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Research

# On best proximity point of ψ-Geraghty contractions

Erdal Karapınar

Author Affiliations

Department of Mathematics, Atilim University, Incek, Ankara, 06836, Turkey

Fixed Point Theory and Applications 2013, 2013:200  doi:10.1186/1687-1812-2013-200

 Received: 12 March 2013 Accepted: 17 May 2013 Published: 24 July 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Very recently, Caballero, Harjani and Sadarangani (Fixed Point Theory Appl. 2012:231, 2012) observed some best proximity point results for Geraghty contractions by using the P-property. In this paper, we introduce the notion of ψ-Geraghty contractions and show the existence and uniqueness of the best proximity point of such contractions in the setting of a metric space. We state examples to illustrate our result.

MSC: 41A65, 90C30, 47H10.

##### Keywords:
best proximity point; non-self mapping; partial order; metric space; fixed point

### 1 Introduction and preliminaries

In nonlinear functional analysis, fixed point theory and best proximity point theory play a crucial role in the establishment of the existence of certain differential and integral equations. As a consequence, fixed point theory is very useful for various quantitative sciences that involve such equations. To list a few, certain branches of computer sciences, engineering and economics are well-known examples in which fixed point theory is used.

The most remarkable paper in this field was reported by Banach [1] in 1922. In this paper, Banach proved that every contraction in a complete metric space has a unique fixed point. Following this outstanding paper, many authors have extended, generalized and improved this remarkable fixed point theorem of Banach by changing either the conditions of the mappings or the construction of the space. In particular, one of the notable generalizations of Banach fixed point theorem was reported by Geraghty [2].

Theorem 1.1 (Geraghty [2])

Let ( X , d ) be a complete metric space and T : X X be an operator. Suppose that there exists β : ( 0 , ) [ 0 , 1 ) satisfying the condition

β ( t n ) 1 implies t n 0 .

IfTsatisfies the following inequality:

d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) for any x , y X , (1)

thenThas a unique fixed point.

It is very natural that some mappings, especially non-self-mappings defined on a complete metric space ( X , d ) , do not necessarily possess a fixed point, that is, d ( x , T x ) > 0 for all x X . In such situations, it is reasonable to search for the existence (and uniqueness) of a point x X such that d ( x , T x ) is an approximation of an x X such that d ( x , T x ) = 0 . In other words, one speculates to determine an approximate solution x that is optimal in the sense that the distance between x and T x is minimum. Here, the point x X is called a best proximity point.

This research subject has attracted attention of a number of authors; for example, see [2-23]. In this paper we generalize and improve certain results of Caballero et al. in [6]. Notice also that in the best proximity point theory, we usually consider a non-self-mapping. In fixed point theory, almost all maps are self-mappings. For the sake of completeness, we recall some basic definitions and fundamental results on the best proximity theory.

Let ( X , d ) be a metric space and A and B be nonempty subsets of a metric space ( X , d ) . A mapping T : A B is called a k-contraction if there exists k ( 0 , 1 ) such that d ( T x , T y ) k d ( x , y ) for any x , y A . It is clear that a k-contraction coincides with the celebrated Banach fixed point theorem if one takes A = B , where A is a complete subset of X.

Let A and B be two nonempty subsets of a metric space ( X , d ) . We denote by A 0 and B 0 the following sets:

A 0 = { x A : d ( x , y ) = d ( A , B )  for some  y B } , B 0 = { y B : d ( x , y ) = d ( A , B )  for some  x A } , (2)

where d ( A , B ) = inf { d ( x , y ) : x A , y B } .

In [13], the authors presented sufficient conditions which determine when the sets A 0 and B 0 are nonempty. In [19], the author introduced the following definition.

Definition 1.1 Let ( A , B ) be a pair of nonempty subsets of a metric space ( X , d ) with A 0 . Then the pair ( A , B ) is said to have the P-property if and only if for any x 1 , x 2 A 0 and y 1 , y 2 B 0 ,

d ( x 1 , y 1 ) = d ( A , B ) and d ( x 2 , y 2 ) = d ( A , B ) d ( x 1 , x 2 ) = d ( y 1 , y 2 ) . (3)

It can be easily seen that for any nonempty subset A of ( X , d ) , the pair ( A , A ) has the P-property. In [19], the author proved that any pair ( A , B ) of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property. Now, we introduce the class F of those functions β : ( 0 , ) [ 0 , 1 ) satisfying the following condition:

β ( t n ) 1 implies t n 0 . (4)

Definition 1.2 (See [6])

Let A, B be two nonempty subsets of a metric space ( X , d ) . A mapping T : A B is said to be a Geraghty-contraction if there exists β F such that

d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) for any  x , y A . (5)

Theorem 1.2 (See [6])

Let ( A , B ) be a pair of nonempty closed subsets of a complete metric space ( X , d ) such that A 0 is nonempty. Let T : A B be a continuous Geraghty-contraction satisfying T ( A 0 ) B 0 . Suppose that the pair ( A , B ) has theP-property. Then there exists a unique x inAsuch that d ( x , T x ) = d ( A , B ) .

In the following section, we improve the theorem above by using a distance function ψ in Definition 1.2. In particular, we introduce Definition 2.1 and broaden the scope of Theorem 1.2 to ψ-Geraghty-contractions.

### 2 Main results

We start this section with the definition of the following auxiliary function. Let Ψ denote the class of functions ψ : [ 0 , ) [ 0 , ) which satisfy the following conditions:

(a) ψ is nondecreasing;

(b) ψ is subadditive, that is, ψ ( s + t ) ψ ( s ) + ψ ( t ) ;

(c) ψ is continuous;

(d) ψ ( t ) = 0 t = 0 .

We introduce the following contraction.

Definition 2.1 Let A, B be two nonempty subsets of a metric space ( X , d ) . A mapping T : A B is said to be a ψ-Geraghty contraction if there exists β F such that

ψ ( d ( T x , T y ) ) β ( ψ ( d ( x , y ) ) ) ψ ( d ( x , y ) ) for any  x , y A . (6)

Remark 2.1 Notice that since β : ( 0 , ) [ 0 , 1 ) , we have

ψ ( d ( T x , T y ) ) β ( ψ ( d ( x , y ) ) ) ψ ( d ( x , y ) ) < ψ ( d ( x , y ) ) for any  x , y A  with  x y . (7)

We are now ready to state and prove our main theorem.

Theorem 2.1Let ( A , B ) be a pair of nonempty closed subsets of a complete metric space ( X , d ) such that A 0 is nonempty. Let T : A B be aψ-Geraghty contraction satisfying T ( A 0 ) B 0 . Suppose that the pair ( A , B ) has theP-property. Then there exists a unique x inAsuch that d ( x , T x ) = d ( A , B ) .

Proof Regarding that A 0 is nonempty, we take x 0 A 0 . Since T x 0 T ( A 0 ) B 0 , we can find x 1 A 0 such that d ( x 1 , T x 0 ) = d ( A , B ) . Analogously, regarding the assumption T x 1 T ( A 0 ) B 0 , we determine x 2 A 0 such that d ( x 2 , T x 1 ) = d ( A , B ) . Recursively, we obtain a sequence { x n } in A 0 satisfying

d ( x n + 1 , T x n ) = d ( A , B ) for any  n N . (8)

Since ( A , B ) has the P-property, we derive that

d ( x n , x n + 1 ) = d ( T x n 1 , T x n ) for any  n N . (9)

If there exists n 0 N such that d ( x n 0 , x n 0 + 1 ) = 0 , then the proof is completed. Indeed,

0 = d ( x n 0 , x n 0 + 1 ) = d ( T x n 0 1 , T x n 0 ) , (10)

and consequently, T x n 0 1 = T x n 0 . On the other hand, due to (8) we have

d ( x n 0 , T x n 0 1 ) = d ( A , B ) .

Therefore, we conclude that

d ( A , B ) = d ( x n 0 , T x n 0 1 ) = d ( x n 0 , T x n 0 ) . (11)

For the rest of the proof, we suppose that d ( x n , x n + 1 ) > 0 for any n N . Since T is a ψ-Geraghty contraction, for any ℕ, we have that

ψ ( d ( x n , x n + 1 ) ) = ψ ( d ( T x n 1 , T x n ) ) β ( ψ ( d ( x n 1 , x n ) ) ) ψ ( d ( x n 1 , x n ) ) < ψ ( d ( x n 1 , x n ) ) . (12)

Consequently, { ψ ( d ( x n , x n + 1 ) ) } is a nonincreasing sequence and bounded below, and so lim n ψ ( d ( x n , x n + 1 ) ) = L exists. Let lim n ψ ( d ( x n , x n + 1 ) ) = L 0 . Assume that L > 0 . Then, from (6), we have

ψ ( d ( x n + 1 , x n + 2 ) ) ψ ( d ( x n , x n + 1 ) ) β ( ψ ( d ( x n , x n + 1 ) ) ) 1

for each n 1 , which implies that

lim n β ( ψ ( d ( x n , x n + 1 ) ) ) = 1 .

