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# Mathematical Induction

by Bertrand Russel

The series of natural numbers, can all be defined if we know what we mean by the three terms "0," "number", and "successor." But we may go a step farther: we can define all the natural numbers if we know what we mean by "0" and "successor." It will help us to understand the difference between finite and infinite to see how this can be done, and why the method by which it is done cannot be extended beyond the finite. We will not yet consider how "0" and "successor" are to be defined: we will for the moment assume that we know what these terms mean, and show how thence all other natural numbers can be obtained.

It is easy to see that we can reach any assigned number, say 30,000. We first define "1" as "the successor of 0," then we define "2" as "the successor of 1," and so on. In the case of an assigned number, such as 30,000, the proof that we can reach it by proceeding step by step in this fashion may be made, if we have the patience, by actual experiment: we can go on until we actually arrive at 30,000. But although the method of experiment is available for each particular natural number, it is not available for proving the general proposition that all such numbers can be reached in this way, i.e. by proceeding from 0 step by step from each number to its successor. Is there any other way by which this can be proved?

Let us consider the question the other way around. What are the numbers that can be reached, given the terms "0" and "successor"? Is there any way by which we can define the whole class of such numbers? We reach 1, as the successor of 0; 2, as the successor of 1; 3, as the successor of 2; and so on. It is this "and so on" that we wish to replace by something less vague and indefinite. We might be tempted to say that "and so on" means that the process of proceeding to the successor may be repeated any finite number of times; but the problem upon which we are engaged is the problem of defining "finite number," and therefore we must not use this notion in our definition. Our definition must not assume that we know what a finite number is.

The key to our problem lies in mathematical induction. It will be remembered that this was the fifth of the five primitive propositions which we laid down about the natural numbers. It stated that any property which belongs to 0, and to the successor of any number which has the property, belongs to all the natural numbers. This was then presented as a principle, but we shall now adopt it as a definition. It is not difficult to see that the terms obeying it are the same as the numbers that can be reached from 0 by successive steps from next to next, but as the point is important we will set forth the matter in some detail.

We shall do well to begin with some dehnitions, which will be useful in other connections also.

A property is said to be "hereditary" in the natural-number series if, whenever it belongs to a number n, it also belongs to n + 1, the successor of n. Similarly a class is said to be "hereditary" if, whenever n is a member of the class, so is n + 1. It is easy to see, though we are not yet supposed to know, that to say a property is hereditary is equivalent to saying that it belongs to all the natural numbers not less than some one of them, e.g. it must belong to all that are not less than 100, or all that are not less than 1000, or it may be that it belongs to all that are not less than 0, i.e. to all without exception.

A property is said to be "inductive" when it is a hereditary property which belongs to 0. Similarly a class is "inductive" when it is a hereditary class of which 0 is a member.

Given a hereditary class of which 0 is a member, it follows that 1 is a member of it, because a hereditary class contains the successors of its members, and 1 is the successor of 0. Similarly, given a hereditary class of which 1 is a member, it follows that 2 is a member of it; and so on. Thus we can prove by a step-by-step procedure that any assigned natural number, say 30,000, is a member of every inductive class.

We will define the "posterity" of a given natural number with respect to the relation "immediate predecessor" (which is the converse of "successor") as all those terms that belong to every hereditary class to which the given number belongs. It is again easy to see that the posterity of a natural number consists of itself and all greater natural numbers; but this also we do not yet officially know.

By the above definitions, the posterity of 0 will consist of those terms which belong to every inductive class.

It is now not difficult to make it obvious that the posterity of 0 is the same set as those terms that can be reached from 0 by successive steps from next to next. For, in the li.rst place, 0 belongs to both these sets (in the sense in which we have defined our terms) ; in the second place, if n belongs to both sets, So does n + 1. It is to be observed that we are dealing here with the kind of matter that does not admit of precise proof, namely, the comparison of a relatively vague idea with a relatively precise one. The notion of "those terms that can be reached from 0 by successive steps from next to next" is vague, though it seems as if it conveyed a dell&e meaning; on the other hand, "the posterity of 0" is precise and explicit just where the other idea is hazy. It may be taken as giving what we meant to mean when we spoke of the terms that can be reached from 0 by successive steps.

We now lay down the following definition:-
The "natural numbers" are the posterity of 0 with respect to the relation "immediate predecessor" (which is the converse of "successor").

We have thus arrived at a definition of one of Peano’s three primitive ideas in terms of the other two. As a result of this definition, two of his primitive propositions-namely, the one asserting that 0 is a number and the one asserting mathematical induction-become unnecessary, since they result from the definition. The one asserting that the successor of a natural number is a natural number is only needed in the weakened form "every natural number has a successor."

We can, of course, easily define "0" and "successor" by means of the definition of number in general. The number 0 is the number of terms in a class which has no members, i.e. in the class which is called the "null-class." By the general detlnition of number, the number of terms in the null-class is the set of all classes similar to the null-class, i.e. (as is easily proved) the set consisting of the null-class all alone, i.e. the class whose only member is the null-class. (This is not identical with the null-class: it has one member, namely, the null-class, whereas the null-class itself has no members. A class which has one member is never identical with that one member, as we shall explain when we come to the theory of classes.) Thus we have the following purely logical detition:- 0 is the class whose only member is the null-class.

It remains to define "successor." Given any number n, let α be a class which has n members, and let x be a term which is not a member of &aplha;. Then the class consisting of α with x added on will have n + 1 members. Thus we have the following detiition:-
The successor of the number of terms in the class α is the number of terms in the class consisting of α together with x,where x is any term not belonging to the class.

Certain niceties are required to make this definition perfect, but they need not concern us. It will be remembered that we have already given a logical detition of the number of terms in a class, namely, we defined it as the set of all classes that are similar to the given class.

We have thus reduced Peano’s three primitive ideas to ideas of logic: we have given definitions of them which make them definite, no longer capable of an insnity of different meanings, as they were when they were only determinate to the extent of obeying Peano’s five axioms. We have removed them from the fundamental apparatus of terms that must be merely apprehended, and have thus increased the deductive articulation of mathematics.

As regards the five primitive propositions, we have already succeeded in making two of them demonstrable by our delition of "natural number." How stands it with the remaining three? It is very easy to prove that 0 is not the successor of any number, and that the successor of any number is a number. But there is a diaculty about the remaining primitive proposition, namely, "no two numbers have the same successor." The difficulty does not arise unless the total number of individuals in the universe is finite; for given two numbers m and n, neither of which is the total number of individuals in the universe, it is easy to prove that we cannot have m + 1 = n + 1 unless we have m = n. But let us suppose that the total number of individuals in the universe were (say) 10; then there would be no class of 11 individuals, and the number 11 would be the null-class. So would the number 12. Thus we should have 11 = 12; therefore the successor of 10 would be the same as the successor of 11, although 10 would not be the same as 11. Thus we should have two merent numbers with the same successor. This failure of the third axiom cannot arise, however, if the number of individuals in the world is not finite. We shall return to this topic at a later stage.

Assuming that the number of individuals in the universe is not tite, we have now succeeded not only in defining Peano’s three primitive ideas, but in seeing how to prove his five primitive propositions, by means of primitive ideas and propositions belonging to logic. It follows that all pure mathematics, in so far as it is deducible from the theory of the natural numbers, is only a prolongation of logic. The extension of this result to those modern branches of mathematics which are not deducible from the theory of the natural numbers offers no difficulty of principle, as we have shown elsewhere.

The process of mathematical induction, by means of which we detlned the natural numbers, is capable of generalisation. We defined the natural numbers as the "posterity" of 0 with respect to the relation of a number to its immediate successor. If we call this relation N, any number m will have this relation to m + 1. A property is "hereditary with respect to N," or simply "N-hereditary," if, whenever the property belongs to a number m, it also belongs to m + 1, i.e. to the number to which m has the relation N. And a number n will be said to belong to the "posterity" of m with respect to the relation N if n has every N-hereditary property belonging to n. These definitions can all be applied to any other relation just as well as to N.

Thus if R is any relation whatever, we can lay down the following definitions:

A property is called "R-hereditary" when, if it belongs to a term x, and x has the relation R to y, then it belongs to y.

A class is R-hereditary when its defining property is R-hereditary.

A term x is said to be an "R-ancestor" of the term y if y has every R-hereditary property that x has, provided x is a term which has the relation R to something or to which something has the relation R. (This is only to exclude trivial cases.)

The "R-posterity" of x is all the terms of which x is an R-ancestor.

We have framed the above definitions so that if a term is the ancestor of anything it is its own ancestor and belongs to its own posterity. This is merely for convenience.

