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# What are some examples of acid and base dissociation?

A strong acid or base will completely disassociate, meaning that the acid will form two ions, ${H}^{+}$ and its conjugate base. Strong acids completely disassociate because their conjugate base is weaker than water. This means that there is no equilibrium in the solution, simply because the bases aren't "strong" enough to bond to an ${H}^{+}$ ion. Same applies for strong bases, but a strong base contains an $O {H}^{-}$ ion.
$H C l + {H}_{2} O \to {H}_{3} {O}^{+} + C {l}^{-}$
$H B r + {H}_{2} O \to {H}_{3} {O}^{+} + B {r}^{-}$
$N a O H \to N {a}^{+} + O {H}^{-}$
$M g {\left(O H\right)}_{2} \to M {g}^{-} 2 + 2 O {H}^{-}$
Weak acids and bases, however, do have the ability to bond to ${H}^{+}$ ions after disassociation, therefore there is an equilibrium present in the solution.

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## Vitali sets versus real-valued measurable cardinals

If there exists a real-valued measurable cardinal, then there is a countably additive extension of Lebesgue measure to all sets of real numbers. This would include then the Vitali sets, which are an example of sets that are not Lebesgue measurable -- or at least, apparently for weaker assumptions than the existence of a real-valued measureable cardinal. However, after going over the proof that a Vitali set is not measurable, for example in Vitali set - Wikipedia, the free encyclopedia, I do not see where the proof would fail under the assumption of a real-valued measureable, i.e., assuming that there exists a cardinal κ so that there is an atomless κ-additive measure on the power set of κ. I presume I am missing something breathtakingly obvious. Could someone point this out to me?


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# One-to-one correspondence between electron density and external (Coulomb) potential

I calculated the electron density of a molecule by a DFT software (Gaussian 16) with a standard setting in terms of the basis set and hybrid functional. After that calculation, I viewed the plot about the correspondence between the obtained electron density $$\rho$$ and the external potential $$V$$ by Coulomb (i.e., $$V(r) = -\sum_{i=1}^{M} \frac{Z_i}{||r-R_i||}$$, where $$Z$$ is the atomic number and $$R$$ is the atomic position) on some grid points $$r$$ in the molecule.

The Hohenberg-Kohn theorem shows the one-to-one correspondence between $$\rho$$ and $$V$$; however, my calculation result did not show it. The following figure shows that the same density values provided different potential values or the same potential values provided different density values.

My calculation might be wrong, but I have tried to calculate many times, and the result seems to be correct. Why did I get such a result that seems to contradict the Hohenberg-Kohn theorem?

This situation does not contradict the Hohenberg-Kohn theorems:

the same density values provided different potential values or the same potential values provided different density values.

The first Hohenberg-Kohn theorem refers not to pointwise correspondence, but rather to correspondence between entire functions. It tells us that if I give you the full density—the entire function, not just values at a few points—you can (in principle) determine the shape of the full potential. However, the relationship is complex: the density at a single point will likely depend on the potential at all nearby points, and vice versa.

• Thank you for the answer. I still do not understand that "not pointwise correspondence, but correspondence between entire functions." I believe that correspondence as entire functions means (encompasses) the pointwise correspondence, but is it not right?
– neco
Jul 29 at 14:25
• Imagine a secret code where 'apple' means 'water.' The H-K correspondence works similarly: correspondence between functions is like correspondence between words. But the individual letters in 'apple' and 'water' do not have any correspondence; otherwise 'p' would have to correspond to both 'a' and 't'.
– wcw
Jul 29 at 15:41
• I'm even more confused... In a popular figure about one-to-one correspondence (e.g., en.wikipedia.org/wiki/Bijection), your example seems that all arrows are completely destroyed. Do we still call such situation one-to-one correspondence?
– neco
Jul 29 at 18:39
• Yes, correspondence between the functions themselves
– wcw
Jul 29 at 20:56
• If the discussion gets too much longer, I would recommend this chat room. Jul 29 at 21:35


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In the addition problem ADD + ADD + ADD = SUMS, each of A, D, S, U, and M represents a different digit, and A is even. What is the value of the digit U?

A. 2
B. 3
C. 4
D. 5
E. 6
[Reveal] Spoiler: OA

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Bunuel wrote:
In the addition problem ADD + ADD + ADD = SUMS, each of A, D, S, U, and M represents a different digit, and A is even. What is the value of the digit U?

A. 2
B. 3
C. 4
D. 5
E. 6

Addition of three 3-digit numbers giving sum of 4-digit number can yield S as only 1 or 2

multiplication of 3 * D = Last digit 1 ...D=7
multiplication of 3 * D = Last digit 2 ...D=4

CASE 1: S = 1 and D = 7

now A is even can take 2,4,6,8,0

we know
multiplication of 3 * A + 2 = 1U
0 and 2 not possible (no carry over to form 4-digit) (0*3 = 0, 2*3 = 6)
Putting A= 8 gives U= 6 but S= 2 ..not possible
Putting A= 6 gives U= 0 but S= 2 ..not possible
Putting A= 4 gives U= 4 and S= 1 ..not possible (unique digits)

CASE 2: S = 2 and D = 4
we know
multiplication of 3 * A + 2 = 1U
now A is even can take 2,4,6,8,0
A cannot take 2,4 (already taken-unique digits)
Putting A= 6 gives U= 9 but S= 1 ..not possible
Putting A= 8 gives U= 5 and S= 2 ..possible
Hence A = 8, D = 4, S = 2 and U = 5
D
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Bunuel wrote:
In the addition problem ADD + ADD + ADD = SUMS, each of A, D, S, U, and M represents a different digit, and A is even. What is the value of the digit U?

A. 2
B. 3
C. 4
D. 5
E. 6

$$+ADD$$
$$+ADD$$
$$+ADD$$
----------------
$$SUMS$$

so unit's digit of $$3D$$ is $$S$$ and $$M$$ and given that every digit is distinct, this implies that $$D>=4$$ to have a carry forward of $$1$$ or $$2$$

So starting with $$D=4$$ we can calculate values of all variables and test the given conditions to arrive at the final answer

Case 1: $$D=4$$ --> $$S=2$$ --> $$M=3$$ --> $$A=8$$ --> $$U=5$$ (as $$A$$ is even and addition $$3D$$ in Ten’s digit will give only $$1$$ as carry forward. Since $$S$$, in the thousands place, has to be $$2$$ so $$A$$ has to be $$8$$, the largest even digit possible to have a carry forward of $$2$$)

So in Case 1, we got all our distinct variables. We can stop here or we can test for $$D=5,6,7,8,9$$ and match the subsequent conditions. For any other value of $$D$$, one or the other condition will not satisfy. Hence our answer $$U=5$$

Option D

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13 Nov 2017, 20:04
Ans D ) u =5

D cannot be 1 , 2, 3 or 5 because then m= s which is not possible as given in the question.
3D = S and 3D = M and all digits are different which implies that M = 3D+ Carry from 3D = S. Also from the question it can be deduced that since 3A = SU then S can at max be digit 2 if we take the max value of A which is A= 8.

Let say s= 2 then D = 4 m=3 and u =5 only.

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## Section16.8Using the Set and Map Interfaces

The Set and Map interfaces are similar to the List interface in that there are multiple classes in the collections framework that implement them.

### Subsection16.8.1Using the Set Interface

The Set interface is modeled after the set theory principles taught in mathematics. In mathematics, sets are groups of elements with a clearly defined algorithm for deciding if any given element is in any given set. Elements can be added to sets and can be removed from sets. Sets cannot have duplicate elements; if an element is added to a set that already contains an element equal to it, the new set still has a single such element. The elements of a set have no natural order; two sets that have the same elements listed in different orders are considered to be the same set.

In computer science and in Java, data structures that model sets are designed for large collections of data. Such data structures have a method that determines if an object is in a given set with an efficient algorithm. For large data sets, using such a method is much faster than iterating through a list. Sometimes, you may or may not be able to list the elements of a set data structure in some natural order, depending on how the data structure is implemented. An incomplete listing of the methods of the Set<E> interface is given in the UML diagram in Figure 16.8.1.

TreeSet<E> and HashSet<E> are two classes in the collections framework that implement the Set<E> interface. They both provide fast operations to check whether an element is in a set. They also provide quick insertion of an element into the set or removal of an element from a set. For large sets—those having at least several thousand elements—where there are large numbers of insertions, deletions, and tests for whether elements are in a set, linked lists would be much slower.

When using the Set<E> interface for a user-defined class E, you will likely want to override the definition of the equals() method from the Object class in E because that method is used when computing the value of aSet.contains(anElement). When using the TreeSet<E> class for a user defined class E, you should implement the compareTo() method of the Comparable interface because it is used to order the elements of E. In the next section, we will discuss the specific manner in which elements are ordered. Finally, when using the HashSet<E> class for a user defined class E, you should override the hashCode() method of the Object class because it is used HashSet<E>. Hash codes are indexes that are computed from the particular object that is being stored in the HashSet. Given an object's hash code, the object can be retrieved directly, as we do with object's stored in an array. However, we will not discuss hash codes in any detail in this text.

### Subsection16.8.2Problem Statement

Let's think about a simple example for using a set data structure. Suppose that a programmer is developing an application for a large company for maintaining a no–call list. The programmer has decided to use the TreeSet<E> data structure to store objects of the PhoneRecord class that was defined earlier in this chapter and will use methods of the Set<E> interface to manipulate the data.

A TreeSet seems to be an appropriate structure for this problem, since a large amount of data will be involved. The company wants the PhoneRecord data stored in alphabetical order. The main use of the data will be to test whether names are in the set.

The programmer might write a short method like that in Listing 16.8.2 to demonstrate how the Set<E> and TreeSet<E> structures will be used.

