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# Proof of the inconsistency of arithmetic

 Part of aconvergent series onMathematics 1+1=11 v - t - e

The proof of the inconsistency of arithmetic is a pseudomathematical proof used by some creationists to prove that 0=1 as a means to prove that something can exist out of nothing. According to believers of this theory it is fundamental to believe in it because it is a proof that a creator exists. The actual inventor of this proof is Luigi Guido Grandi and was developed by him in 1703. [1]

## Proof

The proof can be explained in 4 steps:

1. $0=0+0+0+0+ \ldots$
2. $0=(1-1)+(1-1)+(1-1)+(1-1)+ \ldots$
3. $0=1+(-1+1)+(-1+1)+(-1+1)+ \ldots$
4. $0=1 ~ \blacksquare$

### Problems with the proof

At step one it is stated that zero is the sum of an infinite series whose terms are all zero. This is true, although - to be precise - not a statement on Arithmetic how it is intended nowadays (see Peano axioms), because it involves infinitely many additions.

Anyway, the error is in step three, where the terms of the infinite series

$(1-1)+(1-1)+(1-1)+(1-1)+ \ldots$

are rearranged to get the series

$1+(-1+1)+(-1+1)+(-1+1)+ \ldots$

Indeed, the terms of a series can be rearranged so that the series has still the same value only if some conditions are satisfied (for example, absolute convergence). So the real conclusion is not that 0 = 1, but that manipulations of infinite series are a delicate matter.

The series with elements of (1-1) sums to zero if and only if all terms are evaluated before the sum is evaluated. Removing the parentheses, then creates a series with the sum alternating between 1 for an odd number of terms and 0 for an even number of them. So evaluating the series as finally stated as zero is simply and obviously incorrect. The sum of the series is indeterminate. There are many indeterminate expressions, a common one is 0/0. Any real number is a solution to x = 0/0. Many fallacious proofs depend on slipping in an unrecognized division by zero. This "proof" is similar in that normally one can remove parenthesis without changing the value of an expression. But a series is not a simple expression. The parentheses actually indicate a requirement to evaluate what is within them first, and there are procedures where sequence matters. The series without the parentheses does not converge, so evaluating it by comparing it with a series that does converge is misleading.

## Problems for mathematics

Needless to say, if 0 really were equal to 1, there would be far bigger concerns than the fact that something can come from nothing. Since any equation can have 0 added to it and keep its value, then it follows that any number is equal to itself plus one. And since 0 = ½ follows from 0 = 1, as well as any other fraction, suddenly every number is equal to every other number.

Eventually, absolutely anything becomes true.

## Problems for sanity

Apart from the issue of why nobody else can harness this power of bringing something from nothing, it also raises the concern that nothing can easily arise from something; that is, everything around you could, at a moment's notice, suddenly cease to exist.

The fact that anyone might take this seriously is very concerning.

## Extra notes

The series at step 2 has been named as Grandi's series in his honor. Though Niels Henrik Abel [2] would have probably thought of him as a Satan worshipper.

## Serious attempt to prove the inconsistency of arithmetic

The mathematician Edward Nelson did some serious mathematics trying to prove the inconsistency of Peano arithmetic, motivated by some deep philosophical objections to the principle on induction. In 2011, he claimed to have proved the inconsistency of Peano arithmetic,[3] but later an error was found in the proof, and he retracted his claim.

## References

1. http://www.csus.edu/indiv/d/dowdenb/misc/god.htm
2. N. H. Abel wrote "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever" (Gardner 1984, p. 171; Hoffman 1998, p. 218).
3. Nelson, E. (2011). "Inconsistency of P". Foundations of Mathematics. Archived from the original on 13 May 2017.


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# "Specialisations of algebraically closed fields".

29 May 2014
11:00
Ugur Efem
Abstract

Algebraically closed fields, and in general varieties are among the first examples
of Zariski Geometries.
I will consider specialisations of algebraically closed fields and varieties.
In the case of an algebraically closed field K, I will show that a specialisation
is essentially a residue map, res from K to a residue field k.
In both cases I will show universality of the specialisation is controlled by the
transcendence degree of K over k.

format: plain


text:
# CDF_2000_S4155203

CDF_2000_S4155203
=================

Z pT measurement in CDF Z -> e+ e- events

Status: VALIDATED

Spires ID: 4155203
Spires URL: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=key+4155203
Experiment: CDF Tevatron Run 1
Year of publication: 2000
Authors:
Hendrik Hoeth <hendrik.hoeth@cern.ch>

Description:
Measurement of transverse momentum and total cross section of e^+e^-
pairs in the Z-boson region of $66~\text{GeV}/c^2 < m_{ee} < 116~\text{GeV}/c^2$ from pbar-p collisions at sqrt(s) = 1.8 TeV, with the
Tevatron CDF detector.  The Z pT, in a fully-factorised picture, is
generated by the momentum balance against initial state radiation (ISR)
and the primordial/intrinsic pT of the Z's parent partons in the incoming
hadrons. The Z pT is important in generator tuning to fix the interplay
of ISR and multi-parton interactions (MPI) ingenerating underlying
event' activity.  This analysis is subject to ambiguities in the
experimental Z pT definition,since the Rivet implementation reconstructs
the Z momentum from the dileptonpair with finite cones for photon brem
the Z in the event record.

Run details:
ppbar collisions at 1800 GeV. $Z/\gamma*$ Drell-Yan events with e+ e-
decay mode only.

References:
Phys.Rev.Lett.84:845-850,2000
arXiv:hep-ex/0001021 - http://arxiv.org/abs/hep-ex/0001021
doi:10.1103/PhysRevLett.84.845 - http://dx.doi.org/10.1103/PhysRevLett.84.845

# CDF_2001_S4751469

CDF_2001_S4751469
=================

Field & Stuart Run I underlying event analysis.

Status: VALIDATED

Spires ID: 4751469
Spires URL: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=key+4751469
Experiment: CDF Tevatron Run 1
Year of publication: 2001
Authors:
Andy Buckley <andy.buckley@durham.ac.uk>

Description:
The original CDF underlying event analysis, based on decomposing each
event into a transverse structure with toward'', away'' and
transverse'' regions defined relative to the azimuthal direction of the
leading jet in the event. Since the toward region is by definition
dominated by the hard process, as is the away region by momentum balance
in the matrix element, the transverse region is most sensitive to multi-
parton interactions. The transverse regions occupy $|\phi| \in [60\degree, 120\degree]$ for $|\eta| < 1$. The pT ranges for the leading
jet are divided experimentally into the min-bias' sample from 0--20 GeV,
and the JET20' sample from 18--49 GeV.

Run details:
ppbar QCD interactions at 1800 GeV. The leading jet is binned from 0--49
GeV, and histos can usually can be filled with a single generator run
without kinematic sub-samples.

References:
Phys.Rev.D65:092002,2002
FNAL-PUB 01/211-E

# CDF_2002_S4796047

CDF_2002_S4796047
=================

CDF Run 1 charged multiplicity measurement

Status: VALIDATED

Spires ID: 4796047
Spires URL: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=key+4796047
Experiment: CDF Tevatron Run 1
Year of publication: 2002
Authors:
Hendrik Hoeth <hendrik.hoeth@cern.ch>

Description:
A study of ppbar collisions at sqrt(s) = 1800 and 630 GeV collected using
a minimum bias trigger in which the data set is divided into two classes
corresponding to soft' and hard' interactions. For each subsample, the
analysis includes measurements of the multiplicity, transverse momentum
(pT) spectra, and the average pT and event-by- event pT dispersion as a
function of multiplicity. A comparison of results shows distinct
differences in the behavior of the two samples as a function of the
center of mass energy. The properties of the soft sample are invariant as
a function of c.m. energy.

Run details:
QCD events at sqrt(s) = 630 and 1800 GeV.

References:
Phys.Rev.D65:072005,2002
doi:10.1103/PhysRevD.65.072005 - http://dx.doi.org/10.1103/PhysRevD.65.072005

# CDF_2004_S5839831

CDF_2004_S5839831
=================

Transverse cone and 'Swiss cheese' underlying event studies

Status: UNVALIDATED

Spires ID: 5839831
Spires URL: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=key+5839831
Experiment: CDF Tevatron Run 2
Year of publication: 2004
Authors:
Andy Buckley <andy.buckley@durham.ac.uk>

Description:
This analysis studies the underlying event via transverse cones of  $R = 0.7$ at 90 degrees in \phi relative to the leading (highest E) jet, at
sqrt(s) = 630 and 1800 GeV. This is similar to the 2001 CDF UE analysis,
except that cones, rather than the whole central \eta range are used. The
transverse cones are categorised as TransMIN and TransMAX on an event-by-
event basis, to give greater sensitivity to the UE component. 'Swiss
Cheese' distributions, where cones around the leading $n$ jets are
excluded from the distributions, are also included for $n = 2, 3$.  This
analysis is useful for constraining the energy evolution of the
underlying event, since it performs the same analyses at two distinct CoM
energies. WARNING, the Swiss Cheese plots are not considered validated,
as we have not yet been able to reproduce the MC curves shown in the
paper.

Run details:
QCD events at sqrt(s) = 630 & 1800 GeV Several pTmin cutoffs are probably
required to fill the profile histograms, e.g. 0 (min bias), 30, 90, 150
GeV  at 1800 GeV; and 0 (min bias), 20, 90,  150 GeV } at 630 GeV

References:
Phys. Rev. D70, 072002 (2004)
arXiv:hep-ex/0404004 - http://arxiv.org/abs/hep-ex/0404004

CDF_2008_LEADINGJETS
====================

CDF Run 2 underlying event in leading jet events

Status: VALIDATED

Spires ID: NONE
Spires URL: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=key+NONE
Experiment: CDF Tevatron Run 2
Year of publication: 2008
Authors:
Hendrik Hoeth <hendrik.hoeth@cern.ch>

Description:
Rick Field's measurement of the underlying event in leading jet events.
If the leading jet of the event is within $|\eta| < 2$, the event is
accepted and toward'', away'' and transverse'' regions are defined
in the same way as in the original (2001) CDF underlying event analysis.
The leading jet defines the $\phi$ direction of the toward region. The
transverse regions are most sensitive to the underlying event.