It will be observed that if we take for R the relation "parent," "ancestor" and "posterity" will have the usual meanings, except that a person will be included among his own ancestors and posterity. It is, of course, obvious at once that "ancestor" must be capable of detiition in terms of "parent," but until Frege developed his generalised theory of induction, no one could have defined "ancestor" precisely in terms of "parent." A brief consideration of this point will serve to show the importance of the theory. A person confronted for the fist time with the problem of defining "ancestor" in terms of "parent" would naturally say that A is an ancestor of Z if, between A and Z, there are a certain number of people, B, C, . . . , of whom B is a child of A, each is a parent of the next, until the last, who is a parent of Z. But this definition is not adequate unless we add that the number of intermediate terms is to be finite. Take, for example, such a series as the following:-

- 1, - 1/2, - 1/4, - 1/8, . . . 1/8, 1/4, 1/2, 1.

Here we have first a series of negative fractions with no end, and then a series of positive fractions with no beginning. Shall we say that, in this series, - 1/8 is an ancestor of 1/8 ? It will be so according to the beginner’s definition suggested above, but it will not be so according to any definition which will give the kind of idea that we wish to define. For this purpose, it is essential that the number of intermediaries should be finite. But, as we saw, "finite" is to be defined by means of mathematical induction, and it is simpler to define the ancestral relation generally at once than to define it first only for the case of the relation of n to n + 1, and then extend it to other cases.Here, as constantly elsewhere, generality from the first, though it may require more thought at the start, will be found in the long run to economise thought and increase logical power.

The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid. Some believed it to be really a case of induction, in the sense in which that word is used in logic. Poincare considered it to be a principle of the utmost importance, by means of which an i&rite number of syllogisms could be condensed into one argument. We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle. There are some numbers to which it can be applied, and there are others to which it cannot be applied. We define the "natural numbers" as those to which proofs by mathematical induction can be applied, i.e. as those that possess all inductive properties. It follows that such proofs can be applied to the natural numbers, not in virtue of any mysterious intuition or axiom or principle, but as a purely verbal proposition. If "quadrupeds" are defined as animals having four legs, it will follow that animals that have four legs are quadrupeds; and the case of numbers that obey mathematical induction is exactly similar.

We shall use the phrase "inductive numbers" to mean the same set as we have hitherto spoken of as the "natural numbers." The phrase "inductive numbers" is preferable as affording a reminder that the definition of this set of numbers is obtained from mathematical induction.

Mathematical induction affords, more than anything else, the essential characteristic by which the finite is distinguished from the infinite. The principle of mathematical induction might be stated popularly in some such form as "what can be inferred from next to next can be inferred from first to last." This is true when the number of intermediate steps between first and last is finite, not otherwise. Anyone who has ever watched a goods train beginning to move will have noticed how the impulse is communicated with a jerk from each truck to the next, until at last even the hindmost truck is in motion. When the train is very long, it is a very long time before the last truck moves. If the train were infinitely long, there would be an infinite succession of jerks, and the time would never come when the whole train would be in motion. Nevertheless, if there were a series of trucks no longer than the series of inductive numbers (which, as we shall see, is an instance of the smallest of infinites), every truck would begin to move sooner or later if the engine persevered, though there would always be other trucks further back which had not yet begun to move. This image will help to elucidate the argument from next to next, and its connection with finitude. When we come to infinite numbers, where arguments from mathematical induction will be no longer valid, the properties of such numbers will help to make clear, by contrast, the almost unconscious use that is made of mathematical induction where finite numbers are concerned.

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#### Contributed by

How to use this model directly from the 🤗/transformers library:

tokenizer = AutoTokenizer.from_pretrained("shoarora/electra-small-owt")

Update on GitHub

# ELECTRA-small-OWT

This is an unnoficial implementation of an ELECTRA small model, trained on the OpenWebText corpus.

Differences from official ELECTRA models:

• we use a BertForMaskedLM as the generator and BertForTokenClassification as the discriminator
• they use an embedding projection layer, but Bert doesn't have one

(figure from Clark et al. 2020)

ELECTRA uses discriminative LM / replaced-token-detection for pretraining. This involves a generator (a Masked LM model) creating examples for a discriminator to classify as original or replaced for each token.

## Usage

from transformers import BertForSequenceClassification, BertTokenizer

tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')
electra = BertForSequenceClassification.from_pretrained('shoarora/electra-small-owt')

## Code

The pytorch module that implements this task is available here.

Further implementation information here, and here is the script that created this model.

This specific model was trained with the following params:

• batch_size: 512
• training_steps: 5e5
• warmup_steps: 4e4
• learning_rate: 2e-3

#### GLUE Dev results

Model # Params CoLA SST MRPC STS QQP MNLI QNLI RTE
ELECTRA-Small++ 14M 57.0 91. 88.0 87.5 89.0 81.3 88.4 66.7
ELECTRA-Small-OWT 14M 56.8 88.3 87.4 86.8 88.3 78.9 87.9 68.5
ELECTRA-Small-OWT (ours) 17M 56.3 88.4 75.0 86.1 89.1 77.9 83.0 67.1
ALECTRA-Small-OWT (ours) 4M 50.6 89.1 86.3 87.2 89.1 78.2 85.9 69.6

#### GLUE Test results

Model # Params CoLA SST MRPC STS QQP MNLI QNLI RTE
BERT-Base 110M 52.1 93.5 84.8 85.9 89.2 84.6 90.5 66.4
GPT 117M 45.4 91.3 75.7 80.0 88.5 82.1 88.1 56.0
ELECTRA-Small++ 14M 57.0 91.2 88.0 87.5 89.0 81.3 88.4 66.7
ELECTRA-Small-OWT (ours) 17M 57.4 89.3 76.2 81.9 87.5 78.1 82.4 68.1
ALECTRA-Small-OWT (ours) 4M 43.9 87.9 82.1 82.0 87.6 77.9 85.8 67.5

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# If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then

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If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then  [#permalink]

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Updated on: 15 Feb 2017, 12:14
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If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then the value of the product km is

(A) between 20 and 60
(B) between 60 and 100
(C) between 100 and 150
(D) between 150 and 200
(E) greater than 200

Originally posted by lgarces19 on 15 Feb 2017, 12:08.
Last edited by Bunuel on 15 Feb 2017, 12:14, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then  [#permalink]

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15 Feb 2017, 12:17
lgarces19 wrote:
If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then the value of the product km is

(A) between 20 and 60
(B) between 60 and 100
(C) between 100 and 150
(D) between 150 and 200
(E) greater than 200

Square k + m = 20;

(k + m)^2 = 20^2;

k^2 + 2km + m^2 = 400.

Since given that k^2 + m^2 = 289, then 289 + 2km = 400.

km = 111/2 = 55.5.

P.S. Is this question really from GMAT Prep?
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Re: If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then  [#permalink]

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15 Feb 2017, 14:47
Bunuel wrote:
lgarces19 wrote:
If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then the value of the product km is

(A) between 20 and 60
(B) between 60 and 100
(C) between 100 and 150
(D) between 150 and 200
(E) greater than 200

Square k + m = 20;

(k + m)^2 = 20^2;

k^2 + 2km + m^2 = 400.

Since given that k^2 + m^2 = 289, then 289 + 2km = 400.

km = 111/2 = 55.5.

P.S. Is this question really from GMAT Prep?

clearly, I need to brush up my quant skills...I took into consideration that k and m are integers...damn!
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Re: If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then  [#permalink]

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16 Feb 2017, 10:31
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lgarces19 wrote:
If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then the value of the product km is

(A) between 20 and 60
(B) between 60 and 100
(C) between 100 and 150
(D) between 150 and 200
(E) greater than 200

Bunuel's solution is perfect.
I just want to point out that there's no shortage of questions that test the concepts tested here.

Important expansions: (x + y)² = x² + 2xy + y² and (x - y)² = x² - 2xy + y²
So, if you see questions that include information about x² + y² and 2xy (or just xy), then you should consider their relationship with the above expansions.

Cheers,
Brent
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Re: If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then  [#permalink]

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16 Feb 2017, 11:57
lgarces19 wrote:
If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then the value of the product km is

(A) between 20 and 60
(B) between 60 and 100
(C) between 100 and 150
(D) between 150 and 200
(E) greater than 200

$$( k + m ) ^2= k^2 + m^2 + 2km$$

So, $$( 20 ) ^2= 289 + 2km$$

Or, $$400 = 289 + 2km$$

Or, $$2km = 111$$

Or, $$km = 55.5$$

Thus, km must be between 20 and 60, answer must be option (A) between 20 and 60
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Re: If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then &nbs [#permalink] 16 Feb 2017, 11:57
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# On-target SNR measurement

I want to measure the performance of a receiver demodulation algorithm, something like Bit Error Rate as a function of the power of the incoming signal (in dBm) and compare it to simulation results.

I first simulate it on the computer and get a result of BER as a function of SNR.

After implementation on target, I measure the BER of my algorithm by changing the power of the incoming signal (in dBm).

My architecture is that I have an RF circuit that drives an ADC.

How can I relate the real measures with the theoretical, knowing the noise figure of my RF circuit part?

