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# What are the synchronous counters? Explain with an example.

What are the synchronous counters? Explain with an example. Unlike ripple and asynchronous counters the synchronous counter is a kind of counter wherein the clock signal is all the while given to each flip-flop embedded in the counter circuit. All the more explicitly, we can say that each flip-flop is set off in synchronism with the clock input. The values of data inputs like T or J and K are examined and a final result of whether to complement a flip-flop is executed .This decision is made at the time of clock edge. It complements the flip-flop when T = 1 or J = K = 1. It remains in the same state when T = 0 or J = K = 0. Synchronous counters are also termed as ‘Simultaneous counters’ on some occasions.

In many ways, the synchronous counters are better than the asynchronous counters. They are mentioned below:

1. Easy to design and build compared to asynchronous counter.
2. Logic gates control the Count sequence, resulting in very less chances of error.
3. It performs simultaneous actions.
4. No delay in propagation and no ripple effect are found.
5. Operates at a much higher pace than the Asynchronous counter.

• Synchronous Counters are easy to design than asynchronous counters.
• No propagation delay associated with it.
• Count sequence is controlled by logic gates, error chances are lower.
• Have faster operations than asynchronous.
• Synchronous counters can be operated on the higher frequencies

## Types of Synchronous Counters

There are many types of synchronous counters which we use in daily digital electronics. They are as follows:

• Binary counters
• Synchronous UP counter
• Synchronous DOWN counter
• Binary counter with parallel load
• BCD counters
• Ring counters
• Johnson counters etc.

### Designing of Synchronous Counters

The designing of synchronous counters include the following steps:
• Decide the number of flip-flops.
• Excitation table of flip-flops.
• State diagram and circuit excitation table.
• Obtained simplified equations using K-map.
• Draw the logic diagram.

## What are the Binary Counters?

A binary counter can be developed from J-K flip-flops by taking the output of one cell to the clock input of the following. The J and K inputs of each flip-flop are placed to 1 to deliver a toggle at every cycle of the clock input. For every two toggles of the first cell, a toggle is created in the subsequent cell, etc. down to the fourth cell. This delivers a binary number equivalent to the quantity of cycles of the input clock signal. This device is now and again called a “ripple through” counter. A similar device is valuable as a frequency divider.

Several D flip-flop circuits are connected to each other in a binary counter. The flip-flops change state with each clock pulse. The output is a binary number that can be used for digital clocks or timers. Usually its design is asynchronous, with the output of one flip-flop connected to the next. When one cell is toggled, the next one is toggle. If there are two toggles, the next cell in the row is toggled. This arrangement is also called a “ripple” counter.

## 4-bit Synchronous UP Counter

If each J-K flip-flop is enabled to toggle based on whether all the previous flip-flop outputs (Q) are “high” or not, we can get the same count sequence as provided by the asynchronous circuit without the ripple effect. This is because each flip- The flop in this timed circuit will be clocked at exactly the same time.

It provides us with a four-bit synchronous “up” counter. All of the higher order flip-flops stand ready to toggle both the inputs of J and K as “high”. But this happens when the outputs of all previous flip-flops, denoted as “Q”, are “high”.

Otherwise, the inputs of J and K for this flip-flop are both “low”, which in turn puts it in “latch” mode, where it holds its current output state until the next clock pulse arrives.

From the first (LSB), flip-flop has to toggle with each clock pulse, its J and K inputs are connected to Vcc or Vdd, where they are “high” all the time.

The next flip-flop just needs to “recognize” that the output, Q, of the first flip-flop is “high”. This makes it ready to toggle, so no need of an AND Gate.

The remaining flip-flops should be designed to toggle when all lower-order output bits are “high”, therefore an AND gate is needed is this case.

4-bit synchronous up counter start to count from 0 in binary(0000) and increment or count upwards to 15(1111) in binary and then new counting cycle will start after getting reset.

## 4-bit Synchronous Down Counter

To create a synchronous “down” counter, we need to build the circuit to recognize the appropriate bit patterns that predict each toggle state as it counts down.

Unsurprisingly, when we examine the four-bit binary count sequence, we see that all previous bits are low before a toggle. The sequence if followed from bottom to top.

Since each J-K flip-flop is equipped with output Q and output Q’, we can use the output Q to activate the toggle mode on each subsequent flip-flop, since each Q will be “high” when the respective Q will be “low”.

In 4-bit synchronous down counter, the output of flip-flop changes states on the falling edge (1to zero transition) of CLK input which is triggered by Q output of previous flip-flop rather than by the Q output as in the synchronous up counter.

## 4 bit Up/Down Synchronous Counter

Both synchronous and asynchronous counters are capable of counting up and counting down, but there is another type of universal type of counter that counts in both directions either up or down depending on the state of their input control pin and these are known as Bidirectional Counters. Bidirectional counters are also known as Up/Down Counters.

Both synchronous and asynchronous counters are capable of counting up and counting down, but there is another type of universal type of counter that counts in both directions either up or down depending on the state of their input control pin and these are known as Bidirectional Counters. Bidirectional counters are also known as Up/Down Counters.

DIAGRAM:

## Binary Counter with Parallel Load

A parallel load counter is simply a circuit that uses reverse gates and flip-flops to generate a sequential count. When we have used it to create a 4-bit binary code count, it is simply called a parallel-loaded 4-bit binary counter. The binary counter with parallel load comprises of two basic circuits. One is the simple binary counter circuit and the other is the parallel load circuit.

We create a binary counter using J-K flip-flop where when both inputs are one, the output toggles, so if the output of a flip flop other than the last one is one and the counter is on, then it was toggled and in the last it takes if all flip flops are one, so a simple binary counter works on this.

Now we apply a parallel load circuit with it. For that we place a parallel load at one of the inputs of all the circuit gates. The second inputs of the gate are connected with the actual input. This process is carried out in such a way that if an input is applied to one gate then its complement will be applied as an input to the gate whose output is connected to the flip flop. The complement of the input J of the flip-flop is applied to the input K of the flip-flop. Now the result of the circuit is that when the load is enabled the applied input is transferred to the resistive circuit.

We can set up a binary counter with parallel load by adding both the above mentioned circuits. Also we have to connect the complement of load and the count with an AND gate plus OR gates are placed before each inputs of the resistive circuit. This keeps the purpose of making circuit intact.

In the designed circuit only one out of “Load” and “Count” can be enabled at a time. In this way when Load is enabled, all of the AND gates pass the value of Load to the resistor circuit. Likewise, enabling the Count will pass the values of Count to the circuit. But on the other hand if both Load and Count are 0 then a memory state is developed by J-K flip flop. In this way a binary counter with a parallel load works.

If we construct a Table consisting the values of Clear, Clock, Load, Count and Function, we will find some interesting results. When the Clear is 0, the Clock, Load and Count also falls, resulting the Function to become 0. When Count has the value of 1 the flip flop moves towards the next binary start. When Load is 1, the value of Count is ignored and the flip flop takes the value of parallel load. When both Clear and Load are 0, the output remains unaffected.

## What is the BCD Counter?

A BCD (Binary Coded Decimal) counter, also known as a decade counter, is a serial digital counter designed to count ten digits. It performs the reset process automatically when a new clock input signal is received. Since the counter counts ten unique combinations of the input entered, it is called a decade counter. The values that are counted by a BCD counter are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 in binary format. There are also many others. A 4-bit decade counter works like a BCD counter by skipping all 6 of the 24 outputs.

A BCD counter is designed with the help of a J-K flip flop. The outputs of terminals J and K are connected to a logic ‘1’. The clock signal input in each flip-flop is connected to the subsequent flip-flop with the exception of the last flip-flop. The NAND gate output is connected in parallel to the CLR signal for all flip-flops.

The purpose of using BCD counters is to provide accurate representation of decimal number and it’s conversion to and fro. However, there is also a disadvantage of using this counter. It creates the counting delay and the propagation delay during the counting stage.

related topics:


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# Issue with the proof of PCA

I found a very nice PCA proof over here PCA_proof and I'm trying to understand it (I don't know what Langrange multipliers are so I'm trying my best).

From the second page of the previous link, finding the first principal component $a_1$ comes to studying :

$L(a_1)={a_1}^T \Sigma a_1 - \lambda ({a_1}^T a_1 -1)$

So correct me please if I'm wrong

• Acoording to Wikipedia, studying the Langrangian $L(x,\lambda)=f(x) - \lambda (g(x) -c)$ is useful when we want to maximize a function, say $f(x)$ when the variables are constrained by a certain condition, say $g(x)=c$.

• In the first equation, ${a_1}^T \Sigma a_1$ is the $f(x)$ and it says : "I want the first principal component to contain the maximum possible of the data variance " and the second part represents $g(x)$ as ${a_1}^T a_1=1$ ($c$ is the $1$)

So here come the (may be stupid) questions :

• Am-I understanding this correctly ?
• How does the ${a_1}^T \Sigma a_1$ translate the variance projection on the first component $a_1$ ? I know how to project a vector on an axis using a scalar product but I have no idea how It's done with matrices.

Thanks in advance for all your answers.

• The answer to your first question is yes (and you do realize that $a_1^\top a_1 = 1$ means that the vector $a_1$ has length $1$, right?). To answer your second question, denote the centered data matrix $X$ (rows are samples, columns are variables). Then projection of the data onto the vector $a_1$ is given by $Xa_1$. The variance of this projection is given by $$\frac{1}{n-1}(Xa_1)^\top (Xa_1) = \frac{1}{n-1}a_1^\top X^\top X a_1 = a_1^\top \left(\frac{X^\top X}{n-1}\right)a_1 = a_1^\top \Sigma a_1,$$ QED. – amoeba Jan 24 '15 at 23:41

format: plain


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# Mapping from Set to Ordinal Class is Bounded Above

## Theorem

Let $x$ be a set.

Let $\operatorname{On}$ be the class of all ordinals.

Let $f: x \to \operatorname{On}$ be a mapping.

Then $f$ has an upper bound.

### Corollary

Let $\left\langle{x_n}\right\rangle$ be any ordinal-valued sequence.

Then $\left\langle{x_n}\right\rangle$ is bounded above.