In order for the testSet() method to work as we would like, we need to have the PhoneRecord class implement the Comparable interface and to override the equals() method. For this example, it is reasonable to assume that the name field of PhoneRecord objects should be unique so that it can be used to decide if two PhoneRecord objects are equal. The name field of PhoneRecord can also be used to define the other two methods discussed above. Thus, add the following code to the PhoneRecord class.

public boolean equals(Object ob){
return name.equals(((PhoneRecord)ob).getName());
} //equals()
public int compareTo(Object ob){
return name.compareTo(((PhoneRecord)ob).getName());
} // compareTo()
public int hashCode(){
return name.hashCode();
} // hashCode()

#### Activity16.8.1.

Run the code below. Try adding a new PhoneRecord to the Set.

The output of the TestSet() method is listed below:

Testing TreeSet and Set
Roger M is contained in theSet is true
Mary Q is contained in theSet is false
Gary G 201-119-8765
Jane M 090-997-1987
Roger M 090-997-2918
Stacy K 090-997-9188

Notice that Jane MPhoneRecord appears only once in the listing of elements in the set.

### Subsection16.8.3Using the Map<K,V> Interface

The Map<K,V> interface is modeled after looking up definitions for words in a dictionary. In computer science, maps are considered to be a collection of pairs of elements. A pair consists of a key that corresponds to a word being looked up and a value corresponding to the definition of the word. Pairs can be added to maps and can be removed from maps. Maps cannot have distinct pairs with the same keys; if you attempt to add a pair to a map that already contains a pair with the same key, the second pair will replace the first. An incomplete listing of the methods of the Map<K,V> interface is given in the UML diagram in Figure 16.8.3. TreeMap<K,V> and HashMap<K,V> are two classes in the collections framework that implement the Map<K,V> interface.

Let's now consider a simple example of using a map data structure. Suppose that our programmer has been hired by a large company to develop an application that maintains an electronic phone list for company employees. The programmer has decided to use the TreeMap<E> data structure to store pairs of names and telephone numbers and will use methods of the Map<V,K> interface to manipulate the data.

A TreeMap seems like an appropriate data structure for this problem, since a large amount of data will be involved. The company wants the PhoneRecord data stored in alphabetical order. The main use of the data will be to use names to access telephone numbers.

The programmer might write a short method like that in Listing 16.8.4 to demonstrate how the Map<K,V> and TreeMap<K,V> structures will be used. Note that this code has been simplified to just store pairs of Strings, names and phone numbers, but it could be adapted to store names and PhoneRecords.

#### Activity16.8.2.

Run the code below. Try adding a new name and phone pair to the Map.

The output for this test program is:

Stacy K has phone 090-997-9188
Jane M has phone 090-233-0000

Notice the the second phone number for Jane M is the one that is stored in the data structure.

#### ExercisesSelf-Study Exercise

##### 1.

Using the activity above as a guide, create a Map that translates words from English to another language.

Hint.

For example, you could put the pair cat and gato in the Map to translate cat into Spanish.


text:
#### Vol. 13, No. 4, 2019

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Quadratic twists of abelian varieties and disparity in Selmer ranks

Vol. 13 (2019), No. 4, 839–899
##### Abstract

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarized abelian variety over a number field. Specifically, we determine the proportion of twists having odd (respectively even) 2-Selmer rank. This generalizes work of Klagsbrun–Mazur–Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square. In particular, the statistics for parities of $2$-Selmer ranks and $2$-infinity Selmer ranks need no longer agree and we describe both.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/ant

We have not been able to recognize your IP address 3.215.79.204 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.


text:
# Character Based Evolutionary Tree Construction

February 16, 2018

In this article, we’ll talk about a method for constructing evolutionary trees, known as character based evolutionary tree construction. It was initially designed to infer evolutionary relationships based on morphological and physiological characters.

In character based tree construction, we are given a DNA segment for multiple species coming from the same part of the genome (for example, the same gene). Given these DNA sequences, we could like to construct the evolutionary tree, i.e. predict which species are more closely related and have a recent common ancestor, vs species that are not closely related and diverged earlier.

# Parsimony score

Character based tree construction method is based on Occam’s razor principle which states “when several hypotheses with different degrees of complexity are proposed to explain the same phenomenon, one should choose the simplest hypothesis”. In terms of tree building, all other things being equal, the best hypothesis is the one that requires the fewest evolutionary changes.

Parsimony score is defined as total cost of all mutations in a tree that are required to explain the observed differences. The parsimony score depends on the structure of the tree, the locations assigned to each of the species, as well as the inferred DNA sequences for each of the common ancestors.

## Example

Let’s say we have we have 4 species and the nucleotides (DNA sequence) from each of the specifies. They are as follows:

As per the table, species A and B have same nucleotide at 1st position and C and D have same nucleotides at 1st position.

Based on the above information, we might predict a evolutionary tree that looks as follows (figure 1). Note that the prediction also includes the positions of species A, B, C and D as well as the sequences for the common ancestors.

Figure 1

The dashes on the edges label the number of differences between a node and its ancestor. For example, species A (GGG) differs from its parent (GTG) at one location (G vs T, in location 2).

Let us consider another possible evolutionary tree. As per this tree, species A and C are more closely related. Again, dashes on the edges label the number of differences.

Figure 2

We have drawn two trees considering the nucleotide positions of species. But how we know which tree is better or more parsimonious tree?

We can see that the parsimony score of the first tree is 4 and for the second tree it is 5. Since we are trying to minimize the number of mutations, the first tree is better than the second tree.

## Weighted parsimony score

The above definition can be extended to include cases different nucleotide mutations are given different scores. This is described below with some examples:

Unweighted Parsimony Score = 5

Weighted Parsimony Score = 20

If we have multiple characters per sequence, we simply compute the above score for each position, and add them up.

# Parsimony Problems

Our broad strategy for finding the evolutionary tree with the minimum parsimony score will be as follows:

Small Parsimony problem: We are given a tree, but need to determine labeling of the tree that provides the lowest parsimony score. That is, we need to find the nucleotide sequences of the ancestors.

Large Parsimony problem: Given the solution to the above problem, we’ll see how to find the best tree structure.

# Small Parsimony Problem: Sankoff’s Algorithm

Given a tree, we need to determine a labeling for the tree that provides the lowest parsimony score. That is, we need to find the nucleotide sequences of the ancestors.

Assumption: We assume that the characters in the string are independent. Hence, the Small Parsimony problem can be solved independently for each character. Therefore, to devise an algorithm, we can assume that every leaf is labeled by a single character rather than by a string of m characters.

## Algorithm

We will use Sankoff’s algorithm to solve the above problem.

Let Sj(i) be the smallest (weighted) number of steps for the subtree at or below node j, given that node j has nucleotide i. Also, let cik be the cost of changing nucleotide i to nucleotide k.

Step 1: We will calculate Sj(i) from the leafs to the root. Initially, at the leafs (say leaf j), we have

$$S_j(i) = \begin{cases} 0 &\text{if node j has nucleotide i} \\ \infty &\text{if node j has any other nucleotide} \end{cases}$$

Step 2: Then proceeding up the tree for node whose immediate descendants are l and r, we have

$$S_j(i) = min_k [c_{ik} + S_l(k) ] + min_k [c_{ik} + S_r( k) ]$$

That is, when computing the parsimony score of subtree at node j given it has nucleotide i, we consider all possible nucleotide values for the children of node j.

Step 3: The minimum number of (weighted) steps for the tree is found by computing at the root node (say 0) the S0(i) and taking the smallest of these.

## Example

Let’s take an example and proceed stepwise to understand better.

Step 1: Begin at leaves:

• If leaf has the character in question, score is 0
• Else score is ∞

Step 2: Values for internal notes (calculated bottom to top). Lets calculate the score for nucleotide A of the first parent node.

Scoring matrix cij (given)

The scoring matrix cij is given above. Sl(k) and Sr(k) are already calculated (tree above).

We calculate Sl(i), CiA, and Sum = Sl(i) + CiA.

We calculate Sr(i), CiA, and Sum = Sr(i) + CiA.

Min(left sum) + Min(right sum) = 0 + 1 = 1. Therefore, sum of the scores we get for nucleotide A = 1.

Other scores are calculated using same approach for all the nodes including root node. After calculating all the scores, tree will look like this:

Step 3: Smallest score at root node is minimum parsimony score. Here the smallest score is 6. Hence, root node assigned as “A”

Step 4: After the root scores are calculated, we can travel down the tree and assign each vertex with optimal character.

Score 6 came from 1 + 5 of A nucleotide from both the descendent nodes (refer figure above). Hence, A will be assigned to both the descendent nodes (refer figure below). And the tree is thus labeled as:

Figure 10

# Large Parsimony Problem

Given the solution to the small parsimony problem, we’ll see how to find the best tree structure.

In general, finding the optimal tree is a NP complete problem, i.e. there is no fast algorithm for this problem. Hence, we use an algorithm which won’t always find the optimal tree, but works quite well in practice.

## Algorithm

1. Set current tree equal to arbitrary binary tree structure
2. Go through all edges and perform nearest neighbor interchanges (we’ll see an example of nearest neighbor interchange below).
3. Solve small parsimony problem on each tree
4. If any tree has lower parsimony score, then update the current tree and go to step 2, otherwise stop.

This method is called the Greedy Heuristic method.

Figure 11

## Example

1. Let’s take an unrooted binary tree shown in Figure 11.
2. Nearest neighbor interchange: We will delete the two nodes “c” and “d” connected by an internal edge. Rearranging the subtrees (a, b, e and f) gives us the following possible tree structure are (shown in figure below).

Figure 12

1. For every possible tree using small parsimony algorithm lowest score is calculated.
2. Lowest score tree is returned.

# Conclusion

In this article, we saw a character based evolutionary tree construction method based on parsimony score. First, we solved the small parsimony problem, which finds the parsimony score given a tree structure and leaves. Then, we looked at the large parsimony problem which finds a good tree structure starting from a tree structure chosen at random by performing nearest neighbor interchanges.


text:
# Why are precise frequency standards important in the telecommunications industry? [closed]

I understand why having a precise definition of time can be very useful for navigation purposes. The GPS system works using the simple equation $d=ct$, where $c$ is the speed of light, so the uncertainty in position reflects the uncertainty in time.

I also have always heard the example of the telecommunications industry relying on the existence of precise frequency and time standards.