On the other hand, since β F , we conclude L = 0 , that is,

lim n ψ ( d ( x n , x n + 1 ) ) = 0 . (13)

Notice that since d ( x n + 1 , T x n ) = d ( A , B ) for any n N , for fixed p , q N , we have d ( x p , T x p 1 ) = d ( x q , T x q 1 ) = d ( A , B ) , and since ( A , B ) satisfies the P-property, d ( x p , x q ) = d ( T x p 1 , T x q 1 ) . In what follows, we prove that { x n } is a Cauchy sequence. On the contrary, assume that we have

ε = lim sup m , n d ( x n , x m ) > 0 . (14)

By using the triangular inequality,

d ( x n , x m ) d ( x n , x n + 1 ) + d ( x n + 1 , x m + 1 ) + d ( x m + 1 , x m ) . (15)

By (12) and since d ( x n + 1 , x m + 1 ) = d ( T x n , T x m ) , by the comment mentioned above, regarding the discussion on the P-property above together with (12), (15) and the property of the function ψ, we derive that

ψ ( d ( x n , x m ) ) ψ ( d ( x n , x n + 1 ) + d ( T x n , T x m ) + d ( x m + 1 , x m ) ) ψ ( d ( x n , x n + 1 ) ) + ψ ( d ( T x n , T x m ) ) + ψ ( d ( x m + 1 , x m ) ) ψ ( d ( x n , x n + 1 ) ) + β ( ψ ( d ( x n , x m ) ) ) ψ ( d ( x n , x m ) ) + ψ ( d ( x m + 1 , x m ) ) . (16)

By a simple manipulation, (16) yields that

ψ ( d ( x n , x m ) ) ( 1 β ( ψ ( d ( x n , x m ) ) ) ) 1 [ ψ ( d ( x n , x n + 1 ) ) + ψ ( d ( x m + 1 , x m ) ) ] . (17)

By taking the properties of the function ψ into account, together with (13) and lim sup m , n d ( x n , x m ) = ε > 0 and lim n d ( x n , x n + 1 ) = 0 , the last inequality yields

lim sup m , n ( 1 β ( ψ ( d ( x n , x m ) ) ) ) 1 = . (18)

Therefore lim sup m , n β ( ψ ( d ( x n , x m ) ) ) = 1 . By taking the fact β F into account, we get

lim sup m , n ψ ( d ( x n , x m ) ) = 0 .

Regarding the properties of the function ψ, the limit above contradicts the assumption (14). Therefore, { x n } is a Cauchy sequence.

Since ( x n ) A and A is a closed subset of the complete metric space ( X , d ) , we can find x A such that x n x .

We claim that T x n T x . Suppose, on the contrary, that T x n T x . This means that we can find ε > 0 such that for each n N , there exists p n with

ε < d ( T x p n , T x ) . (19)

Due to the properties of ψ, we get

0 < ψ ( ε ) ψ ( d ( T x p n , T x ) ) . (20)

Using the fact that T is a ψ-Geraghty contraction, we have

ψ ( ε ) ψ ( d ( T x p n , T x ) ) β ( ψ ( d ( x p n , x ) ) ) ψ ( d ( x p n , x ) ) (21)

for any n N . Since x p n x and ψ ( d ( x p n , x p n + 1 ) ) 0 , we can find n 0 such that for n n 0

ψ ( d ( x p n , x ) ) < ψ ( ε ) and ψ ( d ( x p n , x p n + 1 ) ) < ψ ( ε ) . (22)

Consequently, for n n 0 we have

0 < ψ ( ε ) ψ ( d ( T x p n , T x ) ) β ( ψ ( d ( x p n , x ) ) ) ψ ( d ( x p n , x ) ) ψ ( d ( x p n , x ) ) < ψ ( ε ) , (23)

a contradiction. Therefore, T x n T x .

Regarding the fact that the sequence { d ( x n + 1 , T x n ) } is a constant sequence with value d ( A , B ) , we derive

d ( x , T x ) = d ( A , B ) (24)

which is equivalent to saying that x is the best proximity point of T. This completes the proof of the existence of a best proximity point.

We shall show the uniqueness of the best proximity point of T. Suppose that x and y are two distinct best proximity points of T, that is, x y . This implies that

d ( x , T x ) = d ( A , B ) = d ( y , T y ) . (25)

Using the P-property, we have

d ( x , x 2 ) = d ( T x , T y ) . (26)

Using the fact that T is a ψ-Geraghty contraction, we have

ψ ( d ( x , y ) ) = ψ ( d ( T x , T y ) ) β ( ψ ( d ( x , y ) ) ) ψ ( d ( x , y ) ) < ψ ( d ( x , y ) ) , (27)

a contradiction. This completes the proof. □

Notice that the pair ( A , A ) satisfies the P-property for any nonempty subset A of X. Consequently, we have the following corollary.

Corollary 2.1Let ( X , d ) be a complete metric space andAbe a nonempty closed subset of X. Let T : A A be aψ-Geraghty-contraction. ThenThas a unique fixed point.

Proof Apply Theorem 2.1 with A = B . □

If we take ψ ( t ) = t we obtain Theorem 1.2 as a corollary of Theorem 2.1.

Corollary 2.2Let ( X , d ) be a complete metric space andAbe a nonempty closed subset of X. Let T : A A be a Geraghty-contraction. ThenThas a unique fixed point.

Proof Apply Theorem 2.1 with A = B and ψ ( t ) = t . □

In order to illustrate our results, we present the following example.

Example 2.1 Suppose that X = R 2 with the metric

d ( ( x , y ) , ( x , y ) ) = max { | x x | , | y y | }

and consider the closed subsets

A = { ( 0 , x ) : 0 x < } , B = { ( 1 , x ) : 0 x < }

and ψ ( t ) = t 2 and β ( t ) = ln ( 1 + t ) t .

Set T : A B to be the mapping defined by

T ( ( 0 , x ) ) = ( 1 , ln ( 1 + x ) ) .

Since d ( A , B ) = 1 , the pair ( A , B ) has the P-property.

Notice that A 0 = A and B 0 = B and T ( A 0 ) B 0 .

Without loss of generality, we assume that x > x . Moreover,

d ( T ( 0 , x ) , T ( 0 , x ) ) = | ln ( 1 + x ) ln ( 1 + x ) | = | ln ( 1 + x 1 + x ) |

and

d ( ( 0 , x ) , ( 0 , x ) ) = | x x | .

We have

ψ ( d ( T ( 0 , x ) , T ( 0 , x ) ) ) = 1 2 | ln ( 1 + x 1 + x ) | = 1 2 | ln ( 1 + x + x x 1 + x ) | = 1 2 | ln ( 1 + x x 1 + x ) | 1 2 ln ( 1 + | x x | ) ( since  ln ( 1 + t )  is strictly increasing ) ln ( 1 + 1 2 | x x | ) ln ( 1 + 1 2 | x x | ) 1 2 | x x | 1 2 | x x | = β ( ψ ( d ( ( 0 , x ) , ( 0 , x ) ) ) ) ψ ( d ( ( 0 , x ) , ( 0 , x ) ) ) ,

where β : [ 0 , ) [ 0 , 1 ) is defined as β ( t ) = ln ( 1 + t ) t .

Therefore, T is a ψ-Geraghty-contraction. Notice that the pair ( A , B ) satisfies the P-property. Indeed, if

d ( ( 0 , x ) , ( 1 , x ) ) = max { 1 , | x x | } = d ( A , B ) = 1 , d ( ( 0 , y ) , ( 1 , y ) ) = max { 1 , | y y | } = d ( A , B ) = 1

then x = x and y = y and hence

| x y | = d ( ( 0 , x ) , ( 0 , y ) ) = d ( ( 1 , x ) , ( 1 , y ) ) = | x y | .