## Proof

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

Let $I$ be the image of $f$.
By Union of Subset of Ordinals is Ordinal: Corollary, $\bigcup I$ is an ordinal.
But by Union is Smallest Superset, each element of $I$ is a subset of $\bigcup I$.
Thus $\bigcup I$ is an upper bound of $f$.
$\blacksquare$


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# Infrared Spectroscopy

How2: Interpret infrared spectra

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• Infrared (IR) spectroscopy measures the change in dipole moment of molecules due to irradiating them with light at frequencies that trigger transitions between vibrational energy levels.

The peaks that show up on an IR spectrum lie in the range $400 - {\text{4000 cm}}^{- 1}$ or so, and are given by the transition from one vibrational energy level to another.

When we land on a resonant frequency as we scan the range of frequencies, that corresponds to the frequency for a vibrational mode of the molecule. Some examples are shown below of methane:

Typically we measure absorptions. For simplicity, a non-rotating linear anharmonic oscillator has energy levels given to second order by:

$t i l \mathrm{de} {E}_{\upsilon} = t i l \mathrm{de} {\omega}_{e} \left(\upsilon + \frac{1}{2}\right) - t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + \frac{1}{2}\right)}^{2}$

where:

• $t i l \mathrm{de} {\omega}_{e}$ is the fundamental vibrational frequency of the molecule in its equilibrium position (no displacement).
• $t i l \mathrm{de} {\omega}_{e} {\chi}_{e}$ is the anharmonicity constant of the molecule in its equilibrium position (no displacement).

Those absorption transitions are given by:

$\textcolor{b l u e}{t i l \mathrm{de} {\nu}_{\upsilon \to \upsilon + 1}} = \left[t i l \mathrm{de} {\omega}_{e} \left(\upsilon + 1 + \frac{1}{2}\right) - t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + 1 + \frac{1}{2}\right)}^{2}\right] - \left[t i l \mathrm{de} {\omega}_{e} \left(\upsilon + \frac{1}{2}\right) - t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + \frac{1}{2}\right)}^{2}\right]$

$= t i l \mathrm{de} {\omega}_{e} \left(\upsilon + \frac{3}{2} - \upsilon - \frac{1}{2}\right) + t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + \frac{1}{2}\right)}^{2} - t i l \mathrm{de} {\omega}_{e} {\chi}_{e} {\left(\upsilon + \frac{3}{2}\right)}^{2}$

$= t i l \mathrm{de} {\omega}_{e} + t i l \mathrm{de} {\omega}_{e} {\chi}_{e} \left[{\left(\upsilon + \frac{1}{2}\right)}^{2} - {\left(\upsilon + \frac{3}{2}\right)}^{2}\right]$

$= t i l \mathrm{de} {\omega}_{e} + t i l \mathrm{de} {\omega}_{e} {\chi}_{e} \left[{\upsilon}^{2} + \upsilon + \frac{1}{4} - {\upsilon}^{2} - 3 \upsilon - \frac{9}{4}\right]$

= color(blue)(tildeomega_e -2tildeomega_echi_e(upsilon + 1)

• Vibrating bonds absorb infrared energy at a frequency that corresponds to the vibrational frequency of the bond.

In organic chemistry, this corresponds to frequencies of 15 to 120 THz.

These frequencies are expressed as wavenumbers:

"wavenumber" = "frequency"/"speed of light" = ν/c

So the wavenumbers range from 500 to 4000 cm⁻¹.

Molecules do not have rigid bond lengths and bond angles.

Rather, bond lengths and angles represent the average positions about which atoms vibrate.

Each vibration may cause a change in the dipole moment of the molecule. The alternating electrical field of the infrared radiation can interact with the alternating dipole moment.

If the frequency of the radiation matches the vibrational frequency, the bond will absorb the radiation. The amplitude of the vibration will increase.

The spectrometer detects this absorption and records it as a peak in a plot of transmission versus frequency.

Within a narrow range, each type of bond vibrates at a characteristic wavenumber. This makes infrared spectroscopy useful to identify functional groups in a molecule.

Here’s a short table of common absorption frequencies.

Notice how you can identify the important vibrations in the spectrum of ethyl acetate. The C=O and C-O absorptions tell you that this is an ester.

Here’s a video that gives a simple explanation of infrared spectroscopy.

The infrared spectrum tells us what functional groups are present in a molecule.

#### Explanation:

The bonds in molecules are vibrating, and the vibrational energy is quantized.

The bonds can stretch and bend only at certain allowed frequencies.

A molecule will absorb energy from radiation that has the same energy as its vibrational modes.

This energy is in the infrared region of the electromagnetic spectrum.

Each functional group has vibrational frequencies in a small region of the IR spectrum, so IR spectra give us information about the functional groups that are present.

Here is a table that lists the characteristic vibrational frequencies of some functional groups.

## Questions

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# Defining general acidic strength

The $$K_\mathrm{a}$$ or $$K_\mathrm{b}$$ values provide an idea of how likely a specific step is to happen.

However, how do you specify one acid just being stronger than another, as in the case of $$\ce{H2CO3}$$ $$(\mathrm{p}K_\mathrm{a1} = 3.6,$$ but $$\mathrm{p}K_\mathrm{a2} = 10.32)$$ and phenol $$(\mathrm{p}K_\mathrm{a} = 9.95)?$$ How do you compare them?

• Compare pKa1 for both compounds Jul 12 '20 at 10:31
• Isn't it used to measure the likelihood of just the first deprotonation? Jul 12 '20 at 10:54
• Isn't that what acidity is a measure of? Jul 12 '20 at 10:58
• Acidity is, by definition, the tendency of a compound to act as an H+ donor for Bronsted acids. For Lewis acids, acidity relates to the compound's ability to accept an electron pair. That doesn't necessarily mean the first deprotonation every time, does it? Jul 12 '20 at 11:32
• It generally is.. atleast to the best of my knowledge Jul 12 '20 at 12:00

According to the Brønsted-Lowry acid base theory, an acid is a compound that releases a $$\ce{H+}$$ ion to give a conjugate base. For most acids, this reaction exists in a dynamic equilibrium. The equilibrium constant of this reaction is what defines $$K_\mathrm{a}$$. Generally more stable the product, the more of the forward reaction takes place.

Using this last statement, when comparing two compounds, we can say that if the conjugate base of one compound is more stable, it would mean that the value of its $$K_\mathrm{a}$$ would be greater. This implies its $$\mathrm{p}K_\mathrm{a}$$ would be less, meaning that the compound would be more acidic than the other.

For example, taking the case of two compounds, $$\ce{HCOOH}$$ and $$\ce{C6H5-OH}$$: Here formate ion would have resonance via oxygen whereas phenoxide ion would have resonance via carbon. This makes phenol less acidic than formic acid.

$$\mathrm{p}K_\mathrm{a} \ {\text{ of phenol} = 10.0}$$

$$\mathrm{p}K_\mathrm{a} \ {\text{ of formic acid} = 3.75}$$

Now, if we consider polyprotic acids, we can only consider one hydrogen at a time. Successive $$K_\mathrm{a}$$s decrease in magnitudes of very large orders.

For example, $$\ce{H3PO4}$$ has $$\mathrm{p}K_\mathrm{a1} = 2.12$$ whereas $$\mathrm{p}K_\mathrm{a2} = 7.21$$ and $$\mathrm{p}K_\mathrm{a3} = 12.68$$.

• Ah, I didn't noticed it. But, in general, I use $\ce{}$ in most of my equations. I was also suggested in the beginning by some user to use $\ce{} instead of ${}$. Anyways, if @andselisk is using, then it'll be fine. btw, I edited just to make numerals beaultiful. Feel free to make further edits :) Jul 12 '20 at 13:45 • What is the difference between using "dollar\ce{...}dollar", with \ce, and using "dollar{...}dollar" without \ce ? Jul 12 '20 at 20:38 • @Maurice, see this answer for mhchem extension, i.e., $\ce{}\$ Jul 13 '20 at 8:58
$$K_\mathrm{a}$$ is the measurement of equilibrium constant when an acid dissociates as: $$\ce{HA <=> H+ + A-}$$ $$K_\mathrm{a} = \frac{[\ce{H+}][\ce{A-}]}{[\ce{HA}]}$$ The term $$\mathrm{p}K_\mathrm{a}$$ is defined as $$\mathrm{p}K_\mathrm{a} = -\log(K_\mathrm{a})$$. So lower the $$\mathrm{p}K_\mathrm{a}$$, the better is the acidity.
In the example given, if you wish to compare the acidity, you have to mention which proton of $$\ce{H2CO3}$$ you are interested to abstract. In other words, $$\ce{H2CO3}$$ is more acidic then phenol but $$\ce{HCO3-}$$ is less acidic then phenol.


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Mixed-integer linear fractional programming (MILFP)

Author: Xiang Zhao (SysEn 6800 Fall 2020)

Steward: Allen Yang, Fengqi You

Introduction

The mixed-integer linear fractional programming (MILFP) is a kind of mixed-integer nonlinear programming (MINLP) that is widely applied in chemical engineering, environmental engineering, and their hybrid field ranging from cyclic-scheduling problems to the life cycle optimization (LCO). Specifically, the objective function of the MINFP is shown as a ratio of two linear functions formed by various continuous variables and discrete variables. However, the pseudo-convexity and the combinatorial nature of the fractional objective function can cause computational challenges to the general-purpose global optimizers, such as BARON, to solve this MILFP problem. In this regard, we introduce the basic knowledge and solution steps of three algorithms, namely the Parametric Algorithm, Reformulation-Linearization method, and Branch-and-Bound with Charnes-Cooper Transformation Method, to efficiently and effectively tackle this computational challenge.

