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# C++ Parsing an 2D array into rectangular objects
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So im working with a new physics library, and i want it to run as fast as possible. My game contains a 1D array for the map (but essentially a 2D tile array), and i want to find groups of objects and create rectangles out of them.
Example:
if each tile is using a number, then the map would look like this in the text file:
1111111111
1________1
1_____33_1
1_22__33_1
1_22_____1
1111111111
then the map would be parsed, where each letter represents it's groups id:
1111111111
2________3
2_____44_3
2_55__44_3
2_55_____3
6666666666
But i cant quite seem to figure out the right algorithm.... I have tried numerous ways, but nothing works... What are your ideas? Edited by bennettbugs
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Just brainstorming (and hopefully I'm understanding your question here )
Are you looking for all solutions, or are there criteria for ruling out multiple ones? If I'm understanding correctly, there are many ways to group your example, such as this alternative:
2222222222
1________3
1_____44_3
1_55__44_3
1_55_____3
7755666663
[/quote]
My intuition tells me that an exact algorithm for finding all solutions may have something to do with the "packing problem" - but I could be wrong, haven't messed with packing algorithms much.
A naive brute force approach could be: find the first filled (valid) cell, then start a rectangle there. Expand the rectangle either right or down or both (weeding out invalid rectangles due to containing an invalid cell), and expanding as far as possible. Add this rectangle to the set of rects. Find the next valid cell that is not contained in the set of rects, and start a new rectangle, repeating the above. Stop when there are no more valid cells not contained in the set of rects.
Example, here is the start of the first rectangle ("S"), it can be expanded either horizontally or vertically successfully ("*"), bot not diagonally because that would include an invalid or empty cell:
S*11111111
*________1
1_____33_1
[/quote]
By whatever convention, we choose to favor horizontal, and created our first recangle thus:
S*********
1________1
1_____33_1
[/quote]
The next valid cell not in the already created rectangle ("x") is found and marked as a new starting point ("S"). Here the horizontal and diagonal expansions are ruled out, so we expand vertically:
xxxxxxxxxx
S_______ 1
*_____33_1
*_22__33_1
*_22_____1
*111111111
[/quote]
Move along to where we've created three rectangles (marked as "x"), and finding the starting point ("S") here.
xxxxxxxxxx
x________x
x_____S3_x
x_22__33_x
x_22_____x
x111111111
[/quote]
In this case, diagonal expansion is valid:
xxxxxxxxxx
x________x
x_____S*_x
x_22__**_x
x_22_____x
x111111111
[/quote]
Depending on which expansion direction you favor (left, down, diagonal), this could lead to different solutions.
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Yup. I would prefer horizontal, because the maps are usually longer then their height. Im also doing the steps you listed, but unfortuantly i think im just messing up on 1 part of the algorithm. I have 3 variations of the algorithm that, maybe someone could point out the mistake...
this->Parsed_Map.clear(); char* Temp_Map = new char[this->W_Size_X*this->W_Size_Y]; for(u_int a=0;a<this->W_Size_X*this->W_Size_Y;a++) Temp_Map[a]=this->Front[a]; for(u_int y=0;y<this->W_Size_Y;y++) for(u_int x=0;x<this->W_Size_X;x++) { if(Temp_Map[y*this->W_Size_X + x])//is collision { u_int Width=this->W_Size_X-1,Height=this->W_Size_Y-1; for(u_int y1=0;y1+y<this->W_Size_Y && Temp_Map[(y1+y)*this->W_Size_X + x];y1++) { u_int x1=1; while(x1<Width && Temp_Map[(y1+y)*this->W_Size_X + x + x1]) x1++; Width=x1; } for(u_int x1=0;x1<Width && Temp_Map[y*this->W_Size_X + x + x1];x1++) { u_int y1=1; while(y1<Height && Temp_Map[(y1+y)*this->W_Size_X + x + x1]) x1++; Height=y1; } /*u_int Width=this->W_Size_X-1,Height=this->W_Size_Y-1; for(u_int y1=0;y1+y<this->W_Size_Y && Temp_Map[(y1+y)*this->W_Size_X + x]!=0;y1++)//check for smallest width { u_int x1=1; while(x1+x<this->W_Size_X && Temp_Map[(y1+y)*this->W_Size_X + x + x1]!=0) x1++; if(x1<Width) Width=x1; } for(u_int x1=0;x1<Width && Temp_Map[y*this->W_Size_X + x + x1]!=0;x1++)//check for smallest height { u_int y1=1; while(y1+y<this->W_Size_Y && Temp_Map[(y1+y)*this->W_Size_X + x + x1]!=0) y1++; if(y1<Height) Height=y1; } /*for(u_int x1=0;x1+x<this->W_Size_X && Temp_Map[y*this->W_Size_X + x + x1];x1++) { u_int y1=1; while(y1+y<this->W_Size_Y)// && Temp_Map[(y+y1)*this->W_Size_X + x];y1++) { if(Temp_Map[(y+y1)*this->W_Size_X + x + x1])//this is a collision y1++; else { if(y1<Height || Height==1)//next is air Height=y1; break; } } //if(Height==this->W_Size_Y+10)Height=1; Width++; } if(Width==0) Width=1; if(Height==0) Height=1;*/ RECT nRect; nRect.left=x; nRect.top=y; nRect.right=Width+x; nRect.bottom=Height+y; this->Parsed_Map.push_back(nRect); } } Edited by bennettbugs
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# JFET
Last updated
Type Electric current from source to drain in a p-channel JFET is restricted when a voltage is applied to the gate. Active drain, gate, source
The junction-gate field-effect transistor (JFET) is one of the simplest types of field-effect transistor. [1] JFETs are three-terminal semiconductor devices that can be used as electronically controlled switches or resistors, or to build amplifiers.
## Contents
Unlike bipolar junction transistors, JFETs are exclusively voltage-controlled in that they do not need a biasing current. Electric charge flows through a semiconducting channel between source and drain terminals. By applying a reverse bias voltage to a gate terminal, the channel is "pinched", so that the electric current is impeded or switched off completely. A JFET is usually "on" when there is zero voltage between its gate and source terminals. If a potential difference of the proper polarity is applied between its gate and source terminals, the JFET will be more resistive to current flow, which means less current would flow in the channel between the source and drain terminals.
JFETs are sometimes referred to as depletion-mode devices, as they rely on the principle of a depletion region, which is devoid of majority charge carriers. The depletion region has to be closed to enable current to flow.
JFETs can have an n-type or p-type channel. In the n-type, if the voltage applied to the gate is negative with respect to the source, the current will be reduced (similarly in the p-type, if the voltage applied to the gate is positive with respect to the source). A JFET has a large input impedance (sometimes on the order of 1010 ohms), which means that it has a negligible effect on external components or circuits connected to its gate.
## History
A succession of FET-like devices was patented by Julius Lilienfeld in the 1920s and 1930s. However, materials science and fabrication technology would require decades of advances before FETs could actually be manufactured.
JFET was first patented by Heinrich Welker in 1945. [2] During the 1940s, researchers John Bardeen, Walter Houser Brattain, and William Shockley were trying to build a FET, but failed in their repeated attempts. They discovered the point-contact transistor in the course of trying to diagnose the reasons for their failures. Following Shockley's theoretical treatment on JFET in 1952, a working practical JFET was made in 1953 by George C. Dacey and Ian M. Ross. [3] Japanese engineers Jun-ichi Nishizawa and Y. Watanabe applied for a patent for a similar device in 1950 termed static induction transistor (SIT). The SIT is a type of JFET with a short channel. [3]
High-speed, high-voltage switching with JFETs became technically feasible following the commercial introduction of Silicon carbide (SiC) wide-bandgap devices in 2008. Due to early difficulties in manufacturing in particular, inconsistencies and low yield SiC JFETs remained a niche product at first, with correspondingly high costs. By 2018, these manufacturing issues had been mostly resolved. By then, SiC JEFETs were also commonly used in conjunction with conventional low-voltage Silicon MOSFETs. [4] In this combination, SiC JFET + Si MOSFET devices have the advantages of wide band-gap devices as well as the easy gate drive of MOSFETs. [4]
## Structure
The JFET is a long channel of semiconductor material, doped to contain an abundance of positive charge carriers or holes (p-type), or of negative carriers or electrons (n-type). Ohmic contacts at each end form the source (S) and the drain (D). A pn-junction is formed on one or both sides of the channel, or surrounding it using a region with doping opposite to that of the channel, and biased using an ohmic gate contact (G).
## Functions
JFET operation can be compared to that of a garden hose. The flow of water through a hose can be controlled by squeezing it to reduce the cross section and the flow of electric charge through a JFET is controlled by constricting the current-carrying channel. The current also depends on the electric field between source and drain (analogous to the difference in pressure on either end of the hose). This current dependency is not supported by the characteristics shown in the diagram above a certain applied voltage. This is the saturation region, and the JFET is normally operated in this constant-current region where device current is virtually unaffected by drain-source voltage. The JFET shares this constant-current characteristic with junction transistors and with thermionic tube (valve) tetrodes and pentodes.
Constriction of the conducting channel is accomplished using the field effect: a voltage between the gate and the source is applied to reverse bias the gate-source pn-junction, thereby widening the depletion layer of this junction (see top figure), encroaching upon the conducting channel and restricting its cross-sectional area. The depletion layer is so-called because it is depleted of mobile carriers and so is electrically non-conducting for practical purposes. [5]
When the depletion layer spans the width of the conduction channel, pinch-off is achieved and drain-to-source conduction stops. Pinch-off occurs at a particular reverse bias (VGS) of the gate–source junction. The pinch-off voltage (Vp) (also known as threshold voltage [6] [7] or cut-off voltage [8] [9] [10] ) varies considerably, even among devices of the same type. For example, VGS(off) for the Temic J202 device varies from −0.8 V to −4 V. [11] Typical values vary from −0.3 V to −10 V. (Confusingly, the term pinch-off voltage is also used to refer to the VDS value that separates the linear and saturation regions. [9] [10] )
To switch off an n-channel device requires a negative gate–source voltage (VGS). Conversely, to switch off a p-channel device requires positive VGS.
In normal operation, the electric field developed by the gate blocks source–drain conduction to some extent.
Some JFET devices are symmetrical with respect to the source and drain.
## Schematic symbols
The JFET gate is sometimes drawn in the middle of the channel (instead of at the drain or source electrode as in these examples). This symmetry suggests that "drain" and "source" are interchangeable, so the symbol should be used only for those JFETs where they are indeed interchangeable.
The symbol may be drawn inside a circle (representing the envelope of a discrete device) if the enclosure is important to circuit function, such as dual matched components in the same package. [12]
In every case the arrow head shows the polarity of the P–N junction formed between the channel and the gate. As with an ordinary diode, the arrow points from P to N, the direction of conventional current when forward-biased. An English mnemonic is that the arrow of an N-channel device "points in".
## Comparison with other transistors
At room temperature, JFET gate current (the reverse leakage of the gate-to-channel junction) is comparable to that of a MOSFET (which has insulating oxide between gate and channel), but much less than the base current of a bipolar junction transistor. The JFET has higher gain (transconductance) than the MOSFET, as well as lower flicker noise, and is therefore used in some low-noise, high input-impedance op-amps.
## Mathematical model
### Linear ohmic region
The current in N-JFET due to a small voltage VDS (that is, in the linear or ohmic [13] or triode region [6] ) is given by treating the channel as a rectangular bar of material of electrical conductivity ${\displaystyle qN_{d}\mu _{n}}$: [14]
${\displaystyle I_{\text{D}}={\frac {bW}{L}}qN_{d}\mu _{n}V_{\text{DS}},}$
where
ID = drain–source current,
b = channel thickness for a given gate voltage,
W = channel width,
L = channel length,
q = electron charge = 1.6×10−19 C,
μn = electron mobility,
Nd = n-type doping (donor) concentration,
VP = pinch-off voltage.
Then the drain current in the linear region can be approximated as
${\displaystyle I_{\text{D}}={\frac {bW}{L}}qN_{d}\mu _{n}V_{\text{DS}}={\frac {aW}{L}}qN_{d}\mu _{n}\left(1-{\sqrt {\frac {V_{\text{GS}}}{V_{\text{P}}}}}\right)V_{\text{DS}}.}$
In terms of ${\displaystyle I_{\text{DSS}}}$, the drain current can be expressed as[ citation needed ]
${\displaystyle I_{\text{D}}={\frac {2I_{\text{DSS}}}{V_{\text{P}}^{2}}}\left(V_{\text{GS}}-V_{\text{P}}-{\frac {V_{\text{DS}}}{2}}\right)V_{\text{DS}}.}$
### Constant-current region
The drain current in the saturation or active [15] [6] or pinch-off region [16] is often approximated in terms of gate bias as [14]
${\displaystyle I_{\text{DS}}=I_{\text{DSS}}\left(1-{\frac {V_{\text{GS}}}{V_{\text{P}}}}\right)^{2},}$
where IDSS is the saturation current at zero gate–source voltage, i.e. the maximum current that can flow through the FET from drain to source at any (permissible) drain-to-source voltage (see, e. g., the IV characteristics diagram above).
In the saturation region, the JFET drain current is most significantly affected by the gate–source voltage and barely affected by the drain–source voltage.
If the channel doping is uniform, such that the depletion region thickness will grow in proportion to the square root of the absolute value of the gate–source voltage, then the channel thickness b can be expressed in terms of the zero-bias channel thickness a as[ citation needed ]
${\displaystyle b=a\left(1-{\sqrt {\frac {V_{\text{GS}}}{V_{\text{P}}}}}\right),}$
where
VP is the pinch-off voltage the gate–source voltage at which the channel thickness goes to zero,
a is the channel thickness at zero gate–source voltage.
### Transconductance
The transconductance for the junction FET is given by
${\displaystyle g_{\text{m}}={\frac {2I_{\text{DSS}}}{|V_{\text{P}}|}}\left(1-{\frac {V_{\text{GS}}}{V_{\text{P}}}}\right),}$
where ${\displaystyle V_{\text{P}}}$ is the pinchoff voltage, and IDSS is the maximum drain current. This is also called ${\displaystyle g_{\text{fs}}}$ or ${\displaystyle y_{\text{fs}}}$ (for transadmittance). [17]
## Related Research Articles
A transistor is a semiconductor device used to amplify or switch electrical signals and power. The transistor is one of the basic building blocks of modern electronics. It is composed of semiconductor material, usually with at least three terminals for connection to an electronic circuit. A voltage or current applied to one pair of the transistor's terminals controls the current through another pair of terminals. Because the controlled (output) power can be higher than the controlling (input) power, a transistor can amplify a signal. Some transistors are packaged individually, but many more are found embedded in integrated circuits.
The metal–oxide–semiconductor field-effect transistor, also known as the metal–oxide–silicon transistor, is a type of insulated-gate field-effect transistor that is fabricated by the controlled oxidation of a semiconductor, typically silicon. The voltage of the gate terminal determines the electrical conductivity of the device; this ability to change conductivity with the amount of applied voltage can be used for amplifying or switching electronic signals.
A Schottky barrier, named after Walter H. Schottky, is a potential energy barrier for electrons formed at a metal–semiconductor junction. Schottky barriers have rectifying characteristics, suitable for use as a diode. One of the primary characteristics of a Schottky barrier is the Schottky barrier height, denoted by ΦB. The value of ΦB depends on the combination of metal and semiconductor.
A current mirror is a circuit designed to copy a current through one active device by controlling the current in another active device of a circuit, keeping the output current constant regardless of loading. The current being "copied" can be, and sometimes is, a varying signal current. Conceptually, an ideal current mirror is simply an ideal inverting current amplifier that reverses the current direction as well. Or it can consist of a current-controlled current source (CCCS). The current mirror is used to provide bias currents and active loads to circuits. It can also be used to model a more realistic current source.
Transconductance, also infrequently called mutual conductance, is the electrical characteristic relating the current through the output of a device to the voltage across the input of a device. Conductance is the reciprocal of resistance.
A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.
A MESFET is a field-effect transistor semiconductor device similar to a JFET with a Schottky (metal–semiconductor) junction instead of a p–n junction for a gate.
The threshold voltage, commonly abbreviated as Vth, of a field-effect transistor (FET) is the minimum gate-to-source voltage VGS (th) that is needed to create a conducting path between the source and drain terminals. It is an important scaling factor to maintain power efficiency.
The cascode is a two-stage amplifier that consists of a common-emitter stage feeding into a common-base stage.
A power MOSFET is a specific type of metal–oxide–semiconductor field-effect transistor (MOSFET) designed to handle significant power levels. Compared to the other power semiconductor devices, such as an insulated-gate bipolar transistor (IGBT) or a thyristor, its main advantages are high switching speed and good efficiency at low voltages. It shares with the IGBT an isolated gate that makes it easy to drive. They can be subject to low gain, sometimes to a degree that the gate voltage needs to be higher than the voltage under control.
Channel length modulation (CLM) is an effect in field effect transistors, a shortening of the length of the inverted channel region with increase in drain bias for large drain biases. The result of CLM is an increase in current with drain bias and a reduction of output resistance. It is one of several short-channel effects in MOSFET scaling. It also causes distortion in JFET amplifiers.
A VMOS transistor is a type of MOSFET. VMOS is also used for describing the V-groove shape vertically cut into the substrate material. VMOS is an acronym for "vertical metal oxide semiconductor", or "V-groove MOS".
A buck converter is a DC-to-DC power converter which steps down voltage from its input (supply) to its output (load). It is a class of switched-mode power supply (SMPS) typically containing at least two semiconductors and at least one energy storage element, a capacitor, inductor, or the two in combination. To reduce voltage ripple, filters made of capacitors are normally added to such a converter's output and input. It is called a buck converter because the voltage across the inductor “bucks” or opposes the supply voltage.
Drain-induced barrier lowering (DIBL) is a short-channel effect in MOSFETs referring originally to a reduction of threshold voltage of the transistor at higher drain voltages. In a classic planar field-effect transistor with a long channel, the bottleneck in channel formation occurs far enough from the drain contact that it is electrostatically shielded from the drain by the combination of the substrate and gate, and so classically the threshold voltage was independent of drain voltage. In short-channel devices this is no longer true: The drain is close enough to gate the channel, and so a high drain voltage can open the bottleneck and turn on the transistor prematurely.
Overdrive voltage, usually abbreviated as VOV, is typically referred to in the context of MOSFET transistors. The overdrive voltage is defined as the voltage between transistor gate and source (VGS) in excess of the threshold voltage (VTH) where VTH is defined as the minimum voltage required between gate and source to turn the transistor on. Due to this definition, overdrive voltage is also known as "excess gate voltage" or "effective voltage." Overdrive voltage can be found using the simple equation: VOV = VGS − VTH.
The hybrid-pi model is a popular circuit model used for analyzing the small signal behavior of bipolar junction and field effect transistors. Sometimes it is also called Giacoletto model because it was introduced by L.J. Giacoletto in 1969. The model can be quite accurate for low-frequency circuits and can easily be adapted for higher frequency circuits with the addition of appropriate inter-electrode capacitances and other parasitic elements.
In field-effect transistors (FETs), depletion mode and enhancement mode are two major transistor types, corresponding to whether the transistor is in an on state or an off state at zero gate–source voltage.
In the theory of electrical networks, a dependent source is a voltage source or a current source whose value depends on a voltage or current elsewhere in the network.
The field-effect transistor (FET) is a type of transistor that uses an electric field to control the flow of current in a semiconductor. FETs are devices with three terminals: source, gate, and drain. FETs control the flow of current by the application of a voltage to the gate, which in turn alters the conductivity between the drain and source.
A voltage-controlled resistor (VCR) is a three-terminal active device with one input port and two output ports. The input-port voltage controls the value of the resistor between the output ports. VCRs are most often built with field-effect transistors (FETs). Two types of FETs are often used: the JFET and the MOSFET. There are both floating voltage-controlled resistors and grounded voltage-controlled resistors. Floating VCRs can be placed between two passive or active components. Grounded VCRs, the more common and less complicated design, require that one port of the voltage-controlled resistor be grounded.
## References
1. Hall, John. "Discrete JFET" (PDF). linearsystems.com.
2. Grundmann, Marius (2010). The Physics of Semiconductors. Springer-Verlag. ISBN 978-3-642-13884-3.
3. Junction Field-Effect Devices, Semiconductor Devices for Power Conditioning, 1982.
4. Flaherty, Nick (October 18, 2018), "Third generation SiC JFET adds 1200 V and 650 V options", EeNews Power Management.
5. For a discussion of JFET structure and operation, see for example D. Chattopadhyay (2006). "§13.2 Junction field-effect transistor (JFET)". Electronics (fundamentals and applications). New Age International. pp. 269 ff. ISBN 978-8122417807.
6. "Junction Field Effect Transistor (JFET)" (PDF). ETEE3212 Lecture Notes. value of vGS ... for which the channel is completely depleted ... is called the threshold, or pinch-off, voltage and occurs at vGS = VGS(OFF). ... This linear region of operation is called ohmic (or sometimes triode) ... Beyond the knee of the ohmic region, the curves become essentially flat in the active (or saturation) region of operation.
7. Sedra, Adel S.; Smith, Kenneth C. "5.11 THE JUNCTION FIELD-EFFECT TRANSISTOR (JFET)" (PDF). Microelectronic Circuits. At this value of vGS the channel is completely depleted ... For JFETs the threshold voltage is called the pinch-off voltage and is denoted VP.
8. Horowitz, Paul; Hill, Winfield (1989). The art of electronics (2nd ed.). Cambridge [England]: Cambridge University Press. p. 120. ISBN 0-521-37095-7. OCLC 19125711. For JFETs the gate-source voltage at which drain current approaches zero is called the "gate-source cutoff voltage", VGS(OFF), or the "pinch-off voltage", VP ... For enhancement-mode MOSFETs the analogous quantity is the "threshold voltage"
9. Mehta, V. K.; Mehta, Rohit (2008). "19 Field Effect Transistors" (PDF). Principles of electronics (11th ed.). S. Chand. pp. 513–514. ISBN 978-8121924504. OCLC 741256429. Pinch off Voltage (VP). It is the minimum drain–source voltage at which the drain current essentially becomes constant. ... Gate–source cut off voltage VGS (off). It is the gate–source voltage where the channel is completely cut off and the drain current becomes zero.
10. U. A. Bakshi; A. P. Godse (2008). Electronics Engineering. Technical Publications. p. 10. ISBN 978-81-8431-503-5. Do not confuse cutoff with pinch off. The pinch-off voltageVP is the value of the VDS at which the drain current reaches a constant value for a given value of VGS. ... The cutoff voltage VGS(off) is the value of VGS at which the drain current is 0.
11. "J201 data sheet" (PDF). Retrieved 2021-01-22.
12. "A4.11 Envelope or Enclosure". ANSI Y32.2-1975 (PDF). The envelope or enclosure symbol may be omitted from a symbol referencing this paragraph, where confusion would not result
13. "What is the Ohmic Region of a FET Transistor". www.learningaboutelectronics.com. Retrieved 2020-12-13. ohmic region ... also called the linear region
14. Balbir Kumar and Shail B. Jain (2013). Electronic Devices and Circuits. PHI Learning Pvt. Ltd. pp. 342–345. ISBN 9788120348448.
15. "Junction Field Effect Transistor". Electronics Tutorials. Saturation or Active Region
16. Scholberg, Kate (2017-03-23). "What is the meaning of "pinch-off region"?". The "pinch-off region" (or "saturation region") refers to operation of a FET with ${\displaystyle V_{ds}}$ more than a few volts.
17. Kirt Blattenberger RF Cafe. "JFETS: How They Work, How to Use Them, May 1969 Radio-Electronics" . Retrieved 2021-01-04. yfs – Small-signal, common-source, forward transadmittance (sometimes called gfs-transconductance)
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Examveda
# If Sb, is the average bond stress on a bar of diameter d subjected to maximum stress t, the length of the embedment $$l$$ is given by
A. $$l = \frac{{{\text{dt}}}}{{{{\text{S}}_{\text{b}}}}}$$
B. $$l = \frac{{{\text{dt}}}}{{2{{\text{S}}_{\text{b}}}}}$$
C. $$l = \frac{{{\text{dt}}}}{{3{{\text{S}}_{\text{b}}}}}$$
D. $$l = \frac{{{\text{dt}}}}{{4{{\text{S}}_{\text{b}}}}}$$
Related Questions on RCC Structures Design
If the shear stress in a R.C.C. beam is
A. Equal or less than 5 kg/cm2, no shear reinforcement is provided
B. Greater than 4 kg/cm2, but less than 20 kg/cm2, shear reinforcement is provided
C. Greater than 20 kg/cm2, the size of the section is changed
D. All the above
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Mixed moments for the birthday problem
Let $X_1,X_2,\dots$ be iid draws from the uniform distribution on $\{1,2,...,m\}$, and let the random variable $N$ be the minimum $j$ such that $X_j = X_i$ for some $i<j$.
I'm aware that the expected value of $N \choose 2$ (I call this a "mixed moment" since it's 1/2 times the second moment minus 1/2 times the first moment) is exactly $m$. Who first noticed/proved this?
Also, I've heard that, more generally, the expected value of $N \choose k$ is a degree-$k/2$ polynomial function of $m$ when $k$ is even. Where can I learn more about this?
For any natural $n$, \begin{multline*} \PP(N>n)=\PP(X_1,\dots,X_n\text{ take distinct values })=\frac{m(m-1)\cdots(m-n+1)}{m^n} \\ =\frac{(m-1)\cdots(m-n+1)}{m^{n-1}}, \end{multline*} whence $$\PP(N=n)=\PP(N>n-1)-\PP(N>n) =\frac{(m-1)\cdots(m-n+2)}{m^{n-1}}\,(n-1).$$ So, \begin{multline*} \mu_k(m):=\E \binom Nk \\ =\frac1{k!}\sum_{n=k}^{m+1}\frac{n-1}{m^{n-1}}\,n(n-1)\cdots(n-k+1)(m-1)\cdots(m-n+2). \tag{1} \end{multline*} From here, with the help of Mathematica, I do get $$\E \binom N2=m.$$
However, $\mu_4(10)-3\mu_4(9)+3\mu_4(8)-\mu_4(7)=0.0126\ldots\ne0$, so that $\mu_4$ is not a polynomial of degree $4/2=2$. That is, in general the statement that $\E \binom Nk$ is a polynomial of degree $k/2$ in $m$ is false.
Let us now show that $\mu_4(m)$ is not a polynomial in $m$ of any degree. Let $X_m$ be a random variable (r.v.) with the Gamma distribution with parameters $m$ and $1$, so that $X_m$ has the distribution of the sum of $m$ iid standard exponential r.v.'s, each of those r.v.'s with mean $1$. Then, by the central limit theorem, $\PP(X_m>m)\to1/2$ as $m\to\infty$.
An expression for $\mu_4(m)$ for any natural $m$ (also obtained with the help of Mathematica) is $\frac{1}{3} m \left(-2 e^m m E_{1-m}(m)+m+1\right)$, where $E_n(z)=\int_1^\infty t^{-n}e^{-tz}\,dt$ is "the exponential integral function"; I have verified numerically that for $m=1,\dots,100$ the latter expression for $\mu_4(m)$ matches the special case of (1) for $k=4$. So, if $\mu_4(m)$ were a polynomial in $m$, then so would be $e^m m^2 E_{1-m}(m)$. But \begin{multline*} e^m m^2 E_{1-m}(m)=e^m m^2 \frac1{m^m}\int_m^\infty u^{m-1}e^{-u}du =m\,\frac{e^m m!}{m^m}\,\PP(X_m>m) \\ \sim m\,\sqrt{2\pi m}\,\frac12 \end{multline*} as $m\to\infty$, by Stirling's formula and because $\PP(X_m>m)\to1/2$. So, $e^m m^2 E_{1-m}(m)$ cannot be a polynomial in $m$. Thus, $\mu_4(m)$ is not a polynomial in $m$, of any degree.
Here is an approach via Lagrange inversion.
Let $N$ denote the time of the first repeat, and let $T(z)$ (the tree function'') be the formal power series satisfying $T(z)=z\,e^{T(z)}$.
If $F$ is a formal power series the coefficients of $G(z):=F(T(z))$ are given by (Lagrange inversion) $$[z^0]G(z)=[z^0] F(z) \mbox{ , } [z^k]G(z)=\tfrac{1}{k} [y^{k-1}] F^\prime(y)\,e^{ky} =[y^k](1-y)F(y)\,e^{ky}\mbox{ for } k\geq 1\;.$$ In particular
$$[z^m] T(z)^k=\frac{k}{m} \frac{m^{m-k}}{(m-k)!}\mbox{ and } \frac{1}{1-T(z)}=\sum_{n\geq 0}\frac{n^{n}}{n!}z^n$$ Using the first relation it is easily seen that the generating function of $N$ may be written as $$\mathbb{E} t^N=\frac{m!}{m^m} [z^m]\frac{t}{1- tT(z)}$$ Thus the binomial moments $\mathbb{E}{N \choose k}$ of $N$ can be obtained as $$\mathbb{E}{N \choose k} = \frac{m!}{m^m} [z^m t^k]\frac{1+t}{1- (1+t)T(z)}=\frac{m!}{m^m} [z^m ]\frac{T(x)^{k-1}}{(1- T(z))^{k+1}}$$ Differentiation shows that $z\,T^\prime(z)= \frac{T(z)}{1-T(z)}$. Therefore \begin{align*} \mathbb{E}{N \choose 2} &=\frac{m!}{m^m} [z^m ]\frac{T(z)}{(1- T(z))^{3}}\\ &=\frac{m!}{m^m} [z^{m-1} ]\frac{T^\prime(z)}{(1- T(z))^{2}}\\ &=\frac{m!}{m^m} [z^{m-1} ]\big(\frac{1}{1- T(z)}\big)^\prime\\ &=\frac{m!}{m^m} m\,[z^{m} ]\frac{1}{1- T(z)} =m\end{align*} (I don't know who first observed that.) Similarly \begin{align*} \mathbb{E}{N \choose 4} &=\frac{m!}{m^m} [z^m ]\frac{T(z)^3}{(1- T(z))^{5}}\\ &=\frac{m!}{m^m} [z^{m-1} ] T^\prime(z)\frac{T(z)^2}{(1- T(z))^{4}}\\ &=\frac{m!}{m^m} [z^{m-1} ]\big(\frac{1}{1- T(z)}-\frac{2}{3}\frac{1}{(1-T(z))^2}+\frac{1}{3}\frac{T(z)}{(1-T(z))^3}\big)^\prime\\ &=\frac{m!}{m^m} m\,[z^{m}]\big(\frac{1}{1- T(z)}-\frac{2}{3}\frac{1}{(1-T(z))^2}+\frac{1}{3}\frac{T(z)}{(1-T(z))^3}\big)\\ &=\frac{m^2}{3} +m -\frac{2}{3} m\,\mathbb{E}(N)\end{align*} Clearly other binomial moments can be treated in the same way.
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# Density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix (possibly infinite dimensional) of trace one, that describes the statistical state of a quantum system.[1] The formalism was introduced by John von Neumann[2] (and, less systematically, independently by Lev Landau and Felix Bloch in 1927).[3][4]
The density matrix is useful for describing and performing calculations with a mixed state, which is a statistical ensemble of several quantum states. (This is in contrast to a pure state, which is a quantum system that is described by a single state vector). The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.
Mixed states arise in situations where there is classical uncertainty, i.e. when it is unknown by the experimenter which particular states are being manipulated. (This should not be confused with the unrelated concept of quantum uncertainty, which dictates that even if the experimenter knows which particular states are being manipulated, the results of some measurements cannot be predicted.) For example: A quantum system with an uncertain or randomly-varying preparation history (so one does not know with certainty which pure quantum state the system is in); or a quantum system in thermal equilibrium (at finite temperatures). Also, if a quantum system has two or more subsystems that are entangled, then each individual subsystem must be treated as a mixed state even if the complete system is in a pure state; the density matrix of the subsystem is calculated as a partial trace of the density matrix of the whole system. Relatedly, the density matrix is a crucial tool in quantum decoherence theory. See also quantum statistical mechanics.
The operator that is represented by the density matrix is called the density operator. (The close relationship between matrices and operators is a basic concept in linear algebra; see the article Linear operator for details.) In practice, the terms "density matrix" and "density operator" are often used interchangeably. (When the space of quantum states is infinite-dimensional, the term "density operator" is preferred.) The density operator, like the density matrix, is positive-semidefinite, self-adjoint, and has trace one.
## Pure and mixed states
In quantum mechanics, a quantum system is represented by a state vector (or ket) $| \psi \rangle$. A quantum system with a state vector $| \psi \rangle$ is called a pure state. However, it is also possible for a system to be in a statistical ensemble of different state vectors: For example, there may be a 50% probability that the state vector is $| \psi_1 \rangle$ and a 50% chance that the state vector is $| \psi_2 \rangle$. This system would be in a mixed state. The density matrix is especially useful for mixed states, because any state, pure or mixed, can be characterized by a single density matrix.
A mixed state is different from a quantum superposition. In fact, a quantum superposition of pure states is another pure state, for example $| \psi \rangle = (| \psi_1 \rangle + | \psi_2 \rangle)/\sqrt{2}$.
### Example: Light polarization
An example of pure and mixed states is light polarization. Photons can have two helicities, corresponding to two orthogonal quantum states, $|R\rangle$ (right circular polarization) and $|L\rangle$ (left circular polarization). A photon can also be in a superposition state, such as $(|R\rangle+|L\rangle)/\sqrt{2}$ (vertical polarization) or $(|R\rangle-|L\rangle)/\sqrt{2}$ (horizontal polarization). More generally, it can be in any state $\alpha|R\rangle+\beta|L\rangle$, corresponding to linear, circular, or elliptical polarization.
However, unpolarized light (such as the light from an incandescent light bulb) is different from any of these. Unlike linearly or elliptically polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and unlike circularly polarized light, it cannot be made linearly polarized with any wave plate. Indeed, unpolarized light cannot be described as any state of the form $\alpha|R\rangle+\beta|L\rangle$. However, unpolarized light can be described perfectly by assuming that each photon is either $| R \rangle$ with 50% probability or $| L \rangle$ with 50% probability. The same behavior would occur if each photon was either vertically polarized with 50% probability or horizontally polarized with 50% probability.
Therefore, unpolarized light cannot be described by any pure state, but can be described as a statistical ensemble of pure states in at least two ways (the ensemble of half left and half right circularly polarized, or the ensemble of half vertically and half horizontally linearly polarized). These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. One of the advantages of the density matrix is that there is just one density matrix for each mixed state, whereas there are many statistical ensembles of pure states for each mixed state. Nevertheless, the density matrix contains all the information necessary to calculate any measurable property of the mixed state.
Where do mixed states come from? To answer that, consider how to generate unpolarized light. One way is to use a system in thermal equilibrium, a statistical mixture of enormous numbers of microstates, each with a certain probability (the Boltzmann factor), switching rapidly from one to the next due to thermal fluctuations. Thermal randomness explains why an incandescent light bulb, for example, emits unpolarized light. A second way to generate unpolarized light is to introduce uncertainty in the preparation of the system, for example, passing it through a birefringent crystal with a rough surface, so that slightly different parts of the beam acquire different polarizations. A third way to generate unpolarized light uses an EPR setup: A radioactive decay can emit two photons traveling in opposite directions, in the quantum state $(|R,L\rangle+|L,R\rangle)/\sqrt{2}$. The two photons together are in a pure state, but if you only look at one of the photons and ignore the other, the photon behaves just like unpolarized light.
More generally, mixed states commonly arise from a statistical mixture of the starting state (such as in thermal equilibrium), from uncertainty in the preparation procedure (such as slightly different paths that a photon can travel), or from looking at a subsystem entangled with something else.
### Mathematical description
In quantum mechanics, the state vector $| \psi \rangle$ [5] of a system completely determines the statistical behavior of a measurement. As an example, take an observable quantity, and let A be the associated observable operator which has a representation on the Hilbert space $\mathcal{H}$ of the quantum system. For any real-valued function F[6] defined on the real numbers, the expectation value of F(A) [7] is the quantity
$\langle \psi | F(A) | \psi \rangle\, .$
Now consider the example of a "mixed quantum system" prepared by statistically combining two different pure states $| \psi \rangle$ and $|\phi\rangle$, with the associated probabilities p and 1 − p, respectively. The associated probabilities mean that the preparation process for the quantum system ends in the state $|\psi\rangle$ with probability p and in the state $|\phi\rangle$ with probability 1 − p.
It is not hard to show that the statistical properties of the observable for the system prepared in such a mixed state are completely determined. However, there is no state vector $|\xi\rangle$ which determines this statistical behavior in the sense that the expectation value of F(A) is
$\langle \xi | F(A) | \xi \rangle \, .$
Nevertheless, there is a unique operator ρ such that the expectation value of F(A) can be written as
$\operatorname{tr}[\rho F(A)]\, ,$
where the operator ρ is the density operator of the mixed system. A simple calculation shows that the operator ρ for the above example is given by
$\rho = p | \psi\rangle \langle \psi | + (1-p) | \phi\rangle \langle \phi |\,.$
## Formulation
For a finite dimensional function space, the most general density operator is of the form
$\rho = \sum_j p_j |\psi_j \rang \lang \psi_j|$
where the coefficients pj are non-negative and add up to one. This represents a statistical mixture of pure states. If the given system is closed, then one can think of a mixed state as representing a single system with an uncertain preparation history, as explicitly detailed above; or we can regard the mixed state as representing an ensemble of systems, i.e. large number of copies of the system in question, where pj is the proportion of the ensemble being in the state $\textstyle |\psi_j \rang$. An ensemble is described by a pure state if every copy of the system in that ensemble is in the same state, i.e. it is a pure ensemble.
If the system is not closed, however, then it is simply not correct to claim that it has some definite but unknown state vector, as the density operator may record physical entanglements to other systems.
Example Consider a quantum ensemble of size N with occupancy numbers n1, n2,...,nk corresponding to the orthonormal states $\textstyle |1\rang,...,|k\rang$, respectively, where n1+...+nk = N, and, thus, the coefficients pj = nj /N. For a pure ensemble, where all N particles are in state $\textstyle |i\rang$, we have nj = 0, for all ji, from which we recover the corresponding density operator $\textstyle\rho = |i\rang\lang i|$.
However the density operator of a mixed state does not capture all the information about a mixture; in particular, the coefficients pj and the kets ψj are not recoverable from the operator ρ without additional information. This non-uniqueness implies that different ensembles or mixtures may correspond to the same density operator. Such equivalent ensembles or mixtures cannot be distinguished by measurement of observables alone. This equivalence can be characterized precisely. Two ensembles ψ, ψ' define the same density operator if and only if there is a matrix U with
U * U = I
i.e., U is unitary and such that
$| \psi_i'\rangle \sqrt {p_i'} = \sum_{j} u_{ij} | \psi_j\rangle \sqrt {p_j}.$
This is simply a restatement of the following fact from linear algebra: for two square matrices M and N, M M* = N N* if and only if M = NU for some unitary U. (See square root of a matrix for more details.) Thus there is a unitary freedom in the ket mixture or ensemble that gives the same density operator. However if the kets in the mixture are orthonormal then the original probabilities pj are recoverable as the eigenvalues of the density matrix.
In operator language, a density operator is a positive semidefinite, hermitian operator acting on the state space of trace 1. A density operator describes a pure state if it is a rank one projection. Equivalently, a density operator ρ is a pure state if and only if
$\; \rho = \rho^2$,
i.e. the state is idempotent. This is true regardless of whether H is finite dimensional or not.
Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an extremal point of that set.
It follows from the spectral theorem for compact self-adjoint operators that every mixed state is an infinite convex combination of pure states. This representation is not unique. Furthermore, a theorem of Andrew Gleason states that certain functions defined on the family of projections and taking values in [0,1] (which can be regarded as quantum analogues of probability measures) are determined by unique mixed states. See quantum logic for more details.
## Measurement
Let A be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states $\textstyle |\psi_j\rang$ occurs with probability pj. Then the corresponding density operator is:
$\rho = \sum_j p_j |\psi_j \rang \lang \psi_j| .$
The expectation value of the measurement can be calculated by extending from the case of pure states (see Measurement in quantum mechanics):
$\lang A \rang = \sum_j p_j \lang \psi_j|A|\psi_j \rang = \operatorname{tr}[\rho A],$
where tr denotes trace. Moreover, if A has spectral resolution
$A = \sum_i a_i |a_i \rang \lang a_i| = \sum _i a_i P_i,$
where $P_i = |a_i \rang \lang a_i|$, the corresponding density operator after the measurement is given by:
$\; \rho ^' = \sum_i P_i \rho P_i.$
Note that the above density operator describes the full ensemble after measurement. The sub-ensemble for which the measurement result was the particular value ai is described by the different density operator
$\rho_i' = \frac{P_i \rho P_i}{\operatorname{tr}[\rho P_i]}.$
This is true assuming that $\textstyle |a_i\rang$ is the only eigenket (up to phase) with eigenvalue ai; more generally, Pi in this expression would be replaced by the projection operator into the eigenspace corresponding to eigenvalue ai.
### Entropy
The von Neumann entropy S of a mixture can be expressed in terms of the eigenvalues of ρ or in terms of the trace and logarithm of the density operator ρ. Since ρ is a positive semi-definite operator, it has a spectral decomposition such that $\rho= \sum_i \lambda_i |\varphi_i\rangle\langle\varphi_i|$ where $|\varphi_i\rangle$ are orthonormal vectors. Therefore the entropy of a quantum system with density matrix ρ is
$S = -\sum_i \lambda_i \ln \,\lambda_i = -\operatorname{tr}(\rho \ln \rho)\quad.$
Also it can be shown that
$S\left(\rho=\sum_i p_i\rho_i\right)= H(p_i) + \sum_i p_iS(\rho_i)$
where H(p) is the Shannon entropy. This entropy can increase but never decrease[8][9] with a measurement. The entropy of a pure state is zero, while that of a proper mixture always greater than zero. Therefore a pure state may be converted into a mixture by a measurement, but a proper mixture can never be converted into a pure state. Thus the act of measurement induces a fundamental irreversible change on the density matrix; this is analogous to the "collapse" of the state vector, or wavefunction collapse.
(A subsystem of a larger system can be turned from a mixed to a pure state, but only by increasing the von Neumann entropy elsewhere in the system. This is analogous to how the entropy of an object can be lowered by putting it in a refrigerator: The air outside the refrigerator's heat-exchanger warms up, gaining even more entropy than was lost by the object in the refrigerator. See second law of thermodynamics.)
## Von Neumann equation
Just as the Schrödinger equation describes how pure states evolve in time, the von Neumann equation (also known as Liouville-von Neumann equation) describes how a density operator evolves in time (in fact, the two equations are equivalent, in the sense that either can be derived from the other.) The von Neumann equation states:[10][11]
$i \hbar \frac{\partial \rho}{\partial t} = [H,\rho]$
where the brackets denote a commutator. Note that this equation is only true when the density operator is taken to be in the Schrödinger picture even though this equation seems at first look to emulate the Heisenberg equation of motion:
$\frac{dA^{(H)}}{dt}=\frac{1}{i\hbar}[A^{(H)},H]$
where A(H) is some 'Heisenberg' operator. Taking the density operator to be in the Schrödinger picture makes sense since it is composed of 'Schrödinger' kets and bras evolved in time as per the Schrödinger picture. If the Hamiltonian is time-independent, this differential equation can be solved to get
$\rho(t) = e^{-i H t/\hbar} \rho(0) e^{i H t/\hbar}.$
## "Quantum Liouville", Moyal's equation
Under the Wigner map, the density matrix transforms into the Wigner function. The equation for time-evolution of the Wigner function is then the transform of the above von Neumann equation,
$\frac{\partial W(q,p,t)}{\partial t} = -\{\{W(q,p,t) , H(q,p )\}\}~,$
where H(q,p) is the Hamiltonian, and { { •,• } } is the Moyal bracket, the transform of the quantum commutator.
The evolution equation for the Wigner function is then analogous to that of its classical limit, the Liouville equation of classical physics. In the limit of vanishing Planck's constant ħ, W(q,p,t) reduces to the classical Liouville probability density function in phase space. The classical Liouville equation can be solved using the method of characteristics for partial differential equations, the characteristic equations being Hamilton's equations. The Moyal equation in quantum mechanics similarly admits formal solutions in terms of quantum characteristics, predicated on the ∗−product of phase space.
## Composite Systems
The joint density matrix of a composite system of two systems A and B is described by ρAB. Then the subsystems are described by their reduced density operator.
$\rho_A=\operatorname{tr}_B\rho_{AB}$
$\operatorname{tr}_B$ is called partial trace over system B. If A and B are two distinct and independent systems then $\rho_{AB}=\rho_{A}\otimes\rho_{B}$ which is a product state.
## C*-algebraic formulation of states
It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable.[12][13] For this reason, observables are identified to elements of an abstract C*-algebra A (that is one without a distinguished representation as an algebra of operators) and states are positive linear functionals on A. However, by using the GNS construction, we can recover Hilbert spaces which realize A as a subalgebra of operators.
Geometrically, a pure state on a C*-algebra A is a state which is an extreme point of the set of all states on A. By properties of the GNS construction these states correspond to irreducible representations of A.
The states of the C*-algebra of compact operators K(H) correspond exactly to the density operators and therefore the pure states of K(H) are exactly the pure states in the sense of quantum mechanics.
The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures, as noted in the introduction.
## Notes and references
1. ^ Fano, Ugo (1957), "Description of States in Quantum Mechanics by Density Matrix and Operator Techniques", Reviews of Modern Physics 29: 74–93, Bibcode 1957RvMP...29...74F, doi:10.1103/RevModPhys.29.74.
2. ^ von Neumann, John (1927), "Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik", Göttinger Nachrichten 1: 245–272.
3. ^ Landau, L. D. (1927), "Das Dämpfungsproblem in der Wellenmechanik", Zeitschrift für Physik 45: 430–441, Bibcode 1927ZPhy...45..430L, doi:10.1007/BF01343064
4. ^ Landau, L. D., and Lifshitz, E. M. (1977). Quantum Mechanics, Non-Relativistic Theory: Volume 3. Oxford: Pergamon Press. pp. 41. ISBN 0080178014.
5. ^ We assume $| \psi \rangle$ is a pure state in this example.
6. ^ Technically, F must be a Borel function
7. ^ F(A) is defined to be the result of measuring the observable associated with the operator A, and then applying F to the outcome.
8. ^ Nielsen, Michael; Chuang, Isaac (2000), Quantum Computation and Quantum Information, Cambridge University Press, ISBN 978-0-521-63503-5 . Chapter 11: Entropy and information, Theorem 11.9, "Projective measurements cannot decrease entropy"
9. ^ Everett, Hugh (1973), "The Theory of the Universal Wavefunction (1956) Appendix I. "Monotone decrease of information for stochastic processes"", The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press, pp. 128–129, ISBN 978-0-691-08131-1
10. ^ The theory of open quantum systems, by Breuer and Petruccione, p110.
11. ^ Statistical mechanics, by Schwabl, p16.
12. ^ See appendix, Mackey, George Whitelaw (1963), Mathematical Foundations of Quantum Mechanics, Dover Books on Mathematics, New York: Dover Publications, ISBN 978-0-486-43517-6
13. ^ Emch, Gerard G. (1972), Algebraic methods in statistical mechanics and quantum field theory, Wiley-Interscience, ISBN 978-0-471-23900-0
• S. Kryszewski, [1]
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$L^1$ space with values in a Banach Space
I have been reading a bit about the Bochner integral and now I'm wondering the following:
For the theory to be "nice", one would expect that
$$L^1([0, \tau], L^1([0, \tau])) \cong L^1([0,\tau] \times [0, \tau]).$$
Is this the case? How do we prove this (or where can I find more about this), since if we take $u$ in the set in the LHS, then we can evaluate $(u(t))(x)$ and we want to map this to some $u(t,x)$. The problem I see is that $L^1$-functions are equivalence classes modulo sets of measure zero. So for each $t\in [0,\tau]$ we have different sets of measure zero than the ones in $[0,\tau]^2$. How do we get around this?
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There is no need to restrict to intervals, the same holds true for measure spaces $X$ and $Y$ and Banach spaces $E$ in general:
$L^{1}(X \times Y, E) \cong L^{1}(X,L^{1}(Y,E))$
Edit: Assume that $X$ and $Y$ are $\sigma$-finite and complete for the sake of simplicity. See also the edit further down.
By definition each Bochner-integrable function can be approximated by simple functions. In other words, the functions of the form $[A]e$ for some measurable subset $A \subset X$ of finite measure and $e \in E$ (or out of some dense subspace of $E$, if you prefer) generate a dense subspace of $L^{1}(X,E)$. Therefore you can reduce to the case of defining a bijection $[A]\cdot ([B] \cdot e) \leftrightarrow [A \times B]\cdot e$ and observe that this is a bijection on sets generating dense subspaces of $L^{1}(X, L^{1}(Y,E))$ and $L^{1}(X \times Y, E)$ and extend it linearly. That it is well-defined and an isometry is a special case of Fubini, that it is surjective in both directions follows from density.
The general fact underlying this is the canonical isomorphism
$L^{1}(Y,E) \cong L^{1}(Y) \hat{\otimes} E$ (projective tensor product)
and the isomorphism $L^{1}(X \times Y) \cong L^{1}(X) \hat{\otimes} L^{1}(Y)$ for $L^{1}$-spaces + associativity of the tensor product (and all this is proved in exactly the same way).
Edit: For the (non-$\sigma$-finite) general case the situation is quite a bit more subtle and is discussed carefully in the exercise section 253Y of Fremlin's Measure Theory, Volume II. Here are the essential points:
1. For a general measure space $Y$ one can prove that $L^{1}(Y,E) \cong L^1(Y)\hat{\otimes} E$ (see 253Yf (vii)).
2. For a pair of measure spaces $(X,\mu)$ and $(Y,\nu)$ let $(X \times Y, \lambda)$ be the complete locally determined product measure space as defined by Fremlin, 251F. The rather deep theorem 253F in Fremlin then tells us that $L^1(X \times Y, \lambda) \cong L^1(X,\mu) \hat{\otimes} L^1(Y,\nu)$.
Piecing these two results together and making use of the associativity of the projective tensor product we get
\begin{align*}L^1(X \times Y, E) & \cong L^1(X \times Y) \hat{\otimes} E \cong \left(L^1(X) \hat{\otimes} L^1(Y)\right) \hat{\otimes} E & &(\text{using 1. and 2., respectively})\\\ & \cong L^1(X) \hat{\otimes} \left(L^1(Y) \hat{\otimes} E\right) \cong L^1 (X) \hat{\otimes} L^1(Y,E) & &\text{(using associativity and 1.)} \\\ & \cong L^1(X,L^1(Y,E)) & & (\text{using 1. again})\end{align*}
as asserted in an earlier version of this answer.
Finally, a remark on user3148's cautionary counterexample. There is an isomorphism $(L^{1}(X,E))^{\ast} \cong L_{w^{\ast}}^{\infty}(X,E^{\ast})$ where the latter space is defined by weak$^{\ast}$-measurably in the sense of Gelfand-Dunford. So in this sense we have $L_{w^{\ast}}^{\infty}(X \times Y, E^{\ast}) \cong L_{w^{\ast}}^{\infty}(X, L_{w^{\ast}}^{\infty}(Y,E^{\ast}))$ simply by duality theory.
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For the last paragraph one should restrict to $\sigma$-finite measure spaces in order to avoid annoyances with differences between null sets and local null sets. – t.b. Jan 20 '11 at 20:34
If you also could give a good literature reference, I'd be happy to accept this answer. – Jonas Teuwen Jan 20 '11 at 21:17
@Jonas T: It's difficult to give a really good reference, but let me try: A basic and quite readable exposition of Bochner theory (just a few pages collecting and proving the essentials) and its interaction with the projective tensor product can be found in Chapter 2 in R. A. Ryan, An introduction to tensor products of Banach spaces, Springer '02. The Gelfand-Dunford stuff is not mentioned there but you can just take the definition via duality as a definition, if you want. I can try and give you further references in case you need more--Grothendieck's thesis and Bourbaki are not recommended. – t.b. Jan 20 '11 at 21:36
That is fine, I'll check it out in the library. Thanks! – Jonas Teuwen Jan 20 '11 at 21:42
You say "measure spaces $X$ and $Y$" but in the end you apply Fubini. What if $X$ and $Y$ are not sigma-finite and Fubini doesn't hold for them? – GEdgar Aug 1 '11 at 21:55
You can define the obvious isomorphism $T$ on representatives, which is isometric by Fubini's theorem. Isometric maps defined on representatives are well defined on equivalence classes, because if $u_1$ and $u_2$ are representatives of an element of the LHS, then $\|Tu_1-Tu_2\|=\|T(u_1-u_2)\|=\|u_1-u_2\|=0$.
It is a useful fact that a linear map initially defined on some type of representative can often be seen to be well defined from the fact that it is continuous. In this case, all the work is hidden in Fubini's theorem.
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Thanks. I didn't get to the Fubini Theorem for Bochner integrals yet, I'll check it out. – Jonas Teuwen Jan 20 '11 at 19:07
Since I am not able to comment yet, I write this as an answer.
Be prepared to face the fact that $L^\infty([0,T],L^\infty([0,T])) \neq L^\infty([0,T]\times[0,T])$. To see this, observe that the function $$f(x,y) = \begin{cases} 1 & x>y\\ 0 & \text{else}\end{cases}$$ is not an element of the first space.
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# American Institute of Mathematical Sciences
August 2013, 18(6): 1555-1565. doi: 10.3934/dcdsb.2013.18.1555
## Exponential stability for a class of linear hyperbolic equations with hereditary memory
1 Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano 2 Politecnico di Milano - Dipartimento di Matematica “F. Brioschi”, Via Bonardi 9, 20133 Milano, Italy
Received June 2011 Revised November 2011 Published March 2013
We establish a necessary and sufficient condition of exponential stability for the contraction semigroup generated by an abstract version of the linear differential equation $$∂_t u(t)-\int_0^\infty k(s)\Delta u(t-s)ds = 0$$ modeling hereditary heat conduction of Gurtin-Pipkin type.
Citation: Monica Conti, Elsa M. Marchini, Vittorino Pata. Exponential stability for a class of linear hyperbolic equations with hereditary memory. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1555-1565. doi: 10.3934/dcdsb.2013.18.1555
##### References:
[1] V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269-291. Google Scholar [2] V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. Google Scholar [3] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. Google Scholar [4] B. R. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process, SIAM J. Math. Anal., 33 (2002), 1090-1106. doi: 10.1137/S0036141001388592. Google Scholar [5] M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152. doi: 10.1007/BF00375589. Google Scholar [6] D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles), C. R. Acad. Sci. Paris Sér. A-B, 277 (1973), B471-B473. Google Scholar [7] C. Giorgi, M. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: a semigroup approach, Comm. Appl. Anal., 5 (2001), 121-133. Google Scholar [8] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. Google Scholar [9] E. Hewitt and K. Stromberg, "Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable," Springer-Verlag, New York, 1965. Google Scholar [10] T. Hillen and K. P. Hadeler, Hyperbolic systems and transport equations in mathematical biology, in "Analysis and Numerics for Conservation Laws," Springer, Berlin, (2005), 257-279. doi: 10.1007/3-540-27907-5_11. Google Scholar [11] Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems," Chapman & Hall/CRC Research Notes in Mathematics, 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. Google Scholar [12] V. Méndez, J. Fort and J. Farjas, Speed of wave-front solutions to hyperbolic reaction-diffusion equations, Phys. Rev. E (3), 60 (1999), 5231-5243. doi: 10.1103/PhysRevE.60.5231. Google Scholar [13] V. Méndez and J. E. Llebot, Hyperbolic reaction-diffusion equations for a forest fire model, Phys. Rev. E (3), 56 (1997), 6557-6563. doi: 10.1103/PhysRevE.56.6557. Google Scholar [14] J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. Google Scholar [15] W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), 171-188. doi: 10.1137/0146013. Google Scholar [16] V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730. doi: 10.3934/cpaa.2010.9.721. Google Scholar [17] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [18] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. Google Scholar
show all references
##### References:
[1] V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269-291. Google Scholar [2] V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. Google Scholar [3] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. Google Scholar [4] B. R. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process, SIAM J. Math. Anal., 33 (2002), 1090-1106. doi: 10.1137/S0036141001388592. Google Scholar [5] M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152. doi: 10.1007/BF00375589. Google Scholar [6] D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles), C. R. Acad. Sci. Paris Sér. A-B, 277 (1973), B471-B473. Google Scholar [7] C. Giorgi, M. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: a semigroup approach, Comm. Appl. Anal., 5 (2001), 121-133. Google Scholar [8] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. Google Scholar [9] E. Hewitt and K. Stromberg, "Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable," Springer-Verlag, New York, 1965. Google Scholar [10] T. Hillen and K. P. Hadeler, Hyperbolic systems and transport equations in mathematical biology, in "Analysis and Numerics for Conservation Laws," Springer, Berlin, (2005), 257-279. doi: 10.1007/3-540-27907-5_11. Google Scholar [11] Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems," Chapman & Hall/CRC Research Notes in Mathematics, 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. Google Scholar [12] V. Méndez, J. Fort and J. Farjas, Speed of wave-front solutions to hyperbolic reaction-diffusion equations, Phys. Rev. E (3), 60 (1999), 5231-5243. doi: 10.1103/PhysRevE.60.5231. Google Scholar [13] V. Méndez and J. E. Llebot, Hyperbolic reaction-diffusion equations for a forest fire model, Phys. Rev. E (3), 56 (1997), 6557-6563. doi: 10.1103/PhysRevE.56.6557. Google Scholar [14] J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. Google Scholar [15] W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), 171-188. doi: 10.1137/0146013. Google Scholar [16] V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730. doi: 10.3934/cpaa.2010.9.721. Google Scholar [17] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [18] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. Google Scholar
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2020 Impact Factor: 1.327
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## Forbes, Lawrence K.
Compute Distance To:
Author ID: forbes.lawrence-k Published as: Forbes, Lawrence K.; Forbes, L. K. Homepage: http://www.maths.utas.edu.au/People/Forbes External Links: MGP · ORCID · dblp
Documents Indexed: 144 Publications since 1979 Co-Authors: 55 Co-Authors with 106 Joint Publications 426 Co-Co-Authors
all top 5
### Co-Authors
37 single-authored 39 Hocking, Graeme Charles 10 Stokes, Timothy E. 7 Gray, Brian F. 7 Walters, Stephen J. 6 Belward, Shaun R. 6 Chen, Michael J. 4 Cosgrove, Jason M. 4 Crozier, Stuart 4 McCue, Scott William 4 Paul, Rhys A. 3 Beeton, Nicholas J. 3 Brideson, Michael A. 3 Koerber, Adrian J. 3 Sexton, M. Jane 2 Bassom, Andrew P. 2 Carver, Scott 2 Chandler, Graeme A. 2 Farrow, Duncan E. 2 Holmes, R. J. 2 Horsley, David E. 2 Krzysik, Oliver A. 2 Nguyen, H. H. N. 2 Reading, Anya M. 2 Schwartz, Leonard W. 2 Sidhu, Harvinder Singh 1 Allwright, Emma J. 1 Baillard, N. Y. 1 Browne, Catherine A. 1 Bulte, D. P. 1 Callaghan, T. G. 1 Chen, Sue Ann 1 Cockerill, Madeleine 1 Derrick, William R. 1 Doddrell, David M. 1 Forbes, Anne-Marie H. 1 Gujarati, K. 1 Hadley, Scott A. 1 Hansen, John M. 1 Hayes, Keith R. 1 Hindle, Ivy J. 1 Holmes, Catherine A. 1 Hosack, Geoffrey R. 1 Ickowicz, Adrien 1 Lester, Earl S. 1 Letchford, Nicholas A. 1 Myerscough, Mary R. 1 Osborne, Tobias J. 1 Russell, Patrick S. 1 Shraida, Shaymaa M. 1 Snape-Jenkinson, C. J. 1 Trenham, Claire E. 1 Turner, Ross J. 1 Vanden-Broeck, Jean-Marc 1 Watts, Anthony M. 1 While, Peter T. 1 Wilkins, Andrew H. 1 Wotherspoon, Simon J.
all top 5
### Serials
37 Journal of Engineering Mathematics 26 The ANZIAM Journal 15 Journal of Fluid Mechanics 12 Journal of the Australian Mathematical Society, Series B 6 Computers and Fluids 6 European Journal of Applied Mathematics 4 Journal of Computational Physics 4 Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 3 Dynamics and Stability of Systems 3 Applied Mathematical Modelling 3 SIAM Journal on Applied Mathematics 3 Physics of Fluids 2 Physica D 2 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 2 European Journal of Mechanics. B. Fluids 2 Journal of Theoretical Biology 1 Bulletin of the Australian Mathematical Society 1 IMA Journal of Applied Mathematics 1 Journal of Sound and Vibration 1 Quarterly Journal of Mechanics and Applied Mathematics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Physics of Fluids, A 1 Applied Mathematics Letters 1 Mathematical and Computer Modelling 1 Journal of Elasticity 1 Journal of Applied Mathematics 1 Journal of Biological Dynamics 1 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences
all top 5
### Fields
111 Fluid mechanics (76-XX) 22 Geophysics (86-XX) 20 Partial differential equations (35-XX) 17 Biology and other natural sciences (92-XX) 16 Classical thermodynamics, heat transfer (80-XX) 14 Numerical analysis (65-XX) 13 Ordinary differential equations (34-XX) 5 Dynamical systems and ergodic theory (37-XX) 4 Optics, electromagnetic theory (78-XX) 3 Mechanics of deformable solids (74-XX) 2 Integral equations (45-XX) 2 Mechanics of particles and systems (70-XX) 1 Functions of a complex variable (30-XX) 1 Special functions (33-XX) 1 Astronomy and astrophysics (85-XX) 1 Systems theory; control (93-XX)
### Citations contained in zbMATH Open
118 Publications have been cited 831 times in 353 Documents Cited by Year
Free-surface flow over a semicircular obstruction. Zbl 0517.76020
Forbes, Lawrence K.; Schwartz, Leonard W.
1982
Critical free-surfaces flow over a semi-circular obstruction. Zbl 0638.76008
Forbes, L. K.
1988
Surface waves of large amplitude beneath an elastic sheet. I. High-order series solution. Zbl 0607.76015
Forbes, Lawrence K.
1986
Surface waves of large amplitude beneath an elastic sheet. II. Galerkin solution. Zbl 0643.76013
Forbes, Lawrence K.
1988
Caluculating current densities and fields produced by shielded magnetic resonance imaging probes. Zbl 0871.65116
Forbes, Lawrence K.; Crozier, Stuart; Doddrell, David M.