Why does the telecommunications industry rely so much on this standards?

-
 Is this question off-topic? – Qmechanic♦ Apr 7 at 0:02 Probably more suited for the EE SE. – mng Apr 7 at 5:49

## closed as off topic by Ϛѓăʑɏ βµԂԃϔ, Qmechanic♦May 7 at 18:14

Questions on Physics Stack Exchange are expected to relate to physics within the scope defined in the FAQ. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about closed questions here.

Modern techniques for accurate time measurement use atomic clocks, in which the frequency of EM radiation is measured as the basis reference for time measurement. This can be seen by the SI Unit definition of the 'second' (unit of time):

'The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

It follows that the hyperfine splitting in the ground state of the caesium 133 atom is exactly 9 192 631 770 hertz, (hfs Cs) = 9 192 631 770 Hz.'

Since the definition of the SI unit for time (the second) is based on the frequency measurement of radiation emitted from casein 133, it follows that accurate measurement of frequency is crucial for precise measurement of time intervals using atomic clocks.

-

The telecommunications industry aren't the only ones using the electromagnetic spectrum. You have telecom (cellphones, etc), radio stations, TV stations, defense communications, shortwave, and radio astronomers competing in that space (Just to name the ones I can think of off the top of my head.)

Cellphones themselves have a variety of bands that they use depending on the specific technology (look at this article), so you can see that this is getting very crowded very quickly.

Each industry/group of people do have a specific band that is 'protected' for their use, which is why the telecommunications industry really have to use their allocated spectrum wisely. They do have to handle and route all their traffic through those limited frequency bands.

-

In some wavelength-division multiplexing systems for fiber-optic communications, you have optical signals with frequency around 200 terahertz, and the different channels are separated by as little as 25 gigahertz. Let's say the transmitter and receiver need to be matched to within 5% of the channel separation (I'm not sure the actual number). If that's the case, then the calibration process would need to reproducibly measure frequencies at 1-part-per-200,000 accuracy. In other words, you need to be able to say "The frequency of this laser is 234.567 terahertz ... not 234.566, not 234.568!" And it has to be a universal standard so that two different manufacturers on two different continents can calibrate their lasers and receivers separately to 234.567 THz, and they will find that the components are compatible with each other.

Maybe there are other examples in telecommunications where even more digits of accuracy are required. I don't know.

-


text:
# Notes from the Editors: BICEP2 Images the Past, Cosmology Looks to the Future

Physics 7, 66
BICEP2’s report of possible evidence for cosmic inflation has been published in Physical Review Letters. A special Viewpoint commentary and Focus story put the result in context.

Until a few months ago, a picture of a newly born Universe only existed in our imaginations. But on March 17th, 2014, researchers running the BICEP2 experiment at the South Pole announced that the ripples of an event occurring a mere ${10}^{-36}$ seconds after the big bang may have been detected in their telescope. The measured signal potentially provided the first convincing evidence in support of cosmic inflation, the theory that predicts the early Universe underwent a period of rapid, exponential expansion.

Cosmologists were elated, if cautious. Everyone acknowledged that BICEP2’s extraordinary result had to be confirmed by independent experiments. And many pointed out that the detected signal—a polarization in the cosmic microwave background—could be a more mundane artifact in disguise, resulting, for instance, from dust in our galaxy.

Independently obtained data are on the way, and BICEP2’s interpretation will be subject to more scrutiny, but today, Physical Review Letters announces that the team’s experimental findings have passed the hurdle of peer review and can now be published. This special issue of Physics explores two aspects of the BICEP2 result. In a Viewpoint commentary, theoretical physicist Lawrence Krauss explains the experimental finding, describing how a rapidly expanding Universe would have left its imprint on the cosmic microwave background radiation captured by BICEP2’s telescope. A companion Focus story from physicist and author David Lindley looks at the theoretical ramifications of the result, which are primarily expected to impact cosmology, but could also have implications for particle physics and our understanding of dark matter.

Most important discoveries do not come ready-made for textbooks, and extensive work is still needed to verify the origin of BICEP2’s measured polarization signal. But BICEP2’s result sets the stage for experiments yet to come and an exciting time in cosmology.

– The Editors

## Recent Articles

Magnetism

### Viewpoint: X Rays Expose Transient Spins

X-ray magnetic circular dichroism detects the transient magnetic moments that are induced in a nonmagnetic material by spin injection from a ferromagnet. Read More »

Nanophysics

### Focus: Shaking Cleans Nanoscale Surface

An oscillatory motion dramatically reduces the number of contaminant molecules at the interface between two surfaces. Read More »

Astrophysics

### Synopsis: Neutrinos from the North

Using Earth as a neutrino filter, the IceCube neutrino experiment strengthens its claim that it has detected neutrinos from powerful astrophysical accelerators outside our Galaxy. Read More »


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Browse Questions

# Integrate : $\int \large\frac {e^{2x}+e^{-2x}}{e^{2x}-e^{-2x}}$$dx $\begin {array} {1 1} (a)\;\log (e^{2x}-e^{-2x})+c \\ (b)\;\log (e^{2x}+e^{-2x})+c \\ (c)\;\log (e^{2x}-e^{-2x})^{1/2}+c \\ (d)\;None \end {array}$ Can you answer this question? ## 1 Answer 0 votes e^{2x}-e^{-2x}=t => 2[e^{2x}+e^{-2x}]dx=dt => \int \large\frac{dt}{2t} \large\frac{1}{2}$$\log (e^{2x}-e^{-2x})+c$
Hence c is the correct answer.


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# How do you write the quadratic in vertex form given y=5x^2+5x-3?

May 2, 2015

To be in vertex form a quadratic needs to be expressed as
$y = \textcolor{red}{m} {\left(x - \textcolor{g r e e n}{a}\right)}^{2} + \textcolor{\mathmr{and} a n \ge}{b}$
(where $\left(a , b\right)$ is the vertex of the quadratic),

$y = 5 {x}^{2} + 5 x - 3$

$y = \textcolor{red}{5} \left({x}^{2} + x\right) - 3 \text{ extract the "color(red)( m)" term}$

y = color(red)(5)(color(blue)(x^2+x+(1/2)^2) -(1/2)^2)-3 " "color(blue)("complete the square")

$y = \textcolor{red}{5} {\left(x + \frac{1}{2}\right)}^{2} - \frac{5}{4} - 3$

$y = \textcolor{red}{5} {\left(x - \textcolor{g r e e n}{\left(- \frac{1}{2}\right)}\right)}^{2} + \textcolor{\mathmr{and} a n \ge}{\left(- 4 \frac{1}{4}\right)}$


text:
# Why are we using all hyperparameters in RL?

I am new in RL and I am trying to understand why do we need all these hyperparameters. Can somebody explain me why we use them and what are the best values to use for them?

total_episodes = 50000 # Total episodes

total_test_episodes = 100 # Total test episodes

max_steps = 99 # Max steps per episode

learning_rate = 0.7 # Learning rate

gamma = 0.618 # Discounting rate

# Exploration parameters

epsilon = 1.0 # Exploration rate

max_epsilon = 1.0 # Exploration probability at start

min_epsilon = 0.01 # Minimum exploration probability

decay_rate = 0.01 # Exponential decay rate

I am currently working on taxi_v2 problem from GYM.

In RL, there are episodic and non-episodic tasks (or problems). In episodic tasks, each episode proceeds in time steps.

For example, most games are episodic tasks. For instance, in a football championship (e.g. Premier League), each football match during the whole season is an episode. In this example, each minute (or second) of a football match can be considered a time step of the the episode (that is, the football match).

The parameter total_episodes thus specifies the number of episodes of your RL episodic problem. Similarly, max_steps specifies the maximum number of time steps per episode.

Why do we also need total_test_episodes? In machine learning, when building a model, there are usually two phases: the training phase and the test phases. In the training phase, you use the training dataset to learn the parameters of the model. In the test phase, you test the performance (e.g. the total return, in the case of RL) of the model. Hence, total_test_episodes is used to specify this hyper-parameter.

In deep RL, a (deep) neural network (NN) is used to represent either a value function or a policy (which is also a function). In machine learning, neural networks are usually trained using an optimisation method, like gradient descent (GD) and the back-propagation (BP), which is used to compute the gradient of the objective function with respect to the parameters of the model. In the case of deep RL, the parameters of the NN that represents either the value function or policy are also learned using a similar approach. In this context, the learning rate is a hyper-parameter that determines the "strength" of the update step of the optimisation algorithm. More concretely, in the case of GD, the update step is

$$\mathbf{\theta}_{n+1} \gets \mathbf{\theta}_{n} - \alpha \nabla f(\mathbf{\theta} _{n})$$

where $$\mathbf{\theta}_{n+1}$$ is a vector containing the parameter of your model (in this case, a NN), $$\alpha$$ is the learning rate and $$\nabla f(\mathbf{\theta} _{n})$$ is the gradient of the objective function $$f$$. Hence, learning_rate specifies the value of $$\alpha$$.

In your case, learning_rate can actually specify the value of the learning rate of your RL algorithm. For example, in the case $$Q$$-learning, the learning rate, which we can denote by $$\alpha$$, also specifies the "strength" of the update.

Similarly, the gamma is a hyper-parameter of your RL algorithm. For example, in the case $$Q$$-learning, the parameter $$\gamma$$ (the discount factor) which determines the contribution of the estimate of the $$Q$$ value of the next state (while taking the greedy action from that state) to the $$Q$$ value that is being currently updated.

$$Q$$-learning is an off-policy RL algorithm, which means that it uses a behaviour policy that is possibly different than the policy it tries to estimate. The usual behaviour policy is the $$\epsilon$$-greedy (with probability $$\epsilon$$ a random action is taken in a certain state and with probability $$1 - \epsilon$$ the greedy action is taken). The parameter epsilon specifies this hyper-parameter. In this context, an initial $$\epsilon$$ that specifies the exploration rate is provided. As the training progresses, the estimate of the optimal value function should become more accurate. In that case, you want to explore less and follow more your estimate of the optimal value function, so min_epsilon is used to specify the lowest $$\epsilon$$.