Standard Form and Properties

Consider such standard form of the MILFP:

{\displaystyle {\begin{aligned}\max \quad \mathbb {Q} (x,y)={c_{0}+\sum _{i}c_{1,i}m_{i}+\sum _{j}c_{2,j}y_{j} \over d_{0}+\sum _{i}d_{1,i}m_{i}+\sum _{j}d_{2,j}y_{j}}\\s.t.\quad \ a_{0,k}+\sum _{i}a_{1,i}m_{i}+\sum _{j}a_{2,j}y_{j}=0,\quad \forall k\in K\\m_{i}\geq 0,\quad \forall i\in I\\y_{j}\in {0,1},\quad \forall j\in J\end{aligned}}}

The properties of the objective function ${\displaystyle \mathbb {Q} (x,y)}$ are shown as follows:

1. Numbered list item ${\displaystyle \mathbb {Q} (x,y)}$ is (strictly) pseudoconcave and pseudoconvex over its domain.
2. Numbered list item The local optimal of ${\displaystyle \mathbb {Q} (x,y)}$ is the same as its global optimal.

Notably, several nonlinear solvers that can deal with the pseudoconvexity, such as the spatial branch-and-bound (SBB), are capable of solving the MILFP. However, the memory usage of these solvers is enormous when solving a large-scale problem that is applied in industrial scheduling or supply chain optimization project. Hence, we introduce the parametric algorithm, and reformulation-linearization method, which can reformulate the MILFP into the mixed-integer linear programming (MILP) problem, to reduce the memory usage and enhance solution efficiency.

Parametric Algorithm

One way to successively reformulate and solve the MILFP is to apply the parametric algorithm, which can find the global optimality within finite iterations. The linearly parametric form of the reformulated objective function has the advantage of directly finding the global optimality, while the size of the sub-problem remains the same. The solution approach is shown as follows:

The original form of the objective function is: ${\displaystyle \quad \mathbb {Q} (x,y)={c_{0}+\sum _{i}c_{1,i}m_{i}+\sum _{j}c_{2,j}y_{j} \over d_{0}+\sum _{i}d_{1,i}m_{i}+\sum _{j}d_{2,j}y_{j}}}$

We use a parametric parameter ${\displaystyle q}$ to reformulate the objective function ${\displaystyle \mathbb {Q} (x,y)}$ into ${\displaystyle \S (x,y,q)}$:

${\displaystyle \max \quad \mathbb {Q} (x,y)={A(x,y) \over B(x,y)}}$

is reformulated into

${\displaystyle \max \quad \S (x,y,q)=A(x,y)-q*B(x,y)}$

${\displaystyle A(x,y)={c_{0}+\sum _{i}c_{1,i}m_{i}+\sum _{j}c_{2,j}y_{j}}}$

${\displaystyle B(x,y)={d_{0}+\sum _{i}d_{1,i}m_{i}+\sum _{j}d_{2,j}y_{j}}}$


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# Trying to understand an exercise using factorials with induction

Exercise: Prove that (n + 1)! - n! = n(n!) for any n $\ge$ 1

I will skip the basic step since I understand that part.

(n + 2)! - (n + 1)! = (n + 1)!(n + 2) - n!(n + 1) I understand this line
But, I don't understand starting at this next line
= n(n + 1)! + 2(n + 1)! - n(n!) - n!
= n(n + 1)! + (n + 1)! - n(n!) + (n + 1)! - n!
= n(n + 1)! + (n + 1)! - n(n!) + n(n!)
= n(n + 1)! + (n + 1)!
= (n + 1)!(n + 1)

If anyone can help me to understand this then that would be great!

Thanks,

Tony

-
Is any body aware of any good tutorials that show examples of solving factorials by induction? – Tony Oct 21 '13 at 21:02

$(n+2)! = (n+2)(n+1)! = n(n+1)! + 2(n+1)!$, by the distributive property.

The inductive step would be to assume that $(n + 1)! - n! = n(n!)$ holds for some $n > 1$.

We then want to show that $\color{blue}{(n+2)!} -\color{red}{(n+1)!} = (n+1)(n+1)!$

We begin by rearranging...

$$\color{blue}{(n+2)!} -\color{red}{(n+1)!} =\color{blue}{(n+2)}(n+1)! - \color{red}{(n+1)}(n!)$$ $$=\color{blue}{n}(n+1)! + \color{green}{2}(n+1)! - \color{red}{n}(n!) -\color{red}{1}(n!)$$

Notice that I changed the blue $2$ into green. That is so we can have $$=\color{black}{n}(n+1)! + \color{green}{1}(n+1)! - \color{black}{n}(n!) + \color{green}{1}(n+1)! -\color{black}{1}(n!)$$

The two terms on the right are $(n+1)! - n!$, which equals $\color{red}{n(n!)}$ by hypothesis. So now we have... $$=\color{black}{n}(n+1)! + \color{black}{1}(n+1)! - \color{black}{n}(n!) + \color{red}{n(n!)}$$

The two terms on the right cancel. $$=\color{blue}{n}(n+1)! + \color{blue}{1}(n+1)!$$

Using the distributive property (factoring out a $(n+1)!$, if you will) ... $$\color{blue}{(n+1)}(n+1)!$$

So the claim holds for $n+1$.

-
Okay, thanks. I see that now but, what happened to the 2 in the next line and, is n! = (n + 1)! - n!? Hmm, is 2(n + 1)! equal to both of the (n + 1)! in the next line? – Tony Oct 21 '13 at 20:54
The $2(n+1)!$ splits into two terms of $(n+1)!$ in the next line. You would do well to write all this out and use colored pencils to keep track of the various terms/factors. – The Chaz 2.0 Oct 21 '13 at 21:12
Thanks again. I have copied your answer and made some notes for it also. – Tony Oct 21 '13 at 21:48
Hi again. Am I correct in assuming that (n + 1)! - n! is equal to n(n!). If this is correct do you mind showing how this is worked out? I am sorry for all of the questions but I am really struggling with this exercise. The study guide I have does not really show a lot of work on solving factorials with induction. – Tony Oct 21 '13 at 21:58
Hey, that is great. I'll take this and make sure I understand it. Thanks for all of you help. – Tony Oct 21 '13 at 23:02


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# history expansion

Reuse and expand the shell history in sh, bash, zsh, rbash and ksh. More information: https://www.gnu.org/software/bash/manual/html_node/History-Interaction.

• Run the previous command as root (!! is replaced by the previous command):

sudo !!

• Run a command with the last argument of the previous command:

{{command}} !\$

• Run a command with the first argument of the previous command:

{{command}} !^

• Run the Nth command of the history:

!{{n}}

• Run the command n lines back in the history:

!-{{n}}

• Run the most recent command containing string:

!?{{string}}?

• Run the previous command, replacing string1 with string2:

^{{string1}}^{{string2}}^

• Perform a history expansion, but print the command that would be run instead of actually running it:

{{!-n}}:p


text:
Resurrecting the minimal renormalizable supersymmetric SU(5) model - Archive ouverte HAL Access content directly
Journal Articles Journal of High Energy Physics Year : 2016

## Resurrecting the minimal renormalizable supersymmetric SU(5) model

(1) , (2) , (3)
1
2
3
Borut Bajc
• Function : Author
Stéphane Lavignac
• Function : Author
Timon Mede
• Function : Author

#### Abstract

It is a well-known fact that the minimal renormalizable supersymmetric SU(5) model is ruled out assuming superpartner masses of the order of a few TeV. Giving up this constraint and assuming only SU(5) boundary conditions for the soft terms, we find that the model is still alive. The viable region of the parameter space typically features superpartner masses of order $10^2$ to $10^4$ TeV, with tan beta values between 2 and 5, but lighter spectra with single states around 10 TeV are also possible. The main constraints come from proton decay, the Higgs mass, the requirement of the SU(5) spectrum being reasonably below the Planck scale, and the lifetime of the universe. A generic feature of the model is metastability of the electroweak vacuum. In the absence of a suitable dark matter particle in the neutralino sector, a light (order GeV or smaller) gravitino is a natural candidate.

### Dates and versions

cea-01304334 , version 1 (19-04-2016)

### Identifiers

• HAL Id : cea-01304334 , version 1
• ARXIV :
• DOI :

### Cite

Borut Bajc, Stéphane Lavignac, Timon Mede. Resurrecting the minimal renormalizable supersymmetric SU(5) model. Journal of High Energy Physics, 2016, 2016 (1), ⟨10.1007/JHEP01(2016)044⟩. ⟨cea-01304334⟩

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text:
Todd Leen

#### Abstract

Ideally pattern recognition machines provide constant output when the inputs are transformed under a group 9 of desired invariances. These invariances can be achieved by enhancing the training data to include examples of inputs transformed by elements of g, while leaving the corresponding targets unchanged. Alternatively the cost function for training can include a regularization term that penalizes changes in the output when the input is transformed un(cid:173) der the group.

This paper relates the two approaches, showing precisely the sense in which the regularized cost function approximates the result of adding transformed (or distorted) examples to the training data. The cost function for the enhanced training set is equivalent to the sum of the original cost function plus a regularizer. For unbiased models, the regularizer reduces to the intuitively obvious choice - a term that penalizes changes in the output when the inputs are transformed under the group. For infinitesimal transformations, the coefficient of the regularization term reduces to the variance of the distortions introduced into the training data. This correspon(cid:173) dence provides a simple bridge between the two approaches.


text:
Problems

# Sparse Tables

The array with n integers is given. Write a program that answers the question: find the minimum on a segment between u and v inclusively.

#### Input

The first line contains three positive integers n, m (1n105, m107) and a1 (1a1 < 16714589) - the number of elements in array, the number of queries and the first element in an array respectively. The second line contains two positive integers u1 and v1 (1u1, v1n) - the first query.

The elements a2, a3, ..., an are given with the next formula:

ai+1 = (23 · ai + 21563) mod 16714589.

For example, if n = 10, a1 = 12345, we get the next array:

a = (12345, 305498, 7048017, 11694653, 1565158, 2591019, 9471233, 570265, 13137658, 1325095).

The queries are generated this way:

ui+1 = ((17 · ui + 751 + ansi + 2i) mod n) + 1,

vi+1 = ((13 · vi + 593 + ansi + 5i) mod n) + 1,

where ansi is the answer to the i-th query.

#### Output

Print um, vm and ansm (the values for the last query and the answer to it).