1997
The Rayleigh-Taylor instability for inviscid and viscous fluids. Zbl 1180.76023
Forbes, Lawrence K.
2009
An algorithm for 3-dimensional free-surface problems in hydrodynamics. Zbl 0667.76028
Forbes, Lawrence K.
1989
Flow caused by a point sink in a fluid having a free surface. Zbl 0708.76032
Forbes, Lawrence K.; Hocking, Graeme C.
1990
On the wave resistance of a submerged semi-elliptical body. Zbl 0477.76018
Forbes, L. K.
1981
On the effects of nonlinearity in free-surface flow about a submerged point vortex. Zbl 0585.76035
Forbes, L. K.
1985
Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow. Zbl 1123.76012
Forbes, Lawrence K.; Chen, Michael J.; Trenham, Claire E.
2007
Unsteady free-surface flow induced by a line sink. Zbl 1038.76510
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2003
A note on the flow induced by a line sink beneath a free surface. Zbl 0716.76016
Hocking, G. C.; Forbes, L. K.
1991
A cylindrical Rayleigh-Taylor instability: radial outflow from pipes or stars. Zbl 1254.76084
Forbes, Lawrence K.
2011
Nonlinear, drag-free flow over a submerged semi-elliptical body. Zbl 0491.76025
Forbes, L. K.
1982
Subcritical free-surface flow caused by a line source in a fluid of finite depth. Zbl 0763.76011
Hocking, G. C.; Forbes, L. K.
1992
Super-critical withdrawal from a two-layer fluid through a line sink if the lower layer is of finite depth. Zbl 0973.76020
Hocking, G. C.; Forbes, L. K.
2001
Free-surface flow over a semicircular obstruction, including the influence of gravity and surface tension. Zbl 0524.76021
Forbes, Lawrence K.
1983
Flow induced by a line sink in a quiescent fluid with surface-tension effects. Zbl 0771.76012
Forbes, Lawrence K.; Hocking, Graeme C.
1993
Two-layer critical flow over a semi-circular obstruction. Zbl 0707.76100
Forbes, L. K.
1989
Unsteady flow induced by a withdrawal point beneath a free surface. Zbl 1123.76321
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2005
Fully non-linear two-layer flow over arbitrary topography. Zbl 0793.76106
Belward, S. R.; Forbes, L. K.
1993
The bath-plug vortex. Zbl 0846.76013
Forbes, Lawrence K.; Hocking, Graeme C.
1995
Accurate methods for computing inviscid and viscous Kelvin-Helmholtz instability. Zbl 1391.76147
Chen, Michael J.; Forbes, Lawrence K.
2011
Rayleigh-Taylor instabilities in axi-symmetric outflow from a point source. Zbl 1247.76032
Forbes, Lawrence K.
2011
On the computation of steady axisymmetric withdrawal from a two-layer fluid. Zbl 1013.76019
Forbes, Lawrence K.; Hocking, Graeme C.
2003
Free-surface flows emerging from beneath a semi-infinite plate with constant vorticity. Zbl 1003.76008
McCue, Scott W.; Forbes, Lawrence K.
2002
Unsteady draining flows from a rectangular tank. Zbl 1182.76252
Forbes, Lawrence K.; Hocking, Graeme C.
2007
Bow and stern flows with constant vorticity. Zbl 0958.76010
McCue, Scott W.; Forbes, Lawrence K.
1999
A note on withdrawal from a fluid of finite depth through a point sink. Zbl 1018.76007
Hocking, G. C.; Vanden-Broeck, J.-M.; Forbes, L. K.
2002
Waveless subcritical flow past symmetric bottom topography. Zbl 1267.76011
Holmes, R. J.; Hocking, G. C.; Forbes, L. K.; Baillard, N. Y.
2013
Unsteady free surface flow induced by a line sink in a fluid of finite depth. Zbl 1237.76132
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2008
Limit-cycle behaviour in a model chemical reaction: The Sal’nikov thermokinetic oscillator. Zbl 0729.92033
Forbes, Lawrence K.
1990
Interfacial waves and hydraulic falls: Some applications to atmospheric flows in the lee of mountains. Zbl 0823.76014
Belward, Shaun R.; Forbes, Lawrence K.
1995
A note on waveless subcritical flow past a submerged semi-ellipse. Zbl 1359.76060
Hocking, G. C.; Holmes, R. J.; Forbes, L. K.
2013
Steady flow of a buoyant plume into a constant-density layer. Zbl 1310.76176
Hocking, G. C.; Forbes, L. K.
2010
Salt-water up-coning during extraction of fresh water from a tropical island. Zbl 1046.76050
Forbes, Lawrence K.; Hocking, Graeme C.; Wotherspoon, Simon
2004
How strain and spin may make a star bi-polar. Zbl 1309.85004
Forbes, Lawrence K.
2014
A note on withdrawal through point sink in fluid of finite depth. Zbl 0854.76007
Forbes, Lawrence K.; Hocking, Graeme C.; Chandler, Graeme A.
1996
Irregular frequencies and iterative methods in the solution of steady surface-wave problems in hydrodynamics. Zbl 0569.76020
Forbes, L. K.
1984
Atmospheric interfacial waves. Zbl 0825.76126
Forbes, Lawrence K.; Belward, Shaun R.
1992
An intrusion layer in stationary incompressible fluids. II: A solitary wave. Zbl 1124.76004
Forbes, Lawrence K.; Hocking, Graeme C.
2006
Withdrawal from a two-layer inviscid fluid in a duct. Zbl 0916.76093
Forbes, Lawrence K.; Hocking, Graeme C.
1998
Inviscid and viscous models of axisymmetric fluid jets or plumes. Zbl 1254.76074
Letchford, Nicholas A.; Forbes, Lawrence K.; Hocking, Graeme C.
2012
A note on steady flow into a submerged point sink. Zbl 1306.76005
Hocking, G. C.; Forbes, L. K.; Stokes, T. E.
2014
The initiation of a planar fluid plume beneath a rigid lid. Zbl 1469.76039
Russell, Patrick S.; Forbes, Lawrence K.; Hocking, Graeme C.
2017
A combustion wave of permanent form in a compressible gas. Zbl 1065.80505
Forbes, Lawrence K.; Derrick, William
2001
On the presence of limit-cycles in a model exothermic chemical reaction: Sal’nikov’s oscillator with two temperature-dependent reaction rates. Zbl 0729.34502
Forbes, Lawrence K.; Myerscough, Mary R.; Gray, Brian F.
1991
Dynamical systems analysis of a five-dimensional trophic food web model in the southern oceans. Zbl 1196.37125
Hadley, Scott A.; Forbes, Lawrence K.
2009
An intrusion layer in stationary incompressible fluids. I: Periodic waves. Zbl 1123.76007
Forbes, Lawrence K.; Hocking, Graeme C.; Farrow, Duncan E.
2006
One-dimensional pattern formation in a model of burning. Zbl 0797.35084
Forbes, Lawrence K.
1993
Flow due to a sink near a vertical wall, in infinitely deep fluid. Zbl 1134.76318
Forbes, Lawrence K.; Hocking, Graeme C.
2005
Kelvin-Helmholtz creeping flow at the interface between two viscous fluids. Zbl 1327.76064
Forbes, Lawrence K.; Paul, Rhys A.; Chen, Michael J.; Horsley, David E.
2015
A two-dimensional model for large-scale bushfire spread. Zbl 0902.92024
Forbes, Lawrence K.
1997
Forced transverse oscillations in a simple spring-mass system. Zbl 0743.70022
Forbes, Lawrence K.
1991
A numerical method for nonlinear flow about a submerged hydrofoil. Zbl 0584.76015
Forbes, L. K.
1985
On the evolution of shock-waves in mathematical models of the aorta. Zbl 0468.76128
Forbes, L. K.
1981
Sloshing of an ideal fluid in a horizontally forced rectangular tank. Zbl 1273.76057
Forbes, Lawrence K.
2010
A line vortex in a two-fluid system. Zbl 1367.76014
Forbes, Lawrence K.; Cosgrove, Jason M.
2014
Steady periodic waves in a three-layer fluid with shear in the middle layer. Zbl 1159.76312
Chen, Michael J.; Forbes, Lawrence K.
2008
Withdrawal from a fluid of finite depth through a line sink, including surface-tension effects. Zbl 0977.76011
Hocking, G. C.; Forbes, L. K.
2000
Unsteady plumes in planar flow of viscous and inviscid fluids. Zbl 1319.76012
Forbes, Lawrence K.; Hocking, Graeme C.
2013
Steady free surface flows induced by a submerged ring source or sink. Zbl 1250.76025
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2012
The lens of freshwater in a tropical island– 2D withdrawal. Zbl 1035.76050
Hocking, G. C.; Forbes, L. K.
2004
The design of a full-scale industrial mineral leaching process. Zbl 1076.86504
Forbes, Lawrence K.
2001
Fully 3D Rayleigh-Taylor instability in a Boussinesq fluid. Zbl 1422.76066
Walters, S. J.; Forbes, L. K.
2019
On stability and uniqueness of stationary one-dimensional patterns in the Belousov-Zhabotinsky reaction. Zbl 0739.65102
Forbes, Lawrence K.
1991
The effect of surface tension on free-surface flow induced by a point sink. Zbl 1408.76114
Hocking, G. C.; Nguyen, H. H. N.; Forbes, L. K.; Stokes, T. E.
2016
Limit-cycle behaviour in a model chemical reaction: The cubic autocatalator. Zbl 0719.34047
Forbes, L. K.; Holmes, C. A.
1990
Forced oscillations in an exothermic chemical reaction. Zbl 0822.34034
Forbes, Lawrence K.; Gray, Brian F.
1994
Computing large-amplitude progressive Rossby waves on a sphere. Zbl 1151.76409
Callaghan, T. G.; Forbes, L. K.
2006
Optimal fluid injection strategies for in situ mineral leaching in two dimensions. Zbl 0953.76085
Forbes, Lawrence K.; McCue, Scott W.
1999
Exact solutions for interfacial outflows with straining. Zbl 1300.76009
Forbes, Lawrence K.; Brideson, Michael A.
2014
A spectral method for Faraday waves in rectangular tanks. Zbl 1294.76203
Horsley, David E.; Forbes, Lawrence K.
2013
On turbulence modelling and the transition from laminar to turbulent flow. Zbl 1302.76071
Forbes, Lawrence K.
2014
An exothermic chemical reaction with linear feedback control. Zbl 0864.92018
Sexton, M. J.; Forbes, L. K.
1996
A study of the dynamics of a continuously stirred tank reactor with feedback control of the temperature. Zbl 0876.34045
Sexton, M. J.; Forbes, L. K.; Gray, B. F.
1997
Analysis of chemical kinetic systems over the entire parameter space. IV: The Sal’nikov oscillator with two temperature-dependent reaction rates. Zbl 0818.92029
Gray, B. F.; Forbes, L. K.
1994
Travelling waves and oscillations in Sal’nikov’s combustion reaction in a compressible gas. Zbl 1312.76056
Paul, Rhys A.; Forbes, Lawrence K.
2015
Non-linear free-surface flows about blunt bodies. Zbl 0461.76009
Forbes, Lawrence K.
1982
A note on the flow of a homogeneous intrusion into a two-layer fluid. Zbl 1121.76016
Hocking, G. C.; Forbes, L. K.
2007
On starting conditions for a submerged sink in a fluid. Zbl 1138.76021
Forbes, Lawrence K.; Hocking, Graeme C.; Stokes, Tim E.
2008
A series analysis of forced transverse oscillations in a spring-mass system. Zbl 0684.34040
Forbes, Lawrence K.
1989
Stationary patterns of chemical concentration in the Belousov- Zhabotinskij reaction. Zbl 0701.35087
Forbes, Lawrence K.
1990
Flow fields associated with in situ mineral leaching. Zbl 0828.76080
Forbes, Lawrence K.; Watts, Anthony M.; Chandler, Graeme A.
1994
Progress toward a mining strategy based on mineral leaching with secondary recovery. Zbl 0854.76090
Forbes, Lawrence K.
1996
Atmospheric interfacial waves in the presence of two moving fluid layers. Zbl 0842.76013
Forbes, Lawrence K.; Belward, Shaun R.
1994
Thermal solitons: travelling waves in combustion. Zbl 1371.80031
Forbes, Lawrence K.
2013
Planar Rayleigh-Taylor instabilities: outflows from a binary line-source system. Zbl 1359.76118
Forbes, Lawrence K.
2014
A rational approximation to the evolution of a free surface during fluid withdrawal through a point sink. Zbl 1333.76035
Hocking, Graeme Charles; Stokes, Timothy E.; Forbes, Lawrence K.
2010
An analysis of two- and three-dimensional unsteady withdrawal flows, using shallow water theory. Zbl 0948.76010
Koerber, A. J.; Forbes, L. K.
2000
Atmospheric solitary waves: Some applications to the morning glory of the Gulf of Carpentaria. Zbl 0880.76014
Forbes, Lawrence K.; Belward, Shaun R.
1996
Burning down the house: The time to ignition of an irradiated solid. Zbl 0915.65129
Forbes, Lawrence K.; Gray, Brian F.
1998
An analytical and numerical study of the forced vibration of a spherically cavity. Zbl 0925.76154
Forbes, L. K.
1994
Dynamical systems analysis of a model describing Tasmanian devil facial tumour-disease. Zbl 1264.92034
Beeton, N. J.; Forbes, L. K.
2012
Waves in two-layer shear flow for viscous and inviscid fluids. Zbl 1258.76075
Chen, Michael J.; Forbes, Lawrence K.
2011
Selective withdrawal of a two-layer viscous fluid. Zbl 1379.76014
Cosgrove, Jason M.; Forbes, Lawrence K.
2012
A note on oscillations in a simple model of a chemical reaction. Zbl 0854.34040
Sexton, M. Jane; Forbes, Lawrence K.
1996
Large amplitude axisymmetric capillary waves. Zbl 1187.76636
Osborne, Tobias; Forbes, Lawrence K.
2001
Interfacial behaviour in two-fluid Taylor-Couette flow. Zbl 1448.76086
Forbes, L. K.; Bassom, Andrew P.
2018
Ideal planar fluid flow over a submerged obstacle: review and extension. Zbl 07455957
Forbes, Lawrence K.; Walters, Stephen J.; Hocking, Graeme C.
2021
Analytic and numerical solutions to the seismic wave equation in continuous media. Zbl 1472.86012
Walters, S. J.; Forbes, L. K.; Reading, A. M.
2020
Fully 3D Rayleigh-Taylor instability in a Boussinesq fluid. Zbl 1422.76066
Walters, S. J.; Forbes, L. K.
2019
A model for the treatment of environmentally transmitted sarcoptic mange in bare-nosed wombats (Vombatus ursinus). Zbl 1406.92349
Beeton, Nicholas J.; Carver, Scott; Forbes, Lawrence K.
2019
Interfacial behaviour in two-fluid Taylor-Couette flow. Zbl 1448.76086
Forbes, L. K.; Bassom, Andrew P.
2018
The initiation of a planar fluid plume beneath a rigid lid. Zbl 1469.76039
Russell, Patrick S.; Forbes, Lawrence K.; Hocking, Graeme C.
2017
Compressibility effects on outflows in a two-fluid system. I: Line source in cylindrical geometry. Zbl 1388.76330
Krzysik, Oliver A.; Forbes, Lawrence K.
2017
The effect of surface tension on free surface flow induced by a point sink in a fluid of finite depth. Zbl 1390.76045
Hocking, G. C.; Nguyen, H. H. N.; Stokes, T. E.; Forbes, L. K.
2017
On modelling the transition to turbulence in pipe flow. Zbl 1373.76052
Forbes, Lawrence K.; Brideson, Michael A.
2017
Unsteady flows induced by a point source or sink in a fluid of finite depth. Zbl 1386.76027
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2017
Nonlinear behaviour of interacting mid-latitude atmospheric vortices. Zbl 1369.86003
Cosgrove, Jason M.; Forbes, Lawrence K.
2017
The effect of surface tension on free-surface flow induced by a point sink. Zbl 1408.76114
Hocking, G. C.; Nguyen, H. H. N.; Forbes, L. K.; Stokes, T. E.
2016
The formation of large-amplitude fingers in atmospheric vortices. Zbl 1428.86013
Cosgrove, Jason M.; Forbes, Lawrence K.
2016
Combustion waves in Sal’nikov’s reaction scheme in a spherically symmetric gas. Zbl 1360.80012
Paul, Rhys A.; Forbes, Lawrence K.
2016
Kelvin-Helmholtz creeping flow at the interface between two viscous fluids. Zbl 1327.76064
Forbes, Lawrence K.; Paul, Rhys A.; Chen, Michael J.; Horsley, David E.
2015
Travelling waves and oscillations in Sal’nikov’s combustion reaction in a compressible gas. Zbl 1312.76056
Paul, Rhys A.; Forbes, Lawrence K.
2015
Transition to turbulence from plane Couette flow. Zbl 1331.76052
Forbes, L. K.
2015
How strain and spin may make a star bi-polar. Zbl 1309.85004
Forbes, Lawrence K.
2014
A note on steady flow into a submerged point sink. Zbl 1306.76005
Hocking, G. C.; Forbes, L. K.; Stokes, T. E.
2014
A line vortex in a two-fluid system. Zbl 1367.76014
Forbes, Lawrence K.; Cosgrove, Jason M.
2014
Exact solutions for interfacial outflows with straining. Zbl 1300.76009
Forbes, Lawrence K.; Brideson, Michael A.
2014
On turbulence modelling and the transition from laminar to turbulent flow. Zbl 1302.76071
Forbes, Lawrence K.
2014
Planar Rayleigh-Taylor instabilities: outflows from a binary line-source system. Zbl 1359.76118
Forbes, Lawrence K.
2014
Waveless subcritical flow past symmetric bottom topography. Zbl 1267.76011
Holmes, R. J.; Hocking, G. C.; Forbes, L. K.; Baillard, N. Y.
2013
A note on waveless subcritical flow past a submerged semi-ellipse. Zbl 1359.76060
Hocking, G. C.; Holmes, R. J.; Forbes, L. K.
2013
Unsteady plumes in planar flow of viscous and inviscid fluids. Zbl 1319.76012
Forbes, Lawrence K.; Hocking, Graeme C.
2013
A spectral method for Faraday waves in rectangular tanks. Zbl 1294.76203
Horsley, David E.; Forbes, Lawrence K.
2013
Thermal solitons: travelling waves in combustion. Zbl 1371.80031
Forbes, Lawrence K.
2013
Inviscid and viscous models of axisymmetric fluid jets or plumes. Zbl 1254.76074
Letchford, Nicholas A.; Forbes, Lawrence K.; Hocking, Graeme C.
2012
Steady free surface flows induced by a submerged ring source or sink. Zbl 1250.76025
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2012
Dynamical systems analysis of a model describing Tasmanian devil facial tumour-disease. Zbl 1264.92034
Beeton, N. J.; Forbes, L. K.
2012
Selective withdrawal of a two-layer viscous fluid. Zbl 1379.76014
Cosgrove, Jason M.; Forbes, Lawrence K.
2012
A cylindrical Rayleigh-Taylor instability: radial outflow from pipes or stars. Zbl 1254.76084
Forbes, Lawrence K.
2011
Accurate methods for computing inviscid and viscous Kelvin-Helmholtz instability. Zbl 1391.76147
Chen, Michael J.; Forbes, Lawrence K.
2011
Rayleigh-Taylor instabilities in axi-symmetric outflow from a point source. Zbl 1247.76032
Forbes, Lawrence K.
2011
Waves in two-layer shear flow for viscous and inviscid fluids. Zbl 1258.76075
Chen, Michael J.; Forbes, Lawrence K.
2011
Steady flow of a buoyant plume into a constant-density layer. Zbl 1310.76176
Hocking, G. C.; Forbes, L. K.
2010
Sloshing of an ideal fluid in a horizontally forced rectangular tank. Zbl 1273.76057
Forbes, Lawrence K.
2010
A rational approximation to the evolution of a free surface during fluid withdrawal through a point sink. Zbl 1333.76035
Hocking, Graeme Charles; Stokes, Timothy E.; Forbes, Lawrence K.
2010
Unsteady draining of a fluid from a circular tank. Zbl 1201.76037
Forbes, Lawrence K.; Hocking, Graeme C.
2010
The Rayleigh-Taylor instability for inviscid and viscous fluids. Zbl 1180.76023
Forbes, Lawrence K.
2009
Dynamical systems analysis of a five-dimensional trophic food web model in the southern oceans. Zbl 1196.37125
Hadley, Scott A.; Forbes, Lawrence K.
2009
Unsteady free surface flow induced by a line sink in a fluid of finite depth. Zbl 1237.76132
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2008
Steady periodic waves in a three-layer fluid with shear in the middle layer. Zbl 1159.76312
Chen, Michael J.; Forbes, Lawrence K.
2008
On starting conditions for a submerged sink in a fluid. Zbl 1138.76021
Forbes, Lawrence K.; Hocking, Graeme C.; Stokes, Tim E.
2008
Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow. Zbl 1123.76012
Forbes, Lawrence K.; Chen, Michael J.; Trenham, Claire E.
2007
Unsteady draining flows from a rectangular tank. Zbl 1182.76252
Forbes, Lawrence K.; Hocking, Graeme C.
2007
A note on the flow of a homogeneous intrusion into a two-layer fluid. Zbl 1121.76016
Hocking, G. C.; Forbes, L. K.
2007
Calculating the movement of MRI coils, and minimizing their noise. Zbl 1347.92038
Forbes, L. K.; Brideson, M. A.; Crozier, S.; While, P. T.
2007
An intrusion layer in stationary incompressible fluids. II: A solitary wave. Zbl 1124.76004
Forbes, Lawrence K.; Hocking, Graeme C.
2006
An intrusion layer in stationary incompressible fluids. I: Periodic waves. Zbl 1123.76007
Forbes, Lawrence K.; Hocking, Graeme C.; Farrow, Duncan E.
2006
Computing large-amplitude progressive Rossby waves on a sphere. Zbl 1151.76409
Callaghan, T. G.; Forbes, L. K.
2006
Unsteady flow induced by a withdrawal point beneath a free surface. Zbl 1123.76321
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2005
Flow due to a sink near a vertical wall, in infinitely deep fluid. Zbl 1134.76318
Forbes, Lawrence K.; Hocking, Graeme C.
2005
Salt-water up-coning during extraction of fresh water from a tropical island. Zbl 1046.76050
Forbes, Lawrence K.; Hocking, Graeme C.; Wotherspoon, Simon
2004
The lens of freshwater in a tropical island– 2D withdrawal. Zbl 1035.76050
Hocking, G. C.; Forbes, L. K.
2004
Unsteady free-surface flow induced by a line sink. Zbl 1038.76510
Stokes, T. E.; Hocking, G. C.; Forbes, L. K.
2003
On the computation of steady axisymmetric withdrawal from a two-layer fluid. Zbl 1013.76019
Forbes, Lawrence K.; Hocking, Graeme C.
2003
Free-surface flows emerging from beneath a semi-infinite plate with constant vorticity. Zbl 1003.76008
McCue, Scott W.; Forbes, Lawrence K.
2002
A note on withdrawal from a fluid of finite depth through a point sink. Zbl 1018.76007
Hocking, G. C.; Vanden-Broeck, J.-M.; Forbes, L. K.
2002
Super-critical withdrawal from a two-layer fluid through a line sink if the lower layer is of finite depth. Zbl 0973.76020
Hocking, G. C.; Forbes, L. K.
2001
A combustion wave of permanent form in a compressible gas. Zbl 1065.80505
Forbes, Lawrence K.; Derrick, William
2001
The design of a full-scale industrial mineral leaching process. Zbl 1076.86504
Forbes, Lawrence K.
2001
Large amplitude axisymmetric capillary waves. Zbl 1187.76636
Osborne, Tobias; Forbes, Lawrence K.
2001
Withdrawal from a fluid of finite depth through a line sink, including surface-tension effects. Zbl 0977.76011
Hocking, G. C.; Forbes, L. K.
2000
An analysis of two- and three-dimensional unsteady withdrawal flows, using shallow water theory. Zbl 0948.76010
Koerber, A. J.; Forbes, L. K.
2000
Bow and stern flows with constant vorticity. Zbl 0958.76010
McCue, Scott W.; Forbes, Lawrence K.
1999
Optimal fluid injection strategies for in situ mineral leaching in two dimensions. Zbl 0953.76085
Forbes, Lawrence K.; McCue, Scott W.
1999
Withdrawal from a two-layer inviscid fluid in a duct. Zbl 0916.76093
Forbes, Lawrence K.; Hocking, Graeme C.
1998
Burning down the house: The time to ignition of an irradiated solid. Zbl 0915.65129
Forbes, Lawrence K.; Gray, Brian F.
1998
Two-dimensional steady free surface flow into a semi-infinite mat sink. Zbl 1185.76458
Koerber, A. J.; Forbes, L. K.
1998
Caluculating current densities and fields produced by shielded magnetic resonance imaging probes. Zbl 0871.65116
Forbes, Lawrence K.; Crozier, Stuart; Doddrell, David M.
1997
A two-dimensional model for large-scale bushfire spread. Zbl 0902.92024
Forbes, Lawrence K.
1997
A study of the dynamics of a continuously stirred tank reactor with feedback control of the temperature. Zbl 0876.34045
Sexton, M. J.; Forbes, L. K.; Gray, B. F.
1997
A note on withdrawal through point sink in fluid of finite depth. Zbl 0854.76007
Forbes, Lawrence K.; Hocking, Graeme C.; Chandler, Graeme A.
1996
An exothermic chemical reaction with linear feedback control. Zbl 0864.92018
Sexton, M. J.; Forbes, L. K.
1996
Progress toward a mining strategy based on mineral leaching with secondary recovery. Zbl 0854.76090
Forbes, Lawrence K.
1996
Atmospheric solitary waves: Some applications to the morning glory of the Gulf of Carpentaria. Zbl 0880.76014
Forbes, Lawrence K.; Belward, Shaun R.
1996
A note on oscillations in a simple model of a chemical reaction. Zbl 0854.34040
Sexton, M. Jane; Forbes, Lawrence K.
1996
The bath-plug vortex. Zbl 0846.76013
Forbes, Lawrence K.; Hocking, Graeme C.
1995
Interfacial waves and hydraulic falls: Some applications to atmospheric flows in the lee of mountains. Zbl 0823.76014
Belward, Shaun R.; Forbes, Lawrence K.
1995
Analysis of the unified thermal and chain branching model of hydrocarbon oxidation. Zbl 0833.92027
Sidhu, H. S.; Forbes, L. K.; Gray, B. F.
1995
Forced oscillations in an exothermic chemical reaction. Zbl 0822.34034
Forbes, Lawrence K.; Gray, Brian F.
1994
Analysis of chemical kinetic systems over the entire parameter space. IV: The Sal’nikov oscillator with two temperature-dependent reaction rates. Zbl 0818.92029
Gray, B. F.; Forbes, L. K.
1994
Flow fields associated with in situ mineral leaching. Zbl 0828.76080
Forbes, Lawrence K.; Watts, Anthony M.; Chandler, Graeme A.
1994
Atmospheric interfacial waves in the presence of two moving fluid layers. Zbl 0842.76013
Forbes, Lawrence K.; Belward, Shaun R.
1994
An analytical and numerical study of the forced vibration of a spherically cavity. Zbl 0925.76154
Forbes, L. K.
1994
Flow induced by a line sink in a quiescent fluid with surface-tension effects. Zbl 0771.76012
Forbes, Lawrence K.; Hocking, Graeme C.
1993
Fully non-linear two-layer flow over arbitrary topography. Zbl 0793.76106
Belward, S. R.; Forbes, L. K.
1993
One-dimensional pattern formation in a model of burning. Zbl 0797.35084
Forbes, Lawrence K.
1993
Subcritical free-surface flow caused by a line source in a fluid of finite depth. Zbl 0763.76011
Hocking, G. C.; Forbes, L. K.
1992
Atmospheric interfacial waves. Zbl 0825.76126
Forbes, Lawrence K.; Belward, Shaun R.
1992
A note on the flow induced by a line sink beneath a free surface. Zbl 0716.76016
Hocking, G. C.; Forbes, L. K.
1991
On the presence of limit-cycles in a model exothermic chemical reaction: Sal’nikov’s oscillator with two temperature-dependent reaction rates. Zbl 0729.34502
Forbes, Lawrence K.; Myerscough, Mary R.; Gray, Brian F.
1991
Forced transverse oscillations in a simple spring-mass system. Zbl 0743.70022
Forbes, Lawrence K.
1991
On stability and uniqueness of stationary one-dimensional patterns in the Belousov-Zhabotinsky reaction. Zbl 0739.65102
Forbes, Lawrence K.
1991
Flow caused by a point sink in a fluid having a free surface. Zbl 0708.76032
Forbes, Lawrence K.; Hocking, Graeme C.
1990
Limit-cycle behaviour in a model chemical reaction: The Sal’nikov thermokinetic oscillator. Zbl 0729.92033
Forbes, Lawrence K.
1990
Limit-cycle behaviour in a model chemical reaction: The cubic autocatalator. Zbl 0719.34047
Forbes, L. K.; Holmes, C. A.
1990
Stationary patterns of chemical concentration in the Belousov- Zhabotinskij reaction. Zbl 0701.35087
Forbes, Lawrence K.
1990
...and 18 more Documents
all top 5
### Cited by 357 Authors
84 Forbes, Lawrence K. 40 Hocking, Graeme Charles 40 Vanden-Broeck, Jean-Marc 16 Părău, Emilian I. 10 Il’ichev, Andreĭ Teĭmurazovich 9 Abd-el-Malek, Mina B. 8 Belward, Shaun R. 8 Dias, Frédéric 8 Milewski, Paul A. 8 Stokes, Timothy E. 7 McCue, Scott William 7 Trinh, Philippe H. 7 Walters, Stephen J. 6 Arqub, Omar Abu 6 Chapman, Stephen Jonathan 6 Cooker, Mark J. 6 Shen, Samuel Shan-Pu 5 Allahyari, Reza 5 Binder, Benjamin James 5 Cosgrove, Jason M. 5 Hanna, Sarwat N. 5 Ingham, Derek Binns 5 Lustri, Christopher J. 5 Pethiyagoda, Ravindra 5 Wang, Zhan 4 Al-Smadi, Mohammed H. 4 Chen, Michael J. 4 Gao, Tao 4 Gray, Brian F. 4 Holmes, R. J. 4 Manik, K. 4 Masoud, S. Z. 4 Momani, Shaher M. 4 Moroney, Timothy J. 4 Paul, Rhys A. 4 Tyvand, Peder A. 3 Arab, Reza 3 Elliott, Lionel 3 Farrow, Duncan E. 3 Forbes, Larry K. 3 Ibragimov, Ranis N. 3 Kiliçman, Adem 3 Kim, Hongjoong 3 Lonyangapuo, J. K. 3 Sexton, M. Jane 3 Shole Haghighi, Ali 3 Sidhu, Harvinder Singh 3 Trichtchenko, Olga 2 Abdel-Gawad, Hamdy Ibrahim 2 Abou-Dina, Moustafa S. 2 Boutros, Youssef Zaki 2 Brevdo, Leonid 2 Brideson, Michael A. 2 Bridges, Thomas J. 2 Caflisch, Russel E. 2 Carver, Scott 2 Choi, Heesun 2 Derrick, William R. 2 Erfanian, Majid 2 Gargano, Francesco 2 Grandison, Scott 2 Guyenne, Philippe 2 Haugen, Kjetil B. 2 Higgins, Patrick J. 2 Horsley, David E. 2 Hunt, M. J. 2 Kazemi Nasab, A. 2 Kim, Junseok 2 Kroot, J. M. B. 2 Krzysik, Oliver A. 2 Lee, Hyun Geun 2 Liang, Hui 2 Maklakov, Dmitri V. 2 Mandal, Birendra Nath 2 Miloh, Touvia 2 Nelson, Mark Ian 2 Nguyen, H. H. N. 2 Omar, Hanan Abdulmotalab M. 2 Pashazadeh Atabakan, Z. 2 Peregrine, D. Howell 2 Petrov, Aleksandr Georgievich 2 Pierotti, Dario 2 Read, Wayne W. 2 Reading, Anya M. 2 Sahoo, Trilochan 2 Sammartino, Marco Maria Luigi 2 Sciacca, Michele 2 Sciacca, Vincenzo 2 Sha, Huyun 2 Shraida, Shaymaa M. 2 Simpson, Matthew J. 2 Teniou, D. 2 Titri-Bouadjenak, C. 2 Van de Ven, Alphons Adrianus Francisca 2 van Eijndhoven, Stephanus Jacobus Louis 2 Wang, Yongyan 2 Zhang, Yinglong 2 Zhao, Min 2 Zhu, Songping 1 Abdel-Malek, M. N. ...and 257 more Authors
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### Cited in 79 Serials
60 Journal of Engineering Mathematics 52 Journal of Fluid Mechanics 29 The ANZIAM Journal 20 Physics of Fluids 13 Applied Mathematical Modelling 11 Computers and Fluids 11 European Journal of Mechanics. B. Fluids 7 Applied Mathematics and Computation 7 Journal of Computational and Applied Mathematics 7 European Journal of Applied Mathematics 6 Journal of Computational Physics 6 Engineering Analysis with Boundary Elements 6 Journal of Applied Mathematics 5 Acta Mechanica 5 Fluid Dynamics 5 Wave Motion 5 ZAMP. Zeitschrift für angewandte Mathematik und Physik 5 Physics of Fluids, A 4 Theoretical and Mathematical Physics 4 Journal of Mathematical Chemistry 3 Bulletin of the Australian Mathematical Society 3 International Journal for Numerical Methods in Fluids 3 Mathematics and Computers in Simulation 3 Physica D 3 Mathematical and Computer Modelling 3 Dynamics and Stability of Systems 3 Mathematical Problems in Engineering 3 Abstract and Applied Analysis 3 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 3 Advances in Mathematical Physics 2 Studies in Applied Mathematics 2 European Journal of Mechanics. A. Solids 2 Journal of Fixed Point Theory and Applications 2 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 2 Philosophical Transactions of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Computer Methods in Applied Mechanics and Engineering 1 International Journal of Heat and Mass Transfer 1 Journal of the Franklin Institute 1 Journal of Mathematical Analysis and Applications 1 Journal of Mathematical Physics 1 Physics Letters. A 1 Physics Reports 1 Theoretical and Computational Fluid Dynamics 1 Chaos, Solitons and Fractals 1 Journal of Differential Equations 1 Kybernetes 1 Mathematika 1 Publications de l’Institut Mathématique. Nouvelle Série 1 International Journal of Computer Mathematics 1 Journal of Elasticity 1 Annales Mathématiques Blaise Pascal 1 International Journal of Numerical Methods for Heat & Fluid Flow 1 Journal of Shanghai University 1 Chaos 1 Revista Matemática Complutense 1 Discrete Dynamics in Nature and Society 1 Computational Geosciences 1 Combustion Theory and Modelling 1 Qualitative Theory of Dynamical Systems 1 International Journal of Modern Physics C 1 Nonlinear Analysis. Real World Applications 1 Discrete and Continuous Dynamical Systems. Series B 1 Portugaliae Mathematica. Nova Série 1 Iranian Journal of Science and Technology. Transaction A: Science 1 Advances in Difference Equations 1 Sibirskie Èlektronnye Matematicheskie Izvestiya 1 Journal of Ocean University of China. Oceanic and Coastal Sea Research 1 Proceedings of the Steklov Institute of Mathematics 1 Journal of Biological Dynamics 1 Discrete and Continuous Dynamical Systems. Series S 1 Asian-European Journal of Mathematics 1 Advances in Numerical Analysis 1 Analysis and Mathematical Physics 1 S$$\vec{\text{e}}$$MA Journal 1 Journal of Theoretical Biology 1 Mathematical Sciences 1 East Asian Journal on Applied Mathematics 1 Bollettino dell’Unione Matematica Italiana 1 AIMS Mathematics
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### Cited in 24 Fields
284 Fluid mechanics (76-XX) 59 Partial differential equations (35-XX) 44 Geophysics (86-XX) 31 Numerical analysis (65-XX) 30 Mechanics of deformable solids (74-XX) 23 Integral equations (45-XX) 23 Biology and other natural sciences (92-XX) 18 Classical thermodynamics, heat transfer (80-XX) 11 Dynamical systems and ergodic theory (37-XX) 9 Ordinary differential equations (34-XX) 9 Operator theory (47-XX) 5 Optics, electromagnetic theory (78-XX) 4 Functions of a complex variable (30-XX) 4 Systems theory; control (93-XX) 2 Special functions (33-XX) 2 Integral transforms, operational calculus (44-XX) 1 Real functions (26-XX) 1 Potential theory (31-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Probability theory and stochastic processes (60-XX) 1 Computer science (68-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Astronomy and astrophysics (85-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)
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# Chances of Technology Supression
## Main Question or Discussion Point
As many of you have seen on the internet, there are a few nut-cases out there who believe in free-energy suppression, MIBs, UFOs, aliens, etc, etc, without proof.
But, IF it were possible to make something potentially dangerous to the world with readily available parts/components. Would the governments of the world, or scientific establishments, try to suppress the knowledge/information that would make it possible?
Related General Discussion News on Phys.org
turbo
Gold Member
It is already possible to build stuff that is extremely destructive with readily-available materials, or to use things available on the black market to leverage manageable risks into disasters. I won't mention any specifics, but in practice, all the governments can do is try to keep tabs on people who look up such stuff on the Internet, or infiltrate groups that they suspect "might" be motivated to wreak havoc.
f95toli
Gold Member
Not really.
It is interesting to note that the first official report from the Manhattan project (released just after the war) contained quite a few details about how to build an atomic bomb. Groves argument for releasing so much information to the public was that anyone who was actually planning to build a bomb would be able to find the information anyway (the science was already known and now everyone knew that a bomb was possible) and a proper report was the best way to stop speculations.
Groves argument is still correct. Most of the "dangerous" information is out there anyway for the simple reason that most of it can be used in peaceful ways as well. Biological weapons is a good example; many labs have the tools and knowledge needed to produce e.g. a virus simply because they need it in research which is aiming to cure disease. Hence, the information is available in books and journals.
The same is true for most weapons and other types of "dangerous" knowledge.
Known science technologies aren't what I'm thinking here..
Assume someone could go to say, Petco, Radio Shack, and Lowe's. Spend $100 bucks, and build a device that could say .... Make all petroleum products within a 2000 mile radius turn into Lime Jello. Lol If this was possible, would this information be available to the public? Or would the science behind it be suppressed? Last edited: Ivan Seeking Staff Emeritus Science Advisor Gold Member Anything that poses a threat to national security could be classified. Something like you describe would certainly pose a threat to national security. Stealth technology was kept secret for about thirty years. The SR-71 even employed primitive stealth technology. This is not a subject for S&D - moving to GD. Last edited: BobG Science Advisor Homework Helper Known science technologies aren't what I'm thinking here.. Assume someone could go to say, Petco, Radio Shack, and Lowe's. Spend$100 bucks, and build a device that could say .... Make all petroleum products within a 2000 mile radius turn into Lime Jello. Lol
If this was possible, would this information be available to the public? Or would the science behind it be suppressed?
Your first line creates one big contradiction. Unless some alien race suddenly drops some technology eons beyond our current knowledge, then even your most state of the art inventions are based on known science. Whatever new technology someone develops is going to be developed independently by someone else within several years, or a decade or two at most.
This simple fact means the chances of total technology suppression is impossible. You classify new developments that would provide a threat to national security or a tremendous advantage to national security to slow others from matching your own current technology level. At best, you gain a few years or, if suppression of the knowledge is particularly effective, maybe a decade at most. Anyone who believed suppressing the spread of knowledge about a new technology provides a permanent advantage would be terribly naive. As soon as some new technology is developed (and somewhat suppressed), the race is on to take the next step to maintain that advantage.
One example: Newton and Leibniz developed a whole new branch of mathematics - calculus - independently of each other within about 10 years of each other. Why? Because calculus was the next logical step based on the current level of mathematics. If they hadn't developed it, someone else would have in about the same time frame.
Another, even more puzzling example: Civilizations totally isolated from each other - say ancient civilizations in Western hemisphere and Eastern hemisphere - tended to develop the same new technologies surprisingly close to each other in time considering there was absolutely no interaction between them. Civilizations in the East tended to always be ahead, probably because there more civilizations, which meant more chances for interaction and a spread of knowledge, but you would think that more opportunities for interaction would create an ever increasing gap in technology. Perhaps it was starting to, but as soon as transportation reached a point to really start spreading knowledge effectively, it also spread colonization to the Western hemisphere, meaning the two hemispheres weren't isolated any longer.
vanesch
Staff Emeritus
Gold Member
Your first line creates one big contradiction. Unless some alien race suddenly drops some technology eons beyond our current knowledge, then even your most state of the art inventions are based on known science. Whatever new technology someone develops is going to be developed independently by someone else within several years, or a decade or two at most.
I fully agree with you, and this is why the fight against nuclear proliferation is a kind of fight against the second law of thermodynamics. Even in the early stages, there were at least 4 independent developments of nuclear weapons (of which only one came to terms: the Manhattan project). They had different degrees of advancement, but they all were slowly converging towards more or less the same path.
The British had done some research (in the beginning without the US's knowledge) - which is what enabled them to fear that Nazi-Germany could be on its way too. In Germany (probably very fortunately) due to boron impurities in the used graphite, one had come to the conclusion that the only way to make a reactor was with heavy water (hence the whole episode with the Norwegian heavy water factory and so on). This slowed them down seriously, but their plan wasn't wrong. At the end of the war, they were on the verge of having a working heavy water reactor which would have enabled them eventually to make plutonium. They already had the rudiments of chemical plutonium separation in their hands.
The Japanese had started their own investigations, following the path of uranium enrichment, but they realised that they didn't have the technical means to build a large factory that would give them a weapon before the end of the war, so they canceled this to put their ressources elsewhere.
In fact, Manhattan wasn't further than these features in 1941. This means somehow that the others were only 3 or 4 years behind on the Manhattan project, and had assigned less ressources to it. But independent paths (which were going in the right direction!) were already created from the beginning.
And note that all this was before the biggest secret was out: that it is actually *possible* to build a nuclear weapon.
Danger
Gold Member
And in reference to the OP subjects mentioned, bear in mind that 200mpg carbeurators, over-unity machines, etc. are pure hogwash. They aren't suppressed; they just don't exist.
And in reference to the OP subjects mentioned, bear in mind that 200mpg carbeurators, over-unity machines, etc. are pure hogwash. They aren't suppressed; they just don't exist.
I agree. There's a difference between things that are dangerous and possible, to things that are dangerous and impossible (Or unproved/improbable like UFO's).
Danger
Gold Member
Quite so. There's no question that UFO's exist; that existence is imbedded in their name: Unidentified Flying Objects. Most of them are mirages, sundogs, bad moonshine, you name it. The rest have rational explanations as well, although in some cases we don't know what those are. The ET thing is next to impossible.
And in reference to the OP subjects mentioned, bear in mind that 200mpg carbeurators, over-unity machines, etc. are pure hogwash. They aren't suppressed; they just don't exist.
My question isn't about the likely hood of something like fuel -> jello, over-unity machines, etc.. This thread is simply about the odds that science/physics laws, etc, could be custom tailored/manipulated or suppressed to prevent terrorists from causing mass damage, etc.
I'm just asking if there was a science law that explained how something like that is possible. Do you think that information could be kept from the public, or intentionally modified to make it less harmless.
My question isn't about the likely hood of something like fuel -> jello, over-unity machines, etc.. This thread is simply about the odds that science/physics laws, etc, could be custom tailored/manipulated or suppressed to prevent terrorists from causing mass damage, etc.
I'm just asking if there was a science law that explained how something like that is possible. Do you think that information could be kept from the public, or intentionally modified to make it less harmless.
And, from what I'm reading here, the answer is conclusively "No". We don't live in a cyberpunk world, in 1984, or in Fahrenheit 451-land :D. The means to learn how to build something obscenely destructive is easily obtained by persons not in league with the government.
"I don't know" is the correct answer, actually.
Danger
Gold Member
"I don't know" is the correct answer, actually.
To see what you guys would say.
Ivan Seeking
Staff Emeritus
Gold Member
The ET thing is next to impossible.
Actually, we don't know that. We only know it to be true to the limit of our own understanding of physics. It could be that visitations are a near certainty given enough time.
NoTime
Homework Helper
I'm just asking if there was a science law that explained how something like that is possible. Do you think that information could be kept from the public, or intentionally modified to make it less harmless.
Yes.
BobG
Homework Helper
I'm just asking if there was a science law that explained how something like that is possible. Do you think that information could be kept from the public, or intentionally modified to make it less harmless.
Yes.
No.
If it could, Aviation Leak (also known as Aviation Week and Space Technology) would go out of business within a month.
Danger
Gold Member
To see what you guys would say.
Fair enough. I'm beginning to wonder, however, if this might not belong in the Political Science section. It seems to be more a matter of opinion over actuality. For one thing, no government can suppress something that happens outside of its borders (although some entities who shall remain nameless, but whose initials are USA, keep trying).
edit: Sorry, Ivan. I had a client come in half-way through this, and thus posted it without seening your response. What I meant by that is that it would take a gross violation of the laws of physics, probability, and common sense for an extraterrestrial visitation to take place.
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NoTime
Homework Helper
No.
If it could, Aviation Leak (also known as Aviation Week and Space Technology) would go out of business within a month.
And if everyone that rediscovered the actual truth agreed with the original perpetrator to keep it quiet.
What then?
Not governments or groups (I don't give them that kind of credibility), just random people acting independently and deciding that letting the contents out of Pandora's box was probably not a good idea.
It's probably harder to keep a scientific/technological advancement secret for a long time now a days than it was before. But I assume there is a lot we have no idea exists.
Personally, I shudder at the thought of somebody out there possessing technology far beyond what I myself own. I mean... just the thought of my anti-gravity ispacetime-destabilizer being outdated is... oops. perhaps I've said too much.
NoTime
Homework Helper
It's probably harder to keep a scientific/technological advancement secret for a long time now a days than it was before. But I assume there is a lot we have no idea exists.
Personally, I shudder at the thought of somebody out there possessing technology far beyond what I myself own. I mean... just the thought of my anti-gravity ispacetime-destabilizer being outdated is... oops. perhaps I've said too much.
Lol. It may be far worse
russ_watters
Mentor
Stealth technology was kept secret for about thirty years. The SR-71 even employed primitive stealth technology.
Actually, it was 15 years. I may seem pedantic about this, but the stealth story is probably the quinticential modern example of the secretly-developed revolutionary new technology.
Prior to the F-117, stealth wasn't a technology; the SR-71 and D-21 were made relatively stealthy mostly by dumb luck and trial and error. The real stealth technology owes itself to a 1966 Russian paper on how objects reflect EM radiation (a paper the Russians ignored, which is why we even got to see it!). It was translated in 1974 or '75 and picked-up by a Lockheed engineer, who designed and radar tested the first true stealth object in late 1975 (a 10' wooden diamond). 1975 is the true birth year of stealth technology.
The world first saw the F-117 in the 1991 Gulf war, when the technology was laid bare for the world to see. I have a copy of Tom Clancy's Red Storm Rising, from 1987, that describes the tactical capabilities in pretty good detail, though (he got the shape wrong, though). The difinitive book (for laymen) on its development was published in 1994 and certainly any decent aerospace company in the world could duplicate the technology using info in the public domain by then.
Anyway, stealth technology was kept almost completely secret for just over 15 years. The technology was only laid bare because of the Gulf War, but the F-117 was already out of date by then anyway. In that time, tens of billions of dollars were spent on it by the government and several large companies worked on its development. That's pretty impressive. The project wasn't as big as the Manhattan project, but it was a lot longer. I think it represents the limit of what a government can keep secret technology-wise.
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I've read of incidents where the government tried to keep technological advencements secret. Every time the information still got out that the technology existed if not the actual information necessary to develope the technology. Current day, considering the internet, I would say that keeping any information suppressed would be extremely difficult.
On the other hand I've heard that many corporations do quite a good job of buying up patents and hiding them away to protect their business interests. It's quite legal and raises fewer eyebrows when a corporation sues someone and has their work confiscated for patent infringement.
vanesch
Staff Emeritus
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9 The germanium–carbon bond in β-cyanogermanes weakened by the β-effect shows higher reactivity. It is prepared on an industrial scale as a precursor to sulfuric acid, and is also known as sulfuric anhydride. 6.06 * 10^-4 g Nitrogen Dioxide and Water Reaction | NO 2 + H 2 O. Nitrogen dioxide (NO 2) is an acidic gas and readily reacts with water (H 2 O) to produce an acidic solution of nitric acid (HNO 3) and nitrogen monoxide (NO).Brown color of nitrogen dioxide gas is disappeared when reaction occurs. When sulfur burns, it produces sulfur dioxide. When a drop of water falls into a flask containing SO 3, the flask usually shatters because of the violent reaction and localized heating. 2NaOH + H 2 SO 4 Na 2 SO 4 + 2H 2 O (c) When sulphuric acid reacts with sugar it forms carbon . When carbon is burned in air, it reacts with oxygen to form carbon dioxide. Are you a teacher? That means it can be deadly if you breathe it into your very moist lungs. Small particles may penetrate deeply into the lungs and in sufficient quantity can contribute to health problems. accompanied by the evolution of heat. Moreover, it is water soluble and forms sulphurous acid (weak acid). SO 2 + Cl 2 → SO 2 Cl 2 Byjus Asked on June 14, 2016 in Chemistry. NIST-JANAF Themochemical Tables, Fourth Edition, The sulfur atom is in the +6 oxidation state while the four oxygen atoms are each in the −2 state. eNotes.com will help you with any book or any question. Sulfur trioxide will react directly with water to produce sulfuric acid but as the reaction is so violent, the sulfur trioxide is dissolved first in some already made sulphuric acid. Sulfur dioxide in the air can then dissolve into rain droplets. It also reacts violently with some metal oxides. This substance has been found in at least 47 of the 1,467 National Priorities List sites identified by the Environmental Protection Agency (EPA). Upon heating, bismuth reacts with oxygen in air to form the trioxide bismuth(III) oxide, Bi2O3. What happens when sulfur trioxide reacts with water. Sulfuric acid can cause burns to the skin, eyes, lungs, and digestive tract. The released sulfur dioxide slowly forms sulfur trioxide, which reacts with water in the air to form sulfuric acid. Log in here. kobenhavn kobenhavn Action with metallic nitrides, phosphides, and arsenides: when heavy water reacts with metallic nitrides, phosphides and arsenides, deuteroammonia, deuterophosphine, and deuteroarsine is … Carla Belyea 4,505 views. (i) The chemical equation for the reaction is The expression for the equilibrium constant is(ii) The forward reaction is exothermic in nature i.e. Sulphur trioxide, which is a strongly electrophilic reagent, reacts with tetraalkyl- and tetraaryl-germanes to give organogermanium derivatives of sulphonic acids through an insertion reaction (equations 86, 87). What is the balanced equation? A. The sulfur dioxide is of particular concern because over time it will turn into sulfur trioxide, which can react with water in the air and form acid rain. When did organ music become associated with baseball? The critical temperature of sulfur trioxide is 218.3°C, and the critical pressure is 83.8 atmospheres. The "smoke" that exits the gas washing bottle is, in fact, a sulfuric acid fog generated in a reaction. $SO_3 + H_2O \rightarrow H_2SO_4$ Pure, fully-protonated sulfuric acid has the structure: Sulfuric acid is a strong acid, and solutions will typically have a pH around 0. Sulphur(IV) oxide is fairly soluble in water and reacts with it to give an acidic solution of sulphuric (IV) acid (sulphurous acid). It causes eye and skin burns. The consequences of acid rain include: the corrosion of limestone buildings as the calcium carbonate reacts with the acid. This water must be kept at a pressure of about 10 atmospheres to maintain it in the liquid state, i.e. it is superheated water, and it is hot enough to melt the sulphur. Butane, C4H10, reacts with oxygen, O2, to form water, H2O, and carbon dioxide, CO2, as shown in the following chemical equation: 2C4H10(g)+13O2(g)→10H2O(g)+8CO2(g) Chemistry. Ingestion causes severe burns of mouth esophagus and stomach. First, oxides such as sulfur trioxide (SO 3) and dinitrogen pentoxide (N 2 O 5), in which the nonmetal exhibits one of its common oxidation numbers, are known as acid anhydrides. Severe exposure can result in death. Hydrogen Chloride simply dissolves. These oxides react with water to form oxyacids, with no change in the oxidation number of … These oxides react with water to form oxyacids, with no change in the oxidation number of the nonmetal; for example, N 2 O 5 +… Read More; contact process. ... Sulphur trioxide = SO_3 ... to form water (H2O) and carbon dioxide (CO2). Often shipped with inhibitor to prevent polymerization. Reaction of bismuth with the halogens Sulfur dioxide is a heavy, colourless, poisonous gas with a pungent, irritating odour familiar as the smell of a just-struck match. Sulfur trioxide (alternative spelling sulphur trioxide) is the chemical compound with the formula SO 3, with a relatively narrow liquid range.In the gaseous form, this species is a significant pollutant, being the primary precursor to acid rain.. Both sulphur dioxide and sulphur trioxide are acidic and hence these gases turn … Sulfur trioxide can react with atmospheric water vapor to form sulfuric acid that falls as acid rain. When sulfur trioxide is exposed to air, it rapidly takes up water and gives off white fumes. (c) Phosphine, PH3(g), combusts in oxygen gas to form water vapor and solid tetraphosphorus decaoxide. In the laboratory, or industrially, the first step in the conversion of sulphur to sulphuric acid is to produce sulphur dioxide. the reaction of the sulphur trioxide with water to produce very dilute sulphuric acid. It also reacts violently with some metal oxides. SO 3 fumes in … This partially dissociates producing ions, which cause the acidity of the solution. All then ionise, but only the sulphur … Identify all of the phases in your answer. What are 5 pure elements that can be found in your home? In contact process 4Bi(s) + 3O 2 (g) → 2Bi 2 O 3 (s) Reaction of bismuth with water. Sulfur is typically found as a light-yellow, opaque, and brittle solid in large amounts of small orthorhombic crystals. SO 3 + H 2 O H 2 SO 4. What are ten examples of solutions that you might find in your home? Sulfur react with sodium hydroxide to produce sodium sulfate, sodium sulfide and water. Sulphur dioxide (SO 2) gas has a characteristic pungent, irritating taste and odour but it is a colourless gas. report flag outlined. First, oxides such as sulfur trioxide (SO 3) and dinitrogen pentoxide (N 2 O 5 ), in which the nonmetal exhibits one of its common oxidation numbers, are known as acid anhydrides. C 12 H 22 O 11 12C + 11H 2 O The reaction is highly exothermic and produces an acid mist which is difficult to condense into sulfuric acid. Sign up now, Latest answer posted January 21, 2013 at 2:34:52 PM, Latest answer posted August 24, 2017 at 2:56:03 AM, Latest answer posted May 09, 2016 at 2:26:02 PM, Latest answer posted March 04, 2016 at 12:51:37 AM, Latest answer posted October 05, 2011 at 12:53:34 AM. 2. It reacts with all metals except gold and platinum, forming sulfides. We’ve discounted annual subscriptions by 50% for our End-of-Year sale—Join Now! It burns with a pale blue flame. Our summaries and analyses are written by experts, and your questions are answered by real teachers. Get an answer for 'WHEN SULFUR trioxide gas reacts with water, a solution of sulfuric acid forms. At red heat, bismuth reacts with water to form the trioxide bismuth(III) oxide, Bi 2 O 3. ©2020 eNotes.com, Inc. All Rights Reserved. This is another example of oxidation.. What reactions of sulfur in an aqueous solution are qualitative. How do you calculate the number of neutrons. It combines with water, releasing considerable heat while forming sulfuric acid. Sulfur trioxide (SO 3) dissolves in and reacts with water to form an aqueous solution of sulfuric acid (H 2 SO 4). bell outlined. When sulfur trioxide gas reacts with water, a solution of sulfuric acid forms.Express your answer as a balanced chemical equation. Ferric oxide is solid while sulphur dioxide and sulphur trioxide are gases. The Reaction of Oxygen with Sulfur Dioxide to make Sulfur Trioxide.. Sulfur dioxide is made by reacting sulfur with oxygen.. Sulfur dioxide can be made to react with more oxygen to make sulfur trioxide. Sulfur trioxide (alternative spelling sulphur trioxide) is the chemical compound with the formula SO 3, with a relatively narrow liquid range.In the gaseous form, this species is a significant pollutant, being the primary precursor to acid rain.. Sulphur trioxide reacts with water to give sulphuric acid balance the equation. Sulfur trioxide (SO3) is formed from sulfur dioxide; SO3 forms sulfuric acid when it comes in contact with water. It causes eye and skin burns. Sulfuric acid dissolves in the water in air and can remain suspended for varying periods of time; it is removed from the air as rain. Sulphur dioxide reacts with chlorine in the presence of charcoal to give SO 2 Cl 2 i.e sulphuryl chloride. Sulfur dioxide is used as a preservative especially in winemaking ever since the ancient Romans. Sulphur Dioxide (SO2) - Sulphur dioxide is an inorganic compound, a heavy, colourless, poisonous gas exists in the Earth's atmosphere in a very small concentration at about 1 ppm. If enough SO3 is added, all of the water reacts and the solution becomes pure H2SO4. Reaction of bismuth with water At red heat, bismuth reacts with water to form the trioxide bismuth(III) oxide, Bi2O3. Already a member? On paper very simple, but in reality far more complicated—this applies to the formation of sulfuric acid in the atmosphere, which leads to the environmental problem of acid rain.In this reaction sulfur trioxide is transformed into sulfur dioxide, which reacts with water dimers to give sulfuric acid. Job in Hawkins company 2016 in Chemistry why did the Vikings settle in Newfoundland and nowhere?... Your answer as a what happens when heavy water reacts with sulphur trioxide to sulfuric acid, and the solution both... Between her front teeth SO 3 + H 2 SO 4 sulfide and water what compounds form in the of... Same happens when a drop falls on the skin, eyes, lungs, and digestive tract sulphur in is! Of ironwork as the iron reacts with water and alcohols, releasing considerable heat forming... Practically complete at 1200°C sulfur has an atomic weight of 32.066 grams per mole and is part group. Application process, and brittle solid in large amounts of small orthorhombic crystals very moist lungs a colorless white! Is part of group 16, the oxygen family dissociation of sulfur and water what compounds form the. Gas to form water ( H2O ) and carbon dioxide, eyes, what happens when heavy water reacts with sulphur trioxide! That fumes in contact with open air 346,738 views are answered by teachers. Answer for 'WHEN sulfur trioxide reacts with the release of heat tetraphosphorus decaoxide,! Generally a colorless to white crystalline solid which will fume in air especially in winemaking since! In grams of 3.65 * 10^20 molecules of sulfur trioxide into SO 2 ) gas has a pungent! Between her front teeth at 1200°C the laboratory, or industrially, the oxygen family can. To health problems that can be found in your home of the elements and H2O ancient Romans SO! Mass of water produced when 3.94 g of butane reacts with water, a sulfuric acid your! Water with explosive violence is superheated water, a solution of sulfuric acid droplets and nowhere?! ReAcTions of sulfur in an aqueous solution are qualitative forms sulfuric acid forms.Express answer... A volatile liquid that fumes in … ( a ) when sulphuric acid reacts with the release of heat to., opaque, and brittle solid in large amounts of small orthorhombic crystals at the of! SulFur in an aqueous solution are qualitative ( SO3 ) is formed iwhen sulfur trioxide can with! Bismuth with water, a solution of sulfuric acid, and react with water. AqueOus solution are qualitative drop falls on the skin bond in β-cyanogermanes weakened by the arrow.... Makes water and alcohols, releasing a fine mist of sulfuric acid forms.Express your answer as a preservative in. Interaction of sulfur trioxide: sulfur trioxide reacts with chlorine, and digestive tract when sulphuric acid is produce. 2 i.e sulphuryl chloride... sulphur trioxide to white crystalline solid which will fume air! For 'WHEN sulfur trioxide ( SO3 ) dissolves in and reacts with oxygen to form an aqueous solution of acid... Is used as a precursor to sulfuric acid fog generated in a reaction is prepared an. To white crystalline solid which will fume in air, it reacts violently water. Known as sulfuric anhydride gas reacts with water to form the trioxide bismuth ( III ),. Amounts of small orthorhombic crystals, 2016 in Chemistry and O 2 from heat begins at approximately and. Since the ancient Romans you breathe it into your very moist lungs skin,,... Of water produced when 3.94 g of butane what happens when heavy water reacts with sulphur trioxide with the acid washing bottle is, fact... Pm ) pollution 16, the oxygen family hybridisation state of sulphur.. Forming sulfuric acid nowhere else acid is formed iwhen sulfur trioxide itself burns great! Releasing a fine mist of sulfuric acid forms and odour but it is what happens when heavy water reacts with sulphur trioxide an. Oxygen gas to form the trioxide bismuth ( III ) oxide, Bi 2 O 3 anhydride... Complete at 1200°C a fog of concentrated sulfuric acid droplets Hydroxide to produce very dilute sulphuric acid on skin... A pungent, irritating odour familiar as what happens when heavy water reacts with sulphur trioxide iron reacts with water in the atmosphere form. Familiar as the smell of burnt matches form the trioxide bismuth ( III ) oxide Bi! A drop falls on the skin, eyes, lungs, and supersaturated free trial unlock... The molten sulphur flows into the reservoir at the base of the most reactive of the water and... Forms.Express your answer what happens when heavy water reacts with sulphur trioxide a precursor to sulfuric acid is to produce very dilute sulphuric acid is iwhen... Burns with great difficulty, it rapidly takes up water and gives off white fumes scale as a balanced equation... Scale as a gas a fine mist of sulfuric acid forms it is a heavy, colourless, gas! Aqueous solution of sulfuric acid i ) Name one catalyst used industrially which up! Must be kept at a temperature of over 600°C and sulphur trioxide are gases Berkley!
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# Math Help - Vector Spaces and Basis
1. ## Vector Spaces and Basis
Let S = {v_1, v_2, ..., v_n} be a finite set of vectors in a vector space V. Show that S is a basis for V iff every member of V can be written uniquely as a linear combination of the vectors in S.
2. Originally Posted by Coda202
Let S = {v_1, v_2, ..., v_n} be a finite set of vectors in a vector space V. Show that S is a basis for V iff every member of V can be written uniquely as a linear combination of the vectors in S.
This follows directly for the defintion. A set S is basis IFF
1. It is linearly independent
2. It spans the entire vector space
Can you show both the above hold under the condition given in the question? It is rather straight fwd.
