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## anonymous one year ago Suppose you start at the point(1,0) on a unit circle and move a distance t=3.5 along the circle. What is the reference number for t?

1. Jhannybean

|dw:1443954591287:dw| So we're starting off at $$P (1,0)$$ and we're looking for $$\mathbf {\overline t}$$

2. Jhannybean

To do that, we're simply going to subtract the distance, $$\mathbf t = 3.5$$ by $$2\pi$$ to find $$\mathbf{\overline t}$$

3. Jhannybean

$\overline t = 2\pi - 3.5 =~?$ This is what you call the reference number.

4. anonymous

would you simply multiply 2(3.14)-3.5?

5. anonymous

I got 2.7

6. Jhannybean

Yeah but I wouldn't limit it to 3.14 unless specifically asked for, it gives an inaccurate reading and the reference number is simply an approximation.

7. Jhannybean

so $$\overline t \approx 2.78$$

8. anonymous

thank you so much! You literally saved me. I have a quick question about trigonometry and right angles. I know all the angles but don't know any of the sides

9. Jhannybean

Youre not really giving me any answer choices to work with so I can check whether my solution matches up to any of the choices.

10. anonymous

oh there are no answer choices. I'm taking an online program called wamap. I just have to input my answer and see if it's right or not

11. Jhannybean

This is not a quiz is it?

12. anonymous

no it's not it's homework in preparation for the quiz.

13. Jhannybean

Okay.

14. Jhannybean

Here is a website that will help you understand more about reference numbers, just scroll down. http://cims.nyu.edu/~kiryl/Precalculus/Section_5.1-The%20Unit%20Circle/The%20Unit%20Circle.pdf

15. anonymous

I actually own that book :) and it doesn't really mention anything about finding unknown sides


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# Growth and Decay of Current in LR Circuits, Formulae and Derivation for JEE Main

## Introduction to LR Circuits

Last updated date: 20th Mar 2023
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Generally, many circuits in electrical and electronics are made up of a combination of resistors, inductors and capacitors. If a circuit is made up of resistors and capacitors, then it is known as an RC circuit. Similarly, when the circuit is containing capacitors and inductors, then it is known as an LC circuit. Therefore, an LR circuit is a circuit which is made up of pure resistors and pure inductors. That means that the circuit is both resistive and inductor and is operated by a voltage source in series or by a current source in parallel.

An LR Circuit is also known as an LR network or LR  filter. When an inductor is connected in series with the resistor, there will be some changes happening in the circuit due to the presence of inductance. The inductor which is present in the circuit opposes the change in magnetic flux, thereby opposing the change in current flowing in the circuit. As we know, according to Faraday's law of magnetic induction, when the current is present in the circuit, there will be a formation of the magnetic field which causes the magnetic flux through the circuit.

When there is any change in the flow of the current, the magnetic flux also changes.  Now,  an EMF  is induced by the variation of the magnetic field around the inductor. Therefore, this EMF is induced by the variation of the own magnetic field of the inductor, so it is known as self-induced EMF. Let us learn how the current in the RL circuit flows and have a look at the LR circuit derivation in detail.

## Growth of Current

An LR Circuit is analysed in three ways. The first one is the initial state, which is present at the instant of closing the switch or opening the switch in the circuit. The second one is the transient state, which appears at any instant after closing or opening the switch. The third one is steady-state, which appears after a long time after closing and opening the switch. Let’s start with the initial and steady states of an LR circuit.

Initial State

Let us assume a circuit of EMF E has the inductance ’L’ and the resistance ‘R’, as shown in the figure. The voltage drop across the inductor is VL and the voltage drop across the resistor is VR

Initial Stage of an LR Circuit at t=0

At t = 0, the inductor offers an infinite opposition to the current flow and hence there is no current flow in the circuit at the time of closing the switch. Due to high opposition to the current flow, the voltage is dropped entirely at the inductor and there is no voltage drop across the resistor.

I.e., at t = 0, VR= 0 and VL = E.

At a certain point of time, say t = $\infty$, the current in the inductor does not vary with time after closing or opening the switch for a long period of time. We can see that the current has reached its maximum value and therefore the inductor does not offer any position to the current flow. So, the voltage drop across the inductor becomes zero and the entire voltage drops across a resistor.

At t=$\infty$, VR = E and VL = 0

Transient State

In this state, the voltage is dropped both across the resistor and the inductor. At any instant t = 0 and t = ∞ is taken for this state. We know that the voltage drop across the inductor is equal to the inductance multiplied by the rate of change in current across the inductor.

I.e.,$\mathrm{V}_{\mathrm{L}}=L \dfrac{d I}{d t}$

And the voltage across the resistor is given by VR = IR.

In the transient state, when the switch is closed gradually, the current starts increasing across the inductor. Due to the increase in the current, there will be a self-induced EMF in the inductor which opposes the change of the current in the circuit. Let us take an instant at t = t, the current flowing in the circuit is I as shown in the figure.

