceeds the number of equations, and we claim that if I, J are large enough then this is exactly what happens. Hence for large enough I, J the latter would be less than the former. Since the degree in k of each rising factorial and each falling factorial that appears in 4.

A statement that the information in the notice is accurate, and under penalty of perjury, that the complaining party is authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

If you do not include all of the above information, it may invalidate your notification or cause a delay of the processing of the DMCA notification. Please note that, under Section f of the Copyright Act, any person who knowingly materially misrepresents that material or activity is infringing may be subject to liability.

Please also note that the information provided in your notification to us may be forwarded to the person who provided the allegedly infringing content. Company reserves the right to publish Claimant information on the site in place of disabled content. Notice and Take down Procedure Procedure: It is expected that all users of any part of the Company system will comply with applicable copyright laws.

The Company will comply with the appropriate provisions of Class 10 Maths Ch 6 Ex 6.3 Zero the DMCA in the event a counter notification is received. Please note that under Section f of the Copyright Act, any person who knowingly materially misrepresents that material or activity was removed or disabled by mistake or misidentification may be subject to liability.

Fishing Boat Plans Pdf 702 Diy Canoe Rack For Trailer 40