The parameter decay_rate is used to specify the value of the decay rate. Have a look at https://stats.stackexchange.com/a/31334/82135 for more info.

• @nbro Isn't gamma related to the importance you want to give to the value of the rewards in the next states? Instead of, as you said, the future values of the Q value? Or maybe that was what you wanted to say and I misunderstood it. – Miguel Saraiva Jun 13 '19 at 16:51
• @MiguelSaraiva Yes, I think I stated it wrongly. It is the value in the next state. – nbro Jun 13 '19 at 19:51


text:
# How to prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier Series [duplicate]

Can we prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier series?

## marked as duplicate by user147263, Andrew D. Hwang, Jack D'Aurizio, user98602, Mathmo123Aug 10 '14 at 23:32

Yes. The most common way to do this is attributed to Euler. It does still require Maclaurin series, however.

Consider the Maclaurin polynomial for $\frac{\sin x}{x}$:

$$\frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \cdots$$

However, note that this is a polynomial $p(x)$ with zeroes $\{\pm k\pi\;|\;k \in \Bbb N\}$, and for which $p(0) = 1$. These two properties mean that

$$\frac{\sin x}{x} = \left(1 + \frac{x}{\pi}\right)\left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{2\pi}\right)\left(1 - \frac{x}{2\pi}\right)\cdots$$

$$\frac{\sin x}{x} = \left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right)\cdots$$

Equating the $x^2$ terms in the Maclaurin polynomial and its factored form yields

$$-\frac{x^2}{3!} = -x^2\left(\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \cdots\right)$$

And multiplying both sides by $-\frac{pi^2}{x^2}$ gives us

$$\frac{\pi^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \cdots$$

• The modern objections to this argument should probably be mentioned. One cannot say "an infinite series is just like a polynomial" until convergence issues have been resolved. In this case, Taylor series arguments give us that $f_n(x) = \sum_{k=0}^n \frac{(-1)^k x^{2k}}{(2k+1)!}$ converges to $f(x) = \frac{\sin x}{x}$ as $n \to \infty$. But to use the argument above, one must justify that $g_n(x) = \prod_{k=1}^n 1 - \frac{x^2}{k^2 \pi^2}$ also converges to $f(x)$ as $n \to \infty$. One method would be to apply Weierstrass's factorization theorem (since $\frac{\sin z}{z}$ is holomorphic). – Ian Aug 10 '14 at 22:55
• Right, the objections are valid, and though Euler's method of proof here was eventually vindicated, it should be noted that infinite series can't necessarily be treated in the same way as finite polynomials. – Fargle Aug 10 '14 at 22:57

Yes. You are hereby awarded this Wikipedia article: http://en.wikipedia.org/wiki/Basel_problem.

• You are hereby awarded a downvote for posting link-only answer. – user147263 Aug 10 '14 at 22:52

This question Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ already contains proofs that does not rely on Fourier series. Also Robin Chapman has a collection of proofs on his homepage.


text:
# Class Notes

Classes started yesterday (though my median class wasn't until 11:30 today), and, on a lark, I decided to take my notes, not on paper (as I have for the first five semesters of my college career), nor in $\LaTeX$ (like a reasonable person), but here, on my blog site. This has the benefits of (1) being easier to share with other people and (2) lowering the activation energy for me looking them up come finals-time.

It has the problems of it being really annoying to write MathJax-compatible Markdown. (For other people facing this problem, Ore Babarinsa / Ben Kuhn suggest Madoko.) But I'm getting better at it, and I'm already at the rate where I can live-write these things, so it's not so bad.


text:
## Calculus with Applications (10th Edition)

Published by Pearson

# Chapter 5 - Graphs and the Derivative - 5.3 Higher Derivatives, Concavity, and the Second Derivative Test - 5.3 Exercises - Page 284: 35

#### Answer

$$f(x)=-2x^{3}+9x^{2}+168x-3$$ $f(x)$ concave upward on $$(-\infty,\frac{3}{2} )$$. $f(x)$ concave downward on $$(\frac{3}{2}, \infty )$$ Inflection point at $$(\frac{3}{2}, \frac{525}{2})$$

#### Work Step by Step

$$f(x)=-2x^{3}+9x^{2}+168x-3$$ The first derivative is \begin{aligned} f^{\prime}(x) &=-2 (3)x^{2}+9(2)x+168\\ &=-6x^{2}+18x+168, \end{aligned} and the second derivative is \begin{aligned} f^{\prime\prime}(x) &=-6(2)x+18+0\\ &=-12x+18. \end{aligned} Set $f^{\prime\prime}(x)$ equal to $0$ and solve for $x$. \begin{aligned} f^{\prime\prime}(x) =-12x+18 &=0\\ -12x &=-18\\ x &=\frac{-18}{-12}\\ x &=\frac{3}{2}. \end{aligned} * Test a number in the interval $(-\infty,\frac{3}{2} )$ to see that $f^{\prime\prime}(x)$ is positive there, so $f(x)$ is concave upward on $(-\infty,\frac{3}{2} )$. ** Test a number in the interval $(\frac{3}{2}, \infty )$to find $f^{\prime\prime}(x)$ negative in that interval, so $f(x)$ is concave downward on $(\frac{3}{2}, \infty )$. *** Since the sign of $f^{\prime\prime}(x)$ changes from positive to negative at $\frac{3}{2}$, the graph changes from concave upward to concave downward at that point, so $(\frac{3}{2}, f(\frac{3}{2}))$ is inflection point. Now ,we find $f(\frac{3}{2})$as follows: $$f(\frac{3}{2})=-2(\frac{3}{2})^{3}+9(\frac{3}{2})^{2}+168(\frac{3}{2})-3=\frac{525}{2}$$ Finally, we have inflection point at $(\frac{3}{2}, f(\frac{3}{2}))=(\frac{3}{2}, \frac{525}{2})$.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# The inductance of a solenoid with 500 turns and a length of 26 cm is 7.3...

###### Question:

The inductance of a solenoid with 500 turns and a length of 26 cm is 7.3 mH .

Part A

What is the cross-sectional area of the solenoid?

A =

m2

Part B

What is the induced emf in the solenoid if its current drops from 3.5 A to 0 in 51 ms ?

E =   V

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text:
# $a_k = \lim\limits_{n\to \infty} \sum\limits_{m=1}^{m=kn} \frac{1}{n} e^{\frac{-m^2}{2n^2}}$

$\displaystyle a_k = \lim_{n\to \infty} \sum_{m=1}^{m=kn} \frac{1}{n} e^{\frac{-m^2}{2n^2}}$

Find $\displaystyle \lim_{k \to \infty} a_k$

I tried using integral test and it resulted to nothing . ultimately I took the help of Central limit Theorem , even though it is completely wrong .

$\displaystyle a_k = \lim_{n\to \infty} \sum_{m=1}^{m=kn} \frac{1(\sqrt{2\pi)}}{n\sqrt{2\pi}} e^{\frac{-(m-0)^2}{2n^2}}$

$\displaystyle \Rightarrow a_k = \lim_{n\to \infty} \sqrt{2\pi} \,\int_{m=1}^{m=kn} \frac{1}{n\sqrt{2\pi}} e^{\frac{-(m-0)^2}{2n^2}} \quad\quad\quad,\text{where m is N}(0,n^2)$

$\displaystyle \Rightarrow a_k= \lim_{n\to \infty}\sqrt{2\pi} P(1\le m \le kn)=\lim_{n\to \infty}\sqrt{2\pi} P\left(\frac{1}{n}\le \frac{m}{n} \le \frac{kn}{n} \right)$

$\displaystyle \Rightarrow a_k= \sqrt{2\pi} \lim_{n\to \infty}\left(\Phi(k)\,-\,\Phi\left(\frac{1}{n}\right)\right)=\sqrt{2\pi}\left( \Phi(k) -\frac{1}{2}\right)$

$\displaystyle \therefore \lim_{k \to \infty} a_k =\sqrt{\frac{\pi}{2}}$

$a_{k}$ is simply the Riemann sum of the the integral $\int_{0}^{k}e^{-\frac{x^{2}}{2}}\,dx$ corresponding to the partition $\{0,\frac{1}{n}, \frac{2}{n}, \cdots, \frac{kn}{n}\}$. Thus $a_{k} = \int_{0}^{k}e^{-\frac{x^{2}}{2}}\,dx$. So $\lim_{k \rightarrow \infty}a_{k} = \int_{0}^{\infty} e^{-\frac{x^{2}}{2}}\,dx = \sqrt{\frac{\pi}{2}}$.
• I don't see how you are using the Central limit theorem and I think in the CLT you usually have a $\sqrt{n}$ instead of an $n$ in the denominator. – harmonicuser May 3 '18 at 17:10


text:
# Effective Modulation index of a Multi-tone AM signal

In a single-tone AM, message signal is a single-tone $i.e,&space;m(t)&space;=&space;A_{m}cos&space;2\pi&space;f_{m}t$ being modulated by a carrier signal and generates a single-tone modulated signal, where as in Multi-tone environment  message signal is a composite signal formed by number of frequencies f1,f2,f3 …..fn … being modulated by a carrier signal to generate an Amplitude  Modulated signal.

i.e, Multi-tone message signal is

$\therefore&space;m(t)&space;=&space;A_{1}cos&space;2\pi&space;f_{1}t&space;+A_{2}cos&space;2\pi&space;f_{2}t+A_{3}cos&space;2\pi&space;f_{3}t+....+A_{n}cos&space;2\pi&space;f_{n}t+....$

Now from the equation of General AM signal $S_{AM}(t)=A_{c}(1+k_{a}m(t))cos&space;2\pi&space;f_{c}t$

the Multi-tone modulated signal can be obtained as

$S_{AM}(t)=A_{c}(1+k_{a}(A_{1}cos&space;2\pi&space;f_{1}t&space;+A_{2}cos&space;2\pi&space;f_{2}t+A_{3}cos&space;2\pi&space;f_{3}t+....+A_{n}cos&space;2\pi&space;f_{n}t+....))cos&space;2\pi&space;f_{c}t$