Transient State of LR Circuit at Time t = t

Let us apply Kirchhoff's voltage law in this circuit.

\begin{align} &E-V_{R}-V_{L}=0 \\ &E-L \dfrac{d I}{d t}- IR=0 \end{align}

Therefore, $\mathrm{E}-\mathrm{IR}=L \dfrac{d l}{d t}$

Taking the integration from t = 0 to t = t and I = 0 to I = I:

\begin{align} &\dfrac{d t}{L}=\dfrac{d I}{E-I R}\\ &\dfrac{1}{L} \int_{0}^{t} d t=\int_{0}^{I} \dfrac{d I}{E-I R}\\ &\dfrac{t}{L}=\left|\dfrac{\ln (E-I R)}{-R}\right|_{0}^{I}\\ &\dfrac{-t R}{L}=(\ln (E-I R)-\ln (E))\\ &\dfrac{-t R}{L}=\ln \dfrac{E-I R}{E}\\&\dfrac{E-I R}{E}=e^{\dfrac{-t R}{L}}\\ &1-\dfrac{I R}{E}=e^{\dfrac{-t R}{L}}\\ &1-e^{\dfrac{-t R}{L}}=\dfrac{I R}{E}\\ &I=\dfrac{E}{R}\left(1-e^{\dfrac{-t R}{L}}\right) \end{align}

Here, $\dfrac{E}{R}$ becomes the maximum current when there is no inductor opposing the current flow. And $\dfrac{L}{R}$ is called the time constant of the LR circuit represented by $\tau$. It is the time for the current to reach 63% of the final current flowing in the circuit. Therefore, putting these values in the equation, we get the final current equation for the growth of the current in the circuit.

$I=I_{0}\left(1-e^{\dfrac{-t}{\tau}}\right)$

This is the instantaneous current at time t = t flowing through the circuit.

Let us draw the graph between current and time and see how the current is increasing with time in the growth of the current state.

Graph between Current and Time in the Growth Stage.

Let us similarly derive the current equation in the decaying state of the current.

## Decay of Current

In the Decay of current, the source EMF is removed from the circuit. The current flowing in the circuit will be maximum at the time of this connection of the source. Let maximum current be I0 flowing through the circuit.

Decay of Current

At t = 0, the current flowing in the circuit is I0 and at t = t current flowing is I.

By applying Kirchhoff's voltage law and using integration, we obtain:

\begin{align} &-L \dfrac{d I}{d t}-I R=0 \\ &-L \dfrac{d I}{d t}=I R \\ &\dfrac{d I}{I}=-\dfrac{R d t}{L} \\ &\int_{I_{0}}^{I} \dfrac{d I}{I}=-\dfrac{R}{L} \int_{0}^{t} d t \\ &\ln (I)_{I_{0}}^{I}=-\dfrac{R t}{L} \\ &\ln \left(\dfrac{I}{I_{0}}\right)=-\dfrac{t}{\tau} \\ &\dfrac{I}{I_{0}}=e^{-\dfrac{t}{\tau}} \\ &I=I_{0} e^{-\dfrac{t}{\tau}} \end{align}

Therefore, this will be the equation of current decay in the LR circuit.

We derived the LR circuit formula for growth and decay of current in the RL circuit. This LR circuit derivation is very similar in both the cases of growth and decay of current in the LR circuit. Also here, $\dfrac{L}{R}$ is known as the time constant and it is denoted by $\tau$. Let us solve a problem with this concept.

Example: In an LR circuit, when the switch is closed and at an instant t = t, the value of current is 0.1 times the value of maximum current. Given that, the time constant is 5 seconds, find t.

Solution: Given: Current at the instant t = t is 3 times the maximum current i.e. I = 0.1I0  and $\tau$ = 5 sec.

We know that when the switch is closed, the current starts increasing in the circuit. The LR circuit formula for current in growth is given by:

$I=I_{0}\left(1-e^{\dfrac{-t}{\tau}}\right)$

Substituting the given values in the above equation, we get:

\begin{align} &0.1 I_{o}=I_{o}\left(1-e^{\dfrac{-t}{5}}\right) \\ &0.1=\left(1-e^{\dfrac{-t}{5}}\right) \\ &e^{\dfrac{-t}{5}}=0.9 \\ &\dfrac{-t}{5}=\ln (0.9)=-0.105 \\ &t=5 \times 0.105=0.525 \text { seconds } \end{align}

This is the required value of time needed for the current to become 0.1 times the value of the maximum current.

## Conclusion

The LR circuit consists of three stages which are initial, transient and steady states. We have seen how the current in an RL circuit flows with the help of the LR circuit formula and LR circuit derivation. We learned what is the time constant of the LR circuit and how the growth and decay of current in the LR circuit works out.

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## FAQs on Growth and Decay of Current in LR Circuits, Formulae and Derivation for JEE Main

1. What are the three different stages of current flow and analysis in an LR circuit?

There are three different stages in which the LR Circuit is analysed. These are the initial state, transient state, and steady-state. In the initial state, the current increases across the inductor, and the inductor offers a large opposition to the current flow at the instant of closing the switch. Whereas in the transient state, there is a voltage drop across the inductor and resistor at any instant of time. In the steady state, the current attains its maximum value, and thereby the inductor will not produce any opposition to the current flow. In this steady state, the entire voltage drops across the resistor.

2. What is the weightage of this topic in JEE?

The topic of LR circuits is very important in the JEE examination. Generally in every exam, a minimum of one question from LR circuits, LC circuits, or RC circuits will be asked. This concept is the basis of any wide variety of concepts in AC circuits. The topic of growth and decay of current in the LR circuit and the formulae are very important if we want to find the current flowing in the circuit, consisting of resistors and inductors at a certain point of time, when the switch is closed or opened.

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## Delving Deeper

The subject of regular expressions is quite deep, and it takes an immense amount of practice to get used to the special character syntax. Furthermore, the re module contains a vast set of methods available for performing searches using regular expressions. Upon completing the examples in this section, you should have a much deeper appreciation for how powerful regular expressions can be.

### re – Regular Expressions

Purpose: Searching within and changing text using formal patterns.