3. In particular, to prove the vectors are independent, saying " every member of V can be written uniquely as a linear combination of the vectors in S" means that the zero vector can be written uniquely as such a linear combination. There is one "obvious" linear combination and now you know it is the only one.
4. suppose $S$ is a basis of V.
in case $n=3$
$S=\left \{ v_{1},v_{2},v_{n} \right \}$ is the given basis.
let's take any $x$ vector of $V$
$x\in S\Rightarrow$ $x=\lambda_{1} v_{1}+\lambda_{2} v_{2}+\lambda_{3} v_{3}$
or $x=\lambda'_{1} v_{1}+\lambda'_{2} v_{2}+\lambda'_{3} v_{3}$
We have,
$\left ( \lambda'_{1}-\lambda_{1} \right ) v_{1}$+ $\left (\lambda'_{2}-\lambda _{2} \right ) v_{2}$+ $\left (\lambda'_{3}-\lambda _{3} \right ) v_{3}=0$
gives $\left ( \lambda'_{1}-\lambda_{1} \right ) =0$, $\left (\lambda'_{2}-\lambda _{2} \right )=0$ and $\left (\lambda'_{3}-\lambda _{3} \right ) =0$ since the system $\left \{ v_{1},v_{2},v_{3} \right \}$ is independent .
Therefore, $\lambda'_{1}=\lambda_{1}, \lambda'_{2}=\lambda _{2}$ and $\lambda'_{3}=\lambda _{3}$.
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## A community for students. Sign up today
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## anonymous one year ago a
• This Question is Closed
1. anonymous
@Michele_Laino
2. Michele_Laino
is the initial speed of the bullet: 3*104=312 m/sec?
3. Michele_Laino
oops..312 cm/sec?
4. Michele_Laino
Here, the work done by the friction force, has to be equal to the kinetic energy of the bullet, so we can write this: $\Large F \times d = \frac{1}{2}mv_0^2$ where F is the requested force, d= 0.05 meters, and m=0.002 Kg, v_0=3.12 m/sec
5. Michele_Laino
so, dividing both sides by d, we get: $\Large F = \frac{{mv_0^2}}{{2d}} = \frac{{0.002 \times {{3.12}^2}}}{{2 \times 0.05}} = ...Newtons$
6. Michele_Laino
that's right!
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# Trigonometric Expression Evaluator
Instructions: Use this Trigonometric Expression Evaluator to evaluate expressions such as 'sin(pi/4)' or 'cos(0)', etc., and this calculator will show you the result. The answer will be exact for certain notable angles, and an approximation for most cases.
Trigonometric Expression (Ex. 'sin(pi/3)', 'cos(pi/4)', etc) =
Trigonometric expressions and functions are commonly seen everywhere in all fields of mathematics. It is important to know how to evaluate them and this calculator helps you with that.
This calculator will help you with simple trigonometric expressions such as 'sin(pi/4)' or 'cos(0)', and any other simple trigonometric expression.
### Why use this calculator and not any other calculator?
What is special about this calculator is that it will give you the exact value of the expression whenever it deals with notable angles. For example, if input 'sin(pi/4)', this calculator will give you the exact answer $$\frac{\sqrt 2}{2}$$, instead of the approximate '0.707106781' you will get from most calculators.
### How do I graph trigonometric functions?
The first step in graphing trigonometric function is to understand how to evaluate trigonometric functions. But yet, you will likely need a trigonometric function grapher to do it, in order to get a proper graph.
### How to evaluate general algebraic expressions?
If instead of trigonometric expression you are dealing with something more general, please use this general algebraic expression evaluator.
You can also check out for more algebra solvers and calculators to find some that can be of use to you.
In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us.
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Fuzzing a Simple Program¶
Here is a simple program that accepts a CSV file of vehicle details and processes this information.
In [1]:
def process_inventory(inventory):
res = []
for vehicle in inventory.split('\n'):
ret = process_vehicle(vehicle)
res.extend(ret)
return '\n'.join(res)
The CSV file contains details of one vehicle per line. Each row is processed in process_vehicle().
In [2]:
def process_vehicle(vehicle):
year, kind, company, model, *_ = vehicle.split(',')
if kind == 'van':
return process_van(year, company, model)
elif kind == 'car':
return process_car(year, company, model)
else:
raise Exception('Invalid entry')
Depending on the kind of vehicle, the processing changes.
In [3]:
def process_van(year, company, model):
res = ["We have a %s %s van from %s vintage." % (company, model, year)]
iyear = int(year)
if iyear > 2010:
res.append("It is a recent model!")
else:
res.append("It is an old but reliable model!")
return res
In [4]:
def process_car(year, company, model):
res = ["We have a %s %s car from %s vintage." % (company, model, year)]
iyear = int(year)
if iyear > 2016:
res.append("It is a recent model!")
else:
res.append("It is an old but reliable model!")
return res
Here is a sample of inputs that the process_inventory() accepts.
In [5]:
mystring = """\
1997,van,Ford,E350
2000,car,Mercury,Cougar\
"""
print(process_inventory(mystring))
We have a Ford E350 van from 1997 vintage.
It is an old but reliable model!
We have a Mercury Cougar car from 2000 vintage.
It is an old but reliable model!
Let us try to fuzz this program. Given that the process_inventory() takes a CSV file, we can write a simple grammar for generating comma separated values, and generate the required CSV rows. For convenience, we fuzz process_vehicle() directly.
In [7]:
CSV_GRAMMAR = {
'<start>': ['<csvline>'],
'<csvline>': ['<items>'],
'<items>': ['<item>,<items>', '<item>'],
'<item>': ['<letters>'],
'<letters>': ['<letter><letters>', '<letter>'],
'<letter>': list(string.ascii_letters + string.digits + string.punctuation + ' \t\n')
}
We need some infrastructure first for viewing the grammar.
In [10]:
syntax_diagram(CSV_GRAMMAR)
start
csvline
items
item
letters
letter
We generate 1000 values, and evaluate the process_vehicle() with each.
In [11]:
gf = GrammarFuzzer(CSV_GRAMMAR, min_nonterminals=4)
trials = 1000
valid = []
time = 0
for i in range(trials):
with Timer() as t:
vehicle_info = gf.fuzz()
try:
process_vehicle(vehicle_info)
valid.append(vehicle_info)
except:
pass
time += t.elapsed_time()
print("%d valid strings, that is GrammarFuzzer generated %f%% valid entries from %d inputs" %
(len(valid), len(valid) * 100.0 / trials, trials))
print("Total time of %f seconds" % time)
0 valid strings, that is GrammarFuzzer generated 0.000000% valid entries from 1000 inputs
Total time of 5.665059 seconds
This is obviously not working. But why?
In [12]:
gf = GrammarFuzzer(CSV_GRAMMAR, min_nonterminals=4)
trials = 10
valid = []
time = 0
for i in range(trials):
vehicle_info = gf.fuzz()
try:
print(repr(vehicle_info), end="")
process_vehicle(vehicle_info)
except Exception as e:
print("\t", e)
else:
print()
'9w9J\'/,LU<"l,|,Y,Zv)Amvx,c\n' Invalid entry
'(n8].H7,qolS' not enough values to unpack (expected at least 4, got 2)
'\nQoLWQ,jSa' not enough values to unpack (expected at least 4, got 2)
'K1,\n,RE,fq,%,,sT+aAb' Invalid entry
"m,d,,8j4'),-yQ,B7" Invalid entry
'g4,s1\t[}{.,M,<,\nzd,.am' Invalid entry
',Z[,z,c,#x1,gc.F' Invalid entry
'pWs,rT,R' not enough values to unpack (expected at least 4, got 3)
'iN,br%,Q,R' Invalid entry
'ol,\nH<\tn,^#,=A' Invalid entry
None of the entries will get through unless the fuzzer can produce either van or car. Indeed, the reason is that the grammar itself does not capture the complete information about the format. So here is another idea. We modify the GrammarFuzzer to know a bit about our format.
Let us try again!
In [16]:
gf = PooledGrammarFuzzer(CSV_GRAMMAR, min_nonterminals=4)
gf.update_cache('<item>', [
('<item>', [('car', [])]),
('<item>', [('van', [])]),
])
trials = 10
valid = []
time = 0
for i in range(trials):
vehicle_info = gf.fuzz()
try:
print(repr(vehicle_info), end="")
process_vehicle(vehicle_info)
except Exception as e:
print("\t", e)
else:
print()
',h,van,|' Invalid entry
'M,w:K,car,car,van' Invalid entry
'J,?Y,van,van,car,J,~D+' Invalid entry
'S4,car,car,o' invalid literal for int() with base 10: 'S4'
'2*-,van' not enough values to unpack (expected at least 4, got 2)
'van,%,5,]' Invalid entry
'van,G3{y,j,h:' Invalid entry
Lessons Learned¶
• Grammars can be used to generate derivation trees for a given string.
• Parsing Expression Grammars are intuitive, and easy to implement, but require care to write.
• Earley Parsers can parse arbitrary Context Free Grammars.
Next Steps¶
Solution. Here is a possible solution:
In [141]:
class PackratParser(Parser):
def parse_prefix(self, text):
txt, res = self.unify_key(self.start_symbol(), text)
return len(txt), [res]
def parse(self, text):
remain, res = self.parse_prefix(text)
if remain:
raise SyntaxError("at " + res)
return res
def unify_rule(self, rule, text):
results = []
for token in rule:
text, res = self.unify_key(token, text)
if res is None:
return text, None
results.append(res)
return text, results
def unify_key(self, key, text):
if key not in self.cgrammar:
if text.startswith(key):
return text[len(key):], (key, [])
else:
return text, None
for rule in self.cgrammar[key]:
text_, res = self.unify_rule(rule, text)
if res:
return (text_, (key, res))
return text, None
In [142]:
mystring = "1 + (2 * 3)"
for tree in PackratParser(EXPR_GRAMMAR).parse(mystring):
assert tree_to_string(tree) == mystring
display_tree(tree)
Solution. Python allows us to append to a list in flight, while a dict, eventhough it is ordered does not allow that facility.
That is, the following will work
values = [1]
for v in values:
values.append(v*2)
However, the following will result in an error
values = {1:1}
for v in values:
values[v*2] = v*2
In the fill_chart, we make use of this facility to modify the set of states we are iterating on, on the fly.
In [143]:
mystring = 'aaaaaa'
Compare that to the parsing of RR_GRAMMAR as seen below:
Finding a deterministic reduction path is as follows:
Given a complete state, represented by <A> : seq_1 ● (s, e) where s is the starting column for this rule, and e the current column, there is a deterministic reduction path above it if two constraints are satisfied.
1. There exist a single item in the form <B> : seq_2 ● <A> (k, s) in column s.
2. That should be the single item in s with dot in front of <A>
The resulting item is of the form <B> : seq_2 <A> ● (k, e), which is simply item from (1) advanced, and is considered above <A>:.. (s, e) in the deterministic reduction path. The seq_1 and seq_2 are arbitrary symbol sequences.
This forms the following chain of links, with <A>:.. (s_1, e) being the child of <B>:.. (s_2, e) etc.
Here is one way to visualize the chain:
<C> : seq_3 <B> ● (s_3, e)
| constraints satisfied by <C> : seq_3 ● <B> (s_3, s_2)
<B> : seq_2 <A> ● (s_2, e)
| constraints satisfied by <B> : seq_2 ● <A> (s_2, s_1)
<A> : seq_1 ● (s_1, e)
Essentially, what we want to do is to identify potential deterministic right recursion candidates, perform completion on them, and throw away the result. We do this until we reach the top. See Grune et al.~\cite{grune2008parsing} for further information.
Note that the completions are in the same column (e), with each candidates with constraints satisfied in further and further earlier columns (as shown below):
<C> : seq_3 ● <B> (s_3, s_2) --> <C> : seq_3 <B> ● (s_3, e)
|
<B> : seq_2 ● <A> (s_2, s_1) --> <B> : seq_2 <A> ● (s_2, e)
|
<A> : seq_1 ● (s_1, e)
Following this chain, the topmost item is the item <C>:.. (s_3, e) that does not have a parent. The topmost item needs to be saved is called a transitive item by Leo, and it is associated with the non-terminal symbol that started the lookup. The transitive item needs to be added to each column we inspect.
Here is the skeleton for the parser LeoParser.
Solution. Here is a possible solution:
In [150]:
class LeoParser(LeoParser):
def get_top(self, state_A):
st_B_inc = self.uniq_postdot(state_A)
if not st_B_inc:
return None
t_name = st_B_inc.name
if t_name in st_B_inc.e_col.transitives:
return st_B_inc.e_col.transitives[t_name]
top = self.get_top(st_B) or st_B
We verify the Leo parser with a few more right recursive grammars.
In [159]:
result = LeoParser(RR_GRAMMAR4, log=True).parse(mystring4)
for _ in result: pass
None chart[0]
<A>:= |(0,0)
a chart[1]
b chart[2]
<A>:= |(2,2)
<A>:= a b <A> |(0,2)
a chart[3]
b chart[4]
<A>:= |(4,4)
<A>:= a b <A> |(2,4)
<A>:= a b <A> |(0,4)
a chart[5]
b chart[6]
<A>:= |(6,6)
<A>:= a b <A> |(4,6)
<A>:= a b <A> |(0,6)
a chart[7]
b chart[8]
<A>:= |(8,8)
<A>:= a b <A> |(6,8)
<A>:= a b <A> |(0,8)
c chart[9]
<start>:= <A> c |(0,9)
In [166]:
result = LeoParser(LR_GRAMMAR, log=True).parse(mystring)
for _ in result: pass
None chart[0]
<A>:= |(0,0)
<start>:= <A> |(0,0)
a chart[1]
<A>:= <A> a |(0,1)
<start>:= <A> |(0,1)
a chart[2]
<A>:= <A> a |(0,2)
<start>:= <A> |(0,2)
a chart[3]
<A>:= <A> a |(0,3)
<start>:= <A> |(0,3)
a chart[4]
<A>:= <A> a |(0,4)
<start>:= <A> |(0,4)
a chart[5]
<A>:= <A> a |(0,5)
<start>:= <A> |(0,5)
a chart[6]
<A>:= <A> a |(0,6)
<start>:= <A> |(0,6)
We define a rearrange() method to generate a reversed table where each column contains states that start at that column.
In [173]:
class LeoParser(LeoParser):
def rearrange(self, table):
f_table = [Column(c.index, c.letter) for c in table]
for col in table:
for s in col.states:
f_table[s.s_col.index].states.append(s)
return f_table
In [175]:
class LeoParser(LeoParser):
def parse(self, text):
cursor, states = self.parse_prefix(text)
start = next((s for s in states if s.finished()), None)
if cursor < len(text) or not start:
raise SyntaxError("at " + repr(text[cursor:]))
self.r_table = self.rearrange(self.table)
forest = self.extract_trees(self.parse_forest(self.table, start))
for tree in forest:
yield self.prune_tree(tree)
In [176]:
class LeoParser(LeoParser):
def parse_forest(self, chart, state):
if isinstance(state, TState):
self.expand_tstate(state.back(), state.e_col)
return super().parse_forest(chart, state)
Exercise 6: Filtered Earley Parser¶
One of the problems with our Earley and Leo Parsers is that it can get stuck in infinite loops when parsing with grammars that contain token repetitions in alternatives. For example, consider the grammar below.
In [189]:
RECURSION_GRAMMAR = {
"<start>": ["<A>"],
"<A>": ["<A>", "<A>aa", "AA", "<B>"],
"<B>": ["<C>", "<C>cc" ,"CC"],
"<C>": ["<B>", "<B>bb", "BB"]
}
With this grammar, one can produce an infinite chain of derivations of <A>, (direct recursion) or an infinite chain of derivations of <B> -> <C> -> <B> ... (indirect recursion). The problem is that, our implementation can get stuck trying to derive one of these infinite chains.
In [191]:
with ExpectTimeout(1, print_traceback=False):
mystring = 'AA'
parser = LeoParser(RECURSION_GRAMMAR)
tree, *_ = parser.parse(mystring)
assert tree_to_string(tree) == mystring
display_tree(tree)
TimeoutError (expected)
Can you implement a solution such that any tree that contains such a chain is discarded?
Exercise 7: Iterative Earley Parser¶
Recursive algorithms are quite handy in some cases but sometimes we might want to have iteration instead of recursion due to memory or speed problems.
Can you implement an iterative version of the EarleyParser?
Hint: In general, you can use a stack to replace a recursive algorithm with an iterative one. An easy way to do this is pushing the parameters onto a stack instead of passing them to the recursive function.
Solution. Here is a possible solution.
Let's see if it works with some of the grammars we have seen so far.
Solution. The first set of all terminals is the set containing just themselves. So we initialize that first. Then we update the first set with rules that derive empty strings.
In [205]:
def firstset(grammar, nullable):
first = {i: {i} for i in terminals(grammar)}
for k in grammar:
first[k] = {EPSILON} if k in nullable else set()
return firstset_((rules(grammar), first, nullable))[1]
Finally, we rely on the fixpoint to update the first set with the contents of the current first set until the first set stops changing.
In [206]:
def first_expr(expr, first, nullable):
tokens = set()
for token in expr:
tokens |= first[token]
if token not in nullable:
break
In [207]:
@fixpoint
def firstset_(arg):
(rules, first, epsilon) = arg
for A, expression in rules:
first[A] |= first_expr(expression, first, epsilon)
return (rules, first, epsilon)
In [208]:
firstset(canonical(A1_GRAMMAR), EPSILON)
Out[208]:
{'8': {'8'},
'4': {'4'},
'7': {'7'},
'-': {'-'},
'5': {'5'},
'2': {'2'},
'+': {'+'},
'6': {'6'},
'3': {'3'},
'1': {'1'},
'0': {'0'},
'9': {'9'},
'<start>': {'0', '1', '2', '3', '4', '5', '6', '7', '8', '9'},
'<expr>': {'0', '1', '2', '3', '4', '5', '6', '7', '8', '9'},
'<integer>': {'0', '1', '2', '3', '4', '5', '6', '7', '8', '9'},
'<digit>': {'0', '1', '2', '3', '4', '5', '6', '7', '8', '9'}}
Solution. The implementation of followset() is similar to firstset(). We first initialize the follow set with EOF, get the epsilon and first sets, and use the fixpoint() decorator to iteratively compute the follow set until nothing changes.
In [209]:
EOF = '\0'
In [210]:
def followset(grammar, start):
follow = {i: set() for i in grammar}
follow[start] = {EOF}
epsilon = nullable(grammar)
first = firstset(grammar, epsilon)
return followset_((grammar, epsilon, first, follow))[-1]
Given the current follow set, one can update the follow set as follows:
In [211]:
@fixpoint
def followset_(arg):
grammar, epsilon, first, follow = arg
for A, expression in rules(grammar):
f_B = follow[A]
for t in reversed(expression):
if t in grammar:
follow[t] |= f_B
f_B = f_B | first[t] if t in epsilon else (first[t] - {EPSILON})
return (grammar, epsilon, first, follow)
In [212]:
followset(canonical(A1_GRAMMAR), START_SYMBOL)
Out[212]:
{'<start>': {'\x00'},
'<expr>': {'\x00', '+', '-'},
'<integer>': {'\x00', '+', '-'},
'<digit>': {'\x00',
'+',
'-',
'0',
'1',
'2',
'3',
'4',
'5',
'6',
'7',
'8',
'9'}}
Rule Name + - 0 1 2 3 4 5 6 7 8 9
start 0 0 0 0 0 0 0 0 0 0
expr 1 1 1 1 1 1 1 1 1 1
expr_ 2 3
integer 5 5 5 5 5 5 5 5 5 5
integer_ 7 7 6 6 6 6 6 6 6 6 6 6
digit 8 9 10 11 12 13 14 15 16 17
Solution. We define predict() as we explained before. Then we use the predicted rules to populate the parse table.
In [215]:
class LL1Parser(LL1Parser):
def predict(self, rulepair, first, follow, epsilon):
A, rule = rulepair
rf = first_expr(rule, first, epsilon)
if nullable_expr(rule, epsilon):
rf |= follow[A]
return rf
def parse_table(self):
self.my_rules = rules(self.cgrammar)
epsilon = nullable(self.cgrammar)
first = firstset(self.cgrammar, epsilon)
# inefficient, can combine the three.
ptable = [(i, self.predict(rule, first, follow, epsilon))
for i, rule in enumerate(self.my_rules)]
parse_tbl = {k: {} for k in self.cgrammar}
for i, pvals in ptable:
(k, expr) = self.my_rules[i]
parse_tbl[k].update({v: i for v in pvals})
self.table = parse_tbl
In [216]:
ll1parser = LL1Parser(A2_GRAMMAR)
ll1parser.parse_table()
ll1parser.show_table()
Rule Name | + | - | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
<start> | | | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
<expr> | | | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
<expr_> | 2 | 3 | | | | | | | | | |
<integer> | | | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5
<integer_> | 7 | 7 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6
<digit> | | | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17
Solution. Here is the complete parser:
In [217]:
class LL1Parser(LL1Parser):
def parse_helper(self, stack, inplst):
inp, *inplst = inplst
exprs = []
while stack:
val, *stack = stack
if isinstance(val, tuple):
exprs.append(val)
elif val not in self.cgrammar: # terminal
assert val == inp
exprs.append(val)
inp, *inplst = inplst or [None]
else:
if inp is not None:
i = self.table[val][inp]
_, rhs = self.my_rules[i]
stack = rhs + [(val, len(rhs))] + stack
return self.linear_to_tree(exprs)
def parse(self, inp):
self.parse_table()
k, _ = self.my_rules[0]
stack = [k]
return self.parse_helper(stack, inp)
def linear_to_tree(self, arr):
stack = []
while arr:
elt = arr.pop(0)
if not isinstance(elt, tuple):
stack.append((elt, []))
else:
# get the last n
sym, n = elt
elts = stack[-n:] if n > 0 else []
stack = stack[0:len(stack) - n]
stack.append((sym, elts))
assert len(stack) == 1
return stack[0]
In [218]:
ll1parser = LL1Parser(A2_GRAMMAR)
tree = ll1parser.parse('1+2')
display_tree(tree)
`
|
|
As The last ship sailed towards the distant horizon I sat there watching on a rock My mind slowly drifting away Forming into my... Dreamtale
What is the proper way to create a bootable image of my system
From https://unix.stackexchange.com/questions/275772/what-is-the-proper-way-to-create-a-bootable-image-of-my-system
The machine doesn't even always need to be the same, since Linux creates the /dev, /proc and /sys filesystems on the fly as the kernel boots which gives you a lot of freedom to make some pretty drastic hardware changes.
Let's say your OS is installed on disk /dev/sda. You can make an ISO of /dev/sda and all of its partitions, whatever they may be, with the following command:
dd if=/dev/sda of=/path/to/image.iso
The downside to this is that the image will be the full size of the disk you specified as if (input file), even if that disk is not full.
If you'd like to clone the disk directly from /dev/sda, simply insert another disk and use something like:
dd if=/dev/sda of=/dev/sdb
|
|
# Recurrence of 0 in a random walk
Assume $\mathcal{S} := \{0, 1, \cdots \}$, $p(0,1)=1$ and $p(n,0)=p(n,n+1)=\frac{1}{2}$ for $n=1,2, \cdots$. Is $0$ recurrent or transient?
So, basically this is an irreducible, closed but infinite Markov chain. Hence, we know that either all states are recurrent or transient.
If I define $\rho_{n,0} := P(T_0 < \infty \; | \; X_0 = n)$ where $T_0 := \inf \{ n \ge 1 : X_n =0 \}$, we will have the following recursive relation
\begin{align} \rho_{n,0} = \frac{1}{2}(\rho_{n+1,0} + 1) \end{align}
so it seems that there is always positive probability for not coming back to 0 (simply keep moving to right). Can we argue that 0 is transient then?
HINT probability to never return to 0 starting at 0 is to always move to the right, i.e. $$\lim_{n \to \infty} (1/2)^n = 0$$
|
|
# How do you multiply (x - 1)(x + 1)(x - 3)(x + 3)?
Nov 1, 2015
${x}^{4} - 10 {x}^{2} + 9$
#### Explanation:
Frist of all, use twice the formula $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$ to obtain
$\textcolor{g r e e n}{\left(x - 1\right) \left(x + 1\right)} \textcolor{b l u e}{\left(x - 3\right) \left(x + 3\right)} = \textcolor{g r e e n}{\left({x}^{2} - 1\right)} \textcolor{b l u e}{\left({x}^{2} - 9\right)}$
If you're satisfied, you can leave the expression like this. Otherwise, you can go for a full expansion, multiplying term by term, and obtaining
$\left({x}^{2} - 1\right) \left({x}^{2} - 9\right) = {x}^{4} - 9 {x}^{2} - {x}^{2} + 9 = {x}^{4} - 10 {x}^{2} + 9$
|
|
# Butlin:Unix for Bioinformatics - advanced tutorial
(Difference between revisions)
## Revision as of 07:52, 14 August 2013
This tutorial is still under construction. Please come back.
## Overview
This session will be more challenging if you are still new to Unix. However, it’s main aim is to demonstrate that Unix is much more than an environment in which you can execute big programmes like bwa or samtools. Rather, with it’s suite of text searching, extraction and manipulation programmes, it is itself a powerful tool for the many small bioinformatic tasks that you will encounter during everyday work. This module expects that you have done the steps in the Basic Unix module, i. e. it expects that you have already created certain folders and have copied the example data into your account. There are still small (hopefully not large) bugs lurking in this protocol. Please help improve it by correcting those mistakes or adding comments to the talk page. Many thanks.
## Before you start
$qrsh Then change to the directory NGS_workshop, that you created in the Basic Unix module:$ cd NGS_workshop/Unix_module
$zless Unix_tut_sequences.fastq.gz$ zcat Unix_tut_sequences.fastq.gz | wc -l
$man wc gives you the number of lines in your sequence file, which is the number of reads times four for fastq format. Note, you can usually avoid uncompressing data files, which saves disk space. ## TASK 4: I have a .fastq file with raw sequences from a RAD library. How can I find out how many of the reads contain the correct sequence of the remainder of the restriction site? Let's assume you had 5bp long barcode sequences incorporated into the single-end adapters, which should show up at the beginning of each sequence read. Let's also assume that you have used the restriction enzyme SbfI for the creation of the library, which has the following recognition sequence: CCTGCAGG . So the correct remainder of the restriction site that you expect to see after the 5bp barcode is TGCAGG. First have a look at your fastq file again:$ zless -N Unix_tut_sequences.fastq.gz
Each sequence record contains four lines. The actual sequences are on the 2nd, 6th, 10th, 14th, 18th line and so on. The following can give you the count:
$zcat Unix_tut_sequences.fastq.gz | perl -ne 'print unless ($.-2)%4;' | grep -c "^.....TGCAGG"
This is a pipeline which first uncompresses your sequence read file and pipes it into the perl command, which extracts only the DNA sequence part of each fastq record. $. in perl stands for the current line number, %4 returns modulo 4 of the current line number minus 2. If that results in a floating point number (non-interger), then the line is not printed out. Get it?$ zcat Unix_tut_sequences.fastq.gz | perl -ne'print unless ($.-2)%4;' | less Grep searches each line from the output of the perl command for the regular expression given in quotation marks. ^ stands for the beginning of a line. A dot stands for any single character. There are five dots, because the barcodes are all 5 base pairs long. The -c switch makes grep return the number of lines in which it has found the search pattern at least once.$ man grep
$man perlrun$ zcat Unix_tut_sequences.fastq.gz | perl -ne'print unless ($.-2)%4;' | grep "^.....TGCAGG" | less In less, type: /^.....TGCAGG then hit enter. ## TASK 5: I have split my reads by barcode and I have quality filtered them. Now I want to know how many reads I have left from each (barcoded) individual. How can I find that out?$ for file in *.fq; \
do echo -n "$file " >> retained; \ cat$file | perl -ne 'print unless ($.-2)%4;' | wc -l >> retained; \ done &$ less retained
This bash for loop goes sequentially over each file in the current directory which has the file ending .fq. It prints that file name to the output file retained. The >> redirection character makes sure that all output is appended to the file retained. Otherwise only the output from the last command in the loop and from the last file in the list of files would be stored in the file retained. The ampersand & at the end of the command line sends it into the background, which means you get your command line prompt back immediately while the process is running in the background.
## TASK 6: I have 30 output files from a programme, but their names are not informative. I want to insert the keyword cleaned into their names? How can I do this?
Renaming files is a very common and important task. I’ll show you a simple and fast way to do that. There are as always many ways to solve a task. First, you could type 30 mv commands that’s the fastest way to rename a single file but doing that 30 times is very tedious and error prone. Second, you could use a bash for loop as above in combination with echo and the substitution command sed, like this:
$for file in *.out; do mv$file echo $file | sed 's|^$$.*$$$$\.out$$$|\1_cleaned\2|'; done
but that's a bit tricky if you are not using sed every day. Note the two backticks characters: one before echo, the other right before the last semicolon. Everything in backticks will be evaluated by bash and then replaced by it’s output. So in this case it’s the modified file name that is provided as second argument to the mv command. If you want to find out what the sed command does, take a look at the beginning of this sed tutorial.
The best way, however, to solve the task is by downloading the perl rename script (note, this is not the system rename command that comes with Unix, which has very limited capabilities), put it into a directory that is in your PATH (e. g. ~/prog) and make that text file executable with chmod. You can download Larry Wall's rename.pl script from here with wget:
$cd ~/src$ wget http://tips.webdesign10.com/files/rename.pl.txt
or use firefox if you are in an interactive session.
$mv rename.pl.txt ~/prog/rename.pl Note, this cut/pastes and renames at the same time.$ ll ~/prog
Let’s call the programme:
rename.pl
You should get Permission denied. That’s because we first have to tell bash that we permit its execution:
$chmod u+x ~/prog/rename.pl$ man chmod
The u stands for user (that’s you) and x stands for executable. Let’s try again:
$rename.pl How to make the documentation in the script file visible?$ cd ~/prog
$pod2man rename.pl > rename.pl.1$ mv rename.pl.1 ~/man/man1
Then look at the documentation with:
$man rename.pl First let's create 30 empty files to rename:$ cd
$mkdir test$ cd test
$for i in {1..30}; do touch test_$i.out; done
$ll$ rename.pl -nv 's/^(.+)(\.out)/$1_cleaned$2/' *.out
You already know that bash expands *.out at the end of the command line into a list of all the file names in the current directory that end with .out. So the command does the renaming on all the 30 files we created. Let’s look at the stuff in single quotes. s turns on substitution, since we are substituting old file names with new file names. The forward slashes / separate the search and the replacement patterns. The search pattern begins with ^, which stands for the beginning of the file name. The dot stands for any single character. The + means one or more of the preceding character. So .+ will capture everything from the beginning of the file name until .out. In the second pair of brackets we needed to escape the dot with a backslash to indicate that we mean dot in its literal sense. Otherwise, it would have been interpreted as representing any single character. Anything that matches the regular expressions in the brackets will be automatically saved, so that it can be called in the replacement pattern. Everything that matches the pattern in the first pair of brackets is saved to $1, everything that matches the pattern in the second pair of brackets is saved to$2. Fore more on regular expressions, look up:
$man grep$ man perlrequick
## I have got a text file originating from a Windows or an old Mac text editor. But when I open it in less, all lines are concatenated into one line and with repeatedly the strange ^M character in it. How can I fix this?
cd ~/NGS_workshop/Unix_module
ll
cat -v text_file_from_Windows.txt
Unix has only linefeeds, the old Mac’s had only carriage returns and Windows uses both characters together to represent one return character. The following will remove the carriage return from the end of lines, leaving only linefeeds:
tr -d ‘\r’ < text_file_from_Windows.txt > file_from_Windows_fixed.txt
cat -v file_from_Windows_fixed.txt
man tr
Note, most Unix programmes print their output to STDOUT, which is the screen. If you want to save the output, it needs to be redirected into a new file. Never redirect output back into the input file, like this:
tr -d ‘\r’ < text_file_from_Windows.txt > text_file_from_Windows.txt
cat text_file_from_Windows.txt
You’ve just clobbered your input file.
less text_file_from_Mac.txt
The old Mac text file has just one carriage return instead of a linefeed. That’s why, all lines of the file are concatenated into one line. To fix this:
tr ‘\r’ ‘\n’ < text_file_from_Mac.txt > file_from_Mac_fixed.txt
less file_from_Mac_fixed.txt
sed ‘s/\r/\n/g’ < text_file_from_Windows.txt > \ file_from_Windows_fixed.txt
dos2unix text_file_from_Windows.txt
mac2unix text_file_from_Mac.txt
The last two programmes are available on iceberg, but are not by default included in a Unix system. You would have to install them yourself. Note also, that dos2unix and mac2unix do in place editing, i. e. the old version will be overwritten by the new version.
## I have just mapped my reads against the reference sequences and I got a BAM file for each individual. How can I find out the proportion of reads from each individual that got mapped successfully?
Note: it seems that the samtools view command should be able to do that with the -f and -c switches. However, in my trials it only returned the total number of reads in the BAM file, i .e including those that did not get mapped. Fortunately, this is a really easy Unix task. First let’s have a look at the .bam file:
samtools view alignment.bam | less
If that fits on your screen without line wrapping ... lucky bastard! If not, just turn off line wrapping in less:
samtools view alignment.bam | less -S
The second column contains a numerical flag which indicates the result of the mapping for that read. The flag 0 stands for a successful mapping of the read. So in order to get the number of reads that got mapped successfully we need to count the number of lines with zeros in their second column. Let’s cut out the second column of the tab delimited .bam file:
samtools view alignment.bam | cut -f 2 | less
man cut
Next, we have to sort this column numerically in order to collapse it into unique numbers:
samtools view alignment.bam | cut -f 2 | sort -n | uniq -c
man sort
With the -c switch to uniq the output is a contingency table of the flags from column 2 that can be found in your alignment file. However, some reads with a 0 flag in the second column still have a mapping quality of 0 (don’t ask me why), which is in the 5 column. So, in order to get a count of the number of reads that got mapped successfully with a mapping quality above 0, use gawk:
samtools view alignment.bam | gawk '$2==0 &&$5>0' | wc -l
man gawk
Gawk is the GNU version of awk. Both are almost identical except that awk can’t handle large files. $2 stands for the content in the second column,$5 for the fifth and && means logical AND. Gawk prints only those lines to output which match our conditions. We then simply count those lines.
## I have got a big multi-fasta file with thousands of sequences. How can I extract only those fasta records whose sequences contain at least 3 consecutive repeats of the dinucleotide "AG"?
Believe it or not, this task can be solved with a combination of Unix commands, i. e. no programming is required. Let’s approach the task step by step. First have a look at the input file:
less multi_fasta.fa
We could just use the programme grep to search each line of the fasta file for our pattern (i. e. 3 consecutive AG), but since grep is line based we would lose the fasta headers and some part of the sequence in the process. So, we somehow have to get the fasta headers and their corresponding sequences on the same line.
tr '\n' '@' < multi_fasta.fa | less
Everything is on one line now (but don’t do this with really large files as this causes the whole file to be read into memory).
tr '\n' '@' < multi_fasta.fa | sed 's/>/#>/g' | less
Note the g at the end of the sed command, which stands for global. With the global option sed does the replacement for every occurrence of a search pattern in a line, not just for the first.
tr '\n' '@' < multi_fasta.fa | sed 's/>/#>/g' | tr '#' '\n' | less
Ok, we are almost ready to search, but we first need to get rid of the @ sign in the middle of the sequences.
tr '\n' '@' < multi_fasta.fa | sed 's/>/#>/g' | tr '#' '\n'| | sed 's/$$[AGCT]$$@$$[AGCT]$$/\1\2/' | less
In the sed command the search and the replacement patterns are enclosed in forward slashes /. Anything that matches the pattern between $$and$$ will be stored by sed and can be called again in the replacement pattern. The square brackets [ ] mean “match any one character that is enclosed by them”. In the replacement pattern \1 stands for the base before the @ sign, \2 stands for the base after the @ sign. Finally, let’s search for the microsats:
tr '\n' '@' < multi_fasta.fa | sed 's/>/#>/g' | tr '#' '\n'| | \
sed 's/$$[AGCT]$$@$$[AGCT]$$/\1\2/' | egrep "@.*(AG){3,}.*@" | less
Note the use of egrep instead of just grep for extended regular expressions. The search pattern is enclosed in quotations marks. Our sequences are delimited by @’s on each side. The dot stands for any single character. The asterisk * means “zero or more of the preceding character”. So .* could match exactly nothing or anything else. The {3,} means 3 or more times the preceding character. Without the brackets this would only refer to the the G in AG. Now that we have all sequences with 3x AG microsats, let’s get them back into fasta format again:
tr '\n' '@' < multi_fasta.fa | sed 's/>/#>/g' | tr '#' '\n'| | \
sed 's/$$[AGCT]$$@$$[AGCT]$$/\1\2/' | egrep "@.*(AG){5,}.*@" | \
tr '@' '\n' | less
And let’s get rid of empty lines:
tr '\n' '@' < multi_fasta.fa | sed 's/>/#>/g' | tr '#' '\n'| | \
sed 's/$$[AGCT]$$@$$[AGCT]$$/\1\2/' | egrep "@.*(AG){3,}.*@" | \
tr '@' '\n' | grep -v "^$” > AGAGAG.fa The grep search pattern contains the regular expression symbols for the beginning and the end of the line with nothing in between ⇒ an empty line. The -v switch inverts the matching. Open AGAGAG.fa in less, and type /(AG){3,} and hit Enter. Each sequence should have a highlighted match. The syntax of regular expressions for grep, sed and less (which are very similar) is one of the most useful things you can learn. The most flexible regular expressions, however, are provided by Perl. Finally, let’s count how many sequences are in the output file: grep -c “^>” AGAGAG.fa ## I have a large table from the output of the programme stacks containing the condensed information for each reference tag. Each column is tab delimited. The first column contains the tag ids. Another column the reference sequence for the tag. How can I create a multi-fasta file from this table with the tag ids as fasta headers? First, convince yourself that the table is actually tab delimited: cat -T stacks_output.tsv | less -S Tabs will be replaced by ^I. In which column is the “Consensus sequence” of each tag? We want the “Catalog ID” as fasta header. Let’s first extract the two columns we need: cut -f 1,5 stacks_output.tsv | less The first line of the output contains the column headers of the input table. We don’t want them in the fasta file. So let’s remove this line: cut -f 1,5 stacks_output.tsv | tail -n +2 | less man tail Now let’s insert a > in front of the tag ids in order to mark them as the fasta headers. cut -f 1,5 tacks_output.tsv | tail -n +2 | sed ‘s/$$.*$$/>\1/’ | less In the sed command $$.*$$ captures a whole line, which we call in the replacement pattern with \1. Finally we have to replace the tab that separates each header from its sequence by a return character in order to bring each sequence on the line below its header. cut -f 1,5 tacks_output.tsv | tail -n +2 | sed ‘s/$$.*$$/>\1/’ | \ tr ‘\t’ ‘\n’ | less If you’re satisfied with the result, then redirect the output of tr into an output file. But what if you also wanted the multi fasta file to be sorted by tag id? cut -f 1,5 tacks_output.tsv | tail -n +2 | sort -nk 1 | \ sed ‘s/$$.*$$/>\1/’ | tr ‘\t’ ‘\n’ | less With the sort command we turn on numerical sort with -n and the -k switch lets us specify on which column the sorting should be done (by default, sort would use the whole line for sorting). ## I want to map the reads from 96 individuals against a reference sequence (e. g. partial or full reference genome or transcriptome, etc.). A mapping programme takes one of the 96 input files and tries to find locations for each read in the reference sequence. How can I parallelise this task and thus get the job done in a fraction of the time? Current hardware in the Iceberg computer cluster: SGE allows jobs to be submitted to the cluster of 96 Sun X2200s (each with 4 CPUs and 16GB memory) 31 Sun X2200s (each with 8 CPUs and 32GB memory and Infiniband) 96 Dell C6100 nodes (with 12 CPUs and 24GB memory and Infiniband) … that’s $96 \times 4 + 31 \times 8 + 96 \times 12 = 1,784$ CPU’s !!! Let’s try to use up to 96 of them at the same time. cd ~/Genomics_workshop/Unix_module ll ind_seqs There are 96 fastq files with the reads from 96 individuals in this folder. All input files contain a number from 1 to 96. Otherwise, their names are identical. We now want to map the reads from those individuals against the “reference_seq”. We have already prepared a so-called array job submission script for you. Let’s have a look at it. nano array-job.sh The lines starting with #$ are specific to the SGE. The first requests 2 Gigabyte of memory. the second asks for slightly less than 1 hour to complete each task of the job. Any job or task taking less than 8 hours will be submitted to the short queue by the SGE and there is almost no waiting time for this queue. The next two lines specify that you get an email when each task has been begun and ended. -j y saves STDERR and STDOUT from each task in one file. The important line is the following:
#$-t 1-96 This initialises 96 task ids, which we can call in the rest of the array job submission script with$SGE_TASK_ID.
stampy \
-g ../reference_seq \
-h ../reference_seq \
-M ind_$SGE_TASK_ID.fq \ -o ind_$SGE_TASK_ID.sam
This is the actual command line that executes stampy, our mapping programme. The -M switch to stampy takes the input file, the -o switch the output file name. Submitting this array job script to the SGE scheduler is equivalent to submitting 96 different job scripts, each with an explicit number instead of \$SGE_TASK_ID. After exiting nano let’s submit the job.
qsub array_job.sh
Qstat
This is the end of the advanced Unix session. If you’ve made it until here CONGRATULATIONS !
## Notes
Please feel free to post comments, questions, or improvements to this protocol. Happy to have your input!
1. List troubleshooting tips here.
2. You can also link to FAQs/tips provided by other sources such as the manufacturer or other websites.
3. Anecdotal observations that might be of use to others can also be posted here.
## References
Relevant papers and books
1. Goldbeter, A and Koshland, DE (1981) - Proc Natl Acad Sci U S A 78(11) 6840-4 PMID 6947258
2. Jacob, F and Monod, J J (1961) - Mol Biol 3(3) 318-56 PMID 13718526
3. Ptashne, M (2004) Genetic Switch: Phage Lambda Revisited - Cold Spring Harbor Laboratory Press ISBN 0879697164
## Contact
• Who has experience with this protocol?
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# Modeling the Atmosphere, pt. 2
The previous Modeling the Atmosphere post derived an expression for describing the change in pressure $dP$ with respect to change in altitude $dz$ in terms of density $\rho$ and acceleration due to gravity $g$:
$\frac{dP}{dz} = -\rho g$
This equation is short to write, but it is not particularly useful. Before applying it to real-life situations, it must be transformed into the barometric equation.
First, recall that density $\rho$ represents mass per unit volume. In Modeling the Atmosphere, pt. 1, $\rho$ was used to replace the expression $\frac{M}{V}$, where $M$ represents the mass of a thin slice $S$ of air with volume $V$. Now,
$\frac{dP}{dz} = -\frac{Mg}{V}$
Air is not actually composed of slices, so it would be more convenient to represent the mass $M$ in terms of particles. By volume, Earth’s atmosphere is composed of about 78% $N_2$, 21% $O_2$, and 1% $Ar$. Approximating Earth air as an ideal gas and using some information from the periodic table, the mass of one mole ($6.022 \times 10^{23}$) of air particles can be calculated:
$0.78(2 \cdot 14.02 \frac{\text{g}}{\text{mol}}) + 0.21(2 \cdot16.00 \frac{\text{g}}{\text{mol}}) + 0.01(39.95 \frac{\text{g}}{\text{mol}}) \approx 29.98 \frac{\text{g}}{\text{mol}} \approx 0.029 \frac{\text{kg}}{\text{mol}}$
So on average, $6.022 \times 10^{23}$ air particles (a.k.a. one mole of air particles) masses approximately $0.029 \text{ kg}$. Using this knowledge, the mass $M$ of one slice $S$ of air may be re-expressed as $0.029 \frac{\text{kg}}{\text{mol}} \cdot N \text{mol}$, where $N$ represents the number of moles of air particles in the slice $S$ of atmosphere. The same process holds for any other collection of particles that can be modeled as an ideal gas. For a general gas, each mole might mass $m \text{ kg}$ instead of $0.029 \text{ kg}$. Generally:
$\frac{dP}{dz} = \frac{-mNg}{V}$
and for Earth’s atmosphere in particular,
$\frac{dP}{dz} = \frac{-0.029 N g}{V}$
Now that the mass $M$ of the slice $S$ of atmosphere has been re-expressed as $mN$, let’s re-express the volume $V$ of the atmosphere slice using the ideal gas law:
$PV = NkT$
Solving for volume yields:
$V = \frac{NkT}{P}$
Substitute:
$\frac{dP}{dz} = \frac{-mNg}{\frac{NkT}{P}}$
and simplify:
$\frac{dP}{dz} = \frac{-mg}{kT} P$
At this point, we’ve almost transformed the starting equation into the barometric equation. To review, $m$ represents the mass per mole of the ideal gas (for Earth’s atmosphere, $m \approx 0.029 \frac{\text{kg}}{\text{mol}}$ ), $g \approx 9.81 \frac{\text{m}}{\text{s}^2}$ is the acceleration due to gravity, $k \approx 1.38 \times 10^{-23} \frac{\text{J}}{\text{K}}$ is the Boltzmann constant, and $T$ is the temperature in Kelvin (assumed to be constant – regardless of height – in this idealized atmosphere).
The next step requires calculus. First, “multiply both sides” by the differential $dz$:
$dP = \frac{-mg}{kT} P dz$
and divide both sides by $P$:
$\frac{1}{P} dP = \frac{-mg}{kT} dz$
Using the classic technique for solving differential equations, integrate both sides:
$\int \frac{1}{P} dP = \int \frac{-mg}{kT} dz$
$\ln{P} = \frac{-mg}{kT} z + c$
Exponentiate both sides:
$P(z) = e^{\frac{-mg}{kT}z+c} = e^c e^{\frac{-mg}{kT}z}$
Using the boundary condition when $z = 0$, it’s clear that the constant $e^c$ corresponds to the pressure at zero altitude: $P_0 = e^c$. Thus,
$P(z) = P_0 e^{\frac{-mg}{kT}z}$
Finally, we have derived the barometric equation. Using this equation, the pressure $P$ of an idealized atmosphere can be determined as a function of altitude $z$. Interestingly, this derivation is not the only method for obtaining the barometric equation. Check back for an alternate derivation and an application to satellite testing with weather balloons.
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Recombining Compressed PubChem SD Files with Open Babel
While testing ChemPhoto, it became necessary to test the chemical structure imaging application with SD Files containing several hundred thousand records. Although it's tempting to meet this need by constructing "dummy" files with the same record or small set of records repeated, tests are always far more illuminating when real data is used.
PubChem is an excellent source of large molecular datasets, and the entire database can be downloaded by FTP. Because of PubChem's massive size, what's downloadable consists of files broken up into groups of about 25,000 in gzipped SD File format (*.sdf.gz). Although this is an excellent resource, it creates a problem: how can you conveniently recombine this set of compressed SD Files into a single SD File?
You might think about writing some "quick" code in your language of choice. Fortunately, Open Babel gets the job done - without any of the coding or debugging.
The following command will create a single SD File from all of the compressed SD Files in a given directory, while also stripping explicit hydrogens and removing all fields except PUBCHEM_COMPOUND_CID.
babel *.sdf.gz pubchem.sdf -d --delete PUBCHEM_COMPOUND_CANONICALIZED,PUBCHEM_CACTVS_COMPLEXITY,PUBCHEM_CACTVS_HBOND_ACCEPTOR,PUBCHEM_CACTVS_HBOND_DONOR,PUBCHEM_CACTVS_ROTATABLE_BOND,PUBCHEM_CACTVS_SUBSKEYS,PUBCHEM_IUPAC_OPENEYE_NAME,PUBCHEM_IUPAC_CAS_NAME,PUBCHEM_IUPAC_NAME,PUBCHEM_IUPAC_SYSTEMATIC_NAME,PUBCHEM_IUPAC_TRADITIONAL_NAME,PUBCHEM_NIST_INCHI,PUBCHEM_EXACT_MASS,PUBCHEM_MOLECULAR_FORMULA,PUBCHEM_MOLECULAR_WEIGHT,PUBCHEM_OPENEYE_CAN_SMILES,PUBCHEM_OPENEYE_ISO_SMILES,PUBCHEM_CACTVS_TPSA,PUBCHEM_MONOISOTOPIC_WEIGHT,PUBCHEM_TOTAL_CHARGE,PUBCHEM_HEAVY_ATOM_COUNT,PUBCHEM_ATOM_DEF_STEREO_COUNT,PUBCHEM_ATOM_UDEF_STEREO_COUNT,PUBCHEM_BOND_DEF_STEREO_COUNT,PUBCHEM_BOND_UDEF_STEREO_COUNT,PUBCHEM_ISOTOPIC_ATOM_COUNT,PUBCHEM_COMPONENT_COUNT,PUBCHEM_CACTVS_TAUTO_COUNT,PUBCHEM_BONDANNOTATIONS,PUBCHEM_CACTVS_XLOGP
865543 molecules converted
7 info messages 15372962 audit log messages
Apparently, there is no way to tell babel to keep just a particular field in an SD File - they need to be removed individually.
Still, not bad for a few seconds on the command line.
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Browse Questions
A uniformly charged conducting sphere of $2.4\; m$ diameter has a surface charge density of $80.0 \mu C/m^2$ Find the charge on the sphere.
Can you answer this question?
$(A)\;1.45 \times 10^{–3}\; C$
Hence A is the correct answer.
answered Jun 4, 2014 by
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# Indepth Example¶
In this page I will detail the way I created the reports that can be found in the examples directory.
Let’s start with the content of common.py, this file stores the definition of an invoice that will be used to create the different reports. The invoice is a simple python dictionary with some methods added for the sake of simplicity:
from os.path import join, dirname
class Invoice(dict):
@property
def total(self):
return sum(l['amount'] for l in self['lines'])
@property
def vat(self):
return self.total * 0.21
inv = Invoice(customer={'name': 'John Bonham',
'zip': 1000,
'city': 'Montreux'}},
lines=[{'item': {'name': 'Vodka 70cl',
'reference': 'VDKA-001',
'price': 10.34},
'quantity': 7,
'amount': 7 * 10.34},
{'item': {'name': 'Cognac 70cl',
'reference': 'CGNC-067',
'price': 13.46},
'quantity': 12,
'amount': 12 * 13.46},
{'item': {'name': 'Sparkling water 25cl',
'reference': 'WATR-007',
'price': 4},
'quantity': 1,
'amount': 4},
{'item': {'name': 'Good customer',
'reference': 'BONM-001',
'price': -20},
'quantity': 1,
'amount': -20},
],
id='MZY-20080703',
status='late',
bottle=(open(join(dirname(__file__), 'bouteille.png'), 'rb'),
'image/png'))
## Create a simple OpenOffice Writer template¶
Let’s start with the simple template defined in basic.odt.
This report will be created and rendered with the following three line of code:
from relatorio.templates.opendocument import Template
basic = Template(source='', filepath='basic.odt')
file('bonham_basic.odt', 'wb').write(basic.generate(o=inv).render().getvalue())
Notice that the dictionary passed to generate is used to bind names to make them accessible to the report. So you can access the data of the invoice with a Text Placeholder containing o.customer.name. This is where you can see our Genshi heritage. In fact, all reports using relatorio are subclasses of Genshi’s Template. Thus you can use most of the goodies provided by Genshi.
To iterate over a list you must use an hyperlink (created through ‘Insert > Hyperlink’) and encode as the target the Genshi expression to use. The URL-scheme used must be relatorio. You can use whatever text you want as the link text, but we find it way more explicit to display the Genshi code used. Here is the example of the for loop.
And thus here is our invoice, generated through relatorio:
## One step further: OpenOffice Calc and OpenOffice Impress templates¶
Just like we defined a Writer template it is just as easy to define a Calc/Impress template. Let’s take a look at pivot.ods.
As usual you can see here the different way to make a reference to the content of the invoice object:
• through the Text Placeholder interpolation of Genshi
• or through the hyperlink specification I explained earlier.
Note that there is another tab in this Calc file used to make some data aggregation thanks to the data pilot possibilities of OpenOffice.
And so here is our rendered template:
Note that the type of data is correctly set even though we did not have anything to do.
## Everybody loves charts¶
Now we would like to make our basic report a bit more colorful, so let’s add a little chart. We are using PyCha to generate them from our pie_chart template:
options:
width: 600
height: 400
background: {hide: true}
legend: {hide: true}
axis: {labelFontSize: 14}
padding: {bottom: 10, left: 10, right: 10, top: 10}
chart:
type: pie
output_type: png
dataset:
{% for line in o.lines %}
- - ${line.item.name} - - [0,$line.amount]
{% end %}
Once again we are using the same syntax as Genshi but this time this is a TextTemplate. This file follow the YAML format thus we can render it into a data structure that will be sent to PyCha:
• the options dictionary will be sent to PyCha as-is
• the dataset in the chart dictionary is sent to PyCha through its .addDataset method.
And here is the result:
## A (not-so) real example¶
Now that we have everything to start working on our complicated template invoice.odt, we will go through it one step at a time.
In this example, you can see that not only the openoffice plugin supports the for directive, it also supports the if directive and the choose directive that way you can choose to render or not some elements.
The next step is to add images programmatically, all you need to do is to create frame (‘Insert > Frame’) and name it image: expression just like in the following example:
The expression when evaluated must return a couple whose first element is a file object containing the image and second element is its mimetype. Note that if the first element of the couple is an instance of relatorio report then this report is rendered (using the same arguments as the originating template) and used as a the source for the file definition.
This kind of setup gives us a nice report like that:
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# Exchange splitting
1. Dec 6, 2011
### mendes
I would like to understand what is "exchange splitting" in atomic orbitals.
For which orbitals does it happen ? Is there any similarity between this phenomenon and the Zeeman effect (which breaks the degeneracy on the magnetic quantum number and then splits the orbital into sub-orbitals having different energy according to the spin magnetic number) ?
I know that I must be confused...
Thanks.
2. Feb 24, 2012
### M Quack
Exchange splitting is the energy difference of two electronic states due to exchange interactions, i.e. direct overlap of electronic wave functions, rather than a magnetic field.
You can sometimes calculate an equivalent magnetic field that would produce the same splitting via the Zeeman effect. These "effective" fields can be huge, 50T and more.
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The Vulgar fraction reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)
# Vulgar fraction
Videos show Africa through the eyes of children
In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. The fraction "three divided by four" or "three over four" or "three fourths" or "three quarters" can be written as
or 3 ÷ 4
or 3/4
In this article, we will use the last of these notations. The first quantity, the number "on top of the fraction", is called the numerator, and the other number is called the denominator. The denominator can never be zero because division by zero is not defined. All vulgar fractions are rational numbers and, by definition, all rational numbers can be expressed as vulgar fractions.
Table of contents 1 Introduction 2 Arithmetic 3 Other ways of writing fractions 4 See also
## Introduction
To understand the meaning of any vulgar fraction consider some unit (e.g. a cake) divided into an equal number of parts (or slices). The number of slices into which the cake is divided is the denominator. The number of slices in consideration is the numerator. So: Were I to eat 2 slices of a cake divided into 8 equal slices then I would have eaten 2/8 (or two eighths) of the cake. Note that had the cake been divided into 4 slices and had I eaten one of those then I would have eaten the same amount of cake as before. Hence, 2/8 = 1/4. Had I eaten 1 and a half full cakes I would have eaten 12 of the one eighth of a cake slices or 12/8. If the cakes been divided into quarters I would have eaten 6/4 cakes. 12/8 = 6/4 = 3/2 = (1 + 1/2) cakes.
## Arithmetic
Several rules for calculation with fractions are useful:
Cancelling. If both the numerator and the denominator of a fraction are multiplied or divided by the same non-zero number, then the fraction does not change its value. For instance, 4/6 = 2/3 and 1/x = x / x2.
Adding fractions. To add or subtract two fractions, you first need to change the two fractions so that they have a common denominator, for example the lowest common multiple of the denominators which is called the lowest common denominator; then you can add or subtract the numerators. For instance, 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
Multiplying fractions. To multiply two fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. For instance, 2/3 × 1/4 = (2×1) / (3× 4) = 2/12 = 1/6. It is helpful to read "2/3 × 1/4" as "two thirds of one quarter". If I took 2/3 of a cake and gave 1/4 of that part away, the part I gave away would be equivalent to 1/6 of a full cake.
Reciprocal of fractions. To take the reciprocal of fractions, simply swap the numerator and the denominator, so the reciprocal of 2/3 is 3/2. If the numerator is 1, i.e. the fraction is a unit fraction, then the reciprocal is an integer, namely the denominator, so the reciprocal of 1/3 is 3/1 or 3.
Dividing fractions. As dividing is the same as multiplying by the reciprocal, to divide one fraction by another one, flip numerator and denominator of the second one, and then multiply the two fractions. For instance, (2/3) / (4/5) = 2/3 × 5/4 = (2×5) / (3×4) = 10/12 = 5/6.
## Other ways of writing fractions
A fraction greater than 1 can also be written as a mixed number, i.e. as the sum of a positive integer and a fraction between 0 and 1 (sometimes called a proper fraction). For example
23/4 = 11/4.
In general:
This notation has the advantage that one can readily tell the approximate size of the fraction; it is rather dangerous however, because 23/4 risks being understood as 2×3/4, which would equal 3/2, rather than 2+3/4 . To indicate multiplication between an integer and a fraction, the fraction is instead put inside parentheses: 2 (3/4) = 2 × 3/4.
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This is the tale of a really, really inefficient way of printing out the text Hello World.
It was not a simple programming project. It kind of turned into a really slow-burning programming project.
NOTE: In February 2015, the project underwent some major new developments. This page is being re-constructed and rewritten! Meanwhile, here’s a link to the Git repository.
## Some minor background
I hated math in school. There was only one big reason for this: There was no real motivation to learn any of the stuff. This, combined with my depression, didn’t really make it too much fun to learn.
But this little project was one of those things that slooooowly woke me up to brush up at least some basic practical math skills. It got much more interesting once I started tinkering around with Processing – for example, at one point I thought “man, juggling all these numbers with x and y components is really fucking tricky”, then I read tutorials that used vector classes, and lo and behold, I had motivation to dig out my math books and re-read the stuff on vector calculations.
As of writing (in early 2015), the Finnish school system is planning on introducing programming as a mandatory subject. GOOD. Especially if the math classes have something to contribute to the programming classes and vice versa. Give the kids practical ways of using math for good and awesome, I say. I only learnt the practical ways of programming on my own, and not much on the math because there was no motivation to study it.
## The original code
This is how it all began in February 2001, written for the mathematical programming language GNU Octave:
See what I did there?
No? Most of that is just plotting code, so here is the heart of the operation.
Simply put, the whole process is simple:
1. We have a string: “Hello world”
2. We turn it into an array of ASCII values: [72, 101, 108, 108, 111, 32, 119, 111, 114, 108, 100]
3. We use Octave polyfit() function to fit a polynomial which, for x = {1..11}, will evaluate to the ASCII values in the above array.
4. Polyfit will return the coefficients in array “p” and evaluated values in “evald”.
5. We convert the values from “evald” back to character values and print it out. Tadah!
GNUPLOT produced this delightful graph that shows what the polynomial looks like when it’s plotted out:
…I’ve been told that Octave these days has plotting functionality of its own. I don’t know. For practical purposes, I’ve since moved to other software packages. My current choices for number-crunching are R and Maxima.
## A mysterious block of numbers
So Octave creates a polynomial – so far, however, we haven’t seen the coefficients. Here’s how the original dump of the coefficents looked like:
# Created by Octave 2.0.16.92, Wed Feb 21 15:53:56 2001 <wwwwolf@nighthowl>
# name: p
# type: matrix
# rows: 14
# columns: 1
1.99079381095492e-05
-0.00115437249297999
0.0287925989153794
-0.403199605046038
3.46465373237423
-18.6812013367602
61.4275957129813
-109.846789475099
64.1870784258645
69.9070799001331
-43.5100385152995
-77.6672165963919
-1.37969637341796
124.474075996299
Or, if you prefer it to be written out as a real honest-to-goodness mathemathical thingy:
\begin{align} y &=& & 1.99079381095492 \times 10^{-5} x^{13} &-& 0.00115437249297999 x^{12} &+& 0.0287925989153794 x^{11} \\ &&-& 0.403199605046038 x^{10} &+& 3.46465373237423 x^9 &-& 18.6812013367602 x^8 \\ &&+& 61.4275957129813 x^7 &-& 109.846789475099 x^6 &+& 64.1870784258645 x^5 \\ &&+& 69.9070799001331 x^4 &-& 43.5100385152995 x^3 &-& 77.6672165963919 x^2 \\ &&-& 1.37969637341796 x &+& 124.474075996299 && \end{align}
Yep. Now that the smoke clears, we can see that Octave has produced a nice 13th order polynomial.
Here’s a few immediate observations:
1. Precision is going to be a total headache here. We’re operating in realm of binary computers, and here we have gigantic floating point numbers. Shit is going to be rather evil when precision is going to be issue. Rounding errors are going to make us cry.
2. Mathematicians are probably screaming in agony right now. This is computer mathematics, not real mathematics! You’re supposed to prove that the house is on fire, and go back to bed! I’m a computer person so, after proving this is feasible, I rather design a networked smart-home application to automatically forward fire alarms to the fire department.
3. Computer science folks will probably say “oh gosh, I’m so glad that IEEE floats are a thing, because otherwise this would not work.” Such are the mysterious modern ways of the floating point numbers.
4. Most people will look at the bunch of numbers and say “where the FUCK is that ‘Hello World’ hidden, anyway?” MISSION ACCOMPLISHED.
## And we have arrived to a definition
So here we have an actual definition of our Hello World program:
Evaluate $$y$$ of
\begin{align} y &=& & 1.99079381095492 \times 10^{-5} x^{13} &-& 0.00115437249297999 x^{12} &+& 0.0287925989153794 x^{11} \\ &&-& 0.403199605046038 x^{10} &+& 3.46465373237423 x^9 &-& 18.6812013367602 x^8 \\ &&+& 61.4275957129813 x^7 &-& 109.846789475099 x^6 &+& 64.1870784258645 x^5 \\ &&+& 69.9070799001331 x^4 &-& 43.5100385152995 x^3 &-& 77.6672165963919 x^2 \\ &&-& 1.37969637341796 x &+& 124.474075996299 && \end{align}
for $$x = 1 \ldots 11$$ where $$x \in \mathbb{Z}$$, rounding each value of $$y$$ to nearest integer and converting the resulting integer to a character based on the ASCII value.
## Implementation
The above numbers mean that we’re totally at the mercy of IEEE double precision floats.
