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saving . . . saved Regarding the error message while compilation, "Sorry, but pdflatex did not succeed." has been deleted. Regarding the error message while compilation, "Sorry, but pdflatex did not succeed." has been hidden .
Regarding the error message while compilation, "Sorry, but pdflatex did not succeed."
##### Question
Sir, I wrote the first code for "Hello World!!!!!!!!", but after pressing "Ctrl+T", compiler gave the following message without output
"
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7440 64-bit)

Sorry, but pdflatex did not succeed.

The log file hopefully contains the information to get MiKTeX going again:

C:\Users\shri\AppData\Local\MiKTeX\2.9\miktex\log\pdflatex.log

pdflatex: major issue: So far, you have not checked for MiKTeX updates.

Sorry, but "MiKTeX Compiler Driver" did not succeed.

The log file hopefully contains the information to get MiKTeX going again:

C:\Users\shri\AppData\Local\MiKTeX\2.9\miktex\log\texify.log

texify: major issue: So far, you have not checked for MiKTeX updates.  ".

Please kindly give the solution, so that the code will execute and give output as pdf.

LaTeX LaTeX-on-Windows-using-TeXworks 03-04 min 40-50 sec 11-06-20, 9:29 p.m. sumedhmaths@gmail.com

You have probably not installed MikTeX properly.

Also, did you use the file hello.tex that comes with the tutorial?  If not, please use it.
13-06-20, 11:07 p.m. kannan
Respected sir, I have installed MikTeX properly as per the instructions and used the same file mentioned above "hello.tex". But, I am getting the same error message. Even it is not working on TeXmaker and TeXstudio. Please suggest to me regarding the same. Thank you.
16-06-20, 11:26 p.m. sumedhmaths@gmail.com
I don't know, but I think the problem is due to the file named "pdflatex", as it is showing the error message "Sorry, but pdflatex did not succeed." Because of this, I am unable to learn and practice new concepts in LaTeX.
16-06-20, 11:31 p.m. sumedhmaths@gmail.com


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# zbMATH — the first resource for mathematics

Analysis on local Dirichlet spaces. I: Recurrence, conservativeness and $$L^ p$$-Liouville properties. (English) Zbl 0806.53041
Local Dirichlet spaces are used as an appropriate frame to unify and extend various famous results from differential geometry obtained originally by Yau, Cheng/Yau, Karp, Karp/Li, Grigor’yan. Sharp conditions for recurrence as well as for conservativeness and for exponential instability are given. These conditions are in terms of the volume growth $$r \mapsto m(B_ r(x))$$ of metric balls $$B_ r(x) \subset X$$ which are defined intrinsically by the respective Dirichlet form on $$L^ 2(X,m)$$. Also $$L^ p$$-growth conditions for nonnegative sub- or supersolutions on $$X$$ are derived. In particular, $$L^ p$$-Liouville theorems are obtained.

##### MSC:
 31C25 Dirichlet forms 60J45 Probabilistic potential theory 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
Full Text:

format: plain


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# Comparing two models of recurrent neural networks for predicting time series

I'd like to train a RNN to predict a time series (The data set is composed of sequences of cell cycle division times). Iv'e seen two models that seem appropriate. Presented here is my understanding for how they work.

In the first, each training sequence is of some length $$n$$ and the goal of the model is to predict the sequence $$\delta$$ time steps into the future. So the target for each sequence would be that same sequence shifted $$\delta$$ time steps into the future. Then at test time, the input is some seed sequence $$x$$, and the output is $$x'$$, the prediction of $$x$$ shifted $$\delta$$ time steps into the future. We can then feed $$x'$$ into the model and get $$x''$$, and so on, generating a prediction of $$x$$'s future for an arbitrary length.

In the second model, at training time the target of any time step $$t$$, call it $$y^{}$$, is fed into the hidden state of the next time step $$h^{}$$ and at test time, instead of passing the real target $$y^{}$$. we pass the predicted output $$\hat{y}^{}$$, again, enabling us to generate a future prediction of an arbitrary length.

The more I think about it, the more it seems that really, the second model is actually a special case of the first with $$\delta=1$$. Is this observation correct? If not, what is the difference and when should I use each one? Are there other RNN architectures that may be relevant?