By removing the RF signal, I have been able to measure (numerically, after the ADC) the RMS value of the noise in base band (1024 points @240kHz). By applying a RF signal at a good level (-80 dBm), I have measured the RMS value of my baseband signal+noise in the same way as above.

Given these data, I can calculate the (S+N)/N ratio and then the S/N when I apply -80 dBm at the input of my receiver.

My conclusion is that knowing this I can calculate the SNR at every RF level before and after the RF circuit. I can then plot the theorical results, the measurements vs the incoming signal in dBm.

Is it correct? If not why?

Edit: I know the SNR is a ratio of power, but we can do it with voltage (RMS value or amplitude is better?)

Edit2: I just realised that with my method I can only measure a (S+N)/N ratio (as RMS values cannot be added or substracted)...


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# Sonda lambda time btween transitions 0

## Time sonda btween

Add: fezoquxi21 - Date: 2020-12-09 22:56:01 - Views: 7565 - Clicks: 2773
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a, b, c), the closure type includes unnamed non-static data members, declared in unspecified order, that hold copies of all entities that were so captured. Time sonda Converter; HEALPix Pixel Info; Calculators; Coordinate Conversions. The "Coordinate Format" element controls both input. Het werkgeheugen meet 4GB. Get online real time access latest desktopand mobile devices. Add transitions between slides in Keynote on iPad. Transitions XTRActive lenses are uniquely designed to protect your eyes from the brightest sun outdoors and harsh artificial light indoors. 1) matrix form shown in Eqn 11.

com is the guide for paid work, volunteering, interning, study, language learning, travel and living abroad. Python time Module In this article, we will explore time module in detail. De OnePlus 8T is het vlaggenschip van de fabrikant van de tweede helft van. That leaves the slide in view, sonda lambda time btween transitions 0 so I can sonda lambda time btween transitions 0 see the transition’s preview.

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For example, the Push transition makes a slide appear to push the previous slide off the screen. Then, the right hand side of the equation is the probability of being at State $$i$$ sonda lambda time btween transitions 0 at time $$t + 1$$ given the entire past history of the chain, sonda lambda time btween transitions 0 or the location of the particle in every single State (here, the State $$j$$ at transitions time $$t$$, and all the way down to $$x_0$$, which is just arbitrary notation for some constant that $$X_0. The Poisson distribution is used to model the number of events occurring within a given time interval. &169; Matt Bognar Department sonda lambda time btween transitions 0 of Statistics and Actuarial Science sonda lambda time btween transitions 0 University of Iowa. Create speaker notes. &0183;&32;If the lambda-expression captures anything by sonda lambda time btween transitions 0 copy (either implicitly with capture clause = or explicitly with a capture that does not include the character &, sonda lambda time btween transitions 0 e. Lambda forms will contain their own invocation counters which will allow them to delay compilation until they are "hot" sonda lambda time btween transitions 0 enough. The (mathematically sonda lambda time btween transitions 0 equivalent to Eqns 11. You can add a transition between any two slides in your presentation. We examined how a virus, bacteriophage λ, evolved to infect its host, Escherichia coli, through a novel pathway. The \(r_ij$$ : are the correlation coefficients between variable $$i$$ and principal component $$j$$, where $$i$$ ranges from 1 to 4 and $$j$$ ; from 1 to 2. A good value of $$\lambda$$ is one which makes the size of the seasonal variation about the same across the whole series, as that makes the forecasting model simpler. . Het lcd-scherm oogt scherp door de full-hd-resolutie en heeft een kleine inkeping voor de sonda lambda time btween transitions 0 selfiecamera.

2, a 4X1 census vector (LHS) at time t+1 results from the right-multiplication of a 4X4 projection matrix by a 4X1 census vector at time t. sonda lambda time btween transitions 0 Those sonda lambda time btween transitions 0 data members that correspond to captures without initializers. The value transitions returned generally represents the number of seconds since 00:00 hours, UTC (i. It is usually used in scenarios sonda lambda time btween transitions 0 where we are counting the occurrences of certain events sonda lambda time btween transitions 0 that appear to happen at a certain rate, but sonda lambda time btween transitions 0 completely at random (without a certain structure). Although libraries may use a different representation of time: Portable.

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English

# Item

ITEM ACTIONSEXPORT

Released

Paper

#### Controlling a diatomic shape resonance with non-resonant light

##### MPS-Authors
/persons/resource/persons21794

Lemeshko,  Mikhail P.
Molecular Physics, Fritz Haber Institute, Max Planck Society;

/persons/resource/persons21529

Friedrich,  Bretislav
Molecular Physics, Fritz Haber Institute, Max Planck Society;

##### External Resource
No external resources are shared
##### Fulltext (public)

1105.0761v1.pdf
(Preprint), 304KB

##### Supplementary Material (public)
There is no public supplementary material available
##### Citation

Aganoglu, R., Lemeshko, M. P., Friedrich, B., González-Férez, R., & Koch, C. P. (in preparation). Controlling a diatomic shape resonance with non-resonant light.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-EE18-8
##### Abstract
A (diatomic) shape resonance is a metastable state of a pair of colliding atoms quasi-bound by the centrifugal barrier imposed by the angular momentum involved in the collision. The temporary trapping of the atoms' scattering wavefunction corresponds to an enhanced atom pair density at low interatomic separations. This leads to larger overlap of the wavefunctions involved in a molecule formation process such as photoassociation, rendering the process more efficient. However, for an ensemble of atoms, the atom pair density will only be enhanced if the energy of the resonance comes close to the temperature of the atomic ensemble. Herein we explore the possibility of controlling the energy of a shape resonance by shifting it toward the temperature of atoms confined in a trap. The shifts are imparted by the interaction of non-resonant light with the anisotropic polarizability of the atom pair, which affects both the centrifugal barrier and the pair's rotational and vibrational levels. We find that at laser intensities of up to $5\times 10^{9}$ W/cm$^2$ the pair density is increased by one order of magnitude for $^{87}$Rb atoms at $100 \mu$K and by two orders of magnitude for $^{88}$Sr atoms at $20 \mu$K.


text:
## H.1 Assignment: Cross Recurrence Quantification Analysis

1. Create two variables for CRQA analysis:
y1 <- sin(1:900*2*pi/67)
y2 <- sin(.01*(1:900*2*pi/67)^2)
1. You have just created two sine waves. We’ll examine if and how they are coupled in a shared phase space. As a first step plot them.

2. Find an embedding delay (using mutual information) and an embedding dimension (if you calculate an embedding dimension using package fractal for each signal seperately, as a rule of thumb use the highest embedding dimension you find in further analyses).

# General settings for crqa()
par0 <- list(rescale = 0,
normalize = 0,
mindiagline = 2,
minvertline = 2,
tw = 0,
whiteline = FALSE,
recpt = FALSE,
side = "both",
checkl = list(do = FALSE, thrshd = 3, datatype = "categorical",pad = TRUE)
)

# Settings for optimizeParam()
par <- list(lgM =  20, steps = seq(1, 6, 1),
normalize = par0$normalize, rescale = par0$rescale,
mindiagline = par0$mindiagline, minvertline = par0$minvertline,
tw = par0$tw, whiteline = par0$whiteline,
recpt = par0$recpt, fnnpercent = 10, typeami = "mindip") 1. We can now create a cross recurrence matrix. Fill in the values you decided on. You can choose a radius automatically, look in the crqa manual. Note: There is no rescaling of data, the sines were created in the same range. You can plot a matrix using image(). You could also check package nonlinearTseries. If you sourced the nlRtsa library, use plotRP.crqa() • Get the optimal parameters using a radius which will give us 2%-5% recurrent points. (ans <- optimizeParam(ts1 = y1, ts2 = y2, par = par, min.rec = 2, max.rec = 5)) $radius
[1] 0.2128611

$emddim [1] 2$delay
[1] 15
1. Run the CRQA.
crqaOutput <- crqa(ts1 = y1, ts2 = y2,
delay = ans$delay, embed = ans$emddim,
radius = ans$radius, normalize = par0$normalize,
rescale = par0$rescale, mindiagline = par0$mindiagline, minvertline = par0$minvertline, tw = par0$tw,
whiteline = par0$whiteline, recpt = par0$recpt,
side = par0$side, checkl = par0$checkl
)
1. Can you understand what is going on? Explain the the lack of recurrent points at the beginning of the time series.
plotRP.crqa(crqaOutput)

1. Examine the synchronisation under the diagonal LOS. Look in the manual of crqa or Coco & Dale (2014).

2. Make a plot of the diagonal profie. How far is the peak in RR removes from 0 (Line of Synchronisation)?

This is based on the timeseries (without embedding)

win <- 50
(res <-  drpdfromts(y1, y2, ws = win, datatype = 'continuous', radius = ans$radius))[2:3] $maxrec
[1] 0.3080357

$maxlag [1] 55 plot(-win:win,res$profile,type = "l", lwd = 5, xlab = "Delays", ylab = "Recurrence")
abline(v=0)

To get the diagonal profile from the recurrence plot, use spdiags().

dprofile <- spdiags(crqaOutput$RP) plot(-win:win,colMeans(dprofile$B[, between(dprofile\$d, -win,win)]), type="l",
lwd = 5, xlab = "Delays", ylab = "Recurrence")
abline(v=0)

1. Perform the same steps with a shuffled version (or use surrogate analysis!) of the data of timeseries $$y1$$. You can use the embedding parameters you found earlier.


text:
# Birds on the move

More migration last night, but I’ve gotta get out early today so the radar will have to wait. You don’t have to though, as you can get the real-time (and archived) radar for today via this website:

I’ll post an update with the radar and interpretation later today.