You can store the above numbers in doubles, write an evaluator, and you get proper results… as long as you’re using a real, honest-to-goodness, modern programming language. Tried this in Ruby – no problems, works as advertised.
Tried this in C – had some headaches with types, but hey, it works.
Fortran? Oh, why not… …but why not?
My original plan was to write this program in FORTRAN 77. Because I wanted to do a fitting tribute to the grandfather of number-crunching.
My original implementation didn’t work at all, though. First – one of the reasons was that Fortran is a little bit picky about implied number conversions. I had to be damn explicit at every turn that calculations have to be performed as double precision floats at every turn. This confused the hell out of me – modern languages usually have some sort of notion of “the programmer has no fucking clue how to implement this and mixes integers and floating point numbers on willy-nilly, let’s fall back to full-on floating point math to be sure”
Secondly? …uh, I’m not a Fortran expert, but I think Fortran has some really weird concepts of what the hell is a double precision float. Based on a little bit of digging, I think FORTRAN 77’s notion of double precision numbers isn’t the same as IEEE doubles and that explicit support of IEEE floats is a new notion in Fortran land. *ahem* which would make sense, because IEEE floating point standard only came to be in 1985…
So here’s the big damn caveat of the FORTRAN 77 version: Trying to compile this with gfortran (which is primarily a Fortran 9x compiler) will probably screw the end results up because it’s doing whatever passes as doubles in Fortran-double-land.
However, if you compile this using f2c, it will work, because it will essentially dump the things in C doubles as is and everything is smooth sailing in IEEE-double-land.
So we’re basically cheating, I guess.
Here’s the bastard in FORTRAN 77:
…I was particularly keen to do the PROPER CAPITALISATION and GUIDO\tVAN\tROSSUM\tING of the code.
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FAQ question_answer137) Examine the following statements and select the correct answer using the codes given below: 1. MCQ 1 C40 is the first of its kind summit being organised for the first time in Copenhagen It is an initiative of UN Habitat Choose We have launched our mobile APP get it now. Increasing the moisture without increasing the temperature. Which of the above statements is/are correct? 2. Mark the correct order of pressure belts from equator to poles. Water vapour present in the air produces the optical phenomena of red and orange hues in the sky at sunrise and sunset which are known as dawn and dusk respectively. question_answer14) Which of the following fact about dust particles in the atmosphere is/are correct? Methane also affects the degradation of the ozone layer. This area is near the equator. Which of the above statements is/are correct? II. Molecular layer extends from 200 to 1100 km. 2. 4. question_answer102) Consider the following statements, question_answer103) $\frac{\text{Amount}\,\text{of}\,\text{water}\,\text{vapour}\,\text{actually}\,\text{present}\,\text{in}\,\text{the}\,\text{air}}{\text{Amount}\,\text{of}\,\text{water}\,\text{vapour}\,\text{required}\,\text{to}\,\text{saturate}\,\text{the}\,\text{air}}\text{ }\!\!\times\!\!\text{ 100}$ The above equation represents the, question_answer104) Relative humidity is the ratio of the. Trade winds blow between. Which of the following statements is not true regarding Westerly?s? 'C' climate has temperature above ${{18}^{o}}C$ throughout the year. https://www.studyadda.com 2. Ionosphere is the layer of convection currents. 'C' climate has temperature above ${{18}^{o}}C$ throughout the year. Sample Papers Absolute humidity is expressed in terms of grams of moisture per kilogram of air. 4. 1. Mountain breezes generally cause inversion of temperature in the valley Which of the above statements is/are correct? 3. Reason (R): The directions of wind pattern in the northern and the southern hemisphere are governed by the Coriolis Effect. I. 2. 2. The reflective quality of a surface, expressed in percentage of the total energy received known as its albedo. 4. Air is saturated when the relative humidity is 100 percent, 2. A temperate cyclone is formed when a cold air mass meets a warm air mass. Coriolis force is higher in southern hemisphere as compared to northern hemisphere. II. Which of these statements are correct? | What causes high pressure in the sub-tropical zone? The reflective quality of a surface, expressed in percentage of the total energy received known as its albedo. Which of the above statements is/are correct? This area receives very little precipitation. Winter dry season D. 2. 1. Increasing the temperature without increasing the moisture. Normal El Nino forms in the Central Pacific Ocean whereas El Nino Modoki forms in Eastern Pacific ocean. Thermal equator remains at the equator throughout the year. Specific humidity is expressed as grams of moisture per cubic metric of air. They act as hydroscopic nuclei around which water vapour condenses to produce cloud. | Sub-polar low pressure belt 3.? La Nina is characterized by unusually cold ocean temperature in equatorial Indian Ocean whereas El Nino is characterized by unusually warm ocean temperature in the equatorial Pacific Ocean. In the middle latitudes, east coast regions receive more rainfall than their western counterparts. Westerlies in southern hemisphere are stronger and persistent than in northern hemisphere. I. question_answer12) Assertion (A): The thickness of the atmosphere is the maximum over the equator.????? Consider the following statements concerning air masses. II. Impending rain or snow B. Intense cooling of earth's surface due to rapid radiation 2. BWh?????????? 2. Overcast skies, rain and raw weather are generally associated with them. Koppen's classification is based on the precipitation. Air moving from high area to low area along the valley floor 1. question_answer10) Consider the following statements about the atmosphere 1. Atmosphere is held to the earth by its gravitational force and is an integral part of the earth. 2. 3. Sloping boundary surface between contrasting air masses is called a front. 2. Hurricanes cause large scale destruction in the area influenced by them. question_answer168) Consider the following statements: 1. II. Even in summer, the temperature is below freezing point. 3.? Thom thwarts used precipitation and temperature to demarcate climate boundaries. 1. They blow uninterrupted in the Northern and Southern Hemisphere. 1. The Inter-tropical convergence Zone (ITZ) formed by the converging winds maintains its position at the equator throughout the year. Hydrogen is confined to lower layers of the atmosphere, 4. C. 1. Aurora Borealis D. Mesosphere 4. Chemically it is divided into homosphere and heterosphere. Coriolis force is higher in southern hemisphere as compared to northern hemisphere. 3. The ice cap climate or ‘EF’ occurs in this region. Prakash Kumar. Daily Current Affairs MCQ Quiz for UPSC, IAS, UPPSC/UPPCS, MPPSC. They cause heavy destruction to crops and grasslands in the areas affected by them. 3 sec, OTP has been sent to your mobile number and is valid for one hour, The correct sequence of atmospheric gases as expressed in percentage by volume is. Mitron, my [T25] series contains 25 Mock MCQ sets primarily aimed at UPSC IAS/IPS Civil Service exam aspirants. question_answer66) Which of the following statements is not true regarding Westerly?s? Jet Stream Theory?? All the weather phenomena occur in this layer. Balance of heat received and emitted by the earth is called its heat budget. 1. In the context of the above statements, which one of the following is correct? 2. 1. Kharaburan????? 1. question_answer29) Match list I with list II and select the correct answer using the code given below List I?? question_answer39) Consider the following statements:?????????? Practice daily MCQs for the Geography and analyse your strong and weak areas question_answer158) Consider the following statements: 1. Dust particles present in the atmosphere act as hydroscopic matter. 2. North Temperate Zone 2. Thermosphere is divided into ionosphere and exosphere. Tarun Goyal. 2. Organisms at higher trophic level [â¦] Prelims MCQs Quiz 44 : Environment 1. question_answer170) Which one of the following is not correctly matched with reference to Thomthwaite's classification of climates? Rise in temperature of the air after injecting heat in the air. Climate And Weather In Indian Geography MCQ Questions and answers with easy explanations. Rockies of North America C. In Mediterranean region, summers receive more rainfall. Which one of the following pairs is not correctly matched? All you need of UPSC at this link: UPSC â« What is climate change? According to Koeppen, the capital letters A,C,D and E delineate humid climates and B dry climates. question_answer115) Which type of clouds does not indicate fair weather? Rise in temperature of the air after injecting heat in the air. III. III. It preserves earth's radiated heat. Posted on October 18, 2018 Author ias4sure Categories The MCQ Factory Tags Animal Welfare Board of India, Free MCQs for UPSC, Geography MCQs, UPSC MCQs, World Geography MCQs Leave a comment on Topicwise MCQs for Geography for UPSC Water vapour in the atmosphere decreases with altitude. In the northern hemisphere, the isotherms bend equator ward while crossing landmasses and pole ward while crossing the oceans. Which of the following is/are not the features of convectional rainfall? Also, the decade 2011-2020 would be the warmest ever. Thermal equator remains at the equator throughout the year. daily quiz for upsc, daily quiz for ssc, daily quiz for rrb ntpc, daily quiz gk today, daily quiz gradeup, daily quiz hindi, daily quiz hssc, daily quiz in hindi, daily quiz ias, daily quiz ibps guide, daily quiz jagran josh, gk daily quiz, gk daily quiz in hindi, current affairs mcq for upsc, current affairs mcq pdf, current affairs mcq with answer, current affairs mcq 2. question_answer75) Sun's halo is produced by refraction of light in: question_answer76) Which of the following areas are not influenced by the monsoon winds? 2. Regular fall of temperature with height in the troposphere is known as normal lapse rate. Thom thwart?s classification of climate attempts to define the climate boundaries quantitatively. [a] Moscow in the Russian Federation has the highest annual range of temperature in the world. question_answer171) The seasonal reversal of winds is the typical characteristic of, Copyright © 2007-2020 | Air is saturated when the relative humidity is 100 percent 2. question_answer58) Which of the following is/are correctly matched? They are usually found near tropo pause. 3. Hydrogen present in the air is a heavy gas and is confined to the lower layers of the atmosphere only. 3. Convergence of air from the subtropical high pressure belts. | question_answer47) Atmospheric pressure depends on Temperature of the air. A tornado has a large diameter and is very destructive Which of the above statements is/are correct? Increasing use of the chlorofluorocarbons has resulted in depletion of ozone layer in the atmosphere. 10 lessons. Interior parts of the continents experience higher annual range of temperature as compared to the coastal areas. Which one of the following is considered as an interruption of the planetary wind system? It is mainly on eastern and western margins of continents. question_answer20) The earth receives only two billionth part of the total solar radiation emitted by the sun. [b] Singapore has one of the lowest annual ranges of temperature in the world. Indian Geography is an important topic of all competitive exams in India. question_answer99) Match list I with list II and select the correct answer using the code given below: List I?????????????? Factors are correct???????????????., there should be 1 when freezing point Examine the following is not correctly matched Koppen in his of! Define the climate boundaries periodic winds volume is for exams like IBPS, SCC, UPSC NEET! From east to west in the northern hemisphere experiences higher annual range of temperature in Russian! Ii ( Avg Temperature/ moisture ) A. equatorial low pressure area near the equator acquire their basic properties from regions! 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Question_Answer36 ) in Koppen 's classification of world climates everyone being able to in. Deformation of the atmosphere is the maximum velocity of winds is observed over Characteristics! Winter rainfall in northern India causes development of anti-cyclonic conditions with low temperatures question_answer30 ) the 's! A mixture of fog and smoke MCQ questions and answers with easy explanations latitude 4 smaller seaward penetration than landward... Following are true characteristic of Chinooks percentage by volume is angle of incidence of the Amazon, the Congo Malaysia. Question_Answer80 ) La Nina is suspected to have caused recent floods in Australia question_answer142 ) which of the following is. The 'Roaring Forties ' as grams of moisture per cubic metric of air reached which of the cyclone ',... Regions and acquire their basic properties from these regions and E delineate humid climates B. 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Below 6 km, I agree that I am at least 13 years old and read! Central Pacific Ocean whereas El Nino forms in eastern Pacific Ocean whereas El Nino Modoki has been.... Cause: 1 and help in radio transmission the Indian sub-continent in summer, the isotherms bend ward... Way of moisture per kilogram of air are deflected from their path due to the earth 's and! Reflective quality of a surface, expressed in percentage of the Amazon, the ideal conditions for jet... Vapour condenses to produce cloud question_answer47 ) atmospheric pressure in the exam of summer,! Towards the poles ) La Nina is suspected to have caused recent floods in Australia effect the! Question_Answer163 ) in which form of waves, solar radiation received at the equator size of the atmosphere is mixture. Shop for Environment for Civil Services to northern hemisphere and anti-clockwise in the and! Belts from equator towards the poles part I the Hot, wet climate is found in atmosphere... 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Some parts of equatorial Africa experience higher annual range of temperature in the rotation of cyclone... And water particles select the correct sequence of atmospheric layers ) list II ( Country/Region a... Agree to the equatorial low pressure trough 1 depletion of ozone layer the! And they break and short wave radio waves and help in radio transmission the warmest ever: Environment.! Conditions is EF ’ occurs in every month of the air in Koppen 's classification of climate portrayed Plateau... This Mock Test of Test: world climate & climate change quizzes methane discovered! Eastern and western margins of continents B dry climates occurs before sunrise is as. Which wind system fanned around an intensely low pressure system the cold front has a large diameter and confined! Is surrounded by cold air mass 4 dioxide which is released by various human activities for taking some the. Valley which of the following statements:?????????! And southern hemisphere we have launched our mobile APP get it now is/are matched., question_answer95 ) Norwester is experienced in, question_answer96 ) Local wind )???????. Change quizzes world climate upsc mcq all type of El Nino forms in eastern Pacific Ocean whereas El Nino in. Receive maximum solar radiation reaches the earth 's surface due to the lower layers the... Of land does not indicate fair weather help of these winds in the northern?.
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# Sean's SAP post seems oddly shallow-what's happening?
1. Aug 9, 2007
### marcus
Sean's SAP post seems oddly shallow--what's happening?
This post over at Cosmic Variance blog.
BTW currently it says discussion will be closed on 6 September (but those time limits sometimes get extended.)
==quote==
Unusual Features of Our Place In the Universe That Have Obvious Anthropic Explanations
Sean at 5:02 pm, August 7th, 2007
The “sensible anthropic principle” says that certain apparently unusual features of our environment might be explained by selection effects governing the viability of life within a plethora of diverse possibilities, rather than being derived uniquely from simple dynamical principles. Here are some examples of that principle at work.
* Most of the planetary mass in the Solar System is in the form of gas giants. And yet, we live on a rocky planet.
* Most of the total mass in the Solar System is in the Sun. And yet, we live on a planet.
* Most of the volume in the Solar System is in interplanetary space. And yet, we live in an atmosphere.
* Most of the volume in the universe is in intergalactic space. And yet, we live in a galaxy.
* Most of the ordinary matter in the universe (by mass) consists of hydrogen and helium. And yet, we are made mostly of heavier elements.
* Most of the particles of ordinary matter in the universe are photons. And yet, we are made of baryons and electrons.
* Most of the matter in the universe (by mass) is dark matter. And yet, we are made of ordinary matter.
* Most of the energy in the universe is dark energy. And yet, we are made of matter.
* The post-Big-Bang lifespan of the universe is very plausibly infinite. And yet, we find ourselves living within the first few tens of billions of years (a finite interval) after the Bang.
That last one deserves more attention, I think.
==endquote==
For starters here's a little semantic point. By googling "selection effect" one can learn, e.g. from Wikipedia, that:
SELECTION EFFECTS DONT GOVERN the viability of life
on the contrary, THE VIABILITY OF LIFE GOVERNS certain kinds of SELECTION EFFECTS
.
And selection effects are a source of error in human reasoning---they are not physical effects occurring in nature.
I'll get to that in the second post and give some sources.
Then just as a general observation, should this proposed "SAP" be called the Shallow Anthropic Principle, or would it be more accurate to call it the Silly Anthropic Principle? Does this post have any scientific content? Would someone like to interpret?
It begins with a false dichotomy "... explained by selection effects governing the viability of life within a plethora of diverse possibilities, rather than being derived uniquely from simple dynamical principles."
Let's compare apples with apples---don't say "derived uniquely" in one case and simply "explained" in another. Lets put it fairly:
certain apparently unusual features of our environment might be explained by selection effects governing the viability of life within a plethora of diverse possibilities, rather than being explained from simple dynamical principles.
This is still not straight-talk. "Selection effects" as ordinarily understood do not "govern the viability of life". PHYSICS governs the viability of life, however you define it.
Life is a physical phenomenon and, however you define it, will be subject to limitations as to environment.
There is no dichotomy between explaining by whatever effects govern the viability of life, and explaining by simple dynamical principles.
===============
ORDINARY PHYSICS (and derived chemistry) can explain why one can expect to find life more where there is a rich chemistry and phase structure (solid liquid gas). Some places are too hot or chemically too monotonous or too limited in phase structure for one to expect life. If one found some analog of life existing in the sun, it would be unexpected and one would be surprised.
So the beginning of the post has a certain sleaz- or sloppiness----it begins by taking certain cases of explanation by simple dynamical principles (governing viability of life) and RENAMING them explanation by "sensible anthropic principle".
Then it declares a false contrast between the two kinds of explanation (one of which has a phony name and actually a case of the other.)
========================
once past the initial false dichotomy one sees 9 statements. The first 8 seem shallow or trivial to me. Maybe someone would like to explain why they are not explainable by ordinary dynamical principles, to whatever extent they need explanation.
There is, after all, a search for extrasolar planets going on motivated in part by curiosity about extrasolar life. For good physical reasons, one wants especially to find rocky watery exoplanets because one expects them to harbor life with greater liklihood than other planet and nonplanet environments. Whatever life is, people consider it more apt to be found (if at all) in rocky habitable-zone wet places.
And one reads of people already deciding on what signatures to look for.
Each of those first 8 question strikes me as a self-absorbed way of phrasing a practical question. Practical versions of the questions are, for instance:
Why should we look for life on habitable zone planets instead of on the surfaces of stars?
Why would we look for life-signs on rocky terrestrial-size planets instead of on gas giant planets?
Why would we look for life at planetary systems around stars, instead of in interstellar space, or even intergalactic space?
==============
BECAUSE FOR SIMPLE PHYSICAL REASONS WE ARE MORE LIKELY TO FIND IT THERE.
("It's the Chemistry, duh")
Formulated as a question of how you spend research money, and observatory time, there are direct answers.
But the SAP post phrases these questions like the child's question "WHY AM I ME" appealing to everyone's latent narcissism.
"Why do I live here instead of on the sun? The sun is so much bigger, Mommy!"
"Why do I live here instead of on Jupiter? Jupiter is bigger, isn't it, Dad?"
Last edited: Aug 9, 2007
2. Aug 9, 2007
### marcus
I googled "selection effect" and got this Wikipedia entry
http://en.wikipedia.org/wiki/Selection_bias
"Selection bias is a distortion of evidence or data that arises from the way that the data is collected. It is sometimes referred to as the selection effect. The term selection bias is most often refers to the distortion of a statistical analysis, due to the method of collecting samples. If the selection bias is not taken into account then any conclusions drawn may be wrong."
Does anyone have another online dictionary or encyclopedia reference?
I'm not completely satsified with this Wikipedia definition---I might like it better if it explicitly included the UNCONSCIOUS bias arising from the observer's circumstances.
How about this: A selection effect is a skewing of the observer's picture that arises from an unrepresentative sample---which could occur unintentionally because of who and where he happens to be.
I'm not sure. Maybe someone else can offer a better. What I do feel sure about is that you can't honestly say
"selection effects governing the viability of life" in various environments because
SELECTION EFFECTS DONT GOVERN the viability of life
on the contrary, THE VIABILITY OF LIFE GOVERNS certain kinds of SELECTION EFFECTS
.
the physical possibility of the physical phenomenon of life, restricting it with high probability to limited circumstances, will in part govern what we see when we wake up.
we are fairly likely to be on a planet with plenty of different chemical elements and enough gravity to keep an atmosphere so that water can be liquid.
==========================
I also don't like that "apparently unusual" in Sean's SAP post. Why wasn't I born in intergalactic space, Mommy? There is SO MUCH MORE SPACE out there between galaxies!
That kind of reasoning depends so much on the PROBABILITY MEASURE that one puts on the space of possibilities. If one choses a frivolous or naive probability measure to begin with, one gets a silly result. In fact results can vary wildly.
The first 8 questions do not strike me as science---but as a kind of candy for the imagination. What is going on? It sounds like soft-core propaganda for something. For what? Why should Sean (who used to be thought of as a scientist) be peddling kindergarten versions of Anthropery?
Does anyone have any explanation or comment? Am I the only one who is troubled by this CosmicVariance post?
=================
Bee Hossenfelder had a pithy comment here
http://www.math.columbia.edu/~woit/wordpress/?p=582#comment-27314
that I find relevant:
...What I notice - and what I welcome - is that there is more discussion about the question what is science, pseudoscience, and where we are headed. I find that a healthy development. I hope there will be a practical outcome of that which allows researchers to refocus their efforts on real science, and not to waste time on politics, networking, or advertisement. One has to ask why pseudo-scientific ideas gain popularity. Because they are cheap to produce and they sell well. It’s the Walmart of science. You get everything, it looks okay, but if you try to use it will fall into pieces.
================
George Orwell has a great essay called Politics and the English Language about how the corruption of language, the distortion of the meanings of words, can make for a decline of politics.
One thinks of current leaders with their slogans and clichés manufactured by Think (about how to lie effectively) Tanks.
Maybe we need an essay on Science and the English Language.
Bee just needs a little more experience---maybe by the time she is 50 she will be able to write one.
Last edited: Aug 10, 2007
3. Aug 9, 2007
### marcus
the special case of puzzle #9
#9 is different because it is a TIME question. The other questions had obvious practical correlatives "where would you point your telescope to look for life" because they were essentially WHERE problems.
The reason you point your telescope at a wet rocky is not "because we exist" or "because of the Anthropic Principle". That would be a sappy answer. You point your telescope at a wet rocky because that's where you expect to have the most chance of finding life. There are sound dynamical principles underlying this---so it seems like a soft-sell for some grander version of Anthropery to give the Anthropic name to this case of straight physics.
That covers the first 8 examples, but now we have a TIME example, the ninth one.
"The post-Big-Bang lifespan of the universe is very plausibly infinite. And yet, we find ourselves living within the first few tens of billions of years (a finite interval) after the Bang."
Personally I don't find this an unlikely time to be alive. I would say that on good physical grounds.
This is as likely a time as any other and a lot more likely than some.
I expect the universe will be cold dark and virtually uninhabitable in its old age.
So what's the problem? Charles Lineweaver has an interesting recent article that goes quite a bit further than I need to go here. According to him the ILLUSION OF UNEXPECTEDNESS of a certain "cosmic coincidence" was caused by a selection bias. And he explains the coincidence away as something that 68 percent of all observers would see----not unusual after all.
http://arxiv.org/abs/astro-ph/0703429
The Cosmic Coincidence as a Temporal Selection Effect Produced by the Age Distribution of Terrestrial Planets in the Universe
Charles H. Lineweaver, Chas A. Egan
(Submitted on 16 Mar 2007)
"The energy densities of matter and the vacuum are currently observed to be of the same order of magnitude: $$(\Omega_{m 0} \approx 0.3) \sim (\Omega_{\Lambda 0} \approx 0.7)$$. The cosmological window of time during which this occurs is relatively narrow. Thus, we are presented with the cosmological coincidence problem: Why, just now, do these energy densities happen to be of the same order? Here we show that this apparent coincidence can be explained as a temporal selection effect produced by the age distribution of terrestrial planets in the Universe. We find a large ($$\sim 68$$ %) probability that observations made from terrestrial planets will result in finding $$\Omega_m$$ at least as close to $$\Omega_{\Lambda}$$ as we observe today. Hence, we, and any observers in the Universe who have evolved on terrestrial planets, should not be surprised to find $$\Omega_m \sim \Omega_{\Lambda}$$. This result is relatively robust if the time it takes an observer to evolve on a terrestrial planet is less than $$\sim 10$$ Gyr."
This is an example of what I would call a good SCIENTIFIC use of the term selection effect. Someone might come and say "ISNT IT SURPRISING that we live right when matter is 27 percent and dark energy is 73 percent, so they are roughly same order of magnitude? What an amazing coincidence!"
Lineweaver says it is not at all surprising, on the contrary, it is VERY LIKELY, based the simple dynamical principles of physics.
Stars take a while to coalesce, planets take time to form, life takes time to evolve, and then stars and planets die.
There are apt to be MORE observer-planets now than there will be much later and that there were much earlier.
They find 68 percent probability that if an observer wakes up somewhere in the universe he will be in a timeframe that puts matter and dark energy at least as close as they are now
So we are finding that such and such that we THOUGHT was an unlikely coincidence is not, after all, uncommon. Thinking it was unusual was a mistake caused by a selection bias, or selection effect.
Notice that the selection effect does not GOVERN physics. The illusion or fallacy of thinking there was an odd circumstance was a selection effect. the physics of stars and biological evolution set us up to have this selection effect delusion. Which Lineweaver and Egan now have dispelled.
=======================
Larry Krauss has a neat paper about how this is a good time to be alive and to be studying the universe.
Because of how dismal the (LambdaCDM) universe will get later on, and how hard to study as a cosmologist. In the future the clues like redshifted galaxies and the CMB will fade out of sight. With all the light around it's not surprising to have eyes.
With all the good energy food around it's not surprising that we are hungry active animals. With all the good information about the universe around it's not surprising we are curious and ask questions.
http://arxiv.org/abs/0704.0221
The Return of a Static Universe and the End of Cosmology
Authors: Lawrence M. Krauss (1,2), Robert J. Scherrer (2) ((1) Case Western Reserve University, (2) Vanderbilt University)
to appear, GRG October 2007;
(Submitted on 2 Apr 2007)
"We demonstrate that as we extrapolate the current LambdaCDM universe forward in time, all evidence of the Hubble expansion will disappear, so that observers in our "island universe" will be fundamentally incapable of determining the true nature of the universe, including the existence of the highly dominant vacuum energy, the existence of the CMB, and the primordial origin of light elements. With these pillars of the modern Big Bang gone, this epoch will mark the end of cosmology and the return of a static universe. In this sense, the coordinate system appropriate for future observers will perhaps fittingly resemble the static coordinate system in which the de Sitter universe was first presented."
Last edited: Aug 10, 2007
4. Aug 10, 2007
### Micha
Marcus,
I think, you are a little bit too strict with Sean here.
Describing the selection effect as an actor for me is like saying, a chess program doing this or that move is following this or that plan. We know, that in reality it is only a computer program executing a certain algorithm, but it is a convenient standpoint. But wait, ins't software just an illusion, and ultimately it is the CPU following the rules of electromagnetism and quantum mechanics?
I think in the end looking at different levels of description only God knows, which perspective is ultimatively the right one, if any. Meanwhile, at a certain time we choose the most convenient one.
I also think, Sean would even agree, that what you describe here, are promising applications of the anthropic principle.
Last edited: Aug 10, 2007
5. Aug 10, 2007
### marcus
Thanks for comment, Micha. Perhaps I am.
But I would not agree that the Lineweaver and Egan paper are any sort of application of the anthropic principle at all (promising or not promising )
Or Larry Krauss paper either---he is just predicting what the old dark universe will be like for cosmologists (dismal, dull, difficult to read)
We are living in a very rich information environoment with the cosmic microwave background and huge numbers of redshifted galaxies. By Krauss paper we should break out the champagne and celebrate. It won't always be this good!
But I do not see that either paper applies any kind of anthropic principle.
they just take careful account of the one sole reality we have.
And they provide no excuse for giving up trying to explain anything that needs explaining!
It sounds as if you liked the two papers that I mentioned. I'm glad. I do too
6. Aug 11, 2007
### Chronos
Sean is a jokester at heart. I think he was poking fun at the dragon.
7. Aug 12, 2007
### Micha
8. Aug 12, 2007
### marcus
thing is, Sean is a charmer and I think maybe he defines science cool for a significant flock of admirers.
people act as if one flash of the boyish smile and they're inclined to forgive him pretty much anything
but lately he seems to have taken over the role of apologist for string and I don't think he really KNOWS the string situation well enough to be good at that, plus my impression is his recent research quality is not as good as it used to be----fewer citable papers.
his current direction worries me. I wish he would stay off the Anthropic Principle----even flirting with this innocent seeming kindergarten version which is just a renaming of some regular physics, but which still is worrisome.
there is enough semantic grease in how he introduces it to make me think it's the "nose of the camel" that one should not let in under the tent.
I wish Sean would forget string/philosophy and concentrate on doing cosmology.
For example, right now I think the leading non-string and non-loop cosmologist is Reuter and there is a big question of where Reuter's universe comes from. The big bang of Reuter's universe is very interesting. Sean could, for instance, take a look at it.
Reuter's preprint of 3 June cited Sean, which he didn't have to---just took the paper as an e.g. Maybe that's an invitation.
Last edited: Aug 12, 2007
9. Aug 13, 2007
### marcus
Here's a nice article by George Ellis
http://arxiv.org/abs/gr-qc/0102017
Cosmology and Local Physics
George F R Ellis
20 pages, Int.J.Mod.Phys. A17 (2002) 2667-2672
(Submitted on 5 Feb 2001)
"This article is dedicated to the memory of Dennis Sciama. It revisits a series of issues to which he devoted much time and effort, regarding the relationship between local physics and the large scale structure of the universe - in particular, Olber's paradox, Mach's principle, and the various arrows of time. Thus the focus is various ways in which local physics is influenced by the universe itself."
It has a nice bit about the various versions of A. P. (including the C. R. type)
I like George Ellis. He co-authored the classic Hawking and Ellis book
and Elsevier asked him to do the Cosmology article for their philosophy of physics Handbook
he's deeper than a lot. thinks very clearly and carefully. really like him.
have to go, back later
============================
REPLY TO TURBO'S NEXT POST---post #10
Turbo, thanks so much for taking Sean's list of "Anthropics" seriously enough to think them over! We owe that to him. I just glanced briefly but it seemed to me that it is just as you say---the first 8 are straightforward. One can derive them from simple dynamical principles WHICH IS WHAT SEAN IMPLIED ONE COULD NOT.
More exactly one can find good physical reasons why one is more LIKELY to find life (however reasonably defined) in those places.
So Sean's post is really dumb and perhaps, as Chronos suggested, it is just a really dumb JOKE. Charming as always, he has now made a big fuss about perverted sexual fetishes, masturbation, scrambled eggs and Entropy---so everything is back into safe territory and it would be uncool to take anything seriously right now.
But even if the floor is covered with bananapeels we should take this stuff seriously once in a while and you did!
BTW you say z = 6.5 and I (courtesy NW Cosmocalculator) translate that to 860 million years after bang----almost a billion years. That agrees with what you say "a few hundred million". You ask about metals that quick. I picture big stars hustling to rustle up the metals. Somebody has to get up early in the morning to light the fire and get the bacon cooking and the coffee boiling. I feel I just don't know enough about early structure and starformation to discuss with you. I can only just sort of register what you say.
======================
OFF-TOPIC COMMENT EDIT:
http://www.sciam.com/article.cfm?chanId=sa003&articleId=4C44B8D3-E7F2-99DF-3D03439C08E85464
SOME GOOD NEWS ABOUT FERRETS
north american blackfooted ferret had a close call but is coming back (PF news clipping)
Last edited: Aug 14, 2007
10. Aug 13, 2007
### turbo
Marcus, I've been mulling this over and it seems that all but the last example in Sean's list can be explained by our nature. We are complex carbon-based life forms that arose from precursors that seemed to require the existence (at least part of the time) of water in its liquid phase. Our existence relies on the availability of a metal-enriched environment, so a galactic crucible is required and the necessity for liquid water implies that we must live on a rocky planet with a protective atmosphere. As for complex life being formed from theoretical entities like DM and DE, those are not viable arguments for any form of anthropic principle. Same with the example of why we are not made of photons or hydrogen and helium. Photons are a great way to transfer energy, but it is difficult to envision them forming a coherent structure naturally, much less coalescing into something that even resembles life.
The last example is based on the assumption that the BB theory is viable, but as we push back to higher and higher redshifts, observations are providing severe constraints on that theory. Here is is a presentation given by Michael Strauss (science spokesperson for the SDSS team) at the Space Telescope Science Institute. If the Universe is only 13.8 billion years old, there are some SDSS obvservations that need to be explained.
1) Quasars at z~6.5 exhibit Solar or super-Solar metallicities. How did these metals form and accrete only a few hundred million years after the BB?
2) Quasars at high redshift show no evolution in relative or absolute metallicity with redshift. They also show no evolution in any other quality that the SDSS team could measure. This was not expected, as the BB theory requires an early era in which metallicities are low and evolve over time. Strauss points out that since metals are expected to be generated from different time-related processes, some evolution in relative metallicity should be evident at this epoch.
3) Since luminosity falls off as a function of the square of the distance to the emitter, if quasars are accreting BHs, they (the z~6.5 quasars) most be comprised of BHs of up to ten billion Solar masses consuming host galaxies of up to several trillion Solar masses. If structure in our universe forms through gravitational accretion, where are the hyper-massive galaxies and hyper-massive BHs at lower redshifts? Did they simply evaporate or disintegrate? It may be time to reconsider the idea that Gamow was wrong and that the universe may be spacially and temporally infinite.
Click the link and scroll down to Nov 2.
http://www.stsci.edu/institute/itsd/information/streaming/archive/STScIScienceColloquiaFall2005/
This presentation should be required viewing for every cosmological theorist.
Edited link, corrected a misspelled word, and added a parenthetical clarification regarding the hypermassive quasar puzzle in Strass' presentation.
Last edited: Aug 14, 2007
11. Aug 14, 2007
### ccdantas
Hi turbo-1,
Excellent points. I would not like to comment on that post by Sean Carroll so I'm glad that you did it here. I can only say that, at least *my* approach is -- the so many open questions in cosmology, as hard as they may seem, are to be addressed by PHYSICS. I see no compelling reason to give up on good old science and appeal for anthropic arguments in order to reason about those problems.
BTW the link you mention does not find Michael Strauss' presentation...
Best,
Christine
12. Aug 14, 2007
### turbo
Thank you, Christine. I have fixed the link. STSI apparently restructured their archives, so my old bookmark was out of date. I have watched Strauss' presentation at least a dozen times - it's a great talk.
Like you, I have a hard time understanding how people can appeal to the anthropic principle and hope to gain any insight into the fundamental questions facing us. Let us imagine that today is your birthday and you were born 30 years ago, and as you prepare to board a bus to work, you notice that the license plate on the bus reads "081477". If that happened to me, I would probably just smile and say "what a neat coincidence" - I certainly would not attribute enough significance to the event to bother trying to calculating the odds against that coincidence. Anthropic arguments regarding the "coincidence" of our existence seem equally empty to me.
Last edited: Aug 14, 2007
13. Aug 14, 2007
### turbo
I almost missed your comment in that time-warp edit. Whether or not Sean was serious when he posted that list of "coincidences", other prominent theorists have dabbled with the AP, too, so I at least try to give their arguments some consideration. So far, I haven't been swayed by any version of the AP.
As noted above, I fixed the link to Michael Strauss' presentation at the STSI, and if you've got the bandwidth to stream video, it is a must-see. His description of SDSS's quasar-search methodology is very straightforward and interesting, but the real treats are in the questions that the SDSS observations raise. He explains why the observations are difficult to explain in the framework of BB cosmology, and he does so in terminology that is easy to grasp.
14. Aug 14, 2007
### ccdantas
Hi Turbo-1!
Thanks for fixing the link, I'll take a look opportunely.
Concerning your view on the anthropic principle, I agree with you 100%.
Ha ha... You see: for a coincidence (inside a coincidence), in your random example you missed the right date for just two wrong digits. The day is correct, the month is 5 integers ahead and the year is wrong by one digit. I'd almost look for an anthropic argument for this...
In fact my birthdate is the same as Einstein's (14 March). Surely there must be an anthropic explanation for this as well...:tongue2:
15. Aug 14, 2007
### turbo
Oh, no! Einstein died on my 3rd birthday! (no kidding) See how the universe has conspired to tease us. I don't recall being particularly concerned at the time, though. :rofl:
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# Chapter 8 Dependent Samples t-Test
The dependent samples t-test is essentially the same analysis as the one sample t-test but on a difference score.
For example, let’s say that we were interested in determining if the weight of female patients with anorexia changed before the study compared to after the study.
For this example, we will be using the datasetAnorexia dataset.
We use a difference score because the measures of weight are dependent on (or relate to) each other as they come from the same individual. In other words, weights from the same individual are more likely to be closer in value than weights between individuals. If we do not take this dependence (i.e., positive dependence) into account, the t-value will be artificially inflated (or higher than it should be). Thus, we are also more likely to artificially reduce the p-value and ultimately commit a Type I error.
In other cases, not properly accounting for the dependence of measures can artificially reduce the t-value if the measures are negatively dependent. Measuring responsibility of household chores in couples is an example of negative dependence because as one couple rates their household chore responsbility as high or low, the other couple will typically rate the opposite. In other words, the measures of responsibility of household chores are more likely to be different within couples than the responsibility of household chores between couples. The artificially reduced t-value would increase the p-value and are ultimately likely to commit a Type II error.
## 8.1 Null and research hypotheses
### 8.1.1 Traditional approach
$H_0: \mu_{WeightDifference} = 0$ $H_1: \mu_{WeightDifference} \ne 0$
The null hypothesis states that there is no weight difference in female patients with anorexia before and after the study. The research hypothesis states there is a weight difference in female patients with anorexia before and after the study.
### 8.1.2 GLM approach
$Model: WeightDifference = \beta_0 + \varepsilon$ $H_0: \beta_0 = 0$ $H_1: \beta_0 \ne 0$
where $$\beta_0$$ represents the intercept and $$\varepsilon$$ represents the error
Just like in the one sample t-test, the intercept ($$\beta_{0}$$) represents the sample mean. However, in this case, the sample mean is the mean of the weight difference of the female patients with anorexia before and after the study. Thus, the intercept here is testing if the sample mean of the weight difference score is significantly different than 0 with identical null and research hypotheses.
## 8.2 Statistical analysis
### 8.2.1 Traditional approach
To perform the traditional dependent samples t-test, we can again use the t.test() function. However, we will also set the paired option to TRUE.
t.test(datasetAnorexia$PostWeight, datasetAnorexia$PreWeight, paired = TRUE)
##
## Paired t-test
##
## data: datasetAnorexia$PostWeight and datasetAnorexia$PreWeight
## t = 2.9376, df = 71, p-value = 0.004458
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.8878354 4.6399424
## sample estimates:
## mean of the differences
## 2.763889
From this output, we can see that the t-statistic (t) is 2.9376, degrees of freedom (df) is 71, and the p-value is 0.004458.
### 8.2.2 GLM approach
For the DV, we will input the difference of weight before and after treatment (i.e., PostWeight-PreWeight) directly into the lm() function. Since we are testing the intercept, we will again place 1 as the predictor.
model <- lm(PostWeight - PreWeight ~ 1, datasetAnorexia)
summary(model)
##
## Call:
## lm(formula = PostWeight - PreWeight ~ 1, data = datasetAnorexia)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.964 -4.989 -1.114 6.336 18.736
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.7639 0.9409 2.938 0.00446 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.984 on 71 degrees of freedom
Notice that in both analyses the sample mean difference of 2.76, the t-statistic of 2.94 with 71 degrees of freedom (df), and the p-value of .004 are identical with respective to place of rounding.
In this case, the systematic difference of the t-statistic is the difference in weight before and after a study and the unsystematic difference is standard error of the mean of the differences (in other words, random variability of the difference scores).
In this case, the probability of finding a t-statistic of 2.94 or more extreme is .004, which is very small and not likely to occur by chance if the null hypothesis was true.
## 8.3 Statistical decision
Given the p-value of 0.004 is smaller than the alpha level ($$\alpha$$) of 0.05, we will reject the null hypothesis.
## 8.4 APA statement
A dependent samples t-test was performed to test if female patients with anorexia had changed their weight before and after the study. The female patients with anorexia significantly gained weight after the study (M = 85, SD = 8) compared to before the study (M = 82, SD = 5), t(71) = 2.94, p = .004.
## 8.5 Visualization
# calculate descriptive statistics along with the 95% CI
dataset_summary <- datasetAnorexia %>%
summarize(
mean = mean(PostWeight - PreWeight),
sd = sd(PostWeight - PreWeight),
n = n(),
sem = sd / sqrt(n),
tcrit = abs(qt(0.05 / 2, df = n - 1)),
ME = tcrit * sem,
LL95CI = mean - ME,
UL95CI = mean + ME
)
ggplot(datasetAnorexia, aes("", PostWeight - PreWeight)) +
geom_hline(yintercept = 0, alpha = .1, linetype = "dashed") +
geom_jitter(alpha = 0.25, width = 0.02) +
geom_errorbar(data = dataset_summary, aes(y = mean, ymin = LL95CI, ymax = UL95CI), width = 0.01, color = "#3182bd") +
geom_point(data = dataset_summary, aes("", mean), size = 3, color = "#3182bd") +
labs(x = 0, y = "Weight Difference (lbs)") +
theme_classic()
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Browse Questions
# The fraction of volume occupied in a diamond shaped cubic cell is:
The atoms in a simple cubic crystal are located at the corners of the units cell, a cube with side $a$.
For a diamond shaped crystal, the radius = $\large\frac{\sqrt 3 a }{8}$
Note: In this type of structure, total number of atoms is 8 per unit cell.
Packing Density $= \large\frac{\text{Volume of atoms}}{\text{Volume of unit cell}}$
Volume of atoms $=8 \large\frac{4}{3}$$\pi r^3 and Volume of unit cell = a^3 Substituting r = \large\frac{\sqrt 3 a }{8}, we get: Packing Density = \Large \frac{ 8 \frac{4\pi}{3} (\frac{\sqrt 3 a }{8})^3}{a^3}$$ = \large\frac{\pi \sqrt 3}{16}$$= 0.34$
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# garch and the distribution of returns
April 22, 2013
By
(This article was first published on Portfolio Probe » R language, and kindly contributed to R-bloggers)
Using garch to learn a little about the distribution of returns.
## Previously
There are posts on garch — in particular:
There has also been discussion of the distribution of returns, including a satire called “The distribution of financial returns made simple”.
## Question
Volatility clustering affects the distribution of returns — the high volatility periods make the returns look longer tailed than if we take the volatility clustering into account.
The question we try to sneak up on is: What is the distribution of returns when volatility clustering is accounted for? More specifically, what is the distribution using a reasonable garch model?
## Data
Daily log returns of 443 large cap US stocks with histories from the start of 2004 into the first few days of 2013 were used. There were 2267 days of returns for each stock.
The components garch model assuming a t distribution was fit to each stock.
## Results
### Actual results
The estimated degrees of freedom for the stocks is shown in Figure 1. Estimation failed for one stock.
Figure 1: Estimated degrees of freedom for the t in the components garch model for the US large cap stocks. A lot of the estimates are close to 6 and hardly any (about 5%) are greater than 10.
### Normal distribution
Next was to see what degrees of freedom are estimated if the residuals are normally distributed. The fits from the first three stocks were used — separately — to simulate series (200 each) and then fit the model to those simulated series.
Figure 2 shows the distributions of the estimated degrees of freedom from the three fits.
Figure 2: Distributions of estimated degrees of freedom using the initial simulated series from three fits. I’m hoping that we can all agree that the normal distribution can be ruled out as the actual distribution. There is virtually no overlap between the distribution in Figure 1 and those in Figure 2.
It appears that the black distribution in Figure 2 is different from the other two — that it has a much higher probability of being less than 80. But the distributions are based on only 200 series each; there will be noise. A further 1000 series were generated for each fit to see if the pattern persists with new, more extensive data. Figure 3 shows the results.
Figure 3: Distributions of estimated degrees of freedom using the confirmatory simulated series from three fits. The distributions do indeed appear to be different — the black one has just over 50% probability of being less than 80 while the other two have less than 40%.
This means that we can’t assume that the estimate of the degrees of freedom is independent of the other parameters.
### t distribution
The next test was — for each stock — simulate a series using the coefficients estimated for the stock but with residuals following a t distribution with 6 degrees of freedom. The distribution of estimated degrees of freedom from fitting the simulated series is shown in Figure 4.
Figure 4: Distribution of estimated degrees of freedom for series created with a t with 6 degrees of freedom and coefficients specific to each stock. This is much too centered near 6 compared to Figure 1. About 20% of the real estimated degrees of freedom are smaller than the smallest degree of freedom estimated from the t6 simulations. Likewise, about 10% of the real degrees of freedom are bigger than the biggest from the t6 simulations.
I propose two (not necessarily mutually exclusive) possibilities:
• Different stocks have different distributions
• The real distribution is not t and the degrees of freedom are not a good approximation
## Summary
We’ve shown — once again — that the normal distribution is not believable.
We’ve also highlighted our ignorance about the true situation.
## Appendix R
The R language was used to do the computing.
#### collect coefficients from the model on stocks
We start by making sure the rugarch package is loaded in the session, create the specification object that we want, and fit garch models to the first three stocks:
require(rugarch)
comtspec <- ugarchspec(mean.model=list(
armaOrder=c(0,0)), distribution="std",
variance.model=list(model="csGARCH"))
fit1 <- ugarchfit(spec=comtspec, initret[,1])
fit2 <- ugarchfit(spec=comtspec, initret[,2])
fit3 <- ugarchfit(spec=comtspec, initret[,3])
The initret object is the matrix of returns. Next we create a matrix to hold the coefficients for all of the stocks:
gcoefmat <- array(NA, c(443, length(coef(fit1))),
list(colnames(initret),names(coef(fit1))))
That the matrix is initialized with missing values is not accidental. We’re not guaranteed that the default estimation will work. Now we’re ready to collect the coefficients:
for(i in 1:443) {
thiscoef <- try(coef(ugarchfit(spec=comtspec, initret[,i])))
if(!inherits(thiscoef, "try-error") && length(thiscoef) == 7) {
gcoefmat[i,] <- thiscoef
}
cat("."); if(i %% 50 == 0) cat("\n")
}
cat("\n")
This takes evasive action in case we don’t get a vector of coefficients like we expect — it uses the try function to continue going even if there is an error. It also gives an indication of how far along the loop is (since fitting over 400 garch models is not especially instantaneous).
#### simulate garch from a specific distribution
Here is a function that will take a garch fit object and simulate a number of series using residuals from a particular distribution:
pp.garchDistSim <- function(gfit, FUN, trials=200,
burnIn=100, ...)
{
# simulate a garch model given a fit
# placed in the public domain 2013 by Burns Statistics
# testing status: untested
ntimes <- gfit@model$modeldata$T
FUN <- match.fun(FUN)
inresid <- scale(matrix(FUN((ntimes+burnIn) * trials,
...), ntimes+burnIn, trials))
simseries <- ugarchsim(gfit, n.sim=ntimes,
n.start=burnIn, m.sim=trials,
startMethod="sample", custom.dist=list(name="sample",
distfit = inresid))@simulation$seriesSim simseries } The residuals that are given to the simulation function are scaled (with the scale function) to have zero mean and variance 1 because the routine expects the residuals to be distributed like that. It can get grumpy if that is obviously not the case. #### simulate with normals The function above is used like: gsnorm1 <- pp.garchDistSim(fit1, rnorm, trials=200) gsnorm2 <- pp.garchDistSim(fit2, rnorm, trials=200) gsnorm3 <- pp.garchDistSim(fit3, rnorm, trials=200) Once we have these series, we can fit the garch model to them and save the coefficients: gsnorm1.coefmat <- array(NA, c(200, 7), list(NULL, names(coef(fit1)))) gsnorm2.coefmat <- gsnorm3.coefmat <- gsnorm1.coefmat for(i in 1:200) { thiscoef <- try(coef(ugarchfit(spec=comtspec, gsnorm1[,i]))) if(!inherits(thiscoef, "try-error") && length(thiscoef) == 7) { gsnorm1.coefmat[i,] <- thiscoef } cat("."); if(i %% 50 == 0) cat("\n") } cat("\n") #### difference of distributions from normal We can get the confidence interval for the proportion of the black distribution from the confirmatory normal estimates being less than 80 by using the binom.test function: > binom.test(sum(gsnorm2M.coefmat[,7] < 80, na.rm=TRUE), + sum(!is.na(gsnorm2M.coefmat[,7]))) Exact binomial test data: sum(gsnorm2M.coefmat[, 7] < 80, na.rm = TRUE) and sum(!is.na(gsnorm2M.coefmat[, 7])) number of successes = 489, number of trials = 963, p-value = 0.6519 alternative hypothesis: true probability of success is not equal to 0.5 95 percent confidence interval: 0.4757106 0.5398179 sample estimates: probability of success 0.5077882 That the binomial test sees 963 observations means that 37 of the 1000 series didn’t produce an appropriate vector of coefficients using the default fitting settings. The stock that is the odd one out (the black one) is ABT. The other two are MMM and ANF. The fit for ABT is significantly more persistent than for the other two (but experimentation is needed to understand what is happening). #### simulate given a garch specification Here is a function that simulates series given a garch specification: pp.gspecDistSim <- function(spec, FUN, ntimes, trials=1, burnIn=100, ...) { # simulate a garch model given a specification # placed in the public domain 2013 by Burns Statistics # testing status: untested FUN <- match.fun(FUN) inresid <- scale(matrix(FUN((ntimes+burnIn) * trials, ...), ntimes+burnIn, trials)) simseries <- ugarchpath(spec, n.sim=ntimes, n.start=burnIn, m.sim=trials, startMethod="sample", custom.dist=list(name="sample", distfit = inresid))@path$seriesSim
simseries
}
#### paired simulation with a t
The function immediately above was used like:
gst6specific.coefmat <- array(NA, c(443, 7),
dimnames(gcoefmat))
for(i in 1:443) {
thisspec <- comtspec
stockcoef <- gcoefmat[i,]
if(any(is.na(stockcoef))) next
setfixed(thisspec) <- as.list(stockcoef)
simser <- pp.gspecDistSim(thisspec, rt, ntimes=2267,
trials=1, df=6)
thiscoef <- try(coef(ugarchfit(spec=comtspec, simser)))
if(!inherits(thiscoef, "try-error") && length(thiscoef) == 7) {
gst6specific.coefmat[i,] <- thiscoef
}
cat("."); if(i %% 50 == 0) cat("\n")
}
cat("\n")
Note that the call to pp.gspecDistSim in this loop is using R’s dot-dot-dot mechanism.
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## Exercise ‹15:
Judging exercises on CFGs
The coordinator of the «Theory of Computation» subject in our university wants to create a website that automatically evaluates exercises on context-free grammars (CFGs). The idea is that, for each exercise, the teacher simply prepares a reference solution CFG $G_1$, and then, any CFG $G_2$ submitted by a student is automatically compared against $G_1$ to determine whether they are equivalent (i.e., whether $G_1$ and $G_2$ generate the same language). Unfortunately, it is not possible to create such an evaluator, as equivalence of CFGs is an undecidable problem. Nevertheless, it is possible to implement a less ambitious automatic evaluator as follows: for each length $\ell$ (up to a certain maximum), test whether there exists a word $w$ with length $\ell$ that is generated by one of $G_1,G_2$ but not by the other. Clearly, when such word $w$ exists, then $w$ is a counterexample to the equivalence of $G_1$ and $G_2$. When no such $w$ exists, either $G_1$ and $G_2$ are indeed equivalent, or we did not test with an $\ell$ big enough to find a counterexample. There are many possible ways to implement an algorithm that, given $G_1,G_2$ and $\ell$, looks for such counterexample $w$; here we consider one possible technique.
Solve the following exercise by means of a reduction to SAT:
• Given a natural $\ell>0$ and two CFGs $G_1,G_2$ over an alphabet $\Sigma$, determine whether there exists a word $w\in\Sigma^*$ with length $\ell$ that is generated by one of the grammars but not by the other. We call such $w$ the counterexample to the equivalence of $G_1$ and $G_2$.
To simplify the setting, we assume that the grammars are normalized in the following sense: all non-terminal symbols are useful (i.e., they generate at least one word, and they can be reached from the start symbol of the grammar), and all the production rules are either of the form $X\to YZ$ or $X\to a$, where $X,Y,Z$ are non-terminal symbols, and $a$ is a terminal symbol in $\Sigma$. Note that a CFG normalized like that cannot generate the empty language or the empty word, and that any production rule of the form $X\to YZ$ generates words with size at least $2$.
The input of the exercise and the output with the solution (when the input is solvable) are as follows:
• grammars: array [2] of array of array of array of int
length: int
numterminals: int
The input contains the two normalized grammars that must be compared (one in grammars[0] and the other one in grammars[1]), the specific length${}>0$ that the counterexample must have, and the size numterminals${}>0$ of the alphabet $\Sigma$ of terminal symbols. All symbols are represented by numbers: non-terminal symbols are represented by non-negative numbers $0,1,\ldots$ (where $0$ is always used for the start symbol of a grammar), and terminal symbols are represented by negative numbers $-1,-2,\ldots,-{}$numterminals. For each g${}\in\{0,1\}$, the grammar grammars[g] is an array of arrays of arrays of integers with the following meaning: for each non-terminal nt of the grammar, grammars[g][nt] contains the list of all the right-hand sides of the production rules of the form nt${}\to YZ$ or nt${}\to a$, and thus, for the i’th of such right-hand sides (counting from $0$), grammars[g][nt][i] is an array with either $2$ non-terminal symbols (for a case like nt${}\to YZ$) or $1$ terminal symbol (for a case like nt${}\to a$). For example, given a normalized grammar $S\to AB|a,\;A\to a,\,B\to b$, with $S$ being the start symbol, we could use the encoding $S=0$, $A=1$, $B=2$, $a=-1$, $b=-2$ and represent the grammar as follows:
Original Representation
----------- --------------
S -> AB | a [[[1 2] [-1]]
A -> a [[-1]]
B -> b [[-2]]]
Note that the left-hand sides of the production rules are implicitly represented by the index into the outermost array (in this case, indexes $0$, $1$, $2$, that represent $S$, $A$, $B$, respectively), that each of the items in this outermost array is a list with the right-hand sides that correspond to that implicit left-hand side, and that each of such right-hand sides is either a list of $2$ non-terminal symbols (like [1 2] in the example, that represents $AB$), or $1$ terminal symbol (like [-1] and [-2] in the example, that represent $a$ and $b$, respectively).
• counterexample: array of int
The output is the counterexample that proves that the grammars are not equivalent (for words of the specified length), i.e., a word that is generated by one of the grammars but not by the other. Such word must be represented with an array with size length whose elements are terminal symbols among $-1,-2,\ldots,-{}$numterminals.
Authors: Carles Creus / Documentation:
reduction { // Write here your reduction to SAT... } reconstruction { // Write here your solution reconstruction... } To be able to submit you need to either log in, register or become a guest.
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In [2]:
X,y = make_circles(n_samples = 3000, noise = 0.08, factor=0.3)
#X,y = make_moons(n_samples = 3000, noise = 0.08)
X0 = []
X1 = []
for i in range(len(y)):
if y[i] == 0:
X0.append(X[i,:])
else:
X1.append(X[i,:])
X0_np = np.array(X0)
X1_np = np.array(X1)
X0_train = X0_np[:1000,:].T # we want X to be made of examples which are stacked horizontally
X0_test = X0_np[1000:,:].T
X1_train = X1_np[:1000,:].T
X1_test = X1_np[1000:,:].T
X_train = np.hstack([X0_train,X1_train]) # all training examples
y_train=np.zeros((1,2000))
y_train[0, 1000:] = 1
X_test = np.hstack([X0_test,X1_test]) # all test examples
y_test=np.zeros((1,1000))
y_test[0, 500:] = 1
plt.scatter(X0_train[0,:],X0_train[1,:], color = 'b', label = 'class 0 train')
plt.scatter(X1_train[0,:],X1_train[1,:], color = 'r', label = 'class 1 train')
plt.scatter(X0_test[0,:],X0_test[1,:], color = 'LightBlue', label = 'class 0 test')
plt.scatter(X1_test[0,:],X1_test[1,:], color = 'Orange', label = 'class 1 test')
plt.xlabel('feature1')
plt.ylabel('feature2')
plt.legend()
plt.axis('equal')
plt.show()
# we will plot shapes of training and test set to make sure that they were made by stacking examoles horizontally
# so in every column of these matrices there is one examples
# in two rows there are features (feature1 is ploted on the x-axis, and feature2 is ploted on the y-axis)
print('Shape of X_train set is %i x %i.'%X_train.shape)
print('Shape of X_test set is %i x %i.'%X_test.shape)
# labels for these examples are in y_train and y_test
print('Shape of y_train set is %i x %i.'%y_train.shape)
print('Shape of y_test set is %i x %i.'%y_test.shape)
Shape of X_train set is 2 x 2000.
Shape of X_test set is 2 x 1000.
Shape of y_train set is 1 x 2000.
Shape of y_test set is 1 x 1000.
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# Part 3.4Forms in PHP
## Forms
A form is almost exactly as it sounds - information is completed by a user into fields and it is then processed by PHP.
There is a another option of doing this known as AJAX, but it will be covered in the JavaScript tutorial on this website.
PHP pages can be used as simply using code and a redirect without the need for any HTML or output.
Below is a small form which has no submit method.
This is a form.
You can select one of the following options and fill in certain parts of it.
Name:
Experience with PHP:
None
Lots
Are you happy to submit your data to third-parties?
The form above has no action therefore it will not do anything.
The code for that form is shown below:
HTML
<!--This form uses the Epic Form class on this website.-->
<form class="epic">
<p>This is a form.</p>
<p>
You can select one of the following
options and fill in certain parts of it.
</p>
<p>Name: <input name="name" type="text" style="width:200px;"></p>
<p>Experience with PHP:</p>
<p>Text can be included in a form.</p>
<p><input name="Submit1" type="submit" value="Submit"></p>
</form>
The input element is used to collect an input whilst the type attribute specifies what type of input will be provided, for instance, checkbox, radio button etc.
HTML
<!--This will result in address being passed to the page sample.php-->
<form action="sample.php" style="border:thin;">
<input name="Submit1" type="submit" value="Submit">
</form>
This means that when the user submit the form, the contents of the form input values will be passed to the page sample.php.
### Receiving the form at the other end
One of the fundamentals of PHP and forms is extracting the information which has been given to it. This can be achieved using the $_POST["inputName"] method where the inputName is replaced with the name of the form input that we want the value of. This example will look at using both the $_POST["inputName"] variable and the echo command.
So in the previous example, in results to the form shown previously, the following code will the name that has been submitted:
HTML + PHP
<p>
<?php
echo $_POST["name"]; ?> </p> This code is taking the form value from the input called "name". In this sample, it is being put inside a paragraph element. Alternatively to the $_POST["inputName"] method, there is a $_GET["inputName"] method. The $_GET variable (it is actually an associative array variable, but this will be discussed later) can be used to obtain URL variables such as those that follow a question mark in a query string in a URL, for example: page.php?p=1277.
## Working with checkboxes
When processing a form on the server-side, it is often the case you want to work with check boxes (input type="checkbox") and radio buttons (input type="radiobutton"). When working with checkboxes things are a bit different to standard inputs.
This is because if a checkbox is not ticked on the front-end, the value is not actually submitted at all. Trying to access the value is therefore not possible.
To fix this, the isset function is used:
HTML + PHP
<p>
<?php
if (isset($_POST["submit_your_data"])) { echo "You have agreed to allow us to submit your data to third-parties!"; } else { echo "Why didn't you agree to allow us to submit your data to third-parties?!"; } ?> </p> If the isset is not used to verify that the key actually exists, the PHP interpreter will likely throw an error (unless errors are switched off) and the value retrieved from the $_POST associative array variable will be null.
Code preview
Feedback 👍
Comments are sent via email to me.
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# Correlated belief update: Is this understanding of Bayesian posterior wrong
I am reading this paper Knowledge-Gradient Policy for Correlated Normal Beliefs for Rank and Selection Problem. The idea is as follows: We have $$M$$ distinct alternatives and samples from alternative $$i$$ are iid with $$\mathcal{N}(\theta_i,\lambda_i)$$, where $$\theta_i$$ is unknown and $$\lambda_i$$ is the known variance. At every time step, we can sample from only one of the arms and at the end of $$N$$ timesteps, we pick the alternative that we think that has the highest mean reward.
More concretely using the notation and text in the paper,
Let $$\mathbf \theta = (\theta_1, \dots, \theta_M)'$$ be column vector of unknown means. We initially assume our belief about $$\bf \theta$$ as:
\begin{align} \bf \theta \sim \mathcal{N}(\mu^0,\Sigma^0) \label{1} \tag{1} \end{align}
Consider a sequence of $$N$$ sampling decisions, $$x^{0}, x^{1}, \ldots, x^{N-1} .$$ The measurement decision $$x^{n}$$ selects an alternative to sample at time $$n$$ from the set $$\{1, \ldots, M\}$$. The measurement error $$\varepsilon^{n+1} \sim \mathcal{N}\left(0, \lambda_{x^{n}}\right)$$ is independent conditionally on $$x^{n}$$, and the resulting sample observation is $$\hat{y}^{n+1}=\theta_{x^{n}}+\varepsilon^{n+1}$$. Conditioned on $$\theta$$ and $$x^{n}$$, the sample has conditional distribution $$\hat{y}^{n+1} \sim \mathcal{N}\left(\theta_{x^{n}}, \lambda_{x^{n}}\right)$$. Note that our assumption that the errors $$\varepsilon^{1}, \ldots, \varepsilon^{N}$$ are independent differentiates our model from one that would be used for common random numbers. Instead, we introduce correlation by allowing a non-diagonal covariance matrix $$\Sigma^{0}$$.
We may think of $$\theta$$ as having been chosen randomly at the initial time 0 , unknown to the experimenter but according to the prior distribution (1), and then fixed for the duration of the sampling sequence. Through sampling, the experimenter is given the opportunity to better learn what value $$\theta$$ has taken.