Here is a sketch of how these models should look (unrolled)

• Inputs are sequences of real valued numbers. Training data is composed of a few hundred such sequences, of varying length. The goal is to predict the future development of a given seed sequence. – H.Rappeport Sep 17 '18 at 15:10
• The dimension at each time point is 1, a single scalar. I thought RNNs were much better in deducing temporal relations in sequences? Also, the input sequences may be of varying length. – H.Rappeport Sep 17 '18 at 15:52
• My advice is first try traditional time series forecasting. You need this to establish the baseline anyways. I doubt you'll be able to beat TS analysis with RNN. – Aksakal Sep 17 '18 at 16:52
• @Aksakal . Could you elaborate on what you mean by conventional TS analysis? – H.Rappeport Sep 17 '18 at 18:22
• ARIMA, state space analysis, Hodrick Prescott filters, exponential smoothing etc. Are you familiar with time series forecasting? If not then you should get familiar, in my opinion, because for what you describe these things have a better chance to succeed than RNN – Aksakal Sep 17 '18 at 18:25


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# Discussion¶

## Limitations¶

The method we presented just assumed that the roll is zero. Also we did not estimate the height $$h$$ of the camera. In the real world you could estimate the height using a tape measure and will probably only make an error of around 5 percent. Assuming a roll of zero does not seem to lead to practical problems, since this is done in the source code of openpilot, which is known to perfrom really well. As a bonus exercise you can run experiments with code/tests/camera_calibration/carla_sim.py where you change the roll of the camera or you slightly modify the height. Investigate how this affects the control of the vehicle. Regarding estimation of height and roll we also recommend to have a look at this paper.

Another limitation: The method we discussed in this chapter only works if your autonomous vehicle software stack uses lane detection in image space and if it is used in areas with good lane markings. But what if your software doesn’t predict lanes in image space? Maybe it predicts lanes in world space as openpilot, or maybe it doesn’t predict lanes at all and makes predictions end-to-end. In either approaches, this method of camera calibration isn’t going to work. As an alternative, we can use visual odometery (VO) based camera calibration.

## Alternative: VO-based camera calibration¶

In this approach, visual odometery is performed to find the motion of the camera. This approach also needs to be performed when the car is aligned with the lane line, i.e. when it is going straight and fast. Since the output of VO would be motion of the camera, this information can be used to find out the orientation of the camera with respect to the car. See how openpilot performs this in the calibrationd module. The vehicle forwards axis is more or less identical to the direction of the translation vector, since the vehicle is driving straight. But having the vehicle forwards axis with respect to the camera reference frame means that you can estimate how the optical axis (the z-axis) of the camera is tilted with respect to the vehicle forwards direction. Hence you get the extrinsic rotation matrix! However, you also need to assume the roll is zero in this approach. To get started with visual odometery, the fastest way is to get started is PySlam. It has lots of methods for visual odometery including novel approaches based on deep learning.


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Recent Massive Star Formation in NGC1068 Derived from UV/H$\alpha$ Comparisons
Session 77 -- Starbursts
Display presentation, Friday, January 14, 9:30-6:45, Salons I/II Room (Crystal Gateway)

## [77.04] Recent Massive Star Formation in NGC1068 Derived from UV/H$\alpha$ Comparisons

S. G. Neff (LASP/GSFC/NASA), M. N. Fanelli (NRC/LASP/GSFC), L. J. Roberts, K. P. Cheng (HSTX/GSFC), P. M. N. Hintzen (CSULB/LASP/GSFC/NASA), E. P. Smith, A. M. Smith, T. P. Stecher (LASP/GSFC/NASA), R. C. Bohlin (STScI), R. W. O'Connell (UVa), M. S. Roberts (NRAO)

We present broadband ultraviolet images of NGC1068 made at 152 and 249nm with 3 arcsec resolution. The images were made using the Ultraviolet Imaging Telescope (UIT) in December 1990. The UV images are compared with ground-based H$\alpha$ images having comparable resolution.