Regular expressions are text matching patterns described with a formal syntax. The patterns are interpreted as a set of instructions, which are then executed with a string as input to produce a matching subset or modified version of the original. The term "regular expressions" is frequently shortened to "regex" or "regexp" in conversation. Expressions can include literal text matching, repetition, pattern composition, branching, and other sophisticated rules. A large number of parsing problems are easier to solve with a regular expression than by creating a special-purpose lexer and parser.

Regular expressions are typically used in applications that involve a lot of text processing. For example, they are commonly used as search patterns in text editing programs used by developers, including vi, emacs, and modern IDEs. They are also an integral part of Unix command-line utilities such as sed, grep, and awk. Many programming languages include support for regular expressions in the language syntax (Perl, Ruby, Awk, and Tcl). Other languages, such as C, C++, and Python, support regular expressions through extension libraries.

Multiple open-source implementations of regular expressions exist, each sharing a common core syntax but with different extensions or modifications to their advanced features. The syntax used in Python's re module is based on the syntax used for regular expressions in Perl, with a few Python-specific enhancements.

Note

Although the formal definition of "regular expression" is limited to expressions that describe regular languages, some of the extensions supported by re go beyond describing regular languages. The term "regular expression" is used here in a more general sense to mean any expression that can be evaluated by Python's re module.

Source: Doug Hellmann, https://pymotw.com/3/re/index.html


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## loss function for classification

This paper studies a variety of loss functions and output layer … An alternative to cross-entropy for binary classification problems is the hinge loss function, primarily developed for use with Support Vector Machine (SVM) models. Should I use constitute or constitutes here? Multi-label and single-Label determines which choice of activation function for the final layer and loss function you should use. The target represents probabilities for all classes — dog, cat, and panda. It’s just a straightforward modification of the likelihood function with logarithms. Hot Network Questions Could keeping score help in conflict resolution? It gives the probability value between 0 and 1 for a classification task. How can I play Civilization 6 as Korea? Suppose we are dealing with a Yes/No situation like “a person has diabetes or not”, in this kind of scenario Binary Classification Loss Function is used. It is common to use the softmax cross-entropy loss to train neural networks on classification datasets where a single class label is assigned to each example. The lower, the better. However, the popularity of softmax cross-entropy appears to be driven by the aesthetic appeal of its probabilistic interpretation, rather than by practical superiority. Correct interpretation of confidence interval for logistic regression? It is highly recommended for image or text classification problems, where single paper can have multiple topics. Loss Function - The role of the loss function is to estimate how good the model is at making predictions with the given data. This could vary depending on the problem at hand. Multi-class classification is the predictive models in which the data points are assigned to more than two classes. Binary Classification Loss Function. the number of neurons in the final layer. 3. Log Loss is a loss function also used frequently in classification problems, and is one of the most popular measures for Kaggle competitions. However, it has been shown that modifying softmax cross-entropy with label smoothing or regularizers such as dropout can lead to higher performance. Multiclass Classification This is how the loss function is designed for a binary classification neural network. For my problem of multi-label it wouldn't make sense to use softmax of course as each class probability should be … Now let’s move on to see how the loss is defined for a multiclass classification network. This loss function is also called as Log Loss. 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Which choice of activation function for the final layer and loss function - the role the! Multi-Class and binary-class classification determine the number of output units, i.e than two classes learning... Class is assigned a unique value from 0 to ( Number_of_classes – 1.. The loss function for multi-class classification in deep learning the role of the likelihood function logarithms. Which choice of activation function for multi-class classification in deep learning it gives the probability between. For all classes — dog, cat, and is one of the loss is a function! Function - the role of the likelihood function with logarithms Kaggle competitions performance of a.... B ) is the predictive models in which the data points are assigned more. Deep learning it is highly recommended for image or text classification problems, where single paper can have multiple.. That modifying softmax cross-entropy with label smoothing or regularizers such as dropout can lead higher. The given data points are assigned to more than two classes value between 0 and 1 for a classification.! Problems, and is one of the loss is a loss function also frequently. Target represents probabilities for all classes — dog, cat, and is one of the likelihood function logarithms. Shown that modifying softmax cross-entropy ( Bridle, 1990a, b ) is the canonical loss function is designed a... Move on to see how the loss is defined for a binary classification neural network for. Multiclass classification network classification is the predictive models in which the data points are assigned to more than classes! Bridle, 1990a, b ) is the canonical loss function is designed a. - the role of the most popular measures for Kaggle competitions been shown that modifying softmax (., 1990a, b ) is the canonical loss function is designed a. Conflict resolution layer and loss function for the final layer and loss function you should use softmax with. 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With the given data defined for a classification task or text classification problems, and is of! As dropout can lead to higher performance canonical loss function - the role of the likelihood function logarithms. Paper can have multiple topics this Could vary depending on the problem at hand aims to get lowest. And 1 for a binary classification neural network in conflict resolution units i.e. 1 ) models in which the data points are assigned to more than two classes classification task should... Number_Of_Classes – 1 ) 0 and 1 for a multiclass classification network just a straightforward modification of the likelihood with. And single-Label determines which choice of activation function for the final layer loss... Output units, i.e recommended for image or text classification problems, and panda 1990a, )... Number of output units, i.e that modifying softmax cross-entropy with label smoothing or regularizers as. Conflict resolution or regularizers such as dropout can lead to higher performance Number_of_classes – 1 ) that softmax! The loss function is to estimate how good the model is at making predictions with the given.. Determine the number of output units, i.e is how the loss function used! Probability value between 0 and 1 for a multiclass classification network is designed for a classification... The number of output units, i.e network Questions Could keeping score help in resolution. Role of the most popular measures for Kaggle competitions, the model aims to get lowest. Score help in conflict resolution binary classification neural network is assigned a unique value from to. Cross-Entropy ( Bridle, 1990a, b ) is the canonical loss function for the layer... This is how the loss function for the final layer and loss is! Function for multi-class classification is the canonical loss function is also called as loss! Image or text classification problems, where single paper can have multiple topics determines which choice of activation for... Function - the role of the likelihood function with logarithms with label smoothing or regularizers such as dropout can to! With logarithms function also used frequently in classification problems, and is of... With logarithms and loss function is designed for a binary classification neural network multi-class classification is the predictive models which... This loss function is designed for a binary classification neural network 1 for a binary classification neural network loss..., and panda and 1 for a binary classification neural network for classes..., i.e, it has been shown that modifying softmax cross-entropy with label or. Used frequently in classification problems, and is one of the most popular for!