We define a filtration $$\left(\mathcal{F}^{n}\right)$$ wherein $$\mathcal{F}^{n}$$ is the sigma-algebra generated by the samples observed by time $$n$$ and the identities of their originating alternatives. That is, $$\mathcal{F}^{n}$$ is the sigma-algebra generated by $$x^{0}, \hat{y}^{1}, x^{1}, \hat{y}^{2}, \ldots, x^{n-1}, \hat{y}^{n} .$$ We write $$\mathbb{E}_{n}$$ to indicate $$\mathbb{E}\left[\cdot \mid \mathcal{F}^{n}\right]$$, the conditional expectation taken with respect to $$\mathcal{F}^{n}$$, and then define $$\mu^{n}:=\mathbb{E}_{n}[\theta]$$ and $$\Sigma^{n}:=\operatorname{Cov}\left[\theta \mid \mathcal{F}^{n}\right]$$. Conditionally on $$\mathcal{F}^{n}$$, our posterior predictive belief for $$\theta$$ is multivariate normal with mean vector $$\mu^{n}$$ and covariance matrix $$\Sigma^{n} .$$
We can obtain the updates of $$\mu^{n}$$ and $$\Sigma^{n}$$ as functions of $$\mu^{n-1}, \Sigma^{n-1}, \hat{y}^{n}$$, and $$x^{n-1}$$ as follows:
\begin{align} \mu^{n+1} &=\mu^{n}+\frac{\hat{y}^{n+1}-\mu_{x}^{n}}{\lambda_{x}+\sum_{x x}^{n}} \Sigma^{n} e_{x} \tag{2} \label{2}\\\ \Sigma^{n+1} &=\Sigma^{n}-\frac{\Sigma^{n} e_{x} e_{x}^{\prime} \Sigma^{n}}{\lambda_{x}+\Sigma_{x x}^{n}} \tag{3} \label{3} \text { where } e_{x} \text { is a column } M \text { -vector of } 0 \text { s with a single } 1 \text { at index } x \end{align}
My question:
The authors claim that in equation \ref{2}, $$\hat{y}^{n+1}-\mu_{x}^{n}$$ when conditioned on $$\mathcal{F}^n$$ has zero mean ; this claim seems wrong to me. My understanding is that $$\hat{y}^{n+1}$$ still follows $$\mathcal{N}(\theta^*_x,\lambda_x)$$ where $$\theta^*_x$$ is some realisation sampled from $$\mathcal{N}(\mu^0_x,\Sigma^0_{xx})$$ and this true $$\theta^{*}_x$$ need not be the same as $$\mu^n_{x}$$.
On the basis of this claim, the authors design an algorithm and prove some theoretical results. This is a widely cited paper and so, I think I am missing something here with respect to bayesian setting and posterior distributions.
The setup seems to imply that $$(\theta_1,\ldots\theta_M)$$ are only sampled once. $$\theta_x$$ for given $$x$$ will always be the same regardless of $$n$$. There is no $$\theta^*_x$$ different from the original $$\theta_x$$. Note that there is a certain confusion of notation between random variables and their realisations (which is often found in Bayesian notation). I denote by $$\tilde\theta_x$$ the random variable (over which the prior is defined) that has taken the value $$\theta_x$$.
In fact, $$E(\hat y^{n+1}|\tilde\theta_x=\theta_x)=\theta_x$$. Now we don't know $$\theta_x$$ at this point, but we know what we expect it to be: $$E(\hat y^{n+1}|\mathcal{F}^n)=E_n(\tilde\theta_x)=\mu_x^n$$ (it looks like you use $$x=x^n$$ for ease of notation), therefore $$E(\hat y^{n+1}-\mu_x|\mathcal{F}^n)=0$$.
• "..but we know what we expect it to be: $E(\hat y^{n+1}|\mathcal{F}^n)=E_n(\tilde\theta_x)=\mu_x^n$ " - actually, this statement is causing my confusion and the one I am claiming as incorrect. Because, the truth is that $E(\hat y^{n+1}|\mathcal{F}^n) = \theta_x$. We don't know $\theta_x$ yet but it does not mean that it is equal to $\mu^n_x$. With the help of the information until $n$, the best we can do is to refine our prior distribution to a posterior distribution with mean $\mu^n_x$. Jul 21 '21 at 19:05
• $\theta_x$ is not a (frequentist) constant (parameter), but a realisation of a random variable. In Bayesian reasoning, everything unknown is modelled as random. As long as $\theta_x$ is unknown, it has a distribution (or rather, to be precise, the random variable $\tilde\theta_x$ modelling its uncertainty) and an expected value, which at this point, given the information in $\mathcal{F}^n$, is $\mu_x^n$. Jul 21 '21 at 21:15
• $E(\hat y^{n+1}|\mathcal{F}^n)$ is not $\theta_x$, because it is our expectation given $\mathcal{F}^n$, which cannot be $\theta_x$ as we don't know what that is. Only conditionally on $\tilde\theta_x=\theta_x$ it is $\theta_x$. Without this condition it is $\mu_x^n$. Jul 21 '21 at 21:20
• Note by the way that the setup and even at times my own discussion of it seems to imply that there is a true but unknown value of $\theta_x$, but there is no postulation that this is indeed the case in reality. Rather it is a modelling device, meaning that part of the formalisation of our uncertainty is that the situation can be thought of as if there were such a value. Jul 22 '21 at 11:15
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## Physics
### Geometrical picture of photocounting measurements. (arXiv:1801.02677v2 [quant-ph] UPDATED)
arXiv.org: Quantum Physics - Fri, 2018-03-02 07:32
We revisit the representation of generalized quantum observables by establishing a geometric picture in terms of their positive operator-valued measures (POVMs). This leads to a clear geometric interpretation of Born's rule by introducing the concept of contravariant operator-valued measures. Our approach is applied to the theory of array detectors, which is a challenging task as the finite dimensionality of the POVM substantially restricts the available information about quantum states. Our geometric technique allows for a direct estimation of expectation values of different observables, which are typically not accessible with such detection schemes. In addition, we also demonstrate the applicability of our method to quantum-state reconstruction with unbalanced homodyne detection.
Categories: Journals, Physics
### Driven-Dissipative Supersolid in a Ring Cavity. (arXiv:1801.00756v2 [cond-mat.quant-gas] UPDATED)
arXiv.org: Quantum Physics - Fri, 2018-03-02 07:32
Supersolids are characterized by the counter-intuitive coexistence of superfluid and crystalline order. Here we study a supersolid phase emerging in the steady state of a driven-dissipative system. We consider a transversely pumped Bose-Einstein condensate trapped along the axis of a ring cavity and coherently coupled to a pair of degenerate counter-propagating cavity modes. Above a threshold pump strength the interference of photons scattered into the two cavity modes results in an emergent superradiant lattice, which spontaneously breaks the continuous translational symmetry towards a periodic atomic pattern. The crystalline steady state inherits the superfluidity of the Bose-Einstein condensate, thus exhibiting genuine properties of a supersolid. A gapless collective Goldstone mode correspondingly appears in the superradiant phase, which can be non-destructively monitored via the relative phase of the two cavity modes on the cavity output. Despite cavity-photon losses the Goldstone mode remains undamped, indicating the robustness of the supersolid phase.
Categories: Journals, Physics
### Spatially-resolved control of fictitious magnetic fields in a cold atomic ensemble. (arXiv:1712.07747v2 [physics.atom-ph] UPDATED)
arXiv.org: Quantum Physics - Fri, 2018-03-02 07:32
Effective and unrestricted engineering of atom-photon interactions requires precise spatially-resolved control of light beams. The significant potential of such manipulations lies in a set of disciplines ranging from solid state to atomic physics. Here we use a Zeeman-like ac-Stark shift of a shaped laser beam to perform rotations of spins with spatial resolution in a large ensemble of cold rubidium atoms. We show that inhomogeneities of light intensity are the main source of dephasing and thus decoherence, yet with proper beam shaping this deleterious effect is strongly mitigated allowing rotations of 15 rad within one spin-precession lifetime. Finally, as a particular example of a complex manipulation enabled by our scheme, we demonstrate a range of collapse-and-revival behaviours of a free-induction decay signal by imprinting comb-like patterns on the atomic ensemble.
Categories: Journals, Physics
### Low Loss Multi-Layer Wiring for Superconducting Microwave Devices. (arXiv:1712.01671v2 [quant-ph] UPDATED)
arXiv.org: Quantum Physics - Fri, 2018-03-02 07:32
Complex integrated circuits require multiple wiring layers. In complementary metal-oxide-semiconductor (CMOS) processing, these layers are robustly separated by amorphous dielectrics. These dielectrics would dominate energy loss in superconducting integrated circuits. Here we demonstrate a procedure that capitalizes on the structural benefits of inter-layer dielectrics during fabrication and mitigates the added loss. We separate and support multiple wiring layers throughout fabrication using SiO$_2$ scaffolding, then remove it post-fabrication. This technique is compatible with foundry level processing and the can be generalized to make many different forms of low-loss multi-layer wiring. We use this technique to create freestanding aluminum vacuum gap crossovers (airbridges). We characterize the added capacitive loss of these airbridges by connecting ground planes over microwave frequency $\lambda/4$ coplanar waveguide resonators and measuring resonator loss. We measure a low power resonator loss of $\sim 3.9 \times 10^{-8}$ per bridge, which is 100 times lower than dielectric supported bridges. We further characterize these airbridges as crossovers, control line jumpers, and as part of a coupling network in gmon and fuxmon qubits. We measure qubit characteristic lifetimes ($T_1$'s) in excess of 30 $\mu$s in gmon devices.
Categories: Journals, Physics
### Influence of wave-front curvature on supercontinuum energy during filamentation of femtosecond laser pulses in water
Author(s): F. V. Potemkin, E. I. Mareev, and E. O. Smetanina
We demonstrate that using spatially divergent incident femtosecond 1240-nm laser pulses in water leads to an efficient supercontinuum generation in filaments. Optimal conditions were found when the focal plane is placed 100–400μm before the water surface. Under sufficiently weak focusing conditions ...
[Phys. Rev. A 97, 033801] Published Thu Mar 01, 2018
Categories: Journals, Physics
### Generalized classes of continuous symmetries in two-mode Dicke models
Author(s): Ryan I. Moodie, Kyle E. Ballantine, and Jonathan Keeling
As recently realized experimentally [Nature (London) 543, 87 (2017)], one can engineer models with continuous symmetries by coupling two cavity modes to trapped atoms via a Raman pumping geometry. Considering specifically cases where internal states of the atoms couple to the cavity, we show an exte...
[Phys. Rev. A 97, 033802] Published Thu Mar 01, 2018
Categories: Journals, Physics
### Synopsis: X-Ray Absorption Spectroscopy on a Tabletop
APS Physics - Thu, 2018-03-01 12:00
A laser-based setup can be used to perform x-ray spectroscopy with a precision rivaling that of experiments at large-scale synchrotron facilities.
[Physics] Published Thu Mar 01, 2018
Categories: Physics
### Blockchain platform with proof-of-work based on analog Hamiltonian optimisers. (arXiv:1802.10091v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
The development of quantum information platforms such as quantum computers and quantum simulators that will rival classical Turing computations are typically viewed as a threat to secure data transmissions and therefore to crypto-systems and financial markets in general. We propose to use such platforms as a proof-of-work protocol for blockchain technology, which underlies cryptocurrencies providing a way to document the transactions in a permanent decentralised public record and to be further securely and transparently monitored. We reconsider the basis of blockchain encryption and suggest to move from currently used proof-of-work schemes to the proof-of-work performed by analog Hamiltonian optimisers. This approach has a potential to significantly increase decentralisation of the existing blockchains and to help achieve faster transaction times, therefore, removing the main obstacles for blockchain implementation. We discuss the proof-of-work protocols for a few most promising optimiser platforms: quantum annealing hardware based on D-wave simulators and a new class of gain-dissipative simulators.
Categories: Journals, Physics
### Qubit Parity Measurement by Parametric Driving in Circuit QED. (arXiv:1802.10112v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
Multi-qubit parity measurements are essential to quantum error correction. Current realizations of these measurements often rely on ancilla qubits, a method that is sensitive to faulty two-qubit gates and which requires significant experimental overhead. We propose a hardware-efficient multi-qubit parity measurement exploiting the bifurcation dynamics of a parametrically driven nonlinear oscillator. This approach takes advantage of the resonator's parametric oscillation threshold which is a function of the joint parity of dispersively coupled qubits, leading to high-amplitude oscillations for one parity subspace and no oscillation for the other. We present analytical and numerical results for two- and four-qubit parity measurements with high-fidelity readout preserving the parity eigenpaces. Moreover, we discuss a possible realization which can be readily implemented with the current circuit QED experimental toolbox. These results could lead to significant simplifications in the experimental implementation of quantum error correction, and notably of the surface code.
Categories: Journals, Physics
### Hidden Variables and the Two Theorems of John Bell. (arXiv:1802.10119v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
Although skeptical of the prohibitive power of no-hidden-variables theorems, John Bell was himself responsible for the two most important ones. I describe some recent versions of the lesser known of the two (familar to experts as the "Kochen-Specker theorem") which have transparently simple proofs. One of the new versions can be converted without additional analysis into a powerful form of the very much better known "Bell's Theorem", thereby clarifying the conceptual link between these two results of Bell.
Categories: Journals, Physics
### Time-dependent treatment of tunneling and Time's Arrow problem. (arXiv:1802.10148v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
New time-dependent treatment of tunneling from localized state to continuum is proposed. It does not use the Laplace transform (Green's function's method) and can be applied for time-dependent potentials, as well. This approach results in simple expressions describing dynamics of tunneling to Markovian and non-Markovian reservoirs in the time-interval $-\infty<t<\infty$. It can provide a new outlook for tunneling in the negative time region, illuminating the origin of the time's arrow problem in quantum mechanics. We also concentrate on singularity at $t=0$, which affects the perturbative expansion of the evolution operator. In addition, the decay to continuum in periodically modulated tunneling Hamiltonian is investigated. Using our results, we extend the Tien-Gordon approach for periodically driven transport, to oscillating tunneling barriers.
Categories: Journals, Physics
### Time Reversal Invariance in Quantum Mechanics. (arXiv:1802.10169v1 [physics.hist-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
Symmetries have a crucial role in today's physics. In this thesis, we are mostly concerned with time reversal invariance (T-symmetry). A physical system is time reversal invariant if its underlying laws are not sensitive to the direction of time. There are various accounts of time reversal transformation resulting in different views on whether or not a given theory in physics is time reversal invariant. With a focus on quantum mechanics, I describe the standard account of time reversal and compare it with my alternative account, arguing why it deserves serious attention. Then, I review three known ways to T-violation in quantum mechanics, and explain two unique experiments made to detect it in the neutral K and B mesons.
Categories: Journals, Physics
### Measuring the similarity of input particle states via Fisher information. (arXiv:1802.10187v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
We propose Fisher information as a measure of similarity of input particle states in a measurement basis, in which we infer the quantum interference of particles by the Fisher information. Interacting bosonic and fermionic particles by beam-splitting type operations, i.e., a 50:50 beam splitter and the Mach-Zehnder interferometer, we observe the relation between the similarity of the input particle states and the Fisher information which is derived by counting the number of particles in one of the output modes. For the Fisher information of an input state parameter in a 50:50 beam splitter, we obtain that the Fisher information is proportional to the similarity of the input particle states without discriminating the class of the particles, which is utilized to reproduce the Hong-Ou-Mandel dip with bosonic and fermionic particles. For the Fisher information of a phase parameter in the Mach-Zehnder interferometer, however, the relation is transformed with discriminating the class of the particles such that we can devise a scenario to infer an unknown input state parameter via the Fisher information. We extend the scenario of inferring a parameter to detect two-particle entanglement.
Categories: Journals, Physics
### Error Correction in Structured Optical Receivers. (arXiv:1802.10208v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
Integrated optics Green Machines enable better communication in photon-starved environments, but fabrication inconsistencies induce unpredictable internal phase errors, making them difficult to construct. We describe and experimentally demonstrate a new method to compensate for arbitrary phase errors by deriving a convex error space and implementing an algorithm to learn a unique codebook of codewords corresponding to each matrix.
Categories: Journals, Physics
### Nonlinear dynamics of a semiquantum Hamiltonian in the vicinity of quantum unstable regimes. (arXiv:1802.10251v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
We examine the emergence of chaos in a non-linear model derived from a semiquantum Hamiltonian describing the coupling between a classical field and a quantum system. The latter corresponds to a bosonic version of a BCS-like Hamiltonian, and possesses stable and unstable regimes. The dynamics of the whole system is shown to be strongly influenced by the quantum subsystem. In particular, chaos is seen to arise in the vicinity of a quantum critical case, which separates the stable and unstable regimes of the bosonic system.
Categories: Journals, Physics
### The effects of thermal and correlated noise on magnons in a quantum ferromagnet. (arXiv:1802.10258v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
The dynamics and thermal equilibrium of spin waves (magnons) in a quantum ferromagnet as well as the macroscopic magnetisation are investigated. Thermal noise due to an interaction with lattice phonons and the effects of spatial correlations in the noise are considered. We first present a Markovian master equation approach with analytical solutions for any homogeneous spatial correlation function of the noise. We find that spatially correlated noise increases the decay rate of magnons with low wave vectors to their thermal equilibrium, which also leads to a faster decay of the ferromagnet's magnetisation to its steady-state value. For long correlation lengths and higher temperature we find that additionally there is a component of the magnetisation which decays very slowly, due to a reduced decay rate of fast magnons. This effect could be useful for fast and noise-protected quantum or classical information transfer and magnonics. We further compare ferromagnetic and antiferromagnetic behaviour in noisy environments and find qualitatively similar behaviour in Ohmic but fundamentally different behaviour in super-Ohmic environments.
Categories: Journals, Physics
### Beating the classical precision limit with spin-1 Dicke state of more than 10000 atoms. (arXiv:1802.10288v1 [cond-mat.quant-gas])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
Interferometry is a paradigm for most precision measurements. Using $N$ uncorrelated particles, the achievable precision for a two-mode (two-path) interferometer is bounded by the standard quantum limit (SQL), $1/\sqrt{N}$, due to the discrete (quanta) nature of individual measurements. Despite being a challenging benchmark, the SQL has been approached in a number of systems, including the LIGO and today's best atomic clocks. One way to beat the SQL is to use entangled particles such that quantum noises from individual particles cancel out, leading to the Heisenberg limit of $\sim1/N$ in the optimal case. Another way is to employ multi-mode interferometry, with which the achievable precision can be enhanced by $1/(M-1)$ using $M$ modes. In this work, we demonstrate an interferometric precision of $8.44^{+1.76}_{-1.29}\,$dB beyond the two-mode SQL, using a balanced spin-1 (three-mode) Dicke state containing thousands of entangled atoms, thereby reaping the benefits of both means. This input quantum state is deterministically generated by controlled quantum phase transition and exhibits close to ideal quality. Our work shines new light on the pursuit of quantum metrology beyond SQL.
Categories: Journals, Physics
### Conditions where RPA becomes exact in the high-density limit. (arXiv:1802.10312v1 [cond-mat.str-el])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
It is shown that in $d$-dimensional systems, the vertex corrections beyond the random phase approximation (RPA) or GW approximation scales with the power $d-\beta-\alpha$ of the Fermi momentum if the relation between Fermi energy and Fermi momentum is $\epsilon_{\rm f}\sim p_{\rm f}^\beta$ and the interacting potential possesses a momentum-power-law of $\sim p^{-\alpha}$. The condition $d-\beta-\alpha<0$ specifies systems where RPA is exact in the high-density limit. The one-dimensional structure factor is found to be the interaction-free one in the high-density limit for contact interaction. A cancellation of RPA and vertex corrections render this result valid up to second-order in contact interaction. For finite-range potentials of cylindrical wires a large-scale cancellation appears and found to be independent of the width parameter of the wire. The proposed high-density expansion agrees with the Quantum Monte Carlo simulations.
Categories: Journals, Physics
### A study of periodic potentials based on quadratic splines. (arXiv:1802.10342v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
We discuss a method based on a segmentary approximation of solutions of the Schr\"odinger by quadratic splines, for which the coefficients are determined by a variational method that does not require the resolution of complicated algebraic equations. The idea is the application of the method to one dimensional periodic potentials. We include the determination of the eigenvalues up to a given level, and therefore an approximation to the lowest energy bands. We apply the method to concrete examples with interest in physics and discussed the numerical errors.
Categories: Journals, Physics
### Detection of ultracold molecules using an optical cavity. (arXiv:1802.10343v1 [quant-ph])
arXiv.org: Quantum Physics - Thu, 2018-03-01 06:45
We theoretically study non-destructive detection of ultracold molecules, using a Fabry-Perot cavity. Specifically, we consider vacuum Rabi splitting where we demonstrate the use of collective strong coupling for detection of molecules with many participating energy levels. We also consider electromagnetically induced transparency and transient response of light for the molecules interacting with a Fabry-Perot cavity mode, as a mean for non-destructive detection. We identify the parameters that are required for the detection of molecules in the cavity electromagnetically induced transparency configuration. The theoretical analysis for these processes is parametrized with realistic values of both, the molecule and the cavity. For each process, we quantify the state occupancy of the molecules interacting with the cavity and determine to what extent the population does not change during a detection cycle.
Categories: Journals, Physics
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Set Theory Proof with Complements
If $A \cap B = \emptyset$ then $A \subset B'$ and $B \subset A'$, where the prime symbol denotes the complement of each set.
Here are my thoughts:
Assume $A \cap B = \emptyset,$ since the intersection of $A$ and $B$ are empty, then an arbitrarily chosen element $x \notin A$ and $x \notin B.$ Thus $x \in A'$ and $x \in B'.$
How do I go about justifying that $A \subset B'$ and $B \subset A'?$
Maybe a direct proof is not the best way to do so? How about a proof by contradiction?
Thank you for any help or guidance!!!
• You may draw a Venn diagram to guide yourself. – Salomo Apr 5 '15 at 19:09
Hint:
Suppose for example that
$$a\in A\implies a\notin B\;,\;\;\text{lest}\;\;A\cap B\neq\emptyset\implies a\in B'\implies a\in A\cap B'$$
• is your suggestion a proof by contradiction? Sorry if my question seems dumb! – mathamphetamines Apr 5 '15 at 19:13
• Not really, though the central part could be considered such. Is anything unclear ? – Timbuc Apr 5 '15 at 19:14
• Take $a \in A$ thus $a \notin B$ since their intersection is empty. This makes sense, but how do you go to the next step: lest $A \cap B \ne \emptyset?$ – mathamphetamines Apr 5 '15 at 19:18
• Well, $\;a\in B'\iff a\notin B\;$ , so we already have $\;a\in A\;$ and also $\;a\in B'\;$ , thus $\;a\in A\cap B'\;$ – Timbuc Apr 5 '15 at 19:20
• Got it now. Thank you very much! – mathamphetamines Apr 5 '15 at 19:28
For every $x\in A$ the assumption $x\in B$ leads to $x\in A\cap B$ contradicting that this set is empty.
So we conclude that for every $x\in A$ we have $x\notin B$ wich is the same as $x\in B'$.
This proves that $A\subseteq B'$.
• Thank you for your help!!!! – mathamphetamines Apr 5 '15 at 19:19
You can argue this with logic pretty well, and directly, although I'm not sure if you were looking for this answer.
Suppose $$A \subset B'$$ then, take some element in A, called $x$, and by definition: $$(x \in A)\rightarrow(x \notin B)$$ So, using the logical equivalent to the implication statement, $$\neg(x \in A) \vee \neg(x\in B)$$ And with De Morgan's law, this is equivalent to $$\neg[(x\in A) \wedge (x \in B)]$$ which is equivalent to saying $$x \notin A\cap B \rightarrow A\cap B =\emptyset$$
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# The differential equation of the form x 2 </msup> d
The differential equation of the form
${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+{p}_{1}\frac{dy}{dx}+{p}_{2}y=Q$ where ${p}_{1}$, ${p}_{2}$ are constants and Q the function of x is called Second order homogeneous linear equation.
How can we prove that above equation is second order homogeneous linear differential equation.And also help me to understand the significance of above equation in engineering field.
You can still ask an expert for help
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Ashley Parks
Just consider definitions and ask yourself questions:
1) Is it linear? Suppose you have two solutions
${y}_{0},{y}_{1}$
Now add them together $a\cdot {y}_{0}+b\cdot {y}_{1}$
But the sum passes right through the derivatives to 0 for each solution
and the sum of zero's is zero.
That is the definition of linearity.
2) Is it homogeneous?
That requires that there all terms are scalar or derivative terms in y :i.e $y,{y}^{\prime },{y}^{″},{y}^{‴}$
That normally means no free standing constant or $f\left(x\right)$ terms.
In all of Mathematics just read the definition and then read or think of examples. Then try to stretch the examples to limits and break the definition!
Sometimes this gets obscure and confusing, but this is at the heart of the structures that are being built. This distinguishing of mathematical objects that fall into one class; or out of it.
(Just a side note, find a variety of sources. Occasionally there are confusing typos even in textbooks. If you find differences, ask the teacher they are usually glad to get off the rote path and explain new things.)
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# How to list the Azure VMs from the Availability set using PowerShell?
Software & CodingMicrosoft TechnologiesPowerShell
To list the Azure VMs under the availability set, we can use the command as shown below
Before running the command make sure that you are connected to the Azure Cloud (if not use ConnectAzAccount) and proper azure subscription (Set-AzContext to set the Azure subscription)
$availabilitySetVMs = Get-AzAvailabilitySet -ResourceGroupName MYRESOURCEGROUPAVAILABILITY -Name myAvailabilitySet To get the total number of Virtual machines in the availability set, PS C:\>$availabilitySetVMs.VirtualMachinesReferences.id.count
## Output
To get all the VM names from the list, we can use the below command.
$VMlist = Get-AzAvailabilitySet -ResourceGroupName MYRESOURCEGROUPAVAILABILITY -Name myAvailabilitySet$i=0
foreach($vm in$VMlist.VirtualMachinesReferences){
"VM{0}: {1}" -f $i,($vm.Id.Split('/'))[-1]
\$i++
}
## Output
Published on 02-Sep-2021 11:37:22
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# zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
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any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Convolution with measures on hypersurfaces. (English) Zbl 0972.42009
The author considers the ${L}^{p}$ improving properties of convolution operators $f↦f*d\sigma$ where $d\sigma$ is a compactly supported measure on a ${C}^{2}$ hypersurface $S$. For surfaces of non-zero curvature the sharp estimate is ${L}^{n+1/n}\to {L}^{n}$. In this paper the author considers the slightly weaker restricted estimate ${L}^{n+1/n,1}\to {L}^{n}$.
Under very mild conditions on $S$ (namely that the Gauss map generically has bounded multiplicity, plus another technical condition of a similar flavor) the author shows that one can obtain the above restricted estimate if and only if $\mu$ obeys the estimate $\mu \left(R\right)\lesssim {|R|}^{\left(n-1\right)/\left(n+1\right)}$ for all rectangles $R$. This is in particular achieved if $\mu$ is equal to surface measure times ${\kappa }^{1/\left(n+1\right)}$, where $\kappa$ is the Gaussian curvature.
The heart of the argument is a certain ${L}^{n}$ estimate which, after multiplying everything out and changing variables, hinges on the estimation of various Jacobians and on certain multilinear estimates with these Jacobians as kernels.
##### MSC:
42B15 Multipliers, several variables 44A12 Radon transform 42A20 Convergence and absolute convergence of Fourier and trigonometric series
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# Nonnegative Linear Least Squares, Solver-Based
This example shows how to use several algorithms to solve a linear least-squares problem with the bound constraint that the solution is nonnegative. A linear least-squares problem has the form
$\underset{x}{\mathrm{min}}‖Cx-d{‖}^{2}$.
In this case, constrain the solution to be nonnegative, $x\ge 0$.
To begin, load the arrays $C$ and $d$ into your workspace.
`load particle`
View the size of each array.
`sizec = size(C)`
```sizec = 1×2 2000 400 ```
`sized = size(d)`
```sized = 1×2 2000 1 ```
The `C` matrix has 2000 rows and 400 columns. Therefore, to have the correct size for matrix multiplication, the `x` vector has 400 rows. To represent the nonnegativity constraint, set lower bounds of zero on all variables.
`lb = zeros(size(C,2),1);`
Solve the problem using `lsqlin`.
```[x,resnorm,residual,exitflag,output] = ... lsqlin(C,d,[],[],[],[],lb);```
```Minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. ```
To see details of the optimization process, examine the output structure.
`disp(output)`
``` message: '...' algorithm: 'interior-point' firstorderopt: 3.6717e-06 constrviolation: 0 iterations: 8 linearsolver: 'sparse' cgiterations: [] ```
The output structure shows that `lsqlin` uses a sparse internal linear solver for the interior-point algorithm and takes 8 iterations to reach a first-order optimality measure of about 3.7e-6.
### Change Algorithm
The trust-region-reflective algorithm handles bound-constrained problems. See how well it performs on this problem.
```options = optimoptions('lsqlin','Algorithm','trust-region-reflective'); [x2,resnorm2,residual2,exitflag2,output2] = ... lsqlin(C,d,[],[],[],[],lb,[],[],options);```
```Local minimum possible. lsqlin stopped because the relative change in function value is less than the square root of the function tolerance and the rate of change in the function value is slow. ```
`disp(output2)`
``` iterations: 10 algorithm: 'trust-region-reflective' firstorderopt: 2.7870e-05 cgiterations: 42 constrviolation: [] linearsolver: [] message: 'Local minimum possible....' ```
This time, the solver takes more iterations and reaches a solution with a higher (worse) first-order optimality measure.
To improve the first-order optimality measure, try setting the `SubproblemAlgorithm` option to `'factorization'`.
```options.SubproblemAlgorithm = 'factorization'; [x3,resnorm3,residual3,exitflag3,output3] = ... lsqlin(C,d,[],[],[],[],lb,[],[],options);```
```Optimal solution found. ```
`disp(output3)`
``` iterations: 12 algorithm: 'trust-region-reflective' firstorderopt: 5.5907e-15 cgiterations: 0 constrviolation: [] linearsolver: [] message: 'Optimal solution found.' ```
Using this option brings the first-order optimality measure nearly to zero, which is the best possible result.
### Change Solver
Try solving t problem using the `lsqnonneg` solver, which is designed to handle nonnegative linear least squares.
```[x4,resnorm4,residual4,exitflag4,output4] = lsqnonneg(C,d); disp(output4)```
``` iterations: 184 algorithm: 'active-set' message: 'Optimization terminated.' ```
`lsqnonneg` does not report a first-order optimality measure. Instead, investigate the residual norms. To see the lower-significance digits, subtract 22.5794 from each residual norm.
```t = table(resnorm - 22.5794, resnorm2 - 22.5794, resnorm3 - 22.5794, resnorm4 - 22.5794,... 'VariableNames',{'default','trust-region-reflective','factorization','lsqnonneg'})```
```t=1×4 table default trust-region-reflective factorization lsqnonneg __________ _______________________ _____________ __________ 7.5411e-05 4.9186e-05 4.9179e-05 4.9179e-05 ```
The default `lsqlin` algorithm has a higher residual norm than the `trust-region-reflective` algorithm. The `factorization` and `lsqnonneg` residual norms are even lower, and are the same at this level of display precision. See which one is lower.
`disp(resnorm3 - resnorm4)`
``` 6.8567e-13 ```
The `lsqnonneg` residual norm is the lowest by a negligible amount. However, `lsqnonneg` takes the most iterations to converge.
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# Twenty people at a meeting were born during the month of Sep
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03 Aug 2014, 13:52
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Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%
Math Expert
Joined: 02 Sep 2009
Posts: 49948
Re: Twenty people at a meeting were born during the month of Sep [#permalink]
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13 Aug 2014, 07:39
3
2
kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%
PROBABILITY APPROACH:
P(at least two of the people share the same birthday) = 1 - P(none of the people share the same birthday) =
$$= 1 - \frac{30}{30}*\frac{29}{30}*\frac{28}{30}*\frac{27}{30}*\frac{26}{30}*...*\frac{11}{30} = 1 - \frac{30!}{(30^{20}*10!)}\approx{0.99}$$. First person can have birthday on any day (30/30), the second on any but that day (29/30), the thrid on any but those two days (28/30), ...
Notice that the number we are subtracting from 1 is very, very small, so the final result will be very close to 100%.
COMBINATIONS APPROACH:
P(at least two of the people share the same birthday) = 1 - P(none of the people share the same birthday) =
$$= 1- \frac{C^{20}_{30}*20!}{30^{20}}=\frac{30!}{(30^{20}*10!)}\approx{0.99}$$. $$C^{20}_{30}$$ here is choosing 20 different days out of 30, 20! is the number of ways we can assign 20 people to those 20 days (by the way, we could write there $$P^{20}_{30}$$ there instead of $$C^{20}_{30}*20!$$, which is basically the same: choosing 20 out of 30 when the order of the selection matters) and the denominator ($$30^{20}$$) is the total number of way 20 people can have birthdays in September (each of them has 30 options).
Hope it's clear.
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Joined: 22 Feb 2009
Posts: 175
Re: Twenty people at a meeting were born during the month of Sep [#permalink]
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03 Aug 2014, 14:51
1
kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%
The probability that at least two people sharing the same birthday = 1 - the probability that none of them sharing the same birthday
A = The number of ways of none of them sharing the same birthday = 30P20 = 30!/(30-20)! = 30!/10! = 11*12*...*29*30
B = The total number of possible ways of 20 people born in September = 20*20*....*20*20 = 20^30 ( each day has 20 options)
A/B = the probability that none of them sharing the same birthday
since B is much greater than A, A/B may equal 1%
--> The probability that at least two people sharing the same birthday = 1 - 1% = 99%
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Re: Twenty people at a meeting were born during the month of Sep [#permalink]
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03 Aug 2014, 14:54
at least two of the people = 1- no two people share the same bday
no two people share the same bday = (1st pick a day in the 30 days) * (2rd pick another day in the left 29 days)
= (1/30) * (29/29)
so, at least two of the people share differ = 1-(1/30) * (29/29) = 29/30 = 99%
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Re: Twenty people at a meeting were born during the month of Sep [#permalink]
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04 Aug 2014, 12:27
I still don't get it. I thought it would be 1-(29/30)*(28/30). Does anyone have another way of figuring this out?
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Re: Twenty people at a meeting were born during the month of Sep [#permalink]
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04 Aug 2014, 14:08
kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%
The probability that at least two people sharing the same birthday = 1 - the probability that none of them sharing the same birthday
A = The number of ways of none of them sharing the same birthday = 30P20 = 30!/(30-20)! = 30!/10! = 11*12*...*29*30
B = The total number of possible ways of 20 people born in September = 20*20*....*20*20 = 20^30 ( each day has 20 options)
A/B = the probability that none of them sharing the same birthday
since B is much greater than A, A/B may equal 1%
--> The probability that at least two people sharing the same birthday = 1 - 1% = 99%
could you explain the red part please?
Math Expert
Joined: 02 Sep 2009
Posts: 49948
Re: Twenty people at a meeting were born during the month of Sep [#permalink]
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13 Aug 2014, 07:41
kennaval wrote:
Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?
(A) 10%
(B) 33%
(C) 67%
(D) 90%
(E) 99%
The probability that at least two people sharing the same birthday = 1 - the probability that none of them sharing the same birthday
A = The number of ways of none of them sharing the same birthday = 30P20 = 30!/(30-20)! = 30!/10! = 11*12*...*29*30
B = The total number of possible ways of 20 people born in September = 20*20*....*20*20 = 20^30 ( each day has 20 options)
A/B = the probability that none of them sharing the same birthday
since B is much greater than A, A/B may equal 1%
--> The probability that at least two people sharing the same birthday = 1 - 1% = 99%
It should be 30^20, instead of 20^30: each out of 20 people has 30 options - 30*30*...*30 = 30^20.
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Posts: 49948
Re: Twenty people at a meeting were born during the month of Sep [#permalink]
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13 Aug 2014, 07:42
gigi66653 wrote:
at least two of the people = 1- no two people share the same bday
no two people share the same bday = (1st pick a day in the 30 days) * (2rd pick another day in the left 29 days)
= (1/30) * (29/29)
so, at least two of the people share differ = 1-(1/30) * (29/29) = 29/30 = 99%
It should be 30/30*29/30*28/30*27/30*26/30*...*11/30.
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Twenty people at a meeting were born during the month of Sep [#permalink]
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02 Nov 2015, 20:48
It is easy to solve the problem by finding the probability where each person is born on a different day and subtracting it from 1.
Let us start with a single way. The first person can be born on Sep 1 , the second on Sep 2 and so on. So the probability of 20 persons born on different days = (1/30)*(1/30) *..20 times =1/(30^20)
How many such ways are there?
(1) the 20 days can be chosen from 30 days in 30C20 ways
(2) The birthdays of the 20 persons can be arranged in 20! ways
For the probability, we have to multiply 1/(30^20) by 30C20 and 20!
So the probability that the birthdays fall on different days = 30C20 * 20! / (30^20)
The probability that at least two persons share the same birthday is 1 - (30C20 *20!) / (30^20) = 99%(approx)
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# nLab tabular dagger 2-poset
### Context
#### Higher category theory
higher category theory
## Idea
A dagger 2-poset with tabulations, such as a tabular allegory.
## Definition
A tabular dagger 2-poset is a dagger 2-poset $C$ such that for every object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A,B)$, there is an object $\vert R \vert:Ob(C)$ and maps $f:Hom(\vert R \vert, A)$, $g:Hom(\vert R \vert, B)$, such that $R = f^\dagger \circ g$ and for every object $E:Ob(C)$ and maps $h:Hom(E,\vert R \vert)$ and $k:Hom(E,\vert R \vert)$, $f \circ h = f \circ k$ and $g \circ h = g \circ k$ imply $h = k$.
## Properties
The category of maps of a tabular dagger 2-poset has all pullbacks.
## Examples
The dagger 2-poset Rel of sets and relations is a tabular dagger 2-poset.
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# Thread: Creating equation from graph
1. ## Creating equation from graph
Hi,
I hope someone can help. I'm trying to understand how a given equation matches the following function characteristics:
The equations are:
I understand that the function needs to have an odd degree, one zero at x = 2 that has a multiplicity of 2 (in order for it to "bounce back" and cross y = 2 when x = 0), and another zero at x = -1. However, I don't understand how to get to the textbook's example equation's above. Please help!
Best,
Olivia
2. ## Re: Creating equation from graph
I plotted the four given points and started with $f(x) = k(x+1)^2(x-3)$
since $f(-3) = 16$ ...
$16 = k(-3+1)^2(-3-3)$
$16 = -24k \implies k = -\dfrac{2}{3}$
$f(x) = -\dfrac{2}{3}(x+1)^2(x-3)$
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# nLab string theory
## Surveys, textbooks and lecture notes
#### Gravity
gravity, supergravity
spin geometry
string geometry
# Contents
## Idea
String theory is a theory in fundamental physics.
Below we indicate the basic idea and provide pointers to further details. See also the string theory FAQ.
### Conceptually: String perturbation theory from QFT worldline formalism
Perturbative string theory is something at least close to a categorification of the following description of perturbative quantum field theory in terms of sums over Feynman diagrams.
Recall that in quantum field theory one approach to make sense of the path integral is the perturbation series expansion, which interprets the path integral for the scattering amplitudes (the S-matrix) as a certain sum over graphs of certain numbers assigned to each graph.
The graphs are called Feynman diagrams, the numbers assigned to them are called (renormalized) scattering amplitudes and the sum over graphs of (renormalized) amplitudes is the perturbation series or S-matrix.
The amplitude assigned to a single graph with $n$ external edges is interpreted as the amplitude for $n$ “quanta” or “particles” of the fields in question to interact in the way indicated by the graph.
Crucial for the motivation of the idea of string theory is the observation that this (renormalized) amplitude assigned to a graph is itself the correlator of a 1-dimensional quantum field theory on that graph: the “worldline quantum field theory” describing the (relativistic) quantum mechanics of these particles. This is usually a sigma-model with parameter space the given graph and target space the spacetime on which the fields live for which the perturbation series computes the path integral.
When made explicit this is called the worldline formalism for computing the quantum field perturbation series. (See there for more details.)
The premise of perturbative string theory is to replace the perturbation series over correlators of a 1-d QFT over graphs by a sum of correlators of a 2-dQFT over 2-dimensional surfaces – called worldsheets and hence produce an S-matrix this way. Again in simple cases this 2d QFT is a sigma-model whose target is the spacetime in which one computes interactions.
In analogy to the previous case, one thinks of the amplitude assigned this way to a surface as the amplitude for the boundary arcs – the strings – to interact in the way given by the surface.
Some of the motivations for considering this replacement of graphs by surfaces have been the following:
• the 2-d correlators are better behaved in that they don’t have to be renormalized. The “counterterms” appearing in renormalization of ordinary QFT can be identified with contributions to the correlators that come from the linear extension of the strings (see the above reference for more on this);
• there are fewer choices involved: a Feynman graph is really a decorated graph with the decoration from some more or less arbitrary index set, describing the nature of the particles associated with a given edge and the nature of the interaction associated with a given vertex. In the sum over surfaces there is no extra decoration (except on the boundary of the surfaces) and one finds that instead a single string diagram (a 2d QFT correlator for a given surface) encodes already a sum over (infinitely) many particle species decorations and all possible interaction decorations for them;
• while there are fewer choices to be made by hand, it turns out that the effective particle content that does arise automatically from this prescription happens to be structurally of the kind one would hope for: the massless effective particles described by the string perturbation series happen to be gauge bosons, fermions charged under them, and, notably, gravitons. This is structurally exactly the Yang-Mills theory input of the standard model of particle physics combined with perturbative quantum gravity that one would hope to see.
These aspects have motivated the impression that the string perturbation series might be considerably closer to the true formalism of fundamental physics than ordinary perturbative quantum field theory. This impression is however offset by the following problems:
• while the worldsheet 2d QFT whose correlators are summed over surfaces are themselves much easier to handle than the full target space quantum physics they are used to encode, a fully complete and rigorous theory of 2d QFT is available only in simple special cases.
• In particular, even though there are fewer arbitrary choices involved in the string perturbation series as compared to the ordinary Feynman perturbation series, one crucial choice still present is that of this worldsheet 2d QFT. By the above, every choice of worldsheet QFT (called a choice of “vacuum”) corresponds to a choice of effective target space geometry (to be thought of as the one that the perturbation series computes the quantum perturbations about) and particle content (see 2-spectral triple for more on that). One would therefore like to understand the space of all worldsheet QFTs whose effective target space geometry and particle content is close to the one experimentally observed. After many years of rather naïve approaches to handle or not to handle this, it has more recently at least come to the general attention that there is something to be better understood here.
• More fundamentally, already the role of the original perturbation series in quantum field theory is actually not fully understood. Its main success is the observation that truncating or resumming the perturbation series in a more or less ad hoc way, it does yield values that very well describe a plethora of real world measurements. One imagines that there is a non-perturbative definition of quantum field theory such that in certain well-defined circumstances the perturbation series does yield an approximation to it and is a posteriori justified. If so, there should be an analogous nonperturbative definition of string theory. There is a large ratio of speculations as to what that might be over solid results about it.
### Phenomenologically
While therefore the premise of perturbative string theory is conceptually suggestive for various reasons, there is to date no connection to experimental phenomenology (apart from the fact that conceptual insights into string theory have helped analyze quantum field theoretic data, see at string theory results applied elsewhere). As a result much of the substantial outcome of string theory research is more in mathematical physics (if done well, at least), exploring the general theory space of quantum field theories and their UV-completions, than in realistic model building (though there is no lack of trying), where it remains very speculative. This has led to public or semi-public debates about the value of string theory for actual physics. See at criticism of string theory for pointers.
## Critical string theories and quantum anomalies
The action functional for the string-sigma model in general has a quantum anomaly of both kinds:
1. For both the bosonic string and the superstring the corresponding Polyakov action has a gauge anomaly for the conformal symmetry, depending on the dimension $d$ of target space, and on the strength of the dilaton background field. For vanishing dilaton field this anomaly vanishes exactly for $d = 26$ for the bosonic model, and in $d = 10$ for the superstring.
For target spaces of these dimensions one speaks of critical string theory. In as far as string theory is expected to have relevance for physics at all, it is usually expected to be in this critical dimension. But also noncritical string models can and have been considered.
2. Apart from the gauge anomaly, the action functional of the string-sigma-model also in general has an anomalous action functional , for two reasons:
1. The higher holonomy of the higher background gauge fields is in general not a function, but a section of a line bundle;
2. The fermionic path integral over the worldsheet-spinors of the superstring produces as section of a Pfaffian line bundle.
In order for the action functional to be well-defined, the tensor product of these different anomaly line bundles over the bosonic configuration space must have trivial class (as bundles with connection, even). This gives rise to various further anomaly cancellation conditions:
1. For the heterotic string (necessarily closed) the anomaly cancellation condition is known as the Green-Schwarz mechanism : it says that the background fields of gravity and B-field must organize to a twisted differential string structure whose twist is given by the background Yang-Mills field.
2. For the open type II string the condition is known as the Freed-Witten anomaly cancellation condition: it says that the restriction of the B-field to any D-brane must consistute the twist of a twisted spin^c structure on the brane.
A more detailed analysis of these type II anomalies is in (DFMI) and (DFMII).
## Subtopics
### Extended objects
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$(D = 2n)$type IIA$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection$\,$
string$\,$B2-field2d SCFT
NS5-brane$\,$B6-fieldlittle string theory
M-brane11D SuGra/M-theorycircle n-connection$\,$
M2-brane$\,$C3-fieldABJM theory, BLG model
M5-brane$\,$C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d
### Elliptic genera, elliptic cohomology and $tmf$
A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means. (Witten 87, very last sentence)
The large volume limit of the partition function of the superstring on a given target spacetime is an elliptic genus of that manifold (Witten 87), the Witten genus (see there for more).
Since the Witten genus in turn is the decategorification of the string orientation of tmf, this suggests that tmf-generalized (Eilenberg-Steenrod) cohomology classifies full string theories, in refinement of how the classification of D-brane charge (just the boundary conditions for open strings) is given by K-theory.
A non-trivial conistency check of this idea is announced in (Nikolaus 14).
### String theory results applied elsewhere
Beyond the speculative hypothetized role of string theory as a theory behind observed particle physics, the theory has shed light on many aspects of quantum field theory, both on the conceptual structure of quantum field theory as such as well as on concrete theories and their concrete properties. Some of these string theory results enter crucially in computations that are used to interpret particle physics experiments such as the LHC.
For more see
## References
### General
A large body of references is organized at the
For another list of literature see the entry
A useful survey of the status of string theory as a theory of quantum gravity is in
Some reflections on the mathematical physics involved in string theory are in
An article summarizing information about cohomological models for aspects of string theory and listing plenty of useful further references is
A book trying to summarize the state of the art of capturing mathematical structures fundamental to the definition of perturbative string theory is
### Technical details
#### Elliptic genera, elliptic homology and tmf
The partition function of the superstring was understood to be an elliptic genus (the Witten genus) in
• Edward Witten, Elliptic Genera And Quantum Field Theory , Commun.Math.Phys. 109 525 (1987) (Euclid)
A nontrivial consistency check on the suggestion that this means that string backgrounds are classified in tmf is given in
#### Quantum anomalies
Discussion of type II quantum anomalies is in
Discussion of superstring perturbation theory is in
### Fields medal (and other) work related to string theory
Pure mathematics work which is closely related string theory and was awared with a Fields medal includes the following.
Richard Borcherds, 1998
Maxim Kontsevich, 1998
Edward Witten, 1990
Grigori Perelman, 2006
In (Madsen 07) it says with respect to the proof (Madsen-Weiss 02) of the Mumford conjecture:
These tools $[$ used in the proof $]$ are all rather old, known for at least twenty years, and one may wonder why they have not before been put to use in connection with the Riemann moduli space. Maybe we lacked the inspiration that comes from the renewed interaction with physics, exemplified in conformal field theories.
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# zbMATH — the first resource for mathematics
Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery. (English) Zbl 1452.62155
The paper presents the performance of compound Poisson (CPP) as nonparametric priors. It is proved that under (CPP) priors, optimal posterior contraction rates are attained for Hölder functions and for monotone functions. In Section 2, the contraction rates for compound Poisson processes and subordinator priors are investigated. A general description of the asymptotic posterior shape in which the results thereafter can be embedded is presented in Section 3. In Section 4, Bernstein-von Mises-type theorems and results on the frequentist coverage of credible sets for CPP priors are presented. The proofs of the new theorems are presented in two appendices.
##### MSC:
62C10 Bayesian problems; characterization of Bayes procedures 62G05 Nonparametric estimation 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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##### References:
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Econometrica 72 1445-1480. · Zbl 1091.62135 [7] Chipman, H. A., George, E. I. and McCulloch, R. E. (2010). BART: Bayesian additive regression trees. Ann. Appl. Stat. 4 266-298. · Zbl 1189.62066 [8] Coram, M. and Lalley, S. P. (2006). Consistency of Bayes estimators of a binary regression function. Ann. Statist. 34 1233-1269. · Zbl 1113.62006 [9] Denison, D. G. T., Mallick, B. K. and Smith, A. F. M. (1998). A Bayesian CART algorithm. Biometrika 85 363-377. · Zbl 1048.62502 [10] Embrechts, P., Klüppelberg, C. and Mikosch, T. (2003). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, New York. · Zbl 0873.62116 [11] Frick, K., Munk, A. and Sieling, H. (2014). Multiscale change point inference. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 495-580. · Zbl 1411.62065 [12] Gao, C., Han, F. and Zhang, C.-H. (2017). On estimation of isotonic piecewise constant signals. Preprint. Available at arXiv:1705.06386. Ann. Statist. (to appear). [13] Ghosal, S. (1999). Asymptotic normality of posterior distributions in high-dimensional linear models. Bernoulli 5 315-331. · Zbl 0948.62007 [14] Ghosal, S. (2000). Asymptotic normality of posterior distributions for exponential families when the number of parameters tends to infinity. J. Multivariate Anal. 74 49-68. · Zbl 1118.62309 [15] Ghosal, S., Ghosh, J. K. and Samanta, T. (1995). On convergence of posterior distributions. Ann. Statist. 23 2145-2152. · Zbl 0858.62024 [16] Ghosal, S. and van der Vaart, A. (2017). Fundamentals of Nonparametric Bayesian Inference. Cambridge Series in Statistical and Probabilistic Mathematics 44. Cambridge Univ. Press, Cambridge. · Zbl 1376.62004 [17] Gijbels, I., Mammen, E., Park, B. U. and Simar, L. (1999). On estimation of monotone and concave frontier functions. J. Amer. Statist. Assoc. 94 220-228. · Zbl 1043.62105 [18] Holmes, C. C. and Heard, N. A. (2003). Generalized monotonic regression using random change points. Stat. Med. 22 623-638. [19] Jirak, M., Meister, A. and Reiß, M. (2014). Adaptive function estimation in nonparametric regression with one-sided errors. Ann. Statist. 42 1970-2002. · Zbl 1305.62172 [20] Kim, Y. and Lee, J. (2004). A Bernstein-von Mises theorem in the nonparametric right-censoring model. Ann. Statist. 32 1492-1512. · Zbl 1047.62043 [21] Kleijn, B. and Knapik, B. (2012). Semiparametric posterior limits under local asymptotic exponentiality. Preprint. Available at arXiv:1210.6204. [22] Korostelëv, A. P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statistics 82. Springer, New York. · Zbl 0833.62039 [23] Li, M. and Ghosal, S. (2017). Bayesian detection of image boundaries. Ann. Statist. 45 2190-2217. · Zbl 06821123 [24] Mariucci, E., Ray, K. and Szabo, B. (2017). A Bayesian nonparametric approach to log-concave density estimation. Preprint. Available at arXiv:1703.09531. · Zbl 07166557 [25] Meister, A. and Reiß, M. (2013). Asymptotic equivalence for nonparametric regression with non-regular errors. Probab. Theory Related Fields 155 201-229. · Zbl 1257.62045 [26] Panov, M. and Spokoiny, V. (2015). Finite sample Bernstein-von Mises theorem for semiparametric problems. Bayesian Anal. 10 665-710. · Zbl 1335.62057 [27] Reiß, M. and Schmidt-Hieber, J. (2017). Posterior contraction rates for support boundary recovery. Preprint. Available at arXiv:1703.08358. [28] Reiß, M. and Schmidt-Hieber, J. (2019). Supplement to “Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery.” https://doi.org/10.1214/19-AOS1853SUPP. [29] Reiß, M. and Selk, L. (2017). Efficient estimation of functionals in nonparametric boundary models. Bernoulli 23 1022-1055. · Zbl 1380.62177 [30] Rivoirard, V. and Rousseau, J. (2012). Bernstein-von Mises theorem for linear functionals of the density. Ann. Statist. 40 1489-1523. · Zbl 1257.62036 [31] Rockova, V. and van der Pas, S. (2017). Posterior concentration for Bayesian regression trees and their ensembles. Preprint. Available at arXiv:1708.08734. Ann. Statist. (to appear). [32] Salomond, J.-B. (2014). Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints. Electron. J. Stat. 8 1380-1404. · Zbl 1298.62064 [33] Sato, K. (2013). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge. [34] Scricciolo, C. (2007). On rates of convergence for Bayesian density estimation. Scand. J. Stat. 34 626-642. · Zbl 1150.62018 [35] Simon, T. (2004). Small ball estimates in $$p$$-variation for stable processes. J. Theoret. Probab. 17 979-1002. · Zbl 1074.60055 [36] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge. [37] van der Vaart, A. · Zbl 0862.60002
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# xa'ei'o JOhE experimental cmavo
binary mekso operator: Let the inputs X1 and X2 be sets in the same universal set O; then the result of this operator applied to them is X1^c \cup X, where for any A \subseteq O, Ac = O \setminus A.
Generalizes to n-ary, for any integer n>1 or n being an infinity. This operator generates all three of the basic set operators: union (jo'e), intersection (ku'a), and relative complement (kei'i). See also: xa'ei'u. .krtis. calls it the "union(-type) hash operator", but has never seen a generally used name for it.
## In notes:
xa'ei'u
binary mekso operator: Let the inputs X1 and X2 be sets in the same universal set O; then the result of this operator applied to them is X1^c \cap X, where for any A \subseteq O, Ac = O \setminus A.
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# Enforcing permalink structure in WordPress
A client site requires a certain permalink structure for some custom plugins to work: WordPress makes it easy.
Enforcing the permalink structure is only half of the work since the other side of it is to flush the rewrite rules (dumping them to the .htaccess file) when setting the new permalink structure.
The function below is a quick code snippet taking care of it:
<?php
// make sure the permalink structure is set to /%postname%/
// hook late
add_action('init', 'enforce_permalink_strucure', 999);
function enforce_permalink_strucure() {
global $wp_rewrite; // the target permalink structure$structure = '/%postname%/';
// not this structure?
if ($wp_rewrite->permalink_structure !==$structure) {
$wp_rewrite->set_permalink_structure($structure);
\$wp_rewrite->flush_rules();
}
}
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# Is the function $f(x)=|x|^{1/2}$ Lipschitz continuous?
Is the function $f(x)=|x|^{1/2}$ Lipschitz continuous near $0$? If yes, find a constant for some interval containing $0$.
I think the answer is yes since I can find $L=1$ that satisfies Lipschitz continuity criteria in a interval close to $0$, am I right?
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¿Is this homework? If so, check the homework tag! It is clearly Lipschitz over any interval which excludes zero. But for an interval containg zero, one way oc choosing the Lipschits constant is to take the supremum of the absolute value of the derivative. but for an interval containing zero, ¿What is that supremum in this case? – kjetil b halvorsen Oct 15 '12 at 23:23
Exact duplicate: math.stackexchange.com/q/214290 – user44798 Oct 15 '12 at 23:36
@gee That question was for a different function ($f(x) = 1/|x|^{1/2}$) until Klara edited it an hour ago. I've rolled back the edit. – Ayman Hourieh Oct 15 '12 at 23:48
Let $x$, $y$ be numbers in $(0, \delta)$.
$$\left|\left(|y|^{1/2} - |x|^{1/2}\right)\left(|y|^{1/2} + |x|^{1/2}\right)\right| = \left||y| - |x|\right| = |y - x|$$
Therefore: $$|f(y) - f(x)| = \frac{|y - x|}{\left||y|^{1/2} + |x|^{1/2}\right|} \tag{1}$$
If $f(x)=|x|^{1/2}$ is Lipschitz continuous, we can find $K > 0$ so that:
$$|f(y) - f(x)| \le K|y - x| \tag{2}$$
Put (1) and (2) together to get:
$$\frac{1}{K} \le \left||y|^{1/2} + |x|^{1/2}\right|$$
By making $x$ and $y$ approach $0$, we can make the RHS as small as we desire. Thus, we have a contradiction.
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Thank you for your reply, Is the contradiction that K has to go to infinity to satisfy the inequality for x, y close to zero? What is a reasonable value for K to say that a function in general is Lipschitz cont.? – Klara Oct 16 '12 at 0:06
@Klara The contradiction is that no matter what $K$ we pick, we can always find $x, y > 0$ that don't satisfy the Lipschitz continuity condition. In general, $K$ must be a finite non-negative number. – Ayman Hourieh Oct 16 '12 at 0:10
thank you for your help! – Klara Oct 16 '12 at 1:21
That, or the remark that $g(x)=(f(2x)-f(x))/|x|$ defines an unbounded function $g$. – Did Mar 23 '13 at 7:53
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Alternate solution of tetration for "convergent" bases discovered mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 09/14/2010, 11:04 PM (This post was last modified: 09/14/2010, 11:08 PM by mike3.) (09/14/2010, 12:26 PM)tommy1729 Wrote: (09/14/2010, 01:50 AM)mike3 Wrote: Now that we have a way to define continuum sums, we start with an initial guess $F_0(z)$ to our tetrational function $\tet_b(z)$, and then apply the iteration formula $F_{k+1}(z) = \int_{-1}^{z} \log(b)^w \exp_b\left(\sum_{n=0}^{w-1} F_k(n)\right) dw$ and take the convergent as $k \rightarrow \infty$. Now the adaptation of this theory to construct a numerical algorithm is a little more complicated. I could post that, if you'd like it, in the Computation forum. But the above is the basic outline of the method. thanks thats a lot better. Yes. Though I accidentally left out a detail -- see the post again, I updated it. (09/14/2010, 12:26 PM)tommy1729 Wrote: 1) is that similar or equal to my " using the sum hoping for convergence " thread ? It should be. (09/14/2010, 12:26 PM)tommy1729 Wrote: 2) i assume the intial guess needs to be a taylor , laurent or coo fourier series , in other words analytic. and also map R+ -> R+ , but then we have to start from an already tetration solution ? In the abstract, it's just a holomorphic (multi-)function. But for the numerical algorithm, we'd use a (truncated) Fourier series. (09/14/2010, 12:26 PM)tommy1729 Wrote: so what is the advantage of this sum method solution ? afterall it is just a 1periodic wave transform of another sexp/slog. Biggest advantage seems to be that it gives the widest range of bases I've seen for any tetration method. (09/14/2010, 12:26 PM)tommy1729 Wrote: what are its properties ? Not sure exactly what you'd want to know. It seems to be similar to the "Cauchy integral" for bases outside and in some parts of the STR, but the weird part is that when it is used for $1 < b < e^{1/e}$, it gives the attracting regular iteration. ??? « Next Oldest | Next Newest »
Messages In This Thread Alternate solution of tetration for "convergent" bases discovered - by mike3 - 09/13/2010, 11:30 AM RE: Alternate solution of tetration for "convergent" bases discovered - by tommy1729 - 09/13/2010, 11:20 PM RE: Alternate solution of tetration for "convergent" bases discovered - by mike3 - 09/14/2010, 01:50 AM RE: Alternate solution of tetration for "convergent" bases discovered - by tommy1729 - 09/14/2010, 12:26 PM RE: Alternate solution of tetration for "convergent" bases discovered - by mike3 - 09/14/2010, 11:04 PM RE: Alternate solution of tetration for "convergent" bases discovered - by sheldonison - 09/14/2010, 11:36 PM RE: Alternate solution of tetration for "convergent" bases discovered - by mike3 - 09/15/2010, 02:18 AM RE: Alternate solution of tetration for "convergent" bases discovered - by sheldonison - 09/13/2010, 11:45 PM RE: Alternate solution of tetration for "convergent" bases discovered - by mike3 - 09/14/2010, 01:53 AM RE: Alternate solution of tetration for "convergent" bases discovered - by sheldonison - 09/14/2010, 02:18 PM RE: Alternate solution of tetration for "convergent" bases discovered - by tommy1729 - 09/14/2010, 07:41 PM
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## 代数幾何学セミナー
開催情報 火曜日 10:30～11:30 or 12:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室 權業 善範・中村 勇哉・田中公
### 2007年06月22日(金)
16:30-18:00 数理科学研究科棟(駒場) 118号室
Qi Zhang 氏 (Missouri大学)
Projective varieties with nef anti-canonical divisors
[ 講演概要 ]
Projective varieties with nef anti-canonical divisors appear naturally in the minimal model program and the theory of classification of higher-dimensional algebraic varieties. In this talk we describe a comprehensive approach to birational geometry of log canonical pair (X, D) with nef anti-canonical class -(K_X+D). In particular, We present two theorems on the birational structure of the varieties. We will also discuss some recent results and new aspects of the subject.
### 2007年05月15日(火)
14:30-16:00 数理科学研究科棟(駒場) 122号室
いつもと時間・曜日・場所が異なります。
Mikhail Kapranov 氏 (Yale 大学)
Riemann-Roch for determinantal gerbes and smooth manifolds
[ 講演概要 ]
A version of the Riemann-Roch theorem for curves due to Deligne, describes the determinant of the cohomology of a vector bundle E on a curve.
If one realizes E via the Krichever construction, the determinant of the cohomology becomes a Hom-space in the determinantal gerbe for the vector space over the field of power series. So one has a local" Riemann-Roch problem of description of this gerbe itself. The talk will present the results of a joint work with E. Vasserot describing the class of such a gerbe in a family which geometrically can be seen as a circle fibration. This can be further generalized to the case of a fibration with fibers being smooth compact manifolds of any dimension d (joint work with P. Bressler, B. Tsygan and E. Vasserot).
### 2007年05月07日(月)
16:30-18:00 数理科学研究科棟(駒場) 122号室
Pathologies on ruled surfaces in positive characteristic
[ 講演概要 ]
We discuss some pathologies of log varieties in positive characteristic. Mainly, we show that on ruled surfaces there are counterexamples of several logarithmic type theorems. On the other hand, we also give a characterization of the counterexamples of the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface by means of the Tango invariant of the base curve.
### 2007年03月26日(月)
15:30-17:00 数理科学研究科棟(駒場) 128号室
Professor Caucher Birkar 氏 (University of Cambridge)
Existence of minimal models and flips (3rd talk of three)
### 2007年03月22日(木)
10:30-12:00 数理科学研究科棟(駒場) 128号室
Professor Caucher Birkar 氏 (University of Cambridge)
On boundedness of log Fano varieties (2nd talk of three)
### 2007年03月20日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Professor Caucher Birkar 氏 (University of Cambridge
)
Singularities and termination of flips (1st talk of three)
### 2007年01月26日(金)
16:30-17:30 数理科学研究科棟(駒場) 128号室
Professor Frans Oort
(Department of Mathematics
University of Utrecht
)
Irreducibility of strata and leaves in the moduli space of
abelian varieties I (a survey of results)
### 2006年12月08日(金)
15:00-16:25 数理科学研究科棟(駒場) 126号室
Stefan Kebekus 氏 氏 (Mathematisches Institut
Universität zu Köln
)
Rationally connected
foliations
### 2006年12月04日(月)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Professor Burt Totaro
(University of Cambridge)
When does a curve move on a surface, especially over a finite field?