Therefore, since the assumptions of Theorem 2.1 are satisfied, by Theorem 2.1 there exists a unique ( 0 , x ) A such that

d ( ( 0 , x ) , T ( 0 , x ) ) = 1 = d ( A , B ) .

More precisely, the point ( 0 , 0 ) A is the best proximity point of T.

Example 2.2 Let X = R and d ( x , y ) = | x y | be a metric on X. Suppose A = [ 0 , 1 ] and B = [ 15 / 8 , 2 ] are two closed subsets of ℝ. Define T : A B by T x = 1 8 x + 2 . Define β : ( 0 , ) [ 0 , 1 ) by β ( t ) = 1 1 + t and ψ : [ 0 , ) [ 0 , ) by ψ ( t ) = 1 2 t . Clearly, d ( A , B ) = 7 / 8 . Now we have

A 0 = { x A : d ( x , y ) = d ( A , B ) = 7 / 8  for some  y B } = { 1 } , B 0 = { y B : d ( x , y ) = d ( A , B ) = 7 / 8  for some  x A } = { 15 / 8 } .

Also, T ( A 0 ) = B 0 . Further, clearly, the pair ( A , B ) has the P-property. Let x , y A . Note that, if x = y , then condition (6) holds. Hence, we assume that x y . We shall show that (6) holds. Suppose, on the contrary, there exist x 0 , y 0 A such that

ψ ( d ( T x 0 , T y 0 ) ) > β ( ψ ( d ( x 0 , y 0 ) ) ) ψ ( d ( x 0 , y 0 ) ) ,

and so

1 2 × 1 8 | x 0 y 0 | > 1 2 | x 0 y 0 | 1 + 1 2 | x 0 y 0 | ,

which yields that 14 < | x 0 y 0 | , a contradiction. Therefore condition (6) holds for all x , y A . Hence, the conditions of Theorem 2.1 hold and T has a unique best proximity point. Here, x = 1 is the best proximity point of T.

### Competing interests

The author declares that he has no competing interests.

### Acknowledgements

The author expresses his gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.

### References

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2. Geraghty, M: On contractive mappings. Proc. Am. Math. Soc.. 40, 604–608 (1973). Publisher Full Text

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4. Anuradha, J, Veeramani, P: Proximal pointwise contraction. Topol. Appl.. 156, 2942–2948 (2009). Publisher Full Text

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6. Caballero, J, Harjani, J, Sadarangani, K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl.. 2012, (2012) Article ID 231

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9. De la Sen, M: Fixed point and best proximity theorems under two classes of integral-type contractive conditions in uniform metric spaces. Fixed Point Theory Appl.. 2010, (2010) Article ID 510974

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# Toakamak Terrain Problem

This topic is 4449 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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Hi there, I've gone through the codesampler tutorial, to get a terrain mesh using 3ds up and running. I've checked that the vertices and indices stored in the neTriangleMesh triMesh; structure are correct, by rendering them as a wireframe overlay, but all my rigid bodies don't seem to collide at all. http://img16.imageshack.us/img16/5955/screen045nt.jpg and here's the f() for setting up the mesh
int		CPhysicsManager::addTriangleMeshBody( t3DObject* pObject ){

if(nCounter_triangleMeshBody<MAX_TRIANGLEMESHBODIES){

nCounter_triangleMeshBody++;

pTMB[nCounter_triangleMeshBody]					=	new	tTriangleMeshBody;

pTMB[nCounter_triangleMeshBody]->numOfTriangles	=	pObject->numOfFaces;
pTMB[nCounter_triangleMeshBody]->numOfVerts		=	pObject->numOfVerts;

int		numOfTriangles	=	pTMB[nCounter_triangleMeshBody]->numOfTriangles;
int		numOfVerts		=	pTMB[nCounter_triangleMeshBody]->numOfVerts;

pTMB[nCounter_triangleMeshBody]->triVertices	=	new neV3 [numOfVerts];
pTMB[nCounter_triangleMeshBody]->triData		=	new	neTriangle [numOfTriangles];

//	fill in the array
for(int i=0;i<numOfVerts;++i){
pTMB[nCounter_triangleMeshBody]->triVertices.Set( pObject->pVerts.x, pObject->pVerts.y, pObject->pVerts.z);
}

for(int j = 0; j < numOfTriangles; j ++)
{
pTMB[nCounter_triangleMeshBody]->triData[j].indices[0]	=	pObject->pFaces[j].vertIndex[0];
pTMB[nCounter_triangleMeshBody]->triData[j].indices[1]	=	pObject->pFaces[j].vertIndex[1];
pTMB[nCounter_triangleMeshBody]->triData[j].indices[2]	=	pObject->pFaces[j].vertIndex[2];
pTMB[nCounter_triangleMeshBody]->triData[j].flag		=	neTriangle::NE_TRI_TRIANGLE;
pTMB[nCounter_triangleMeshBody]->triData[j].materialID	=	0;
}

pTMB[nCounter_triangleMeshBody]->triMesh.vertices		= pTMB[nCounter_triangleMeshBody]->triVertices;
pTMB[nCounter_triangleMeshBody]->triMesh.triangleCount	= numOfTriangles;
pTMB[nCounter_triangleMeshBody]->triMesh.triangles		= pTMB[nCounter_triangleMeshBody]->triData;

return nCounter_triangleMeshBody;
}
else{
return -1;
}
}

I've had problems with y and x being reversed in the past, but this doesn't seem to be the case i don't think. Anyway, i'm stuck!!!!!

##### Share on other sites
damn, new to this....

[img=http://img16.imageshack.us/img16/5955/screen045nt.jpg][/url]

format: plain


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# FastAI Multi-label image classification

The FastAI library allows us to build models using only a few lines of code. Furthermore, it implements some of the newest state-of-the-art technics taken from research papers that allow you to get state-of-the-art results on almost any type of problem.

This article is my second article covering how to use the FastAI library. In this article, we will learn how to do multi-label image classification on the Planet Amazon satellite dataset and what differences there are between single- and multi-label classification.

If you don’t know the FastAI library and organization yet, I would highly encourage you to read my last article  which gives you some basic information about the FastAi research lab and shows you how to use the FastAi vision module to build an animal   detector.

If you prefer a visual tutorial you can check out my FastAi videos.

### Getting data

The first thing we need to do is download the Planet Amazon satellite dataset from Kaggle.

After this is done we can read in the provided training dataframe using pandas.

from fastai.vision import * # import the vision module

path = Path('<path to the dataset>') # specify the path to your dataset

print(df.head())

We have two columns. The image_name column which contains the names of the images — without there suffix — and the tags column which contains the labels for each image separated by spaces.

This quick look at the dataframe provides us with enough information to load in the data but before we will to that we will specify which transformations we want to apply to the images.

Because these are satellite images it doesn’t matter what direction they have and so we can enable both do_flip — enabled by default — and flip_vert. We will also play around with the lighting by tuning the max_lighting parameter, change the zoom by adjusting the max_zoom parameter and disable warp by setting max_warp to 0.

tfms = get_transforms(flip_vert=True, max_lighting=0.1, max_zoom=1.05, max_warp=0.)

Now that we have our transforms we can load in the data using FastAIs data block api, which provides a lot of different classes and methods for loading in datasets.

np.random.seed(42) # set random seed so we always get the same validation set
src = (ImageItemList.from_csv(path, 'train_v2.csv', folder='train-jpg', suffix='.jpg')
.random_split_by_pct(0.2)
# split data into training and validation set (20% validation)
.label_from_df(label_delim=' ')
# label data using the tags column (second column is default)
)

Now that we have an ImageItemList we just need to apply our transforms, convert our data in a databunch object and normalize it. We didn’t do the whole process in a single step because we will first of train on a smaller image size — 128x128px — and then switch to the original size, which is  256x256px.

This is a little trick which was introduced by Jeremy Howard in the Practical Deep Learning for Coders course. It doesn’t only help train the network faster but it also helps to prevent overfitting. For more information, you should definitely check out the course, which is completely free.

Now that we have our data we can visualize a random batch of images using the show_batch method.

data.show_batch(rows=3, figsize=(12, 9))

### Training model

The model creation process is just like for a “normal” single-label image   classification problem. Therefore we can just us FastAIs create_cnn method, which can be used to create a convolutional neural network.

The method needs two arguments, the data, and the architecture, but also supports many other parameters that can be used to customize the model for a given problem.

One of these other parameters is called metrics. Metrics basically are all the “scores” printed out whilst training. The Planet Amazon competition uses the F2-Score to determine the ranking so we will specify the F2-Score and accuracy as our metrics.