Time limit 2 seconds
Memory limit 128 MiB
Input example #1
10 8 12345
3 9

Output example #1
5 3 1565158

Author В.Гольдштейн
Source Зимние сборы в Харькове 2010 День 2


text:
## pushing events

Questions about the LÖVE API, installing LÖVE and other support related questions go here.
Forum rules
omarshariff
Prole
Posts: 3
Joined: Mon Aug 30, 2010 11:14 pm

### pushing events

Anyone have any ideas why the following simple code doesn't work? Is there something I'm missing regarding pushing events and the love.handlers table?

Code: Select all

function love.load()
love.handlers["walk"] = function(d) print("awesome " .. d) end
end

function love.update(dt)
if love.keyboard.isDown('a') then
love.event.push("walk", 'left')
elseif love.keyboard.isDown('d') then
love.event.push("walk", 'right')
end
end
Cheers!

Tesselode
Party member
Posts: 555
Joined: Fri Jul 23, 2010 7:55 pm

### Re: pushing events

So you made a function? I don't think you need to use event.push. Just say love.handlers["walk"]("left").

bartbes
Sex machine
Posts: 4946
Joined: Fri Aug 29, 2008 10:35 am
Location: The Netherlands
Contact:

### Re: pushing events

Because the event can't be mapped to an inbuilt type and hence is discarded, love.graphics.push should be returning false.
But why this rape of the even system, I ask?

omarshariff
Prole
Posts: 3
Joined: Mon Aug 30, 2010 11:14 pm

### Re: pushing events

"But why this rape of the even system, I ask?"

I didn't realize that the event system was restricted to love only, the wiki doesn't make this distinction. In fact, the wiki appears to suggest that the love.event objects and functions are available to regular developers for use. Could you please expand on why you think this is an abuse of the event system? I thought that my approach was quite clean and elegant, unless I have misunderstood the purpose of the event system. Could you explain?

And to answer Tesselodes question, I was hoping to take advantage of the event system so that I could ensure a seperation of concerns. You see, in my program I am attempting to seperate the input controllers (joysticks) and AI controllers (AI inputs) so that my characters can be controled by either. I've done this by reducing the various outputs from each to simple instructions (MOVELEFT, FIRERIGHT, etc). I was hoping that I would be able to simplify things and ensure this speration by having the various characters/sprites listen for events. That way, I should be able to swap an AI controller character with a joystick controlled character at runtime.

This is mere exploratory work at the moment, hence my first initial investigations into the love.event system. Looking at the contents of the default love.run function, the event loop there seemed to duplicate what I wanted, which led me to the love.event wiki entry.

Again, if anyone has more information on the love.event system, especially as to why I should not use it for such things, I would appreaciate it (or even better, update the wiki with these reasons!)

Robin
The Omniscient
Posts: 6506
Joined: Fri Feb 20, 2009 4:29 pm
Location: The Netherlands
Contact:

### Re: pushing events

omarshariff wrote:I didn't realize that the event system was restricted to love only, the wiki doesn't make this distinction. In fact, the wiki appears to suggest that the love.event objects and functions are available to regular developers for use. Could you please expand on why you think this is an abuse of the event system? I thought that my approach was quite clean and elegant, unless I have misunderstood the purpose of the event system. Could you explain?
love.event is for passing events such as key down, a mouse click or quitting the game from the lower depths to their handler (love.keypressed, love.mousepressed, etc.)

The only thing you'll usually push is 'q', to quit the game.

What you're trying to do is normally done by functions.

Code: Select all

function walk(direction)
-- do something
end
omarshariff wrote:And to answer Tesselodes question, I was hoping to take advantage of the event system so that I could ensure a seperation of concerns. You see, in my program I am attempting to seperate the input controllers (joysticks) and AI controllers (AI inputs) so that my characters can be controled by either. I've done this by reducing the various outputs from each to simple instructions (MOVELEFT, FIRERIGHT, etc). I was hoping that I would be able to simplify things and ensure this speration by having the various characters/sprites listen for events. That way, I should be able to swap an AI controller character with a joystick controlled character at runtime.
Still, plain old functions do this excellently.

Otherwise, I think there are libraries that implement something like you want. (kikito, one of yours maybe?)

I don't think we ever had anyone else misunderstand the purpose of love.event, but perhaps it would be good to explain on the wiki that it mostly takes a passive role, rather than an active one, and only in relation to system events, as opposed to game events.

bartbes
Sex machine
Posts: 4946
Joined: Fri Aug 29, 2008 10:35 am
Location: The Netherlands
Contact:

### Re: pushing events

omarshariff wrote:Could you please expand on why you think this is an abuse of the event system?
Well, Robin had a nice explanation:
Robin wrote:only in relation to system events, as opposed to game events.
There we go.

omarshariff
Prole
Posts: 3
Joined: Mon Aug 30, 2010 11:14 pm

### Re: pushing events

I don't think we ever had anyone else misunderstand the purpose of love.event
raises hand weakly...

Cheers for the info guys. I'll look into those suggestions tonight. Thanks!

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Users browsing this forum: Geecko, Google [Bot], steVeRoll and 16 guests


text:
kyamagu/mexopencv

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Collection and a development kit of matlab mex functions for OpenCV library
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mexopencv

Collection and a development kit of matlab mex functions for OpenCV library

The package provides matlab mex functions that interface a hundred of OpenCV APIs. Also the package contains C++ class that converts between Matlab's native data type and OpenCV data types. The package is suitable for fast prototyping of OpenCV application in Matlab, use of OpenCV as an external toolbox in Matlab, and development of an original mex function.

Contents

The project tree is organized as follows.

+cv/            directory to put compiled mex files, wrappers, or help files
Doxyfile        config file for doxygen
Makefile        make script
doc/            directory for documentation
lib/            directory for compiled c++ library files
samples/        directory for sample application codes
src/            directory for c++ source files
src/+cv/        directory for mex source files
src/+cv/private directory for private mex source files
test/           directory for test scripts and resources
utils/          directory for utilities

Compile

Prerequisite:

• Unix: matlab, opencv (>=2.3.1), g++, make, pkg-config
• Windows: matlab, opencv (>=2.3.1), supported compiler

Note: opencv (>=2.1.0) is partially supported.

Unix

First make sure you have OpenCV installed in the system. If not, install the package available in your package manager (e.g., libopencv-dev in Debian/Ubuntu, opencv-devel in Fedora, opencv in Macports), or install the source package from http://opencv.willowgarage.com/wiki/ . Make sure pkg-config command can identify opencv path. If you have all the prerequisite, go to the mexopencv directory and type:

$make  This will build and place all mex functions inside +cv/. Specify your matlab directory if you install matlab other than /usr/local/matlab $ make MATLABDIR=/Applications/MATLAB_R2012a.app

Optionally you can test the library functionality

$make test  If matlab says 'Library not loaded' or any other error in the test, it's likely the compatibility issue between a system library and matlab's internal library. You might be able to fix this issue by preloading the library file. On linux, set the correct library path in LD_PRELOAD environmental variable. For example, if you see GLIBCXX_3.4.15 error in mex, use the following to start matlab. $ LD_PRELOAD=/usr/lib/libstdc++.so.6 matlab

Note that you need to find the correct path to the shared object. For example, /usr/lib64/ instead of /usr/lib/. You can use locate command to find the location of the shared object.

On Mac OS X, this variable is named DYLD_INSERT_LIBRARIES. You can check ldd command line tool to check the dependency of the mex file in linux. On mac, you can use otool -L command.

Developer documentation can be generated with doxygen if installed

\$ make doc

This will create html and latex files under doc/.

Windows

Make sure you have OpenCV installed in the system and correctly set up Path system variable. e.g., c:\opencv\build\x86\vc10\bin. See http://opencv.willowgarage.com/wiki/ for the instruction. Also make sure you install a compiler supported by Matlab. See http://www.mathworks.com/support/sysreq/previous_releases.html for the list of supported compilers for different versions of Matlab. Windows 64-bit users need to install Windows SDK.

Once you satisfy the above requirement, in the matlab shell, type

>> cv.make

to build all mex functions. By default, mexopencv assumes the OpenCV library is installed in C:\opencv. If this is not the case, specify the path as an argument.

>> cv.make('opencv_path', 'c:\your\path\to\opencv')

Note that if you build OpenCV from source, this path specification does not work. You need to replace dll files in the OpenCV package with newly built binaries. Or, you need to modify +cv/make.m to correctly link your mex files with the library.

To remove existing mexopencv binaries, use the following command.

>> cv.make('clean')

Visual Studio 2008 compatibility issue

Users report incompatibility with Visual Studio 2008. Try not to use Visual Studio 2008 with mexopencv. For this reason, mexopencv on Windows platform do not work with Matlab R2009b or earlier.

Nevertheless, if you want to try using Visual Studio 2008, obtain stdint.h and use cv.make to compile the package. Visual Studio 2008 or earlier does not comply C99 standard and lacks stdint.h header file. Luckily, the header file is available on the Web. For example, http://msinttypes.googlecode.com/svn/trunk/stdint.h

Place this file under include directory in the mexopencv package.

Error: Invalid MEX file or Segmentation fault

The OpenCV windows package contains c++ binary files compiled with _SECURE_SCL=1 flag, but mex command in Matlab does not use this option by default, which results in Invalid MEX file or segmentation fault on execution. The current version of cv.make script adds _SECURE_SCL=1 flag in the build command and should have no problem with the distributed binary package.

If you see Invalid MEX file or segmentation fault with manually built OpenCV dll's, first make sure you compile OpenCV with the same _SECURE_SCL flag to the mex command. The default mex configuration, which is created with the mex -setup command in matlab, is located in the following path in recent versions of Windows.

C:\Users\(Username)\AppData\Roaming\MathWorks\MATLAB\(version)\mexopts.bat

Open this file and edit /D_SECURE_SCL option.

If you see Invalid MEX file error even when having the matched _SECURE_SCL flag, it probably indicates some other compatibility issues. Please file a bug report at http://github.com/kyamagu/mexopencv .