Run details:
ppbar QCD interactions at 1960 GeV. Particles with  $c \tau > {}$10 mm
should be set stable. Several $p_\perp^\text{min}$  cutoffs are
probably required to fill the profile histograms. $p_\perp^\text{min} = {}$ 0 (min bias), 10, 20, 50, 100, 150 GeV. The corresponding merging
points are at $p_T =$ 0, 30, 50, 80,  130, 180 GeV

# CDF_2008_NOTE_9351

CDF_2008_NOTE_9351
==================

CDF Run 2 underlying event in Drell-Yan

Status: VALIDATED

Spires ID: NONE
Spires URL: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=key+NONE
Experiment: CDF Tevatron Run 2
Year of publication: 2008
Authors:
Hendrik Hoeth <hendrik.hoeth@cern.ch>

Description:
Deepak Kar and Rick Field's measurement of the underlying event in Drell-
Yan events. Z -> ee and Z -> $\mu\mu$ events are selected using a Z mass
window cut between 70 and 110 GeV. Toward'', away'' and
transverse'' regions are defined in the same way as in the original
(2001) CDF underlying event analysis. The reconstructed Z defines the
$\phi$ direction of the toward region. The leptons are ignored after the
Z has been reconstructed. Thus the region most sensitive to the
underlying event is the toward region (the recoil jet is boosted into the
away region).

Run details:
* ppbar collisions at 1960 GeV. * Drell-Yan events with $Z/\gamma* -> e e$ and $Z/\gamma* -> \mu\mu$. * A mass cut $m_{ll} > 70~\text{GeV}$ can
be applied on generator level. * Particles with $c \tau > 10~\text{mm}$
should be set stable.

References:
CDF public note 9351

# D0_2004_S5992206

D0_2004_S5992206
================

Run II jet azimuthal decorrelation analysis

Status: VALIDATED

Spires ID: 5992206
Spires URL: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=key+5992206
Experiment: D0 Tevatron Run 2
Year of publication: 2004
Authors:
Lars Sonnenschein <lars.sonnenschein@cern.ch>

Description:
Correlations in the azimuthal angle between the two largest pT jets have
been measured using the D0 detector in ppbar collisions at 1960 GeV. The
analysis is based on an inclusive dijet event sample in the central
rapidity region. The correlations are determined for four different pT
intervals.

Run details:
QCD events in ppbar interactions at sqrt(s) = 1960 GeV.

References:
Phys. Rev. Lett., 94, 221801 (2005)
arXiv:hep-ex/0409040 - http://arxiv.org/abs/hep-ex/0409040
`


text:
## 2012 practice final#2A

Volume: $\Delta S = nR\ln \frac{V_{2}}{V_{1}}$
Temperature: $\Delta S = nC\ln \frac{T_{2}}{T_{1}}$

Lily Wei 1D
Posts: 10
Joined: Fri Sep 25, 2015 3:00 am

### 2012 practice final#2A

For this problem, why is the change in entropy calculated using only $\Delta S = nR ln(\frac{V2}{V1})$
and not $\Delta S = nR ln\frac{P1}{P2}$ ? The problem states that "The He in the metal case is initially held at a pressure of 1.0 atm", so doesn't that mean that the pressure is going to change and we should take that into account?

Irena Roy 4H
Posts: 40
Joined: Fri Sep 25, 2015 3:00 am

### Re: 2012 practice final#2A

The question gives you no other information that could lead you to conclude that the pressure did actually change in the process of He leaking.

Alex Nguyen 3I
Posts: 100
Joined: Fri Sep 25, 2015 3:00 am

### Re: 2012 practice final#2A

Using the equation with volume change already accounts for the pressure change because of PV=nRT. Therefore you can use one or the other when finding the entropy. Only when there's a temperature change with a volume/pressure change do you need 2 equations with a combined entropy.

Suraj Palaparty 2F
Posts: 27
Joined: Fri Sep 25, 2015 3:00 am

### Re: 2012 practice final#2A

The case is initially at 1.00 atm, taking into account the change in pressure later in the problem, while the safe is isolated.

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text:
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The weight per cubic foot of air (0.075 lbs) is needed to convert between the air volume and weight. Generally, the most constant parameter is notably the volumetric heat capacity (at least for … If We Inject The Same Energy AU Into 1 Kg Of Each Substance (holding Their Volumes Constant), Which One Will Feature A Larger Temperature Rise? The atmosphere has an equivalent of 5 km height at sea level pressure. Consider for a moment two side-by-side cubic meters of material — one cube is water, the other air. Heat capacity is an extensive property, i.e., it depends on the amount and size of the substance. Ideal gas The cooling air will normally be supplied at 55°F. Answer to The (constant volume) heat capacity of air is much smaller than that of water (1J/gK) vs 4.18 (1 J/gK ). Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications! In particular it is dependent on temperature itself, as well as on the pressure and the volume of the system. BTU can also be used pragmatically as a point of reference for the amount of heat that an appliance generates; the higher the BTU rating of an appliance, the greater the heating capacity. If mass is used for one component, it must be used for all components, similarly for volume. ΔT = 10°C. It depends on the temperature, pressure, and volume of the system under consideration. The standard heat capacity equation is of the form Q = mcΔT. The units of mass or volume don’t matter, as long as they match; eg: if ml are used for one component, they must be used for all components. You can target the Engineering ToolBox by using AdWords Managed Placements. Commons Attribution-Noncommercial-Share Alike 3.0 United States The heat capacity at constant volume is three halves nR, and the heat capacity at constant pressure is five halves nR. From … Specific heat capacity units are usually joules per gram-kelvin, or J/g⋅K, even though the kilogram (kg) is the SI unit of mass. Specifc heat capacity of air … The specific heat capacity of air depends on its temperature, moisture content, and the altitude at which you measure it. AddThis use cookies for handling links to social media. K). Example 1 Determine the heat capacity of copper of mass 70 g and the temperature difference is 20 o C if 300 J of heat is lost. This may influence on the accuracy of air conditioning and industrial air handling process calculations. License. The values up to 1000 K were originally published in "Tables A heat capacity is the temperature change per unit heat absorbed by a system during a reversible process: . Tables are available recommending minimum air change rates per volume of space. 21.0 - … Specific heat at constant volume: .715 Joules per gram per degree Kelvin or .17 BTU's per pound per degree Rankine. 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Ch 3, Lesson C, Page 2 - Definition of Constant V and P Heat Capacities. So this quantity, heat capacity, is an important property of materials for anyone interested in energy efficiency or heating and cooling. Note that the nominal specific heat capacity values used for air at 300K are C P = 1.00 kJ/kg.K, C v = 0.717 kJ/kg.K,, and k = 1.4. Contributors and Attributions. The new heat capacity depends on the proportion of each component, which can be calculated from mass or volume. m = mass of air = volume x density = (413.34 x 1.205) kg (see link) c = specific heat capacity of air (at constant pressure) = 1005 J/ (kgK) (see link) deltaT = temperature change; e.g. The branch of physics called statistical mechanics tells us, and experiment confirms, that CV of any ideal gas is given by this equation, regardless of the number of degrees of freedom. This quantity is known as the specific heat capacity (or simply, the specific heat), which is the heat capacity per unit mass of a material. - other definitions of STP and NTP are widely used. This is the number of Joules (energy) to raise temperature 1 degree Kelvin which is the same as 1 degree Celcius. Therefore we are looking for a heat capacity of the form ε = C (T), where C is some mathematical function. We now introduce two concepts useful in describing heat flow and temperature change. It's Specific heat is 1.005kJ/kg/°C. = 1.00 kJ/kg.K, Cv = The heat input (Q) required to raise the temperature of n moles of gas from T 1 to T 2 depends not only on ΔT but also on how the pressure and volume of the gas are changed.. Note that if we take 1 kg of solvent, then m is the moles of solute and the total heat capacity is 1000 c p (1 + mM 2). Of course, to end up with BTU per hour on one side and cubic feet per minute on the other, we need to throw in a factor of 60. 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Energy cannot be created or destroyed to my knowledge so these are physically knowable values. It is straightforward to then … It is a property solely of the substance of which the system is composed. The air of such a composition is named dry. The SI unit of specific heat is Joule per kelvin per kg (J kg-1 K-1). A one square meter column has 6,150 kj/°C (1x1.23x5,000=6,150). If the amount is taken to be the volume of the sample (as is sometimes done in engineering), one gets the volumetric heat capacity (whose SI unit is joule per kelvin per cubic meter, J/K/m 3 ). Heat capacity has units of J K -1. The movement of molecules also depends on a range of other factors, such as pressure, the viscosity of the substance, its concentration, resistance to diffusion, the distance that a molecule travels in order for the diffusion to occur, and the mass of a molecule. 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The nominal values used for air at 300 K are C P = 1.00 … Thus summing all the moles of the products we have: Using the tables of Sensible Enthalpy vs Temperature we evaluated the enthalpy of all four products at a temperature of 1280K. These applications will - due to browser restrictions - send data between your browser and our server. Air has a heat capacity of about 700 Joules per kg per °K and a density of just 1.2 kg/m 3, so its initial energy would be 700 x 1 x 1.2 x 293 = 246,120 Joules — a tiny fraction of the thermal energy stored in the water. The table following gives the values of The atmosphere has an equivalent of 5 km height at sea level pressure. Determine the temperature and pressure of the air at the end of each process, the net work output per cycle [kJ/kg], and the thermal efficiency. If you want to promote your products or services in the Engineering ToolBox - please use Google Adwords. Heat causes molecules to move, and this movement is called molecular diffusion. Cp is the heat capacity of a substance undergoing a process at constant pressure and Cv is the heat capacity of a substance undergoing a process at constant molar volume. Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the SketchUp Pro Sketchup Extension Warehouse! Experiments show that the transferred heat depends on three factors: (1) The change in temperature, (2) the mass of the system, and (3) the substance and phase of … There are two important heat capacities: Heating or cooling at constant volume: Q v = nC v ΔT, where C v is the molar heat capacity at constant volume.. The heat capacity of most systems is not a constant. Altitude and Specific Volume Correction mcΔT = Q. Heat capacity of air: 1005 J/kg/K . The last Also, air is dispersed as a gas so that in any given volume the mass of air that would fill it is only a fraction of the mass of water that would fill the same volume. By weight (isobaric mass) the difference is only about 4x. Remember that is the change in the air parcel’s internal energy. The air is measured in CFM, yet the specific heat is per pound of air. m = mass of air = volume x density = (413.34 x 1.205) kg (see link) c = specific heat capacity of air (at constant pressure) = 1005 J/ (kgK) (see link) deltaT = temperature change; e.g. Key Terms. The density of air is 1.23 kg/m³. We now introduce two concepts useful in describing heat flow and temperature change. british thermal unit(International table) = [Btu(IT)], degree celcius = [°C], degree fahrenheit = [°F], degree kelvin = [K], degree rankin = [°R], joule = [J], kilocalorie(International table) = [kcal(IT)], kilogram = [kg], kilojoule = [kJ], kilowatthour = [kWh], mole = [mol], pound =[lb]. of Thermal Properties of Gases", NBS Circular 564,1955. The density of air at room temperature is 1.2kg/m³. According to the first law of thermodynamics, for constant volume process with a monatomic ideal gas the molar specific heat will be: Cv= 3/2R = 12.5 J/mol K. because. mcΔT = Q. A chilled water coil can be selected to deliver 4.78 tons of cooling at 1,500 cfm, a flow-to-capacity ratio of 314 cfm per ton. Determine enough values of C ¯ p.2 so that you can plot them on the graph of ϕC p and compare the values. specific heat capacities of air. Only emails and answers are saved in our archive. We are all aware that pressure and temperature (and density) of the air depend on your location on the earth and the season of the year. (b) Make a graph of ϕC p against molality. The heat capacity, CV, depends on the mass and the type of material. An acceptable temperature rise through the server is about 20°F. The calculator below can be used to estimate the specific heat of air at constant volum or constant pressure and at given temperature and pressure.The output heat capacity is given as kJ/(kmol*K), kJ/(kg*K), kWh/(kg*K), kcal/(kg*K), Btu(IT)/(mol*°R) and Btu(IT)/(lbm*°R). °R in the units can be replaced by °F, and vise versa. Urieli is licensed under a Creative The heat capacity of most systems is not constant (though it can often be treated as such). 5.6695E-1 . With better technology, some machines are able to remove 10,000 BTU/hr of heat with the same capacity. Heat capacity is an extensive property, i.e., it depends on the amount and size of the substance. In comparison, volumetric heat capacity of water is 4.2 MJ/m 3 K. A rock volume with a typical volumetric heat capacity of 2.2 MJ/m 3 … Transferring 0.2°C leaves 1,230 kj available (6,150x0.2=1,230). Heat capacity of air: 1005 J/kg/K. The heat cap acity ($$C$$) of a body of matter is the quantity of heat ($$q$$) it absorbs or releases when it experiences a temperature change ($$ΔT$$) of 1 degree Celsius (or equivalently, 1 kelvin) $C=\dfrac{q}{ΔT} \label{12.3.1}$ Heat capacity is determined by both the type … Air has a heat capacity of about 700 Joules per kg per °K and a density of just 1.2 kg/m 3, so its initial energy would be 700 x 1 x 1.2 x 293 = 246,120 Joules — a tiny fraction of the thermal energy stored in the water. When calculating mass and volume flow of air in heated or cooled systems with high accuracy - the specific heat (= heat capacity) should be corrected according values in the figures and table below, or found by use of the calculator. See also subsection on Joule-Thomson expansion here. When choosing an air conditioner, usually a 1 HP (horse power) equipment is able to remove 9,000 BTU/hr of heat. It is important to mention that the specific heat capacity of water is 4.186 joule/gram o C which is higher than any other common substance. CV, the constant relating Q to temperature change, is called the heat capacity at constant volume. It's usually easier to talk about specific heat capacity because the Q in the equation above will change with different amounts of air, the substance of interest here. But, remember, the air through the heating system is flowing and the heat exchange rate will be different to when the volume of air is static.