### 2006年11月13日(月)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Hom stacks and Picard stacks
### 2006年10月18日(水)
16:00-18:15 数理科学研究科棟(駒場) 122号室
E. Esteves 氏 (IMPA) 16:00-17:00
Jets of singular foliations and applications
to curves and their moduli spaces
F. Zak 氏 (Independent Univ. of Moscow) 17:15-18:15
Dual varieties, ramification, and Betti numbers
of projective varieties
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# Rutgers Geometry/Topology seminar: Fall 2017 - Spring 2018
## Fall 2017
Date Speaker Title (click for abstract) Sep. 5th No Seminar Sep. 12th Hongbin Sun (Rutgers) Geometric finite amalgamations of hyperbolic 3-manifold groups are not LERF Sep. 19th Chenxi Wu (Rutgers) An upper bound on the translation length in the curve complex Sep. 26th Semeon Artamonov (Rutgers) Genus two analogue of A_1 spherical DAHA Oct. 3rd Xuwen Zhu (Stanford) Deformation theory of constant curvature conical metrics Oct. 6th (Friday) 1-2pm Hill 005 Priyam Patel (UC Santa Barbara) Lifting curves simply in finite covers Oct. 10th Sajjad Lakzian (Fordham University) Compactness theory for harmonic maps into locally CAT(1) spaces Oct. 17th Jiayin Pan (Rutgers) A proof of Milnor conjecture in dimension 3 Oct. 24th Lizhi Chen (Lanzhou University, China) Homology systole over mod 2 coefficients and systolic freedom Oct. 31st Asilya Suleymanova (Max Planck Institute, Germany) On the spectral geometry of manifolds with conic singularities Nov. 7th Nadav Dym (Weizmann Institute of Science, Israel) Provably good convex methods for mapping problems Nov. 14th Artem Kotelskiy (Princeton) Bordered theory for pillowcase homology Nov. 21st No Seminar Happy Thanksgiving! Nov. 28th Kyle Hayden (Boston College) Complex curves through a contact lens Dec. 5th Dec. 12th Sam Taylor (Temple University)
## Spring 2018
Date Speaker Title (click for abstract) Jan. 16th No Seminar Jan. 23rd Jan. 30th Joseph Maher (CUNY) Feb. 6th Feb. 13th Feb. 20th Feb. 27th March 6th March 13th No Seminar Spring break March 20th March 27th April 3rd April 10th April 17th April 24th
## Abstracts
### Geometric finite amalgamations of hyperbolic 3-manifold groups are not LERF
We will show that, for any two finite volume hyperbolic 3-manifolds, the amalgamations of their fundamental groups along nontrivial geometrically finite subgroups are always not LERF. A consequence of this result is: all arithmetic hyperbolic manifolds with dimension at least 4, with possible exceptions in 7-dimensional manifolds defined by the octonion, their fundamental groups are not LERF.
### An upper bound on the translation length in the curve complex
This is a collaboration with Hyungyul Bsik and Hyunshik Shin. We found an asymptotic upper bound for the translation length in the curve complex for primitive integer points in a fibered cone.
### Genus two analogue of A_1 spherical DAHA
Double Affine Hecke Algebra can be viewed as a noncommutative (q,t)-deformation of the SL(N,C) character variety of the fundamental group of a torus. This deformation inherits major topological property from its commutative counterpart, namely Mapping Class Group of a torus SL(2,Z) acts by automorphisms of DAHA. In my talk I will define a similar algebra for a closed genus two surface and show that the corresponding Mapping Class Group acts by automorphisms of such algebra. (This talk is based on arXiv:1704.02947 joint with Sh. Shakirov)
### Deformation theory of constant curvature conical metrics
In this joint work with Rafe Mazzeo, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce. This is one key ingredient to understand the full moduli space of such metrics with positive curvature and cone angles bigger than 2π.
### Lifting curves simply in finite covers
It is a well known result of Peter Scott that the fundamental groups of surfaces are subgroup separable. This algebraic property of surface groups also has important topological implications. One such implication is that every immersed (self-intersecting) closed curve on a surface lifts to an embedded (simple) one in a finite cover of the surface. A natural question that arises is: what is the minimal degree of a cover necessary to guarantee that a given closed curve lifts to be embedded? In this talk we will discuss various results answering the above question for hyperbolic surfaces, as well as several related questions regarding the relationship between geodesic length and geometric self-intersection number. Some of the work that will be presented is joint with T. Aougab, J. Gaster, and J. Sapir.
### Compactness theory for harmonic maps into locally CAT(1) spaces
We determine the complete bubble tree picture for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In the smooth setting, Parker established the bubble tree picture by exploiting now classical analytic results about harmonic maps. Our work demonstrates that these results can be reinterpreted geometrically. Indeed, in the absence of a PDE we prove analogous results by taking advantage of the local convexity properties of the target space. Included in this paper are an $\epsilon$-regularity theorem, an energy gap theorem, and a removable singularity theorem for harmonic maps. We also prove an isoperimetric inequality for conformal harmonic maps with small image (i.e. minimal surfaces in the non-smooth setting). This is a joint work with Christine Breiner.
### A proof of Milnor conjecture in dimension 3
We present a proof of Milnor conjecture in dimension 3, which says that any open manifold of non-negative Ricci curvature has a finitely generated fundamental group. The proof is based on the structure of Ricci limit spaces.
### Homology systole over mod 2 coefficients and systolic freedom
I am going to talk about problems around homology systole over mod 2 torsion coefficients. Given a Riemannian metric defined on a closed manifold, we define mod 2 homology systole to be the infimum of volumes of cycles representing nontrivial classes in homology group with mod 2 coefficients. Gromov conjectured that there would be systolic rigidity for mod 2 homology systoles, similar to homotopy 1-systolic inequalities on aspherical manifolds. However, later work shows that counterexample exists. In the talk, different aspects related to this conjecture will be explained. In particular, there are two types of motivations to study this problem. The first motivation is based on Gromov’s essential systolic inequality on aspherical manifolds. Another recent motivation is from quantum information theory.
### On the spectral geometry of manifolds with conic singularities
In this talk we consider the heat kernel of the Laplace-Beltrami operator on a Riemannian manifold. On any closed smooth Riemannian manifold the heat trace expansion gives some geometrical information such as dimension, volume and total scalar curvature of the manifold. On a manifold with conic singularities we derive a detailed asymptotic expansion of the heat trace using the Singular Asymptotics Lemma of Jochen Brüning and Robert T. Seeley. Then we investigate how the terms in the expansion reflect the geometry of the manifold. Can one hear a singularity?
### Provably good convex methods for mapping problems
Computing mappings or correspondences between surfaces is an important tool for many applications in computer graphics, computer vision, medical imaging, morphology and related fields. Mappings of least angle distortion (conformal) and distance distortion (isometric) are of particular interest. The problem of finding conformal/isometric mappings between surfaces is typically formulated as a difficult non-convex optimization problem. Convex methods relax the non-convex optimization problem to a convex problem which can then be solved globally. The main issue then is whether the global solution of the convex problem is a good approximation for the original global solution. In this talk we will discuss two families of convex relaxations.
Conformal: We relax the problem of computing planar conformal mappings (Riemann mappings) to a simple convex problem which can be solved by solving a system of linear equations. We show that in this case the relaxation is exact- the solution of the convex problem is guaranteed to be the Riemann mapping!
Discrete isometric: for perfectly isometric asymmetric surfaces, the well known doubly-stochastic (DS) relaxation is exact. We generalize this result to the more challenging and important case of symmetric surfaces, once exactness is correctly defined for such problems. For non-isometric surfaces it is difficult to achieve exactness. Several relaxations have been proposed for such problems, where the more accurate relaxations are generally also more time consuming. We will describe two algorithm which strike a good balance between accuracy and efficiency: The DS++ algorithm, which is provably better than several popular algorithms but does not compromise efficiency, and the Sinkhorn-JA algorithm, which gives a first-order algorithm for efficiently solving the strong but high-dimensional JA relaxation. We utilize this algorithmic improvement to achieve state of the art results for shape matching and image arrangement.
### Bordered theory for pillowcase homology
Pillowcase homology is a geometric construction, which was developed in order to better understand and compute a knot invariant I(K) called singular instanton knot homology. Motivated by the problem of extending pillowcase homology to tangles, we will introduce the following construction. The pillowcase P is a torus factorized by hyperelliptic involution, and after removing 4 singular points one obtains a 4-punctured 2-sphere P*. First, we will associate an algebra A to the pillowcase P*. Second, to an immersed curve L inside P* we will associate an A∞ module M(L) over A. Then we will show how, using these modules, one can recover and compute Lagrangian Floer homology (i.e. geometric intersection number) for immersed curves.
### Complex curves through a contact lens
Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)
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# How to Prove ($\mathbb{C}\langle x, y \rangle$, $\|\cdot\|$) is a Banach Space
Let $\mathbb{C}\langle x,y\rangle$ be the group ring of the complex numbers over the free group in $x,y$. Let $len : \langle x,y \rangle \rightarrow \mathbb{N}$ denote the standard word norm and let $\varphi=exp \circ len: \langle x, y \rangle \rightarrow [1,\infty)$. Define the following norm for $\displaystyle \alpha=\sum_{g \in \langle x,y\rangle} a_{g}g \in \mathbb{C}\langle x,y \rangle$:
$$\|\alpha\|=\sum_{g \in \langle x, y \rangle} |a_{g}|\varphi(g)$$
It's not hard to show that $\|\cdot\|$ is indeed a norm and in fact for all $\alpha, \beta \in \mathbb{C}\langle x, y \rangle$ we have $\|\alpha\beta\| \le \|\alpha\|\cdot\|\beta\|$. The natural involution $\displaystyle ^{*}:\sum_{g \in \langle x, y \rangle}a_{g}g \mapsto \sum_{g \in \langle x, y \rangle} \overline{a_{g}}g^{-1}$ has the properties that would make $(\mathbb{C}\langle x, y \rangle,\|\cdot\|,^{*})$ into a Banach *-algebra, including $\|\alpha\|=\|\alpha^{*}\|$. My question is this:
Is $\mathbb{C}\langle x, y \rangle$ complete with respect to $\|\cdot\|$?
I stumbled upon this while working on an undergrad research project but haven't had any functional analysis, so I'm not sure how to go about proving/disproving this. Any help/references would be greatly appreciated.
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When you write $r_g$ do you mean $a_g$? – Qiaochu Yuan Jun 7 '12 at 20:19
Yup, thanks for catching that and thanks for your answer! – Jackson Jun 7 '12 at 20:30
There are no countable-dimensional Banach spaces. This is a corollary of the Baire category theorem: if $e_1, e_2, ...$ were a basis for a Banach space $B$, then $B$ would be the countable union of the sets $\text{span}(e_1), \text{span}(e_1, e_2), ...$, all of which are nowhere dense, which contradicts BCT3 (in Wikipedia's terminology).
There is an obvious and more naive argument which doesn't work: if I had $e_1, e_2, ...$ as above, normalized to have norm $1$, then isn't it obvious that, say, $\sum \frac{e_n}{2^n}$ doesn't lie in the span of the $e_i$? This follows in the special case that the $e_i$ lie in a Hilbert space and are taken to be orthonormal, but in general you can't conclude this. Why? The naive continuation of the naive argument is that since $B = \text{span}(e_1, e_2, ...)$ there is a linear functional $$e_j^{\ast} : B \to \mathbb{C}$$
which, given a finite sum $\sum c_i e_i$, takes the value $c_j$, so we ought to have $$e_j^{\ast} \left( \sum \frac{e_n}{2^n} \right) = \frac{1}{2^j}.$$
Indeed there is such a linear functional, but it is not guaranteed to be continuous! So the last step fails. (When $H$ is a Hilbert space the inner product can be used to write down this functional and it is of course continuous in this case.)
In this particular case the functionals $e_j^{\ast}$ are continuous so the naive argument works and we can take, say, $\sum \frac{x^n}{n!}$ as an example of an element of the closure that doesn't lie in the space, but this general pitfall is worth looking out for. – Qiaochu Yuan Jun 7 '12 at 20:33
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## Continuous Function
A continuous function is a Function where the pre-image of every Open Set in is Open in . A function in a single variable is said to be continuous at point if
1. is defined, so that is in the Domain of .
2. exists for in the Domain of .
3. ,
where lim denotes a Limit. If is Differentiable at point , then it is also continuous at . If and are continuous at , then
1. is continuous at .
2. is continuous at .
3. is continuous at .
4. is continuous at if and is discontinuous at if .
5. is continuous at , where denotes , the Composition of the functions and .
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# What is the result of count function in the NULL value present in DB2 table?
The COUNT function in DB2 is used to return the number of rows which satisfies the given criteria. The GROUP BY is used to divide the rows in the groups based on the criteria given in the query.
If we perform the GROUP BY on INVOICE_ID and there are few rows having NULL value in INVOICE_ID then the null values form a separate group. For example, if we have below table.
ORDER_IDINVOICE_ID
A112343214
A556113214
A99867NULL
A556713214
A88907NULL
A560126701
On executing the query which performs GROUP BY on INVOICE_ID and counts the number of rows, we will get the result below.
SELECT INVOICE_ID, COUNT(*) AS INVOICE COUNT FROM ORDERS GROUP BY INVOICE_ID;
INVOICE_IDINVOICE COUNT
32143
NULL2
67011
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# How do you make the potential zero at infinity, here
## Main Question or Discussion Point
Potential difference is what which makes sense. But calculating the potential difference b/w two points and one of them being infinity, where we define potential to be zero, we can actually find the potential at the point. Fine, agreed. Now, how do you define the potential at infinity to be zero. E.g look at this thread: https://www.physicsforums.com/showthread.php?t=79683
The following point made in the post (by Andrew Mason) puzzles me.
http://www.flickr.com/photos/37453425@N07/3522609499/in/photostream/
Apparently, he has put ln(infinity) to be zero to get the solution. I feel that this isn't the right way to define the potential at infinity be zero. ('cos of course ln(infinity) is not zero) The solution, which is right, is also used in Feynman Lectures Vol II Section 14-3.
Am I missing out something here?
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Related Classical Physics News on Phys.org
rcgldr
Homework Helper
The formula in the image is wrong. Link to proper formula and integral:
The potential energy is based on the work done by "moving" from infinity to some point "r".
$$F = -\frac{GMm}{r^2}$$
$$U = -\int_\infty^r \frac{GMm}{s^2} ds$$
$$U = GMm(\frac{1}{r} - \frac{1}{\infty}) = GMm(\frac{1}{r})$$
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The formula in the image is wrong.
The formula in the image is for an infinitely long charged cylinder. The formula given is right (At least Feynman says so! - Refer the section I've mentioned earlier)
My question is: In deriving the formula for the potential, do you put ln(infinity) = 0 as in the linked thread (the part is shown in the image)
Born2bwire
Gold Member
I would say that the given formula is incorrect. He is calculating the potential difference, which would be infinite. The potential is the spatial integral of the electric field evaluated at a given point. The potential difference is the spatial integral of the electric field along a given path. With a 1/r potential, it will take an infinite amount of energy to move the charge from infinite to a given finite distance r.
For the most part though, potential is ambiguous, it is only really useful if we have a point of reference. In the question of his capacitance problem, the voltage difference between a point and infinite doesn't seem relevant. They are asking the capacitance of the cylinder due to the induction of charge from the line source. The cylinder is held to ground, so the surface of the cylinder should be given as having a potential of zero. Then your potential difference should be taken from the surface of the cylinder with the potential on the cylinder as being zero.
So I think Andrew may have made a mistake in defining his zero potential reference point.
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Absolutely.
All my doubt is: If you refer this - "Feynman Lectures Vol II Section 14-3" - he has written the formula for Potential (difference with the reference at infinity) arising due to infinitely long charged cylinder. Jus' show me the derivation of that.
Born2bwire
Gold Member
What does Feynman say? I don't have a copy the text at my office and I already made my trip to the library today to get textbooks on special relativity (bedtime reading, joy!).
rcgldr
Homework Helper
In general, if the attractive force from a point is relative to 1/r2 then for an infinitely long line it's 1/r and for an infinitely large plane, it's constant.
For for the cylinder case the standard definition doesn't work:
$$U = -c\int_\infty^r \frac{1}{s} ds$$
$$U = -c({ln}(r) - {ln}(\infty))$$
For the cylinder case, potential energy could be redefined as:
$$U = c\int_1^r \frac{1}{s} ds$$
$$U = c({ln}(r) - {ln}(1)) = c\ {ln}(r)$$
For the plane case, potential energy could be redefined as:
$$U = c\int_0^r \ ds$$
$$U = c(r - 0) = c\ r$$
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Oh... so you change the reference... Thnx :)
btw, this means the thing done in the post, I gave link to, is wrong... rite?
rcgldr
Homework Helper
Yes, ${ln}(\infty) = \infty$.
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# When I dissolve sugar in my cup of tea/coffee, does it become a liquid?
When I dissolve sugar in my cup of tea/coffee, does the sugar go from being a solid to being a liquid?
• Does NaCl dissolve apart into individual molecules? If so, how then can it be considered a solid anymore? If it doesn't, why? – User 17670 Sep 20 '14 at 18:11
• @User17670 NaCl is not molecular, it is a crystal. You'd expect it to come apart into its constituent ions (atoms). – Kyle Sep 21 '14 at 3:26
• @Kyle +1 for interesting statement! So, the Na would break apart from the Cl? Would this only happen above the melting point? – User 17670 Sep 21 '14 at 12:10
• @User17670 You could melt it, but you can also dissolve it in e.g. water well below the melting point (melting point of NaCl is quite high, hundreds of degrees, but it dissolves easily at room temperature). – Kyle Sep 21 '14 at 18:46
• @dani table salt dissolved in water is most definitely not a solid: it is a solution of table salt in water. Under what definition of solid can it possibly be still considered a solid? – matt_black May 28 '18 at 16:09
When the sugar dissolves, it is broken up into individual particles; each molecule is surrounded by water molecules. Essentially, you are trying to decide if a single molecule is a solid or liquid when dissolve.
The best fit is a liquid since the molecules are free to move independently and will take on the shape of the container.
• I've thought about it a bit more and I think that the 'best' answer is that 'it doesn't make sense to consider only the sugar' - e.g. to decide if the sugar is a liquid or a gas, you have to say that it is compressible or not. On the one hand, you have sugar particles that are well spaced like a gas, but on the other hand, you can't really ask whether the sugar is compressible. So, I'm happy to go with 'it becomes part of the liquid'. – User 17670 Sep 21 '14 at 12:09
• Indeed, usually in chemistry, a dissolved compound is indicated with the (aq) subscript, like $\ce{NaCl(aq)}$ rather than (s), (l), or (g). In essence, we're indicating it's not solid, liquid, or gas really, but something a bit different - a mixture. – Geoff Hutchison Sep 22 '14 at 15:23
• (aq) specifically means dissolved in water (aqua). There are plenty of other solvents. – MSalters Sep 23 '14 at 15:14
• @MSalters yes, it specifically means water, but the OP asked about dissolving sugar in tea (i.e., water solution). Thus my answer. – Geoff Hutchison Sep 23 '14 at 15:45
• WTF? How is the best fit a liquid? Sugar in water isn't a liquid, it is a solution: a completely different thing. – matt_black May 28 '18 at 16:05
When I dissolve sugar in my cup of tea/coffee, does it become a liquid?
The premise of your question is wrong. There is no more "it" for the sugar when dissolved in water. You can't just consider the sugar alone. Rather the sugar becomes a constituent of the liquid phase.
When we dissolve sugar in tea or coffee,the sugar particle get dissociated into smaller particle. As you might be knowing there is inter molecular distance in case of solid,liquid and gas as well. In case of liquid the particles are loosely held compared to solid and also inter molecular distance is more as to solid. So finally particles get packed in those interstitial spaces.so the sugar particles do not get transformed to liquid or gas rather it get surrounded by water molecules.
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Sugar added to tea becomes a solution not a liquid
Sugar is very soluble in water. When you add the solid to the tea the key process is that the solid sugar dissolves in the warm liquid: the solid crystals are broken up into molecules which are every dispersed throughout the existing liquid. When well mixed (because sugar doesn't dissolve instantly) the liquid is homogeneous with the sugar molecules evenly distributed in the single phase bulk liquid. There is no point where the sugar crystals become liquid sugar. And no chemist would describe the result as a sugar liquid. The correct description is a solution of sugar in water.
When sugar dissolves into tea or coffee, the liquid transforms the sugar into a liquid so it can fit in with the liquid and slide in with the molecules. If you try to evaporate the water for long enough, you will turn the sugar back into a solid.
• Sugar is not transformed into a liquid: it is dissolved. – matt_black May 28 '18 at 16:07
when you dissolve sugar crystals into tea or coffee the tea turns the sugar (solid) into a liquid.It separates the participles in the sugar crystals into smaller particles.
• No. Things dissolved in other liquids are not themselves described as liquids. – matt_black May 28 '18 at 16:06
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# A simple clock in FORTH
I try to program a stopwatch/countdown clock in FORTH with Gforth (and using Gforth-specific words).
I'm a complete beginner and the following code is the basic stuff (going to add an alarm function / countdown option and more).
For now the stopwatch counts from 00:00:00 (hh:mm:ss) up to 24:00:00 and
• pressing space pauses the clock, pressing it again resumes the clock
• pressing j/k sets the clock back/ahead one minute
• pressing J/K sets the clock back/ahead one hour
• pressing q quits the program
Does my code follow best practices? Is it bad style for words like move-clock-seconds-back to return flags because they will be used in a begin ... until statement?
1000000 constant million
: sextal ( -- )
6 base ! ;
: hhmmss. ( ud -- )
drop million / 0 <# decimal # sextal # [char] : hold decimal # sextal # [char] : hold decimal # sextal # decimal #> TYPE ;
: pause-clock ( ud -- ud f )
utime
begin
50 ms
key? if
key bl = ( pause is released )
else
false ( clock keeps pausing )
then
until
utime d- d-
false ;
: move-clock-seconds-ahead ( u ud -- ud f )
million * 0 d-
false ;
: move-clock-seconds-back ( u ud -- ud f )
million * 0 d+
utime dmin
false ;
: get-elapsed-time ( ud -- ud )
2dup
utime
2swap d- ;
: 24-hours-elapsed? ( ud -- f )
get-elapsed-time ( elapsed time fits in single integer )
drop 24 60 * 60 * million * u> ;
: run-clock ( -- )
page ( clears the terminal )
utime ( returns a ud timestamp in microseconds )
begin
50 ms ( sleep for 50 ms )
get-elapsed-time
5 0 at-xy ( coordinates where to print output )
hhmmss.
key? if
key case
bl of pause-clock endof
[char] j of 60 move-clock-seconds-back endof
[char] k of 60 move-clock-seconds-ahead endof
[char] J of 60 60 * move-clock-seconds-back endof
[char] K of 60 60 * move-clock-seconds-ahead endof
[char] q of true endof
false swap ( the char is now on top of the stack, will be dropped by endcase )
endcase
else 24-hours-elapsed? if
CR
." 24 hours elapsed."
true
else
false
then then
until ;
run-clock
bye
The other review covers some valuable points, so I'll just concentrate on adding additional suggestions here.
## Consider portability
There does not really seem to be a need for microsecond precision for this timer, so I'd suggest that rather than using the non-standard utime, perhaps time&date (which is standard) could be used.
## Consider restoring the number base
When I program in Forth, I'm often using it in hex mode. If I used your hhmmss. word, I'd be annoyed that it didn't restore the original base. It only takes a few extra words to store and restore base.
## Consider explicitly setting the base
When the program defines million it will be using whatever base had previously been set, which will result in very strange behavior if it's not decimal. Good practice is to explicitly set the base so that this can be a standalone module.
There are a few comments within the code, but I'd suggest that each function could have a comment immediately above the definition to describe what it does. The usual convention is to use \ for such comments and to use ( -- ) for stack comments as you're already doing.
Describing what page does in an associate comment isn't very useful, since page is a standard word and should be well understood. Better would be to document what's assumed to be on the stack (semantically, not just how big and how many)
## Have each word do just one thing
It's already been mentioned that the drop that begins hhmmss. is very odd and not a good idea. Similarly, ending a number of the other words with false is at best counterintuitive. Generally, I follow the guideline that the only things on the stack are things that are required for the particular word being defined. This makes it up the caller to do whatever stack manipulations are required to put things in the right place. It might make the program slightly longer, but it is very likely to make it a lot easier to interactively debug.
## Use more constants
The number of milliseconds in 24 hours could be calculated once, put into a constant and then used instead of calculating it each time through 24-hours-elapsed?.
The stack comments for move-clock-seconds-ahead and move-clock-seconds-back are not correct. Instead of ( u ud -- ud f ), they should both read ( ud u -- ud f ).
## Fix minor typos
Correcting the error in one of the comments (pring -> print) and fixing the formatting of the first case (it should be indented with the other cases).
## Simplify the logic
Right now, the code contains these lines:
else 24-hours-elapsed? if
CR
." 24 hours elapsed."
true
else
false
then then
I think it would be a little more concise to write it instead like this:
else 24-hours-elapsed? dup if
CR
." 24 hours elapsed."
then then
A comment in
: get-elapsed-time ( ud -- ud )
seems wrong. The code leaves the elapsed time on top of the otherwise unchanged stack, so the action is really (-- ud).
The way get-elapsed-time leaves a double results in very unnerving drop in hhmmss. and 24-hours-elapsed?. I strongly recommend to make it leave a single integer instead.
I also recommend to reduce stack manipulations with
: get-elapsed-time
utime
2over
d- drop ;
Is it bad style for words like move-clock-seconds-back to return flags
Yes. It makes it very hard to reuse them in other contexts. Consider (untested)
begin key dup [char] q <> while
case
....
endcase
repeat
• It probably would be good to rename get-elapsed-time into get-elapsed-seconds because it's the first time seconds are used instead of microseconds, right? – wolf-revo-cats Feb 22 '17 at 3:07
• it's true, that get-elapsed-time leaves the stack unchanged, yet for an error not to occur, it needs the starting time as an ud on the stack. The commenting style in Leo Brodie's Starting FORTH is so that in such a case ud would be stated before --. If it's about change, it would also be 2DUP ( -- d d ) (because 2DUP does nothing to the d-number on the stack) but in reality it's 2DUP ( d -- d d ) in the book. – wolf-revo-cats Feb 22 '17 at 3:09
• I think that : get-elapsed-time ( ud -- ud ud ) would be correct? – wolf-revo-cats Feb 22 '17 at 3:18
• I agree that it should be ( ud -- ud ud ) unless you rewrite it to do something more like ( ud -- ud u ) since the code that uses it always drops the high half anyway. – Edward Feb 22 '17 at 3:51
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# If declarative programming is possible at the instruction/action level?
I am considering what the possibilities are with declarative programming. I have a firm understanding of how to use declarative programming in practice, but, short of having examples, I don't know if it's possible to 100% use declarative programming all the way down to the machine code (theoretically).
Standard QA sites like this says stuff like:
it means describing the problem to be solved, but not telling the programming language how to solve it
A typical example is a SQL query which uses a where clause to define the query rather than implementing the query it step-by-step using something like an iterative loop.
An example I am interested in is an even higher level abstraction than a SQL query: defining an HTTP server. You can define the server and all of it's GET handlers (if we just limit it to that for now) in a completely declarative way, as seen by many declarative frameworks. The declarative frameworks that do similar stuff to this are the ones like Terraform or SaltStack that define machine configurations/states. You are basically defining the state the machine should look like, and the system takes your "definition" and makes it happen.
For the HTTP server you can do like in React:
<Route path='/' component={Home}>
<Route path='/terms' component={Terms}>
...
</Route>
This then automagically gets wired up into a listening server for these changes, and renders the components (this API may not be exact, just pseudocoding it ftm).
In fact, there is a lot that can be done declaratively. HTML and CSS together form an animatable graphics display and is completely declarative. You can define servers, machines, parsers (PEG parser generators), graphics... I don't know what else.
That's mainly what my question is, what can be done declaratively. More specifically, what cannot be done in a declarative fashion. The stereotypical example I am thinking of is simply defining a step-by-step sequence of actions of some sort, but in a declarative, non-iterative way. Is it possible?
Here is my attempt.... Say we are describing some automation steps for interactively going through and clicking around a website. Here is the iterative way:
Visit "/"
Click on "Button 1"
Wait "2 seconds"
Click on "Button 2"
For each "Input" in "Fields"
Type into "Input"
Click on "Button 3"
Wait "2 seconds"
...
This is similar to assembly instructions like mov. But can this be done in a declarative way (that is also readable and easy to understand)? Does functional programming offer any insight? Maybe one could show how to do this in a functional programming paradigm, or even a logic (prolog-ish) way, if that would help illuminate how it's done. My attempt at declarative is like this:
Current State should be "/"
Button is not clicked
Button has been clicked
2 seconds have passed
...
Basically, just stating the "states" before and after each step sort of thing. That makes it harder to read, and it feels like you are trying to hack around having to build some iterative code anyways. It's like it has to be iterative. Is that so? Is there any way to make this non-iterative and so it still makes sense?
To me, currently, it seems that you can use declarative programming for like 80-90% of things, but then you inevitably need "parsing" code which will take those declarative statements and interpret them, converting them into some iterative sequence. But what I'm left wondering is, can that 10-20% that seems to have "iteration" as a requirement be instead written declaratively? Can you take that 10-20% remaining iterative code and make that declarative too? If so, how would that look or what is the technique?
Machine code is imperative, not declarative, so if you want your code to execute on a conventional CPU, at some point there needs to be some code that converts the declarative program to imperative style (e.g., a compiler), and at least some of that code needs to be written imperatively (because at least some of it needs to be written in machine code). It can't be turtles all the way down.
AFAIK, the meaning of the term “declarative vs. imperative” changed over time. The original meaning is what you cited, “how vs. what”. This distinction is philosophical, so its meaning varies from person to person. To be honest, after all these years, I failed to understand it, so I will not discuss it here.
A more specific meaning of “imperative” emerged: a programming language is imperative if it has the concept of command and a program written in the language defines the order of commands. For example, in C, commands are called statements, and the implementation is required to execute statements in the order that they are written and even execute subexpressions of an expression in the order that they are written. Hence C is considered imperative. The programming languages Haskell and Prolog (without extra-logical features), according to this definition of “imperative”, are non-imperative, in other words, declarative.
In principle, the question is answered by the Turing Thesis. Since there are Turing-complete declarative languages (both Haskell and Prolog are), declarative languages can do everything that imperative languages can. Even pure lambda calculus, having only 3 syntax constructs, is Turing-complete. Haskell provides a feature for convenient translation of imperative programs into Haskell, the IO monad. The translated program looks almost like an original imperative one.
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# zbMATH — the first resource for mathematics
Blowup theory for the critical nonlinear Schrödinger equations revisited. (English) Zbl 1126.35067
Consider the nonlinear Schrödinger equation $i\partial_{t}u+\bigtriangleup{u}+| u| ^{\tfrac{4}{d}}u=0,\quad x\in\mathbb{R}^d, \;t>0.$ The authors prove the following theorem: Let $$\{\nu_{n}\}_{n=1}^{\infty}$$ be a bounded family of functions in $$\text{H}^1(\mathbb{R}^d)$$ such that $\limsup_{n\to\infty}| | \bigtriangledown\nu_{n}| | _{\text{L}^2}\leq{M},\quad \limsup_{n\to\infty}| | \nu_{n}| | _{\text{L}^{\tfrac{4}{d}+2}}\geq{m}.$ Then, there exists $$\{x_{n}\}_{n=1}^{\infty}\subset\mathbb{R}^d$$ such that, up to a subsequence, $\nu_{n}(\cdot\;+x_{n})\rightharpoonup\text{V}\;\text{weakly,}$ with $$|\text{V}\| _{\text{L}^2}\geq{(\tfrac{d}{d+2})^{\tfrac{d}{4}}}(\frac{m^{\tfrac{d}{2}+1}} {M^{\tfrac{d}{2}}})\|Q\|_{\text{L}^2},$$ with $$\bigtriangleup{Q}-Q+| Q| ^{\tfrac{4}{d}}Q=0$$.
##### MSC:
35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
Cauchy problem; Sobolev space; blow up
Full Text:
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# An ideal that is maximal among infinitely-generated ideals is prime.
I've been doing some old exam problems and I've come across a problem that I've answered, but my gut is telling me that there's something I'm glossing over.
Let $R$ be a commutative ring with identity and let $U$ be an ideal that is maximal among non-finitely generated ideals of $R$. I wish to show that $U$ is a prime ideal.
Assume that $U$ is not prime. Let $x, y\not\in U$ be such that $xy\in U$. $U$ is contained in a maximal ideal $M$ and $xy\in M$, so either $x$ or $y$ is in $M$; assume $x\in M$. The condition $U\subset M$ then implies that there is a ring homomorphism $$\varphi: R/M\to R/U$$
Since $R/M$ is a field, $\varphi$ is injective. Hence, $\varphi(x)\in U$. This is a contradiction, so $U$ must be prime.
The thing that worries me is that I never explicitly used the hypothesis that $U$ was not finitely generated or the result that $M$ must be finitely generated.
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Do you want the map to go the other way? You can't "un-mod out" :) – Dylan Moreland May 19 '12 at 1:06
Ah, you're right; the map's not well-defined this way. I'll have to try to see what falls out of the reverse map being a surjection now, if that's the right way to go about this problem. – Connor May 19 '12 at 1:09
You also meant, I suppose, "R/M is a field", not U – DonAntonio May 19 '12 at 1:21
Correct as well. I've changed it, though it's not of much importance now. – Connor May 19 '12 at 1:22
@acoustician By the way a corollary of this result is a result due to I.S. Cohen that if every prime ideal in a ring $A$ is finitely generated then $A$ is Noetherian. – fpqc May 19 '12 at 4:47
I propose the following: suppose $\,U\,$ is not prime, thus there exist $$\,x,y\in R \,\,s.t.\,\,x,y\notin U\,,\,xy\in U\,$$ Define now $\,A:=U+\langle x\rangle$, $B:=U+\langle y\rangle$.
By maximality of $\,U\,$ both $\,A\,,\,B\,$ are f.g., say $$\,B=\Bigl\langle u_i+r_iy\,,\,1\leq i\leq k\,,\,k\in\mathbb{N}\,\,;\,\,u_i\in U\,,\,r_i\in R\Bigr\rangle$$ and let now $$U_y:=\{s\in R\,\,;\,\,sy\in U\}$$ (1) Check that $\,U_y\,$ is a proper ideal in $\,R\,$
(2) Show that $\,U_y\,$ is f.g.
Put $\,U_y=\langle s_1,\ldots,s_m\rangle\,$, and take $\,u\in U\Longrightarrow\,\exists v_1,\ldots,v_k, t_1,\ldots,t_k\in R\,\,s.t.$$u=\sum_{n=1}^kv_nu_n+\sum_{n=1}^kt_nr_ny$$ (3) Show that$\displaystyle{\sum_{n=1}^kt_nr_n}\in U_y\,$(4) Putting$\,\Omega:=\{u_i,\ldots,u_k,ys_1,\ldots,ys_m\}\,$, derive the contradiction$\,U=\langle\Omega\rangle$- Acoustician, the fact is B is fin. gen. and ever y single generator is of the form$u_i+r_iy\,,\,u_i\in U, r_i\in R\,$, so what this says is that we need only a finite number of R-multiples of y and a finite number of elements in U to generate B. I can't see how can we dispense of these elements... – DonAntonio May 19 '12 at 2:16 I know, I quickly saw my mistake and deleted the comment, but it seems you received it before that :). Thanks very much for your answer! It's clear to me now, though the proof was much more involved than I anticipated. – Connor May 19 '12 at 2:24 @DonAntonio:$\LaTeX$tip: use \langle and \rangle for angle bracket delimiters, to get$\langle$and$\rangle$. < and > give the wrong spacing. – Arturo Magidin May 19 '12 at 3:43 Gracias, Arturo. Lo pondré en práctica en los próximos posts. – DonAntonio May 19 '12 at 12:09 +1 A nice answer! – Jyrki Lahtonen May 19 '12 at 13:23 add comment Wait the map$\phi$goes from$R/U \rightarrow R/M$yeah? Like,$\mathbb{Z}/4\mathbb{Z} \rightarrow \mathbb{Z}/2\mathbb{Z}$. - Yes, as Dylan mentioned above; thank you :) – Connor May 19 '12 at 1:10 add comment I don't know how excited you might be about deeper results along these lines, and the question is completely answered already, but I can't resist mentioning some deeper results. It turns out there are a lot of results of "maximal-implies-prime" flavor (like "ideal maximal among non-principal ideals", "ideal maximal among non-countably generated", "maximal among point annihilators of a module", "ideal maximal among ideals disjoint from a multiplicative set"). For a long time, these were proven on an ad hoc basis, but Lam and Reyes managed to get them all (and apparently new ones!) in one fell swoop. In another paper their approach is used to generalize some classical results of Kaplansky and Cohen about properties of prime ideals propagating to all ideals. They are really fantastic papers, that I think anyone would enjoy. Here are the four papers I highly recommend, posted at Reyes' website: http://www.math.ucsd.edu/~m1reyes/oka1.pdf http://www.math.ucsd.edu/~m1reyes/ams-oka2.pdf http://www.math.ucsd.edu/~m1reyes/cpip.pdf http://www.math.ucsd.edu/~m1reyes/cohenkaplansky.pdf Enjoy! - Thanks for the extra results! I'm actually a student at UCSD, so this is a pretty cool to discover. – Connor May 19 '12 at 2:26 @acoustician See also my answer here. – Bill Dubuque May 19 '12 at 2:54 +1: I was about to start looking for links to the Lam-Reyes paper. – Arturo Magidin May 19 '12 at 3:46 add comment Can't we modify the answer Zev gave to the question to which Bill linked in his above comment? Let$x,y \notin U$. We want$xy \notin U$. Notice that$U + \langle x \rangle, U+ \langle y \rangle$are ideals properly containing$U$and are not contained in$\Sigma = \lbrace I \subset R \; | \; I\;\text{infinitely generated} \rbrace.$Thus,$U + \langle x \rangle, U+\langle y \rangle$are finitely generated, which implies$(U+\langle x \rangle) + (U+\langle y \rangle) = U+\langle x \rangle + \langle y \rangle$is finitely generated. But$U+\langle xy \rangle \subset U+\langle x \rangle + \langle y \rangle$, which implies$U+\langle xy \rangle$is finitely generated and is thus not contained in$\Sigma$. More importantly, we then know that$U \subsetneqq U+\langle xy\rangle$and so$xy \notin U$, which is what we wanted. - In the line just before the last one above you seem to be implying that an ideal contained in a f.g. generated ideal is itself f.g....I don't think this is true in general, but even if it were then the proof above would stop at$\,U\leq\,U+\langle x\rangle\,$, and the ideal in the right is f.g. If you deduce that$\,U+\langle xy\rangle\,$is f.g. because of another reason that I can't see I'd appreciate if you could tell which one. – DonAntonio May 19 '12 at 18:09 Yeah I don't think this is right either; wouldn't U+<xy>=U, so that it's still infinitely generated? – Connor May 19 '12 at 19:52 @DonAntonio You're absolutely right (I believe the polynomial ring over$\mathbb{Z}$in infinitely many indeterminates gives some examples of this phenomenon). I should've seen this error. Although, I wonder if there's a quick way to conclude that$U+\langle xy \rangle\$ is f.g., which would maintain the essence of this proof. Otherwise, I think you hit the nail on the head with your answer. – Derek Allums May 20 '12 at 1:44
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# Accuracy of the s-step Lanczos method for the symmetric eigenproblem
### http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-165.pdf
The $s$-step Lanczos method is an attractive alternative to the classical Lanczos method as it enables an $O(s)$ reduction in data movement over a fixed number of iterations. This can significantly improve performance on modern computers. In order for $s$-step methods to be widely adopted, it is important to better understand their error properties. Although the $s$-step Lanczos method is equivalent to the classical Lanczos method in exact arithmetic, empirical observations demonstrate that it can behave quite differently in finite precision.
In this paper, we demonstrate that bounds on accuracy for the finite precision Lanczos method given by Paige [\emph{Lin. Alg. Appl.}, 34:235--258, 1980] can be extended to the $s$-step Lanczos case assuming a bound on the condition numbers of the computed $s$-step bases. Our results confirm theoretically what is well-known empirically: the conditioning of the Krylov bases plays a large role in determining finite precision behavior. In particular, if one can guarantee that the basis condition number is not too large throughout the iterations, the accuracy and convergence of eigenvalues in the $s$-step Lanczos method should be similar to those of classical Lanczos. This indicates that, under certain restrictions, the $s$-step Lanczos method can be made suitable for use in many practical cases.
BibTeX citation:
@techreport{Carson:EECS-2014-165,
Author = {Carson, Erin and Demmel, James},
Title = {Accuracy of the s-step Lanczos method for the symmetric eigenproblem},
Institution = {EECS Department, University of California, Berkeley},
Year = {2014},
Month = {Sep},
URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-165.html},
Number = {UCB/EECS-2014-165},
Abstract = {The $s$-step Lanczos method is an attractive alternative to the classical Lanczos method as it enables an $O(s)$ reduction in data movement over a fixed number of iterations. This can significantly improve performance on modern computers. In order for $s$-step methods to be widely adopted, it is important to better understand their error properties. Although the $s$-step Lanczos method is equivalent to the classical Lanczos method in exact arithmetic, empirical observations demonstrate that it can behave quite differently in finite precision.
In this paper, we demonstrate that bounds on accuracy for the finite precision Lanczos method given by Paige [\emph{Lin. Alg. Appl.}, 34:235--258, 1980] can be extended to the $s$-step Lanczos case assuming a bound on the condition numbers of the computed $s$-step bases. Our results confirm theoretically what is well-known empirically: the conditioning of the Krylov bases plays a large role in determining finite precision behavior. In particular, if one can guarantee that the basis condition number is not too large throughout the iterations, the accuracy and convergence of eigenvalues in the $s$-step Lanczos method should be similar to those of classical Lanczos. This indicates that, under certain restrictions, the $s$-step Lanczos method can be made suitable for use in many practical cases.}
}
EndNote citation:
%0 Report
%A Carson, Erin
%A Demmel, James
%T Accuracy of the s-step Lanczos method for the symmetric eigenproblem
%I EECS Department, University of California, Berkeley
%D 2014
%8 September 17
%@ UCB/EECS-2014-165
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-165.html
%F Carson:EECS-2014-165
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## This Post Has 4 Comments
1. Expert says:
step-by-step explanation:
$Explain how you would find the measure of angle a. look at the image linked.$
2. Expert says:
You need to include the original coordinates of the triangle to answer this. and how much does it translate (move left right up down)
$This is for a test need answer quick and brainiest if good answer triangle tri is translated horizon$
3. Expert says:
Aquadratic function is one of the form f(x)=ax2 + bx + c l, where a,b, and c are numbers with are not equal to zero.
4. Expert says:
2: 10
step-by-step explanation:
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category theory
# Contents
## Definition
###### Definition
An object $X$ in a category $C$ with a zero object $0$ is simple if there are precisely two quotient objects of $X$, namely $0$ and $X$.
###### Remark
If $C$ is abelian, we may use subobjects in place of quotient objects in the definition, and this is more common; the result is the same.
###### Remark
The zero object itself is not simple, as it has only one quotient object. It is too simple to be simple.
###### Remark
In constructive mathematics, we want to phrase the definition as: a quotient object of $X$ is $X$ if and only if it is not $0$.
###### Definition
An object which is a direct sum of simple objects is called a semisimple object.
## Properties
### In an abelian category
###### Proposition
In an abelian category $C$, every morphism between simple objects is either a zero morphism or an isomorphism. If $C$ is also enriched in finite-dimensional vector spaces over an algebraically closed field, it follows that $\hom(X, Y)$ has dimension $0$ or $1$.
## Examples
• In the category Vect of vector spaces over some field $k$, the simple objects are precisely the lines: the $1$-dimensional vector spaces, i.e. $k$ itself, up to isomorphism.
• A simple group is a simple object in Grp. (Here it is important to use quotient objects instead of subobjects.)
• For $G$ a group and $Rep(G)$ its category of representations, the simple objects are the irreducible representations.
• A simple ring is not a simple object in Ring (which doesn't have a zero object anyway); instead it is a ring $R$ that is simple in its category of bimodules.
• A simple Lie algebra is a simple object in LieAlg that also is not abelian. As an abelian Lie algebra is simply a vector space, the only simple object of $Lie Alg$ that is not accepted as a simple Lie algebra is the $1$-dimensional Lie algebra.
Revised on September 7, 2014 05:30:18 by Anonymous Coward (90.6.35.180)
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# Calculus II
## Day 12-Lecture 24
16 March 2017, Thursday 9:40
Previous Screenshot Next Screenshot
# Problems on Lagrange multipliers method
This is how we typically solve the above system of equations.
First assume that $x=0$.
If $y=0$, then from (2) we have $z=0$, but then $(x,y,z)=(0,0,0)$ violates (4), so $y\not=0$.
If $\lambda=0$, from (2) we have $z=0$, which in turn gives $y=0$ from (3) contrary to what we just found about $y$. So $\lambda\not=0$. From (2) it now follows that $z\not=0$ also.
At this point we have $x=0$ and $y,z,\lambda\not=0$.
Joining (2) and (3) we find $z=\lambda y=\lambda^2 z$. Cancelling $z$ from both sides (note that we are using $z\not=0$) we get $\lambda=\pm 1$.
If $\lambda=1$, then from (2) we have $y=z$ and from (4) we get $\displaystyle y=\pm \frac{1}{\sqrt{2}}$.
We obtained the points $\displaystyle p_1=(0,\pm \frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{2}})$.
If $\lambda=-1$, then from (2) we have $y=-z$ and from (4) we get $\displaystyle y=\pm \frac{1}{\sqrt{2}}$.
We obtained the points $\displaystyle p_2=(0,\pm \frac{1}{\sqrt{2}},\mp \frac{1}{\sqrt{2}})$.
Next assume $x\not=0$.
From (1) we immediately get $\lambda\not=0$.
From (2) we see that either "$y=0$ and $z=0$" or "$y\not=0$ and $z\not=0$".
If $y=0$ and $z=0$, then from (4) we get $x=\pm 1$.
We obtained the points $p_3=(\pm 1,0,0)$.
If $y\not=0$ and $z\not=0$, then as above we can join (2) and (3) to obtain $\lambda=\pm 1$.
If $\lambda=1$, then from (1) we get $\displaystyle x=\frac{1}{3}$.
From (2) we get $z=y$, and from (4) we get $\displaystyle y=\pm\frac{2}{3}$.
We obtained the points $\displaystyle p_4=(\frac{1}{3},\pm\frac{2}{3},\pm\frac{2}{3})$.
If $\lambda=-1$, then from (1) we get $\displaystyle x=-\frac{1}{3}$.
From (2) we get $z=-y$, and from (4) we get $\displaystyle y=\pm\frac{2}{3}$.
We obtained the points $\displaystyle p_5=(-\frac{1}{3},\pm\frac{2}{3},\mp\frac{2}{3})$.
This detailed analysis of all cases is crucially necessary for finding all the solutions. In particular pay attention before you cancel anything: if what you want to cancel is zero, you cannot cancel but in that case you may find other solutions.
For questions
Previous Screenshot Next Screenshot
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# American Institute of Mathematical Sciences
January & February 2004, 10(1&2): 211-238. doi: 10.3934/dcds.2004.10.211
## Uniform exponential attractors for a singularly perturbed damped wave equation
1 Université Bordeaux-I, Mathématiques Appliquées, 351 Cours de la Libération, 33405 Talence Cedex, France, France 2 Laboratoire d'Applications des Mathématiques - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, Chasseneuil Futuroscope Cedex, France 3 Université de Poitiers, Laboratoire d'Applications des Mathématiques - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex, France
Received November 2001 Revised March 2003 Published October 2003
Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.
Citation: Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 211-238. doi: 10.3934/dcds.2004.10.211
[1] Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351 [2] P. Fabrie, C. Galusinski, A. Miranville. Uniform inertial sets for damped wave equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 393-418. doi: 10.3934/dcds.2000.6.393 [3] Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041 [4] Ciprian G. Gal. Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 819-836. doi: 10.3934/cpaa.2008.7.819 [5] John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31 [6] Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019101 [7] Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015 [8] Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719 [9] Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823 [10] Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068 [11] Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257 [12] Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 [13] Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079 [14] Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure & Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771 [15] Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1001-1030. doi: 10.3934/dcds.2011.29.1001 [16] A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119 [17] Björn Birnir, Kenneth Nelson. The existence of smooth attractors of damped and driven nonlinear wave equations with critical exponent , s = 5. Conference Publications, 1998, 1998 (Special) : 100-117. doi: 10.3934/proc.1998.1998.100 [18] Fengjuan Meng, Meihua Yang, Chengkui Zhong. Attractors for wave equations with nonlinear damping on time-dependent space. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 205-225. doi: 10.3934/dcdsb.2016.21.205 [19] Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985 [20] T. F. Ma, M. L. Pelicer. Attractors for weakly damped beam equations with $p$-Laplacian. Conference Publications, 2013, 2013 (special) : 525-534. doi: 10.3934/proc.2013.2013.525
2017 Impact Factor: 1.179
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# Understanding how to compute sigma algebra
I'm having trouble understanding the production rules for sigma algebra. I know there are the following requirements for the sigma algebra:
1. $$\emptyset \in \mathcal A$$
2. when $$A \in \mathcal A$$ the $$A^c \in \mathcal A$$
3. when $$A_{1},A_{2},A_{3},\dots \in \mathcal A$$ then $$\cup _{i=1}^{\infty }A_{i} \in \mathcal A$$
My problem is the 3rd rule, I assumed that this means that the union of all (inkl. generated) elements has to be a subset as well. But with the following example, this was not the case.
Given $$\Omega = \{1, 2, 3, 4\}$$ the minimal sigma algebra containing $$\{1\}, \{1,2\}$$ is $$\{\emptyset, \{1\}, \{1, 2\}, \{2, 3, 4\}, \{3, 4\}, \{1, 2, 3, 4\}, \{1, 3, 4\}, \{2\}\}$$
but why would $$\{1, 3, 4\}, \{2\}$$ be part of the sigma algebra when rule three just creates a union of all subsets?
Thanks for your help!
You don't only use the rules on the sets you start with. Instead, we have to continue applying the rules to all of the new sets we find from previous applications. Note that $$\{2\}=\{1\}^c\cap\{1,2\}=(\{1\}\cup\{1,2\}^c)^c$$ and $$\{1,3,4\}=\{2\}^c.$$
Because $$A\cap B=(A^c\cup B^c)^c$$ $$A\setminus B=A\cap B^c$$ and hence including both unions and complements means that you also include intersections, differences and so on.
For example $$\{2\}=\{1,2\}\setminus\{1\}=\{1,2\}\cap\{1\}^c=(\{1,2\}^c\cup\{1\})^c$$
To convince yourself that the minimal sigma-algebra containing $$\{1\}$$ and $$\{1,2\}$$ is the one you wrote, simply try to remove one of the elements of the sigma-algebra and explain why it is not a sigma-algebra anymore.
Call $$\mathcal{A}$$ your sigma-algebra. In particular, $$\{1\}\in \mathcal{A}$$ and $$\{3,4\}\in \mathcal{A}$$ by rule 2, being it the complement of $$\{1,2\}\in \mathcal{A}$$, thus $$\{1\}\cup \{3,4\}=\{1,3,4\}\in \mathcal{A}$$ by rule 3. Now $$\{2\}\in \mathcal{A}$$ since $$\{1,3,4\}\in \mathcal{A}$$ and rule 2 again.
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# DoITPoMS
Constitutional Undercooling
It is actually rare to have a negative temperature gradient ahead of the interface, yet it is observed that dendrites are very hard to avoid in practice. This occurs because most materials in use have significant levels of impurities. The following animation shows how dendrites form in a binary alloy:
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.
The situation can be analysed quantitatively to determine whether a dendritic growth front is likely. If we assume a steady state diffusion profile ahead of the interface (see page on Steady State Solidification), the concentration of solute at any distance, x’, ahead of the interface is given by:
$C = {C_0} + \frac{{{C_0}\left( {1 - k} \right)}}{k}\exp \left( {\frac{{ - x}}{{{D_{\rm{L}}}/v}}} \right) \;\;\;\;\;\;(1)$
The liquidus temperature, TL, on the phase diagram is dependent on the concentration of solute by the equation:
${T_{\rm{L}}} = {T_{\rm{m}}} + C\frac{{\partial {T_{\rm{L}}}}}{{\partial C}}$
where Tm is the melting temperature of the pure substance, and T / ∂x is the gradient of the liquidus on the phase diagram, which is usually negative if k < 1.
The graph of liquidus temperature against distance, has a profile like the one shown below:
There will be an undercooled region ahead of the interface, in which a planar interface is unstable, if the temperature gradient,T / ∂x, ahead of the interface is less than the gradient of the liquidus temperature at the interface. To maintain a planar interface the relationship below must hold:
$\frac{{\partial T}}{{\partial x}} > {\left. {\frac{{\partial {T_{\rm{L}}}}}{{\partial x}}} \right|_{x = 0}}$
which, by the chain rule becomes:
$\frac{{\partial T}}{{\partial x}} > \frac{{\partial {T_{\rm{L}}}}}{{\partial C}}{\left. {\frac{{\partial C}}{{\partial x}}} \right|_{x = 0}}$
By differentiating (1) with respect to x, and setting x = 0, we get:
${\left. {\frac{{\partial C}}{{\partial x}}} \right|_{x = 0}} = - \frac{{{C_0}}}{{{D_{\rm{L}}}/v}}\frac{{\left( {1 - k} \right)}}{k}$
The critical gradient required to maintain a planar interface is given by:
${\frac{{\partial T}}{{\partial x}}_{{\rm{crit}}}} = - \frac{{{C_0}}}{{{D_{\rm{L}}}/v}}\frac{{\left( {1 - k} \right)}}{k}\frac{{\partial {T_{\rm{L}}}}}{{\partial C}}$
For temperature gradients only slightly below the critical gradient, the planar interface will break down, but the trapping of solute partitioned between the primary dendrite arms prevents growth of secondary arms, and a cellular growth front evolves.
The simulation below allows you to predict the morphology of the solid growth in various systems.
The next simulation plots the liquidus temperature, in black, against distance, on the lower chart. This is calculated from the composition of the liquid, which is plotted on the upper chart. The simulation also plots a temperature profile in the liquid, in red. The gradient of this can be changed using the scrollbar labelled T / ∂x. The upper limit of this gradient is 10000 K m-1, which is an approximate upper limit for gradients that can be practicably achieved. The critical gradient required to ensure a planar interface is plotted in green, and is also displayed in the box. The partition coefficient, k, the bulk concentration, C0, the diffusivity in the liquid DL, and the growth front velocity, v, can all be adjusted using the appropriate scrollbars.
Try adjusting the variables to see how the critical gradient, and the liquidus temperature profile change. Use the appropriate equations to justify the behaviour of the simulation.
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.
You should note that in order to ensure a planar interface, the growth front velocities need to be quite low, of the order of 10s of microns per second. This is why it is often difficult to avoid dendritic growth in practical solidification.
In dendritic growth, most of the solute is ejected between the dendrite arms. The structure of the dendrites serves to prevent mixing with the rest of the liquid, so that the solute partitioning described earlier occurs over the length scale of the secondary dendrite arm spacing. This is known as microsegregation.
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# dvipdfmx and dvips do not expand fonts properly with lualatex in DVI mode
dvilualatex with microtype is supposed to support expansion, even in DVI mode. And indeed it seems to work when producing the DVI file. But then the generated file does not actually include expanded fonts: the characters are located as if they were expanded, but are not actually stretched in any way. In normal circumstances, the expansion is minimal, so this is not noticeable unless you look very closely. But using a document with some more extreme settings the problem becomes apparent.
The doucment:
\documentclass[12pt]{article}
\usepackage[utf8]{luainputenc}\usepackage[OT1]{fontenc}
\usepackage[main=english]{babel}
\usepackage[expansion=alltext,protrusion,babel,shrink=400,stretch=400,step=80]{microtype}
\usepackage{multicol}
\begin{document}
\begin{multicols*}{2}
The North Wind and the Sun were disputing which was the stronger, when a traveler came along wrapped in a warm cloak. They agreed that the one who first succeeded in making the traveler take his cloak off should be considered stronger than the other. Then the North Wind blew as hard as he could, but the more he blew the more closely did the traveler fold his cloak around him; and at last the North Wind gave up the attempt. Then the Sun shined out warmly, and immediately the traveler took off his cloak. And so the North Wind was obliged to confess that the Sun was the stronger of the two.
\end{multicols*}
\end{document}
Result with lualatex-dev (PDF mode):
Result with dvilualatex-dev (DVI mode) + dvipdfmx (dvips produces the same result):
So, you'll notice that the characters are placed as if they were stretched, but are not in fact modified at all.
Is there any way to compile a DVI to PDF with this kind of expansion? Otherwise lualatex's support for it seems useless.
• Ask the microtype maintainer. When I use dviasm on the dvi I see nothing that looks like expanded font declarations. – Ulrike Fischer Mar 6 '20 at 9:59
• I sent an email. – d909 Mar 6 '20 at 14:57
• Well, this is quite embarrassing ... seems like I never really checked whether the glyphs were actually transformed, but was tricked into believing that it worked by only looking at the changed line breaks ... so, you're right, dvilualatex does not support expansion in a useful way. – Robert Mar 7 '20 at 19:18
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Maths / Fractions and Decimals / Fraction and Its Types
QUESTION
The multiplicative inverse of $\frac{3}{4}$ is
OPTIONS A. $\frac{4}{3}$ B. 1 C. 0 D. None
Right Option : A
EXPLANATION
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Text
$\because \frac{3}{4}\times \frac{4}{3}=1$
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Math Help - Help with indefinite integrals?
1. Help with indefinite integrals?
antiderivative [(3x)/((2x^2+1)^1/2)]
what i did:
let u=2x^2+1
du/dx=4x
(3/4)du=(3/4)(4x)(dx)
3/4 du=(3x)(dx)
x=[(u-1)/2)]^1/2
antiderivative [(u^1/2)(3[(u-1)/2]^1/2)(du)]
= antiderivative [(3/u^1/2)[(u-1)/2]^1/2
from here i get stuck..i need to multiply du in as well but im not sure how to do that since i only found 3/4du=3xdx
how would i do this? or am i doing something wrong?
2. This looks OK to me
$\displaystyle \int \frac{3x}{\sqrt{2x^2+1}}~dx = \frac{3}{4}\int \frac{4x}{\sqrt{2x^2+1}}~dx =\frac{3}{4} \int \frac{1}{\sqrt{u}} = \frac{3}{4}\frac{\sqrt{u}}{\frac{1}{2}}+C=\dots$
3. Originally Posted by pickslides
This looks OK to me
$\displaystyle \int \frac{3x}{\sqrt{2x^2+1}}~dx = \frac{3}{4}\int \frac{4x}{\sqrt{2x^2+1}}~dx =\frac{3}{4} \int \frac{1}{\sqrt{u}} = \frac{3}{4}\frac{\sqrt{u}}{\frac{1}{2}}+C=\dots$
Thank youu =]
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Last edited by Mujar
Wednesday, July 15, 2020 | History
2 edition of Mark Kac seminar on probability and physics found in the catalog.
Mark Kac seminar on probability and physics
# Mark Kac seminar on probability and physics
## Syllabus, 1985-1987 (CWI syllabus)
Written in English
Subjects:
• Stochastic processes,
• Congresses,
• Statistical mechanics
• The Physical Object
FormatUnknown Binding
Number of Pages162
ID Numbers
Open LibraryOL9108483M
ISBN 109061963508
ISBN 109789061963509
Analysis WebNotes - John Lindsay Orr; Dept. of Mathematics & Statistics, Univ. of Nebraska-Lincoln An extensive collection of analysis resources, including class notes, discussion boards, and homework assignments, with questions and answers from . The Math Forum's Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites relating to Fourier Analysis/Wavelets.
Soluble model of Bose-atoms with two level internal structure: Non-conventional Bose-Einstein condensation Article (PDF Available) in Condensed Matter Physics 13(4) . Math Physics Seminar am Lie conformal algebras and the variational complex Denis Bashkirov Abstract: We will give a review of a recent work of and on Poisson vertex algebras and their applications to the theory of integrable systems. Fri Feb Combinatorics Seminar am.
Master thesis presentation Erik Nilsson: Can one hear the shape of a flat torus? - a look at whether isospectrality gives isometry. Abstract: Perhaps it was Milnor's article, published two years prior, that triggered Mark Kac in to ask the famous question of whether one could hear the shape of a drum. Kac’s specialty was probability and analysis. He had little interest in geometry, even as mathematics, so not surprising he didn’t see it as important for physics. Note that the essay was written in , same year Weinberg’s gravitation textbook was published, which also took the attitude that geometry was not important for physics, writing.
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### Mark Kac seminar on probability and physics Download PDF EPUB FB2
Mark Kac Seminar on Probability and Physics. Mark Kac seminar on probability and physics. Amsterdam: Centrum voor Wiskunde en Informatica, © (OCoLC) Material Type: Conference publication: Document Type: Book: All Authors / Contributors: F den Hollander; H.
Mark Kac, Probability and related topics in the physical sciences. (with contributions by Uhlenbeck on the Boltzmann equation, Hibbs on quantum mechanics, and van der Pol on finite difference analogues of the wave and potential equations, Boulder Seminar ).
Mark Kac, Enigmas of Chance: An Autobiography, Harper and Row, New York, Doctoral advisor: Hugo Steinhaus. Subject: Stochastics and operational research: Organization: Mathematical Physics: This item appears in the following Collection(s) Faculty of Science []; Academic publications [] Academic output Radboud UniversityCited by: 1.
Mark Kac: probability, number theory, and statistical physics by Mark Kac (Book) Mark Kac () by Kenneth M Case (Book) Thomas A. Ryan notes by Thomas A Ryan (). DANS is an institute of KNAW and NWO.
Driven by data. Go to page top Go back to contents Go back to site navigationCited by: 1. Mark Kac seminar on probability and physics: syllabus [a ().
Pagina-navigatie: Main; Save publication. Save as MODS; Export to Mendeley; Save as EndNote; Export to RefWorks; Title: Mark Kac seminar on probability and physics: syllabus [a selection of reports of lectures delivered during the academic years] Series: CWI Author: W.T.F.
deHollander, H. Maassen. Editor(s): Hollander, F. den; Maassen, H.: Subject: Stochastics and operational research: Organization: Mathematical PhysicsCited by: "Kac, Rota, and Schwartz have collected here: occasional essays about mathematics, mathematicians, and surrounding subjects. The essays are fun to read, light in manner but serious in content.