Although the overall brightness distributions are similar in the UV and in H$\alpha$, we find significant differences when we compare the images in detail. In particular, the extremely luminous UV region 10 arcsec NW of the active nucleus is unremarkable in H$\alpha$. Several other regions are observed with differing UV and H$\alpha$ appearances.

We use the UV/H$\alpha$ ratio in different regions to explore the recent star formation history of NGC1068. Although both the UV light and the H$\alpha$ emission trace the young stellar population, the H$\alpha$ tends to fade more quickly as the ionizing stars evolve off the main sequence; the UV continuum emission lasts longer, as it originates from a broader range of stellar ages and temperatures. Thus, the knots' which are UV-bright but H$\alpha$-weak are probably regions currently containing young stars with intermediate masses (5-15 $M_{\sun}$). Alternatively, the knots' may have formed with a truncated IMF.

We compare our observations to population synthesis models of Fanelli \it et al. \rm to determine ages of the star- forming regions and to constrain the IMF in selected regions of NGC1068.


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Image Details

Choose export citation format:

Optical Continuum Reverberation in the Dwarf Seyfert Nucleus of NGC 4395

• Authors: John W. Montano, Hengxiao Guo, 恒潇 郭, Aaron J. Barth, Vivian U, Raymond Remigio, Diego H. González-Buitrago, Juan V. Hernández Santisteban

John W. Montano et al 2022 The Astrophysical Journal Letters 934 .

• Provider: AAS Journals

Caption: Figure 2.

Lags as a function of wavelength in the rest frame of NGC 4395, from the combined Night 1 + Night 2 data. Horizontal error bars denote the width of each filter. The β = 4/3 power-law fit to the riz data points is shown as a red curve. Blue dashed and dotted curves represent SSD lag–wavelength models calculated for BH masses of 1.7 × 104 and 4 × 105 M , respectively.


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# Tag Info

19

The subtropical jetstream The subtropical jetstream can be explained with angular momentum arguments. In an idealized circulation model of the atmosphere, there is a hadley cell that circulates upward at the equator, poleward along the tropical tropopause, downward around 30 N/S and equatorward at the surface. Parcels rising at the equator have a fair bit ...

10

As your diagram shows, the jet stream actually has two circulating components: one is "around the earth" (which is why the transatlantic crossing is so much faster going East than going West), while the second is more like a vortex, with a circulating flow "around" the main direction. This circulating flow is vertical in two places: North of the jet stream ...

9

I wouldn't characterize the movement of the jetstream as a "less well known" problem, at least not to anyone who has sat through a synoptic meteorology class. The jetstream from a dynamic point of view is just an enhanced gradient in the heights on a constant pressure surface (or pressures on a constant height surface) and there are many factors the ...

8

Looking at the annual climatology of the 200 hPa winds as shown below from NCAR NCEP reanalysis it is noted that the mid latitudes have by and large westerly winds (for a planet that rotates west to east). On the other hand averaged over a latitude circle(zonal average) the low latitude tropics and equator have by and large easterly winds. The reference ...

5

Jet streams are caused by a combination of a planet's rotation on its axis and atmospheric heating. These form air masses near each other of different temperatures, which cause jet streams. Also, a links that might help you: What causes the jet stream?

4

FlightAware suggests there are no non-stop flights from LA to Delhi. Was certainly going to point out that we're pretty limited on ultra long-haul flights still, and it'd take a very special flight setup to make your scenario. You'd need both: Two locations about 180° longitude apart so that the distance going east vs. the distance going west is ...

4

Keep in mind that your jetstream map is only a schematic description. In reality you will experience shear (strong wind along jet axis, decreasing outwards). Assuming that the airship would drift with the wind, the shear would throw you off the jetstream and you would sooner or later been taken to convergent areas in the upper troposphere (poles, or ...

3

In theory, maybe you could do it, but some of the jet stream loops are even tighter than in Aabaakawad's examples. The wind shear and associated turbulence in and around the jet stream's are so severe that no 'light plane' could survive. Any glider-like aircraft would break up almost immediately.

3

I have gone through both the references - The effect of jet streak curvature on kinematic fields and the background reference in that paper - Isolation of the inertial gravity component in a nonlinear atmospheric model and neither of those references mention any connection to a Cartesian based NS and defining curvature in that coordinate space. To me the ...