heat capacity of air per volume


text:
## Chemistry and Chemical Reactivity (9th Edition)

The atomic weight of gallium (Ga) is 69.723, which is $x=\frac{69.723-69}{71-69}=36.15\%$ of the way between the two approximate isotope's weights (their mass numbers). Thus, B is the correct answer.


text:
# Smooth Density Histogram bounded on non-rectangular domain

I'm trying to plot the Smooth Density Histogram of a double pendulum (particularly the location of the bottom pendulum). Since a pendulum has physical constraints, when shown in the Smooth Density Histogram there are unrealistic locations of probabilities (i.e. there is a probability at a location where it is impossible for the pendulum to reach because the smoothing function "smoothed" out to zero).

Is there a way to have a type of exclusion that dictates the physical constraints of the system, when applying the smoothing function.

For example the following code:

deqns = {Subscript[m, 1] x1''[
t] == (\[Lambda]1[t]/Subscript[l, 1]) x1[
t] - (\[Lambda]2[t]/Subscript[l, 2]) (x2[t] - x1[t]),
Subscript[m, 1] y1''[
t] == (\[Lambda]1[t]/Subscript[l, 1]) y1[
t] - (\[Lambda]2[t]/Subscript[l, 2]) (y2[t] - y1[t]) -
Subscript[m, 1] g,
Subscript[m, 2] x2''[
t] == (\[Lambda]2[t]/Subscript[l, 2]) (x2[t] - x1[t]),
Subscript[m, 2] y2''[
t] == (\[Lambda]2[t]/Subscript[l, 2]) (y2[t] - y1[t]) -
Subscript[m, 2] g};

aeqns = {x1[t]^2 + y1[t]^2 ==
Subscript[l, 1]^2, (x2[t] - x1[t])^2 + (y2[t] - y1[t])^2 ==
Subscript[l, 2]^2};

ics = {x1[0] == 1, y1[0] == 0, y1'[0] == 0, x2[0] == 1, y2[0] == -1,
y2'[0] == 0};

params = {g -> 9.81, Subscript[m, 1] -> 1, Subscript[m, 2] -> 1,
Subscript[l, 1] -> 1, Subscript[l, 2] -> 1};

soldp = First[
NDSolve[{deqns, aeqns, ics} /. params, {x1, y1, x2,
y2, \[Lambda]1, \[Lambda]2}, {t, 0, 15000},
Method -> {"IndexReduction" -> {"Pantelides",
"ConstraintMethod" -> "Projection"}}]];

will give the solution to a double pendulum (where each pendulum is of length one). ($$t$$ is really large so if you want to run it yourself feel free to bring it down an order of magnitude).

Looking at the smooth histogram,

SmoothDensityHistogram[
Map[Function[Evaluate[{x2[#], y2[#]} /. soldp]],
Range[0, 15000, 0.025]], Mesh -> 5,
PlotRange -> {{-2, 2}, {-2, 0.1}}]

you can see I only plotted to $$y=-2$$ to emphasis that there should not exist a probability after this point. So is there a way to force the smoothing function to not smooth anything outside of the boundary conditions i.e. Exclusions -> Function[{x, y}, x^2 + y^2 > 2^2]

• does adding the option RegionFunction -> Function[{x, y}, x^2 + y^2 <= 4]  give what you need? – kglr Feb 19 '19 at 18:32
• or play with the option MaxExtraBandwidths using SmoothKernelDistribution on your data and DensityPlot its pdf (PDF[SmoothKernelDistribution[ Map[Function[Evaluate[{x2[#], y2[#]} /. soldp]], Range[0, 2 1500, 0.025]], MaxExtraBandwidths ->0], {x, y}]). – kglr Feb 19 '19 at 18:47
• @kglr I think the RegionFunction just cuts off what is being plotted and doesn't actually fix the smoothing problem. (I tried this) – Josh Feb 19 '19 at 19:07
• For sharper boundaries sometimes an "adaptive bandwidth" produces more desirable results. However, it does not appear that SmoothHistogramDensity accepts anything but fixed bandwidths. SmoothKernelDistribution does allow for adaptive bandwidths. An article that might be useful is sciencedirect.com/science/article/pii/S0047259X09001651. – JimB Feb 19 '19 at 20:04

I'm not so sure you can get there from here using the current nonparametric density capabilities of Mathematica. (That is not to say that what you want isn't desirable.)

First, you really don't have independent random samples from a bivariate distribution. (You really don't have samples at all but observations along a very long but deterministic process.) However, if you consider the data generation process as a sort of random number generator, then using 0.025 as the step function results in high correlations between neighboring steps. Why is this potentially bad? The automatic bandwidth selected is estimated using sample size assuming that observations are independent (with a few other assumptions) which results in undersmoothing.

Second, there are not only external boundaries but there are apparent discontinuities (essentially big jumps in "density") associated with internal borders. Fixed-width kernel density estimation don't handle such things very well. (Adaptive kernel density estimation is much better but for your case I'm not so sure and the adaptive bandwidth that one can use in SmoothKernelDistribution is not lightning fast.)

I'm a great believer in getting rid of histograms in favor of nonparametric density estimation but for this case, I think you might be much better off using a very densely gridded histogram with a very large sample size (because of the external borders and the internal jumps).