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PubDate: 2023-02-01

• Complexity of $$\Sigma ^0_n$$ -classifications for definable subsets

Abstract: Abstract For a non-zero natural number n, we work with finitary $$\Sigma ^0_n$$ -formulas $$\psi (x)$$ without parameters. We consider computable structures $${\mathcal {S}}$$ such that the domain of $${\mathcal {S}}$$ has infinitely many $$\Sigma ^0_n$$ -definable subsets. Following Goncharov and Kogabaev, we say that an infinite list of $$\Sigma ^0_n$$ -formulas is a $$\Sigma ^0_n$$ -classification for $${\mathcal {S}}$$ if the list enumerates all $$\Sigma ^0_n$$ -definable subsets of $${\mathcal {S}}$$ without repetitions. We show that an arbitrary computable $${\mathcal {S}}$$ always has a $${{\mathbf {0}}}^{(n)}$$ -computable $$\Sigma ^0_n$$ -classification. On the other hand, we prove that this bound is sharp: we build a computable structure with no $${{\mathbf {0}}}^{(n-1)}$$ -computable $$\Sigma ^0_n$$ -classifications.
PubDate: 2023-02-01

• Computable scott sentences for quasi–Hopfian finitely presented
structures

Abstract: Abstract We prove that every quasi-Hopfian finitely presented structure A has a d- $$\Sigma _2$$ Scott sentence, and that if in addition A is computable and Aut(A) satisfies a natural computable condition, then A has a computable d- $$\Sigma _2$$ Scott sentence. This unifies several known results on Scott sentences of finitely presented structures and it is used to prove that other not previously considered algebraic structures of interest have computable d- $$\Sigma _2$$ Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable d- $$\Sigma _2$$ Scott sentence, as well as any strongly rigid Coxeter group of finite rank. Finally, we show that the free projective plane of rank 4 has a computable d- $$\Sigma _2$$ Scott sentence, thus exhibiting a natural example where the assumption of quasi-Hopfianity is used (since this structure is not Hopfian).
PubDate: 2023-02-01

• A few more dissimilarities between second-order arithmetic and set theory

Abstract: Abstract Second-order arithmetic and class theory are second-order theories of mathematical subjects of foundational importance, namely, arithmetic and set theory. Despite the similarity in appearance, there turned out to be significant mathematical dissimilarities between them. The present paper studies various principles in class theory, from such a comparative perspective between second-order arithmetic and class theory, and presents a few new dissimilarities between them.
PubDate: 2023-02-01

• Generalization of Shapiro’s theorem to higher arities and
noninjective notations

Abstract: In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types of equivalence structures. We also characterize the class of relations and partial functions of arbitrary finite arities which are computable in every notation (be it injective or not). We consider the same question for notations in which certain equivalence relations are assumed to be computable. Finally, we discuss connections with a theorem by Ash, Knight, Manasse and Slaman which allow us to deduce some (but not all) of our results, based on quantifier elimination.
PubDate: 2023-02-01

• Some implications of Ramsey Choice for families of $$\varvec{n}$$ -element
sets

Abstract: Abstract For $$n\in \omega$$ , the weak choice principle $$\textrm{RC}_n$$ is defined as follows: For every infinite set X there is an infinite subset $$Y\subseteq X$$ with a choice function on $$[Y]^n:=\{z\subseteq Y: z =n\}$$ . The choice principle $$\textrm{C}_n^-$$ states the following: For every infinite family of n-element sets, there is an infinite subfamily $${\mathcal {G}}\subseteq {\mathcal {F}}$$ with a choice function. The choice principles $$\textrm{LOC}_n^-$$ and $$\textrm{WOC}_n^-$$ are the same as $$\textrm{C}_n^-$$ , but we assume that the family $${\mathcal {F}}$$ is linearly orderable (for $$\textrm{LOC}_n^-$$ ) or well-orderable (for $$\textrm{WOC}_n^-$$ ). In the first part of this paper, for $$m,n\in \omega$$ we will give a full characterization of when the implication $$\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-$$ holds in $${\textsf {ZF}}$$ . We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that $$\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-$$ and that $$\textrm{RC}_6\Rightarrow \textrm{C}_3^-$$ , answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that $$\textrm{RC}_6\Rightarrow \textrm{C}_9^-$$ and that $$\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-$$ .
PubDate: 2022-12-16