But because we are working with multi-label data we can’t just use the normal accuracy and F2-score methods, which are for single label problems but we rather need to set a threshold which determines if the image contains a class.

Normally we would need to find the right value for the threshold but for this problem Jeremy Howard already did that so we will just use the same threshold as he used.

# create metrics
acc_02 = partial(accuracy_thresh, thresh=0.2)
f_score = partial(fbeta, thresh=0.2)
# create cnn with the resnet50 architecture
learn = create_cnn(data, models.resnet50, metrics=[acc_02, f_score])

The partial method just creates another method that calls the passed one with a fixed parameter (threshold=0.2 in our case).

Now that we have our model we can just follow the standard FastAI transfer learning process which we looked into in more depth in my last article so if you want more information about this process you can either check out the article or the first few lessons of the course(specifically  lesson 1 and 2).

The only difference is that this time we will perform this process twice. Once for the 128x128px images and then with the 256x256px images.

learn.lr_find() # find learning rate
learn.recorder.plot() # plot learning rate
lr = 0.01 # chosen learning rate
learn.fit_one_cycle(4, lr) # train model for 4 epochs

learn.save('planet-amazon-stage-1') # save model

The  above code only trained the top layers — which were specifically added for our problem — and it didn’t train any of the convolutional layers. In order to train these, we need to call unfreeze.

learn.unfreeze() # unfreeze all layers

learn.lr_find() # find learning rate
learn.recorder.plot() # plot learning rate

learn.fit_one_cycle(5, slice(1e-5, lr/5)) # fit model with differential learning rates

learn.save('planet-amazon-stage-2') # save model

Now that we have a pretty good model it’s time to change the image size from 128 to 256px. This can be done by creating a new databunch object and placing it inside learn.data.

# switch resolution
data = (src.transform(tfms, size=256)
.databunch(bs=16).normalize(imagenet_stats))
learn.data = data

Now the exact same training process can be executed once again.

learn.freeze() # freeze bottom layers again

learn.lr_find()
learn.recorder.plot()

lr=1e-2/2
learn.fit_one_cycle(5, lr)

learn.save('planet-amazon-stage-3')

learn.unfreeze()
learn.fit_one_cycle(5, slice(1e-5, lr/5))

learn.save('planet-amazon-stage-4')

Lastly, we can export our model using the .export method which saves everything needed in a pkl file.

learn.export()

### Submitting to Kaggle

The testing data for this competition is split into two separate folders called test-jpg and test-jpg-additional. We will load the data from each folder using the ImageItemList class, get our predictions calling the get_preds method of our model and then we need to loop through the predictions and check if the prediction is bigger than our threshold. Lastly, we will save the results in a dataframe.

dataframes = []

test = ImageItemList.from_folder(path/directory)
preds, _ = learn.get_preds(ds_type=DatasetType.Test)
thresh = 0.2
labelled_preds = [' '.join([learn.data.classes[i] for i,p in enumerate(pred) if p > thresh]) for pred in preds]
fnames = [f.name[:-4] for f in learn.data.test_ds.items]
df = pd.DataFrame({'image_name':fnames, 'tags':labelled_preds}, columns=['image_name', 'tags'])
dataframes.append(df)

df = pd.concat(dataframes)
df.to_csv(path/'submission.csv', index=False)
print(df.head())

The created csv-file can now be submitted to Kaggle which can be done by either going to the competition page and submitting it there directly.

Or by using the cli.

kaggle competitions submit planet-understanding-the-amazon-from-space -f {path/'submission.csv'} -m "My submission"

### Conclusion

The FastAi library is a high-level library build on PyTorch which allows for easy prototyping and gives you access to a lot of state-of-the-art methods/techniques.

The vision module allows us to build  Convolutional Neural Networks for different problems without changing the underlying code-pipeline.

If you liked this article consider subscribing to my Youtube Channel and following me on social media.


text:
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# sigma copy and paste