Usage

Once mex functions are compiled, you can add path to the project directory and call mex functions within matlab using package name cv.

addpath('/path/to/mexopencv');
result = cv.filter2D(img, kern);  % with package name 'cv'
import cv.*;
result = filter2D(img, kern);     % no need to specify 'cv' after imported

Note that some functions such as cv.imread overload Matlab's builtin function when imported. Use the scoped name when you need to avoid name collision. It is also possible to import individual functions. Check help import in matlab.

Check a list of functions available by help command in matlab.

>> help cv; % shows list of functions in package 'cv'

Contents of cv:

GaussianBlur                   - Smoothes an image using a Gaussian filter
Laplacian                      - Calculates the Laplacian of an image
VideoCapture                   - VideoCapture wrapper class
...

>> help cv.VideoCapture; % shows documentation of VideoCapture

VIDEOCAPTURE  VideoCapture wrapper class

Class for video capturing from video files or cameras. The class
provides Matlab API for capturing video from cameras or for reading
video files. Here is how the class can be used:
...

Look at the samples/ directory for an example of an application.

Also mexopencv includes a simple documentation utility that generates HTML help files for matlab. The following command creates a user documentation under doc/matlab/ directory.

addpath('utils');
MDoc;

Online documentation is available at http://www.cs.stonybrook.edu/~kyamagu/mexopencv

You can test the functionality of compiled files by UnitTest class located inside test directory.

addpath('test');
UnitTest;

Developing a new mex function

All you need to do is to add your mex source file in src/+cv/. If you want to add a mex function called myfunc, create src/+cv/myfunc.cpp. The minimum contents of the myfunc.cpp would look like this:

#include "mexopencv.hpp"
void mexFunction( int nlhs, mxArray *plhs[],
int nrhs, const mxArray *prhs[] )
{
// Check arguments
if (nlhs!=1 || nrhs!=1)
mexErrMsgIdAndTxt("myfunc:invalidArgs", "Wrong number of arguments");

// Convert MxArray to cv::Mat
cv::Mat mat = MxArray(prhs[0]).toMat();

// Do whatever you want

// Convert cv::Mat back to mxArray*
plhs[0] = MxArray(mat);
}

This example simply copies an input to cv::Mat object and then copies again to the output. Notice how the MxArray class provided by mexopencv converts mxArray to cv::Mat object. Of course you would want to do something more with the object. Once you create a file, type make to build your new function. The compiled mex function will be located inside +cv/ and accessible through cv.myfunc within matlab.

The mexopencv.hpp header includes a class MxArray to manipulate mxArray object. Mostly this class is used to convert between opencv data types and mxArray.

int i            = MxArray(prhs[0]).toInt();
double d         = MxArray(prhs[0]).toDouble();
bool b           = MxArray(prhs[0]).toBool();
std::string s    = MxArray(prhs[0]).toString();
cv::Mat mat      = MxArray(prhs[0]).toMat();   // For pixels
cv::Mat ndmat    = MxArray(prhs[0]).toMatND(); // For N-D array
cv::Point pt     = MxArray(prhs[0]).toPoint();
cv::Size siz     = MxArray(prhs[0]).toSize();
cv::Rect rct     = MxArray(prhs[0]).toRect();
cv::Scalar sc    = MxArray(prhs[0]).toScalar();
cv::SparseMat sp = MxArray(prhs[0]).toSparseMat(); // Only double to float

mxArray* plhs[0] = MxArray(i);
mxArray* plhs[0] = MxArray(d);
mxArray* plhs[0] = MxArray(b);
mxArray* plhs[0] = MxArray(s);
mxArray* plhs[0] = MxArray(mat);
mxArray* plhs[0] = MxArray(ndmat);
mxArray* plhs[0] = MxArray(pt);
mxArray* plhs[0] = MxArray(siz);
mxArray* plhs[0] = MxArray(rct);
mxArray* plhs[0] = MxArray(sc);
mxArray* plhs[0] = MxArray(sp); // Only 2D float to double

Check MxArraay.hpp for a complete list of the conversion API.

If you rather want to develop a matlab function that internally calls a mex function, make use of the +cv/private directory. Any function placed under private directory is only accessible from +cv/ directory. So, for example, when you want to design a matlab class that wraps the various behavior of the mex function, define your class at +cv/MyClass.m and develop a mex function dedicated for that class in src/+cv/private/MyClass_.cpp. Inside of +cv/MyClass.m, you can call MyClass_() without cv namescope.

Testing

You can optionally add a testing script for your new function. The testing convention in mexopencv is that testing scripts are all written as a static function in a matlab class. For example, test/unit_tests/TestFilter2D.m is a class that describes test cases for filter2d function. Inside of the class, a couple of test cases are written as a static function whose name starts with 'test'.

If there is such a class inside test/unit_tests/, typing make test would invoke all test cases and show your result. Use test/ directory to place any resource file necessary for testing. An example of testing class is shown below:

classdef TestMyFunc
methods (Static)
function test_1
ref = [1,2,3];                  % reference output
dst = cv.myfunc(src);           % execute your function
assert(all(dst(:) == ref(:)));  % check the output
end

function test_error_1
try
cv.myfunc('foo');           % myfunc should throw an error
error('UnitTest:Fail','myfunc incorrectly returned');
catch e
assert(strcmp(e.identifier,'mexopencv:error'));
end
end
end
end

In Windows, add path to the test directory and invoke UnitTest to run all the test routines.

Documenting

You can create a Matlab help documentation for mex function by having the same file with '.m' extension. For example, a help file for filter2D.mex* would be filter2D.m. Inside the help file should be only matlab comments. An example is shown below:

%MYFUNC  brief description about myfunc
%
% Detailed description of function continues
% ...


text:
# Linear equation Numerical analysis consistency

\begin{align*} x - \alpha y &= 1\\ \alpha x - y &= 1 \end{align*}

For which values of alpha does the system have an infinite number of solutions, no solutions and one solution.

Find the solution when it is unique.

My attempt:

$-\alpha \cdot \mathrm{eqn}_1 + \mathrm{eqn}_2$ resulting in $(\alpha^2 - 1)y = 1-\alpha$.

then we get $y = (1-\alpha)/(\alpha^2-1)$

so, $y = -1/(1+\alpha)$, but I am trying to proceed

-
Is there a second half to your question? It looks like it got cut off mid sentence – Willie Wong May 3 '12 at 9:30
thanks for noticing & editing – mary May 3 '12 at 9:39
@mary: Note that you can't divide by $\alpha^2 - 1$ if this term is 0. – martini May 3 '12 at 9:39

\begin{align*} x - \alpha y &= 1\\ \alpha x - y &= 1 \end{align*}

By simplifying, we have: $$\frac{x-1}{\alpha}=\alpha x-1$$

Which is:

$$(\alpha^2-1)x-(\alpha+1)=0$$

$$x=\frac{\alpha+1}{\alpha^2-1}$$

Assuming $\alpha \neq \pm 1$, $x=\frac{1}{\alpha-1}$ is your unique solution. For $\alpha=1$ and $\alpha=-1$ you have many and no solution, respectively.

-

What you did looks good so far. If every single step you made is legitimate you have found your unique solution: $y=-1/(1+\alpha)$ and $x=1-\alpha/(1+\alpha)$. What do I mean by legitimate? When you are dealing with this kind of exercise what usually "goes wrong" is that you want to divide by $0$ somewhere. You divided by $\alpha^2-1$ at one point, this doesn't work when $\alpha=\pm1$. What happens in these two cases?

-


text:
Threading with Ligand

2 posts / 0 new
Threading with Ligand
#1

Hi everyone,

I'm trying to use the partial_thread application to fill in some gaps in a PDB crystal structure, 2hko.  This structure has an FAD ligand and a key water I'm trying to keep during threading.  I've cleaned the file as recommended, changed the FAD ligand atom types to match the canonical Rosetta atom types, and made similar changes to my water.  While the threading runs, the ligand gets thrown out.

This is the command I used:

partial_thread.static.linuxgccrelease -database \$ROSETTA_HOME/main/database -in:file:fasta lsd1.fasta -in:file:alignment alignment.aln -in:file:template_pdb 2hko_cleaned.pdb -verbose

I'm pretty new to Rosetta so please bear with me!  Thanks,

-Eugene

AttachmentSize
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Post Situation:
Thu, 2018-12-06 15:10
ekwan

The output of the partial_thread application will only contain the contents of the input fasta file. If you don't have a ligand in the fasta file, it won't be present in the output.

I think you can take advantage of the fact that the partial_thread application actually takes "annotated sequence" fasta files, rather than strictly one letter ones. As such, you can add an entry like Z[FAD] to the sequence where you want your ligand to be. The Z comes from the one letter code specified in the IO_STRING line of the params file for your ligand (normally Z), and the entry in square brackets is the full name of the ligand (the name from the NAME line of the params file). You should then also include the ligand as a position in your alignment file, so partial_thread knows it's supposed to copy it over.

Sometimes this is too much hassle to get right, though. The potentially easier way of doing things is to do the partial threading on just the protein sequence, and then copy over the ligand coordinates after the fact. (You'll still need to do adjustments to the input fasta file to the hybridize stage, though, to make sure you get the ligand there.)

Fri, 2019-03-29 13:21
rmoretti


text:
# Closed form to an interesting series: $\sum_{n=1}^\infty \frac{1}{1+n^3}$

Intutitively, I feel that there is a closed form to $$\sum_{n=1}^\infty \frac{1}{1+n^3}$$

I don't know why but this sum has really proved difficult. Attempted manipulating a Mellin Transform on the integral solution:

$$\int_0^\infty \frac{\text{d}x}{1+x^3}=\frac{\pi}{3}\csc \frac{\pi}{3}$$ But to little avail.

Checking W|A gives the austere solution: $$\frac{1}{3}\sum_{\{x|x^3+1=0\}} x \space\text{digamma}(1-x)$$

Which I completely don't understand. Thank you for any help.