Until then, good birding!

David

This entry was posted in Birds, Fall Migration 2011, Forecast, Migration, Migration Radar, NEXRAD Migration Study. Bookmark the permalink.


text:
# IC Schematics - Convention and Reality

#### teliocide

Joined Sep 26, 2013
55
In the following each of the inverters are shown with ins and outs but no connection with Vdd or Vss

In a functioning circuit like this

They are shown as being connected op amp 6.

I do not get it.
I am also at a loss how to set this up in LtSpice

Thanks

Last edited by a moderator:

#### WBahn

Joined Mar 31, 2012
24,854
In the following each of the inverters are shown with ins and outs but no connection with Vdd or Vss

In a functioning circuit like this

They are shown as being connected op amp 6.

I do not get it.
I am also at a loss how to set this up in LtSpice

Thanks
"op amp 6"???

I see six inverters. Inverters are NOT op amps.

The power connections to the six inverters on the die are via pins 7 (Vss) and 14 (Vdd). Since all of the inverters on on a single die, there no need for separate power/ground connections to each one.

In your second drawing, the power ground connections are shown on just one of the six inverters -- the rest are inferred via the netlist.

How you set it up in a particular simulator depends both on the simulator and on the parts library involved.

#### teliocide

Joined Sep 26, 2013
55
So on the 4069 circuit the connections to Pin 7 and Pin 14 on N6 is just the convention for indicating a common power source to all the inverters in the circuit. (not actually connected to N6 as the inverter only has an IN and an OUT.

#### crutschow

Joined Mar 14, 2008
23,544
I am also at a loss how to set this up in LtSpice
The CD4000 and 74HCxx digital models I have for LTspice do not show power and ground connections on the schematic.
They are hidden connections.
They require a power supply connected to ground on the schematic with a Vdd node name on the supply plus node.

Here's an example of a CD4017 circuit in LTspice:

Last edited:

#### WBahn

Joined Mar 31, 2012
24,854
So on the 4069 circuit the connections to Pin 7 and Pin 14 on N6 is just the convention for indicating a common power source to all the inverters in the circuit. (not actually connected to N6 as the inverter only has an IN and an OUT.
ALL of the physical inverters on the chip have power and ground connections. Just as all logic gates and opamps and comparators and other electronic components do. Logic devices have to be powered in order to work properly.

But drawing all of those power and ground connections significantly clutters the diagram. Plus, if each gate has a power and ground node, what happens if someone attempts to power four of the inverters from one supply and the remaining two from another?

There are many approaches to dealing with this. Some put explicit power/ground connections on just one of the gates and use the library tools to share them with other gates. Some make the power and ground connections hard tied to specific global net names. There are several other approaches as well. They are each a compromise between simplicity and flexibility for the users of the library.

#### MisterBill2

Joined Jan 23, 2018
4,066
ALL of the physical inverters on the chip have power and ground connections. Just as all logic gates and opamps and comparators and other electronic components do. Logic devices have to be powered in order to work properly.

But drawing all of those power and ground connections significantly clutters the diagram. Plus, if each gate has a power and ground node, what happens if someone attempts to power four of the inverters from one supply and the remaining two from another?

There are many approaches to dealing with this. Some put explicit power/ground connections on just one of the gates and use the library tools to share them with other gates. Some make the power and ground connections hard tied to specific global net names. There are several other approaches as well. They are each a compromise between simplicity and flexibility for the users of the library.
Power and ground (return) are understood, and if the simulation software does not understand that then it is of marginal value. I have designed a lot of digital circuits that worked the first time that they were wired like the drawing. Why waste time with a simulator???

#### crutschow

Joined Mar 14, 2008
23,544
Why waste time with a simulator?
Well obviously not all of us are as good at designing mistake free designs as you.

#### MisterBill2

Joined Jan 23, 2018
4,066
Well obviously not all of us are as good at designing mistake free designs as you.
A timing chart showing output changes based on input changes is a fairly simple way of validating logic. It does consume a bit of paper, but it is quite powerful.It has the advantage of often making errors visible earlier and with less effort, and it can be verified as correctly representing a circuit, while taking less time than entering a model in a simulator. Even better, it is a useful tool for verifying logic as a circuit is created.

#### WBahn

Joined Mar 31, 2012
24,854
Power and ground (return) are understood, and if the simulation software does not understand that then it is of marginal value.
Really? Let's see, I've designed circuits for systems that had 2.5 V, 3.3 V, and 5.0 V logic all in the same circuit. A simulator that only offered a single, fixed logic power wouldn't have been marginal, it would have been useless. Most of the circuits I've designed had a different return node than most of the rest of the circuit and further needed the substrate bulk connected tied to a separate net from the return power rail. A simulator that assumed that these were tied together would have been useless.

We actually had three levels of symbols for our standard logic cells. The lowest had everything pinned out, including the substrate bulk connection. The next level had these tied to parameterized signals so that we could define which power/ground/bulk nets they went to in the netlist without having to clutter up the schematics with the pins (but rather by setting attributes on the symbols), and the top level had power tied to the global VddD, ground tied to VssD, and the bulk tied to Vss. The latter were used when we only had a single digital supply rail.

I have designed a lot of digital circuits that worked the first time that they were wired like the drawing. Why waste time with a simulator???
And were you willing to guarantee, to a customer about to pay the better part of a million dollars to have chips fabricated with your design, that it would work the first time over all supply voltages and temperatures, even down to liquid nitrogen, and across all four of the foundry's process corners and, if they didn't, that you would do the redesign and submission for free? We were -- that was our standard guarantee.

#### MisterBill2

Joined Jan 23, 2018
4,066
WBahn, Oh Wow No, I don't design chips, and I am glad of that. For our company an order of 10 machines was a huge order, and each machine might have 2 or three custom, or standard, circuit boards. I was always careful to design with logic and analog chips that were available from multiple sources so that nothing was ever dependent on a single sourced part. Just totally different from your line of business, because those complex chips have so much NRE that production runs have to be really big. So we work in two different worlds, it seems. And that is fine, because who would need a thousand specialized machines to check products coming off the end of a production line one a minute?
We never would consider custom ICs, because they would never be cost effective. One person did point out that he could get all of the functionality into one IC and save us almost $5 per unit, for a cost of about$25,000. That was far greater than our entire IC budget for the year. And that included a large production run of 35 laser speed measuring systems, whose IC count was 5 for the sensor and three for the display.
So for us and for our operation a timing chart was the way to go. IC design, especially the big ones, is an entirely different world, not just a different ball game. Of course our products did all come with a similar guarantee.

#### DickCappels

Joined Aug 21, 2008
5,952
This might useful to you, it is an expansion of crutschow's explanation in post #4.

Right-mouse-click on the device in your schematic. A dialog will be shown.
Enter the parameter(s) into any field from "Value" to "SpiceLine2".

Vhigh=5 Vlow=0 Ref=1.5
Trise=5n Tfall=5n Td=5n

"Ref" is the threshold. Default is (Vhigh+Vlow)/2

The above is from Google Groups

#### BR-549

Joined Sep 22, 2013
4,938
It might be different today.....but 20 yrs.ago I could order our own designed IC chips. If I recall, the min order was 500 units. It was a wired headset transceiver with vox....for aircraft helmets.
This was analog.....we would solder a breadboard working prototype circuit to spec. Send this circuit with details to a job contracted IC designer. After IC design approval, the IC guy would get bids from several chip manufacturers. So we had a design cost and manufacture cost.

But the beauty was we had control of price. It was custom work. So we might use 300 units out of 500.....but the chip "NUT" would be in 300 unit cost.

If I recall correctly, there were many custom chip manFs. Maybe not now.

#### WBahn

Joined Mar 31, 2012
24,854
It might be different today.....but 20 yrs.ago I could order our own designed IC chips. If I recall, the min order was 500 units. It was a wired headset transceiver with vox....for aircraft helmets.
This was analog.....we would solder a breadboard working prototype circuit to spec. Send this circuit with details to a job contracted IC designer. After IC design approval, the IC guy would get bids from several chip manufacturers. So we had a design cost and manufacture cost.