Here's what I get using 50,000 rather than 15,000 for the maximum t value:

n = 50000;
soldp = First[
NDSolve[{deqns, aeqns, ics} /. params, {x1, y1, x2, y2, λ1, λ2}, {t, 0, n},
Method -> {"IndexReduction" -> {"Pantelides", "ConstraintMethod" -> "Projection"}}]];

data = Map[Function[Evaluate[{x2[#], y2[#]} /. soldp]], Range[0, n, 0.025]];

hdata = HistogramList[data, 200];
x = Table[(hdata[[1, 1, i]] + hdata[[1, 1, i - 1]])/2, {i, 2, Length[hdata[[1, 1]]]}];
y = Table[(hdata[[1, 2, i]] + hdata[[1, 2, i - 1]])/2, {i, 2, Length[hdata[[1, 1]]]}];
htable = Flatten[Table[{x[[i]], y[[j]], hdata[[2, i, j]]}, {i, Length[x]}, {j, Length[y]}], 1];

ListContourPlot[htable, PlotRange -> All,
Contours -> {10, 20, 30, 40, 50, 100, 150, 200, 300, 400, 500, 600, 700, 800},
AspectRatio -> 1, ImageSize -> Medium,
PlotLegends -> Automatic]

Looking at the figure, there's still a lot of bumpiness and besides having external borders, the count (I just used the histogram count rather than converting to density) jumps dramatically near the borders (both internal and external borders) which just makes this a very difficult surface to fit. Maybe a much larger sample size will make things better.

Here's an alternative visualization that uses ListPlot with a very low Opacity:

ListPlot[Table[Evaluate[{x2[t], y2[t]} /. soldp], {t, 0, 15000, 0.01}],
PlotStyle -> {Black, Opacity[0.002], PointSize[0.005]}]

Otherwise, SmoothKernelDistribution has two options for smoothing kernels that seem potentially useful if they could only be combined -- "Bounded" and "Radial".

d = SmoothKernelDistribution[
Map[Function[Evaluate[{x2[#], y2[#]} /. soldp]], Range[0, 15000, 0.025]],
Automatic, {"Bounded", {{-2, 2}, {-2, 0}}, "Gaussian"}];

ContourPlot[PDF[d, {x, y}], {x, -2, 2}, {y, -2, 0.1}, MaxRecursion -> 3, PlotPoints -> 50]

d = SmoothKernelDistribution[
Map[Function[Evaluate[{x2[#], y2[#]} /. soldp]], Range[0, 15000, 0.025]],

ContourPlot[PDF[d, {x, y}], {x, -2, 2}, {y, -2, 0.1}, MaxRecursion -> 3, PlotPoints -> 50]

Those are both kind of horrifying. Maybe the general kernel specification func could be pressed into service by someone smarter than me!

Maybe a trip through polar coordinates makes the "Bounded" option more useful. (I wonder if this needs to be corrected for the effect of changing the area of different volumes of phase space...)

dat = Map[Function[Evaluate[{x2[#], y2[#]} /. soldp]], Range[0, 15000, 0.025]];
pdat = ToPolarCoordinates[dat];
d = SmoothKernelDistribution[pdat, Automatic,
{"Bounded", {{0, 2}, {0, 2 π}}, "Gaussian"}];
ContourPlot[Evaluate[PDF[d, {r, θ}] /. {r -> Sqrt[x^2 + y^2], θ -> ArcTan[x, -y]}],
{x, -2, 2}, {y, -2, 2}, PlotPoints -> 200, Contours -> 10, PlotRange -> {0, All}]

Getting better, but still kind of hideous! Polar ContourPlot thanks to this old answer by @rcollyer (with an extra minus sign before y, because ???)

• I like the work you did, but you are right, I think their could be something better. I was wondering this morning if mirroring locally around the boundary conditions (obviously this is an estimation as it is impossible to exactly mirror around the boundary condition), run the smoothing function, then go in and change all values outside of the boundary to zero (thus a discontinuity would exist around the boundary). I have no idea how to do that or if it is even a good idea, it was just a thought I had. – Josh Feb 20 '19 at 16:31


text:
# Creating a Series¶

There are couple of ways to create a new Series from scratch.

Let's explore.

In [1]:
# Just like how NumPy is almost always abbreviated as np...
import numpy as np
#  pandas is usually shortened to pd
import pandas as pd

## Creating from a dictionary¶

Let's use this sample data here. In our example, test_balance_data is just a standard Python dictionary the key is username, and the value is that user's current account balance.

In [2]:
test_balance_data = {
'pasan': 20.00,
'treasure': 20.18,
'ashley': 1.05,
'craig': 42.42,
}

The Series constructor accepts any dict-like object

In [3]:
balances = pd.Series(test_balance_data)

Notice that labels have been set from the test_balance_data.keys() and the values are set from test_balance_data.values()

In [4]:
balances

Out[4]:
pasan       20.00
treasure    20.18
ashley       1.05
craig       42.42
dtype: float64

## Creating from an Iterable¶

You can pass any iterable as the first argument

In [5]:
unlabeled_balances = pd.Series([20.00, 20.18, 1.05, 42.42])

NOTE: When labels are not present they're defaulted to incremental integers starting at 0

In [6]:
unlabeled_balances

Out[6]:
0    20.00
1    20.18
2     1.05
3    42.42
dtype: float64

You can also provide the index argument which requires an iterable the same size as your data.

In [7]:
labeled_balances = pd.Series(
[20.00, 20.18, 1.05, 42.42],
index=['pasan', 'treasure', 'ashley', 'craig']
)

Note, the order of the labels is guaranteed.

In [8]:
labeled_balances

Out[8]:
pasan       20.00
treasure    20.18
ashley       1.05
craig       42.42
dtype: float64

One thing to remember is that a NumPy array is also iterable. In fact, you'll find NumPy and Pandas get along really well together.

In [9]:
ndbalances = np.array([20.00, 20.18, 1.05, 42.42])
pd.Series(ndbalances)

Out[9]:
0    20.00
1    20.18
2     1.05
3    42.42
dtype: float64

## Creating from a scalar and an index¶

If you pass in a scalar that value will be broadcasted to the keys specified in the index argument

In [10]:
pd.Series(20.00, index=["guil", "jay", "james", "ben", "nick"])

Out[10]:
guil     20.0
jay      20.0
james    20.0
ben      20.0
nick     20.0
dtype: float64


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

Last week we made an impassioned plea for attention to style in formatting R and SAS code.

While it’s always better to adopt a consistent style and use it whenever you write code, the reality is that sometimes formatting slips (or you end up repurposing code that others wrote. In those situations, the formatR package (due to Yihui Xie) is a useful tool to help tidy up code.

Imagine that the poorly formatted and somewhat obfuscated code from the last entry has been copied to the clipboard:
ds=read.csv("http://www.math.smith.edu/r/data/help.csv");attach(ds)
fOo=ks.test(age[female==1],age[female==0],data=ds)
plotdens=function(x,y,mytitle, mylab){densx = density(x)
densy = density(y);plot(densx,main=mytitle,lwd=3,xlab=mylab,
bty="l");lines(densy,lty=2,col=2,lwd=3);xvals=c(densx$x, rev(densy$x));yvals=c(densx$y,rev(densy$y));polygon(xvals,
yvals,col="gray")};mytitle=paste("Test of ages: D=",round(fOo$statistic,3), " p=",round(fOo$p.value,2),sep="");plotdens(age[female==1],
age[female==0],mytitle=mytitle,mylab="age (in years)")
legend(50,.05,legend=c("Women","Men"),col=1:2,lty=1:2,lwd=2)

Running the commands:
library(formatR)
tidy.source()

will generate the following output:
ds = read.csv("http://www.math.smith.edu/r/data/help.csv")
attach(ds)
fOo = ks.test(age[female == 1], age[female == 0], data = ds)
plotdens = function(x, y, mytitle, mylab) {
densx = density(x)
densy = density(y)
plot(densx, main = mytitle, lwd = 3, xlab = mylab, bty = "l")
lines(densy, lty = 2, col = 2, lwd = 3)
xvals = c(densx$x, rev(densy$x))
yvals = c(densx$y, rev(densy$y))
polygon(xvals, yvals, col = "gray")
}
mytitle = paste("Test of ages: D=", round(fOo$statistic, 3), " p=", round(fOo$p.value, 2), sep = "")
plotdens(age[female == 1], age[female == 0], mytitle = mytitle,
mylab = "age (in years)")

The cleaned up code is much easier to parse, with minimal effort. Some quibbles: I’m not as fond of using 4 spaces of indentation. And I always worry what’ll happen if there are syntax errors (yipes!). But this is a useful tool to have in your box.


text:
Tiếng Anh 12

# Đề số 12 - Đề thi thử THPT Quốc Gia môn Tiếng Anh

Bình chọn:
4.9 trên 7 phiếu

## Đáp án và lời giải chi tiết Đề số 12 - Đề thi thử THPT Quốc Gia môn Tiếng Anh

Đề bài

Mark the letter A, B, C or D on your answer sheet to indicate the word whose underlined part differs from the other three in pronunciation in each of the following questions.

Question 1.

A. expands                       B. installs

C. swallows                       D. rejects

Question 2.

A. purpose                         B. compose

C. suppose                         D. propose

Mark the letter A, B, C or D on your answer sheet to indicate the word that differs from the other three in the position of primary stress in each of the following questions.

Question 3.

A. compulsory                 B. certificate

C. category                      D. accompany

Question 4.

A. decent                    B. secure

C. maintain                 D. confide

Mark the letter A, B, C or D on your answer sheet to indicate the correct answer to each of the following question.

Question 5. I _____ the garden. No sooner had I finished watering it than it came down in torrents.

A. shouldn’t have watered

B. mustn’t have watered

C. can’t have watered

D. needn’t have watered

Question 6. A true relationship is thought to be a friendship that may _____ through changes in the lives of the friends.

A. entail                       B. remain

C. endure                    D. continue

Question 7. I’m not going to go ice skating! I’d only fall over and _____ a fool of myself.

A. create                      B. show

C. do                            D. make

Question 8. Silence _____ the theatre as the audience awaited the opening curtain with expectation and excitement.

A. fell in with                B. came between

C. hung over                D. dropped out of

Question 9. The scientists _____ on this project try to find the solutions to air pollution.