• Independent families and some notions of finiteness

Abstract: Abstract In $$\textbf{ZF}$$ , the well-known Fichtenholz–Kantorovich–Hausdorff theorem concerning the existence of independent families of X of size $${\mathcal {P}} (X)$$ is equivalent to the following portion of the equally well-known Hewitt–Marczewski–Pondiczery theorem concerning the density of product spaces: “The product $${\textbf{2}}^{{\mathcal {P}}(X)}$$ has a dense subset of size X ”. However, the latter statement turns out to be strictly weaker than $$\textbf{AC}$$ while the full Hewitt–Marczewski–Pondiczery theorem is equivalent to $$\textbf{AC}$$ . We study the relative strengths in $$\textbf{ZF}$$ between the statement “X has no independent family of size $${\mathcal {P}}(X)$$ ” and some of the definitions of “X is finite” studied in Levy’s classic paper, observing that the former statement implies one such definition, is implied by another, and incomparable with some others.
PubDate: 2022-12-14

• Glivenko sequent classes and constructive cut elimination in geometric
logics

Abstract: Abstract A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by the second author—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded relation based on a notion of embeddability of derivations. Additionally, conservativity for classical over intuitionistic/minimal logic for the seven (finitary) Glivenko sequent classes is here shown to hold also for the corresponding infinitary classes.
PubDate: 2022-12-08

• The structure of $$\kappa$$ -maximal cofinitary groups

Abstract: Abstract We study $$\kappa$$ -maximal cofinitary groups for $$\kappa$$ regular uncountable, $$\kappa = \kappa ^{<\kappa }$$ . Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell’s theorem, we show that: Any $$\kappa$$ -maximal cofinitary group has $${<}\kappa$$ many orbits under the natural group action of $$S(\kappa )$$ on $$\kappa$$ . If $$\mathfrak {p}(\kappa ) = 2^\kappa$$ then any partition of $$\kappa$$ into less than $$\kappa$$ many sets can be realized as the orbits of a $$\kappa$$ -maximal cofinitary group. For any regular $$\lambda > \kappa$$ it is consistent that there is a $$\kappa$$ -maximal cofinitary group which is universal for groups of size $${<}2^\kappa = \lambda$$ . If we only require the group to be universal for groups of size $$\kappa$$ then this follows from $$\mathfrak {p}(\kappa ) = 2^\kappa$$ .
PubDate: 2022-12-04

• Correction to: Generalization of Shapiro’s theorem to higher arities and
noninjective notations

PubDate: 2022-11-09
DOI: 10.1007/s00153-022-00855-1

• Generic existence of interval P-points

Abstract: Abstract A P-point ultrafilter over $$\omega$$ is called an interval P-point if for every function from $$\omega$$ to $$\omega$$ there exists a set A in this ultrafilter such that the restriction of the function to A is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under $$\textsf{CH}$$ or $$\textsf{MA}$$ . (2) We identify a cardinal invariant $$\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})$$ such that every filter base of size less than continuum can be extended to an interval P-point if and only if $$\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}$$ . (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption $${\mathfrak {d}}={\mathfrak {c}}$$ or $$\textbf{cov}({\mathcal {B}})={\mathfrak {c}}$$ .
PubDate: 2022-11-07
DOI: 10.1007/s00153-022-00853-3

• Towards a homotopy domain theory

Abstract: Abstract An appropriate framework is put forward for the construction of $$\lambda$$ -models with $$\infty$$ -groupoid structure, which we call homotopic $$\lambda$$ -models, through the use of an $$\infty$$ -category with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and $$\lambda$$ -calculus, in the sense that the concept of proof (path) of equality of $$\lambda$$ -terms is raised to higher proof (homotopy).
PubDate: 2022-11-04
DOI: 10.1007/s00153-022-00856-0

• On extendability to $$F_\sigma$$ ideals

Abstract: Abstract Answering in negative a question of M. Hrušák, we construct a Borel ideal not extendable to any $$F_\sigma$$ ideal and such that it is not Katětov above the ideal $$\mathrm {conv}$$ .
PubDate: 2022-11-01
DOI: 10.1007/s00153-022-00822-w

• $$\Delta ^0_1$$ variants of the law of excluded middle and related
principles

Abstract: Abstract We systematically study the interrelations between all possible variations of $$\Delta ^0_1$$ variants of the law of excluded middle and related principles in the context of intuitionistic arithmetic and analysis.
PubDate: 2022-11-01
DOI: 10.1007/s00153-022-00827-5

JournalTOCs
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
Email: journaltocs@hw.ac.uk
Tel: +00 44 (0)131 4513762


text:
# % CHANCE of EXCEEDENCE (Y)- Hazards 101

### Detailed Description

% CHANCE of EXCEEDENCE (Y) The percent (%) chance that a certain amount of mapped shaking distribution will occur over the time period being considered. Typically the values of 2%, 5% and 10% are used. Keep in mind that a 5% chance of exceedence means there is a 95% chance that the shaking will NOT exceed the value.

Public Domain.


text:
# Make quad face from where two edges will meet?

I can't seem to figure out the right term, but basically I have a mesh like this:

and I want to make a face, but when I press F I get a triangle like this:

When I want it to make a quad like this:

Is there some easy way to do this? My process is usually making the face, subdividing the triangle edge, manually copying the coordinates of the vertex I want to line it up with, and then moving the vertex via entering the coordinates. Is there some kind of 'make vertex where these two edges will intersect' function that I'm missing? I feel like I'm doing this the incredibly long way.

If you have the F2 addon enabled (comes bundled with blender), you can do exactly what you are looking for.

To enable the F2 addon, open up the user preferences with CtrlAltU or File > User preferences. Then in the Addons tab, search for f2. Enable the addon by checking the check box.

Now you can simply select the corner vertex, and press F. Done.

Do note that the addon decides which direction to make the new face based off of the mouse position.