But the beauty was we had control of price. It was custom work. So we might use 300 units out of 500.....but the chip "NUT" would be in 300 unit cost.

If I recall correctly, there were many custom chip manFs. Maybe not now.
The number of players in the ASIC world has definitely dwindled. As the design rules have gone down, the cost of the fab have gone through the roof. Multi-billions of dollars for a fab that has to recoup most of that investment in just a few years time.

There are a handful of places that will do multi-project wafer (MPW) runs where your design is placed on a reticle with many others. You normally get five to ten die from one of those. I've been out of that world for about a decade now (still do some design work for my old employer from time to time, but don't get involved with the fab interface at all any more) and at that time about the cheapest you could even do that was about $20k. When I first started there in 1995, you could actually do it for as low as$800.

Now, even if you have a design for which 2 um (or larger) rules would be adequate, it's pretty much impossible to find a fab that is still using those rules and even harder to find one that will do MPW runs.

#### teliocide

Joined Sep 26, 2013
55
My God this has really gone awry ........

I ask questions because I lack knowledge and experience but have interest.
I have numerous qualifications but none that relate to electronics.

Electronics is ubiquitous and inescapable.

It is a shame that life is so short so I can never become an expert in everything.
There would be no need to annoy people with my ignorant questions.

So you (plural) can puff up your feathers and demonstrate your intellectual supremacy. .

A true expert no matter how gifted has had to do some of the hard yards.
Satisfaction awaits you - just allow a few crumbs of your wisdom to drop from the table.
Then us Plebs have something tangible to fight over.

#### BR-549

Joined Sep 22, 2013
4,938
Did you understand posts 1 thru 5? On a schematic.....on IC, OP-AMP, and other circuits........the power pins and power supply bypassing are not shown at symbol location. This is for clarity. When looking at a schematic....we are tracing the signal.....not the power supply. The power supply is considered house keeping. We don't analyze IC circuits like we would a transistor circuit.

You will find representative power rails and bypassing caps.....usually in the power supply section of schematic.

Does that make any sense to you?

#### crutschow

Joined Mar 14, 2008
23,544
My God this has really gone awry ........
Posters here often get into side quibbles about the thread subject and sometimes sort of go off-rail, but don't let that affect your posts.
Just ignore the irrelevant.

#### Bordodynov

Joined May 20, 2015
2,430
See

#### MisterBill2

Joined Jan 23, 2018
4,066
The beauty of this site is that civil discourse is not stomped on nor forbidden. Certainly it went a bit off the original question, and certainly my remark about simulation was challenged, but in a decent manner, pointing out that what does not fit in some worlds is vital in others..
BUT, returning to the original question, I think, the device model that would include the power connections would be quite large, and totally distract from the common use of simulations. Most digital designs include enough bypassing and voltage stablizing effort that power connections are not included in the logic simulation. Of course, that does not include any design work within a specific integrated circuit, which, again, is a totally different world.

#### teliocide

Joined Sep 26, 2013
55
Thanks to all
Primarily I am here to learn.

I have a story.

20 years ago I went to the Hanoi university to learn Vietnamese.
For white devils, like myself, this is very very difficult.

After a lot of hard study I realized that the only Vietnamese people that could understand what I spoke were Vietnamese people that spoke English. Extremely useful NOT!

I see an analogy with this forum.

When a novice asks a questions the experts here have no real idea of the position of ignorance the novice is speaking from.
The answers provided tend to be "club speak" ie expert to expert. The poor old novice is drowned in unfamiliarity and alien terms.

I understand that the quality of the answer is determined by the clarity of the question.

But when you don't speak the language how can one ask precise questions.

#### eetech00

Joined Jun 8, 2013
1,726
Hi

There are many ways to do this in LTspice. Here's another way.
This approach requires a special library "CD4000_v.lib" from the LTspice Yahoo group. This library supports symbols with or without supply pins.
It also requires special symbols, one with supply pins, and one without supply pins.

See image below.

eT


text:
Recognitions:
Gold Member

## Alternative theories being tested by Gravity probe B

Indeed, Garth. Let the data speak for itself. I do not lean either way, and I am certain you feel the same way. It will be difficult to sieve through the data . . . I hope you will be critical of that process.
 Has anyone done a parameterized post-Newtonian analysis? Can one express the expected results in terms of the usual Eddington alpha, beta gamma and higher order parameters? Any refs? Best, Jim
 I should have googled first. Apparently it tests gamma and alpha-one ( a non-conservative parameter), according to Will. No doubt that is why Nordstrom thinks the money has been wasted, as gamma has already been strongly constrained and most people believe in the conservation laws. Best, Jim

Recognitions:
Gold Member
 Quote by jgraber I should have googled first. Apparently it tests gamma and alpha-one ( a non-conservative parameter), according to Will. No doubt that is why Nordstrom thinks the money has been wasted, as gamma has already been strongly constrained and most people believe in the conservation laws. Best, Jim
You'll find quite an exchange on the use of the word(s) "believe" (actually belief) in the dark matter, dark energy & gravity thread!

The fact that other viable alternative gravitational/cosmological theories are also being tested by GP-B, such as SCC, makes the enterprise worthwhile.

This is especially so in the light of persistent problems with the standard model, even if we gloss over the fact that the Higgs boson/inflaton, the DM particle and DE have not been identified in the laboratory.

A recent paper examines a link between DM and baryonic matter Cold Dark Matter as Compact Composite Objects
 Some of the observations that may be in conflict with the standard viewpoint are: • The density profile is too cuspy, [4], [5], [6]. The disagreement of the observations with high resolution simulations is alleviated with time, but some questions still remain [5], [6]. • The number of dwarf galaxies in the Local group is smaller than predicted by CCDM simulations, [4], [5], [6]. This problem is also becoming less dramatic with time [5], [6]. • CCDM simulations produce galaxy disks that are too small and have too little angular momentum, [4], [5], [6]; • There is a close relation between rotation curve shape and light distribution. This implies that there is a close coupling between luminous and dark matter which is difficult to interpret, see e.g. [7]; • There is a correlation in early-type galaxies supporting the hypothesis that there is a connection between the DM content and the evolution of the baryonic component in such systems, see e.g.[8]; • The order parameter (either the central density or the core radius) correlates with the stellar mass in spirals[9]. This suggests the existence of a well-defined scale length in dark matter haloes, linked to the luminous matter, which is totally unexpected in the framework of CDM theory, but could be a natural consequence of DM and baryon interaction. • There is a mysterious correlation between visible and DM distributions on log−log scale, which is very difficult to explain within the standard CCDM model [10]; • A recent analysis of the CHANDRA image of the galactic center finds that the intensity of the diffuse X-ray emission significantly exceeds the predictions of a model which includes known Galactic sources [11]. The spectrum is consistent with hot 8 KeV spatially uniform plasma. The hard X-rays are unlikely to result from undetected point sources, because no known population of stellar objects is numerous enough to account for the observed surface brightness.
It also seems that an Age Problem is raising its head again as observations of old evolved objects are being made at z > 4.

All the more reason to keep an open mind and continue to confirm our "beliefs" with experimental verification.

We live in interesting times!

Garth

Recognitions:
Gold Member
 Quote by jgraber Has anyone done a parameterized post-Newtonian analysis? Can one express the expected results in terms of the usual Eddington alpha, beta gamma and higher order parameters? Any refs? Best, Jim
Try Will's: The Confrontation between General Relativity and Experiment or my: Resolving the Degeneracy: Experimental tests of the New Self Creation Cosmology and a heterodox prediction for Gravity Probe B for an alternative model.

They both use the parameterized post-Newtonian (PPN) analysis.

Garth

Recognitions:
Gold Member
Halfway through Phase II!