A. working                       B. worked

C. are working                 D. who working

Question 10. If Nam had passed the GCSE examination, he _____ to take the entrance examination to the university.

A. would have been allowed

B. would have allowed

C. will have allowed

D. would allow

Question 11. When the kids asked him about his girlfriend, he’d go as red as a _____.

A. tomato                     B. chili

C. strawberry                D. beetroot

Question 12. My brothers are often very ____ to what I say. They are really lovable.

A. obedience                B. obedient

C. obey                         D. obstacle

Question 13. I must warn you I am not used to _____ so rudely

A. speak                         B. be spoken

C. speak to me               D. being spoken to

Question 14. He is tired now. He _____ the gardening for hours.

A. has done                   B. is doing

C. did                             D. has been doing

Question 15. Let me explain everything to you, _____?

A. shall we                      B. will you

C. will we                        D. do you

Question 16. The trouble with James is that he never _____ on time for a meeting.

A. turns up                   B. takes off

C. takes up                  D. turns down

Mark the letter A, B, C or D on your answer sheet to indicate the word(s) CLOSEST in meaning to the underlined word(s) in each of the following questions.

Question 17. As orders are becoming more and more, we’ve been rushed off our feet all week.

A. very angry                   B. very busy

C. very worried                D. very happy

Question 18. The atomic bomb annihilated the whole city, so nothing was left standing.

A. disturbed                     B. constructed

C. converted                    D. destroyed

Mark the letter A, B, C or D on your answer sheet to indicate the word(s) OPPOSITE in meaning to the underlined word(s) in each of the following questions.

Question 19. The minister came under fire for his rash decision to close the factory.

A. was dismissed            B. was acclaimed

C. was criticized              D. was penalized

Question 20. Those who advocate for doctor-assisted suicide say the terminally ill should not have to suffer.

A. support                               B. oppose

C. annul                                  D. convict

Mark the letter A, B, C or D on your answer sheet to indicate the most suitable response to complete each of the following exchanges.

Question 21. Mr. Collin is speaking to his student, Brian.

Mr. Collin: “You’ve been making very good progress. I’m proud of you!”

Brian: “_____”

A. No problem

C. Everything’s alright. Thank you.

D. I really appreciate you saying that.

Question 22. Peter: “Could you fill it up, please?”

Ivan: “_____”

A. Sure. Shall I check the oil as well?

B. Not at all. I’ll fill the oil for you.

C. Certainly. My job is to fill in the application.

D. Where? Of course not.

Mark the letter A, B, C or D on your answer sheet to indicate the sentence that is closest in meaning to each of the following questions.

Question 23. People say that at least ten applicants have been selected for the job interview.

A. It is said that well over ten people are interested in having an interview for the job.

B. It is said that no fewer than ten people are going to be interviewed for the job.

C. People say that fewer than ten job interviews have been held so far.

D. I have heard that only ten people have been chosen to have the job interviewed.

Question 24. Eating with chopsticks feels strange to Jonathan.

A. Eating with chopsticks isn’t what Jonathan used to.

B. Jonathan is not used to eating with chopsticks.

C. Not feeling strange, Jonathan tries eating with chopsticks.

D. Jonathan didn’t use to eat with chopsticks.

Question 25. “Why don’t you have your room repainted?” said Robert to Lan.

A. Robert suggested having Lam’s room repainted.

C. Robert wanted to know why Lam doesn’t have his room repainted.

D. Robert suggested that Lam should have his room repainted.

Mark the letter A, B, C, or D on your answer sheet to indicate the sentence that best combines each pair of sentences in the following questions.

Question 26. In your country, schools are open to all. The accept children of any race, color or creed.

A. In our country, schools are open to all children irrespective of race, color or creed.

B. In our country, schools are open to all children except for race, color or creed.

C. Apart from  race, color or creed, schools are open to all children in your country.

D. Children of any race, color or creed can be admitted to our schools when they are opened.

Question 27. He felt very tired. He was determined to continue to climb up the mountain.

A. He felt so tired that he was determined to continue to climb up the mountain.

B. Feeling very tired, he was determined to continue to climb up the mountain.

C. Tired as he might, he was determined to continue to climb up the mountain.

D. As result of his tiredness, he was determined to continue to climb up the mountain.

Mark the letter A, B, C or D in your answer sheet to indicate the underlined part that needs correction in each of the following questions.

Question 28. Thomas refused to join the school basketball team, this was not good news at all.

A. to join                     B. school

C. this                         D. good

Question 29. I am now in my attempt winning a place at the same university to my brother.

A. now in                      B. winning

C. at the same              D. to

Question 30. However type of raw materials is used in making paper, the process is essentially the same.

A. However                  B. is used

C. paper                       D. essentially

Read the following passage and mark A, B, C, or D to indicate the correct answer to each of the blanks.

British families started going on holiday to the seaside around the middle of the 19th century. The invention of the railways made this possible. The first holidaymakers were quite rich and went for their health and education. The seaside was a place to be (31) ______ of illnesses, and doctors recommended bathing in the sea and drinking sea water. Also to increase their knowledge, families attended concerts and read books from libraries.

At that time, ordinary working people had very little time (32) ______. However, in 1871, the government introduce four “Bank Holidays” – national holiday days. This allowed people to have a day or two out, which now and then gave them a taste for leisure and the seaside. At first they went on day-trips, taking (33) ______ of special cheap tickets on the railways. By the 1880s, rising incomes meant many ordinary workers and their families could have a week’s holiday at the seaside. Rail fares were reduced and cheap hotels were built to (34) ______ them. Holidaymakers enjoyed being idle, sitting on the beach, bathing in the sea, and eating ice-cream. Cheap entertainment was (35) ______ offer and holidaymakers went to have fun.

Today, the English seaside remained popular, with more than 18 million holidays taken there each year.

Question 31.

A. recovered                 B. cured

C. improved                   D. remedied

Question 32.

A. out                            B. off

C. from                          D. away

Question 33.

A. opportunity              B. benefit

Question 34.

A. cater                         B. board

C. lodge                         D. accommodate

Question 35.

A. for                            B. on

C. in                              D. to

Read the following passage and mark the letter A, B, C, or D on your answer sheet to indicate the correct answer to each of the questions.

William Sydney Porter (1862-1910), who wrote under the pseudonym of O. Henry, was born in North Carolina. His only formal education was to attend his Aunt Lina’s school until the age of fifteen, where he developed his lifelong love of books. By 1881 he was a licensed pharmacist. However, within a year, on the recommendation of a medical colleague of his Father’s, Porter moved to La Salle County in Texas for two years herding sheep. During this time, Webster’s Unabridged Dictionary was his constant companion, and Porter gained a knowledge of ranch life that he later incorporated into many of his short stories. He then moved to Austin for three years, and during this time the first recorded use of his pseudonym appeared, allegedly derived from his habit of calling “Oh, Henry” to a family cat. In 1887, Porter married Athol Estes. He worked as a draftsman, then as a bank teller for the First National Bank.

In 1894 Porter founded his own humor weekly, the “Rolling Stone”, a venture that failed within a year, and later wrote a column for the Houston Daily Post. In the meantime, the First National Bank was examined, and the subsequent indictment of 1886 stated that Porter had embezzled funds. Porter then fled to New Orleans, and later to Honduras, leaving his wife and child in Austin. He returned in 1897 because of his wife’s continued ill-health, however she died six months later. Then, in 1898 Porter was found guilty and sentenced to five years imprisonment in Ohio. At the age of thirty five, he entered prison as a defeated man; he had lost his job, his home, his wife, and finally his freedom. He emerged from prison three years later, reborn as O. Henry, the pseudonym he now used to hide his true identity. He wrote at least twelve stories in jail, and after re-gaining his freedom, went to New York City, where he published more than 300 stories and gained fame as America’s favorite short Story writer. Porter married again in 1907, but after months of poor health, he died in New York City at the age of forty-eight in 1910. O. Henry’s stories have been translated all over the world.

Question 36. According to the passage, Porter’s Father was _____.

A. the person who gave him a life-long love of books

B. a medical doctor

D. responsible for his move to La Salle County in Texas

Question 37. Why did the author write the passage?

A. to outline the career of a famous American

B. because of his fame as America’s favorite short story writer

C. because it is a tragic story of a gifted writer

D. to outline the influences on O. Henry’s writing

Question 38. The word “imprisonment” in paragraph 2 is closet in meaning to ____________.

A. captivity                     B. escape

C. insult                         D. punishment

Question 39. What is the passage primarily about?

A. The life and career of William Sydney Porter.

B. The way to adopt a nickname.

C. O.Henry’s influence on American literature.

Question 40. The author implies which of the following is true?

A. Porter’s wife might have lived longer if he had not left her in Austin when he fled.

B. Porter was in poor health throughout his life.

C. O. Henry is as popular in many other countries as he is in America.

D. Porter would probably have written less stories if he had not been in prison for three years.

Question 41. Which of the following is true, according to the passage?

A. Porter left school at 15 to become a pharmacist

B. Porter wrote a column for the Houston Daily Post called “Rolling Stone”

C. The first recorded use of his pseudonym was in Austin

D. Both of Porter’s wives died before he died

Question 42. The word “pseudonym” in the passage refers to _____.

A. William Sydney Porter

B. O. Henry

C. Athol Estes

D. the Aunt Lina

Read the following passage and mark the letter A, B, C, or D on your answer sheet to indicate the correct answer to each of the questions.

INTERNET JOBS

Contrary to popular belief, one does not have to be a trained programmer to work online. Of course, there are plenty of jobs available for people with high-tech computer skills, but the growth of new media has opened up a wide range of Internet career opportunities requiring only a minimal level of technical expertise. Probably one of the most well-known online job opportunities is the job of Webmaster. However, it is hard to define one basic job description for this position. The qualifications and responsibilities depend on what tasks a particular organization needs a Webmaster to perform.