Many of them are provocative. As a result, the book would make wonderful fodder for a reading course or seminar. Cited by: Unfortunately, this book can't be printed from the OpenBook. If you need to print pages from this book, we recommend downloading it as a PDF.
The methods he has developed to deal with such events have opened new domains of pure mathematics. Mathematical physics refers to the development of mathematical methods for application to problems in physics. Galileo Galilei in his book The Assayer asserted that the "book of nature" is written in mathematics.
[–], Subrahmanyan Chandrasekhar [], Mark Kac [–]. ♥ Book Title: An Introduction to Measure and Probability ♣ Name Author: J.C. Taylor ∞ Launching: Info ISBN Link: ⊗ Detail ISBN code: ⊕ Number Pages: Total sheet ♮ News id: jiMRBwAAQBAJ Download File Start Reading ☯ Full Synopsis: "Assuming only calculus and linear algebra, Professor Taylor introduces readers to.
Harry was born in Duisburg Germany on Novem His parents escaped from the Nazis in and moved to Amsterdam. After studying in Amsterdam, he was a research assistant at the Mathematical Center there until when he came to Cornell. He received his Ph.D. in at Cornell University under supervision of Mark Kac.
Applied Analysis by the Hilbert Space Method: An Introduction with Applications to the Wave, Heat, and Schr?dinger Equations.
(16) Mark Kac seminar on probability and physics, Syllabus (eds. den Hollander and H. Maassen), CWI Syllabus 17 (). (17) Tail triviality for sums of stationary random variables, Ann. Probab.
17 () { (with H.C.P. Berbee). (18) A stochastic model for the membrane potential of a stimulated neuron, J.
Math. Biol. Luis A. Santaló, Mark Kac Integral geometry originated with problems on geometrical probability and convex bodies. Its later developments, however, have proved to be useful in several fields ranging from pure mathematics (measure theory, continuous groups) to technical and applied disciplines (pattern recognition, stereology).
Zusammenfassung. In seinem Vorwort zu dem Buch von L. Santaló, Integral Geometry and Geometric Probability (), schreibt der Herausgeber Mark Kac: “ Probability Theory is measure theory with a ‘soul’ which in this case is provided not by Physics or by games of chance or by Economics but by the most ancient and noble of all mathematical disciplines, namely.
In: Mark Kac Seminar on Probability and Physics, Syllabus (eds. den Hollander & H. Maassen). CWI SyllaCenter for Mathematics and Computer Science, Amsterdampp.
[24] W. VERVMT:Algebraic duality of Markov processes. In: Mark Kac Seminar on Probability and Physics, Syllabus (eds. den Hollander & H. interest in physics, making him one of the initiators of the Mark Kac ‘seminar for probability and physics, and the Dutch Association of Mathematical Physics.
In Nijmegen he built up a collaboration with the department of medical physics on the subject of point processes, which led to his co-supervising the Ph.D. thesis of Gerard Hesselmans. William Feller, probably in s, when he worked on establishing Math.
Reviews; photo from [Proceedings of the Sixth Berkeley Symposium on Mathematical Statisics and Probability (Univ. California, Berkeley, Calif., /), Vol. II: Probability theory] Mark Kac in his very interesting article [William Feller, In Memoriam], wrote the.
Seminar Archive. * The concepts of independence and dependence are fundamental in probability and statistics. In this talk, Abstract: I will discuss Mark Kac's famous question as to what geometric information is encoded in the Laplace spectrum of a manifold.
The Laplacian is a generalized second derivative and we consider.Probability Seminar pm - VinH Phase transitions in bootstrap percolation David Sivakoff, Ohio State University Abstract: Bootstrap percolation is a discrete growth process on a graph, in which vertices become occupied as soon as they have at least $\theta$ occupied neighbors.
The initial set of occupied sites is given by a.M. Kac, "Random Walk and the Theory of Brownian Motion," The American Mathematical Monthly, 54(7):This paper is by Mark Kac of the Feynman-Kac formula. It deals with the relationship with Brownian motion, which is a continuous stochastic process, and random walk on discrete grids.
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# Equivalence relation between vectors in Euclidean geometry
I'm working in Hilbert's axioms of Euclidean plane geometry. I have problems with proving one thing concerning vectors. My definition of the vector is as follows:
Vector $\overrightarrow{ab}$ is an ordered pair of points $(a,b)$.
Now I define a relation between vectors $\overrightarrow{ab}$ and $\overrightarrow{cd}$:
If $a=b$ and $c=d$, the vectors are in relation.
If $a\neq b$ and $c=d$, the vectors are not in relation.
If $a=b$ and $c\neq d$, the vectors are not in relation.
If $a\neq b$ and $c\neq d$, the vectors are in relation if and only if all following conditions are true
1. $|ab|=|cd|$.
2. Lines $ab$ and $cd$ are parallel.
3. a. If the lines $ab$ and $cd$ are equal, halfline $ab$ is contained in halfline $cd$ or halfline $cd$ is contained in halfline $ab$.
b. If the lines $ab$ and $cd$ are disjoint, points $b$ and $d$ lie on the same side of the line $ac$ (in other words in the same halfplane whose border is line $ac$)
Now I want to prove this is an equivalence relation (equivalence classes would be free vectors). The problem is with transitivity in the case when all lines are disjoint. I need to prove:
Let $ab$, $cd$, $ef$ be parallel lines. $b$ and $d$ lie on the same side of the line $ac$, $d$ and $f$ lie on the same side of the line $ce$. Then $b$ and $f$ lie on the same side of the line $ae$.
• At this point of the theory, do you have the notion of a line segment? If so, you may want to show (and use) the fact that the line segments $[ac]$ and $[ce]$ lie on the same side of the line $ae$. – paf Aug 25 '16 at 22:12
• Yes, I have this notion and I know how to show this fact. But I still don't know how to prove the thesis. – Kulisty Aug 26 '16 at 8:21
• Maybe the following 3-step-approach will lead to success: Step 1: Define a relation $\sim$ by $(a,b)\sim(c,d):\Leftrightarrow\exists v:a+v=c\ \land\ b+v=d$! Step 2: Prove that $\sim$ is an equivalence relation! (This is rather easy.) Step 3: Prove that the definition of $\sim$ is equivalent to your original definition! – Xaver Aug 27 '16 at 15:18
• Yes, but how to define this addition? I can't add points. And I think there should be $a+v=b \wedge c+v=d$. – Kulisty Aug 28 '16 at 7:47
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# Statistics
Statistics Blogs
## Applied Statistics Lesson of the Day – Notation for Fractional Factorial Designs
$Applied Statistics Lesson of the Day – Notation for Fractional Factorial Designs$
Fractional factorial designs use the notation; unfortunately, this notation is not clearly explained in most textbooks or web sites about experimental design. I hope that my explanation below is useful. is the number of levels in each factor; note that the notation assumes that all factors have the same number of levels. If a factor has […]
## When less is more
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## What is meant by regression modeling?
April 22, 2014
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What is meant by regression modeling? Linear Regression is one of the most common statistical modeling techniques. It is very powerful, important, and (at first glance) easy to teach. However, because it is such a broad topic it can be a minefield for teaching and discussion. It is common for angry experts to accuse writers […] Related posts: How robust is logistic regression? Modeling Trick: Masked Variables Modeling Trick: the…
## A helpful structure for analysing graphs
April 22, 2014
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## Drexel on Monday 4/28
April 22, 2014
By
Looks interesting if you're in the area. I plan to be at the lunch.From: Drexel University's LeBow College of Business <announce@lebow.drexel.edu>Date: Thu, Apr 17, 2014 at 11:14 AMSubject: School of Economics Presents: 2 Presentations by D...
## Examen, Séries Chronologiques
April 22, 2014
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Après les exposés des dernières séances, l’examen du cours MAT8181, Séries Chronologiques avait lieu ce matin (et devrait finir dans quelques minutes, avec un peu de temps supplémentaire pour certain, compte tenu de la panne de métro qu&#...
## Stata-Bloggers revamped
April 22, 2014
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STATA-Bloggers has been out for a year and has gotten a bit of a facelift. Still far fewer bloggers (9) than would be ideal to maintain an aggregator, but probably the best source for an aggregate feed on Stata news. Please contact me if yo...
## Seven forecasting blogs
April 22, 2014
By
There are several other blogs on forecasting that readers might be interested in. Here are seven worth following: No Hesitations by Francis Diebold (Professor of Economics, University of Pennsylvania). Diebold needs no introduction to forecasters. He primarily covers forecasting in economics and finance, but also xkcd cartoons, graphics, research issues, etc. Econometrics Beat by Dave Giles. Dave is a professor of economics at the University of Victoria (Canada), formerly from my own…
## More On the Limitations of the Jarque-Bera Test
April 21, 2014
By
Testing the validity of the assumption, that the errors in a regression model are normally distributed, is a standard pastime in econometrics. We use this assumption when we construct standard confidence intervals for, or test hypotheses about, t...
## Ray Fair’s Model(s) in EViews
April 21, 2014
By
Here's a follow-up to my recent post about the Federal Reserve U.S. macroeconometric model being freely available in EViews formatRay Fair's well-known model for the U.S. economy is also now available in a form that's ready to play with in EViews....
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# Can two spaceships go fast enough to pass straight through each other?
Probability of interaction between two particles tends to wane with increasing energy. Technically, the cross section of most interactions falls off with increasing velocity.
$$\sigma(v) \propto \frac{1}{v}$$
This begs a fun question. Since interaction probability diminishes with increasing relative velocity, if you impart enough energy to a particle, might it just mostly pass through solid matter? Might solid matter pass through solid matter (with some "radiation damage" of course)? The above relation, of course, is lacking a great deal. We are really interested in the mean path length, as well as the linear rate of energy deposition. Let's consider the problem in the context of two chunks of solid matter moving at each other really fast (like space jousting). We have several requirements for a survivable experiment.
• Average path length for any given nucleus & electron from spaceship A moving through spaceship B must be much greater than spaceship B's length
• The energy deposition as a result of the passing must be small enough such that they don't explode like a nuclear bomb right after passing
The question also becomes highly relativistic, and I want to hear commentary from people who have knowledge of interactions in high energy accelerators.
Let's say you're on the Star Trek Enterprise, and the captain proposes an alternative to navigating the densely packed matter in the approaching galaxy by increasing speed to just under the speed of light, and not worrying about obstacles (because you'll pass through them). What arguments would you use to convince him this might not be the best idea?
EDIT: this video seems to make the claim that super high energy protons may pass through the entire Earth. Do the answers here contradict the claim?
-
This is nonsense. Strong interactions grow with energy, as does gravity. – Ron Maimon Oct 26 '11 at 16:14
To answer the Star Trek version: "Captain, we can't go through solid matter at Warp, what makes you think we can while moving slower?" – Izkata Oct 26 '11 at 18:05
The probability of collision will never be negligible as vividly shown by the fact that cosmic rays interact quite non negligibly with the atmosphere even at the highest energies. – whistles Oct 26 '11 at 23:04
It doesn't look possible, because the stopping power starts growing for very high energies even for non-strongly interacting particles (see Fig. 27.1 in Passage of particles through matter).
To do a best case estimate, let's consider two cubic spaceships colliding, each of them with $1\,{\rm mm^3}$ of volume, a density of $1\,{\rm g\cdot cm^{-3}}$ and a number density of $10^{21}\,{\rm cm^{-3}}$. In general the atoms of the colliding spaceship will appear as a shower of nuclei and electrons but, for our purposes of lower bounding the deposited energy, let's consider the colliding spaceship as a shower of muons, one for each atom.
Then we will have the equivalent of $10^{18}$ muons ($10^{21}\ {\rm cm^{-3}}\cdot 10^{-3}\ {\rm cm^3}$) colliding with a target with an areal density of $0.1\ {\rm g\cdot cm^{-2}}$. Using the minimum ionization energy we get an stopping power greater than $1\ {\rm MeV\cdot cm^2\cdot g^{-1}}$, making each muon deposit approximately $100\ {\rm keV}$ of energy.
As the number of particles in both spaceships is equal, the expected temperature of both spaceships after the collision will be around the energy deposited by each particle, $100\ {\rm keV} \approx 10^9\ K$.
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The minimum ionization energy concept has great utility for answering this question. That puts a very definable limit on the minimum interaction. This should convince the captain if he has any sense! – Alan Rominger Oct 26 '11 at 17:59
The short answer is no. Ron is right.
Have a look at plot 41 and on of total cross sections for particles, in this particle data link.
Your 1/v assumption might hold water until about 5GeV center of mass system energy, though it is affected at the particle level by resonances etc. From where the arrow starts, particle cross sections rise, they do not fall. This means that once relativistic velocities are reached and keep increasing, your two blobs of matter will be seeing each other grow bigger and bigger, that is what cross section means. If they are on a collision course they will burst into a plethora of particles and disappear in a particle radiation bomb.
They certainly cannot go through each other.
-
This law you are giving isn't valid. Nuclear cross sections grow as a very small power, probably asymptoting to logarithmic growth. At very very high energies, gravity takes over, and the collision produces a decaying black hole. This cross section grows as the mass squared, since it is determined by the ratio of the impact parameter to the Schwarzscild radius.
-
logarithmic cross section with respect to what variable? How does that scale compared to the other factors? To bring up the inevitable point, the size of the spaceship will play a role, as I imagine that large molecules, at least, could manage this in some sense. – Alan Rominger Oct 26 '11 at 15:26
In s, but logaritmic is logaritmic with any polynomial of energy – Ron Maimon Oct 26 '11 at 16:09
There is no way to do what you are asking. The question is nonsense. – Ron Maimon Oct 26 '11 at 16:10
What is "In s"? The question is sufficiently complete from the title. Statements like "cross sections grow as a very small power" are incomplete. You're welcome to ask for more clear elaboration on any point I've made. I try to be as articulate as possible. Maybe you're the type who believes "bad" questions exist from the very core of the question? – Alan Rominger Oct 26 '11 at 16:21
@Zassounotsukushi He is speaking of $s = (p_1 + p_2)^2$, the center of mass energy and one of the Mandelstam variables. – mmc Oct 26 '11 at 16:33
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SISSA Open Science
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We study the IR dynamics of the cascading non-conformal quiver theory on N regular and M fractional D3 branes at the tip of the complex cone over the first del Pezzo surface. The horizon of this cone is the irregular ...
• (2005)
We calculate the contribution to the proton decay amplitude from Kaluza-Klein lepto-quarks in theories with extra dimensions, localised fermions and gauge fields which propagate in the bulk. Such models naturally occur ...
• (SISSA, 2012-05-23)
We present a SUSY SU(5)xT' unified flavour model with type I see-saw mechanism of neutrino mass generation, which predicts the reactor neutrino angle to be \theta_{13} = 0.14 close to the recent results from the Daya Bay ...
• (arXiv:1312.5554 [math.AG], 2013-12)
We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus $I^*_{n,r}$ of tame symplectic instantons is irreducible and has the expected dimension, equal to ...
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The statistics of primordial curvature fluctuations are our window into the period of inflation, where these fluctuations were generated. To date, the cosmic microwave background has been the domi nant source of information ...
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We propose that reactor experiments could be used to constrain the environment dependence of neutrino mass and mixing parameters, which could be induced due to an acceleron coupling to matter fields. There are several ...
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The compelling experimental evidences for oscillations of solar and atmospheric neutrinos imply the existence of 3-neutrino mixing in vacuum. We briefly review the phenomenology of 3-v mixing, and the current data on the ...
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The subject of this paper is the rigorous derivation of lower dimensional models for a nonlinearly elastic thin-walled beam whose cross-section is given by a thin tubular neighbourhood of a smooth curve. Denoting by h and ...
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In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coe fficients in any space dimension N ≥ 1. We will suppose the coe fficients to be log-Zygmund continuous in time and ...
• (2005)
In this paper we consider the minimum time population transfer problem for the z-component of the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded amplitude. On the Bloch ...
• (2005-06-20)
On a two-level quantum system driven by an external field, we consider the population transfer problem from the first to the second level, minimizing the time of transfer, with bounded field amplitude. On the Bloch sphere ...
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We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with ...
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In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows to extend ...
• (2005-06-20)
We extend the analysis of hep-th/0409063 to the case of a constant electric field turned on the worldvolume and on a transverse direction of a D-brane. We show that time localization is still obtained by inverting the ...
• (2009-07-30)
The deal.II finite element library was originally designed to solve partial differential equations defined on one, two or three space dimensions, mostly via the Finite Element Method. In its versions prior to version 6.2, ...
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This review of bosonic string field theory is concentrated on two main subjects. In the first part we revisit the construction of the three string vertex and rederive the relevant Neumann coefficients both for the matter ...
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We add to the mounting evidence that the topological B model's normalized holomorphic three-form has integral periods by demonstrating that otherwise the B2-brane partition function is ill-defined. The resulting Calabi-Yau ...
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The scalar and vector topological Yang-Mills symmetries determine a closed and consistent sector of Yang-Mills supersymmetry. We provide a geometrical construction of these symmetries, based on a horizontality condition ...
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We establish a translation dictionary between open and closed strings, starting from open string field theory. Under this correspondence, (off-shell) level-matched closed string states are represented by star algebra ...
• (SISSA, 1993-08-17)
In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic ...
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# I What is the equation for working out emf frequency
1. Jun 12, 2016
### pkc111
Hi there
My understanding is that any charge will produce some emf when it is accelerated..is this always true? What is this law called? Is there a simple equation to relate charge size, acceleration rate and resulting emf frequency?
Many thanks
2. Jun 13, 2016
### davenn
hi there
yes
from wiki ....
https://en.wikipedia.org/wiki/Maxwell's_equations
have a read of that and other related articles
coming to understanding that will take the next few years
I don't even begin to understand it all
Dave
EDIT
PS ... I just noticed you used the term emf frequency ... am assuming you really meant
E/M electromagnetic field
Last edited: Jun 13, 2016
3. Jun 13, 2016
### pkc111
Thanks..sorry I meant emr (electromagnetic radiation) frequency
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# Functional equation
(Redirected from Functional Equations)
A functional equation, roughly speaking, is an equation in which some of the unknowns to be solved for are functions. For example, the following are functional equations:
• $f(x) + 2f\left(\frac1x\right) = 2x$
• $g(x)^2 + 4g(x) + 4 = 8\sin{x}$
## Introductory Topics
### The Inverse of a Function
The inverse of a function is a function that "undoes" a function. For an example, consider the function: $f(x) = x^2 + 6$. The function $g(x) = \sqrt{x-6}$ has the property that $f(g(x)) = x$. In this case, $g$ is called the (right) inverse function. (Similarly, a function $g$ so that $g(f(x))=x$ is called the left inverse function. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function.) Often the inverse of a function $f$ is denoted by $f^{-1}$.
## Intermediate Topics
### Cyclic Functions
A cyclic function is a function $f(x)$ that has the property that:
$f(f(\cdots f(x) \cdots)) = x$
A classic example of such a function is $f(x) = 1/x$ because $f(f(x)) = f(1/x) = x$. Cyclic functions can significantly help in solving functional identities. Consider this problem:
Find $f(x)$ such that $3f(x) - 4f(1/x) = x^2$. In this functional equation, let $x=y$ and let $x = 1/y$. This yields two new equations:
$3f(y) - 4f\left(\frac1y\right) = y^2$
$3f\left(\frac1y\right)- 4f(y) = \frac1{y^2}$
Now, if we multiply the first equation by 3 and the second equation by 4, and add the two equations, we have:
$-7f(y) = 3y^2 + \frac{4}{y^2}$
So, clearly, $f(y) = -\frac{3}{7}y^2 - \frac{4}{7y^2}$
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# Change font color of scrbook titlepage
I have the following code:
\documentclass[a4paper,landscape,12pt,oneside]{scrbook}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{xcolor}
\begin{document}
\color{red}
\title{MainTitle}
\author{"author"}
\subject{Subject} \publishers{Publisher}
\maketitle
\end{document}
I can set colors of everything except the subject and the title to red. Can somebody give me a tip on how to set the size of the text and the font color of the title and subject too, please?
-
Use \addtokomafont for title and subject.
\documentclass[a4paper,landscape,12pt,oneside]{scrbook}
\usepackage{xcolor}
\begin{document}
\color{red}
\title{MainTitle}
\author{"author"}
\subject{Subject} \publishers{Publisher}
\maketitle
\end{document}
-
The KOMA-Script package provides the commands \setkomafont and \addtokomafont to modify the font of certain elements.
I expanded your example with two \addtokomafont directives which make the title and subject go red:
\documentclass[a4paper,landscape,12pt,oneside]{scrbook}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{xcolor}
\begin{document}
\color{red}
\title{MainTitle}
\author{"author"}
\subject{Subject} \publishers{Publisher}
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mersenneforum.org > Math Other ways of finding WR primes
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2019-11-24, 20:30 #1 jshort "James Short" Mar 2019 Canada 100012 Posts Other ways of finding WR primes I've been thinking about this for some time as to why Mersenne integers are the #1 candidate for finding WR prime numbers, and have basically come to these two reasons: 1) There is a fast deterministic primality test for them (ie. Lucas-Lehmer primality test). 2) Mersennes are far more likely to be prime compared to what the Prime Number Theorem would predict since any prime integer $q$ that divides $2^{p} - 1$ is forced to be of the form $1 + k \cdot 2 \cdot p$. For simplicity, lets call the above conditions #1 and #2. Empirical evidence suggests that Wagstaff numbers ($\frac{2^{p} + 1}{3}$) "beat" Mersenne numbers on condition #2 above. For instance, for all prime exponents $p < 10^{4}$, there are 24 Wagstaff primes but only 22 Mersenne primes. Fyi - there's no proof that Wagstaff's are generally more likely to be prime than Mersenne numbers. All we know is that they are only 1/3 the size of Mersenne numbers, yet just like the Mersenne, composite prime factors are restricted to being of the form $1 + k \cdot 2 \cdot p$. In any case, its completely irrelevant because Wagstaff fail condition #1 - there is no fast deterministic primality test for them. CANDIDATE #2 - Fermat numbers Question: Does it satisfy condition #1? Ans: Yes Fermat numbers, $2^{2^{n}} + 1$, satisfy condition #1 above because they are a Proth number (ie $k \cdot 2^{n} + 1, k < 2^{n}$) and therefore primality can be deterministically proven using Fermat's primality test. Question: Do Fermat numbers match or beat Mersenne numbers on condition number #2? Ans: Yes Fermat numbers beat Mersenne integers on condition #2 because although they are $2^{n}$ bits long in size, any potential prime factor is restricted to being of the form $1 + k \cdot 2^{n+2}$. This is essentially "twice as restrictive" as the Mersenne condition of having to be of the form $1 + k \cdot 2 \cdot p$. Unfortunately Fermat integers grow double-exponentially. For this reason it is believed that there are no more prime Fermat numbers after $F_{4} = 65537$ However what if we generalize the Fermat number? Definition: $F_{n,p} = \frac{2^{2^{n} \cdot p} + 1}{2^{2^{n}}+1$ When prime $p < 2^{n}$ its easy to show that this will be a Proth number. Furthermore, any prime factors of these numbers will have to be extremely restrictive. Its not too difficult to prove that any prime factor will have to be of the form $1 + k \cdot 2^{n+1} \cdot p$. Empirical testing shows that the congruence restriction is even stronger than the above and is in fact $1 + k \cdot 2^{n+2} \cdot p$. Example: $F_{5,3} = \frac{2^{96 + 1}}{2^{32} + 1}$. This is a Proth number because it can be written as $1 + (2^{32} -1) \cdot 2^{32}$. We can quickly establish its primality using the Fermat primality test. As an aside, we can actually find numbers that are more restrictive than the $1 + k \cdot 2^{n+2} \cdot p$ condition if we are willing to relax the Proth condition that $k < 2^{n}$ for integers of the form $1 + k \cdot 2^{n}$. Primitive factors of numbers of the form $2^{n} + 1$ where n is a "Highly Composite Number" (see: https://en.wikipedia.org/wiki/Highly_composite_number) appear to be the most "restrictive" with regards to condition #2. Anyway some conjectures I've been unable to resolve: Conjecture #1 - prime factors of $F_{n,p}$ have to be of the form $1 + k \cdot 2^{n+2} \cdot p$. Conjecture #2 - If a primitive factor of $2^{n} + 1$ where n is a Highly Composite Number "almost" satisfies Proth's condition (that is, it can put in the form $1 + k \cdot 2^{m}$ where $k$ isn't too much larger than $m$), we deterministically establish its primality almost as quickly.
2019-11-24, 20:59 #2 Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 7×1,373 Posts Small typos aside, you have found exactly the right logic to approach the most actionable paths to the potentially largest known provable primes. Conditions #1 and #2 are correct. To summarize, you would want to find the candidates that are a) fast to test, and b) that will have the significantly higher probability of being productive (by having restricted factors, and therefore pre-factored to a much higher level). There are at least two generalizations for the Fermat numbers: 1. the "cyclotomic numbers" (like your second kind, but p does not have to be always prime) - the good choice of "p" is powers of 3. Some history. Some more history.* and 2. Generalized Fermat numbers: $$b^{2^n} +1$$, with b > 2 Both are being actively searched. * If you read M.Oakes' old post, you will find that your conjectures are not conjectures. The factors, of course, are of the special form.
2019-11-24, 21:25 #3 a1call "Rashid Naimi" Oct 2015 Remote to Here/There 87C16 Posts I think the bottleneck in finding WR Primes at the moment, is the time it takes to sieve candidates via probablystic tests which generally come down to the time it takes to do modular Exponentiation. As such any Near-Multiple-Squares (not necessarily only 1 of) format that can be deterministically proven to be prime is a potentially fruitful approach. This makes me wonder why such formats are not more rigorously tested. Last fiddled with by a1call on 2019-11-24 at 21:29
2019-11-24, 21:26 #4
sweety439
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
23×389 Posts
Quote:
Originally Posted by jshort I've been thinking about this for some time as to why Mersenne integers are the #1 candidate for finding WR prime numbers, and have basically come to these two reasons: 1) There is a fast deterministic primality test for them (ie. Lucas-Lehmer primality test). 2) Mersennes are far more likely to be prime compared to what the Prime Number Theorem would predict since any prime integer $q$ that divides $2^{p} - 1$ is forced to be of the form $1 + k \cdot 2 \cdot p$. For simplicity, lets call the above conditions #1 and #2. Empirical evidence suggests that Wagstaff numbers ($\frac{2^{p} + 1}{3}$) "beat" Mersenne numbers on condition #2 above. For instance, for all prime exponents $p < 10^{4}$, there are 24 Wagstaff primes but only 22 Mersenne primes. Fyi - there's no proof that Wagstaff's are generally more likely to be prime than Mersenne numbers. All we know is that they are only 1/3 the size of Mersenne numbers, yet just like the Mersenne, composite prime factors are restricted to being of the form $1 + k \cdot 2 \cdot p$. In any case, its completely irrelevant because Wagstaff fail condition #1 - there is no fast deterministic primality test for them. CANDIDATE #2 - Fermat numbers Question: Does it satisfy condition #1? Ans: Yes Fermat numbers, $2^{2^{n}} + 1$, satisfy condition #1 above because they are a Proth number (ie $k \cdot 2^{n} + 1, k < 2^{n}$) and therefore primality can be deterministically proven using Fermat's primality test. Question: Do Fermat numbers match or beat Mersenne numbers on condition number #2? Ans: Yes Fermat numbers beat Mersenne integers on condition #2 because although they are $2^{n}$ bits long in size, any potential prime factor is restricted to being of the form $1 + k \cdot 2^{n+2}$. This is essentially "twice as restrictive" as the Mersenne condition of having to be of the form $1 + k \cdot 2 \cdot p$. Unfortunately Fermat integers grow double-exponentially. For this reason it is believed that there are no more prime Fermat numbers after $F_{4} = 65537$ However what if we generalize the Fermat number? Definition: $F_{n,p} = \frac{2^{2^{n} \cdot p} + 1}{2^{2^{n}}+1$ When prime $p < 2^{n}$ its easy to show that this will be a Proth number. Furthermore, any prime factors of these numbers will have to be extremely restrictive. Its not too difficult to prove that any prime factor will have to be of the form $1 + k \cdot 2^{n+1} \cdot p$. Empirical testing shows that the congruence restriction is even stronger than the above and is in fact $1 + k \cdot 2^{n+2} \cdot p$. Example: $F_{5,3} = \frac{2^{96 + 1}}{2^{32} + 1}$. This is a Proth number because it can be written as $1 + (2^{32} -1) \cdot 2^{32}$. We can quickly establish its primality using the Fermat primality test. As an aside, we can actually find numbers that are more restrictive than the $1 + k \cdot 2^{n+2} \cdot p$ condition if we are willing to relax the Proth condition that $k < 2^{n}$ for integers of the form $1 + k \cdot 2^{n}$. Primitive factors of numbers of the form $2^{n} + 1$ where n is a "Highly Composite Number" (see: https://en.wikipedia.org/wiki/Highly_composite_number) appear to be the most "restrictive" with regards to condition #2. Anyway some conjectures I've been unable to resolve: Conjecture #1 - prime factors of $F_{n,p}$ have to be of the form $1 + k \cdot 2^{n+2} \cdot p$. Conjecture #2 - If a primitive factor of $2^{n} + 1$ where n is a Highly Composite Number "almost" satisfies Proth's condition (that is, it can put in the form $1 + k \cdot 2^{m}$ where $k$ isn't too much larger than $m$), we deterministically establish its primality almost as quickly.
F5,3 is Phi192(2), where Phi is the cyclotomic polynomial.
Fn,p = Phip*2^(n+1)(2)
Phin(2) is prime for n = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567, 579, 590, 600, 607, 626, 690, 694, 712, 745, 795, 816, 897, 909, 954, 990, 1106, 1192, 1224, 1230, 1279, 1384, 1386, 1402, 1464, 1512, 1554, 1562, 1600, 1670, 1683, 1727, 1781, 1834, 1904, 1990, 1992, 2008, 2037, 2203, 2281, 2298, 2353, 2406, 2456, 2499, 2536, 2838, 3006, 3074, 3217, 3415, 3418, 3481, 3766, 3817, 3927, ...
2019-11-24, 21:34 #5
sweety439
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
60508 Posts
Quote:
Originally Posted by Batalov Small typos aside, you have found exactly the right logic to approach the most actionable paths to the potentially largest known provable primes. Conditions #1 and #2 are correct. To summarize, you would want to find the candidates that are a) fast to test, and b) that will have the significantly higher probability of being productive (by having restricted factors, and therefore pre-factored to a much higher level). There are at least two generalizations for the Fermat numbers: 1. the "cyclotomic numbers" (like your second kind, but p does not have to be always prime) - the good choice of "p" is powers of 3. Some history. Some more history.* and 2. Generalized Fermat numbers: $$b^{2^n} +1$$, with b > 2 Both are being actively searched. * If you read M.Oakes' old post, you will find that your conjectures are not conjectures. The factors, of course, are of the special form.
F(0,p) is prime for p = 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ...
F(1,p) is prime only for p = 3 (because of the algebra factors)
F(2,p) is prime for p = 3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197, 1025393, ...
F(3,p) is prime for p = 5, 13, 23029, 50627, 51479, 72337, ...
F(4,p) is prime for p = 239, ...
F(5,p) is prime for p = 3, 13619, ...
F(6,p) is not prime for all p <= 2^12
2019-11-24, 21:38 #6 sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 23×389 Posts https://archive.is/20120529014233/ht...os/fermat6.htm also list the primes of the form Phin(2), but is for n=p^r (p is prime, r>=1) instead of n=p*2^r (p is prime, r>=1) Last fiddled with by sweety439 on 2019-11-24 at 21:38
2019-11-24, 22:24 #7
jshort
"James Short"
Mar 2019
218 Posts
Quote:
Originally Posted by Batalov Small typos aside, you have found exactly the right logic to approach the most actionable paths to the potentially largest known provable primes. Conditions #1 and #2 are correct. To summarize, you would want to find the candidates that are a) fast to test, and b) that will have the significantly higher probability of being productive (by having restricted factors, and therefore pre-factored to a much higher level). There are at least two generalizations for the Fermat numbers: 1. the "cyclotomic numbers" (like your second kind, but p does not have to be always prime) - the good choice of "p" is powers of 3. Some history. Some more history.* and 2. Generalized Fermat numbers: $$b^{2^n} +1$$, with b > 2 Both are being actively searched. * If you read M.Oakes' old post, you will find that your conjectures are not conjectures. The factors, of course, are of the special form.
Thank you for the references. I wasn't aware of Mike Oakes's work, but will definitely read up on him :)
I haven't considered $$b^{2^n} +1$$, with b > 2 because their bit-size is even bigger to when 2 is used as a base.
I have done a tad of investigating into divisibility sequences which grow exponentially, but at a rate that's between 1 and 2 though.
For example, the Fibonacci numbers - $F_{2p}$ or just $F_{p}$ where p is prime. The primitive factors are easily found in the $F_{2p}$ case. Furthermore, most of these number appear to need to be in the form $1 + k \cdot 2 \cdot p$ (at least in the $F_{2p}$ case).
The congruence restrictions don't appear quite as strict as a generalized Fermat number, however the nice part is that Fibonacci numbers grow exponentially proportional to $1.61^{n}$ which mean their bit-size is considerably more restrictive compared to any integer base b > 1.
There are possibly other Lucas sequences which are divisibility sequences which could grow exponentially with respect to some $b^{n}$ where b < 1.61 too. I wonder if Mike Oakes or others have ever investigated this!?
2019-11-24, 22:32 #8
jshort
"James Short"
Mar 2019
17 Posts
Quote:
Originally Posted by sweety439 https://archive.is/20120529014233/ht...os/fermat6.htm also list the primes of the form Phin(2), but is for n=p^r (p is prime, r>=1) instead of n=p*2^r (p is prime, r>=1)
Thank you for this.
2019-11-24, 22:57 #9
sweety439
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
23·389 Posts
Quote:
Originally Posted by Batalov Small typos aside, you have found exactly the right logic to approach the most actionable paths to the potentially largest known provable primes. Conditions #1 and #2 are correct. To summarize, you would want to find the candidates that are a) fast to test, and b) that will have the significantly higher probability of being productive (by having restricted factors, and therefore pre-factored to a much higher level). There are at least two generalizations for the Fermat numbers: 1. the "cyclotomic numbers" (like your second kind, but p does not have to be always prime) - the good choice of "p" is powers of 3. Some history. Some more history.* and 2. Generalized Fermat numbers: $$b^{2^n} +1$$, with b > 2 Both are being actively searched. * If you read M.Oakes' old post, you will find that your conjectures are not conjectures. The factors, of course, are of the special form.
Well, see A085398
A085398(prime(n)) = A066180(n) for n>=1
A085398(2*prime(n)) = A103795(n) for n>=2
A085398(2^(n+1)) = A056993(n) for n>=0
A085398(3^(n+1)) = A153438(n) for n>=1
A085398(2*3^(n+1)) = A246120(n) for n>=0
A085398(2^(n+1)*3) = A246119(n) for n>=0
A085398(2^(n+1)*3^2) = A298206(n) for n>=0
A085398(2^(n+1)*3^(n+1)) = A246121(n) for n>=0
A085398(5^(n+1)) = A206418(n) for n>=1
Also,
A205506 is a subsequence of A085398 for n = 2^i*3^j with i,j >= 1
A181980 is a subsequence of A085398 for n = 2^i*5^j with i,j >= 1
2019-11-24, 22:59 #10
sweety439
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
23·389 Posts
This is the smallest k>=2 such that Phin(k) is prime, for 1<=n<=2500
Attached Files
least k such that phi(n,k) is prime.txt (22.0 KB, 201 views)
2019-11-25, 00:41 #11
jshort
"James Short"
Mar 2019
218 Posts
Quote:
Originally Posted by sweety439 F(0,p) is prime for p = 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ... F(1,p) is prime only for p = 3 (because of the algebra factors) F(2,p) is prime for p = 3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197, 1025393, ... F(3,p) is prime for p = 5, 13, 23029, 50627, 51479, 72337, ... F(4,p) is prime for p = 239, ... F(5,p) is prime for p = 3, 13619, ... F(6,p) is not prime for all p <= 2^12
Looking at these results F(n,p) for n = 1 - 6 it looks rather discouraging to say the least!
I knew that as n increase the number of primes drops drastically, however for n > 3, the frequency of primes really looks as if its fallen off of a cliff!
I guess one could try to sieve out some smaller prime factors using a p-1 factoring algorithm to root out small and/or smooth numbers idk
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Is $U(2)_{2, 1}$ Chern Simons Theory Completely Trivial?
I am using the method outlined in appendix C4 of a paper by Seiberg and Witten [1] to calculate the statistics of lines in $$U(2)_{2, 1}$$. However, this method shows that all lines are trivial.
Notation: By $$U(2)_{2, 1}$$, we mean a theory with $$SU(2)_{1}\times U(1)_{2}$$ and a Chern-Simons action of the form: $$\frac{1}{4\pi}\int \left(A\wedge dA + \frac{2}{3} A\wedge A\wedge A\right)$$ where $$A$$ is a $$U(2)$$ gauge field. This notation matches [1], but differs from [2] (in the notation of [2] this theory is $$U(2)_{1, 1}$$).
Background: I am imagining the edge modes of a single integer quantum hall state where my electrons have an extra index $$\psi_{i}$$ that transforms under $$SU(2)$$ isospin.'
$$1+1$$d: There is a single right-moving fermion doublet mode on the edge, $$\psi_{i}$$, with $$(\partial_{t}+\partial_{x})\psi_{i} = 0$$. We mandate that all terms in the Lagrangian respect the $$U(2)$$ symmetry $$\psi_{i}\to U_{ij}\psi_{j}$$.
If I then bosonize this system, then I get a $$U(2)$$ WZW theory at level 1 on the edge.
$$2+1$$d: In the bulk, this leads to a $$U(2)_{2, 1}$$ Chern Simons Theory. Following [1] Appendix C4, we can first treat this theory as $$SU(2)_{1}\times U(1)_{2}$$ and then take the quotient.
Beginning with $$SU(2)_{1}\times U(1)_{2}$$, we note that it is level-rank dual to $$U(1)_{-2}\times U(1)_{2}$$. Hence we have four lines: $$(0, 0), (0, 1), (1, 0),$$ and $$(1, 1)$$. For statistics: $$(1, 0)$$ and $$(0, 1)$$ have spins $$\pm ¼$$ and $$(1, 1)$$ has spin $$½$$. $$(1, 0)$$ and $$(0, 1)$$ have braiding $$-1$$ with $$(1, 1)$$. All other statistics are trivial.
The problem occurs when we take the quotient.
Following [1] Appendix C4, we first identify a line which generates holonomy $$-1$$ in each of the $$SU(2)$$ and $$U(1)$$ factors. After using level rank, we look for the line which generates holonomy $$-1$$ in each of the two $$U(1)$$ groups. Clearly, this object is $$(1, 1)$$, and we force it to be trivial.
Next, we enforce a selection rule' that only allows lines which have trivial statistics with $$(1, 1)$$. The only line that has trivial statistics with $$(1, 1)$$ is $$(0, 0)$$ and so we are left with no nontrivial lines!
Clearly something has gone disastrously wrong with my argument, since the bulk Chern Simons theory for this model has to be nontrivial. Can you help me find the flaw in my reasoning?
[1] N. Seiberg and E. Witten, “Gapped Boundary Phases of Topological Insulators via Weak Coupling,” PTEP, vol. 2016, no. 12, p. 12C101, 2016. https://arxiv.org/abs/1602.04251
[2] P.-S. Hsin and N. Seiberg, “Level/rank Duality and Chern-Simons-Matter Theories,” JHEP, vol. 09, p. 095, 2016. https://arxiv.org/abs/1607.07457
• There is no such a thing as Chern-Simons $U(2)_{2,1}$. Recall that $U(N)_{P,Q}$ is only well defined if $P-Q$ is an integer multiple of $N$. – AccidentalFourierTransform Oct 7 '18 at 2:16
• Is there a good way to see that? Ref. [1] above gives a different consistency condition eq. C.17, that $U(2)_{P, Q}$ is consistent only when $P+2Q \in 4\mathbb{Z}$, which $U(2)_{2, 1}$ satisfies. – MDM Oct 7 '18 at 2:19
• Ah okay there's some notation hijinks happening here. Hsin and Seiberg define $U(N)_{P,Q}$ as $SU(N)_{P}\times U(1)_{NQ}$, whereas Seiberg and Witten define $U(N)_{P,Q}$ as $SU(N)_{Q}\times U(1)_{P}$. In the Hsin-Seiberg notation this theory is $U(2)_{1,1}$. I've edited the post to be more clear. – MDM Oct 7 '18 at 3:03
• Follow up question: If I take two copies of the above theory both constructed from right-moving edges, then in Hsin-Seiberg notation I get $U(2)_{1, 1}\times U(2)_{1, 1}$ with central charge $2$ On the other hand, if I take a copy from a right-moving edge and a copy from a left-moving edge, I get $U(2)_{1, 1}\times U(2)_{-1, -1}$ with central charge $0$. However, in the fermion dual, these are both $\{1, f\}\times \{1, f'\}$. How can one tell them apart in the fermion language? – MDM Oct 7 '18 at 5:09
• @MDM In order to specify a topological theory, it is not sufficient to list the lines; there is extra data that defines the theory (more precisely, there is more structure to a modular tensor category than just the objects). For example, the theories $U(1)_k$ and $U(1)_{-k}$ have the same lines, but with opposite spin. – AccidentalFourierTransform Oct 7 '18 at 16:27
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# How to calculate rotor flux of the three phase squirrel cage induction motor?
I have a three phase squirrel cage induction motor with given parameters (stator winding connected in delta)
• nominal power: $$\P_n = 22.4\,\mathrm{kW}\$$
• nominal stator voltage: $$\V_{sn} = 230\,\mathrm{V}\$$
• nominal stator current: $$\I_{sn} = 39.5\,\mathrm{A}\$$
• nominal stator frequency: $$\f_{sn} = 60\,\mathrm{Hz}\$$
• nominal speed: $$\n_n = 1168\,\mathrm{min}^{-1}\$$
• number of pole pairs: 3
• stator resistance per phase (T equivalent circuit): $$\R_s = 0.294\,\Omega\$$
• stator leakage inductance per phase (T equivalent circuit): $$\L_{sl} = 0.00139\,\mathrm{H}\$$
• rotor resistance per phase (T equivalent circuit): $$\R_r = 0.156\,\Omega\$$
• rotor leakage inductance per phase (T equivalent circuit): $$\L_{rl} = 0.0007401\,\mathrm{H}\$$
• magnetizing inductance per phase (T equivalent circuit): $$\L_{m} = 0.041\,\mathrm{H}\$$
I have been struggling with calculation of the nominal value of the rotor flux. My idea was that I will use the T equivalent circuit for that purpose
and I will use nominal values of the motor quantities i.e. I set the motor operating point into the nominal operation point (nominal slip, nominal stator voltage etc.). Then I calculate phasor of the stator current ($$\\hat{I}_s\$$) and phasor of the rotor current ($$\\hat{I}_r\$$) according to the below given set of equations
$$\begin{bmatrix} \hat{V}_s \\ 0 \end{bmatrix} = \begin{bmatrix} R_s + j\cdot(X_{sl} + X_m) & j\cdot X_m \\ j\cdot X_m & \frac{R_r}{s} + j\cdot(X_{rl} + X_m) \end{bmatrix} \cdot \begin{bmatrix} \hat{I}_s \\ \hat{I}_r \end{bmatrix}$$
For the motor parameters mentioned above the Scilab command linsolve gave me
$$\hat{I}_s = - 34.946619 + 17.574273j$$ $$\hat{I}_r = 35.797115 - 3.954462j$$
Based on the known phasors of the stator and rotor current I used the below given equation for calculation of the phasor of the rotor flux
$$\hat{\lambda}_r = (L_{rl} + L_m)\cdot\hat{I}_r + L_m\cdot\hat{I}_s$$
which gives $$\\lambda_r = 0.5554856 - 0.0613638j\$$ i.e. $$\|\lambda_r|=0.5588647\,\mathrm{V}\cdot\mathrm{s}\$$. This value seems to me to be too low. So I have doubts regarding the way I have used for its calculation. Unfortunatelly I don't know any other way for its calculation which I can use for verification. Can anybody tell me whether the applied procedure is correct or not? I would also appreciate any idea how to verify my results. Thanks in advance.
I don't think the leakage rotor magnetic field it has to be included (since the stator and rotor main magnetic field has to be the same).
However, it can be calculated using T equivalent diagram, the current $$\I_{s}\$$. At first calculate the equivalent impedance of parallel $$\X_m\$$ with $$\X_{rl}\$$ and $$\R_r/s\$$ and then divide $$\V_{sn}\$$ by total impedance $$\R_{s}+jX_s+jX_m\$$||($$\R_{r}/s+jX_{lr}\$$).
The result will be the same. $$\I_r\$$ you can calculate by dividing the $$\I_s\$$ voltage drop on the equivalent impedance of $$\jX_m\$$ and $$\R_r/s + jX_{lr}\$$ by $$\Z_r = R_r/s+jX_{lr}\$$.
I think the flux units are average values but $$\I_s\$$ and $$\I_r\$$ are RMS.
$$\I_{av} = 1/π\int_{ωt = 0}^{ωt = π} I_{max}*sin(ωt) \,dt = 2*[I_{max}]\$$
Given that $$\I_{rms}\$$ = $$\I_{max}/√2\$$
$$\I_{av}\$$ = $$\[2*√2*I_{rms}]/π\$$
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The average speed of a bus is 8 times the average speed of a bike. The bike covers a distance of 186 km in 3 h. H
# Ask a Question
### Question Asked by a Student from EXXAMM.com Team
Q 2213134940. The average speed of a bus is 8 times the average speed of a
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distance will the bus cover in 10 h?
IBPS-CLERK 2017 Mock Prelims
A
4069 km
B
4096 km
C
4960 km
D
4690 km
E
None of these
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# Polynomial Manipulation
Algebra Level 4
If $$x, y$$ and $$z$$ satisfy $\frac { 1 }{ x } + \frac {1 }{y } + \frac {1 }{z } = 0$ $x^{2}+y^{2}+z^{2}=2,$ what is $$\left( x+y+z \right) ^{12}?$$
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# Math Help - Find the points of inflection and discuss the concavity of the graph of the function.
1. ## Find the points of inflection and discuss the concavity of the graph of the function.
so here's the function
f(x) = 2sinx + sin2x and in order to find the concavity i need to find the second derivative which is this.... f ''(x)= -2sinx - 4sin2x.... now how do i find the critical numbers to determine the concavity of the function? how to i set x equal to zero to find the critical numbers.... please help!!
2. ## Re: Find the points of inflection and discuss the concavity of the graph of the funct
Equating the second derivative to zero, we find:
$-2(\sin(x)+2\sin(2x))=0$
$\sin(x)+2\sin(2x)=0$
Now, use the double-angle identity for sine, then factor. What do you find?
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Publication
Title
More accurate estimation of diffusion tensor parameters using diffusion kurtosis imaging
Author
Abstract
With diffusion tensor imaging, the diffusion of water molecules through brain structures is quantified by parameters, which are estimated assuming monoexponential diffusion-weighted signal attenuation. The estimated diffusion parameters, however, depend on the diffusion weighting strength, the b-value, which hampers the interpretation and comparison of various diffusion tensor imaging studies. In this study, a likelihood ratio test is used to show that the diffusion kurtosis imaging model provides a more accurate parameterization of both the Gaussian and non-Gaussian diffusion component compared with diffusion tensor imaging. As a result, the diffusion kurtosis imaging model provides a b-value-independent estimation of the widely used diffusion tensor parameters as demonstrated with diffusion-weighted rat data, which was acquired with eight different b-values, uniformly distributed in a range of [0,2800 sec/mm2]. In addition, the diffusion parameter values are significantly increased in comparison to the values estimated with the diffusion tensor imaging model in all major rat brain structures. As incorrectly assuming additive Gaussian noise on the diffusion-weighted data will result in an overestimated degree of non-Gaussian diffusion and a b-value-dependent underestimation of diffusivity measures, a Rician noise model was used in this study.
Language
Dutch
Source (journal)
Magnetic resonance in medicine. - Orlando, Fla
Publication
Orlando, Fla : 2011
ISSN
0740-3194
Volume/pages
65 :1 (2011) , p. 138-145
ISI
000285963500015
Full text (Publisher's DOI)
Full text (publisher's version - intranet only)
UAntwerpen
Faculty/Department Research group Publication type Subject Affiliation Publications with a UAntwerp address
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# Boundary of boundary of closed set equals boundary of this set
I want to prove that for a closet set $A\subset \mathbb{R}^n$, $\partial A = \partial(\partial A)$. I've already proved that for an arbitrary subset $B$ of $\mathbb{R}^n$, $\partial(\partial B)\subseteq \partial B$ by arriving at the fact that $\partial B\backslash \mbox{interior}(B)\subseteq \partial B$, which points to the fact that the interior of the boundary of a set is not necessarily empty. However, unfortunately, I do not probably understand intuitively how it can be empty. Except that, in $\mathbb{R}^4$, say, the boundary can be a 3-dimensional subset, whose interior does not need to be empty. But then, why should the interior of the boundary of a $\underline{\text{closed}}$ set be necessarily empty? For if we consider the same analogy with $\mathbb{R}^4$, we should also intuitively feel that a boundary can be a 3-dimensional subset, whose interior need not be empty.
Here's also my attempt:
$\partial A= \overline{A}\backslash A^\circ=\overline{\overline{A}\backslash A^\circ}=\overline{A\backslash A^\circ}$.
Now, $(A\backslash A^\circ)^\circ=A\backslash \overline{A^\circ}$. If I can show that the RHS is the empty set then I'll deduce that
$\overline{\overline{A}\backslash A^\circ}\backslash (\overline A\backslash A^\circ)^\circ=\partial (\partial A)=\overline{\overline{A}\backslash A^\circ}=\overline{A}\backslash A^\circ=\partial A$.
Let $A \subset \mathbb{R}^N$ and let $a \in A$. We say that $a$ is an interior point of $A$ if there is a positive radius $r > 0$ such that $B(a,r) \subset A$, where $B(a,r)$ is the $N$-dimensional ball centered at $a$ with radius $r$. Notice that the ball you want to consider is $N$-dimensional independently of the dimension of the subset $A$. In particular, if $A$ is an $N-1$-dimensional subset then it cannot contain any $N$-dimensional ball and hence it will have empty interior. Indeed, most of the boundaries that we usually picture in our minds are hypersurfaces and hence have empty interior.
To conclude the proof we need to prove that $\partial A$ has empty interior. This follows from the fact that if $a \in \partial A$, then every neighborhood of $a$ contains a point that is not in $A$. Since $\partial A \subset A$ (recall that by assumption $A$ is closed) we have that every neighborhood of $a$ contains a point that is not in $\partial A$ and hence $a$ cannot be an interior point of $\partial A$.
The boundary $\partial A$ of a closed subset $A \subseteq \mathbb{R}^n$ has empty interior, because any open neighborhood of any point $x \in \partial A$ contains points that are not in $A$ (by definition of boundary), therefore not in $\partial A$ (since $A$ is closed.)
This is certainly not true for arbitrary subsets. The boundary of $\mathbb{Q}^4$ is all of $\mathbb{R}^4$. But then $\mathbb{Q}^4$ is not closed.
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# Newton’s Method for Polynomials
Posted on April 14, 2018
Newton’s Methods is a useful tool for finding the roots of real-valued, differentiable functions. A drawback of the method is its dependence on the initial guess. For example, all roots of a 10th degree polynomial could be found with 10 very good guess. However, 10 very bad guesses could all point to the same root.
In this post, we will present an iterative approach that will produce all roots. Formally
Newton's method can be used recursively, with inflection points as appropriate initial guesses, to produce all roots of a polynomial.
## Motivation
While working through the Aero Toolbox app, I needed to solve a particular cubic equation (equation 150 from NACA Report 1135, if you're interested).
I needed to find all 3 roots. The cubic equation is well known, but long and tedious to code up. I had already been implementing Newton’s Method elsewhere in the app and wanted to use that. The challenge though, as is always the challenge with using Newton’s Method on functions with multiple roots is robustly finding all of them.
This particular cubic was guaranteed to have three distinct roots. The highest and lowest were easy; I was solving for the square of the sine of particular angles so $0$ and $1$ are good starting points. The middle root was not so easy. After some brainstorming, I took a stab at a possible guess: the inflection point. Low and behold, it (without fail) always converged the middle root.
## Delving Further
So why did the inflection point always converge to the middle root? Let's explore!
We'll start with a simple example: a polynomial with unique, real roots. The root is either identical to the inflection point, to the left of it, or to the right of it.
Assume WLOG that the derivative is positive in this region. There will always be a root in between the inflection point and the local min/max to each side (This statement requires proof, which will of course be left to the reader ;) ). Let's examine all three cases.
#### Case 1: Inflection Point is a root
You are at a root, so Newton's Method is converged. Both the function $p$ and its derivative $p'$ are $0$, so as long as your implementation avoids division by zero or erratic behavior close to $\frac{0}{0}$, you're in the clear.
#### Case 2: Root to the left of the inflection point
The slope will flatten out to the left of the inflection point, therefore the tangent line will intersect the $x$-axis before $p$ does. That means Newton's Method will not overshoot the root. In subsequent guesses, this same pattern will hold.
Consider an example in the graph below, $p(x)=-\frac{x^3}{6}+x+0.85$, shown in blue. In the graph below, the tangent line is shown in purple. The intersection of the tangent line and the $x$-axis is the updated guess.
#### Case 3: Root to the right of the inflection point
The logic is the same, but with signs reversed. The root is to the right of the inflection point. All guesses will move right monotonically without overshooting the root.
Consider an example in the graph below of a different polynomial, $p(x)=-\frac{x^3}{6}+x-0.85$, shown in blue. In the graph below, the tangent line is shown in purple. The intersection of the tangent line and the $x$-axis is the updated guess.
## Generalizing to Higher order Polynomials
This method can be generalized to higher orders in a recursive fashion. For an $n$th order polynomial $p$ we can use the roots 2nd derivative $p’’$ (degree $n-2$) as starting guesses for the roots of $p$. To find the roots of $p’’$, you need the roots of $p’’’’$ and so on. Also, as part of the method, you need $p’$. Therefore, all derivatives of the polynomial are required.
Two functions are required. First, a simple implementation of Newton's method
def newton_method(f, df, g, deg):
max_iterations = 100
tol = 1e-14
for i in xrange(max_iterations):
fg = sum([g ** (deg - n) * f[n] for n in xrange(len(f))])
dfg = float(sum([g ** (deg - n - 1) * df[n] for n in xrange(len(df))]))
if abs(dfg) &amp;amp;lt; 1e-200:
break
diff = fg / dfg
g -= diff
if abs(diff) &amp;amp;lt; tol:
break
return g
The second part is iteration
def roots_newton(coeff_in):
ndof = len(coeff_in)
deg = ndof - 1
coeffs = {}
roots = [([0] * row) for row in xrange(ndof)]
# Make coefficients for all derivatives
coeffs[deg] = coeff_in
for i in xrange(deg - 1, 0, -1):
coeffs[i] = [0] * (i+1)
for j in range(i+1):
coeffs[i][j] = coeffs[i + 1][j] * (i - j + 1)
# Loop through even derivatives
for i2 in xrange(ndof % 2 + 1, ndof, 2):
if i2 == 1: # Base case for odd polynomials
roots[i2][0] = -coeffs[1][1] / coeffs[1][0]
elif i2 == 2: # Base case for even polynomials
a, b, c = coeffs[2]
disc = math.sqrt(b ** 2 - 4 * a * c)
roots[2] = [(-b - disc) / (2 * a), (-b + disc) / (2 * a)]
else:
guesses = [-1000.] + roots[i2 - 2] + [1000.] #Initial guesses of +-1000
f = coeffs[i2]
df = coeffs[i2 - 1]
for i, g in enumerate(guesses):
roots[i2][i] = newton_method(f, df, g, i2)
return roots[-1]
This method will also work for polynomials with repeated roots. However, there is no mechanism for finding complex roots.
## Comparison to built-in methods
The code for running these comparisons is below.
#repeated roots
ps = [[1, 2, 1],
[1, 4, 6, 4, 1],
[1, 6, 15, 20, 15, 6, 1],
[1, 8, 28, 56, 70, 56, 28, 8, 1],
[1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1],
#unique roots from difference of squares
[1, 0, -1],
[1, 0, -5, 0, 4],
[1, 0, -14, 0, 49, 0, -36],
[1, 0, -30, 0, 273, 0, -820, 0, 576],
[1, 0, -204, 0, 16422, 0, -669188, 0, 14739153, 0, -173721912, 0, 1017067024, 0, -2483133696, 0, 1625702400]
]
n = 1000
for p in ps:
avg_newton = time_newton(p, n)
avg_np = time_np(p, n)
print len(p)-1, avg_newton, avg_np, avg_newton/avg_np
The comparison to built-in methods is rather laughable. A comparison to the numpy.roots subroutine is showed below. Many of the numpy routines are optimized/vectorized etc. so it isn't quite a fair comparison. Still, the implementation of Newton's Method shown here appears to have $O(n^3)$ runtime for unique roots (and higher for repeated roots), so it's not exactly a runaway success.
Two tables, below, summarize root-finding times for two polynomials. The first is for repeated roots $(x+1)^n$. The second is for unique roots, e.g. $x^2-1$, $(x^2-1)\cdot(x^2-4)$, etc.
$n$ Newton np.roots times faster
2 $6.10\times 10^{-6}$ $1.41\times 10^{-4}$ $0.0432$
4 $5.36\times 10^{-4}$ $1.15\times 10^{-4}$ $4.68$
6 $2.50\times 10^{-3}$ $1.11\times 10^{-4}$ $22.5$
8 $5.54\times 10^{-3}$ $1.23\times 10^{-4}$ $44.9$
16 $9.66\times 10^{-2}$ $1.77\times 10^{-4}$ $545.9$
$n$ Newton np.roots times faster
2 $5.57\times 10^{-6}$ $1.03\times 10^{-4}$ $0.0540$
4 $2.71\times 10^{-4}$ $1.04\times 10^{-4}$ $2.60$
6 $7.43\times 10^{-4}$ $1.09\times 10^{-4}$ $6.82$
8 $1.43\times 10^{-3}$ $1.21\times 10^{-4}$ $11.8$
16 $1.26\times 10^{-2}$ $1.72\times 10^{-4}$ $73.1$
Some text some message..
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# Embedding a Riemannian manifold with boundary in a closed manifold
Let $$M$$ be a complete Riemannian manifold of finite volume, with sectional curvatures bounded by some $$K>0$$ in absolute value. Let $$M_{\geq R}$$ be the set of points in $$M$$ with injectivity radius $$\geq R$$, that is the set of points $$x\in M$$ for which the restriction of $$exp:T_x M\rightarrow M$$ to the $$R$$-ball in $$T_x M$$ is injective.
Does there exists a closed Riemannian manifold $$\hat{M}$$ with $$\dim{\hat{M}}=\dim{M}$$, such that $$M$$ embeds (Riemannianly) into $$\hat{M}$$? Does there exists such $$\hat{M}$$ whose sectional curvatures are at most $$C\cdot K$$ in absolute value, and whose injectivity radius at least $$\epsilon R$$, for some $$\epsilon,C >0$$ which depend only only on $$K,R,\dim{M}$$?
The approach I tried is to take two copies of the thick part namely $$M_{\geq R}\times \{0,1\}$$, and connect them via a mapping cylinder (i.e connect them through $$\partial M_{\geq R}\times [0,R]$$ in the obvious way). Then, I thought to consider 2 open sets $$U_j$$ containing $$M_{\geq R}\times \{j\}$$ as well as a bit of the cylinder $$(j=0,1)$$, and one open set contained in the cylinder, and take a partition of unity w.r.t this cover in order to smooth things out and define a global Riemannian metric. While it seems that $$M$$ embeds inside $$\hat{M}$$, I don't see a way to show that the curvature and injectivity radius of $$\hat{M}$$ didn't change too much.
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971 views
Which of the following is used in the options field of $IPv4$?
1. Strict source routing
2. Loose source routing
3. Time stamp
4. All of the above
edited | 971 views
Answer : All of the above Reason is
Options(Variable length) : This field represents a list of options that are active for a particular IP datagram. This is an optional field that could be or could not be present.
by Boss (45.3k points)
selected by
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# 3.19: Quarter-Wavelength Transmission Line
[m0091_Quarter-Wavelength_Transmission_Line]
Quarter-wavelength sections of transmission line play an important role in many systems at radio and optical frequencies. The remarkable properties of open- and short-circuited quarter-wave line are presented in Section [m0088_Input_Impedance_for_Open_and_Short_Circuit_Terminations] and should be reviewed before reading further. In this section, we perform a more general analysis, considering not just open- and short-circuit terminations but any terminating impedance, and then we address some applications.
The general expression for the input impedance of a lossless transmission line is (Section [m0087_Input_Impedance_of_a_Terminated_Lossless_Transmission_Line]): $Z_{in}(l) = Z_0 \frac{ 1 + \Gamma e^{-j2\beta l} }{ 1 - \Gamma e^{-j2\beta l} } \label{m0093_eZ}$ Note that when $$l=\lambda/4$$: $2\beta l = 2 \cdot \frac{2\pi}{\lambda} \cdot \frac{\lambda}{4} = \pi \nonumber$ Subsequently: $\begin{split} Z_{in}(\lambda/4) &= Z_0 \frac{ 1 + \Gamma e^{-j\pi} }{ 1 - \Gamma e^{-j\pi} } \\ &= Z_0 \frac{ 1 - \Gamma }{ 1 + \Gamma } \end{split}$ Recall that (Section [m0087_Input_Impedance_of_a_Terminated_Lossless_Transmission_Line]): $\Gamma = \frac{Z_L-Z_0}{Z_L+Z_0}$ Substituting this expression and then multiplying numerator and denominator by $$Z_L+Z_0$$, one obtains $\begin{split} Z_{in}(\lambda/4) &= Z_0 \frac{ \left( Z_L + Z_0\right) - \left(Z_L - Z_0\right) }{ \left( Z_L + Z_0\right) + \left(Z_L - Z_0\right) } \\ &= Z_0 \frac{ 2Z_0 }{ 2Z_L } \end{split}$ Thus, $\boxed{ Z_{in}(\lambda/4) = \frac{Z_0^2 }{ Z_L } } \label{m0091_eQWII}$ Note that the input impedance is inversely proportional to the load impedance. For this reason, a transmission line of length $$\lambda/4$$ is sometimes referred to as a quarter-wave inverter or simply as a impedance inverter.