The latest release from the Gravity Probe B website.
 GP-B DATA ANALYSIS & RESULTS ANNOUNCEMENT STATUS During the 50-week science phase of the GP-B mission and the 7-week instrument calibration phase, which lasted from August 2004 - Septermber 2005, we collected over a terabyte of experimental data. Analysis has been progressing through a 3-phase plan, each subsequent phase building on those preceding it. In Phase I, which lasted from the end of September 2005 through February 2006, the analysis focused on a short term—day-by-day or even orbit-by-orbit—examination of the data. The overall goals of this phase were to optimize the data analysis routines, calibrate out instrumentation effects, and produce initial "gyro spin axis orientation of the day" estimates for each gyro individually. At this stage, the focus was on individual gyro performance; there was no attempt to combine or compare the results of all four gyros, nor was there even an attempt to estimate the gyro drift rates. We are currently progressing through Phase II of the data analysis process, which began at the beginning of March and is scheduled to run through mid-August 2006. During Phase II, our focus is on understanding and compensating for certain long-term systematic effects in the data that span weeks or months. The primary products of this phase will be monthly spin axis drift estimates for each gyro, as well as refined daily drift estimates. In this phase, the focus remains on individual gyro performance. In Phase III, which is scheduled to run from late August 2006 through December 2006, data from all four gyros will be integrated over the entire experiment. The results of this phase will be both individual and correlated gyro drift rates covering the entire 50-week experimental period for all four gyros. These results will be relative to the position of our guide star, IM Pegasi, which changed continually throughout the experiment. Thus, the final step in the analysis, currently scheduled to occur in January 2007, will be to combine our gyro drift results with data mapping the proper motion of IM Pegasi relative to the unchanging position of a distant quasar. The proper motion of IM Pegasi has been mapped with unprecedented precision using a technique called Very Long Baseline Interferometry (VLBI) by Irwin Shapiro and his team at the Harvard-Smithsonian Center for Astrophysics (CfA), in collaboration with Norbert Bartel at York University in Toronto and French astronomer Jean-Francois Lestrade. Playing the role of our own harshest critic, our science team will then perform a careful and thorough final review of the analysis and results, checking and cross-checking each aspect to ensure the soundness of our procedures and the validity of our outcomes. We will then turn the analysis and results over to our GP-B Science Advisory Committee (SAC), that has been closely monitoring our experimental methods, data analysis procedures, and progress for 11 years, to obtain its independent review. In addition, we will seek independent reviews from a number of international experts. Throughout phases II and III, members of our team will be preparing scientific and engineering papers for publication in late 2006-2007. At the same time, we will be working with NASA to plan a formal public announcement of the results of this unprecedented test of General Relativity. We expect to make this announcement of the results in April 2007.
Less than a year to go and counting!

Garth
 Recognitions: Homework Help Science Advisor Thanks for the list of off Broadway gravitation theories. Here's another, alternative theorist with a prediction (0.000): http://www.mass-metricgravity.net/ By the way, I'm working on a flat space gravitation simulator. My original purpose was to show how standard GR differed from the Cambridge gauge gravity version of GR. The Cambridge guys say that their version works on flat space and test particles therefore cross the event horizon in finite coordinate time. Their website is http://www.mrao.cam.ac.uk/~clifford/ . For reasons having to do with elementary particles, I find the Cambridge theory convincing, and I thought an animation showing the GR particles getting stuck on the event horizon while the Cambridge particles went on through to the singularity would be convincing. Now so far I've only got the Newtonian gravity running: http://www.gaugegravity.com/testappl...etGravity.html but I should get GR running this weekend, and the Cambridge version (which amounts to allowing a non diagonal metric) soon after. Where this all gets back to this forum is that I would like to include as many gravity theories as possible, and you've listed quite a few. In order for a theory to be used, I have to be able to write the acceleration in terms of position and velocity. Carl

 Quote by Garth The Gravity Probe B satellite has placed four (over redundant) gyroscopes in low polar Earth orbit to primarily test two predictions of General Relativity. The first effect being tested is (for the GP-B polar orbit) a N-S geodetic precession, caused by the amount a gyro 'leans' over into the slope of curved space. The second effect being tested is the E-W frame-dragging, Lense-Thirring, or gravitomagnetic effect, caused by the spinning Earth dragging space-time around with it. Some researchers, such as Kenneth Nordtvedt, have said that the experiment was worth doing when it was first proposed but that now GR has been verified beyond resonable doubt the result of GP-B is a foregone conclusion. I have now discovered several theories competing with General Relativity(GR) that are being tested and falsified by this experiment: my Self Creation Cosmology).(SCC), Moffat's Nonsymmetric Gravitational Theory (NGT), Hai-Long Zhao's mass variance SR theory (MVSR), Stanley Robertson's Newtonian Gravity theory (NG), and Junhao & Xiang's Flat space-time theory (FST). As the results will be published in the not too distant future they could be interesting!! (Note if anybody knows of any other theories with alternative predictions for GP-B please post them as well for comparison.) 1. GPB Geodetic precession GR = 6.6144 arcsec/yr SCC = 4.4096 arcsec/yr NGT = 6.6144 - a small $\sigma$ correction arcsec/yr MVSR = 6.6144 arcsec/yr NG = 1.6536 arcsec/yr FST = 4.4096 arcsec/yr 2. GPB gravitomagnetic frame dragging precession GR = 0.0409 arcsec/yr SCC = 0.0409 arcsec/yr NGT = 0.0409 arcsec/yr MVSR = 0.0102 arcsec/yr NG = 0.0102 arcsec/yr FST = 0.0000 arcsec/yr I cannot vouch for these other theories, they may well be considered 'crackpot' by some, however all these theories have the advantage, together with GR, that they are able to be falsified by the GP-B results. We continue to wait and see! Garth
Mass-metric relativity is a scalar theory of gravity, and is based on the increase of mass with speed and with gravitational potential. Its predictions for the gpb are: geodetic rate -6.56124 arcsec/yr. Note the sign, indicating that the precession is backward instead of forward as in GR. Lense-Thirring rate -.01924 arcsec/yr. Actually, the Lense-Thirring rate is zero but a geodetic perturbation caused by the yearly orbit of earth about the sun induces a geodetic precession in the opposite direction. Let the experiment decide. A basic paper on mass-metric relativity is the lasl arXiv 0012059 paper, by R.L. Collins.

R.L. Collins
 Recognitions: Homework Help Science Advisor Professor Collins, Please allow me to be the first to welcome you to physics forums. Here are links to your three very fascinating papers on gravitation, in the order I think they should be read: Changing Mass Corrects Newtonian Gravity Newton's inverse-square law of universal gravitation assumes constant mass. But mass increases with speed and perhaps with gravity. By SR, mass is increased over the rest mass by gamma. Rest mass is here postulated to increase under gravity, by $$1/\alpha =1+GM/rc^2$$. We examine the consequences of introducing this changing mass into Newton's law in flat spacetime. This variable mass affects the metric, relative to an observer away from the influence of gravity, contracting both lengths and times (as measured) by alpha/gamma. The gravitational force, as in orbital calculations, differs from Newton's law by the factor $$(\gamma/\alpha)^3$$, and is not quite inverse square. Without adjustable parameters, this accounts fully for the classical tests of GR. The postulated "fifth force" appears at the $$10^-9$$ g level. Gravitationally-influenced space remains Euclidean, but the mass-metric changes make it seem curved when measured. http://www.arxiv.org/abs/physics/0012059 SN1a Supernova Red Shifts http://www.arxiv.org/abs/physics/0101033 The shrinking Hubble constant http://www.arxiv.org/abs/physics/0601013 By the way, I've just got a first cut of a GR simulating program done. I'm not very sure of it, but it seems like it works okay (but I'm not much of a gravity guy): http://www.gaugegravity.com/testappl...etGravity.html I've set the initial conditions to illustrate a fairly extreme case of precession. When I get this program running satisfactorily, I will include your equation of motion. I can hardly wait, but ethanol is keeping me busy right now. Carl

Recognitions:
Gold Member
 Quote by rusty Mass-metric relativity is a scalar theory of gravity, and is based on the increase of mass with speed and with gravitational potential. Its predictions for the gpb are: geodetic rate -6.56124 arcsec/yr. Note the sign, indicating that the precession is backward instead of forward as in GR. Lense-Thirring rate -.01924 arcsec/yr. Actually, the Lense-Thirring rate is zero but a geodetic perturbation caused by the yearly orbit of earth about the sun induces a geodetic precession in the opposite direction. Let the experiment decide. A basic paper on mass-metric relativity is the lasl arXiv 0012059 paper, by R.L. Collins. R.L. Collins
Thank you rusty, the line up is now:

Note:
1. The first effect being tested is (for the GP-B polar orbit) a N-S geodetic precession, caused by the amount a gyro 'leans' over into the slope of curved space.

2. The second effect being tested is the E-W frame-dragging, Lense-Thirring, or gravitomagnetic effect, caused by the spinning Earth dragging space-time around with it.

Einstein's General Relativity(GR)
Barber's Self Creation Cosmology).(SCC),
Moffat's Nonsymmetric Gravitational Theory (NGT),
Hai-Long Zhao's mass variance SR theory (MVSR),
Stanley Robertson's Newtonian Gravity theory (NG), and
Junhao & Xiang's Flat space-time theory (FST).
R. L. Collin's Mass-metric relativity (MMR)

The predictions are:

1. GPB Geodetic precession
GR = 6.6144 arcsec/yr
SCC = 4.4096 arcsec/yr
NGT = 6.6144 - a small $\sigma$ correction arcsec/yr
MVSR = 6.6144 arcsec/yr
NG = 1.6536 arcsec/yr
FST = 4.4096 arcsec/yr
MMR = -6.56124 arcsec/yr

2. GPB gravitomagnetic frame dragging precession
GR = 0.0409 arcsec/yr
SCC = 0.0409 arcsec/yr
NGT = 0.0409 arcsec/yr
MVSR = 0.0102 arcsec/yr
NG = 0.0102 arcsec/yr
FST = 0.0000 arcsec/yr
MMR = -0.01924 arcsec/yr

Garth

Recognitions:
Gold Member
 Quote by rusty Mass-metric relativity is a scalar theory of gravity, and is based on the increase of mass with speed and with gravitational potential. Its predictions for the gpb are: geodetic rate -6.56124 arcsec/yr. Note the sign, indicating that the precession is backward instead of forward as in GR. Lense-Thirring rate -.01924 arcsec/yr. Actually, the Lense-Thirring rate is zero but a geodetic perturbation caused by the yearly orbit of earth about the sun induces a geodetic precession in the opposite direction. Let the experiment decide. A basic paper on mass-metric relativity is the lasl arXiv 0012059 paper, by R.L. Collins. R.L. Collins
rusty has MMR been published in a peer reviewed journal?