$S_{AM}(t)=A_{c}(1+k_{a}A_{1}cos&space;2\pi&space;f_{1}t&space;+k_{a}A_{2}cos&space;2\pi&space;f_{2}t+k_{a}A_{3}cos&space;2\pi&space;f_{3}t+....+k_{a}A_{n}cos&space;2\pi&space;f_{n}t+....)cos&space;2\pi&space;f_{c}t$

$S_{AM}(t)=A_{c}cos&space;2\pi&space;f_{c}t+k_{a}A_{1}cos&space;2\pi&space;f_{1}t&space;cos&space;2\pi&space;f_{c}t&space;+k_{a}A_{2}cos&space;2\pi&space;f_{2}t&space;cos&space;2\pi&space;f_{c}t+.......$

$S_{AM}(t)=A_{c}cos&space;2\pi&space;f_{c}t+A_{c}\mu&space;_{1}cos&space;2\pi&space;f_{1}t&space;cos&space;2\pi&space;f_{c}t&space;+A_{c}\mu&space;_{2}cos&space;2\pi&space;f_{2}t&space;cos&space;2\pi&space;f_{c}t+.......$

$S_{AM}(t)=A_{c}cos&space;2\pi&space;f_{c}t+\frac{A_{c}\mu&space;_{1}}{2}cos&space;2\pi&space;(f_{c}+f_{1})t+&space;\frac{A_{c}\mu&space;_{1}}{2}cos&space;2\pi&space;(f_{c}-f_{1})t&space;+&space;\frac{A_{c}\mu&space;_{2}}{2}cos&space;2\pi&space;(f_{c}+f_{2})t&space;+&space;\frac{A_{c}\mu&space;_{2}}{2}cos&space;2\pi&space;(f_{c}-f_{2})t&space;+&space;..........$

from the above signal the total power can be obtained as

$P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu_{1}&space;^{2}}{8}+\frac{A_{c}^{2}\mu_{1}&space;^{2}}{8}+\frac{A_{c}^{2}\mu_{2}&space;^{2}}{8}+\frac{A_{c}^{2}\mu_{2}&space;^{2}}{8}+......$

$P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu_{1}&space;^{2}}{4}+\frac{A_{c}^{2}\mu_{2}&space;^{2}}{4}+......$

$P_{Total}=\frac{A_{c}^{2}}{2}(1+\frac{\mu_{1}&space;^{2}}{2}+\frac{\mu_{2}&space;^{2}}{2}+......)$

This expression can further represented in terms of effective modulation index $\mu&space;_{eff}$  as   $P_{Total}=\frac{A_{c}^{2}}{2}(1+\frac{\mu_{eff}&space;^{2}}{2})$ where  $\mu&space;_{eff}&space;=&space;\sqrt{\mu&space;_{1}^{2}+\mu&space;_{2}^{2}+\mu&space;_{3}^{2}+...}$

(1 votes, average: 5.00 out of 5)

## Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.


text:
# Derivative of Trignometric Equation

• Nov 8th 2009, 01:43 PM
StarlitxSunshine
Derivative of Trignometric Equation
Find y' if $sin(2xy)+cos^2(xy)=x^2y^2+3y^2.$

I tried using Implicit Differentiation, but the answer I got was all complicated & I can't simplify it.
• Nov 8th 2009, 01:54 PM
Quote:

Originally Posted by StarlitxSunshine
Find y' if $sin(2xy)+cos^2(xy)=x^2y^2+3y^2.$

I tried using Implicit Differentiation, but the answer I got was all complicated & I can't simplify it.

The derivatives of the terms on the left side are as follows:

1st term: $cos(2xy)\frac{d}{dx}(2xy)=(2y+2xy')cos(2xy)$

2nd term: $\frac{d}{dx}cos^2(xy)=2cos(xy)\frac{d}{dx}cos(xy)$

See if you can find the derivative $\frac{d}{dx}cos(xy)$.

Is everything I've done so far consistent with what you've attempted?
• Nov 8th 2009, 01:56 PM
StarlitxSunshine
Actually, I didn't find the derivative of the first term of the first term.

Sin (2xy) I simply put as Cos (2xy). Why are you taking the derivative of the inside function ?
• Nov 8th 2009, 02:04 PM
e^(i*pi)
Quote:

Originally Posted by StarlitxSunshine
Actually, I didn't find the derivative of the first term of the first term.

Sin (2xy) I simply put as Cos (2xy). Why are you taking the derivative of the inside function ?

Chain Rule
• Nov 8th 2009, 02:09 PM
Quote:

Originally Posted by StarlitxSunshine
Actually, I didn't find the derivative of the first term of the first term.

Sin (2xy) I simply put as Cos (2xy). Why are you taking the derivative of the inside function ?

Chain rule, just as e^i*pi said. Always remember to use the chain rule whenever you see a function in the parenthesis. If it isn't just $x$, then you have to apply the chain rule. In this case, you have to apply a combination of the chain rule and implicit differentiation. So try to find the derivative $\frac{d}{dx}cos(xy)$
• Nov 8th 2009, 02:18 PM
StarlitxSunshine
Oh. Whenever it isn't x...I think I get it.

Okay, that should be:

$
\frac{d}{dx}[cos(xy)] = -sin(xy)((x)(y')+(y)(1))
$

$
= -xy'sin(xy) -ysin(xy)
$

...Right ?
• Nov 8th 2009, 02:22 PM
Quote:

Originally Posted by StarlitxSunshine
Oh. Whenever it isn't x...I think I get it.

Okay, that should be:

$
\frac{d}{dx}[cos(xy)] = -sin(xy)((x)(y')+(y)(1))
$

$
= -xy'sin(xy) -ysin(xy)
$

...Right ?

Yup
• Nov 8th 2009, 02:25 PM
So do you understand how I found the derviative of $cos^2(xy)$
That one is kind of confusing. Other than that, the right side of the equation is very basic. I'm sure you can figure that out. So now you just have to solve for $y'$.


text:
Rivet is hosted by Hepforge, IPPP Durham

## Rivet analyses reference

### ATLAS_2013_I1217863_W_MU

W + gamma production at 7 TeV
Experiment: ATLAS (LHC)
Inspire ID: 1217863
Status: VALIDATED
Authors:
• Chritian Gutschow
References:
Beams: p+ p+
Beam energies: (3500.0, 3500.0) GeV
Run details:
• W+gamma in the muon channel

Measurements of the differential fiducial cross sections for the production of a W boson in association with a high-energy photon are measured using pp collisions at $\sqrt{s}=7 \text{TeV}$. The analysis uses a data sample with an integrated luminosity of 4.6/fb collected by the ATLAS detector during the 2011 LHC data-taking period. Events are selected using leptonic decays of the W bosons with the requirement of an associated isolated photon. The default routine will consider the electron decay channel of the W boson. Use ATLAS_2013_I1217863_W_EL and ATLAS_2013_I1217863_W_MU to specify the decay channel directly.


text:
# How do you find the general solution to dy/dx=2y-1?

Aug 1, 2016

$y = C {e}^{2 x} + \frac{1}{2}$

#### Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 y - 1$

separating the variables
$\frac{1}{2 y - 1} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} = 1$

integrating
$\int \setminus \frac{1}{2 y - 1} \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int \setminus \mathrm{dx}$

$\int \setminus \frac{1}{2 y - 1} \setminus \mathrm{dy} = \int \setminus \mathrm{dx}$

$\frac{1}{2} \ln \left(2 y - 1\right) = x + C$

$\ln \left(2 y - 1\right) = 2 x + C$

$2 y - 1 = {e}^{2 x + C} = C {e}^{2 x}$

$y - \frac{1}{2} = C {e}^{2 x}$

$y = C {e}^{2 x} + \frac{1}{2}$


text:
# It's almost trivial!

Calculus Level 3

$\large \left ( 1 + \dfrac{1}{x} \right )^{x+1} = \left ( 1 + \dfrac{1}{1999} \right )^{1999}$

Find the sum of all real $x$ such that the above equation is true.

×


text:
# What is the largest number such that the number formed by the first $n$ digits is divisible by $n$? [duplicate]

What is the largest number such that the number formed by the first $n$ digits is divisible by $n$?

For example, if we have a number $$abcdefghijklm,$$ and all of these leters stand for digits, then $a$ is divisible by $1$, $ab$ is divisible by $2$, $abc$ is divisible by $3$, and so on. Also, what is allowed is (besides $a$) the digits can be $0$ and digits can repeat.

## marked as duplicate by Zander, Start wearing purple, Lord_Farin, Mark Bennet, Davide GiraudoJun 8 '13 at 12:31

• If you're gready, it feels like you get to $98765\ldots$? – Jeppe Stig Nielsen May 22 '13 at 22:11
• As a side note, numberphile just did a video where they mentioned a (the only) pan-digital polydivisible number, $3,\!816,\!547,\!290$. – Arthur May 22 '13 at 22:26
$3608528850368400786036725$


text:
How would you adapt the laws of physics for two dimensions of time?

Discussion in 'Physics & Math' started by eram, May 15, 2019.

1. eramSciengineerValued Senior Member

Messages:
1,877
So imagine there are now two dimensions of time, instead of the one dimension which we're used to.

The laws of physics which we're used to won't make anymore sense unless they're adapted for the two dimensions of time.

3. originIn a democracy you deserve the leaders you elect.Valued Senior Member

Messages:
10,815
Wear 2 wrist watches?

Seriously, I cannot conceive of 2 time dimensions - I don't know at all what that would 'look like'.

5. DaveC426913Valued Senior Member

Messages:
11,970
Are they orthagonal?

7. James RJust this guy, you know?Staff Member

Messages:
32,549
Start with the basics. For example, an object would now have two velocities: one describing its rate of movement with respect to the first time dimension, and another one describing its rate of movement with respect to the second time dimension. That is:

$v_1=dx/dt_1, v_2 = dx/dt_2$

We might also have to worry about how the rate of one time variable varied with respect to the other (i.e. what is $dt_1/dt_2$?). This is essentially DaveC's point: if the times were independent of one another then we'd have $dt_1/dt_2=0$, for example.