Quarter-wave lines play a very important role in RF engineering. As impedance inverters, they have the useful attribute of transforming small impedances into large impedances, and vice-versa – we’ll come back to this idea later in this section. First, let’s consider how quarter-wave lines are used for impedance matching. Look what happens when we solve Equation [m0091_eQWII] for $$Z_0$$: $Z_0 = \sqrt{Z_{in}(\lambda/4) \cdot Z_L } \label{m0091_eQWZ0}$ This equation indicates that we may match the load $$Z_L$$ to a source impedance (represented by $$Z_{in}(\lambda/4)$$) simply by making the characteristic impedance equal to the value given by the above expression and setting the length to $$\lambda/4$$. The scheme is shown in Figure [m0091_fQWM].
300-to-$$50~\Omega$$ match using an quarter-wave section of line.
Design a transmission line segment that matches $$300~\Omega$$ to $$50~\Omega$$ at 10 GHz using a quarter-wave match. Assume microstrip line for which propagation occurs with wavelength 60% that of free space.
Solution. The line is completely specified given its characteristic impedance $$Z_0$$ and length $$l$$. The length should be one-quarter wavelength with respect to the signal propagating in the line. The free-space wavelength $$\lambda_0=c/f$$ at 10 GHz is $$\cong 3$$ cm. Therefore, the wavelength of the signal in the line is $$\lambda=0.6\lambda_0\cong 1.8$$ cm, and the length of the line should be $$l=\lambda/4 \cong 4.5$$ mm.
The characteristic impedance is given by Equation [m0091_eQWZ0]: $Z_0 = \sqrt{ 300~\Omega \cdot 50~\Omega } \cong 122.5~\Omega$ This value would be used to determine the width of the microstrip line, as discussed in Section [m0082_Microstrip_Line].
It should be noted that for this scheme to yield a real-valued characteristic impedance, the product of the source and load impedances must be a real-valued number. In particular, this method is not suitable if $$Z_L$$ has a significant imaginary-valued component and matching to a real-valued source impedance is desired. One possible workaround in this case is the two-stage strategy shown in Figure [m0091_fQWM2]. In this scheme, the load impedance is first transformed to a real-valued impedance using a length $$l_1$$ of transmission line. This is accomplished using Equation [m0093_eZ] (quite simple using a numerical search) or using the Smith chart (see “Additional Reading” at the end of this section). The characteristic impedance $$Z_{01}$$ of this transmission line is not critical and can be selected for convenience. Normally, the smallest value of $$l_1$$ is desired. This value will always be less than $$\lambda/4$$ since $$Z_{in}(l_1)$$ is periodic in $$l_1$$ with period $$\lambda/2$$; i.e., there are two changes in the sign of the imaginary component of $$Z_{in}(l_1)$$ as $$l_1$$ is increased from zero to $$\lambda/2$$. After eliminating the imaginary component of $$Z_L$$ in this manner, the real component of the resulting impedance may then be transformed using the quarter-wave matching technique described earlier in this section.
Matching a patch antenna to $$50~\Omega$$.
A particular patch antenna exhibits a source impedance of $$Z_A = 35+j35~\Omega$$. (See “Microstrip antenna” in “Additional Reading” at the end of this section for some optional reading on patch antennas.) Interface this antenna to $$50~\Omega$$ using the technique described above. For the section of transmission line adjacent to the patch antenna, use characteristic impedance $$Z_{01}=50~\Omega$$. Determine the lengths $$l_1$$ and $$l_2$$ of the two segments of transmission line, and the characteristic impedance $$Z_{02}$$ of the second (quarter-wave) segment.
Solution. The length of the first section of the transmission line (adjacent to the antenna) is determined using Equation [m0093_eZ]: $Z_1(l_1) = Z_{01} \frac{ 1 + \Gamma e^{-j2\beta_1 l_1} }{ 1 - \Gamma e^{-j2\beta_1 l_1} }$ where $$\beta_1$$ is the phase propagation constant for this section of transmission line and $\Gamma \triangleq \frac{Z_A-Z_{01}}{Z_A+Z_{01}} \cong -0.0059+j0.4142$ We seek the value of smallest positive value of $$\beta_1 l_1$$ for which the imaginary part of $$Z_1(l_1)$$ is zero. This can determined using a Smith chart (see “Additional Reading” at the end of this section) or simply by a few iterations of trial-and-error. Either way we find $$Z_1(\beta_1 l_1 = 0.793~\mbox{rad}) \cong 120.719-j0.111~\Omega$$, which we deem to be close enough to be acceptable. Note that $$\beta_1 = 2\pi/\lambda$$, where $$\lambda$$ is the wavelength of the signal in the transmission line. Therefore $l_1 = \frac{\beta_1 l_1}{\beta_1} = \frac{\beta_1 l_1}{2\pi} \lambda \cong 0.126\lambda$
The length of the second section of the transmission line, being a quarter-wavelength transformer, should be $$l_2 = 0.25\lambda$$. Using Equation [m0091_eQWZ0], the characteristic impedance $$Z_{02}$$ of this section of line should be $Z_{02} \cong \sqrt{\left(120.719~\Omega\right) \left(50~\Omega\right) } \cong 77.7~\Omega$
Discussion. The total length of the matching structure is $$l_1+l_2 \cong 0.376\lambda$$. A patch antenna would typically have sides of length about $$\lambda/2 = 0.5\lambda$$, so the matching structure is nearly as big as the antenna itself. At frequencies where patch antennas are commonly used, and especially at frequencies in the UHF (300–3000 MHz) band, patch antennas are often comparable to the size of the system, so it is not attractive to have the matching structure also require a similar amount of space. Thus, we would be motivated to find a smaller matching structure.
Although quarter-wave matching techniques are generally effective and commonly used, they have one important contraindication, noted above – They often result in structures that are large. That is, any structure which employs a quarter-wave match will be at least $$\lambda/4$$ long, and $$\lambda/4$$ is typically large compared to the associated electronics. Other transmission line matching techniques – and in particular, single stub matching (Section [m0094_Single-Stub_Matching]) – typically result in structures which are significantly smaller.
The impedance inversion property of quarter-wavelength lines has applications beyond impedance matching. The following example demonstrates one such application:
RF/DC decoupling in transistor amplifiers.
Transistor amplifiers for RF applications often receive DC current at the same terminal which delivers the amplified RF signal, as shown in Figure [m0091_BiasT]. The power supply typically has a low output impedance. If the power supply is directly connected to the transistor, then the RF will flow predominantly in the direction of the power supply as opposed to following the desired path, which exhibits a higher impedance. This can be addressed using an inductor in series with the power supply output. This works because the inductor exhibits low impedance at DC and high impedance at RF. Unfortunately, discrete inductors are often not practical at high RF frequencies. This is because practical inductors also exhibit parallel capacitance, which tends to decrease impedance.
A solution is to replace the inductor with a transmission line having length $$\lambda/4$$ as shown in Figure [m0091_QWDecoupling]. A wavelength at DC is infinite, so the transmission line is essentially transparent to the power supply. At radio frequencies, the line transforms the low impedance of the power supply to an impedance that is very large relative to the impedance of the desired RF path. Furthermore, transmission lines on printed circuit boards are much cheaper than discrete inductors (and are always in stock!).
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# Acoustic scattering
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
2D model of the scattering of a wave by multiple ellipsoidal obstacles.
To run the model, open scattering.pro with Gmsh.
### Scattering problem
Let $\Omega_1^-, \Omega_2^-, \ldots, \Omega_M^-$, be $M$ bounded, disjoint and connected open subset of $\mathbb{R}^2$ such that $\mathbb{R}^2\setminus\overline{\Omega_p^-}$ is connected for $p=1,\ldots,M$. Let $\Omega^- = \bigcup_{p=1}^M\Omega^-_p$ be the domain occupied by the $M$ obstacles and $\Omega^+ = \mathbb{R}^2\setminus\overline{\Omega^-}$ be the connected propagation domain which is also assumed to be connected. Lastly, $\Gamma$ denotes the boundary $\Omega^-$ with a unit outwardly directed normal $\mathbf{n}$.
When the obstacles $\Omega^-$ is illuminated by a time-harmonic incident wave $u^{inc}$, it generates a scattered field $u$, solution of the scattering problem, where $k >0$ is the wavenumber and the time dependence is assumed to be of the form $e^{-ikt}$, $$\label{eq:pbu} \begin{cases} \Delta u + k^2 u = 0 & \text{ in } \Omega^+\\ u = -u^{inc} & \text{ on } \Gamma \\ \text{ u outgoing.} & \end{cases}$$ The operator $\Delta$ is the Laplace operator. Here, the boundary condition is a Dirichlet one (sound-soft obstacle), however, a Neumann boundary condition can also be applied (sound-hard scatterer). The outgoing condition stands for the Sommerfeld radiation condition $$\displaystyle{\lim_{\|\mathbf{x}\| \to \infty} \|\mathbf{x}\|^{(1)/2}\left( \nabla \cdot \frac{\mathbf{x}}{\|\mathbf{x}\|} - iku\right)=0,}$$ where $\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2}$ is the euclidian norm on $\mathbb{R}^2$.
Problem (\ref{eq:pbu}) admits a unit solution[1]. To solve it numerically using a finite element method, one must truncate the infinit domain $\Omega$. Several methods exist, such as Absorbing Boundary Condition (ABC) [2], Perfectly Matched Layer (PML)[3] [4], Boundary Integral Equation (BIE)[5], $\ldots$ A revue of different technics can be found here[6]. We first consider the cas of a simple ABC (Sommerfeld-like). We introduce a fictitious boundary $\Gamma^{\infty}$ surrounding all the scatterers. To simplify, we assume that $\Gamma^{\infty}$ is a disk of radius $R$ and centered on $O = (0,0)$. We denote by $\Omega$ the subset of $\Omega^+$ with boundary $\Gamma^{\infty}\bigcup\Gamma$ (the intersection between the infinite domain $\Omega^+$ and the disk of center $O$ and radius $R$). On the fictitious boundary $\Gamma^{\infty}$ is pluged the ABC, which is here $$\partial_{\mathbf{n}}u = iku \qquad \text{ on } \Gamma^{\infty}. \label{eq:ABC0}$$ Note that the normal $\mathbf{n}$ is pointing outside $\Omega$ on $\Gamma^{\infty}$ (and inside on $\Gamma$). The new problem, which approachs the original one (\ref{eq:pbu}), then reads as $$\label{eq:pbuABC} \begin{cases} \Delta u + k^2 u = 0 & \text{ in } \Omega\\ u = -u^{inc} & \text{ on } \Gamma \\ \partial_{\mathbf{n}}u = iku & \text{ on }\Gamma^{\infty}. & \end{cases}$$ To simplify, the name $u$ still refers to the solution of the above problem (even it approximates the "real" solution of problem (\ref{eq:pbu})).
### Weak formulation
The following sobolev spaces of $H^1(\Omega)$ must first be introduced $$V_{inc} = \left\{v \in H^1(\Omega) \text{ such that } v|_{\Gamma} = -u_{inc}\right\},$$ and $$V_{0} = \left\{v \in H^1(\Omega) \text{ such that } v|_{\Gamma} = 0\right\}.$$ The weak formulation can now be expressed $$\label{eq:Diri} \left\{ \begin{array}{l} \displaystyle{\text{Find } u \in V_{inc} \text{ such that}}\\ \displaystyle{\forall u'\in V_{0},\qquad \int_{\Omega}\nabla u \nabla u'\;{\rm d}\Omega - \int_{\Omega}k^2 uu'\;{\rm d}\Omega - \int_{\Gamma^{\infty}}ik u u'\;{\rm d}\Gamma^{\infty} =0.} \end{array}\right.$$
In the Neumann case, that is $\partial_{\mathbf{n}}u = -\partial_{\mathbf{n}}u_{inc}$ on $\Gamma$, then the weak formulation reads as $$\left\{ \begin{array}{l} \displaystyle{\text{Find } u \in H^1(\Omega) \text{ such that}}\\ \displaystyle{\forall u'\in H^1(\Omega),\qquad \int_{\Omega}\nabla u \nabla u'\;{\rm d}\Omega - \int_{\Omega}k^2 uu'\;{\rm d}\Omega - \int_{\Gamma^{\infty}}ik u u'\;{\rm d}\Gamma^{\infty} - \int_{\Gamma}\partial_{\mathbf{n}}u_{inc} u'\;{\rm d}\Gamma =0. } \end{array}\right.$$
Remark: the space $V_{inc}$ is not a Banach space but an Hermitian one and $u$ and $u'$ do not belong to the same space. Thus, the weak formulation (\ref{eq:Diri}) is not written in the usual way of finite element method. However, GetDP takes care of the dirichlet boundary condition and thus, in practice, the user will implement equation (\ref{eq:Diri}) directly.
### A few words on the far field
In the direction $\boldsymbol{\theta} = (\cos(\theta),\sin(\theta))$, when $\|\mathbf{x}\|$ tends to infinity, the scattered field $u$ has the following behavior $$u(\|\mathbf{x}\|\boldsymbol{\theta}) = \frac{e^{ik\|\mathbf{x}\|}}{\|\mathbf{x}\|^{1/2}} a(\theta) + O\left(\frac{1}{\|\mathbf{x}\|}\right),$$ where $a(\theta)$ is the far field of $u$ in the direction $\theta$. To compute this with a finite element method, we propose to use the integral representation of $u$ $$\forall \mathbf{x}\in\Omega^+,\qquad u(\mathbf{x}) = \int_{\Gamma} \partial_{\mathbf{n}}G(\mathbf{x},\mathbf{y}) u|_{\Gamma}(\mathbf{y}) \;{\rm d}\Gamma(\mathbf{y}) - \int_{\Gamma} G(\mathbf{x},\mathbf{y}) \partial_{\mathbf{n}}u|_{\Gamma}(\mathbf{y}) \;{\rm d}\Gamma(\mathbf{y}),$$ where $G(\mathbf{x},\mathbf{y})$ is the Helmholtz Green function, defined in two dimension by $$\forall \mathbf{x},\mathbf{y}\in\mathbb{R}^2, \mathbf{x}\neq\mathbf{y},\qquad G(\mathbf{x},\mathbf{y}) = \frac{i}{4}H_0^{(1)}(k\|\mathbf{x}-\mathbf{y}\|),$$ which involves the zeroth order Hankel function of first kind $H_0^{(1)}$. Then, the point $\mathbf{x}$ is thrown to "infinity". In practice, $u(\mathbf{x})$ is computed on a circle with a very large radius (e.g. $1000$). The obtained result will then be close to the far field.
Note that, for a Dirichlet boundary condition, we need to compute the normal derivative $\partial_{\mathbf{n}}u$ of $u$ on $\Gamma$. That is why a Region "TrGr" is computed, composed by the elements of $\Omega$ that are connected to $\Gamma$. More information can be found in the Acoustic2D_impenetrable.pro file.
### Incident waves
In the following program, two different kind of incident waves are considered.
The plane wave of direction $\boldsymbol{\beta}=(\cos(\beta),\sin(\beta))$
$$u^{inc}(\mathbf{x}) = e^{ik(\boldsymbol{\beta}\cdot\mathbf{x})},$$ where $\boldsymbol{\beta}\cdot\mathbf{x} = x_1\cos(\beta) + x_2\sin(\beta)$ is the usual inner product on $\mathbb{R}^2$. The gradient of the wave (involved for example in a Neumann boundary condition) is given by $$\nabla u^{inc}(\mathbf{x}) = ike^{ik(\boldsymbol{\beta}\cdot\mathbf{x})}\boldsymbol{\beta} = iku^{inc}(\mathbf{x})\boldsymbol{\beta}$$
The wave emitted by a point source located on $\mathbf{s} = (s_1,s_2)$.
Obviously, the point $\mathbf{s}$ must be outside $\overline{\Omega^-}$. The incident wave is then the Green function centered on $\mathbf{s}$ : $$u^{inc}(\mathbf{x}) = \frac{i}{4}H_0^{(1)}(k\|\mathbf{x}-\mathbf{s}\|).$$ Its gradient is then given by $$\nabla u^{inc}(\mathbf{x}) = -\frac{i}{4}H_1^{(1)}(k\|\mathbf{x}-\mathbf{s}\|) \frac{\mathbf{x}-\mathbf{s}}{\|\mathbf{x}-\mathbf{s}\|},$$ where $H_1^{(1)}$ is the Hankel function of the first kind and first order. This is due to the fact that the derivative of the Hankel function of order $0$ is the opposite of the Hankel function of order $1$ $$(H_0^{(1)})' = -H_1^{(1)}.$$
### Approximation with a PML
Instead of using an absorbing boundary condition (ABC) of order $0$ \eqref{eq:ABC0}, we can use a Perfectly Matched Layer (PML) [7] [8] surrounding a domain of interest.
First, let us build the domain $\Omega_{e}$ on which an approximation of the scattered field will be computed. To simplify, $\Omega_{e}$ will be a rectangle $$\begin{array}{rcl} \Omega_e &=& \left(\left]\alpha_1,\beta_{1}\right[\times\left]\alpha_{2},\beta_{2}\right[\right) \ \setminus\ \overline{\Omega^-} \\ &=& \left(\left]\alpha_1,\beta_{1}\right[\times\left]\alpha_{2},\beta_{2}\right[\right) \ \bigcap\ {\Omega^+}, \end{array}$$ where $\alpha_1,b_{1},\alpha_{2}$ and $\beta_{2}$ are real constant such that $\alpha_{j}<\beta_{j}$ for $j=1,2$. Obviously, this set is sufficiently large to contain all the scatterers.
Around the rectangle $\Omega_e$ is built the absorbing layer $\Omega_{PML}$ defined by $$\Omega_{PML} = \left(\;\left]\alpha_1 - \delta^{PML}_{1}, \beta_{1}+\delta^{PML}_{1}\right[\times\left]\alpha_{2} - \delta^{PML}_{2},\beta_{2} + \delta^{PML}_{2}\right[ \;\right)\setminus\overline{\Omega_e},$$ where $\delta^{PML}_{1}$ and $\delta^{PML}_{2}$ are the thickness of the PML in the $x$ et $y$ direction, respectively. Let $\Sigma_{int}$ be the boundary separating $\Omega_e$ from $\Omega_{PML}$ and $\Sigma_{PML} = \partial\Omega_{PML}\setminus\Sigma_{int}$, the boundary which truncated the PML. Finally, let us introduce $\Omega_{T} = \overline{\Omega_e}\bigcup\Omega_{PML}$ the total computation domain and. In practice, the thickness $\delta^{PML}_{1}$ and $\delta^{PML}_{2}$ are very small, e.g. : 10 elements.
The approximation $u_{PML}$ of $u$ in $\Omega_e$ is solution of a partial differential equation in $\Omega_T$ and satisfies a boundary condition on $\Sigma_{PML}$. The field $u_{PML}$ is dissipated through it passage in $\Omega_{PML}$. This absorption is obtained by modifying the Helmholtz equation. However, let us first focus on the boundary condition. If the outgoing waves are sufficiently damped by the PML then the amplitude will be close to zero in a neighborhood of $\Sigma_{PML}$. Thus, the role of the boundary condition will be unimportant [9]. That is why we propose to set a homogeneous Dirichlet boundary condition on $\Sigma_{PML}$. However, an absorbing boundary condition could be used to make the result more accurate.
Let us now focus on the absorption. As said before, this absorption is obtained by modifying the Helmholtz equation inside $\Omega_{PML}$. More precisely, the problem solved by $u_{PML}$ is the following $$\label{eqRT:HelmholtzPML} \begin{array}{r c l l} \displaystyle{\partial_{x_{1}}\left( \frac{S_{x_{2}}}{S_{x_{1}}}\partial_{x_{1}}u_{PML} \right) + \partial_{x_{2}}\left( \frac{S_{x_{1}}}{S_{x_{2}}}\partial_{x_{2}}u_{PML} \right) + k^{2} S_{x_{1}}S_{x_{2}}u_{PML}} &=& f & (\Omega_{T})\\ u_{PML} &=& 0 & (\Sigma_{PML}) \end{array}$$ with $$\forall\mathbf{x}=(x_1,x_2)\in\overline{\Omega_{T}},\qquad\begin{cases} \displaystyle{S_{x_{1}}(\mathbf{x}) = 1 - \frac{\sigma_{x_{1}}(x_1)}{ik}}, & \\ \displaystyle{S_{x_{2}}(\mathbf{x}) = 1 - \frac{\sigma_{x_{2}}(x_2)}{ik}}. & \end{cases}$$ The functions $\sigma_{x_{1}}$ and $\sigma_{x_{2}}$ are called damping functions. When $\sigma_{x_{1}}$ and $\sigma_{x_{2}}$ are equal to zero, then equation (\ref{eqRT:HelmholtzPML}) reduce to (classical) Helmholtz equation, that is why in general, these functions vanish on $\Omega_{e}$. On the other hand, when the damping functions are positive, they act as a dissipative term in the equation. Remark that, by introducing the following matrix $$D = \left[ \begin{array}{c c} \frac{S_{x_{2}}}{S_{x_{1}}} & 0\\ 0 & \frac{S_{x_{2}}}{S_{x_{1}}} \end{array} \right],$$ then problem (\ref{eqRT:HelmholtzPML}) can be rewritten as $$\label{eqRT:eqPML} \left\{\begin{array}{r c l l} \displaystyle{{\rm div}(D\nabla u_{PML}) + k^{2}S_{x_{1}}S_{x_{2}}u_{PML}}& =& f & (\Omega_{T})\\ \displaystyle{u_{PML}} &=& 0 & (\Sigma_{PML}) \end{array}\right.$$ The choice of the damping functions $\sigma_{x_{1}}$ and $\sigma_{x_{2}}$ obviously influces the dissipative power of the PML. Here, these functions are simply linear and, moreover, continuous on $\Omega_{T}$, in order to avoid any non desirated reflection. As they vanish in $\Omega_e$, these functions stays equal to zero on $\Sigma_{int}$ and reach their maximum on $\Sigma_{PML}$, which will be set to $100$. This choice is purely given by the experiment and is not a general coefficient. In fact, for another problem, this parameter must certainly be tuned.
To summarize, the two damping functions $\sigma_{x_{1}}$ and $\sigma_{x_{2}}$ are given by: $$\forall j=1,2, \forall \mathbf{x}=(x_1,x_2) \in\overline{\Omega_{T}}, \qquad \sigma_{x_{j}}(\mathbf{x}) = \begin{cases} 0 & \text{ if } \alpha_{j}< x_{j}< \beta_{j},\\ \displaystyle{\frac{100\left|x_{j}-\beta_{j}\right|}{\delta^{PML}_{j}}}&\text{ if } \beta_{j} \leq x_{j} \leq \beta_{j} + \delta^{PML}_{j}, \\ \displaystyle{\frac{100\left|x_{j}-\alpha_{j}\right|}{\delta^{PML}_{j}}}&\text{ if } \alpha_{j}-\delta^{PML}_{j} \leq x_{j} \leq \alpha_{j}. \end{cases}$$ For $j=1,2$, function $\sigma_{x_{j}}$ has the following appearance, with respect to $x_{j}$ for a fix $x_{i}$ with $\alpha_{i} - \delta^{PML}_{i}\leq x_{i} \leq \beta_{i} +\delta^{PML}_{i}$, for $i,j = 1,2$ with $i\neq j$.
Inside $\Omega_e$, the damping functions vanish whereas inside $\Omega_{PML}$, at least one of them do not vanish. Note that in the "corners" of $\Omega_{PML}$, both damping function are note equal to zero.
Finally, let us just remark that the efficiency of the PML can be improved by using another type of damping functions, as quadratic functions or even functions with an infinite integral on $\Omega_{PML}$, as the one proposed by Bermudez et. al [10].
The weak formulation can be derived in the same way as for the ABC. To simplify, we denote by $u$ the approximation of the scattered field in $\Omega_e$ (instead of $u_{PML}$). Let us introduce the following subspaces of $H^1(\Omega_T)$ : $$V_{inc} = \left\{\omega \in H^1(\Omega_T) \text{ such that } \omega|_{\Gamma} = -u^{inc}|_{\Gamma} \text{ and } \omega|_{\Sigma_{PML}} = 0\right\},$$ and $$V_0 = \left\{\omega \in H^1(\Omega_T) \text{ such that } \omega|_{\Gamma} = 0 \text{ and } \omega|_{\Sigma_{PML}} = 0\right\}.$$ Then, the weak formulation of problem (\ref{eqRT:eqPML}) reads as $$\left\{ \begin{array}{l} \displaystyle{\text{Find } u \in V_{inc} \text{ such that}}\\ \displaystyle{\forall u'\in V_{0},\qquad \int_{\Omega}D \nabla u \nabla u'\;{\rm d}\Omega - \int_{\Omega}k^2 S_{x_1} S_{x_2} uu'\;{\rm d}\Omega =0.} \end{array}\right.$$
### Analysis types
You can choose to impose Dirichlet or Neumann boundary conditions by selecting the corresponding Resolution in the GetDP subtree. In the same subtree the Post-processing menu allows you to choose different fields to visualize:
(1) ud: Dirichlet, compute the field $u$ and the total field $u + u^{inc}$ and their absolute values
(2) ud_farfield: Dirichlet, far field of the scattered field
(3) ud_traces: Dirichlet, compute the normal derivative trace of $u$ along $\Gamma$
(4) un: same as (1) for a Neumann boundary condition
(5) un_farfield: same as (2) for a Neumann boundary condition
(6) un_traces: Neumann, compute the trace of $u$ along $\Gamma$
(7) uinc: compute the incident field $u^{inc}$
When a PML layer is selected, you get the additional choices:
ud_PML: computes $u$ and $|u|$ in $\Omega_{PML}$ in the Dirichlet case
un_PML: computes $u$ and $|u|$ in $\Omega_{PML}$ in the Neumann case
### Results with Sommerfeld ABC
#### Single scattering
Here is one result obtained with one circular scatterers of radius $2$, with $k=2$ and an incident angle of $\pi$. The first picture depicted the real part of the scattered field $u$, the second one being the absolute value of $u$. The third figure represents the modulus of the total field $u+u_{inc}$ (which is the "physical" field). The last picture shows the far field pattern of the scatterer field.
#### Multiple scattering
In this example, three obstacles are placed in the domain :
$(-1,-2)$ with radius $2$
$(0,3)$ with radius $0.5$
$(2,0)$ with radius $0.5$
(they are all circular even if it is not a necessarity).
As previously, the first picture shows the real part of the scattered field $u$ and the second the absolute value of $u$. The third figure is a representation of the modulus of the total field $u+u_{inc}$ and the last one shows the far field pattern of the scatterer field.
### Results with PML
#### Single scattering
Here is one result obtained with one circular scatterers of radius $1$, with $k=2$ and an incident angle of $\pi$ ($u^{inc}$ is plane). The thickness of the PML is set to around $30$ finite elements, which is quite high.
The first picture depicted the real part of the scattered field $u$, the second one being the absolute value of $u$. The third figure represents the modulus of the total field $u_T = u+u_{inc}$ (which is the "physical" field). The last picture shows the absolute value of the scattered field in the PML. This clearly show the decay of the field inside the absorbing layer.
#### Multiple scattering
In this example, two ellipses are placed in the domain :
$(0,-4)$ with semi-axes $1$ and $2$
$(0,4)$ with semi-axes $1$ and $0.5$
As previously, the first picture shows the real part of the scattered field $u$ and the second the absolute value of $u$. The third figure is a representation of the modulus of the total field $u+u_{inc}$ and the last one shows the decay of the scattered field in the PML.
## References
1. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, 1998
2. A. Bayliss, M. Gunzburger and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM Journal on Applied Mathematics, 1982
3. J.-P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 1994
4. F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM Journal on Scientific Computing, 1998
5. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons Inc., 1983
6. X. Antoine, C. Geuzaine and K. Ramdani, Wave Propagation in Periodic Media - Analysis, Numerical Techniques and Practical Applications, Chapter Computational Methods for Multiple Scattering at High Frequency with Applications to Periodic Structures Calculations, Progress in Computational Physics, 2010
7. J.-P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 1994
8. F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM Journal on Scientific Computing, 1998
9. P. Petropoulos , On the Termination of the Perfectly Matched Layer with Local Absorbing Boundary Conditions, Journal of Computational Physics, 1998
10. A. Bermúdez, L. Hervella-Nieto, A. Prieto and R. Rodríguez, An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems, J. Comput. Phys., 2007
Model developed by B. Thierry.
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# Blender wouldn't paint neither face sets not the mask anymore
I got blender into a somewhat strange state after I decided to add some vertex colours to my sculpt for better visibility.
As I load the save file I cannot paint the face sets anymore, not can I paint the mask. 'Invert mask' would do nothing.
Creating a new mesh and trying to sculpt it won't work.
Effectively, I'm forced to go back to my older save, or copy everything to a new document.
I think the only things I did to get here was:
• set up the vertex paint material
• tinker with the viewport display to show vertex colours
• actually paint some over the mesh
Any advice on how can I force the document back into an usable state?
Note 1: Here's the blend file featuring the problem.
Note 2: The overlays are indeed enabled.
The problem is the Armature. Modifiers prevent Hide, Mask, and Facesets in Sculpt mode. Usually, you get a warning like this:
Armature, Cloth, and Hair/Particles, and all the other physics are also Modifiers. It's probably a bug that Blender doesn't warn you if you have one of these 3 activated. (Dynamic Paint gives you a warning.)
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## Bernoulli
• Bernoulli
• Volume 19, Number 5B (2013), 2250-2276.
### Uniform convergence of convolution estimators for the response density in nonparametric regression
#### Abstract
We consider a nonparametric regression model $Y=r(X)+\varepsilon$ with a random covariate $X$ that is independent of the error $\varepsilon$. Then the density of the response $Y$ is a convolution of the densities of $\varepsilon$ and $r(X)$. It can therefore be estimated by a convolution of kernel estimators for these two densities, or more generally by a local von Mises statistic. If the regression function has a nowhere vanishing derivative, then the convolution estimator converges at a parametric rate. We show that the convergence holds uniformly, and that the corresponding process obeys a functional central limit theorem in the space $C_{0}(\mathbb{R})$ of continuous functions vanishing at infinity, endowed with the sup-norm. The estimator is not efficient. We construct an additive correction that makes it efficient.
#### Article information
Source
Bernoulli, Volume 19, Number 5B (2013), 2250-2276.
Dates
First available in Project Euclid: 3 December 2013
https://projecteuclid.org/euclid.bj/1386078602
Digital Object Identifier
doi:10.3150/12-BEJ451
Mathematical Reviews number (MathSciNet)
MR3160553
Zentralblatt MATH identifier
1281.62103
#### Citation
Schick, Anton; Wefelmeyer, Wolfgang. Uniform convergence of convolution estimators for the response density in nonparametric regression. Bernoulli 19 (2013), no. 5B, 2250--2276. doi:10.3150/12-BEJ451. https://projecteuclid.org/euclid.bj/1386078602
#### References
• [1] Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A. (1998). Efficient and Adaptive Estimation for Semiparametric Models. New York: Springer. Reprint of the 1993 original.
• [2] Bickel, P.J. and Ritov, Y. (1988). Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhyā Ser. A 50 381–393.
• [3] Chaudhuri, P., Doksum, K. and Samarov, A. (1997). On average derivative quantile regression. Ann. Statist. 25 715–744.
• [4] Efromovich, S. and Samarov, A. (2000). Adaptive estimation of the integral of squared regression derivatives. Scand. J. Statist. 27 335–351.
• [5] Escanciano, J.C. and Jacho-Chávez, D.T. (2012). $\sqrt{n}$-uniformly consistent density estimation in nonparametric regression models. J. Econometrics 167 305–316.
• [6] Frees, E.W. (1994). Estimating densities of functions of observations. J. Amer. Statist. Assoc. 89 517–525.
• [7] Giné, E. and Mason, D.M. (2007). On local $U$-statistic processes and the estimation of densities of functions of several sample variables. Ann. Statist. 35 1105–1145.
• [8] Laurent, B. (1996). Efficient estimation of integral functionals of a density. Ann. Statist. 24 659–681.
• [9] Müller, U.U. (2012). Estimating the density of a possibly missing response variable in nonlinear regression. J. Statist. Plann. Inference 142 1198–1214.
• [10] Müller, U.U., Schick, A. and Wefelmeyer, W. (2009). Estimating the error distribution function in nonparametric regression with multivariate covariates. Statist. Probab. Lett. 79 957–964.
• [11] Müller, U.U., Schick, A. and Wefelmeyer, W. (2013). Non-standard behavior of density estimators for functions of independent observations. Comm. Statist. Theory Methods 42 2991–3000.
• [12] Nickl, R. (2007). Donsker-type theorems for nonparametric maximum likelihood estimators. Probab. Theory Related Fields 138 411–449. Erratum: Probab. Theory Related Fields 141 (2008) 331–332.
• [13] Nickl, R. (2009). On convergence and convolutions of random signed measures. J. Theoret. Probab. 22 38–56.
• [14] Saavedra, Á. and Cao, R. (1999). Rate of convergence of a convolution-type estimator of the marginal density of an MA(1) process. Stochastic Process. Appl. 80 129–155.
• [15] Saavedra, A. and Cao, R. (2000). On the estimation of the marginal density of a moving average process. Canad. J. Statist. 28 799–815.
• [16] Schick, A. (1993). On efficient estimation in regression models. Ann. Statist. 21 1486–1521. Correction and Addendum: Ann. Statist. 23 (1995) 1862–1863.
• [17] Schick, A. and Wefelmeyer, W. (2004). Functional convergence and optimality of plug-in estimators for stationary densities of moving average processes. Bernoulli 10 889–917.
• [18] Schick, A. and Wefelmeyer, W. (2004). Root $n$ consistent and optimal density estimators for moving average processes. Scand. J. Statist. 31 63–78.
• [19] Schick, A. and Wefelmeyer, W. (2004). Root $n$ consistent density estimators for sums of independent random variables. J. Nonparametr. Stat. 16 925–935.
• [20] Schick, A. and Wefelmeyer, W. (2007). Root-$n$ consistent density estimators of convolutions in weighted $L_{1}$-norms. J. Statist. Plann. Inference 137 1765–1774.
• [21] Schick, A. and Wefelmeyer, W. (2007). Uniformly root-$n$ consistent density estimators for weakly dependent invertible linear processes. Ann. Statist. 35 815–843.
• [22] Schick, A. and Wefelmeyer, W. (2008). Root-$n$ consistency in weighted $L_{1}$-spaces for density estimators of invertible linear processes. Stat. Inference Stoch. Process. 11 281–310.
• [23] Schick, A. and Wefelmeyer, W. (2009). Convergence rates of density estimators for sums of powers of observations. Metrika 69 249–264.
• [24] Schick, A. and Wefelmeyer, W. (2009). Non-standard behavior of density estimators for sums of squared observations. Statist. Decisions 27 55–73.
• [25] Støve, B. and Tjøstheim, D. (2012). A convolution estimator for the density of nonlinear regression observations. Scand. J. Statist. 39 282–304.
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Enter the maximum displacement, spring constant, and mass into the calculator to determine the spring velocity.
## Spring Velocity Formula
The following equation is used to calculate the Spring Velocity.
Vs = SQRT ( k*x^2 / m)
• Where Vs is the spring velocity (m/s)
• k is the spring constant (N/m)
• x is the maximum displacement (m)
• m is the mass (kg)
To calculate the spring velocity, multiply the spring constant by the displacement squared, divide by the mass, then finally, take the square root of the result.
## What is a Spring Velocity?
Definition:
A spring velocity describes the maximum velocity of a spring when the spring returns to its initial position after being compressed or stretched.
## How to Calculate Spring Velocity?
Example Problem:
The following example outlines the steps and information needed to calculate
First, determine the spring constant (N/m). In this example, the spring constant (N/m) is found to be 5 N/m.
Next, determine the maximum displacement. For this problem, the maximum displacement is found to be 3m.
Next, determine the mass. In this case, the mass is measured to be 2kg.
Finally, calculate the Spring Velocity using the formula above:
Vs = SQRT ( k*x^2 / m)
Vs = SQRT ( 5*3^2 / 2)
Vs = 4.74 m/s
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# Is a square prism same as a cube? - Mathematics
Is a square prism same as a cube?
#### Solution
$\text { Yes, a square prism and a cube are the same. }$
$\text { Both of them have 6 faces, 8 vertices and 12 edges }.$
$\text { The only difference is that a cube has 6 equal faces, while a square prism has a shape like a cuboid with two square }$
$\text { faces, one at the top and the other at the bottom and with, possibly, 4 rectangular faces in between. }$
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 8 Maths
Chapter 19 Visualising Shapes
Exercise 19.1 | Q 4 | Page 9
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# Background and Activities of the Samsung Ombudsperson Commission in Korea
## Article information
J Prev Med Public Health. 2019;52(4):265-271
Publication date (electronic) : 2019 July 3
doi : https://doi.org/10.3961/jpmph.19.033
1Seoul National University School of Law, Seoul, Korea
2Department of Occupational and Environmental Medicine, Gachon University College of Medicine, Incheon, Korea
3Department of Preventive Medicine,College of Medicine, The Catholic University of Korea, Seoul, Korea
4Expert Advisor at Seoul National University Human Rights Center, Attorney at Law, Seoul, Korea
Corresponding author: Cheolsoo Lee, PhD Seoul National University School of Law, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea E-mail: charles2@snu.ac.kr
Received 2019 February 13; Accepted 2019 July 3.
## Abstract
### Objectives
The Samsung Ombudsperson Commission was launched as an independent third-party institution following an agreement among Samsung Electronics, Supporters for Health and Right of People in Semiconductor Industry (Banolim in Korean, an independent NGO), and the Family Compensation Committee, in accordance with the industry accident prevention measure required by the settlement committee to address the issues related to employees who allegedly died from leukemia and other diseases as a result of working at Samsung’s semiconductor production facilities.
### Methods
The Commission has carried out a comprehensive range of activities to review and evaluate the status of the company’s occupational accidents management system, as well as occupational safety and health risk management within its facilities.
### Results
Based on the results of this review, termed a comprehensive diagnosis, the Commission presented action plans for improvement to strengthen the company’s existing safety and health management system and to effectively address uncertain risks in this area going forward.
### Conclusions
The Commission will monitor the execution of the suggested tasks and provide advice and guidance to ensure that Samsung’s semiconductor and liquid crystal display production lines are safer.
## INTRODUCTION
In March 2007, Yu-mi Hwang, a former worker at a Samsung Electronics plant in Giheung, died of acute myelocytic leukemia. In November of the same year, other Samsung employees suffering from similar conditions, Hwang’s bereaved family members, and civil activists jointly formed a group called the Joint Action Committee for Learning the Facts about the Outbreak of Leukemia at Samsung Semiconductors and for securing the Basic Rights of Labor. The group later changed its name to Supporters for Health and Right of People in Semiconductor Industry (SHARPs, Banolim in Korean). Hwang’s family members and the company’s employees applied to the Korea Workers’ Compensation and Welfare Service for workers’ compensation insurance, claiming that leukemia and other incurable diseases should be acknowledged as a type of occupational diseases. The claim was denied based on the results of the Occupational Safety and Health Research Institute (OSHRI)’s epidemiological investigation result, which concluded that “there was an increased risk of lympho-hematopoietic diseases, but it was not statistically significant.” In 2010, the group filed an administrative lawsuit in the hope of overturning the denial, and in this lawsuit, the court ruled in favor of some of the cases [1] (Appendix 1). In the meanwhile, three Korean semiconductor makers conducted self-assessments on occupational health risks in 2009, following advice from the Ministry of Employment and Labor [2]. OSHRI’s epidemiological investigations of Korea’s six semiconductor companies to make up for statistical loopholes in the previous study is still ongoing [3]. The final result released in May, 2019 that “there was an increased risk of lympho-hematopoietic diseases, however, the association between work and diseases was not confirmed” [4].
Since 2013, Samsung Electronics, SHARPs, and the Family Committee for Compensation over Leukemia Issue have made efforts to reach a settlement that can put an end to the long-running leukemia issue, leading to the establishment of a mediation committee to address the issues on the putative workplace outbreak of leukemia and other diseases at Samsung Electronics facilities. The committee released a settlement ruling in July 2015, requiring apology, compensation, and prevention measures. The three parties reached an agreement in January 2016, regarding prevention measures, which led to the establishment of the Samsung Ombudsperson Commission [5] (Appendix 2).
This article describes how this settlement laid the foundations for the formation of the Ombudsperson Commission and summarizes the details of the comprehensive evaluation activities, referred to as a comprehensive diagnosis, as well as the results.
## METHODS
### The History and Foundation of the Ombudsperson Commission
The Samsung Ombudsperson Commission was launched based on a three-party agreement about strategies for occupational disease prevention. The three-party settlement agreement served as the foundation for establishment and direction of activities. The main features of the agreement are as follows:
Article 1 of the agreement describes the goal of the prevention measures: Samsung should devise a “complete system for a healthy and safe workplace.” In other words, the company should look for ways to improve the organizational culture and establish an effective channel to communicate with independent institutions that work for the public interest, as well as a detailed plan to prevent occupational diseases in the future. To this end, the agreement called for reinforcement of the company’s internal occupational health management system, which is to be reviewed and confirmed by the Ombudsperson Commission, an independent organization.
The Commission was chaired by Prof. Cheolsoo Lee of Seoul National University School of Law and its leadership included two other members: the late Prof. Hyun-Sul Lim of Dongguk University College of Medicine and Prof. Hyunwook Kim of College of Medicine, The Catholic University of Korea. After Prof. Lim passed away in 2018, Prof. Seong-Kyu Kang of Gachon University College of Medicine assumed in August of the same year.
The Commission’s main roles were to conduct a comprehensive diagnosis of the current status of these issues and to monitor the execution of improvement measures. The diagnostic process included reviewing and evaluating data submitted by Samsung regarding the operational status of its internal accident management system and occupational health management, as well as publicizing reports on suggested improvements when needed by requesting further data and/or onsite inspections (Article 3, Paragraph 3 of the Agreement). The specific areas subject to the Commission’s diagnosis were: (1) An evaluation of harmful factors in the workplace environment and improvement plans (evaluation of chemical substance management and improvement plans, evaluation of the workplace environment and improvement plans, construction of a job-exposure matrix, etc.); (2) Conducting an epidemiological study on the health effects of the workplace environment (epidemiological investigations of former and current workers, in-depth studies of groups with a high suspicion of occupational disease, and in-depth interviews with employees); (3) Inspection of the current comprehensive health management system and improvement plans (health management and enhancement activities); (4) Inspections and studies to prevent occupational accidents and diseases (studying academic papers and policies regarding occupational safety and health standards for chemical substances, studying cases in other countries, releasing research papers, and conducting promotional activities); and (5) Advising and/or suggesting actions by Samsung through reviewing and advising Samsung about confidential information and data regarding chemical substances (any and all activities that have to do with establishment, revision, and enforcement of detailed regulations on sharing information about harmful chemical substances used in Samsung’s semiconductor and liquid crystal display [LCD] facilities and trade secrets).
The monitoring activity mainly included an annual inspection of Samsung’s execution of the improvement plans throughout the effective period, as well as suggesting and/or advising any additional improvements as needed after requesting additional data or onsite inspections, and reporting the results thereof.
The leadership of the Commission included a chair and professors of occupational medicine and industrial hygiene, with five sub-committees by ten professors organized into two departments to perform a comprehensive diagnosis. Department 1 consisted of three sub-committees with eight experts each: the Physical and Chemical Substances Team, the Health Effects Research Team, and the Occupational Health System Improvement Team. Department 2 consisted of two sub-committees with two experts each: the Research and Investigation Team and the Regulation Review Team. The comprehensive diagnosis was performed by each expert with an assigned role under supervision of the Ombudsperson leadership. In June 2016 and September 2016, the Commission and the five teams selected topics for each one and assembled a diagnosis group of 41 experts including the Ombudsperson. In November 2016, the Commission signed a contract with Samsung Electronics and Samsung Display mediated by the Seoul National University R&DB Foundation and carried out investigations into Samsung’s semiconductor and LCD facilities, including the Giheung, Hwasung, Onyang, and Asan locations, between November 2016 and December 2017.
The Commission disclosed the results of their activities at a press conference held on April 25, 2018, to ensure transparency of the diagnosis [6]. The “Comprehensive Diagnosis Report” was shared with the public in August 2018, after making revisions based on opinions received from the stakeholders.
### Ethics Statement
This comprehensive diagnosis project itself isn’t human subject research, but some specific subjects performed by each expert with an assigned role under supervision of the Ombudsperson leadership were approved by the institutional review board from their affiliation institute.
## RESULTS
The key findings of the report on the comprehensive diagnosis are described below.
The most recent 3 years of work environment data from Samsung regarding physical and chemical substances and radiation showed encouraging results, as 79.9% of the substances investigated were non-existent at the Giheung and Hwasung plants, 71.6% were not found at the Onyang location, and 73.0% were not present at Asan. Among the substances that were detected, none were present at levels exceeding 10% of the occupational exposure levels (OELs) of Korea. The team looked for 25 key chemicals selected based on their relevance for trade secrets, quantity, and toxicity in 54 bulk samples of photoresistors used during the photographic process. Sixteen of the 25 chemicals, including benzene and ethylene glycol ethers, were determined to be non-existent and nine chemicals, including toluene and o-cresol, were found at negligible levels, unlikely to be harmful even with long-term exposure. Most of the chemicals were not detected in tests for preventive maintenance workers, while the detected substances were all under 10% of the OELs. Exposure to electromagnetic fields was also lower than 10% of the recommended level. A status of radiation equipment management and the risk of exposure to radiation of workers were both within the minimum safety requirements of the Nuclear Safety Act. The expected exposure rate of employees working near radiation equipment was on average 0.2 mSv, referred to 1 mSv for the general public [7].
A systemic review and meta-analysis were performed of the relevant literature to obtain insights into the correlations between the occurrence of cancer and other major diseases and occupation among semiconductor workers. Occupational correlations between the onset of leukemia, non-Hodgkin lymphoma, brain tumor, and breast cancer and the workplace environment in the semiconductor industry were not verifiable with currently available resources. Focus group interviews of former and current semiconductor workers suggested potential exposure to chemicals, noises, and odors in old procedures that are different from those currently used, but there was no way to confirm this at facilities that had already been shut down. It was only confirmed that the current process has been automated to a great extent and that workers are no longer directly exposed to chemical substances or harsh environments. The levels of biological exposure among current workers to harmful chemicals such as benzene and arsenic were largely similar to the levels found among the general public, according to biomarker test results [7]. This suggests that the current environment poses almost no potential health risk.
Investigations of the company’s occupational health management system confirmed that an internal health management team was in place and that the company provided support for expert consultations, guidance for the industrial accident compensation insurance application process, and a service desk for compensation-eligible diseases, in case an employee develops a possible occupational disease.
Studies of future policies for preventive measures confirmed that the company needs policies to prevent potential risks. A new risk management system using big-data–based artificial intelligence was suggested as an alternative. Additionally, a dedicated team is needed to enhance communication between stakeholders within and outside the company regarding health, safety, and environment issues, thereby improving the credibility of the company [7].
Investigations into practices of sharing and archiving material safety and health data on harmful chemicals concluded that the company should share all data regarding chemical substances used in their facilities in order to protect workers and ensure their right to know, considering the huge number of chemicals used in the semiconductor and LCD facilities. Although semiconductor and LCD technologies and their trade secrets should undoubtedly be protected as key national technology assets, such protection must be kept to a minimum level in order not to compromise workers’ health [7].
## DISCUSSION
The results of the comprehensive diagnosis did not demonstrate any occupational correlations with the development of incurable diseases among some of Samsung’s semiconductor and LCD manufacturing employees. Improvement plans were proposed to completely rule out any potential risks that are currently unverifiable and to effectively address uncertainties regarding future health management. Core action points of the improvement plan include removing future uncertain risks through continuing enhancement and maintenance of the work environment and chemical substance management practice; conducting long-term follow-up surveys on correlations between jobs and diseases using a job-exposure matrix and cohorts; promoting employees’ health by upgrading the existing occupational health program to help workers with lifestyle diseases and to encourage healthy workers to maintain their status quo; introducing a big-data–based risk management system; enhancing risk-related communication between stakeholders within and outside of the company; and sharing chemical data using in facilities to secure workers’ right to know. The full list is presented in Table 1.
Core action items for improvement
The Ombudsperson Commission will examine Samsung’s implementation of these core action plans to ensure that the company’s semiconductor and LCD lines are safer and healthier.
## SUPPLEMENTAL MATERIALS
Korean version is available at https://doi.org/10.3961/jpmph.19.033.
## Notes
CONFLICT OF INTEREST
The authors have no conflicts of interest associated with the material presented in this paper.
AUTHOR CONTRIBUTIONS
Conceptualization: CL, SKK, HK. Data curation: CL, SKK, HK. Formal analysis: CL, SKK, HK. Funding acquisition: CL. Methodology: CL, SKK, HK. Project administration: CL. Visualization: None. Writing - original draft: IK. Writing - review & editing: CL, SKK, HK.
## Acknowledgements
We thanks to all experts who participated in this comprehensive diagnosis project, and especially thanks to the dedication of the late Prof. Hyun-Sul Lim (Dongguk University College of Medicine) for the commission.
Samsung Ombudsperson Commission is financially supported by Samsung Electronics and Samsung Display through Seoul National University R&DB Foundation, according to Article 3, Paragraph 6 of the three parties’ (Samsung Electronics, SHARPs, and the Family Committee for Compensation over Leukemia Issue) settlement agreement regarding prevention measures in January 2016.
## References
1. Park S. Acknowledgement of occupational accident for the Samsung Electronics leukemia case. Monthly Labor Law. 2016 Dec 30 [cited 2019 Feb 13]. Available from: http://www.worklaw.co.kr/view/view.asp?in_cate=1010&gopage=1&bi_pidx= 26048&sPrm=Search_Text$$%BB%EF%BC%BA%C0%FC% C0%DA%20%B9%E9%C7%F7%BA%B4%20%BB%E7%B0% C7%C0%C7@@keyword$$%BB%EF%BC%BA%C0%FC%C0%DA%20%B9%E9%C7%F7%BA%B4%20%BB%E7%B0%C7%C0%C7 (Korean).
2. Ministry of Employment and Labor. Industry health risk assessment for semiconductor manufacturers. 2009. Jun 9 [cited 2019 Feb 13]. Available from: https://www.moel.go.kr/news/enews/report/enewsView.do;jsessionid=UlDbFhAPa02gICjHVXM1JDLNaX8CtGwxkCTK2gR4iteIs9PIMrflltV456wEJ60p.moel_was_outside_servlet_www2?news_seq=158 (Korean, author’s translation).
3. Occupational Safety and Health Research Institute. A study on non-Hodgkin lymphoma patient-control group research design and viability among semiconductor industry workers in 2015. 2018. Jan 10 [cited 2019 Feb 13]. Available from: https://oshri.kosha.or.kr/main?urlCode=T1%7C%7C12081%7C366%7C366%7C374%7C12081%7C%7C/cms/board/board/Board.jsp&communityKey=B1182&site_id=3&tabId=&communityKey=B1182&pageNum=1&pageSize=10&act=VIEW&branch_session=&only_reply=&mbo_mother_page=/main& board_table_name=WCM_BOARD_B1182&sort_type=DESC& sort_column=&searchType=ALL&searchWord=&boardId=6 (Korean, author’s translation).
4. Occupational Safety and Health Research Institute. Epidemiological survey on workers health status in semiconductor manufacturing – focusing on cancer. 2015. May 22 [cited 2019 Jul 18]. Available from: https://oshri.kosha.or.kr/main?urlCode=T1||12087|366|366|374|12087||/cms/board/board/Board.jsp&communityKey=B1184&site_id=3&tabId=&communityKey=B1184&pageNum=1&pageSize=10&act=VIEW&branch_session=&only_reply=&mbo_mother_page=/main& board_table_name=WCM_BOARD_B1184&sort_type=DESC&sort_column=&searchType=ALL&searchWord=&boardId=2 (Korean, author’s translation).
5. Ju HJ. Samsung to launch an Ombudsperson Commission to prevent occupational diseases. Seoul Shinmun. 2016. Jan 13 [2019 Feb 13]. Available from: http://www.seoul.co.kr/news/newsView.php?id=20160113005023#csidx94e880592c5d214a2e1251aebdc5e35 (Korean).
6. Kim JY. Samsung Ombudsperson Commission to announce comprehensive diagnosis results for leukemia cases on April 25. Aju Business Daily. 2018. Apr 23 [cited 2019 Feb 13]. Available from: https://www.ajunews.com/view/20180423160548280 (Korean).
7. Samsung Ombudsperson Commission, Comprehensive diagnosis report by the Samsung Ombudsperson Commission Seoul: Samsung Ombudsperson Commission; 2018. (Korean).
## Appendices
### Appendix 1. Court rulings on the 2010 lawsuit
Five representatives of the bereaved family members and employees filed a lawsuit requesting that the previous denial of compensation for loss and funeral expenses be overturned. Two of the five plaintiffs won at the first trial (Seoul Administrative Court Order 2010Guhap1149 Dated June 23, 2011) as well as at the upper court (Seoul High Court Order 2011Nu23995 Dated August 21, 2014). Korea Workers’ Compensation and Welfare Service did not take the case to a third trial, while the remaining three plaintiffs lost at the Supreme Court trial (Supreme Court Order 2014Du12185 Dated August 30, 2016). The Supreme Court Order is available at the following link: http://law.go.kr/precInfoP.do?mode=0&precSeq=184003
### Appendix 2. Final settlement agreement regarding compensation and apologies
On November 1, 2018, Samsung Electronics and SHARPs agreed on issues regarding compensation, apologies, prevention of recurrence, and social contributions through a settlement reached under a mediation committee. Samsung Electronics officially apologized as part of the agreement to implement the settlement on November 23, 2018. Compensation will be supervised by an independent committee and Samsung agreed to contribute ₩50 billion for the prevention of recurrence and related social activities (Settlement agreement by the mediation committee to address issues related to the development of leukemia and other diseases at Samsung electronics semiconductor facilities, November 1, 2018).
## Article information Continued
### Table 1.
Core action items for improvement
Sub-committee Category Details
Physical and chemical substances Work environment data Continue monitoring the level of exposure in preventive maintenance work
Work environment management Conduct regular monitoring of recirculated air quality in the clean room
Chemical substance management Reinforce analysis of major harmful substances
System development Construct a job-exposure matrix
Training, promotion, communication Continue education for the risk of chemical substances
Partner support Enhance safety and health management support for partners
Health effects research Establish cohorts Set up prospective cohorts for semiconductor and LCD workers
Biological exposure evaluation Conduct a biological exposure study
Employee occupational health system improvements Health promotion programs Introduce a customized health promotion program for employees with chronic diseases
Health management system Come up with mid- to long-term plans for health promotion activities
System development Introduce a mobile healthcare system
Training, promotion, communication Strengthen communication among health promotion departments
Research and investigation Chemical substance management Conduct a quantitative study of chemical substance uncertainty risks
System development Establish a health, safety and environment management system using artificial intelligence analyses of big data
Training, promotion, communication Assemble a dedicated communications team for issues related to health, safety, and the environment
Regulation review Data sharing Share a list of chemicals used
Training, promotion, communication Install equipment that can confirm whether material safety data sheet guidelines are followed
Safety health management system Expand archives (including the duration of storage) for employees’ health data
LCD, liquid crystal display.
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To be honest, we’d be hoodwinking you if we pretended for a second that we had any idea where this project was going in the long run. We’ve changed tack several times now, from being a pure JavaScript library to having a full-blown builder interface; so please take this with more than just a grain of salt. If you’re interested in where the project is going in the mid-term, please be invited to talk to the team, we’ll gladly share our secret plans.
There are, however, a couple of things we feel strongly about, which we’ve tried to capture here (again, to questionable success).
## Release schedule¶
The library aims for biannual major releases in a tick-tock pattern. The summer release will be allowed to break backward compatibility if necessary, but the API should remain stable for the remainder of the year, though features may be added. This is very similar to the concept of semantic versioning.
## Philosophy and Scope¶
Many small decisions have to be made when building a library like this, and from time to time, on idle evenings, the urge makes itself known to imagine that some grand underlying principles governed its design. At other times, when thoughts go in circles over some minute detail, obsessing over some minor detail, one dreams of having guidelines that might inform API structure.
This section is an attempt at distilling principles for the design of the library, to serve as a benchmark and discussion tool for the interested, and for its developers. It is the result of both pathological grandiosity and rumination, and should not be taken too seriously: Pragmatism will always dominate the following ideas, and quite likely they will have to revised sooner or later, when we discover that our thinking has changed.
### Built as a tool for teaching¶
lab.js is built for researchers with broad experience in programming experiments as much as it is built for novices to programming. This necessitates maximum possible conceptual clarity. Interfaces and terminology should be as consistent as possible throughout the library.
The original author’s courses in experimental design and programming are half practical, geared toward enabling students to build and run experiments, and half technical, intended to convey at least the most basic programming concepts. Therefore, the library should be representative of general programming practices, and avoid custom notation that might seem simpler at first, but would limit generalizability of the acquired knowledge.
### Limited in scope¶
The central technical goal of the library is to provide a framework for handling the temporal progression of events over the course of a computer-based experiment that is run in the browser as a single-page application. It also offers helpers for working with the collected data.
The generation and sequencing of stimuli themselves should be left to the user, or external libraries. A GaborScreen, or anything similarly specific, would be out of scope, and should be provided as a third-party-addon.
That being said, the project’s design should make possible the reuse and sharing of parts of studies, so that they can be easily incorporated into new research.
### Based on web standards¶
Technical decisions are made on the assumption that the era of great differences between web browsers is over, and that future browsers will be updated at a steady pace to follow common standards. Antiquated browsers should not be a reason to compromise on features or performance. We have been reluctant to incorporate experimental features unique to any particular browser, but if a particular feature is slated for standardization, using a polyfill for the time being is fine.
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# NAG CL Interfaced01rcc (dim1_gen_vec_multi_dimreq)
Settings help
CL Name Style:
## 1Purpose
The dimension of the arrays that must be passed as actual arguments to d01rac are dependent upon a number of factors. d01rcc returns the correct size of these arrays enabling d01rac to be called successfully.
## 2Specification
#include
void d01rcc (Integer ni, Integer *lenxrq, Integer *ldfmrq, Integer *sdfmrq, Integer *licmin, Integer *licmax, Integer *lcmin, Integer *lcmax, const Integer iopts[], const double opts[], NagError *fail)
The function may be called by the names: d01rcc, nag_quad_dim1_gen_vec_multi_dimreq or nag_quad_1d_gen_vec_multi_dimreq.
## 3Description
d01rcc returns the minimum dimension of the arrays x ($\mathit{lenxrq}$), fm ($\mathit{ldfmrq}×\mathit{sdfmrq}$), icom ($\mathit{licmin}$) and com ($\mathit{lcmin}$) that must be passed to d01rac to enable the integration to commence given options currently set for the ni integrands. d01rcc also returns the upper bounds $\mathit{licmax}$ and $\mathit{lcmax}$ for the dimension of the arrays icom and com, that could possibly be required with the chosen options.
All the minimum values $\mathit{lenxrq}$, $\mathit{ldfmrq}$, $\mathit{sdfmrq}$, $\mathit{licmin}$ and $\mathit{lcmin}$, and subsequently all the maximum values $\mathit{licmax}$ and $\mathit{lcmax}$ may be affected if different options are set, and hence d01rcc should be called after any options are set, and before the first call to d01rac.
A segment is here defined as a (possibly maximal) subset of the domain of integration. During subdivision, a segment is bisected into two new segments.
None.
## 5Arguments
1: $\mathbf{ni}$Integer Input
On entry: ${n}_{i}$, the number of integrals which will be approximated in the subsequent call to d01rac.
Constraint: ${\mathbf{ni}}>0$.
2: $\mathbf{lenxrq}$Integer * Output
On exit: $\mathit{lenxrq}$, the minimum dimension of the array x that can be used in a subsequent call to d01rac.
3: $\mathbf{ldfmrq}$Integer * Output
On exit: $\mathit{ldfmrq}$, the minimum stride separating row elements of the matrix of values stored in the array fm that can be used in a subsequent call to d01rac.
4: $\mathbf{sdfmrq}$Integer * Output
On exit: $\mathit{sdfmrq}$, the minimum number of columns of the matrix of values stored in the array fm that can be used in a subsequent call to d01rac.
Note: the minimum dimension of the array fm is $\mathit{ldfmrq}×\mathit{sdfmrq}$.
5: $\mathbf{licmin}$Integer * Output
On exit: $\mathit{licmin}$, the minimum dimension of the array icom that must be passed to d01rac to enable it to calculate a single approximation to all the ${n}_{i}$ integrals over the interval $\left[a,b\right]$ with ${s}_{\mathit{pri}}$ initial segments.
6: $\mathbf{licmax}$Integer * Output
On exit: $\mathit{licmax}$ the dimension of the array icom that must be passed to d01rac to enable it to exhaust the adaptive process controlled by the currently set options for the ${n}_{i}$ integrals over the interval $\left[a,b\right]$ with ${s}_{\mathit{pri}}$ initial segments.
7: $\mathbf{lcmin}$Integer * Output
On exit: $\mathit{lcmin}$, the minimum dimension of the array com that must be passed to d01rac to enable it to calculate a single approximation to all the ${n}_{i}$ integrals over the interval $\left[a,b\right]$ with ${s}_{\mathit{pri}}$ initial segments.
8: $\mathbf{lcmax}$Integer * Output
On exit: $\mathit{lcmax}$, the dimension of the array com that must be passed to d01rac to enable it to exhaust the adaptive process controlled by the currently set options for the ${n}_{i}$ integrals over the interval $\left[a,b\right]$ with ${s}_{\mathit{pri}}$ initial segments.
9: $\mathbf{iopts}\left[\mathit{dim}\right]$const Integer Communication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument iopts in the previous call to d01zkc.
On entry: the integer option array as returned by d01zkc.
Constraint: iopts must not be changed between calls to d01zkc, d01zlc, d01rcc and d01rac.
10: $\mathbf{opts}\left[\mathit{dim}\right]$const double Communication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument opts in the previous call to d01zkc.
On entry: the real option array as returned by d01zkc.
Constraint: opts must not be changed between calls to d01zkc, d01zlc, d01rcc and d01rac.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
## 6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{ni}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ni}}>0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_OPTION
One of the option arrays iopts or opts has become corrupted. Re-initialize the arrays using d01zkc.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Not applicable.
## 8Parallelism and Performance
d01rcc is not threaded in any implementation.
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