To specify the job description of a Webmaster, one needs to identify the hardware and software the website the Webmaster will manage is running on. Different types of hardware and software require different skill sets to manage them. Another key factor is whether the website will be running internally or externally (renting shared space on the company servers). Finally, the responsibilities of a webmaster also depend on whether he or she will be working independently, or whether the firm will provide people to help. All of these factors need to be considered before one can create an accurate webmaster job description.

Webmaster is one type of Internet career requiring in-depth knowledge of the latest computer applications. However, there are also online jobs available for which traditional skills remain in high demand. Content jobs require excellent writing skills and a good sense of the web as a “new media’.

The term “new media” is difficult to define because it compasses a constantly growing set of new technologies and skills. Specifically, it includes websites, email, internet technology, CD-ROM, DVD, streaming audio and video, interactive multimedia presentations, e-books, digital music, computer illustration, video games, virtual reality, and computer artistry.

Additionally, many of today’s Internet careers are becoming paid-by-the-job professions. With many companies having to downsize in tough economic items, the outsourcing and contracting of freelance workers online has become common business practice. The Internet provides an infinite pool of buyers from around the world with whom freelancers can contract their services. An added benefit to such online jobs is that freelancers are able to work on projects with companies outside their own country.

How much can a person make in these kinds of careers? As with many questions related to today’s evolving technology, there is no simple answer. There are many companies willing to pay people with Internet skills salaries well over $70,000 a year. Generally, webmasters start at about$30,000 per year, but salaries can vary greatly. Freelance writers working online have been known to make between $40,000 to$70,000 a year.

Question 43. Which of the followings is true about the job of the freelance writers?

A. They may work with others in the company.

B. It is considered a “content” job.

C. They manage hardware and software.

D. They require in-depth knowledge of applications.

Question 44. The word “vary” in paragraph 6 can be best replaced by _________

A. decrease                             B. change

C. differ                                    D. increase

Question 45. It can be inferred from the passage that _________.

A. only skilled workers make good money.

B. online workers can work full-time online.

C. it is easy to become a webmaster.

D. workers with limited computer skills can not work online.

Question 46. According to the passage, which of the following is TRUE of webmasters?

A. They never work independently.

B. The duties they perform depend on the organization they work for.

C. They require a minimal level of expertise.

D. They do not support software products.

Question 47. What is the purpose of the passage?

A. To inform people about the tasks and the roles of a webmaster.

B. To inform people about employment related to the Internet.

C. To inform people about the computer industry.

D. To explain why webmasters make a lot of money.

Question 48. According to the passage, all of the followings are TRUE except _________.

A. Webmasters must have knowledge of the latest computer applications.

B. Online workers can not free themselves from the office.

C. “New media” is not easy to define.

D. There are online jobs available for workers with minimal computer skills.

Question 49. Which of the followings is not mentioned as part of the “new media”?

A. Internet technology

B. writing skills

C. websites

D. video games

Question 50. The word “identify” in paragraph 2 is closest in meaning to __________.

A. name                                 B. discover

C. encounter                          D. estimate

Lời giải chi tiết

 1 2 3 4 5 D A C A D 6 7 8 9 10 B D C A A 11 12 13 14 15 D B D D A 16 17 18 19 20 A B D C B 21 22 23 24 25 D B B B D 26 27 28 29 30 A C C D A 31 32 33 34 35 B B D D B 36 37 38 39 40 B A A A C 41 42 43 44 45 C B B C B 46 47 48 49 50 C B B B A

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text:
# Math Help - Math B Regents Multiple Choice Questions

1. ## Math B Regents Multiple Choice Questions

I'm doing review for the MAth B Regents as some of you know and there were a couple in the multiple choice section i just didn't get. Here is the link to the the answers:http://www.nysedregents.org/testing/mathre/b-key605.pdf
And here is the link to the questions:http://www.nysedregents.org/testing/mathre/b605.pdf

I just need to know the work necessary to figure out these problems. The numbers i need help in are 4,12,13,15,16, and 20. Thanks guys.

2. Originally Posted by badandy328
I'm doing review for the MAth B Regents as some of you know and there were a couple in the multiple choice section i just didn't get. Here is the link to the the answers:http://www.nysedregents.org/testing/mathre/b-key605.pdf
And here is the link to the questions:http://www.nysedregents.org/testing/mathre/b605.pdf

I just need to know the work necessary to figure out these problems. The numbers i need help in are 4,12,13,15,16, and 20. Thanks guys.
Didn't you post this paper already? why didn't you ask to do these questions in the last post?

4)

$\frac {x^2 - 9x}{45x - 5x^2}$ ....what are the largest common factors in the top and bottom? pull them out

$= \frac {x(x - 9)}{5x(9 - x)}$

$= \frac {x - 9}{5(9 - x)}$

do you think you could take it from here?

3. Originally Posted by badandy328
I'm doing review for the MAth B Regents as some of you know and there were a couple in the multiple choice section i just didn't get. Here is the link to the the answers:http://www.nysedregents.org/testing/mathre/b-key605.pdf
And here is the link to the questions:http://www.nysedregents.org/testing/mathre/b605.pdf

I just need to know the work necessary to figure out these problems. The numbers i need help in are 4,12,13,15,16, and 20. Thanks guys.
The formula for the equation of an ellipse is $\frac {(x - h)^2}{a^2} + \frac {(y - k)^2}{b^2} = 1$

the choice is obvious here, there is only one choice that resembles that form

4. Originally Posted by badandy328
I'm doing review for the MAth B Regents as some of you know and there were a couple in the multiple choice section i just didn't get. Here is the link to the the answers:http://www.nysedregents.org/testing/mathre/b-key605.pdf
And here is the link to the questions:http://www.nysedregents.org/testing/mathre/b605.pdf

I just need to know the work necessary to figure out these problems. The numbers i need help in are 4,12,13,15,16, and 20. Thanks guys.
13)

for this you need to remember that $i^2 = -1$

$\frac {2 + i}{3 + i} = \frac {2 + i}{3 + i} \cdot \frac {3 - i}{3 - i}$ ...rationalize the denomenator

......... $= \frac {6 + i - i^2}{9 - i^2}$

......... $= \frac {6 + i - (-1)}{9 - (-1)}$

......... $= \frac {7 + i}{10}$

5. This should make it a bit more easy for us to help you.

4. Written in simplest form, the expression $\frac{x^2-9x}{45x-5x^2}$ is equivalent to:

12. Which equation, when graphed on a Cartesian coordinate plane, would best represent an elliptical racetrack?
(1) $3x^2 + 10y^2 = 288,000$
(2) $3x^2-10y^2=288,000$
(3) $3x+10y=288,000$
(4) $30xy=288,000$

13. The expression $\frac{2+i}{3+i}$ is equivalent to:

15. A crate weighing $w$ pounds sits on a ramp positioned at an angle $\theta$ with the horizontal. The forces acting on this crate are modeled by the equation $Mw\cos\theta = w\sin \theta$, where $M$ is the coefficient of friction. What is an expression for M in terms of $\theta$?
(1) $M=\tan \theta$
(2) $M=\cot \theta$
(3) $M=\sec \theta$
(4) $M=\csc \theta$

16. If $(a^x)^\frac{2}{3}=\frac{1}{a^2}$, what is the value of x?

20. In the accompanying diagram, $\overline {PR}$ is tangent to circle $O$ at $R$, $\overline {QS} \perp \overline{OR}$, and $\overline{PR}\perp\overline{OR}$. What measure represents $sin\theta$?

[btw, I'm bored.]

6. Originally Posted by badandy328
I'm doing review for the MAth B Regents as some of you know and there were a couple in the multiple choice section i just didn't get. Here is the link to the the answers:http://www.nysedregents.org/testing/mathre/b-key605.pdf
And here is the link to the questions:http://www.nysedregents.org/testing/mathre/b605.pdf

I just need to know the work necessary to figure out these problems. The numbers i need help in are 4,12,13,15,16, and 20. Thanks guys.
i'm a bit rusty on number 15) i think i'll come back to it, if someone doesn't answer it

16)
$\left( a^x \right)^{ \frac {2}{3}} = \frac {1}{a^2}$

when we raise a base with a power to another power, we multiply the powers.

1 over something is the inverse of the something, we undo it by taking the negative of the power of the base

$\Rightarrow a^{ \frac {2x}{3}} = a^{-2}$

equating the powers we obtain.

$\frac {2x}{3} = -2$

$\Rightarrow x = -3$

7. Originally Posted by ecMathGeek
15. A crate weighing $w$ pounds sits on a ramp positioned at an angle $\theta$ with the horizontal. The forces acting on this crate are modeled by the equation $Mw\cos\theta = w\sin \theta$, where $M$ is the coefficient of friction. What is an expression for M in terms of $\theta$?
(1) $M=\tan \theta$
(2) $M=\cot \theta$
(3) $M=\sec \theta$
(4) $M=\csc \theta$
Since $Mwcos\theta=wsin\theta$, dividing by $wcos\theta$, we get:

$M=tan\theta$, which is answer (1).

8. Originally Posted by badandy328
I'm doing review for the MAth B Regents as some of you know and there were a couple in the multiple choice section i just didn't get. Here is the link to the the answers:http://www.nysedregents.org/testing/mathre/b-key605.pdf
And here is the link to the questions:http://www.nysedregents.org/testing/mathre/b605.pdf

I just need to know the work necessary to figure out these problems. The numbers i need help in are 4,12,13,15,16, and 20. Thanks guys.
20)

Okay, again i think i'm thinking too hard about this, but here is the way i see to do it. (save me ecMathGeek if it's too much work).

$\sin(180 - \theta) = \frac {QS}{QO}$ ............ $180 - \theta$ is the angle under line $QO$

remember, $QO = 1$

So, $\sin (180 - \theta) = QS$

Now $\sin (180 - \theta) = \sin ( \theta)$ ....but let's prove this

$\sin (180 - \theta) = \sin (180) \cos ( \theta) - \sin ( \theta) \cos (180) = QS$

$\Rightarrow 0 + \sin ( \theta) = QS$

$\Rightarrow \sin ( \theta) = QS$

9. Originally Posted by Jhevon
20)

Okay, again i think i'm thinking too hard about this, but here is the way i see to do it. (save me ecMathGeek if it's too much work).