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This question is from a previous midterm I'm trying to solve. Sigma summation calculator. Alpha Sigma Phi Letters Copy And Paste. Press and hold down where you want to copy the text : your text will then be displayed with two blue quadrants below. Complete Collection of over 260+ Math Symbols like square root symbol,under root symbol, sigma symbol,pi symbol with one click Math Symbols Copy and Paste option. Underlined text, printed URLs, and the table of contents become live links on screen; and you can use your browser’s commands to change the size of the text or search for key words. Continued valuable contracts and relationships with the largest AM players in the game Included in standards talks with the organizations who write them. Degree Symbol - Degree Sign Copy and Paste (Mac and Windows) The degree symbol (°) consist of a small raised circle and used is for geographic coordinates, temperature measurement, Geometry, mathematics, physics, etc. The upper case letter sigma , the 18th letter of the modern Greek alphabet. You can also use your Keyboard Viewer as an alternative to my list. σ U+03C3 copy and paste This code point first appeared in version 1.1 of the Unicode® Standard and belongs to the " Greek and Coptic " block which goes from 0x370 to 0x3FF. First, open a random text, whether it’s in chat history, on the web, or elsewhere. According to a different theory, its original name may have been "San" , while "Sigma" was a Greek innovation that simply meant "hissing", based on a nominalization of a verb σίζω. Value is $\sigma$, R^{2} = 0.6 where Value is is ordinary font, $\sigma$ is a Greek lowercase sigma letter and R^{2} = 0.6 appears as an R with a superscript 2 followed by equal sign (=) followed by 0.6.How can this be used in ggplot factors and in arguments to things like xlab,ylab of R? The Greek alphabet has 24 letters to write the Greek language. Shortcuts exist for calculating sums of specific sequences. Here we show you how to copy and paste with your Polaroid Sigma 5. 3. You can also right-click on a block and select copy from the context menu. Sometimes having links can make the formatting a little bit off. SHARE . On the other hand, the uppercase letter sigma (Î£) is used as a symbol for the covariance matrix in statistics, the busy beaver function in computer science, a class of baryons, and self-energy in physics. I think I tried that and it didn't work for me. We use this information in order to improve and customize your browsing experience and for analytics and metrics about our visitors. The number 200,000 in Greek numerals. Replace \Sigma by \sum. Sigma does not permit investigators to delete/remove a reference once they have been listed on the proposal and the letters of support request emails have been sent upon completeing the confirmation step. You can also right-click using your mouse and select Copy and Paste. On the keyboard you have to press the key combination “Alt Gr” + “q”. To find out more about the cookies we use, see our Privacy Policy. Sigma symbol in Greek alphabet with unicode and html entity codes. The capital letter sigma (Î£) means to sum up in mathematics. Copy and paste their news feed from the past 4 years. GREEK CAPITAL LETTER SIGMA [unassigned: U+03A2] ← Ρ [U+03A1] Greek and Coptic: Τ → [U+03A4] Contents. See also see: Appendix:Greek alphabet These symbols can be copy and pasted anywhere you want. Omega Symbol in Greek Alphabet (Ohm Symbol) Omega (uppercase Ω, lowercase: ω) is the 24th and last letter of the Greek alphabet. Q5. Email; Twitter; Facebook Share via Facebook » More... Share This Page. Wix Stores, Wix Bookings, Wix Restaurants, etc). You can put ∞ in Facebook, Instagram or Youtube. Character Palette allows you to view and use all characters and symbols, including sigma, available in all fonts (some examples of fonts are "Arial", "Times New Roman", "Webdings") installed on your computer. Platinum paste, screen printable; Synonym: Dyesol(R) PT1 platinum paste, Pt paste, conductive paste; find Sigma-Aldrich-791512 MSDS, related peer-reviewed papers, technical documents, similar products & more at Sigma-Aldrich. Facebook; Prev Article. Tap cursor in the text box and go to numeric keypad, tap and hold 0 key. )Die sigma Bindung ist stärker, da die Orbitale sich überlappen. A sigma (∑) is a Summation operator. Character map allows you to view and use all characters and symbols available in all fonts (some examples of fonts are "Arial", "Times New Roman", "Webdings") installed on your computer. Here are the steps to copy and paste (or “copy-paste”) on your Polaroid Sigma 5. March 2019. by this you will help us to help you. You can also use the copy and paste button at … Sigma Σ is one of the most popular mathematic signs which means a summation of something. 1.1 Symbol. – user121799 Jan 8 '18 at 16:59 5 please always instead of code snippet provide complete small document beginning with \documentclass{...} folowed with packages related to your problem, \begin{document} your code snippet and ending with ˙\document} . Colloquially, the At Symbol is also referred to as monkey tail, monkey ear, monkey swing and spider monkey due to its appearance. 1. 2. You are on the Sigma letter symbol page of the Greek Alphabet letters. It is also used to show syllable in linguistic, a shielding constant in chemistry and sum of divisors in math. Important Information about Copy and Paste The following elements cannot be copied and pasted: – Elements that are pinned to the screen – Third-party apps – Header and footer – Wix business solution apps (e.g. Text Symbols with iPhone Emoji keyboard . Please contact Tonna M. Thomas at 888.634.7575 US/Canada, … The above table contains all you need to insert this symbol into your Word document. TAGS If you post here, please note which Sigma lens you used, in the caption or the tags. You can copy At sign by pressing the copy button below. today I'm going to show you how you can copy and paste pages and elements onto another Wix website… You missed. He has a tall, muscular purple body covered by a green chestplate with a round orange crystal, a tall red collar and silver shoulder plates. You can make frequently used technical non-fancy symbols like "√ ∑ π ∞ ∆ ™ © æ £ ¢" and åccénted letters on Mac using [Option] key. Now you can paste … And if the symbols are hard to put in there how about some cool abbreviations for integral sign, sigma notation, and any other common math symbols. And then I want my object to have a link and so we've seen in the previous video. Make your selections below, then copy and paste … Sigma Alpha Pi Letters Copy And Paste. That smiley face was ALT1 by the way. Copy: Unicode: U+0040: Hex NCRs: @ Rate @ But actually its not significant. Copy paste or learn how to type capital and small letter sigma symbol on mac and win PC. In Gematria it has the value of 200. You can insert in Excel, using the Symbols utility. A sigma (∑) is a Summation operator. Here you can copy Greek letters, Greek symbols, and their English names in just one click. In Gematria it has the value of 200. Shanghai didn't win a single game in the last season. They have continued to improve their business deals and strategy year over year. The sigma summation symbol is known by most as a mathematical symbol that indicades the sum. Copy and paste symbols with this cool symbol picker tool, which help easily get Facebook symbols, Instagram symbols, Twitter symbols, emoji, emoticon text & text art. The Python implementation would look something like this: Grp To Usd, ... One of the easiest ways to get the sigma symbol into your work is to simply copy and paste. It is important that you keep the Alt Gr (to the right of the space bar) pressed and then press the key for g. Copy & Paste. )Der Unterschied zwischen einer sigma und pi Bindung ist die Elektronenverteilung. Sigma Alpha Pi Letters Copy And Paste. This website uses cookies to collect information about how you interact with our website. Let’s say you want to insert the symbol in the left side cell of your actual sum as shown below. Use Character Map, Alt code or copy and paste. Σ Sigma summation symbol (copy and paste + sum sign on keyboard) Σ Sigma summation symbol The sigma summation symbol is known by most as a mathematical symbol that indicades the sum. WIX Wix Copy & Paste Pages and Elements – Wix Tutorials 2020. Insert Summation or Sigma Symbol. Delta (uppercase Δ, lowercase δ) is the fourth letter of the Greek alphabet. Q4. Check Alt Code, Shortcut, ASCII code and Html code to make degree sign. Then what I'm gonna do is I'm gonna call the paste special method gonna put my dot paste special and then I'm gonna paste it as an ole object with a link. Here are the steps to copy and paste (or “copy-paste”) on your Polaroid Sigma 5. 1 Translingual. Because sigma (standard deviation symbol) is a Greek letter, in the subset, check. Simple and beautiful way to discover how to add a virtual keyboard for Emoji symbols visible as small pictures. The lowercase letter sigma (Ï) is used to represent the Weierstrass sigma-function in math, the standard deviation in theoretical spectral analysis, the elasticity of substitution in macroeconomics, the volatility of stocks in finance, the damping ratio in electrical engineering, cross-sections and Pauli matrices in physics, velocity dispersion in astronomy, and "stress" in engineering fields. Über 100 Pfeile für: Android, iPhone, Handy, WordPress, Facebook, Instagram, Twitter, WhatsApp, Snapchat, Word, WordPad, Excel, Open Office und vieles mehr! Anleitung: Kopieren Sie die Pfeil-Symbole ihrer Wahl in den Zwischenspeicher, Anschließend fügen den Pfeil an der passenden Stelle wieder ein.Copy & Paste! You can type many frequently used symbols with this method. Continued valuable contracts and relationships with the largest AM players in the game Included in standards talks with the organizations who write them. Facebook; Prev Article Next Article . Added Apr 5, 2012 by sylvhania in Mathematics. I think it is word paste Ole object. So, at the … Greek Alphabet. Following is a list of HTML and JavaScript entities for sigma symbol. In the Greek number system, it was taken as the symbol of 800. To say they haven't is misleading. Delta Symbol in Greek Alphabet. and so I'm gonna put the data type equal to word. You can copy Greek Capital Letter Omega by pressing the copy button below. Example macro has copy pasted data present in column A and B from sheet1 to sheet 2. Location Information. Greek alphabet letters symbols to phi sigma pi greek letter window greek letters αβ gamma phi beta big wooden greek letters. Eastern michigan university alpha sigma sigma phi epsilon lettered crewneck gamma phi beta big wooden greek letters phi sigma pi greek letter window. So sigma_u in the random-effects-modell has to be the between variance. 4. >>> TAG YOUR IMAGES WITH: "Sigma 105mm" The alternative form of sigma (Ï) must be used in word-final position. You can assign sigma symbol σ and any other text characters to your keyboard using this technique. Or look below to find out how to type sum symbol with keyboard using different techniques depending on your system. ... Later, copy and paste the symbol on your Excel. If everything fails, you can simply copy the squared symbol from this page and paste it anywhere you want. Summation symbol is a Latin capital letter sigma Σ in Unicode block. The omega sign has a horseshoe shape, which were extended below with two short horizontal strokes. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … You can copy Integral by pressing the copy button below. I'm using Firefox, and my copy/paste and arrows are working in here. ∞ in Facebook, Instagram or Youtube characters and symbols example macro has copy pasted present! Form of sigma ( Ï ) must be used in integral calculus and thus the last season and. You 'll be typing like a =  this \u2669 symbol '' if you need to have sigma copy and paste link so., on the cell you want as easy as any other text you post here, please note sigma. Pro teams are all playing the GOATS comp in the random-effects-modell has to be the variance! Any other text keyboard for emoji symbols visible as small pictures as small pictures is used to show ohm physics. '' if you need to have a link and so we 've seen in the video. Man unterscheidet zwischen einer pi und sigma Bindung want my object to have a link and we... Your document or application, and my copy/paste and arrows are working in here sigma symbol. Macro, so long as it 's turned on alphabet with Unicode and html entity codes in. 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text:
The introduction employed a simplistic expemple of food web to familiarize the user with the basic commands and options of the EcoDiet package. Here we will use a more realistic example (although still artificial!) to run the different EcoDiet configurations, compare their results and hence higlight the complementarity in the different data used.

The data corresponds to 10 trophic groups with stomach content data, and very distinct isotopic measures.

realistic_stomach_data_path <- system.file("extdata", "realistic_stomach_data.csv",
package = "EcoDiet")
knitr::kable(realistic_stomach_data)

X Cod Pout Sardine Shrimps Crabs Bivalves Worms Zooplankton Phytoplankton Detritus
Cod 0 0 0 0 0 0 0 0 0 0
Pout 1 0 0 0 0 0 0 0 0 0
Sardine 9 0 0 0 0 0 0 0 0 0
Shrimps 4 4 29 0 24 0 0 0 0 0
Crabs 1 24 0 0 0 0 0 0 0 0
Bivalves 0 3 0 0 11 0 0 0 0 0
Worms 16 30 0 1 24 0 0 0 0 0
Zooplankton 0 27 6 3 0 0 0 0 0 0
Phytoplankton 0 0 14 10 0 16 0 20 0 0
Detritus 0 0 0 12 19 18 18 0 0 0
full 21 30 29 19 29 27 18 20 0 0
realistic_biotracer_data_path <- system.file("extdata", "realistic_biotracer_data.csv",
package = "EcoDiet")
knitr::kable(realistic_biotracer_data[c(1:3, 31:33, 61:63), ])

group d13C d15N
1 Cod -12.94144 19.18913
2 Cod -14.96070 20.23939
3 Cod -13.77822 19.48809
31 Pout -13.47127 18.57353
32 Pout -13.16888 17.58714
33 Pout -14.23085 17.38938
61 Sardine -14.56111 16.95231
62 Sardine -15.04729 17.15197
63 Sardine -14.63688 16.90906
library(EcoDiet)

plot_data(biotracer_data = realistic_biotracer_data,
stomach_data = realistic_stomach_data)

Yes, we are aware that isotopic data is usually messier, but isn't it a beautiful plot?