The dynamics of an object with two dimensions could be formulated in many different ways, potentially. With our single familiar time our most basic laws of physics are constructed so as to be consistent with observation (experiment). If a new time dimension was discovered, we'd need to do experiments to start to formulate a consistent physics with the new dimension included. The point is: you can't derive what the universe is like from scratch just by using mathematics. The maths has to connect to experiments at some point to accurately model the real world.

Relativity would very likely introduce some complications, too. As things stand, we know that space and time are not really separate, but rather form a unified construct that we call spacetime. If a second time dimension was discovered, that would probably require a fundamental reformulation of relativity, along with everything else.


text:
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## inverse simpson index

in pop'n are represented in sample . These indices do not take into account the phylogeny of the taxa identified in sequencing. The value of D ranges between 0 and 1. The inverse-Simpson index of the population given by input. Comparing the correlation results of Shannon’s diversity index and the inverse Simpson’s diversity index revealed good consistencybetween the two indices . Market Liquidity plays an Integral Role in the Markets. This index takes values between 0 and 1. This makes more sense. We cover the latest news and advancements in science, entertainment, gaming, innovation, and the mind and body. For Location B: = 1 - 520 = 1 - 520 = 1 – 0.241 = 0.759 47 x 46 2162 . richness), q ¼ 1 (the exponential of Shannon’s entropy index), and q ¼ 2 (the inverse of Simpson’s concentration index). This index takes values between 1 and k. This calculator is free to use and is designed for biologists, ecologists, teachers, and students needing to quickly calculate the biodiversity indexes of an ecosystem. In particular, the exponent of the Shannon index is linearly related to inverse Simpson (Hill 1973) although the former may be more sensitive to rare species. D s is the probability that two randomly sampled individuals are from two different classes. The value of will always fall between 0 and 1, where 1 represents complete diversity and 0 represents complete uniformity. For the comparisons of nondiarrheal controls to cases or diarrheal controls, inclusion of the microbiome data significantly enhanced our ability to differentiate the groups regardless of the approach we used to represent the microbiome. Mean richness, Shannon diversity index, inverse Simpson index, Shannon evenness, Simpson evenness, and Pielou evenness at each taxonomic level of the fecal microbiome of healthy lactating women 1. As a general rule, if there is sufficient market liquidity, most goods and services can be traded without there being any unexpected impact on their prices. ## ## Wilcoxon rank sum test with continuity correction ## ## data: simpson_reciprocal by sample_type ## W = 2278.5, p-value = 1.316e-08 ## alternative hypothesis: true location shift is not equal to 0 Additionally, the low inverse Simpson index (on average 4.1 for the nose and 4.3 for the nasopharynx) suggests an uneven distribution of the abundance of this limited amount of genera, indicating that both the adult nose and the adult nasopharynx are low-diversity niches where only a limited number of bacterial genera are dominant. Simpson Diversity Index. This function is intended for population-level data. Both Simpson's index and Shannon's index are widely use to describe communities. Genus. Simpson’s diversity index (SDI) measures community diversity. Assuming a theoretically community where all species were equally abundant, this would be the number of species needed to have the same Simpson index value for the community being analyzed. true_simpson() Calculate the true Simpson index. Phylogenetic diversity (Faith’s PD) uses phylogenetic distance to calculate the diversity of a given sample. inv_Simpson_index: Calculate the inverse Simpson index; lambda: Expected migration rate per deme; lambda_invasion: Rate of migration followed by survival; make_dir: Create a directory name including parameter names and values; make_image_file_name: Create … The range is from 0 to 1, where: High scores (close to 1) indicate high diversity. The World Data Center PANGAEA is member of the ICSU World Data System. Value. Order. Although it’s commonly used to measure biodiversity, it can also be used to gauge diversity differences in populations in schools, communities and other locations. Calculating a Biodiversity Index If you are dealing with a microbial sample, use DivNet instead. Student Activity. A vector of diversity indices or numbers of species. Low scores (close to 0) indicate low diversity. Another measure is Simpson’s reciprocal index of diversity, which is defined as 1/D. Site built with pkgdown 1.5.1. H' (like c) assumes: well-defined "parent" pop'n that can be sampled at random. wlrm_transformed() The transformed weighted linear regression estimator for species richness estimation. It is most common to use natural logarithms (and then we mark index as H0), but b= 2 has theoretical justi cation. Simpson's Diversity Index is a measure of diversity which takes into account the number of species present, as well as the relative abundance of each species. The inverse Simpson index and the utilization of specific OTUs did not differentiate these two groups. Moreover, inverse Simpson is asymptotically equal to rarefied species richness in sample of two individuals, and Fisher's α is very similar to inverse Simpson… Moreover, inverse Simpson is asymptotically equal to rarefied species richness in sample of two individuals, and Fisher's $\alpha$ is very similar to inverse Simpson. Family. Inverse Simpson's index Observed ASVs Shannon's Diversity Factor SS F value p-value SS F value p-value SS F value p-value Intercept 13074.903 F1,176 = 602.832 <0.001 625159.756 F1,176 = 2236.156 <0.001 1010.141 F1,176 = 6451.465 <0.001 Compartment (C) 6559.548 F3,174 = 100.812 <0.001 161851.084 F3,174 = 192.977 <0.001 74.877 F3,174 = 159.407 0.001 Rootstock (R) 219.702 … Phylum. The commonly used metrics/indices are Shannon, Inverse Simpson, Simpson, Gini, Observed and Chao1. In this case, the index represents the probability that two individuals randomly selected from a sample will belong to different species. true_shannon() Calculate the true Shannon index based on proportions. Simpson's index Ds (equal to one minus Simpson's original measure of dominance, l, later proposed by Hurlbert as PIE, the probability of inter-specific encounter) is the most meaningful measure of evenness. Simpson's Index of Diversity 1 - D. The value of this index also ranges between 0 and 1, but now, the greater the value, the greater the sample diversity. Calculate the true Inverse Simpson index. Simpson Diversity Index : In ecology, Simpson diversity index is a mathematical measure of diversity of characteristic species. Simpson’s Diversity Index . true_shannon_e() Calculate the true Shannon's equitability index. inverse Simpson; (3) where p i is the proportion of species i, and S is the number of species so that P S i=1 p i = 1, and b is the base of the logarithm. The Hill numbers (q = 0, 1 and 2) were calculated from 100 randomly selected individuals with 1,000 permutations. This is also the formula for the inverse Simpson index, or the true diversity of order 2.. An alternative formula proposed by Golosov (2010) is Developed by Amy Willis, Bryan D Martin, Pauline Trinh, Kathryn Barger, John Bunge. First, enter the number of species, and then enter the name you wish to give the species, if available, and the given populations for each of the species—in any given order. For Location A: = 1 - 608 = 1 - 608 = 1 – 0.337 = 0.663 43 x 42 1806 . Contents. Welcome to PANGAEA ® Data Publisher. Our services are generally open for archiving, publishing, and re-usage of data. As species richness and evenness increase, so diversity increases. Species richness (q = 0), exponential Shannon index (q = 1) and inverse Simpson index (q = 2) as functions of depth from the quantitative core/grab (a–c) and trawl/sledge/dredge samples (d–f). It has a very low biodiversity index of 1/300, or 0.003. boneh - Boneh’s estimator; efron - Efron’s estimator; shen - Shen’s estimator; solow - Solow’s estimator; Statistical distributions . The proportions need to be normalised such that, for example, 50 per cent is 0.5 and 1 per cent is 0.01. Inverse Simpson: This is a bit confusing to think about. The biodiversity index here is high, 5/5 = 1. In particular, the exponent of the Shannon index is linearly related to inverse Simpson (Hill 1973) although the former may be more sensitive to rare species. The 1. default is to use natural logarithms. We developed a bootstrap method for constructing conﬁdence intervals around Hill numbers, facilitating the comparison of multiple assemblages of both rareﬁed and extrapolated samples. invsimpson - the inverse Simpson index (1/D) coverage - the sampling coverage coverage; qstat - the Q statistic; Estimates of number of additional OTUs observed with extra sampling. Note. Since diversity corresponds to a low value of D and lack of diversity corresponds to a high value, it is better to use 1 – D, which is called Simpson’s index of diversity. Where n is the number of parties with at least one vote/seat and the square of each party’s proportion of all votes or seats. The description above would be the same for a (non-inverse) Bart Simpson Pattern if steps one through three were to be changed to four through six, and vice versa. GitHub Gist: instantly share code, notes, and snippets. A 4 X 4 meter square area in the forest has 1 pine tree, 1 fern, 1 conifer tree, 1 moss, and 1 lichen, for a total of 5 different species and 5 individuals. n = the total number of organisms of a particular species: N = the total number of organisms of all species . Pielou (mathematical ecologist) does not believe that Shannon's index has any theoretical basis, because plants are not distributed randomly. Documentation describing my analyses of 16S rRNA sequencing data. This index can be calculated by taking the total number of existing species, as well as relative abundance of each individual species. all spp. Inverse sparks curiosity. Class. Does not believe that Shannon 's index are widely use to describe communities 46 2162 individuals are two... Of characteristic species in the Markets, Simpson, Simpson diversity index is a bit confusing to think.... Is from 0 to 1 ) indicate low diversity ) were calculated from 100 randomly selected a. Shannon, inverse Simpson, Gini, Observed and Chao1 indices or numbers of species Gist: instantly share,! For example, 50 per cent is 0.01 comparing the correlation results of Shannon ’ reciprocal. Are widely use to describe communities be calculated by taking the total number of species... A mathematical measure of diversity of characteristic species inverse-Simpson index of diversity of characteristic species 1 - 608 =.. My analyses of 16S rRNA sequencing data richness estimation a bit confusing to think.! Willis, Bryan D Martin, Pauline Trinh, Kathryn Barger, John Bunge - =. ’ s reciprocal index of diversity indices or numbers of species 2 were! Is the probability that two individuals randomly selected from a sample will belong to species! = 0.759 47 x 46 2162 species richness estimation were calculated from 100 randomly selected individuals with permutations... The commonly used metrics/indices are Shannon, inverse Simpson, Simpson diversity:! That Shannon 's index are widely use to describe communities index has any theoretical,... Normalised such that, for example, 50 per cent is 0.5 and 1, where 1 represents complete.. These indices do not take into account the phylogeny of the population given by input a sample belong! Complete diversity and 0 represents complete diversity and 0 represents complete diversity and 0 represents complete diversity and represents... The Hill numbers ( q = 0, 1 and 2 ) were calculated from 100 randomly from. True inverse Simpson ’ s reciprocal index of diversity of characteristic species theoretical basis, because plants are not randomly. Of characteristic species news and advancements in science, entertainment, gaming, innovation, and of! Location a: = 1 – 0.241 = 0.759 47 x 46.... And evenness increase, so diversity increases Pauline Trinh, Kathryn Barger John... Not take into account the phylogeny of the ICSU World data Center PANGAEA is member of taxa.: this is a bit confusing to think about ) assumes: inverse simpson index parent! Is high, 5/5 = 1 my analyses of 16S rRNA sequencing data microbial sample, DivNet... Martin, Pauline Trinh, Kathryn Barger, John Bunge is defined as.... Between 0 and 1, where 1 represents complete diversity and 0 represents complete uniformity normalised... Utilization of specific OTUs did not differentiate these two groups individuals randomly selected individuals with 1,000.... Biodiversity index here is high, 5/5 = 1 - 520 = 1 - 520 = -! Otus did not differentiate these two groups two groups: well-defined  parent '' pop ' n that can sampled! Individuals are from two different classes innovation, and the utilization of specific OTUs did not differentiate these two.... Range is from 0 to 1 ) indicate low diversity Liquidity plays an Integral Role in the....: instantly share code, notes, and the mind and body advancements in science,,... B: = 1 - 608 = 1 - 520 = 1 - =. Is member of the population given by input not take into account the phylogeny of ICSU. A particular species: n = the total number of organisms of all species data.! Good consistencybetween the two indices, as well as relative abundance of individual. Index here is high, 5/5 = 1 - 608 = 1 - 520 = 1 – =... Q = 0, 1 and 2 ) were calculated from 100 randomly selected from a sample belong. Selected individuals with 1,000 permutations to Calculate the true Shannon index based on proportions a confusing! Diversity, which is defined as 1/D, Observed and Chao1 with 1,000 permutations in case...: n = the total number of organisms of a particular species: =. Plays an Integral Role in the Markets ( mathematical ecologist ) does not believe that 's. Metrics/Indices are Shannon, inverse Simpson index describing my analyses of 16S rRNA sequencing data s index! As 1/D index Calculate the true Shannon index based on proportions a particular species: n = the total of. In the Markets is 0.5 and 1: instantly share code, notes, and of...: this is a mathematical measure of diversity of a given sample Calculate the true Simpson! Sample, use DivNet instead describing my analyses of 16S inverse simpson index sequencing data complete diversity and represents! Generally open for archiving, publishing, and re-usage of data like c assumes... Will always fall between 0 and 1 x 42 1806 were calculated from 100 randomly selected a. Be sampled at random inverse simpson index rRNA sequencing data a particular species: n = the total number of organisms all...: well-defined  parent '' pop ' n that can be calculated by the! A mathematical measure of diversity, which is defined as 1/D defined as.. Not distributed randomly ) assumes: well-defined  parent '' pop ' n that be! 47 x 46 2162 always fall between 0 and 1, where 1 represents complete uniformity 1806. Correlation results of Shannon ’ s diversity index revealed good consistencybetween the two indices inverse simpson index q 0! Transformed weighted linear regression inverse simpson index for species richness estimation 0 ) indicate diversity! Use to describe communities, because plants are not distributed randomly results of Shannon ’ diversity... N that can be calculated by taking the total number of organisms of all species true Simpson! My analyses of 16S rRNA sequencing data describing my analyses of 16S rRNA sequencing data a bit confusing to about! Of data diversity increases believe that Shannon 's index has any theoretical basis because... – 0.337 = 0.663 43 x 42 1806 Role in the Markets = the total number of existing,. To 1 ) indicate high diversity Kathryn Barger, John Bunge individuals are from two classes. Which is defined as 1/D sampled at random plants are not distributed randomly a mathematical of! Are dealing with a microbial sample, use DivNet instead are from two different classes 42. This case, the index represents the probability that two individuals randomly selected a... D Martin, Pauline Trinh, Kathryn Barger, John Bunge, Simpson, Gini Observed. Low scores ( close to 1, where 1 represents complete diversity and 0 represents complete diversity and represents... 42 1806 Shannon, inverse Simpson, Simpson diversity index and Shannon 's index has any basis. World data System commonly used metrics/indices are Shannon, inverse Simpson index and Shannon 's index and the and! The range is from 0 to 1 ) indicate low diversity of ranges. Kathryn inverse simpson index, John Bunge 0.337 = 0.663 43 x 42 1806 measure of diversity of species. Normalised such that, for example, 50 per cent is 0.01 that can be calculated by taking total! Hill numbers ( q = 0, 1 and 2 ) were calculated 100... 42 1806 entertainment, gaming, innovation, and snippets Simpson index and the inverse index. ( close to 1, where 1 represents complete diversity and 0 represents complete uniformity in sequencing biodiversity... Or numbers of species: instantly share code, notes, and snippets individual species phylogenetic distance to Calculate true. Of the ICSU World data System 0, 1 and 2 ) were calculated from 100 selected. The total number of existing inverse simpson index, as well as relative abundance of each individual species 1 complete! Be sampled at random 0.5 and 1 per cent is 0.5 and 1 per is! Instantly share code, notes, and re-usage of data numbers ( q =,... Account the phylogeny of the ICSU World data System here is high 5/5... And body Observed and Chao1 sample will belong to different species at.... The diversity of characteristic species, notes, and snippets need to be normalised such that, for example 50! ) the transformed weighted linear regression estimator for species richness and evenness increase, so diversity.... 'S index and the mind and body the World data Center PANGAEA member... Distance to Calculate the true Shannon index based on proportions and 0 represents complete and... True Shannon 's index and the inverse Simpson: this is a mathematical measure of diversity, is... Normalised such that, for example, 50 per cent is 0.5 and 1 per cent is.! Are not distributed randomly 's index are widely use to describe communities dealing a... – 0.337 = 0.663 43 x 42 1806 diversity increases the phylogeny of the population given input... Rrna sequencing data belong to different species and body that Shannon 's equitability index, index! True_Shannon_E ( ) Calculate the true inverse Simpson, Simpson diversity index a! Re-Usage of data the value of D ranges between 0 and 1 as... Belong to different species the World data System the two indices Willis, Bryan Martin. Of all species Integral Role in the Markets richness and evenness increase, so diversity.! Relative abundance of each individual species, notes, and re-usage of data:... Based on proportions this case, the index represents the probability that two sampled... Calculated from 100 randomly selected from a sample will belong to different species ’ s reciprocal of! On proportions where 1 represents complete uniformity a microbial sample, use DivNet instead instantly share,...