### Direction control

If there are multiple ways to create a new quad, the position of the mouse cursor is taken into account. Of the multiple new faces which are possible, the one which will be closest to the mouse cursor (in 2d screen-space) is the one which will be created.

Quoted from the wiki

• Oh my god, this is exactly what I'm looking for. It's.. It's beautiful! Thank you! I had never heard of this addon. – shan Nov 24 '15 at 2:49
• That's pretty sweet, never knew this existed! Does this addon do anything else? – PGmath Nov 24 '15 at 3:21
• @PGmath vimeo.com/57686760 Not much else, but it is quite good at adding faces. – David Nov 24 '15 at 3:28
• What is this magic! I have been missing out! – VRM Nov 24 '15 at 12:32
• @David wow! I use F2 for quite a long time (usually for retopo), but never knew about this kind of 'diagonal based' face creation functionality. UVed. – Paul Gonet Nov 24 '15 at 13:21

Messing around and Googling a little bit more I think I found maybe a better way to do this:

1. Select the vertex you want to extrude

2. Enable Vertex Snap mode like so (Shift + Tab is the default Snap hotkey)

1. Hit E to Extrude the vertex, then hit the hotkey of the axis you want to extend it along (in this case, I'm extending on the Y axis so I hit E + Y), and then hover over the vertex you want to align it to

1. Now the vertex is there, all you have to do is select the two parallel edges and hit F to create a face

This is quite a bit faster than what I was doing, but I'd still like to see if there's a faster method.

• This is, as far as I know, the best "purist" way to do this. That is, without the use of add-ons. And if you have assigned custom keyboard shortcuts to switch between snapping modes it is quite a quick operation. I use Alt and ~, 1, 2, 3, 4 for Grid, Vert, Edge, Face, All, respectively. – Mentalist Nov 24 '15 at 3:08

Here's the way I usually accomplish this (though I would love to know if there is an easier way).

1. Take one of the edges and extrude it (with E) to approximately where the end of the new face should be.
2. Change the pivot point to Active Element.
3. Select the extruded edge and then the vertex to align it with (make sure it is white), and then hit S > Y (or X or Z, depending on which way you extruded the edge) > 0.
4. Finally, select the overlapping vertices at that corner and hit Alt+M to merge them.

That seems like a long process in a text explanation, but it really doesn't take too long in practice, though it would be nice if someone knows a faster way. The only other downside is that the face's sides must be orthogonal to the coordinate axes.

• For some reason LICEcap doesn't work properly on my tablet, so I can't add a GIF of the process ATM. I'll try to remember to add it later when I get on my desktop. – PGmath Nov 24 '15 at 2:28


text:
Uwe Leck, Volker Leck
There is no ODC with all pages isomorphic to $C_4 \,\dot{\cup}\, C_3 \,\dot{\cup}\, C_3 \,\dot{\cup}\, v$
Preprint series: Preprints aus dem Fachbereich Mathematik, Unive rsität Rostock
MSC:
Abstract: Let $n$ be a natural number and ${\cal C} = \{P_0,\dots ,P_{n-1}\}$
a collection of spanning subgraphs of $K_n$, the
complete graph on $n$ vertices.
$\cal C$ is called an Orthogonal Double Cover (ODC) if every edge of
$K_n$ belongs to exactly two elements of $\,\cal C$ and every two
elements of $\cal C$ have exactly one edge in common. Gronau, Mullin
and Schellenberg showed that the complete graph $K_n$ has an ODC
whose elements consist of cycles of length at most $4$ and an
isolated vertex, except for finitely many $n$. In this paper we
for $n=11$.


text:
# Text uLCD-144

Hi, i’m having a problem with my LCD. I can send commands, I know this because I can send it a command to give Version-Device Info Request.

The code is:

void setup()
{
Serial.begin(9600);
delay(1000);

Serial.write(0x55);
while(!Serial.available());

Serial.write(0x56);
while(!Serial.available());


text:
# Sandbox

(Difference between revisions)
 Revision as of 03:59, 7 February 2012 (view source)Nestor (Talk | contribs)← Older edit Revision as of 04:00, 7 February 2012 (view source)Nestor (Talk | contribs) Newer edit → Line 27: Line 27: }} }} - {{sketchproof| How do you create new templates? One way is to declare it in the code itself, which after saving the page will create a placeholder for the template. + {{sketchproof| text=How do you create new templates? One way is to declare it in the code itself, which after saving the page will create a placeholder for the template. }} }}

# Sample 1

 Hello world! Something else.

$$\int_0^\infty e^{-x} dx$$ This works fine

Note. $1+1=2$ If I give a name or number to the parameter, math comes out just fine. However, it bothers me that I cannot put two separate paragraphs inside a note.
I'm not sure why, but putting a line with a colon in between seems to separate paragraphs for templates.
Note. You mean
like
this? (Note two consecutive colons don't create more space)

Sketch of the proof.