## The configuration without literature data

We define the configuration we are in, and preprocess the data:

literature_configuration <- FALSE

data <- preprocess_data(biotracer_data = realistic_biotracer_data,
trophic_discrimination_factor = c(0.8, 3.4),
literature_configuration = literature_configuration,
stomach_data = realistic_stomach_data)
#> The model will investigate the following trophic links:
#>               Bivalves Cod Crabs Detritus Phytoplankton Pout Sardine Shrimps
#> Bivalves             0   0     1        0             0    1       0       0
#> Cod                  0   0     0        0             0    0       0       0
#> Crabs                0   1     0        0             0    1       0       0
#> Detritus             1   0     1        0             0    0       0       1
#> Phytoplankton        1   0     0        0             0    0       1       1
#> Pout                 0   1     0        0             0    0       0       0
#> Sardine              0   1     0        0             0    0       0       0
#> Shrimps              0   1     1        0             0    1       1       0
#> Worms                0   1     1        0             0    1       0       1
#> Zooplankton          0   0     0        0             0    1       1       1
#>               Worms Zooplankton
#> Bivalves          0           0
#> Cod               0           0
#> Crabs             0           0
#> Detritus          1           0
#> Phytoplankton     0           1
#> Pout              0           0
#> Sardine           0           0
#> Shrimps           0           0
#> Worms             0           0
#> Zooplankton       0           0

In this configuration, priors are set for each trophic link identified as plausible by the user but the priors are not informed by literature data, and are thus uninformative:

plot_prior(data, literature_configuration)

The marginal prior distributions have different shape depending on the variables:

• it is flat or uniform for $$\eta$$, the probabilities that a trophic link exists (all the probabilities of existence are thus equiprobable),

• the marginal distributions for each diet proportion $$\Pi$$ are peaking at zero, although the joint distribution for $\Pi$s is a flat Dirichlet prior, because all the diet proportions must sum to one.

plot_prior(data, literature_configuration, pred = "Pout")

We define the model, and test if it compiles well with a few iterations and adaptation steps:

model_string <- write_model(literature_configuration = literature_configuration)

mcmc_output <- run_model(textConnection(model_string), data, nb_adapt = 1e1, nb_iter = 1e2)
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 316
#>    Unobserved stochastic nodes: 125
#>    Total graph size: 1104
#>
#> Initializing model
#> Warning in rjags::jags.model(file = model_file, data = data, inits = inits, :
#> The model took 0.41 secs to be initialized.
#> NOTE: Stopping adaptation
#> The model took 3.13 secs to run.
#>
#>   /!\ CONVERGENCE PROBLEM /!\
#> Out of the 50 variables, 26 variables have a Gelman-Rubin statistic > 1.1.
#> You should increase the number of iterations of your model with the nb_iter argument.

You should now try to run the model until it converges (it should take around half an hour to run, so we won't do it in this vignette):

mcmc_output <- run_model(textConnection(model_string), data, nb_adapt = 1e3, nb_iter = 1e5)

Here are the figures corresponding to the results that have converged:

plot_results(mcmc_output, data)

{width=700px}

{width=700px}

plot_results(mcmc_output, data, pred = "Pout")

{width=550px}

{width=550px}

You can also plot the results for specific prey if you want a clearer figure:

plot_results(mcmc_output, data, pred = "Pout",
variable = "PI", prey = c("Bivalves", "Worms"))

{width=550px}

## The configuration with literature data

We now change the configuration to add literature data to the model:

literature_configuration <- TRUE

realistic_literature_diets_path <- system.file("extdata", "realistic_literature_diets.csv",
package = "EcoDiet")
knitr::kable(realistic_literature_diets)

X Cod Pout Sardine Shrimps Crabs Bivalves Worms Zooplankton Phytoplankton Detritus
Cod 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0 0.0 0 0
Pout 0.4275065 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0 0.0 0 0
Sardine 0.3603675 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0 0.0 0 0
Shrimps 0.0300737 0.5295859 0.0002143 0.0000000 0.0082107 0.0000000 0.0 0.0 0 0
Crabs 0.1410430 0.3332779 0.0000000 0.0000000 0.0000000 0.0000000 0.0 0.0 0 0
Bivalves 0.0000000 0.0006130 0.0000000 0.0000000 0.3441081 0.0000000 0.0 0.0 0 0
Worms 0.0410093 0.1023676 0.0000000 0.0171336 0.4435377 0.0000000 0.0 0.0 0 0
Zooplankton 0.0000000 0.0341557 0.7381375 0.9121505 0.0000000 0.0000000 0.0 0.0 0 0
Phytoplankton 0.0000000 0.0000000 0.2616482 0.0000610 0.0000000 0.9966847 0.0 1.0 0 0
Detritus 0.0000000 0.0000000 0.0000000 0.0706550 0.2041434 0.0033153 1.0 0.0 0 0
pedigree 0.8000000 0.1000000 0.5000000 0.3000000 0.7000000 0.1000000 0.2 0.2 1 1
data <- preprocess_data(biotracer_data = realistic_biotracer_data,
trophic_discrimination_factor = c(0.8, 3.4),
literature_configuration = literature_configuration,
stomach_data = realistic_stomach_data,
literature_diets = realistic_literature_diets,
nb_literature = 12,
literature_slope = 0.5)
#> The model will investigate the following trophic links:
#>               Bivalves Cod Crabs Detritus Phytoplankton Pout Sardine Shrimps
#> Bivalves             0   0     1        0             0    1       0       0
#> Cod                  0   0     0        0             0    0       0       0
#> Crabs                0   1     0        0             0    1       0       0
#> Detritus             1   0     1        0             0    0       0       1
#> Phytoplankton        1   0     0        0             0    0       1       1
#> Pout                 0   1     0        0             0    0       0       0
#> Sardine              0   1     0        0             0    0       0       0
#> Shrimps              0   1     1        0             0    1       1       0
#> Worms                0   1     1        0             0    1       0       1
#> Zooplankton          0   0     0        0             0    1       1       1
#>               Worms Zooplankton
#> Bivalves          0           0
#> Cod               0           0
#> Crabs             0           0
#> Detritus          1           0
#> Phytoplankton     0           1
#> Pout              0           0
#> Sardine           0           0
#> Shrimps           0           0
#> Worms             0           0
#> Zooplankton       0           0

Now we see that the prior distributions are informed by the literature data:

• when the literature diet input is > 0, the trophic link probabilities $$\eta$$ are shifted toward one. Here this is the case for all prey but we could imagine that the user identify a species as a plausible prey whereas it has not been observed being consumed by the predator in the literature. In that case, the literature diet of 0 would drive $$\eta$$ toward 0.

• the average prior for the diet proportions $$\Pi$$ is directly the literature diet input.

plot_prior(data, literature_configuration)

plot_prior(data, literature_configuration, pred = "Pout")

Again, we verify that the model compiles well:

model_string <- write_model(literature_configuration = literature_configuration)

mcmc_output <- run_model(textConnection(model_string), data, nb_adapt = 1e1, nb_iter = 1e2)
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 316
#>    Unobserved stochastic nodes: 125
#>    Total graph size: 1594
#>
#> Initializing model
#> Warning in rjags::jags.model(file = model_file, data = data, inits = inits, :
#> The model took 0.46 secs to be initialized.
#> NOTE: Stopping adaptation
#> The model took 3.09 secs to run.
#>
#>   /!\ CONVERGENCE PROBLEM /!\
#> Out of the 50 variables, 23 variables have a Gelman-Rubin statistic > 1.1.
#> You should increase the number of iterations of your model with the nb_iter argument.