3

Based on the state of the discussion in the comments I was not sure whether this has an answer now, but I stumbled over it, and I think I can give a short answer from which one can learn a lot. I'll base my discussion on polar $(\rho,\phi)$ coordinates, for the sake of clarity. In classical mechanics we use for the position vector \vec r= r \vec{e}_r = r \...

3

My answer will only relate to the question in the title: Why is pressure on the poles higher in summer and lower in winter? Due to the axis tilt of the earth the winter hemisphere gets less heated by solar radiation. This leads to a high temperature gradient from equator to the winter hemisphere pole (strong baroclinicity). In contrast we have a low ...

3

I'll look at the climates of the town of Inuvik In Canada's Northwest Territory at a latitude of 68°21′42″N against that of Mexico City at a latitude of 19°26'N. In July, the daily mean temperatures are 14.1°C in Inuvik and 18.2°C in Mexico City — a difference of 4.1°C. Compare that small difference with the daily mean temperatures in January of -26.9°C in ...

1

Because in summer the Arctic and sub-Arctic are quite warm, some days even hot, so winds blowing north from the Med or North Africa will not produce so great a contrast as they would in the winter. The reason for the Arctic's hot summers is that for a while the sun is in the sky 24 hours a day.

1

It can't be done. That is my guess. I think you would need a pressure suit or cabin. I am a former USAF weather officer with flight experience. I have a Masters Degree in Meteorology and a BS in Space Physics. I have working experience in analyzing the jet stream and also using data for vertical airspeed (or omega, as some call it). That does not give me ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# Math Help - Integral 2x^2 ln(x^2)

1. ## Integral 2x^2 ln(x^2)

I got problem to integral ln (x^2)...

As the primitive of In (x) = x(ln (x) -1)
I was wondering... Since ln (x^2) = 2 ln (x), could it work in this way to solve an integral???

The answer is: e^2 + 1

2. Originally Posted by dianjoey
I got problem to integral ln (x^2)...

As the primitive of In (x) = x(ln (x) -1)
I was wondering... Since ln (x^2) = 2 ln (x), could it work in this way to solve an integral???

The answer is: e^2 + 1

$2x\ln(x^2) = 2x \cdot 2\ln{x} = 4x \ln{x}$

$4\int_1^e x\ln{x} \, dx$

integration by parts ...

$u = \ln{x}$

$dv = x \, dx$

proceed ...

3. Hi Skeeter,

But the answer is $e^2+1$...

4. Originally Posted by dianjoey
Hi Skeeter,

But the answer is $e^2+1$...
I agree ... you need to finish it.

5. Skeeter,

I am just re-doing my old exercises, I have all steps to get the answer, but I just don't understand how to integral ln (x^2)

6. I will show...

I provide the answer I want those who is willing to explain can check.

7. Originally Posted by skeeter
$2x\ln(x^2) = 2x \cdot 2\ln{x} = 4x \ln{x}$

$4\int_1^e x\ln{x} \, dx$

integration by parts ...

$u = \ln{x}$

$dv = x \, dx$

proceed ...
Now we have $4\cdot\frac{x^2}{2}\ln(x)-4\int\frac{x^2}{2}\cdot\frac{1}{x}\,dx = 2x^2\ln(x)-4\int\frac{x}{2}\,dx$

Can you finish?

Spoiler:
$=2x^2\ln(x)-x^2 = \left[x^2(2\ln(x)-1)\right]_1^e$ $= e^2(2\ln(e)-1) - 1^2(2\ln(1)-1) = e^2(2-1) - 1(0-1) = e^2+1$, as desired.

8. Thx Redsoxfan325, very clear step and you way of solving the integral is easier to understand.

9. Originally Posted by dianjoey
[snip]
Since ln (x^2) = 2 ln (x),

[snip]
As small point ..... ln(x^2) = 2 ln |x|.

But since you're working with x > 0 there is no difference between ln x and ln |x| in this question.