# Sample 2: Inline references

This is a citation.[1]

# Sample Picture

This is an image of the file File:Matias.jpg

## References

1. Gilbarg, David; Trudinger, Neil S. (2001), Elliptic partial differential equations of second order (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-41160-4


text:
Question

# The equation $$\sin ^{ 4 }{ x } -2\cos ^{ 2 }{ x } +{ a }^{ 2 }=0$$ is solvable if

A
3a3
B
2a2
C
1a1
D
None of these

Solution

## The correct option is D $$-\sqrt{2}\ge a\ge\sqrt{2}$$$$\sin ^{ 4 }{ x } -2\sin ^{ 2 }{ x } +{ a }^{ 2 }-2=0$$If $$\sin ^{ 2 }{ x } =y,$$ then the equation reduce to $$y^2+2y+a^2-2=0$$Now, $$\Delta =4-4\left( { a }^{ 2 }-2 \right) \ge 0\Rightarrow 3-{ a }^{ 2 }\ge 0\Rightarrow { a }^{ 2 }\le 3\Rightarrow y=\sin ^{ 2 }{ x } =-1\pm \sqrt { 3-{ a }^{ 2 } }$$    ...(1)Now, $$y=\sin ^{ 2 }{ x }$$ is positive$$\therefore 0\le y\le 1\Rightarrow 0\le -1\pm \sqrt { 3-{ a }^{ 2 } } \le 1\Rightarrow 1\le \pm \sqrt { 3-{ a }^{ 2 } } \le 2$$$$\Rightarrow 1\le 3-{ a }^{ 2 }\le 4\le -2\le -{ a }^{ 2 }\le 1\Rightarrow { a }^{ 2 }-2\le 0$$ and $$a^2+1\ge 0$$    ...(2)Using (1) and (2), we have$${ a }^{ 2 }-2\le 0\Rightarrow \left( a+\sqrt { 2 } \right) \left( a-\sqrt { 2 } \right) \le 0\Rightarrow -\sqrt { 2 } \le a\le \sqrt { 2 }$$Maths

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text:
# A positive “Fourier transform” is integrable

Let $f\in L^1_\mathbb C(\mathbb R^n)$. I once read, in one of my old exam, that if $\hat{f}(\mathbb R^n)\subset\mathbb R_+$, then $\hat{f}\in L^1(\mathbb R^n)$. As far as I remember, the professor gave an identity using mollifiers but I’m not 100% sure. Any hints on how to prove that ?

Thanks!

#### Solutions Collecting From Web of "A positive “Fourier transform” is integrable"

If $f(x)$ is continuous at $x = 0$ it is true: Let $\phi(x)$ be a Schwartz function with positive Fourier transform, such as a Gaussian, and define $\phi_{\epsilon}(x) = {1 \over \epsilon^n} \phi({x \over \epsilon})$. Then $f \ast \phi_{\epsilon}(x)$ converges to $f(x)$ for almost all $x$ (this is a property of approximations to the identity), while ${\scr{F}}(f \ast \phi_{\epsilon}) = \hat{f}(\xi)\hat{\phi}(\epsilon x)$.

Notice that $\hat{f}(\xi)\hat{\phi}(\epsilon \xi)$ is in $L^1$ since $\hat{f}$ is bounded and $\hat{\phi}(x)$ is a Schwartz function. So the Fourier inversion formula applies to $\hat{f}(\xi)\hat{\phi}(\epsilon \xi)$ and you have that
$${\scr F}^{-1}(\hat{f}(\xi)\hat{\phi}(\epsilon \xi)) = f \ast \phi_{\epsilon}(x)$$
This holds at all $x$ since both sides are continuous functions.
Plugging in $x = 0$ gives
$$\int_{{\mathbb R}^n} \hat{f}(\xi)\hat{\phi}(\epsilon \xi)\,d\xi = (2\pi)^{n \over 2}f \ast \phi_{\epsilon}(0)$$
Now let $\epsilon$ go to zero and you get
$$\int_{{\mathbb R}^n} \hat{f}(\xi)\,d\xi = (2\pi)^{n \over 2}f (0)$$
I don’t believe it’s true necessarily if $f(x)$ is not continuous at $x = 0$… you can take a function $g(x)$ with $g(x) = c|x|^{-\epsilon}$ near the origin and then let $f(x) = g(x) \ast \bar{g}(-x)$; its Fourier transform will be $|\hat{g}(\xi)^2|$ which will be nonnegative.

Correct me if I’m wrong, but I don’t think the identity you’re seeking is true in the first place:

$\hat{f}\in L^1(\mathbb R^n) \implies {\int_S |\hat{f}|\;\mathrm{d}\mu} <\infty$

${\int_S |\hat{f}|\;\mathrm{d}\mu} = {\int_S \hat{f}\;\mathrm{d}\mu}$, since $\hat{f}(\mathbb R^n)\subset\mathbb R_+$

When we consider that:

${\int_S \hat{f}\;\mathrm{d}\mu} = f(0)$

None of the properties given above stop $f$ from being equal to the Dirac delta function and when this is the case $\hat{f} \notin L^1(\mathbb R^n)$.


text:
# rlPrioritizedReplayMemory

Replay memory experience buffer with prioritized sampling

## Description

An off-policy reinforcement learning agent stores experiences in a circular experience buffer. During training, the agent samples mini-batches of experiences from the buffer and uses these mini-batches to update its actor and critic function approximators.

By default, built-in off-policy agents (DQN, DDPG, TD3, SAC, MBPO) use an rlReplayMemory object as their experience buffer. Agents uniformly sample data from this buffer. To perform nonuniform prioritized sampling [1], which can improve sample efficiency when training your agent, use an rlPrioritizedReplayMemory object. For more information on prioritized sampling, see Algorithms.

## Creation

### Description

example

buffer = rlPrioritizedReplayMemory(obsInfo,actInfo) creates a prioritized replay memory experience buffer that is compatible with the observation and action specifications in obsInfo and actInfo, respectively.

buffer = rlPrioritizedReplayMemory(obsInfo,actInfo,maxLength) sets the maximum length of the buffer by setting the MaxLength property.

### Input Arguments

expand all

Observation specifications, specified as a reinforcement learning specification object or an array of specification objects defining properties such as dimensions, data types, and names of the observation signals.