• There is no "n" in the summand. Do you mean $\frac{1}{1+n^3}$? – Eleven-Eleven Dec 24 '15 at 1:50
• @Eleven-Eleven Yea, allow me to fix... – user266519 Dec 24 '15 at 1:51
• Look up Apery's constant. I don't think you are going to find a closed form. This is related to $\zeta(3)$. – Eleven-Eleven Dec 24 '15 at 1:54
• Could you please explain the purpose of applying a Mellin Transform? I know about Mellin Transfrom, but its application. – robit Dec 24 '15 at 3:58

Hint. You may use the following series representation of the digamma function $$\psi(z+1) + \gamma = \sum_{n=1}^{\infty}\left( \frac{1}n - \frac{1}{n+z}\right).\tag1$$ Then your goal is to rewrite the general term of your series in a form allowing to use $(1)$. You may start with \begin{align} \frac{1}{1+n^3}=\frac{1}{(n+1)(n-z_0)(n-\bar{z}_0)} \end{align} where $\displaystyle z_0=\frac{1+i\sqrt{3}}2$, then make a partial fraction decomposition giving
$$\frac{1}{1+n^3}=a_1\left(\frac{1}n - \frac{1}{n+1}\right)+a_2\left(\frac{1}n - \frac{1}{n-z_0}\right)+a_3\left(\frac{1}n - \frac{1}{n-\bar{z}_0}\right). \tag2$$
By summing $(2)$ you get a closed form of your initial series.


text: Now imagine that the brass disk is placed horizontally and that it is set to rotate without friction about its (vertical) axle with angular speed ?0. Then, the aluminum disk is dropped on top of the rotating brass disk, and due to the friction between both, after a very short time they are both rotating together with angular speed ?f. You can solve this question using conservation of angular momentum. Why is that?


text:
# HMatrix¶

class HMatrix(*args)

Hierarchical matrices.

Hierarchical matrices (or HMatrix) are a compressed representation of dense matrices. In many applications, matrix coefficients represent an interaction between two degrees of freedom; when these interactions are smooth, it is possible to approximate sub-blocks by a local low-rank approximation B =~ UV^T where B has dimension (m,n), U (m,k), and V (n,k). Of course, this is interesting only if k is much lower than m and n.

In order to obtain this compressed representation, several different steps must be performed:

1. Clustering: creation of rows and columns cluster trees Vertices where interactions are computed are reordered to improve locality. A binary space partition algorithm is used to recursively divide vertex set. Root cell contains all vertices. At each recursion step, a cell is divided into two new cells until it contains less than a given number of vertices. Space partition is performed orthogonally to original axis, by cutting its longest dimension.
• The ‘median’ clustering algorithm divides a cell into two cells containing the same number of degrees of freedom.
• The ‘geometric’ clustering algorithm divides a cell into two cells of the same geometric size
• The ‘hybrid’ clustering algorithm is a mix. It first performs a ‘median’ bisection; if volumes of these new cells are very different, a ‘geometric’ clustering is performed instead.
2. Admissibility: creation of an empty HMatrix structure The first step created a full binary tree for rows and columns degrees of freedom. We will now create a hierarchical representation of our matrix by checking whether some blocks can be replaced by low-rank approximations. The whole matrix represents the interactions of all rows degrees of freedom against all columns degrees of freedom. It can not be approximated by a low-rank approximation, and thus it is replaced by 4 blocks obtained by considering interactions between rows and columns children nodes. This operation is performed recursively. At each step, we compute axis aligned bounding boxes rows_bbox and cols_bbox: if min(diameter(rows_bbox), diameter(cols_bbox)) <= eta*distance(rows_bbox, cols_bbox) then we consider that interaction between rows and columns degrees of freedom can have a local low-rank approximation, and recursion is stopped. Otherwise, we recurse until bottom cluster tree is reached. The whole matrix is thus represented by a 4-tree where leaves will contain either low-rank approximation or full blocks. The eta parameter is called the admissibility factor, and it can be modified.
3. Assembly: coefficients computations The hierarchical structure of the matrix has been computed during step 2. To compute coefficients, we call the assemble method and provide a callable to compute interaction between two nodes. Full blocks are computed by calling this callable for the whole block. If compression method is ‘SVD’, low-rank approximation is computed by first computing the whole block, then finding its singular value decomposition, and rank is truncated so that error does not exceed assemblyEpsilon. This method is precise, but very costly. If compression method is a variant of ACA, only few rows and columns are computed. This is much more efficient, but error may be larger than expected on some problems.
4. Matrix computations Once an HMatrix is computed, usual linear algebra operations can be performed. Matrix can be factorized in-place, in order to solve systems. Or we can compute its product by a matrix or vector. But keep in mind that rows and columns are reordered internally, and thus results may differ sensibly from standard dense representation (for instance when computing a Cholesky or LU decomposition).

Methods

 assemble(*args) Assemble matrix. assembleReal(callable, symmetry) Assemble matrix. assembleTensor(callable, outputDimension, …) Assemble matrix by block. compressionRatio() Compression ratio accessor. copy() Copy matrix. dump(name) Save matrix to a file. factorize(method) Factorize matrix. fullrkRatio() Block ratio accessor. gemm(transA, transB, alpha, a, b, beta) Multiply matrix in-place self=alpha*op(A)*op(B)+beta*self. gemv(trans, alpha, x, beta, y) Multiply vector in-place y=alpha*op(A)*x+beta*y. getClassName() Accessor to the object’s name. getDiagonal() Diagonal values accessor. getId() Accessor to the object’s id. getImplementation(*args) Accessor to the underlying implementation. getName() Accessor to the object’s name. getNbColumns() Accessor to the number of columns. getNbRows() Accessor to the number of rows. norm() Compute norm value. scale(alpha) Scale matrix in-place A=alpha*A. setName(name) Accessor to the object’s name. solve(*args) Solve linear system op(A)*x=b, after A has been factorized. solveLower(*args) Solve lower linear system op(L)*x=b, after A has been factorized. transpose() Transpose matrix in-place.
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

assemble(*args)

Assemble matrix.

Parameters: f : HMatrixRealAssemblyFunction or HMatrixTensorRealAssemblyFunction Assembly function. symmetry : str Symmetry flag, either N or L
assembleReal(callable, symmetry)

Assemble matrix.

Parameters: f : assembly function Callable that takes i,j int parameters and returns a float symmetry : str Symmetry flag, either N or L
assembleTensor(callable, outputDimension, symmetry)

Assemble matrix by block.

Parameters: f : assembly function Callable that takes i,j int parameters and returns a Matrix outputDimension : int Block dimension symmetry : str Symmetry flag, either N or L
compressionRatio()

Compression ratio accessor.

Returns: ratio : 2-tuple of int Numbers of elements in the compressed and uncompressed forms.
copy()

Copy matrix.

As factorization overwrites matrix contents, this method is useful to get a copy of assembled matrix before it is factorized.

Returns: matrix : HMatrix Matrix copy.
dump(name)

Save matrix to a file.

Parameters: fileName : str File name to save to.
factorize(method)

Factorize matrix.

Parameters: method : str Factorization method, either one of: LDLt, LLt or LU
fullrkRatio()

Block ratio accessor.

Returns: ratio : 2-tuple of int Numbers of elements in full blocks and low rank blocks.
gemm(transA, transB, alpha, a, b, beta)

Multiply matrix in-place self=alpha*op(A)*op(B)+beta*self.

Parameters: transA : str Whether to use A or A^t: either N or T. transB : str Whether to use B or B^t: either N or T. alpha : float Coefficient a : HMatrix Multiplied matrix A. b : HMatrix Multiplied matrix B. beta : float Coefficient.
gemv(trans, alpha, x, beta, y)

Multiply vector in-place y=alpha*op(A)*x+beta*y.

Parameters: trans : str Whether to use A or A^t: either N or T. alpha : float Coefficient x : sequence of float Vector to multiply. beta : float Coefficient. y : Point Vector multiplied in-place.
getClassName()

Accessor to the object’s name.

Returns: class_name : str The object class name (object.__class__.__name__).
getDiagonal()

Diagonal values accessor.

Returns: diag : Point Diagonal values.
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getImplementation(*args)

Accessor to the underlying implementation.

Returns: impl : Implementation The implementation class.
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getNbColumns()

Accessor to the number of columns.

Returns: nbColumns : int Number of columns.
getNbRows()

Accessor to the number of rows.

Returns: nbRows : int Number of rows.
norm()

Compute norm value.

Returns: norm : float Frobenius norm.
scale(alpha)

Scale matrix in-place A=alpha*A.

Parameters: alpha : float Coefficient.
setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
solve(*args)

Solve linear system op(A)*x=b, after A has been factorized.

Parameters: b : sequence of float or Matrix Second term of the equation, vector or matrix. trans : bool Whether to solve the equation with A (False) or A^t (True). Defaults to False. x : Equation solution, vector or matrix.
solveLower(*args)

Solve lower linear system op(L)*x=b, after A has been factorized.

Parameters: b : sequence of float or Matrix Second term of the equation, vector or matrix. trans : bool Whether to solve the equation with L (False) or L^t (True). Defaults to False. x : Equation solution, vector or matrix.
transpose()

Transpose matrix in-place.


text:
• Corpus ID: 119096562

Trialities and Exceptional Lie Algebras: DECONSTRUCTING the Magic Square

@article{Evans2009TrialitiesAE,
title={Trialities and Exceptional Lie Algebras: DECONSTRUCTING the Magic Square},
author={Jonathan M. Evans},
journal={arXiv: High Energy Physics - Theory},
year={2009}
}
• Jonathan M. Evans
• Published 9 October 2009
• Mathematics
• arXiv: High Energy Physics - Theory
A construction of the magic square, and hence of exceptional Lie algebras, is carried out using trialities rather than division algebras. By way of preparation, a comprehensive discussion of trialities is given, incorporating a number of novel results and proofs. Many of the techniques are closely related to, or derived from, ideas which are commonplace in theoretical physics. The importance of symmetric spaces in the magic square construction is clarified, allowing the Jacobi property to be…
8 Citations

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References

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• Mathematics
• 2000
This paper is an investigation of the relation between Tit's magic square of Lie algebras and certain Lie algebras of 3 ×3 and 6 × 6 matrices with entries in alternative algebras. By refor- mulating
The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper
• Mathematics
• 2009
Supersymmetry is deeply related to division algebras. Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10.
The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics.
? 1. The object of this paper is to develop some of the simpler properties of division algebras, that is to say, linear associative algebras in which division is possible by any element except zero.
The groups SO( nu -1), SO( nu ), SO( nu +1), SO( nu +1, 1) and SO( nu +2, 2) ( nu =1, 2, 4, 8) and their spin representations are described in terms of the division algebras R, C, H and O.
• Physics
• 2008
We study SU(2) Yang-Mills quantum mechanics with N = 2, 4, 8 and 16 supercharges. This describes the non-relativistic dynamics of a pair of D0-branes moving in d = 3, 4, 6 and 10 spacetime dimensions


text:
# How are power and focal length of a lens related ? You are provided with two lenses of focal length 20 cm and 40 cm respectively. Which lens will

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How are power and focal length of a lens related ? You are provided with two lenses of focal length 20 cm and 40 cm respectively. Which lens will you use to obtain more convergent light ?

by (71.8k points)
selected by

P = (1)/(f).
For obtaining more convergent light, power of lens must be larger. Therefore, its focal length must be smaller, Therefore, the lens of f = 20 cm will provide more convergent light.