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MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

the existence of compact Kahler manifolds satisfying some Hodge numbers' restrictions

Given any $n\geq 2$, is there an example of $n$-dimensional compact Kahler manifold such that its Hodge numbers satisfy $h^{1,1} = h^{2,2} < h^{3,3} = h^{4,4} < h^{5,5} = h^{6,6} < \cdots h^{[\frac{n}{2}],[\frac{n}{2}]}$.

Note that 2-plane complex Grassmannian's Hodge numbers satisfy $h^{0,0} = h^{1,1} < h^{2,2} = h^{3,3} < h^{4,4} = h^{5,5} < h^{6,6}=\cdots h^{[\frac{n}{2}],[\frac{n}{2}]}$.

-


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## Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 3 Algebra Ex 3.7

9th Maths Algebra Exercise 3.7 Question 1.
Find the quotient and remainder of the following.
(i) (4x3 + 6x2 – 23x + 18) ÷ (x + 3)
(ii) (8y3 – 16y2 + 16y -15) ÷ (2y – 1)
(iii) (8x3 – 1) ÷ (2x – 1)
(iv) (-18z + 14z2 + 24z3 + 18) ÷ (3z + 4)
Solution:
(i) (4x3 + 6x2 – 23x + 18) ÷ (x + 3)

Quotient = 4x2 – 6x – 5
Remainder = 33

(ii) (8y3 – 16y2 + 16y -15) ÷ (2y – 1)

Quotient = 4y2 – 6y + 5
Remainder = -10

(iii) (8x3 – 1) ÷ (2x – 1)

Quotient = 4x2 + 2x + 1
Remainder = 0

(iv) (-18z + 14z2 + 24z3 + 18) ÷ (3z + 4)

Quotient = 8z2 – 6z + 2
Remainder = 10

9th Maths Exercise 3.7 Question 2.
The area of rectangle is x2 + 7x + 12. If its breadth is (x + 3), then find its length.
Solution:
Area of a rectangle = x2 + 7x + 12
Its breadth = x + 3
x2 + 7x + 12 = (x + 3) × length
∴ Length = (x2 + 7x + 12) ÷ (x + 3)

Quotient = x + 4; Remainder = 0;
∴ Length = x + 4

Exercise 3.7 Class 9 Question 3.
The base of a parallelogram is (5x + 4). Find its height, if the area is 25x2 – 16.
Solution:
The base of a parallelogram is (5x + 4)
Area = 25x2 – 16
Area = b × h = 25x2 – 16
base = 5x + 4

∴ height of the parallelogram = 5x – 4.

9th Standard Maths Exercise 3.7 Question 4.
The sum of (x + 5) observations is (x3 + 125). Find the mean of the observations.
Solution:
The sum of (x + 5) observations is (x3 + 125)

Quotient = x2 – 5x + 25
Remainder = 0
∴ Mean of the observations = x2 – 5x + 25

9th Maths 3.7 Question 5.
Find the quotient and remainder for the following using synthetic division:
(i) (x3 + x2 – 7x – 3) ÷ (x – 3)
(ii) (x3 + 2x2 – x – 4) ÷ (x + 2)
(iii) (3x3 – 2x2 + 7x – 5) ÷ (x + 3)
(iv) (8x4 – 2x2 + 6x + 5) ÷ (4x + 1)
Solution:
(i) (x3 + x2 – 7x – 3) ÷ (x – 3)
Let p (x) = x3 + x2 – 7x – 3
q (x) = x – 3 To find the zero of x – 3
p(x) in standard form ((i.e.) descending order)
x3 + x2 – 7x – 3

Quotient is x2 + 4x + 5
Remainder is 12

(ii) (x3 + 2x2 – x – 4) + (x + 2)
p (x) = x3 + 2x2 – x – 4
Co-efficients are 1 2 -1 – 4
To find zero of x + 2, put x + 2 = 0; x = -2

∴ Quotient is x2 – 1
Remainder is -2

(iii) (3x3 – 2x2 + 7x – 5) ÷ (x + 3)
To find zero of the divisor (x + 3), put x + 3 = 0; ∴ x = – 3
Dividend in Standard form 3x3 – 2x2 + 7x – 5
Co-efficients are 3 -2 7 – 5
Synthetic Division