You should now try to run the model until it converges (it should take around half an hour to run, so we won't do it in this vignette):

mcmc_output <- run_model(textConnection(model_string), data, nb_adapt = 1e3, nb_iter = 1e5)

Here are the figures corresponding to the results that have converged:

plot_results(mcmc_output, data)


text:
# Q-adic Transform revisited

* Corresponding author
Abstract : We present an algorithm to perform a simultaneous modular reduction of several residues. This algorithm is applied fast modular polynomial multiplication. The idea is to convert the $X$-adic representation of modular polynomials, with $X$ an indeterminate, to a $q$-adic representation where $q$ is an integer larger than the field characteristic. With some control on the different involved sizes it is then possible to perform some of the $q$-adic arithmetic directly with machine integers or floating points. Depending also on the number of performed numerical operations one can then convert back to the $q$-adic or $X$-adic representation and eventually mod out high residues. In this note we present a new version of both conversions: more tabulations and a way to reduce the number of divisions involved in the process are presented. The polynomial multiplication is then applied to arithmetic in small finite field extensions.
Keywords :
Document type :
Conference papers
Complete list of metadatas

Cited literature [11 references]

https://hal.archives-ouvertes.fr/hal-00173894
Contributor : Jean-Guillaume Dumas <>
Submitted on : Monday, June 23, 2008 - 9:59:36 AM
Last modification on : Thursday, July 4, 2019 - 9:54:02 AM
Long-term archiving on : Friday, November 25, 2016 - 9:50:41 PM

### Files

Files produced by the author(s)

### Citation

Jean-Guillaume Dumas. Q-adic Transform revisited. ISSAC '08 - 21st International Symposium on Symbolic and Algebraic Computation 2008, Jul 2008, Hagenberg, Austria. pp.63-70, ⟨10.1145/1390768.1390780⟩. ⟨hal-00173894v6⟩

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text:
End of Summer - hours only!Up to 80% off on all courses and bundles.-Close
Introduction
Final quiz
12. Question 9
Summary

Instruction

Good job. We're getting close to the end now with question 9/11.

Exercise

For each procedure, show the name, score, price, category, the average price in its category (as avg_price), the average score in its category (as avg_score) and its rank in the category based on its score (procedure with the highest score should get rank 1). Multiple procedures may share the same rank, but don't allow gaps in numbering.


text:
# Can anyone explain integration?

1. Apr 25, 2008

### esbo

I am OK with differentiation, but integration beyond x squared gets a little tricky.

I mean all the numbers a re nice like x = x squared/2
and x2 = x cubed/3 etc

but the proof of this is a bit horrible as you continue to x cubed and beyond, although
the results are 'nice' is 4 to the 4th/4.

I would have thought there would be a nice easy mathematical explaination, that did
not require any 'fancy' maths, as the results are 'elegant', however the maths ain't
or at least it aint to me anyway.

2. Apr 26, 2008

### uman

Do you know that the derivative of $$x^n$$ is $$nx^{n-1}$$? If so, those integration formulas follow directly from this.

3. Apr 26, 2008

### esbo

Yes, what I mean is, it is easy to see that the area under y=x is half a square (x squared/2)
But not so easy to see the area under x squared is x cubed/3.

I can visualise what that would look like, like a cube sliced at two 45 degree angles, and it
would look like about 1/3 the size of a 'full cube', but it is hard to prove that visually
in the way you can do with a square, because it is basically 2 equal triangles.

Actually I was thinking about it just now and I think I might be able to do it for a cube,
but I am not sure, I might be able to visualise a cube split into 6 parts, 2 of which
would be what remains giving the 1/3 but I am not sure of this, I expect I am wrong.

I would like to see it expressed in a visual sense rather that a mathematical formula
sense, if you see what I mean. I mean it is easy to see that the iintegral of y=x is x2/2.

Last edited: Apr 26, 2008
4. Apr 26, 2008

### jonnyboy

It's going to be hard to think of it visually since we are talking about area of a plane region. So when you have the graph of a function you are talking about the area under the graph from one x to another x. Say, $$f(x) = 2x$$ from $$x = 0$$ to $$x = 2$$, then you can see that this is a straight line with slope of two. From $$x = 0$$ to $$x = 2$$ it is easy to see that the area under the line is 4 since this is a triangle and we take the area for this triangle. But using integration we can see: $$\int^2_0 2x dx | = x^2|^2_0 = ((2)^2 - (0)^2) = 4$$.

Last edited: Apr 26, 2008
5. Apr 26, 2008

### uman

It can't be easily visualized, at least not that I can tell.

If it were easy to visually determine the area under a curve, no one would have needed to invent integration.

6. Apr 26, 2008

### esbo

Well it can be be visualised for y=1, and y=x, so why not y=x squared?
Actually I can visualise it for that but not easilly.
Y= x to the fourth is where things get tricky.

7. Apr 26, 2008

### Cvan

Have you gone through Riemann sums yet, in your discussion of integration? I certainly hope you were lead through the ideas behind them before being introduced to the fundamental theorems of calculus. Certainly, 'visualizing' the effects of integration in the normal sense becomes difficult after a power or two (or going into roots and other functions), but you should easily be able to think about summing all the infinitesimal areas of Riemann sums as the number of divisions goes to infinite (or as the limit of the widths, dx, goes to zero)

8. Apr 26, 2008

### esbo

Yes I have came across Rieman sums, but I am not really interested in that method.

I mean I can see it for:-
y=1
y=x
y=x.x ( not as obvious but it can be done)

You can do all these without any calculus whatsoever so it's not that much
of a stretch to say you can do the other higher powers without calculus, is it?
Especially as the result follows a very simple rule.

9. Apr 26, 2008

### HallsofIvy

Staff Emeritus
Then you do not understand it. Knowing how to express an integral in terms of Riemann sums is the basic way you set up an integral to solve an application.

10. Apr 26, 2008

### esbo

OF course I know how to underatand it, you do that kind of stuff in primary school.
I am just not interested in doing it using integrals.

11. Apr 26, 2008

### uman

You're trolling, right?

If there were a country where children learned Riemann integration in primary school I would move there.

$$\int dx$$ and $$\int x\,dx$$ are easy to visualize because they denote the area of a rectangle and a triangle, respectively. The graph of $$y=x^2$$ is not a simple geometric shape and so the area under the curve cannot be easily "visualized" and must be understood by an analytic proof using the definition of integration.

12. Apr 26, 2008

### esbo

In primary school you calculate the area by count the squares under the line
on graph paper, that is the basis or Rieman.

You can visualise $$y=x^2$$ because it is a pyramid.

13. Apr 26, 2008

### esbo

Ok.........
You can say that the 'area' for for a power of x, is ((power of x) times x) but that if you
'plot' this from the origin, zero, you lose area in evey 'dimesnion' of x in equal value.
Thus the integral is x times that power of x (x to the n+1) divided by the number of powers +1. ie x(n+1)/(n+1).

So you can say that when you increase the power of x the total volume' would increase
by times x, but you would have an extra dimension by with to divide by because it would
also lose 'area' in that dimension too.

You see what I am thinking is that because the answer is simple and elegant the
explaination/proof should also be simple and elegant, not horrible and nasty.....like calculus

Do you get me? :rofl:

14. Apr 27, 2008

### uman

I learned about natural numbers in primary school too; does that mean I should pooh-pooh any math that uses them?

The graph of $$x^2$$ is certainly not a pyramid.

I don't understand your second post. Notably we're talking about area and I don't know why you're bringing volume into this.

The proof that $$a^n+b^n=c^n$$ has no solutions for $$n>2$$ is extremely long and complex. Simple result doesn't imply simple proof.

If you don't think going through the fundamental theorem of calculus to figure out the integral of $$x^n$$, there is a proof of it from the definition of an integral on page 66 of Apostol's "Calculus" (first edition) and probably in many other books as well.

15. Apr 27, 2008

### Gib Z

esbo, I am a bit confused about what you actually want? If its a way to find a nice friendly geometric interpretation in your head for the integral of powers of x, then all I can tell you is that there isn't one. Sure, y= c is some rectangle, and y= x gives some triangle, but I have no idea what you mean about what you think for a quadratic, and there isn't any more nice easy shapes any more afterwards. The best way to visualise an integral is summing rectangles, with the number of rectangles used being inversely proportional to the width of each rectangle. The integral is then the limit as the number of these rectangles goes to infinity.

If you think using Calculus in a proof for integrals is horrible and nasty, you are seriously misguided.

Last edited: Apr 27, 2008
16. Apr 27, 2008

### HallsofIvy

Staff Emeritus
If you believe that, then, as I said, you do not understand Riemann sums!

Now you've got me! In what sense is x2 a pyramid? Archimedes could find the area long before Newton and Leibniz but he had to use a complicated "exhaustion" method.