If not you can publish it here in the Independent Research Forum and we can discuss it.

Garth

Recognitions:
Gold Member
Now into Phase III of the data analysis of Gravity Probe B.
 We are now beginning Phase III—the final phase-of the data analysis—which will last until January-February, 2007. Whereas in Phases I and II the focus was on individual gyro performance, during Phase III, the data from all four gyros will be integrated over the entire experiment. The results of this phase will be both individual and correlated changes in gyro spin axis orientation covering the entire 50-week experimental period for all four gyros. These results will be relative to the position of our guide star, IM Pegasi, which changed continually throughout the experiment. Thus, the final step in the analysis, currently scheduled to occur early in the spring of 2007, will be to combine our gyro spin axis orientation results with data mapping the proper motion of IM Pegasi relative to the unchanging position of a distant quasar. The proper motion of IM Pegasi has been mapped with unprecedented precision using a technique called Very Long Baseline Interferometry (VLBI) by Irwin Shapiro and his team at the Harvard-Smithsonian Center for Astrophysics (CfA), in collaboration with Norbert Bartel at York University in Toronto and French astronomer Jean-Francois Lestrade. At the end of Phase III, playing the role of our own harshest critic, our science team will then perform a careful and thorough final review of the analysis and results, checking and cross-checking each aspect to ensure the soundness of our procedures and the validity of our outcomes. We will then turn the analysis and results over to the SAC, which has been closely monitoring our experimental methods, data analysis procedures, and progress for 11 years, to obtain its independent review. Moreover, we will seek independent reviews from a number of international experts. In addition to analyzing the data, members of our team are now in the process of preparing scientific and engineering papers for publication in late 2006-2007. We have also begun discussions with NASA to plan a formal public announcement of the results of this unprecedented test of General Relativity. We expect to make this announcement of the results in April 2007.
Still April 2007, and counting!

Garth
 Recognitions: Gold Member Science Advisor While we are waiting you may be interested in Francis Everitt's lecture:Testing Einstein in Space: The Gravity Probe B Mission dated 18 May 2006. Garth

Recognitions:
Gold Member
Gravity Probe B Update -- December 22, 2006
 ============== GP-B MISSION NEWS ============== A recent story about GP-B in Nature ========================= The December 21-28 2006 issue of Nature (v. 444, p. 978-979) contains a short news article stating that Nature has learned that "two unanticipated effects are clouding the [GP-B] team's frame-dragging results" and also that "results were expected by last summer but the announcement never came." The two issues referred to in Nature have been regularly reported to NASA and our GP-B Science Advisory Committee (SAC) and publicly via these status updates. They are: 1) The effect of polhode motion of the gyros on readout calibration (see the polhode story in last month's update, http://einstein.stanford.edu/highlig...ode_story.html) and 2) misalignment torques observed and calibrated during the post-science instrument calibration phase in August-September 2005 (see the four weekly updates of September 2005, http://einstein.stanford.edu/highlig...indexmain.html. In August 2005, a three-phase data analysis plan was devised in order to properly handle these and other issues. As first reported in May 2006, our intent--reached in agreement with NASA--has been to make the first science announcement in April 2007. This is still our plan.
If you want to know more about the Polhode motion see Polhode Behavior in GP-B’s Gyros

Roll on April!

Garth

 Quote by Garth The Gravity Probe B satellite has placed four (over redundant) gyroscopes in low polar Earth orbit to primarily test two predictions of General Relativity. The first effect being tested is (for the GP-B polar orbit) a N-S geodetic precession, caused by the amount a gyro 'leans' over into the slope of curved space. The second effect being tested is the E-W frame-dragging, Lense-Thirring, or gravitomagnetic effect, caused by the spinning Earth dragging space-time around with it. Some researchers, such as Kenneth Nordtvedt, have said that the experiment was worth doing when it was first proposed but that now GR has been verified beyond resonable doubt the result of GP-B is a foregone conclusion. I have now discovered several theories competing with General Relativity(GR) that are being tested and falsified by this experiment: my Self Creation Cosmology).(SCC), Moffat's Nonsymmetric Gravitational Theory (NGT), Hai-Long Zhao's mass variance SR theory (MVSR), Stanley Robertson's Newtonian Gravity theory (NG), and Junhao & Xiang's Flat space-time theory (FST). As the results will be published in the not too distant future they could be interesting!! (Note if anybody knows of any other theories with alternative predictions for GP-B please post them as well for comparison.) 1. GPB Geodetic precession GR = 6.6144 arcsec/yr SCC = 4.4096 arcsec/yr NGT = 6.6144 - a small $\sigma$ correction arcsec/yr MVSR = 6.6144 arcsec/yr NG = 1.6536 arcsec/yr FST = 4.4096 arcsec/yr 2. GPB gravitomagnetic frame dragging precession GR = 0.0409 arcsec/yr SCC = 0.0409 arcsec/yr NGT = 0.0409 arcsec/yr MVSR = 0.0102 arcsec/yr NG = 0.0102 arcsec/yr FST = 0.0000 arcsec/yr I cannot vouch for these other theories, they may well be considered 'crackpot' by some, however all these theories have the advantage, together with GR, that they are able to be falsified by the GP-B results. We continue to wait and see! Garth
Thanks Garth for this interesting overview. I printed out your table and will look at it again when the Gravity Probe B results are available .
What is actually the main motivation for inventing alternative theories to GR ?
What are their main "advantages" ?

Recognitions:
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 Quote by notknowing What is actually the main motivation for inventing alternative theories to GR ? What are their main "advantages" ?
First to 'push the envelope', the concept of scientific truth is that it is a process, one never should believe that the 'final truth' has been found but that the present best theories are always open to experimental testing and theoretical questioning.

Viable alternative theories are important to test the standard theory against, partly to justify and motivate such difficult experiments as Gravity Probe B.

As I said in your quote "Some researchers, such as Kenneth Nordtvedt, have said that the experiment was worth doing when it was first proposed but that now GR has been verified beyond reasonable doubt the result of GP-B is a foregone conclusion." The existence of these other theories argues for a more positive attitude to the experiment.

There are always questions to be asked of the standard theory that other approaches seek to answer. The main questions about the standard $\Lambda$CDM model IMHO are its necessity to invoke Inflation, exotic non-baryonic DM and DE, while the Higgs Boson/Inflaton the DM particle(s) and DE have not been discovered in laboratory experiments. The existence of the PA and other anomalies are also intriguing.

Different alternative theories have different advantages, but to be viable contenders they must not only predict accurately the outcomes of all the experiments and observations predicted by the standard theory but also have a greater explanatory power by doing so more simply.

Garth
 Dear all, Just to mention that there is another alternative theory of gravity (mine: gr-qc/0610079) with predictions different from the ones that you have listed. This is a DArk Gravity theory: DG predicts: 1) The same geodetic effect as in GR 2) No frame dragging 3) A small (but hopefully within the GP-B accuracy) angular deviation during the year but with a one year period (related to the the speed of earth about the sun). regards, F H-C


text:
### Sidebar

Reference

analysis:rhythms

This is an old revision of the document!

Welcome! This is the home page for the optional hands-on (heads-on) 2016 edition of the “Rhythmic Brain” course.

### Contents

#### Installing MATLAB

Make sure you are on a Dartmouth network, such as Dartmouth Secure. Not KAF, Starbucks or eduroam. Then go here, click “Site License Downloads” and keep clicking the link at the bottom of the page until you get to the page for “R2014a”: this is the version you want. Follow the instructions in the PDF.

When the installation is done, see if you can run MATLAB. You should be faced with a screen containing a bunch of windows, including one called “Command Prompt” containing ».

If your MATLAB runs, great! Now decide if you'd rather:

1. Familiarize yourself with MATLAB first. Go here.
2. Dive right in, working with Emotiv data. Read on.

#### Obtaining some sample Emotiv data

Click here. Move this file to a location that shows you are careful with how you handle data. Something like \Documents\PSYC50\Data\TestData is a good choice, because it is clearly named and anticipates that you may have other data in the future beyond the file you grabbed just now. Something like \Downloads or \Desktop is bad because it'll get mixed in with lots of other files and maybe get deleted accidentally.