Quotient is 3x2 – 11x + 40
Remainder is -125

(iv) (8x4 – 2x2 + 6x + 5) ÷ (4x + 1)
To find zero of the divisor 4x + 1, put 4x + 1 = 0 ; 4x = -1; x = $$-\frac{1}{4}$$
Dividend in Standard form 8x4 + 0x3 – 2x2 + 6x + 5
Co-efficients are 8 0 -2 6 5
Synthetic Division

Samacheer Kalvi 9th Maths Chapter 3 Question 6.
If the quotient obtained on dividing (8x4 – 2x2 + 6x – 7) by (2x + 1) is (4x3 + px2 -qx + 3), then find p, q and also the remainder.
Solution:
Let p (x) = 8x4 – 2x2 + 6x – 7
Standard form = 8x4 + 0x3 – 2x2 + 6x – 7

Quotient 4x3 – 2x2 + 3 is compared with the given quotient 4x3 + px2 – qx + 3
Co-efficients of x2 is p = – 2
Co-efficients of x is q = 0
Remainder is – 10

Samacheer Kalvi 9th Maths Book Answers Question 7.
If the quotient obtained on dividing 3x3 + 11x2 + 34x + 106 by x – 3 is 3x2 + ax + b, then find a, b and also the remainder.
Solution:
Let p (x) = 3x3 + 11x2 + 34x + 106
p (x) in standard form
Co-efficients are 3 11 34 106
q(x) = x – 3, its zero x = 3
Synthetic division

Quotient is 3x2 + 20x + 94, it is compared with the given quotient 3x2 + ax + b
Co-efficient of x is a = 20
Constant term is b = 94
Remainder r = 388


text:
Question
Jack has $14. Dylan has$4g more than Jack. Sean has $2g less than Dylan. 1. Find the amount of money that Sean has in terms of g. 2. Find the total amount of money that the three boys have in terms of g. 3. If g = 4, how much does Sean have? 3 m (a) Click button first when a symbol is required. X (b) Click button first when a symbol is required. X (c) Click button first when a symbol is required. X Jack has$14. Dylan has $4g more than Jack. Sean has$2g less than Dylan.
1. Find the amount of money that Sean has in terms of g.
2. Find the total amount of money that the three boys have in terms of g.
3. If g = 4, how much does Sean have?


text:
» » » The intrusive landform produced when magma is cooled within the earth surface is...

# The intrusive landform produced when magma is cooled within the earth surface is...

### Question

The intrusive landform produced when magma is cooled within the earth surface is?

A)
cinder
B)
batholith
C)
composite
D)
geyser


text:
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# Electrical detection of hyperbolic phonon-polaritons in heterostructures of graphene and boron nitride

## Abstract

Light properties in the mid-infrared can be controlled at a deep subwavelength scale using hyperbolic phonons-polaritons of hexagonal boron nitride. While propagating as waveguided modes hyperbolic phonons-polaritons can concentrate the electric field in a chosen nano-volume. Such a behavior is at the heart of many applications including subdiffraction imaging and sensing. Here we employ HPPs in heterostructures of hexagonal boron nitride and graphene as new nano-optoelectronic platform by uniting the benefits of efficient hot-carrier photoconversion in graphene and the hyperbolic nature of hexagonal boron nitride. We demonstrate electrical detection of hyperbolic phonons-polaritons by guiding them towards a graphene pn-junction. We shine a laser beam onto a gap in metal gates underneath the heterostructure, where the light is converted into hyperbolic phonons-polaritons. The hyperbolic phonons-polaritons then propagate as confined rays heating up the graphene leading to a strong photocurrent. This concept is exploited to boost the external responsivity of mid-infrared photodetectors, overcoming the limitation of graphene pn-junction detectors due to their small active area and weak absorption. Moreover this type of detector exhibits tunable frequency selectivity due to the hyperbolic phonons-polaritons, which combined with its high responsivity paves the way for efficient high-resolution mid-infrared imaging.