$\sin(180 - \theta) = \frac {QS}{QO}$ ............ $180 - \theta$ is the angle under line $QO$

remember, $QO = 1$

So, $\sin (180 - \theta) = QS$

Now $\sin (180 - \theta) = \sin ( \theta)$ ....but let's prove this

$\sin (180 - \theta) = \sin (180) \cos ( \theta) - \sin ( \theta) \cos (180) = QS$

$\Rightarrow 0 + \sin ( \theta) = QS$

$\Rightarrow \sin ( \theta) = QS$
You did it as easily as it can be done, with the exception that you demonstrated why your answer is true.

10. I said it once and I'll say it again, YOU GUYS ARE THE BEST!!!! The reason why i didn't ask this in the last topic about the Math B Regents is becuase I was afraid Perfect Hacker would give me an infraction to add to my collection for begging or something like that. I tend to get a lot of those. Anyway thanks again.


text:
?

Free Version
Easy

# Sequential Search Worst Case Number of Comparisons

APCSA-TAX1LL

Using a sequential search, what is the maximum number of comparisons needed to find an element in an unsorted array of length N?

A

$log_2N$

B

$N/2$

C

$N$

D

$N + 1$

E

$2N$


text:
# Problem using Event Trigger Unity to change bool , but bool keeps setting back to false

OK so I have used the Unity Event Trigger many times to check for on-screen button presses for Android games. One example I have made many times is a left and right arrow button, which moves the character. There is OnPointerDown() and OnPointerUp() methods which i am utilising (or trying to at least!).

So I have a bool called isLeftToggled and its initialised as 'false'. OnPointerDown/Up both link to a method called ToggleButtonLeft():

    public void ToggleButtonLeft(){
Debug.Log ("LEFT TOGGLED" + isLeftToggled);

if (isLeftToggled)
{
isLeftToggled = false;
}
else
{
isLeftToggled = true;
}
}

That Debug.Log above reads out ok, it toggles between true and false (although puzzlingly the first log i get is 'True').

However in my update method I have it constantly inform me if its true or false, and it is false for every single call.

I verified this because the ship (game character) doesnt turn when I try to use the bool to turn it. If I initialise the bool to 'true' and remove the toggle by comments, the ship turns fine.

I have the Event Trigger on the button game object, I didnt remember if I had to add something else to make it work. But I am fairly sure the button is working, because it always prints the Debug.Log thats inside the method being called by the two buttons.

Any ideas why the bool is 'false' for every frame. Shouldn't my code above toggle it to true whilst the finger is held on the button?

Here is the whole class (nb. i thought about trimming out the irrelevant parts but didnt want to miss something important out so decided to post whole thing):

public class Player : MonoBehaviour {
float rotationSpeed = 360f;
float currentThrust = 0f;
float thrustDecay = 0.02f;
float accel = 0.25f;
float maxThrust = 150f;

float thrustAngle, prevThrustAngle;
float maxTurningThrust;
public GameObject laserStartPoint;
float laserFireRate = 0.5f;
private float laserFireTimer = 5f; // the 5 is a random amount just to make sure no delay at the beginning

public GameObject laserPrefab;

private List<GameObject> lasers = new List<GameObject>();

private Collider gameArea;

private GameObject afterburner;
public GameObject explosionPrefab;

AudioSource audioSource;
public AudioClip laserSound, deathSound, deathExplosionSound, thrusterSound;
float deathWaitTimeBeforeDestroy = 2f;
bool isDying;
GameObject shipModel;

private bool isLeftToggled = false, isRightToggled;

void Start () {
isDying = false;
gameArea = GameObject.FindGameObjectWithTag("GameArea").GetComponent<Collider>();
maxTurningThrust = maxThrust * 0.012f;
afterburner = GameObject.FindGameObjectWithTag("Afterburner");
afterburner.SetActive(false);
audioSource = GetComponent<AudioSource>();
shipModel = GameObject.FindGameObjectWithTag("ShipModel");
}

void Update () {
if (isDying)
{
shipModel.SetActive(false);
deathWaitTimeBeforeDestroy -= Time.deltaTime;
if (deathWaitTimeBeforeDestroy <= 0)
{
Destroy(this.gameObject);
GameController.instanceOf.PlayerLoseLife ();
}
}
else
{
HandleInput();
CheckScreenBounds();

laserFireTimer += Time.deltaTime;
}

Debug.Log("is left toggled: " + isLeftToggled);
}

void HandleInput(){
// activate weapon
if (Input.GetKey(KeyCode.Space))
{
FireWeapon();
}
// Rotate left
if (Input.GetKey(KeyCode.A) || isLeftToggled)
{

transform.Rotate(0, -rotationSpeed * Time.deltaTime, 0);
}
// Rotate right
if (Input.GetKey(KeyCode.D))
{
transform.Rotate(0, rotationSpeed * Time.deltaTime, 0);
}
// activate thrusters
if (Input.GetKey(KeyCode.W))
{
//audioSource.clip = thrusterSound;
audioSource.PlayOneShot(thrusterSound, 0.25f);
afterburner.SetActive(true);
prevThrustAngle = thrustAngle;
thrustAngle = transform.eulerAngles.y;
if (currentThrust < maxThrust)
{
if (prevThrustAngle != thrustAngle)
{
if (currentThrust > maxTurningThrust) {
currentThrust = maxTurningThrust;
}
}
currentThrust += accel;
}
}
else
{
afterburner.SetActive(false);
if (currentThrust > 0f)
{
currentThrust -= thrustDecay;
}
}

// Move the ship in the direction it was facing at point when thruster button was last pressed
transform.position += new Vector3(Mathf.Sin(thrustAngle * Mathf.Deg2Rad), 0, Mathf.Cos(thrustAngle * Mathf.Deg2Rad)) * currentThrust * Time.deltaTime;

if (currentThrust < 0f)
{
currentThrust = 0f;
}

}

void CheckScreenBounds()
{
if (transform.position.x - (GetComponent<Collider>().bounds.size.x / 2) > gameArea.bounds.size.x / 2)
{
transform.position = new Vector3(-gameArea.bounds.size.x / 2, transform.position.y, transform.position.z);
}
if (transform.position.x + (GetComponent<Collider>().bounds.size.x / 2) < -gameArea.bounds.size.x / 2)
{
transform.position = new Vector3(gameArea.bounds.size.x / 2, transform.position.y, transform.position.z);
}
if (transform.position.z - (GetComponent<Collider>().bounds.size.z / 2) > gameArea.bounds.size.z / 2)
{
transform.position = new Vector3(transform.position.x, transform.position.y, -gameArea.bounds.size.z / 2);
}
if (transform.position.z + (GetComponent<Collider>().bounds.size.z / 2) < -gameArea.bounds.size.z / 2)
{
transform.position = new Vector3(transform.position.x, transform.position.y, gameArea.bounds.size.z / 2);
}
}

void FireWeapon()
{
if (laserFireTimer >= laserFireRate)
{
GameObject tempLaser = laserPrefab;

tempLaser.transform.position = laserStartPoint.transform.position;
tempLaser.transform.rotation = laserStartPoint.transform.rotation;
Instantiate(tempLaser);

laserFireTimer = 0f;
tempLaser = null;

audioSource.PlayOneShot(laserSound);
}

}

public void Die()
{
GameObject explosion = GameObject.Instantiate(explosionPrefab);
explosion.transform.position = transform.position;
audioSource.PlayOneShot(deathSound);
audioSource.PlayOneShot(deathExplosionSound);
isDying = true;
}

private void OnTriggerEnter(Collider other)
{
if (other.tag == "Asteroid" || other.tag == "Saucer" || other.tag == "LaserEnemy")
{
Die();
}
}

public void ToggleButtonLeft(){
Debug.Log ("LEFT TOGGLED" + isLeftToggled);

if (isLeftToggled)
{
isLeftToggled = false;
}
else
{
isLeftToggled = true;
}
}
public void ToggleButtonRight()
{
Debug.Log("RIGHT TOGGLED");

if (!isLeftToggled)
isRightToggled = true;
else
isRightToggled = false;
}

}

here is screenshot of unity. I think it sums up the problem quite well, the Event Trigger on the right shows its calling the method in the class I posted above, the log even shows its getting toggled to True when i push the button, but the ship doesnt rotate, and as you see in the log the bool is 'false' on every frame:

• "isLeftToggled" is likely an instance variable, and you're liking executing ToggleButtonLeft on a different instance than the one you're logging. Post the definition of the class including the declaration of isButtonLeft and the code where you log the value. – user1430 Nov 15 '17 at 3:05
• @JoshPetrie Sure I will grab it now and adjust my question . thanks for looking pal – Big T Larrity Nov 15 '17 at 3:07
• Also if it fixes itself when you remove code, be sure you didn't write if (isLeftToggled = false) – Jay Nov 15 '17 at 3:11
• @JoshPetrie thanks again Josh. ive posted the class now, hope its not too long to see the part you mean – Big T Larrity Nov 15 '17 at 3:12
• I don't see the "update method that constantly informs you" of anything about isLeftToggled? I see Update() calling HandleInput() which rotates something if isLeftToggled is true, but that could easily be misread. – user1430 Nov 15 '17 at 3:17


text:
J. Spelling Check
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Petya has noticed that when he types using a keyboard, he often presses extra buttons and adds extra letters to the words. Of course, the spell-checking system underlines the words for him and he has to click every word and choose the right variant. Petya got fed up with correcting his mistakes himself, that’s why he decided to invent the function that will correct the words itself. Petya started from analyzing the case that happens to him most of the time, when all one needs is to delete one letter for the word to match a word from the dictionary. Thus, Petya faces one mini-task: he has a printed word and a word from the dictionary, and he should delete one letter from the first word to get the second one. And now the very non-trivial question that Petya faces is: which letter should he delete?