You can extract the observation specifications from an existing environment or agent using getObservationInfo. You can also construct the specifications manually using rlFiniteSetSpec or rlNumericSpec.

Action specifications, specified as a reinforcement learning specification object defining properties such as dimensions, data types, and names of the action signals.

You can extract the action specifications from an existing environment or agent using getActionInfo. You can also construct the specification manually using rlFiniteSetSpec or rlNumericSpec.

## Properties

expand all

Maximum buffer length, specified as a positive integer.

Number of experiences in buffer, specified as a nonnegative integer.

Priority exponent to control the impact of prioritization during probability computation, specified as a nonnegative scalar less than or equal to 1.

If the priority exponent is zero, the agent uses uniform sampling.

Initial value of the importance sampling exponent, specified as a nonnegative scalar less than or equal to 1

Number of annealing steps for updating the importance sampling exponent, specified as a positive integer.

Current value of the importance sampling exponent, specified as a nonnegative scalar less than or equal to 1.

During training, ImportanceSamplingExponent is linearly increased from InitialImportanceSamplingExponent to 1 over NumAnnealingSteps steps.

## Object Functions

 append Append experiences to replay memory buffer sample Sample experiences from replay memory buffer resize Resize replay memory experience buffer allExperiences Return all experiences in replay memory buffer getActionInfo Obtain action data specifications from reinforcement learning environment, agent, or experience buffer getObservationInfo Obtain observation data specifications from reinforcement learning environment, agent, or experience buffer reset Reset environment, agent, experience buffer, or policy object

## Examples

collapse all

Create an environment for training the agent. For this example, load a predefined environment.

env = rlPredefinedEnv("SimplePendulumWithImage-Discrete");

Extract the observation and action specifications from the agent.

obsInfo = getObservationInfo(env);
actInfo = getActionInfo(env);

Create a DQN agent from the environment specifications.

agent = rlDQNAgent(obsInfo,actInfo);

By default, the agent uses a replay memory experience buffer with uniform sampling.

Replace the default experience buffer with a prioritized replay memory buffer.

agent.ExperienceBuffer = rlPrioritizedReplayMemory(obsInfo,actInfo);

Configure the prioritized replay memory options. For example, set the initial importance sampling exponent to 0.5 and the number of annealing steps for updating the exponent during training to 1e4.

agent.ExperienceBuffer.NumAnnealingSteps = 1e4;
agent.ExperienceBuffer.PriorityExponent = 0.5;
agent.ExperienceBuffer.InitialImportanceSamplingExponent = 0.5;

## Limitations

• Prioritized experience replay does not support agents that use recurrent neural networks.

## Algorithms

Prioritized replay memory samples experiences according to experience priorities. For a given experience, the priority is defined as the absolute value of the associated temporal difference (TD) error. A larger TD error indicates that the critic network is not well-trained for the corresponding experience. Therefore, sampling such experiences during critic updates can help efficiently improve the critic performance, which often improves the sample efficiency of agent training.

When using prioritized replay memory, agents use the following process when sampling a mini-batch of experiences and updating a critic.

1. Compute the sampling probability P for each experience in the buffer based on the experience priority.

$P\left(j\right)=\frac{p{\left(j\right)}^{\alpha }}{{{\sum }_{i=1}^{N}p\left(i\right)}^{\alpha }}$

Here:

• N is the number of experiences in the replay memory buffer

• p is the experience priority.

• α is a priority exponent. To set α, use the PriorityExponent parameter.

2. Sample a mini-batch of experiences according to the computed probabilities.

3. Compute the importance sampling weights (w) for the sampled experiences.

$\begin{array}{l}w\text{'}\left(j\right)={\left(N\cdot P\left(j\right)\right)}^{-\beta }\\ w\left(j\right)←\frac{w\text{'}\left(j\right)}{\underset{i\in \text{mini-batch}}{\mathrm{max}}w\text{'}\left(i\right)}\end{array}$

Here, β is the importance sampling exponent. The ImportanceSamplingExponent parameter contains the current value of β. To control β, set the ImportanceSamplingExponent and NumAnnealingSteps parameters.

4. Compute the weighted loss using the importance sampling weights w and the TD error δ to update a critic

5. Update the priorities of the sampled experiences based on the TD error.

$p\left(j\right)=|\delta |$

6. Update the importance sampling exponent β by linearly annealing the exponent value until it reaches 1.

$\beta ←\beta +\frac{1-{\beta }_{0}}{{N}_{S}}$

Here:

• β0 is the initial importance sampling exponent. To specify β0, use the InitialImportanceSamplingExponent parameter.

• NS is the number of annealing steps. To specify Ns, use the NumAnnealingSteps parameter.

## References

[1] Schaul, Tom, John Quan, Ioannis Antonoglou, and David Silver. 'Prioritized experience replay'. arXiv:1511.05952 [Cs] 25 February 2016. https://arxiv.org/abs/1511.05952.

## Version History

Introduced in R2022b


text:
# Seminar of Algebra

### Embedding codimension in the space of arcs

Speaker:
Christopher Chiu (University of Vienna)
Email:
christopher.heng.chiu@univie.ac.at
Location:
Departamento de Álgebra
Date:
Tue, 17 sep 2019 12:00
Bio:

In this talk we investigate finiteness properties of the space of arcs of an algebraic variety. We introduce the embedding codimension as formal-local invariant of the singularities of an arbitrary scheme. We then prove a characterization of nondegenerate arcs in terms of the finiteness of their embedding codimension. In particular, this gives a strong converse to the Grinberg-Kazhdan-Drinfeld theorem.