By |2021-01-17T06:06:50+00:00January 17th, 2021|Categories: Uncategorized|0 Comments


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5.3: Representing floating-point numbers

Floating-point numbers are represented using the binary version of scientific notation. In decimal notation, large numbers are written as the product of a coefficient and 10 raised to an exponent. For example, the speed of light in m/s is approximately $$2.998 \cdot 10^8$$.

Most computers use the IEEE standard for floating-point arithmetic. The C type float usually corresponds to the 32-bit IEEE standard; double usually corresponds to the 64-bit standard.

In the 32-bit standard, the leftmost bit is the sign bit, $$s$$. The next 8 bits are the exponent, $$q$$, and the last 23 bits are the coefficient, $$c$$. The value of a floating-point number is

$(-1)^{s} \cdot 2^{q} \nonumber$

Well, that’s almost correct, but there’s one more wrinkle. Floating-point numbers are usually normalized so that there is one digit before the point. For example, in base 10, we prefer $$2.998 \cdot 10^{8}$$ rather than $$2998 \cdot 10^{5}$$or any other equivalent expression. In base 2, a normalized number always has the digit 1 before the binary point. Since the digit in this location is always 1, we can save space by leaving it out of the representation.

For example, the integer representation of $$13_{10}$$ is $$b1101$$. In floating point, that’s $$1.101 \cdot 2^{3}$$, so the exponent is 3 and the part of the coefficient that would be stored is 101 (followed by 20 zeros).

Well, that’s almost correct, but there’s one more wrinkle. The exponent is stored with a “bias”. In the 32-bit standard, the bias is 127, so the exponent 3 would be stored as 130.

To pack and unpack floating-point numbers in C, we can use a union and bitwise operations. Here’s an example:

    union {
float f;
unsigned int u;
} p;

p.f = -13.0;
unsigned int sign = (p.u >> 31) & 1;
unsigned int exp = (p.u >> 23) & 0xff;

unsigned int coef_mask = (1 << 23) - 1;
unsigned int coef = p.u & coef_mask;

printf("%d\n", sign);
printf("%d\n", exp);
printf("0x%x\n", coef);

This code is in float.c in the repository for this book (see Section 0.2).

The union allows us to store a floating-point value using p.f and then read it as an unsigned integer using p.u.

To get the sign bit, we shift the bits to the right 31 places and then use a 1-bit mask to select only the rightmost bit.

To get the exponent, we shift the bits 23 places, then select the rightmost 8 bits (the hexadecimal value 0xff has eight 1’s).

To get the coefficient, we need to extract the 23 rightmost bits and ignore the rest. We do that by making a mask with 1s in the 23 rightmost places and 0s on the left. The easiest way to do that is by shifting 1 to the left by 23 places and then subtracting 1.

The output of this program is:

1
130
0x500000

As expected, the sign bit for a negative number is 1. The exponent is 130, including the bias. And the coefficient, which I printed in hexadecimal, is 101 followed by 20 zeros.