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 1.2: Lesson Two

Difficulty Level: At Grade Created by: CK-12

## Sometimes a Vowel, Sometimes a Consonant: <y>

1. Fill in the blanks. Don't forget the pointed brackets: The letters <a>\begin{align*}\underline{<\mathrm{a} >}\end{align*}, <e>, <i>\begin{align*}\underline{<\mathrm{i}>}\end{align*}, and <o> are always vowels.

2. We can use the same word in different ways. For example, the word blue sometimes means a color, and sometimes it means “sad.” We can also use the same letter in different ways. For example, three letters are sometimes used as vowels and sometimes as consonants. One of them is the letter <y>.

The letter <y> is a consonant when it spells the sound it spells in the word yes. When it spells any other sound, it is a vowel.

3. Listen to the sound the <y> is spelling or helping to spell in these words. Then sort the words into the two groups below:

gymtypebeyondyardyousomedayyearstheypuppyeverywhyyellow\begin{align*}&\text{gym} && \text{yard} && \text{years} && \text{every}\\ &\text{type} && \text{you} && \text{they} && \text{why}\\ &\text{beyond} && \text{someday} && \text{puppy} && \text{yellow}\end{align*}

Words in which the <y> is ...
a consonant a vowel
beyond gym
yard type
you someday
years they
yellow puppy
every
why

4. Fill in the blanks: The four letters that are always vowels are <a>\begin{align*}\underline{<\mathrm{a}>}\end{align*}, <e>, <i>\begin{align*}\underline{<\mathrm{i}>}\end{align*}, and <o>.

5. One letter that is sometimes a vowel and sometimes a consonant is <y>.

Watch the Middles! Fill in the blanks the way we have with beyond. As you read and write the word parts, spell them out to yourself, letter by letter.

beyond
be yond
be yond
be yond
beyond
years
year s
year s
year s
years
seventy
seven ty
seven ty
seven ty
seventy
away
a way
a way
a way
away
holiday
holi day
holi day
holi day
anyone
any one
any one
any one
anyone

Teaching Notes

1. The basic pattern underlying the vowel and consonant uses of <y> is that <y> is a consonant at the beginning of a word, a vowel at the end. In the middle of words it is a consonant only if it is the first letter of a base element and is spelling the first sound of a syllable, as in beyond; otherwise it is a vowel.

2. You will notice that the text pesters the youngsters to remember the pointed brackets that are used to indicate spelled-out letters as opposed to the square brackets that they will later learn to use to indicate spelled-out sounds. This pestering is part of the attempt to keep clear the distinction between sounds and letters.

3. This “Watch the Middles!” is the first of the reinforcers that occur at the end of many of the lessons. Reinforcers are game-like activities that are designed to reinforce some of the concepts or information that are important to the lesson. The immediate tie-in here is that all six of the words contain the letter <y>, sometimes as vowel, sometimes as consonant. But “Watch the Middles!” has more general objectives, as well: (i) It is intended as a rather passive exercise that gives the students practice with hard words in the hope that the repetition will enhance their remembering, (ii) It calls attention to the middle of words, where research indicates most spelling errors occur, (iii) It introduces the students, without calling attention to doing so, to the analysis of words into their elements, something that is very important later in this spelling program.

This Middles contains some word parts about which students may well ask. In the Teaching Notes we will refer to these parts as prefixes, bases, or suffixes. But since the discussion of prefixes, suffixes, and bases comes later in the program, for now it is probably best simply to speak of these with the students as word parts that are important because they appear in other words. The yond in beyond is the same word-part that is in yonder and is related to yon. Be- is a prefix that shows location at or near. The -ty in seventy is a suffix that means “times ten,” as in thirty, forty, sixty, etc. It is related to teen, which means “plus ten” as in fourteen, sixteen, etc. The holi in holiday is a form of the word holy. A holiday was, originally, a holy-day. In away the prefix a- is a reduced form of the Old English preposition on, so the original meaning of away was “on (one's) way.”

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# Irrlicht problem

## Recommended Posts

u235    359
Attention all Irrlicht users. I have hit a somewhat major snag in my development process. I have a map for my first level and I wanted to test it out so banged out a few lines of code and tried to load it up. Program compiles fine and runs, but doesn't show anything because none of the textures are being loaded. I've tried it in two different formats: CShop native CSM and X formats both give the same results. Mesh loads fine, all textures fail to load. I have verified directory structure as far how it's referenced in the mesh file and placed my textures accordingly. What do you think could be the problem? Thanks. -AJ

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Sol462    240
Have you run it in debug (with console output)? You should see "Could not load texture xxx"
Also the X file has to be text and not binary. Maybe they're in the wrong directory or something.

Can we see the code or a screenie?

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Gorax    202
I'm guessing the textures are in the wrong directory. Unless you specify a texture directory, you usually have to keep the textures in the same directory you keep the map in. At least that's the way it works with DeleD maps, so I assume it's the same.

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u235    359
Quote:
 Originally posted by Sol462Have you run it in debug (with console output)? You should see "Could not load texture xxx"Also the X file has to be text and not binary. Maybe they're in the wrong directory or something.Can we see the code or a screenie?

Yes I have run it in debug and got the error message you are referring to. The X file is in text and all textures are in the same directory as the X file. Screens of what it is supposed to look like are here(taken from inside the world editor) and here is a screen of what comes up:

And here is the code:

#include "main.h"int main() {	IrrlichtDevice* device=0;	IVideoDriver* vidDriver = 0;	ISceneManager* sceneMan=0;	device = createDevice(EDT_DIRECTX9, dimension2d<s32>(1024, 768), 32, false, false, false, 0);	vidDriver = device->getVideoDriver();	sceneMan = device->getSceneManager();	IAnimatedMesh *mesh = sceneMan->getMesh("castle.csm");	ICameraSceneNode* camera = 0;	camera = sceneMan->addCameraSceneNodeFPS(0, 100.0f, 300.0f);	device->getCursorControl()->setVisible(false);	vidDriver->setAmbientLight(SColorf(64,64,64));	while(device->run()) {		vidDriver->beginScene(true, true, SColor(200, 130, 36, 95));		sceneMan->drawAll();		vidDriver->endScene();	}	device->drop();	return 0;}

Thanks for the help and any that this extra info illicits.

-AJ

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Gorax    202
You loaded the mesh, but you didn't add it to the scene. Lines 309 to 313 inclusive of ISceneManager.h:

		 *  To load and display a mesh quickly, just do this:		 *  \code		 *  SceneManager->addAnimatedMeshSceneNode(		 *		SceneManager->getMesh("yourmesh.3ds"));		 *  \endcode

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depending on how you exported the model, with or without textures applied. I have exported with materials applied and this is the code:

IAnimatedMeshSceneNode* model;

add this line if you havn't exported with texture;

model->setMaterialTexture(1, m_Video->getTexture("Data/Media/Models/modelTexture.bmp"));

might also add this, or the model may be pitch black
model->setMaterialFlag(EMF_LIGHTING, false);

try that and let us know how you go, hopefully goes good

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u235    359
Well well what do ya know....it really was as simple as I thought it was going to be. Thanks guys for all the help. It doesn't render perfectly, the two polys that make up the side of the roof aren't rendering but I don't think that will be too big a deal to fix. Other than that it renders beautifully and once I get some lights in there and some shading, boy it's going to look good [smile] Thanks again.

-AJ


text:
# Nonlinear elliptic equations with natural growth in general domains

created on 05 Oct 2004

[BibTeX]

Published Paper

Inserted: 5 oct 2004

Journal: Annali di Matematica Pura e Applicata
Volume: 181
Pages: 407-426
Year: 2002

Abstract:

We prove the existence of solutions of nonlinear elliptic equations with first order terms having natural growth'' with respect to the gradient. The assumptions on the source terms lead to the existence of possibly unbounded solutions (though with exponential integrability). The domain $\Om$ is allowed to have infinite Lebesgue measure.

Credits | Cookie policy | HTML 5 | CSS 2.1


text:
# MAGNETIC CIRCULAR DICHROISM OF MATRIX-ISOLATED COPPER ATOMS

Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/11754

Files Size Format View
1981-WH-11.jpg 57.88Kb JPEG image

 Title: MAGNETIC CIRCULAR DICHROISM OF MATRIX-ISOLATED COPPER ATOMS Creators: Zeringue, K. J.; Shaksemampour, Jalal; Vala, Martin Issue Date: 1981 Abstract: The absorption and magnetic circular dichroism spectra of copper atoms in a krypton matrix have been measured as a function of temperature. Focusing on the $^{2}P\leftarrow ^{2} S$ transition and applying a moment analysis we have learned that 1) the observed triplet of bands arises from simultaneous spin-orbit and Jahn-Teller interactions in the excited state, 2) the metal-matrix cage interactions in the large enough to invert the sign of the spin-orbit coupling constant and 3) the orbital g factor of the $^{2}P$ state is significantly reduced due to the Jahn-Teller effect. Discussion of other atomic and cluster transitions will also be given. URI: http://hdl.handle.net/1811/11754 Other Identifiers: 1981-WH-11


text:
# Every closed subset of $\Bbb R$ has countably many isolated points

Every closed subset of $$\Bbb R$$ has countably many isolated points.

Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help!

My attempt:

Let $$A \subseteq \Bbb R$$ be a closed set and $$I$$ be the set of all isolated points of $$A$$.

If $$a \in I$$, then there exists $$\delta > 0$$ such that $$x \neq a$$ and $$|x-a|<\delta$$ implies $$x \notin A$$. Since $$\Bbb Q$$ is dense in $$\Bbb R$$, there exist $$r,s \in \Bbb Q$$ such that $$a-\delta.

As a result, if $$a \in I$$, then there exist $$r,s \in \Bbb Q$$ such that $$a$$ is the only element of $$A$$ in $$(r,s)$$.

Let $$\langle r_n,s_n \rangle_{n \in \Bbb N}$$ be an enumeration of the set $$\{\langle r,s \rangle \mid r,s \in \Bbb Q\}$$.

We define a mapping $$\mathcal F:I \to \Bbb N$$ by $$\mathcal F(a) = \min \{n \in \Bbb N \mid a \text{ is the only element of } A \text{ in } (r_n,s_n)\}$$

Clearly, $$\mathcal F$$ is injective and thus $$|I| \le |\Bbb N| = \aleph_0$$. Hence $$I$$ is countable.

• I think real-analysis is a better tag because you're using two facts: 1- $\mathbb{R}$ is a metric space. 2- It has a dense subset $\mathbb{Q}$. Topology can also deal with spaces that are not metrizable which is way more abstract than $\mathbb{R}$. You can also add 'metric spaces' to your tags but I'm still not sure that the result holds in general metric spaces. (+1) – stressed out Feb 14 at 14:01
• Also, your proof looks correct for $\mathbb{R}$. Well done. – stressed out Feb 14 at 14:05
• I suggest you also edit the title and the first sentence: Every closed subset of $\mathbb R$ has countably many isolated points. – Lee Mosher Feb 14 at 14:06
• And while I agree with @stressedout that your proof is correct, it is just a little bit too complicated. You don't need two points $r,s \in \mathbb Q \cap (a-\delta,a+\delta)$, you only need one point $r \in \mathbb Q \cap (a-\delta,a+\delta)$ to make the proof work. – Lee Mosher Feb 14 at 14:08
• Thank you @stressedout! – MadnessFor MATH Feb 14 at 14:08

## 2 Answers

Your proof looks correct for $$\mathbb{R}$$ with its standard topology. Although, as Lee Mosher pointed out, it could be simplified.

To see that your statement is something that depends on our knowledge of $$\mathbb{R}$$ with its usual metric, and it is not necessarily true in general topological spaces, or even metric spaces, consider the same set, the good old $$\mathbb{R}$$ with the discrete topology this time. That is, consider $$\mathbb{R}$$ equipped with the discrete metric:

$$d(x,y) = \begin{cases}0 & x=y \\ 1 & x\neq y\end{cases}$$

Then all points of $$\mathbb{R}$$ will become isolated points (take $$0<\delta<1$$), proving that your statement is not necessarily true about $$\mathbb{R}$$ with a topology other than the one coming from the Euclidean distance and it's a very specific statement in real analysis.

Also, your statement is true about separable metric spaces, i.e. any metric space that has a countable, dense subset. And the proof is similar to what you wrote taken Lee Mosher's comment into account.

• Is it that, instead of choosing $(r,s)$, we can accomplish the task by choosing $(r, 2a-r)$ where $r \in \Bbb Q \cap (a-\delta,a)$? – MadnessFor MATH Feb 14 at 14:16
• @LeAnhDung I think what Lee Mosher meant is that you need only one point in each set to construct your function $\mathcal{F}$ and you can find it because $\mathbb{Q}\cap (a-\delta, a+\delta) \neq \emptyset$, because rationals are dense in real numbers. I think you don't even need to think about the order on real numbers for this to work, just the fact that rationals are dense suffices. Although, if you have found a relation between $r$ and $s$ depending on $a$, then that's good too. – stressed out Feb 14 at 14:20
• Hi @stressedout, after looking back at my proof, i don't see the closeness of $A$ is used. So i think that the theorem generalizes to Every subset of $\Bbb R$ has countably many isolated points. – MadnessFor MATH Feb 17 at 9:40
• @LeAnhDung Hi again. You're absolutely correct. What you have proved works for any subset of $\mathbb{R}$ and also implies that if $A$ is uncountable, then $A'$ must be uncountable too. Beautiful. – stressed out Feb 17 at 9:46
• Thank you so much for your prompt reply ^^ – MadnessFor MATH Feb 17 at 9:47

Thank to @Lee Mosher and @stressed out's suggestions, I presented a simpler proof here.

Let $$A \subseteq \Bbb R$$ be a closed set and $$I$$ be the set of all isolated points of $$A$$ .

If $$a \in I$$ , then there exists $$\delta > 0$$ such that $$x \neq a$$ and $$|x-a|<\delta$$ implies $$x \notin A$$ . Since $$\Bbb Q$$ is dense in $$\Bbb R$$ , there exist $$r \in \Bbb Q$$ such that $$a-\delta .

As a result, if $$a \in I$$ , then there exist $$r \in \Bbb Q$$ such that $$a$$ is the only element of $$A$$ in $$(r,2a-r)$$ .

Let $$\langle r_n \rangle_{n \in \Bbb N}$$ be an enumeration of $$\Bbb Q$$ .

We define a mapping $$\mathcal F:I \to \Bbb N$$ by $$\mathcal F(a) = \min \{n \in \Bbb N \mid a \text{ is the only element of } A \text{ in } (r_n,2a-r_n)\}$$

Clearly, $$\mathcal F$$ is injective and thus $$|I| \le |\Bbb N| = \aleph_0$$ . Hence $$I$$ is countable.

• Yes. This is correct. You asked for more explanation about you need only one point in each set to construct your function F and you can find it because $\mathbb{Q}\cap(a-\delta,a+\delta) \neq \emptyset$. Well, the idea, as you know, is that you want to represent isolated points by elements of a countable set to find a function $\mathcal{F}$ from $I$ to $\mathbb{N}$. This is essentially what you did in your first solution. You considered all pairs $(r,s)$ which is a subset of $\mathbb{Q} \times \mathbb{Q}$, which is enumerable and then you considered an enumeration. To be continued... – stressed out Feb 14 at 16:54
• Thank you so much @stressedout ^^ I got it. – MadnessFor MATH Feb 14 at 16:57
• OK. I was writing the second part but I'm glad that you got it already. (+1) – stressed out Feb 14 at 16:58


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# If = 3, which of the following could be the value of x – 4?

Author Message
Manager
Joined: 11 Aug 2009
Posts: 121

Kudos [?]: 196 [0], given: 3

If = 3, which of the following could be the value of x – 4? [#permalink]

### Show Tags

18 Nov 2009, 20:38
1
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Difficulty:

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Question Stats:

69% (00:38) correct 31% (00:32) wrong based on 77 sessions

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If $$\sqrt{(x+4)^2} = 3$$, which of the following could be the value of $$x-4$$?

A. -11
B. -7
C. -4
D. -3
E. 5

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-x-4-2-1-2 ... 43886.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 25 Dec 2017, 21:56, edited 3 times in total.

Kudos [?]: 196 [0], given: 3

Manager
Joined: 28 Jul 2009
Posts: 121

Kudos [?]: 38 [0], given: 12

Re: If = 3, which of the following could be the value of x – 4? [#permalink]

### Show Tags

18 Nov 2009, 21:40
The question seems to be incomplete. What is equal to 3?

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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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19 Nov 2009, 05:29
Acc3ss wrote:
The question seems to be incomplete. What is equal to 3?

sorry, just edited it. ^^

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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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19 Nov 2009, 07:47
No worries kairoshan,

\sqrt{((x+4)^2)}=3

Then
x+4=-3 OR x+4=3
x=-7 or -1

-7-4=-11
-1-5=-5

What is the OA?

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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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20 Nov 2009, 06:32
the given expression reduces to
(x+4)^2 - 3^2=0
(x+4+3)(x+4-3)=0
(x-4+11)(x-4+5)=0
x-4=-11 or -5
Ans. A

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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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13 May 2011, 04:30
x = -7

=> x-4 = -11

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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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13 May 2011, 12:03
Square root of a square is +ve or -ve.

But it is given that Square of a square has +ve value

then how can x+4 = -3 ?????
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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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13 May 2011, 19:43
x =-1 or -7

=>x-4 can be -5 or -11

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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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16 May 2011, 10:33
The equation reduces to:
(x+4)^2 = 3^2

So taking the square root on the RHS we get:
x+4 = +/- 3

Now there are 2 possible values:
x+ 4 = 3 and x+4 = -3

You can directly subtract 8 from both the equations to get the answer:
x-4 = (3-8) and x-4 = (-3-8)

Therefore (x-4) has possible values -5 and -11.

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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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08 Oct 2016, 13:26
kairoshan wrote:
If $$\sqrt{(x+4)^2} = 3$$, which of the following could be the value of $$x-4$$?

A. -11
B. -7
C. -4
D. -3
E. 5

$$(X+4)^2 = 9$$

$$(X-4+8)^2 = 9$$
let X-4 = A
$$(A+8)^2 = 9$$
Now, sub answer choices in A
A) $$(-11+8)^2 = (-3)^2 = 9$$
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Re: If = 3, which of the following could be the value of x – 4? [#permalink]

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25 Dec 2017, 21:56
If $$\sqrt{(x+4)^2} = 3$$, which of the following could be the value of x - 4?

A. -11
B. -7
C. -4
D. -3
E. 5

$$\sqrt{(x+4)^2} = 3$$ --> $$(x+4)^2=9$$ --> $$x+4=3$$ or $$x+4=-3$$ --> $$x=-1$$ or $$x=-7$$ --> $$x-4=-5$$ or $$x-4=-11$$.

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-x-4-2-1-2 ... 43886.html
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Re: If = 3, which of the following could be the value of x – 4?   [#permalink] 25 Dec 2017, 21:56
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