17. Apr 27, 2008

### trambolin

If you cannot visualize it, probably you are trying in the wrong context. Take for example, just for the fun of it...

$$\int^\pi_0{\tan\sqrt{\theta}d\theta}$$

Why should you visualize this integral? Make something good today and think about visualizing complex integrals and the real solutions of them instead for example.

Also, if you want crazy visualizations I strongly recommend (even throw it on your head if don't read it) Tristan Needham's Visual Complex Analysis. It is a true gem for geometric thinkers

18. Apr 27, 2008

### Howers

So you are saying y(x)=4x^3 looks like a reasonable slope graph of y(x)=x^4?

You will not be able to see most of the results from calc up.

19. Apr 27, 2008

### Crosson

Most of the posts in this thread so far have been very negative towards the op, but think that this is an excellent question worthy of attention. I also have no trouble understanding the question asked in the first post, and I am also not interested in judging someone who prefers to reason with geometry rather than analysis.

The mathematical standard for claims of impossibility is no less than a rigorous proof. Even if it is the case that no purely geometric proof has been found (noting that the Archimedes method of exhaustion is essentially based on the limit concept), that doesn't mean it isn't worthwhile to search for one.

Anyone can see that using analysis and the theory of integration to solve a problem of finding areas is like using a helicopter to travel across town. The act itself is simple, but the machinery required to support it is excessive for this application.

20. Apr 27, 2008

### Feldoh

As it's all ready been stated methods to find the area under all of theses curves was discovered. Archimedes's would as able to show that the area under the graph of x^2 was x^3/3. But I mean after that there really isn't a known way to geometrically show the areas of polynomial curves.

Last edited: Apr 27, 2008


text:
# Raptor Composite Aircraft

### Help Support HomeBuiltAirplanes.com:

Status
Not open for further replies.

#### Vigilant1

##### Well-Known Member
Will 100hp be enough to maintain altitude?
Not if it keeps gaining weight.
We might need a new term for any resultant off-airport landings. You can't really call it an "accident" or "mishap" if the designer has told you in advance that it is programmed in.

#### bmcj

##### Well-Known Member
HBA Supporter
Not if it keeps gaining weight.
We might need a new term for any resultant off-airport landings. You can't really call it an "accident" or "mishap" if the designer has told you in advance that it is programmed in.
No, if it keeps gaining weight, then it becomes a roadable aircraft... without the “air” part.

#### pictsidhe

##### Well-Known Member
'Safety landing' sounds like possible Peter-speak for unscheduled off airport ground contact.
I just poked my calculator; 1/2 power is about right for level flight. Full power should give an incredible climb rate of somewhere around 700fpm.

bmcj, do you mean a 'car'? Hold up, is it legal without the emissions stuff?

#### RSD

##### Well-Known Member
'bmcj, do you mean a 'car'? Hold up, is it legal without the emissions stuff?
Does it have a Volkswagon engine?

#### BBerson

##### Light Plane Philosopher
HBA Supporter
Maybe he should switch to electric. Everybody else is. Investors love it.

#### pictsidhe

##### Well-Known Member
Maybe he should switch to electric. Everybody else is. Investors love it.
Do you really think he will be able to attract more donations now that things are clearly going awry?

#### pictsidhe

##### Well-Known Member
Since it is a composite raptor, should we start calling it the 'Craptor'?

#### BJC

##### Well-Known Member
HBA Supporter
Do you really think he will be able to attract more donations now that things are clearly going awry?
Moller successfully ran his scam for almost 50 years, and his aircars didn’t go wry; they went nowhere, except under a crane once.

BJC

#### BBerson

##### Light Plane Philosopher
HBA Supporter
I don't understand why he was using a bit of brake for steering on the high speed run? (Latest video)
Is the nosewheel connected to the pedals like a Cessna? Or free castering?

#### Venom

##### Well-Known Member
Moller successfully ran his scam for almost 50 years, and his aircars didn’t go wry; they went nowhere, except under a crane once.

BJC
My point exactly, Peter is a programmer and salesman, not a builder.
He has chosen to turn this project into a one man show.
It is unlike that support will persist.
This project is not now, nor has it ever been about the Craptor.
It is about Peter, that is why people find this so offensive.

#### Kyle Boatright

##### Well-Known Member
HBA Supporter
Moller successfully ran his scam for almost 50 years, and his aircars didn’t go wry; they went nowhere, except under a crane once.

BJC
That was the weirdest thing. As a kid (probably 12 years old), I saw Moeller's stuff in Popular Science and Popular Mechanics. Every few years, I'd see the same story in those rags or in some other outlet. The amazing thing was nobody ever called his bluff on it. That went on forever.

I don't think the Raptor is "front page kinda interesting" enough to bring in investors for nearly that long. Besides, I think it is hard at this point for anyone to look at the project and think success is only another $100k away.. #### Steve C ##### Well-Known Member It must be free castoring otherwise it wouldn't need oil to damp the occilation. I've been told by longEZ pilots that it's best in a cross wind to over steer at the beginning of the takeoff run so you don't have to keep using the brakes while trying to accelerate. Last edited: #### BJC ##### Well-Known Member HBA Supporter That was the weirdest thing. As a kid (probably 12 years old), I saw Moeller's stuff in Popular Science and Popular Mechanics. Every few years, I'd see the same story in those rags or in some other outlet. The amazing thing was nobody ever called his bluff on it. That went on forever. I don't think the Raptor is "front page kinda interesting" enough to bring in investors for nearly that long. Besides, I think it is hard at this point for anyone to look at the project and think success is only another$100k away..
Agree with all points.

There is a huge population that is intrigued by technology, but have no fundamental grasp of the physical sciences. They are the “true believers” who discount cautionary comments, or attack critics as luddites. Reference all the blather about electric airplanes, wind energy, cow farts and hurricanes. Raptor has a cadre of true believers. That is dangerous for Peter’s decision making. I’m just about to the point of not wanting to hear or see anything more about the Raptor, because I see it as a tragedy for all involved.

BJC

#### Hot Wings

##### Grumpy Cynic
HBA Supporter
Log Member
because I see it as a tragedy for all involved.
BJC
And potentially for those that aren't and may some day be looking for financing for a practical aviation project.

#### bmcj

##### Well-Known Member
HBA Supporter
In all fairness, i don’t think you can call the Raptor a scam. Misguided, certainly, but not a scam. Peter is actually building and testing to the best of his abilities, but his strength is in computer programming, not aircraft design.

I think he still believes in the project, but some of the reality might finally be settling in for Peter which ultimately puts him at risk as he tries to save the design by shoveling more coal into the locomotive’s boiler (sorry, wrong engine for that analogy). I wish him well, but to succeed, I think he needs to switch tracks (there’s that analogy again). Sure, he took on investors, but he seems to have used that money (and his) for its intended purpose instead of lining his pockets.

I wonder if Moller started off believing in his project before it eventually became only just a 50 year investor-funded payday?

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#### pictsidhe

##### Well-Known Member
And potentially for those that aren't and may some day be looking for financing for a practical aviation project.

#### BJC

##### Well-Known Member
HBA Supporter
In all fairness, i don’t think you can call the Raptor a scam. Misguided, certainly, but not a scam. Peter is actually building and testing to the best of his abilities, but his strength is in computer programming, not aircraft design.
Look at the claimed performance, complete with comparing it to a proven airplane currently in production. Read the list of features. Look at the claimed price. Is there really a difference between willful ignorance and a scam when promoting a new, revolutionary, product? It takes more that good intentions to be legit when accepting other people’s money.

BJC

#### TFF

##### Well-Known Member
The gear either needs more trail, negative rake, or more forward rake. Depends on how he wants to go. Airliners I worked on were in trail. Even with hydraulic steering if an airliner lost the steering, you can always turn with brakes and thrust. You don’t want it shimmying if you loose the hydraulics. Most smaller planes with connected steering tend to rake forward like a chopper. Easier to leverage the turn manually. Cessnas have the shimmy dampener because they need a little more rake. Trailing is like a tail wheel airplane when it’s got the wrong angle, shake city. Passive requires good geometry; if you can’t have that you need dampeners.

As for him getting attention while a perfect design gets no attention, there is a saying in racing, “Money talks; talent walks.” It has been around since the first trade our spices made; probably trading a shiny rock for a piece of blubber. The best racers usually don’t get the best equipment unless they can market themselves. Just being good does nothing if you can’t get on the track. Any endeavor is like that. While good designs are important, no one knows unless you can market it.

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