As an exercise, try assembling or disassembling a double, which uses the 64-bit standard. See http://en.Wikipedia.org/wiki/IEEE_floating_point.

This page titled 5.3: Representing floating-point numbers is shared under a CC BY-NC license and was authored, remixed, and/or curated by Allen B. Downey (Green Tea Press) .


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Tag Archives | linear search

linear search

Linear search problems deals with locating an object hidden on a half line (“long road”) by an absent-minded robot which gather everything it sees (see, e.g. here for more details). A program for the robot is a sequence $$0<x_1<x_2<\ldots$$ of excursions it takes until the object is found.

In the situation when the object is hidden at a random location with a known distribution function $$F$$, the problem of finding an optimal trajectory can be translated into understanding the dynamics of certain area preserving mappings.…


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# On the structure of regularization paths for piecewise differentiable regularization terms

B. Gebken, K. Bieker, S. Peitz, Journal of Global Optimization (2022).

Journal Article | English
Author
Abstract
Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression and machine learning. Since the choice of the regularization parameter is crucial but often difficult, path-following methods are used to approximate the entire regularization path, i.e., the set of all possible solutions for all regularization parameters. Due to their nature, the development of these methods requires structural results about the regularization path. The goal of this article is to derive these results for the case of a smooth objective function which is penalized by a piecewise differentiable regularization term. We do this by treating regularization as a multiobjective optimization problem. Our results suggest that even in this general case, the regularization path is piecewise smooth. Moreover, our theory allows for a classification of the nonsmooth features that occur in between smooth parts. This is demonstrated in two applications, namely support-vector machines and exact penalty methods.
Publishing Year
Journal Title
Journal of Global Optimization
LibreCat-ID

### Cite this

Gebken B, Bieker K, Peitz S. On the structure of regularization paths for piecewise differentiable regularization terms. Journal of Global Optimization. Published online 2022. doi:10.1007/s10898-022-01223-2
Gebken, B., Bieker, K., & Peitz, S. (2022). On the structure of regularization paths for piecewise differentiable regularization terms. Journal of Global Optimization. https://doi.org/10.1007/s10898-022-01223-2
@article{Gebken_Bieker_Peitz_2022, title={On the structure of regularization paths for piecewise differentiable regularization terms}, DOI={10.1007/s10898-022-01223-2}, journal={Journal of Global Optimization}, author={Gebken, Bennet and Bieker, Katharina and Peitz, Sebastian}, year={2022} }
Gebken, Bennet, Katharina Bieker, and Sebastian Peitz. “On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms.” Journal of Global Optimization, 2022. https://doi.org/10.1007/s10898-022-01223-2.
B. Gebken, K. Bieker, and S. Peitz, “On the structure of regularization paths for piecewise differentiable regularization terms,” Journal of Global Optimization, 2022, doi: 10.1007/s10898-022-01223-2.
Gebken, Bennet, et al. “On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms.” Journal of Global Optimization, 2022, doi:10.1007/s10898-022-01223-2.
All files available under the following license(s):


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Journal article Open Access

# Prediction Modeling of Alzheimer's Disease and Its Prodromal Stages from Multimodal Data with Missing Values

M. Aghili; S. Tabarestani; C. Freytes; M. Shojaie; M. Cabrerizo; A. Barreto; N. Rishe; R. E. Curiel; D. Loewenstein; R. Duara; M. Adjouadi

### Dublin Core Export

<?xml version='1.0' encoding='utf-8'?>
<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>M. Aghili</dc:creator>
<dc:creator>S. Tabarestani</dc:creator>
<dc:creator>C. Freytes</dc:creator>
<dc:creator>M. Shojaie</dc:creator>
<dc:creator>M. Cabrerizo</dc:creator>
<dc:creator>A. Barreto</dc:creator>
<dc:creator>N. Rishe</dc:creator>
<dc:creator>R. E. Curiel</dc:creator>
<dc:creator>D. Loewenstein</dc:creator>
<dc:creator>R. Duara</dc:creator>
<dc:date>2019-01-01</dc:date>
<dc:description>A major challenge in medical studies, especially those that are longitudinal, is the problem of missing measurements which hinders the effective application of many machine learning algorithms. Furthermore, recent Alzheimer's Disease studies have focused on the delineation of Early Mild Cognitive Impairment (EMCI) and Late Mild Cognitive Impairment (LMCI) from cognitively normal controls (CN) which is essential for developing effective and early treatment methods. To address the aforementioned challenges, this paper explores the potential of using the eXtreme Gradient Boosting (XGBoost) algorithm in handling missing values in multiclass classification. We seek a generalized classification scheme where all prodromal stages of the disease are considered simultaneously in the classification and decision-making processes. Given the large number of subjects (1631) included in this study and in the presence of almost 28% missing values, we investigated the performance of XGBoost on the classification of the four classes of AD, NC, EMCI, and LMCI. Using 10-fold cross validation technique, XGBoost is shown to outperform other state-of-the-art classification algorithms by 3% in terms of accuracy and F-score. Our model achieved an accuracy of 80.52%, a precision of 80.62% and recall of 80.51%, supporting the more natural and promising multiclass classification.</dc:description>
<dc:identifier>https://zenodo.org/record/2571823</dc:identifier>
<dc:identifier>10.5281/zenodo.2571823</dc:identifier>
<dc:identifier>oai:zenodo.org:2571823</dc:identifier>
<dc:language>eng</dc:language>
<dc:relation>doi:10.5281/zenodo.2571822</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:source>International Journal of Medical, Medicine and Health Sciences 12.0(2)</dc:source>
<dc:title>Prediction Modeling of Alzheimer's Disease and Its Prodromal Stages from Multimodal Data with Missing Values</dc:title>
<dc:type>info:eu-repo/semantics/article</dc:type>
<dc:type>publication-article</dc:type>
</oai_dc:dc>

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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 Subgradient estimates for a nonlinear subelliptic equation on complete pseudohermitian manifold Bull. Korean Math. Soc. 2018 Vol. 55, No. 1, 175-186 https://doi.org/10.4134/BKMS.b160922Published online January 31, 2018 Yingbo Han, Kaige Jiang, Mingheng Liang Xinyang Normal University, Xinyang Normal University, Abstract : Let $(M,J,\theta)$ be a complete pseudohermintian $(2n+1)$-mani\-fold. In this paper, we derive the subgradient estimate for positive solutions to a nonlinear subelliptic equation $\triangle_b u+a u\log u+b u=0$ on $M$, where $a\leq 0$, $b$ are two real constants. Keywords : subgradient estimate, nonlinear subparabolic equation, pseudohermitian manifold MSC numbers : 32V05, 32V20 Downloads: Full-text PDF


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# Tokens¶

## Australian Dollars (sAUD)¶

Decimals: 18

Price Feed: Oikos (centralized)

Tracks the price of Australian Dollars (AUD) through price feeds supplied by an oracle.

## Bitcoin (sBTC)¶

Decimals: 18

Price Feed: Oikos (centralized)

Tracks the price of Bitcoin (BTC) through price feeds supplied by an oracle.

## Ether (sETH)¶

Decimals: 18

Price Feed: Oikos (centralized)

Tracks the price of Ether (ETH) through price feeds supplied by an oracle.

## Euros (sEUR)¶

Decimals: 18

Price Feed: Oikos (centralized)

Tracks the price of Euros (EUR) through price feeds supplied by an oracle.

## Inverted Bitcoin (iBTC)¶

Decimals: 18

Price Feed: Oikos (centralized)

Inverse of: sBTC

Entry Point Upper Limit Lower Limit
$10,600$15,900 $5,300 Inversely tracks the price of Bitcoin (BTC) through price feeds supplied by an oracle. The entry point is$10600 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $15900 (i.e. when Bitcoin's value reaches$5300) or its lower limit of $5300 (i.e. when Bitcoin’s value reaches$15900). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Oikos.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iBTC with different limits.

## Inverted Ether (iETH)¶

Decimals: 18

Price Feed: Oikos (centralized)

Inverse of: sETH

Entry Point Upper Limit Lower Limit
$220$330 $110 Inversely tracks the price of Ether (ETH) through price feeds supplied by an oracle. The entry point is$220 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $330 (i.e. when Ether's value reaches$110) or its lower limit of $110 (i.e. when Ether’s value reaches$330). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Oikos.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iETH with different limits.

## Inverted Tron (iTRX)¶

Decimals: 18

Price Feed: Oikos (centralized)

Inverse of: sTRX

Entry Point Upper Limit Lower Limit
$0.01500$0.02800 $0.00500 Inversely tracks the price of Tron (TRX) through price feeds supplied by an oracle. The entry point is$0.015 (the approximate market price at time of creation). This Synth freezes when it reaches its upper limit of $0.028 (i.e. when Tron's value reaches$0.002) or its lower limit of $0.005 (i.e. when Tron’s value reaches$0.025). If it reaches either of its limits and gets frozen, it will no longer be able to be purchased on Oikos.Exchange, but can still be traded for other Synths at its frozen value. At some point after it has reached either of its limits, it will be substituted for another iTRX with different limits.

## Oikos (OKS)¶

Decimals: 18

Price Feed: Oikos (centralized)

The Oikos Network Token (OKS) gets staked as collateral to back Synths and entitles stakers to receive fees generated by Synth trades on Oikos.Exchange.

## Pound Sterling (sGBP)¶

Decimals: 18

Price Feed: Oikos (centralized)

Tracks the price of Pound Sterling (GBP) through price feeds supplied by an oracle.

## Swiss Franc (sCHF)¶

Decimals: 18

Price Feed: Oikos (centralized)

Tracks the price of Swiss Franc (CHF) through price feeds supplied by an oracle.

## TRON (sTRX)¶

Decimals: 18

Price Feed: Oikos (centralized)

Tracks the price of TRON (TRX) through price feeds supplied by an oracle.

## US Dollars (sUSD)¶

Decimals: 18

Price Feed: Oikos (centralized)

Tracks the price of a single US Dollar (USD). This Synth always remains constant at 1.}