Input

The input data contains two strings, consisting of lower-case Latin letters. The length of each string is from 1 to 106 symbols inclusive, the first string contains exactly 1 symbol more than the second one.

Output

In the first line output the number of positions of the symbols in the first string, after the deleting of which the first string becomes identical to the second one. In the second line output space-separated positions of these symbols in increasing order. The positions are numbered starting from 1. If it is impossible to make the first string identical to the second string by deleting one symbol, output one number 0.

Examples
Input
abdrakadabraabrakadabra
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text:
How does one derive the equation for the speed of sound?

In my acoustics books I see

$$c^2 = \frac{\mathrm{d}P}{\mathrm{d}\rho}$$

where $c$ is the speed of sound, $P$ is the pressure and $\rho$ is the density. Where does this equation come from? In my books it appears almost as a definition. Could you explain this, or at least point me to an article or a book that addresses this question? Thanks.

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It's from the ideal gas law and the defn of specific heats = Newton-Laplace formula. the derivation isn't hard but is lots of latex and i have to go in a minute –  Martin Beckett Apr 11 '12 at 2:43
@MartinBeckett: It is not. You are just quoting the suboptimal presentation on Wikipedia. The ideal gas law has nothing to do with it. –  Ron Maimon Apr 11 '12 at 7:18
It starts with PV^(Cp/Cv) = constant, which is a pretty fundemental gas law and then considering a small change in volume –  Martin Beckett Apr 11 '12 at 14:55
@MartinBeckett: Except it works in more generality than ideal gasses-- it works in any material where there is a $dP/d\rho$, and this is what OP is asking. –  Ron Maimon Apr 11 '12 at 19:14

This derivation is often neglected, because it is slightly involved (for an ungergraduate presentation) in Newton's way of thinking, with explicit forces, although this is how Newton did it, and it is a little too trivial if you use stress tensor concepts. I will use the stress tensor concept.

Momentum is a conserved quantity, and you should be familiar with the conservation law in differential form:

$$\partial_t \rho + \partial_i J^i = 0$$

Where the repeated i index is summed (Einstein convention). This is clearest in space-time, where the density $\rho$ becomes the time component of a 4-vector, but it is just as true in Galilean Newtonian mechanics.

For momentum, you have three separate conserved momentum densities $p^i$ which obey a conservation law:

$$\partial_t p^j + \partial_i T^{ij} = 0$$

Where the interpretation of $T^{ij}$ is that it is the flow of the i'th component of p in the j direction. There are three equations, since there are three separate conserved quantities--- the x,y,z momentum.

If you compress a region of air, you get a little bit more pressure. The form of pressure is a diagonal stress density, by rotational symmetry (it is also intuitive--- pressure pushed out--- so x-momentum goes in the x direction, y-momentum goes in the y-direction and so on).

The stress T for a pressure is therefore

$$T^{ij} = P(\rho) \delta^{ij}$$

If the material starts out at position x (at each x there is a different infinitesimal volume of material) and the material which would be at x when everything is still is moved at time t to x+\delta X(x,t), then the momentum density is

$$p^{i} = \rho {\partial_t X(x,t)}$$

Assuming $\rho = \rho_0(1 - \delta v)$ where the infinitesimal volume compression, which gives the change in the density, is given by the divergence of X (this is geometrically clear if you draw a picture, or just from the definition of divergence)

$$\delta v = \partial_j X^j$$

To first infinitesimal order, then the pressure is

$$P (\rho) = P(\rho_0) - C\delta v = P(\rho_0) - C \partial_i X^i$$

Where $C= {dP\over d\rho}$

Now, for mathematical convenience (I do not want to deal with transverse sound), consider reducing to one dimension. In this case X(x) is just a one-dimensional function that tells you the displacement, and the stress is just $T^{11} = \rho_0 \partial_x X$ (the flow of x-momentum in the x-direction).

The continuity equation in 1d gives you

$$\rho_0 \partial^2_t X - \rho_0 C \partial^2_x X = 0$$

And this is the wave equation with the speed of sound squared equal to C. You can repeat the 1d analysis with detailed forces, not using the slightly more abstract continuity equation, and this is what Newton did to find the speed of sound way back when.

To see that the one dimensional equation above describes waves moving with a speed $\pm \sqrt{C}$, consider the functional form of such a wave moving with velocity c:

$$\phi(x,t) = f(x-ct)$$

where f is a function of one dependent variable. Differentiate twice in time

$$\partial^2_t \phi = c^2 f''$$

and twice in space:

$$\partial^2_x \phi = f''$$

and you see that it satisfies the wave equation:

$$\partial^2_t \phi - c^2 \partial^2_x \phi = 0$$

The general solution to the 1d equation can be expressesed as the sum of a left-moving and a right-moving wave. This can be used to match any initial conditions of $\phi$ and $\partial_t\phi$, and is therefore the general form. You can derive the general solution from a systematic general theory using Fourier transforms, by considering plane waves.

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Elegant enough, Ron, +1. Maybe one should also explain why the wave equation at the end implies the right speed, e.g. by taking the solutions $\cos(kx-\omega t)$ and showing that $\omega/k=v$ has to agree with the coefficient obtained from $C$ for the equation to hold. –  Luboš Motl Apr 11 '12 at 8:33
@LubošMotl: Ok, I'll do it, if you think its necessary –  Ron Maimon Apr 11 '12 at 15:38
Stop mucking with this--- $\delta v$ is not a change in volume, it is a dimensionless factor defined in the answer. I do not like $\delta v\over v$, this is introducing an extra symbol for no reason. –  Ron Maimon Sep 25 '12 at 8:46


text:
On two-dimensional Gaussian integrals

Fix $\epsilon, 0\leq \epsilon\leq 1/2.$ Let $Z_1,Z_2$ be zero mean, unit variance Gaussian random variables which are jointly Gaussian with $\mathbb{E}Z_1Z_2=-(1-2\epsilon)\leq 0.$

Then,

$$P(Z_1Z_2>0)\geq \epsilon.$$

This curious fact popped out of some calculations I was doing using the Central Limit Theorem. I wonder if it can be proved in some easy way by directly working with the double integral.

I also believe that the inequality is strict when $0<\epsilon<1/2.$ Can this be shown?

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Since $P(Z_1Z_2>0)=\frac12-\frac1{\pi}\arcsin(1-2\epsilon)$, indeed $P(Z_1Z_2>0)\geqslant\epsilon$, and the inequality is strict except when $\epsilon=0$, $\epsilon=\frac12$ and $\epsilon=1$.
@DidierPiau: I think the case of $\epsilon = 1/2$ should be added to the values at which equality is attained. –  cardinal Feb 15 '12 at 22:08
@Hedonist: Here's one way. $Z_1 Z_2 = \rho X_1^2 + \sqrt{1-\rho^2} X_1 X_2$ in distribution where $X_1$ and $X_2$ are iid standard normal and $\rho = -(1-2\epsilon)$. Then $\mathbb P(Z_1 Z_2 > 0 ) = \mathbb P(X_2/|X_1| > -\rho/\sqrt{1-\rho^2})$ and $X_2/|X_1|$ is standard Cauchy. A tiny bit of algebra then gets you the conclusion. –  cardinal Feb 15 '12 at 22:25


text:
1

### IIT-JEE 1989

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle of area 154 sq. units. Then the equation of this circle is
A
$${x^2} + {y^2} + 2x - 2y = 62$$
B
$${x^2} + {y^2} + 2x - 2y = 47$$
C
$${x^2} + {y^2} - 2x + 2y = 47$$
D
$${x^2} + {y^2} - 2x + 2y = 62$$c
2

### IIT-JEE 1989

If the two circles $${(x - 1)^2} + {(y - 3)^2} = {r^2}$$ and $${x^2} + {y^2} - 8x + 2y + 8 = 0$$ intersect in two distinct points, then
A
2 < r < 8
B
r < 2
C
r = 2
D
r > 2
3

### IIT-JEE 1988

If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2}\, = \,{k^2}$$ orthogonally, then the equation of the locus of its centre is
A
$$2\,ax\, + \,2\,by\, - \,({a^2}\, + \,{b^2}\, + \,\,{k^2})\, = \,0$$
B
$$2\,ax\, + \,2\,by\, - \,({a^2}\, - \,\,{b^2}\, + \,\,{k^2})\, = \,0$$
C
$${x^2}\, + \,{y^2}\, - \,3\,\,ax\, + \,4\,by\, + \,\,({a^2}\, + \,\,{b^2}\, - \,\,{k^2})\, = \,0$$
D
$${x^2}\, + \,{y^2}\, - \,2\,\,ax\, - \,4\,by\, + \,\,({a^2}\, - \,\,{b^2}\, - \,\,{k^2})\, = \,0$$.
4

### IIT-JEE 1984

The locus of the mid-point of a chord of the circle $${x^2} + {y^2} = 4$$ which subtends a right angle at the origin is
A
x + y = 2
B
$${x^2} + {y^2} = 1$$
C
$${x^2} + {y^2} = 2$$
D
$$x + y$$=1

### Joint Entrance Examination

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text:
## Two researchers share Abel prize for work in discrete mathematics and computer science

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The Abel Prize

The 2021 Abel Prize, arguably the closest equivalent in mathematics to the Nobel prize, has been awarded jointly to Avi Widgerson of the Institute for Advanced Study in Princeton, and László Lovász of the Eötvös Loránd University in Budapest, for their research linking discrete mathematics and computer science. The recipients will split the award, which is approximately USD\$880,000.

According to Hanz Munthe-Kaas of the University of Bergen in Norway, who chaired the Abel Prize committee, Widgerson and Lovász “really opened up the landscape

## Pandemics, misinformation and pseudoscience

Credit: U.S. Center for Disease Control and Prevention

Covid-19 and the misinformation pandemic

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