Notice that this file has the extension .edf. File extensions typically tell you about what type of file you are dealing with. In this case, our filetype is “EDF” (meaning it contains data arranged according the rules laid out in the European Data Format specification, which you can be happily ignorant about). All this means for you is that you need a file loader that can accept files of the EDF format, such as the one introduced in the next section.

#### Obtaining the code you need to load the Emotiv data

Pick one: the quick way, or the future-proof way.

The quick way: download the MATLAB loader (from here), and place it in a sensible location like \Documents\PSYC50\Code\Loading. This will work for now. However, if you ever want to access an updated version of this file, or other code that I put up for use in the course, you'll have to do more downloading. The future-proof way, below, addresses this.

The future-proof way: I have added the code you'll need for loading EDF data, as well as for doing some other things, on a “GitHub repository”. Git is a system for “distributed version control”: put simply, a way to collaborate on a set of files that makes it easy to share and receive updates to those files. GitHub is a website that hosts those files in a convenient place so that the Git software can do its job.

Let's make sure you have Git on your computer. If you are a Mac user, you should already have git installed. Open up a Terminal, and type git to verify that your computer recognizes it. If you are on Windows, I recommend you download the GitHub desktop. Then, open a “Git shell” (you should have an icon and start menu item) and type git to check this command is recognized.

In your terminal/shell, navigate to a good place to put your GitHub code.

#### Getting started with MATLAB

Depending on your background and programming experience you might find the following resources helpful:

If you are unsure, take a look at the table of contents of the MATLAB Primer in the link above. If there are things you don't recognize, use the Primer itself, or Chapter 2 of the MATLAB for Neuroscientists book to get up to speed. If you've never used MATLAB, definitely start with the steps in the Primer.

Regardless of your MATLAB abilities, some great ways to keep learning are:

If you have no formal training in computer programming (i.e. you have never taken a “Intro to Computer Science” or “Introductory Programming” type course) a great introduction to the “Zen of programming” is to do the pen-and-paper exercises in this short chapter by Daniel Dennett (“The Secrets of Computer Power Revealed”) before you embark on the MATLAB primer linked to above. Most people find it frustrating at first, but rewarding if they stick with it for a few hours.

#### Resources

The following textbooks provide more in-depth treatment of some of the topics we touch on in the course:

These are selected modules from a graduate course I teach on neural data analysis:

#### Note for Linux users

The lab codebase is set up for machines running 64-bit Windows 7 or Mac OS X. If you want to use Linux or some other OS you will probably need to compile some of the low-level loading functions yourself. Some pointers for this are provided in subsequent modules when loading is introduced.

## Discussion

, 2021/06/05 00:33

, 2021/06/06 06:56

, 2021/06/06 13:01

, 2021/06/06 13:53

, 2021/06/06 20:03


text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Paper:

TR98-076 | 13th November 1998 00:00

#### Attribute Efficient PAC Learning of DNF with Membership Queries under the Uniform Distribution

TR98-076
Authors: Nader Bshouty, Jeffrey J. Jackson, Christino Tamon
Publication: 18th December 1998 14:59
Keywords:

Abstract:

We study attribute efficient learning in the PAC learning model with
membership queries. First, we give an {\it attribute efficient}
PAC-learning algorithm for DNF with membership queries under the
uniform distribution. Previous algorithms for DNF have sample size
polynomial in the number of attributes $n$. Our algorithm is the
first attribute efficient learning for DNF, i.e., that runs in polynomial
time and uses sample of size polynomial in $\log n$.
We also develop lower bound techniques for
PAC learning with membership queries under a fixed distribution
and
show that the sample size of our algorithm
for learning DNF under the uniform distribution
is almost optimal in terms of $n$.
Finally, we present a learning algorithm for DNF that is attribute
efficient in its use of random bits.

ISSN 1433-8092 | Imprint


text:
# Transfer of a msg to gazebo from ROS to control the simulated turtlebot3 [closed]

Hey,

I am new to ROS and Gazebo and just try to understand how I can control and command the turtlebot3. I followed the instructions of the common tutorial website and everything went nice. I also built up new worlds etc. Also, when I roslaunch the gazebo world and the corresponding dqn file (for Reinforcement Learning and controlling), a cmd_vel topic is used which is published by the dqn node and subscribed by Gazebo.

Node Overview: C:\fakepath\Bildschirmfoto vom 2019-06-12 13-28-26.png

Here I get two questions. The topic only contains data for linear and angular position

self.pub_cmd_vel = rospy.Publisher('cmd_vel', Twist, queue_size=5)

Twist contains:

marcel@marcel-macbookpro:~\$ rosmsg show geometry_msgs/Twist
geometry_msgs/Vector3 linear
float64 x
float64 y
float64 z
geometry_msgs/Vector3 angular
float64 x
float64 y
float64 z

Anyway, this topic is filled with velocity informations and subscribed by gazebo to control the robot:

def cmd_vel(self, action):
max_angular_vel = 1.5
ang_vel = ((self.action_size - 1)/2 - action) * max_angular_vel * 0.5

vel_cmd = Twist()
vel_cmd.linear.x = 0.15
vel_cmd.angular.z = ang_vel
self.pub_cmd_vel.publish(vel_cmd)

My two big questions are now, how can Twist be filled with velocity information and how is gazebo using this information to control the turtlebot. I mean, there is no node between the dqn_node and gazebo that translates those linear and angular information in roation for the wheels i.e.. Is Gazebo doing this by itself?


text:
# FLT_ROUNDS Macro

Represents the rounding mode for floating-point operations.

Include

<float.h>

Definition

The value 1 for all compilers, implying that rounding is to the nearest representable value.

Remarks

The meaning of values for this macro as defined by the C standard are as follows.

 -1 indeterminable 0 toward zero 1 to nearest representable value 2 toward positive infinity 3 toward negative infinity


text:
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# If n is an integer, then n divisible by how many positive

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If n is an integer, then n divisible by how many positive integers?

(1) n is the product of two different prime numbers.
(2) n and 2^3 are each divisible by the same number of positive integers.
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Baten80 wrote:
If n is an integer, then n divisible by how many positive integers?
(1) n is the product of two different prime numbers.
(2) n and 2^3 are each divisible by the same number of positive integers.

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.
For more on number properties check: math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION:

If n is an integer, then n divisible by how many positive integers?

(1) n is the product of two different prime numbers --> n=ab, where a and b are primes, so # of factors is (1+1)(1+1)=4. Sufficient.

(2) n and 2^3 are each divisible by the same number of positive integers --> 2^3 has 4 different positive factors (1, 2, 4, and 8) so n has also 4. Sufficient.

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if n is an integer, then n is divisible by how many positive integers?

1) n is the product of two different integers
2) n and 2^3 are each divisible by the same number of positive integers.

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30 Jan 2013, 21:33
It took me just 15 seconds to solve this..
N is a product of 2 different prime nos.......then 1,n and dose two prime nos. are divisible by n ...hence 4 nos.
agen, 2^3 = 8, has 4 nos. from which it can be divided...agen n is divisible by 4 nos.
Hence, D
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Re: If n is an integer, then n divisible by how many positive [#permalink]

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31 Jan 2013, 12:10
@ Fozzzy
Statement 1 - n is the product of two different integers . They may be 2*3 or 3*7 or any two integers. Since they yield different products. We cannot determine the # of factors for n. Hence Statement 1 - Insufficient.
Statement 2 - n and 2^3 are each divisible by the same number of positive integers. 2^3 = 8. Having 4 factors (1,2,4,8) . Since the statement says n and 8 are divisible by the same num of integers. n=4. Hence Statement 2 - Sufficient

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if n is an integer then n is divisble by how many positive i [#permalink]

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17 Jul 2013, 00:45
if n is an integer then n is divisble by how many positive integers?

1) n is the product of two different prime numbers.
2) n and $$2^ {-3}$$ are each divisible by the same number of positive integers

[Reveal] Spoiler:
Don't have OA. Please provide explanations! Thanks

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Re: if n is an integer then n is divisble by how many positive i [#permalink]

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17 Jul 2013, 00:50
fozzzy wrote:
if n is an integer then n is divisble by how many positive integers?

1) n is the product of two different prime numbers.
2) n and $$2^ {-3}$$ are each divisible by the same number of positive integers

[Reveal] Spoiler:
Don't have OA. Please provide explanations! Thanks

Merging similar topics. Please refer to the solution above and ask if anything remains unclear.

P.S. You've posted this question before: if-n-is-an-integer-then-n-divisible-by-how-many-positive-109670.html#p1175381
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Re: If n is an integer, then n divisible by how many positive [#permalink]

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26 Aug 2014, 19:29
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Re: If n is an integer, then n divisible by how many positive [#permalink]

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14 Sep 2016, 04:50
Hello from the GMAT Club BumpBot!

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Re: If n is an integer, then n divisible by how many positive   [#permalink] 14 Sep 2016, 04:50
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