## Introduction

Hexagonal boron nitride has found multiple uses in van der Waals heterostructures1,2,3,4,5,6,7,8, such as a perfect substrate for graphene,9, 10 a highly uniform tunnel barrier,11, 12 and an environmentally robust protector. In particular, h-BN substrates enable one to achieve high carrier mobility and homogeneity in graphene.9, 10 In addition, h-BN is a natural hyperbolic material as in the two so called reststrahlen bands (760–825 cm−1 and 1360–1610 cm−1) the in plane ($${\epsilon _{{\rm{x,y}}}}$$) and the out of plane ($${\epsilon _{\rm{z}}}$$) permittivity are of opposite sign.1, 3, 4 As a consequence, h-BN supports propagating hyperbolic phonon-polaritons (HPPs) which are electromagnetic modes1, 3 originating in the coupling of photons to optical phonons. Because of their unique physical properties such as long lifetime, tunability,2 slow propagation velocity,8 and strong field confinement the HPPs have a great potential for applications in nanophotonics. The capability to concentrate light into small volumes can also have far-reaching implications for opto-electronic technologies, such as mid-infrared photodetection,13,14,15,16,17,18,19 on-chip spectroscopy and sensing. These concepts, however, remain underexplored.

Here we present a hyperbolic opto-electronic device that takes taking advantage of that fact that h-BN is at the same time an ideal substrate for graphene as well as an excellent waveguide for HPPs. We show how HPPs can be exploited to concentrate the electric field of incident mid-infrared beam towards a graphene pn-junction, where it is converted to a photovoltage. The impact of the HPPs leads to a strongly increased responsivity of the graphene pn-junction in the mid-infrared up to 1 V/W, with zero bias applied.

In previous studies graphene pn-junctions have shown very high internal efficiencies20,21,22,23,24,25 due to the strong photo-thermoelectric effect in graphene. However, the active area of this type of devices is extremely small, leading to poor light collection. For detecting mid-infrared light this issue is even more acute. By exciting hyperbolic phonon polaritons we strongly enhance the effective absorption. We compare our experimental results with FDTD simulations and an analytical model, providing insight into the underlying physical processes and the frequency tunability of our novel mid-infrared detectors.

## Results

The investigated devices consist of heterostructures of monolayer graphene encapsulated in h-BN, obtained by the polymer-free van der Waals assembly technique,10 and placed on top of two metal gates separated by a narrow gap (Fig. 1a). The graphene layer has a mobility of ~30,000 cm2 Vs−1. It is electrically connected to the source and drain electrodes by edge contacts10 (Methods section). An optical micrograph of a typical device is shown in Fig. 1b. The individually tunable carrier density on both sides of the split gate is used to tune the photosensitivity of our device.20,21,22, 24

The operation of our device is as follows. HPPs are launched at the sharp gold edges of the split gate when the laser beam illuminates the sample under normal incidence with the polarization perpendicular to the gap between the gate electrodes.3, 8, 26 While the HPPs propagate as highly directional rays in both bottom and top h-BN slabs of the stack, they are absorbed when they pass through graphene (Fig. 1c) creating hot carriers. The hot carriers diffuse over a length scale of the electron cooling length (about 0.5–1 μm) and generate a temperature increase peaking at the graphene junction defined by the position of the gap in the metal gates. This inhomogeneous temperature distribution induces a photovoltage due to the Seebeck effect. Thus, all HPPs absorbed within approximately one cooling length from the junction contribute to the photovoltage. Similar HPPs lauching presumably also occurs at the source and drain gold contacts. However, they do not contribute to the photovoltage because the electrodes are situated much further than the electron cooling length from the junction.

The measured spatially resolved photoresponse of the device is shown in Fig. 1d. The photoresponse arises mainly at the junction (shown as a dashed line). In such a graphene junction the photovoltage V ph is generated by the photo-thermoelectric effect:20,21,22, 24, 27

$${V_{{\rm{ph}}}} = \Delta S\Delta T$$
(1)

where, ΔS = S 1 − S 2 is the difference between the Seebeck coefficients of graphene on the left and right side of the junction and ΔT is the difference in electronic temperature at the junction and at the source/drain contacts. The photo-thermoelectric effect dominates over other possible mechanisms of photovoltage generation due to the high Seebeck coefficient of graphene (S ~ 100 μV K−1), which is in-situ tunable by gating.21, 28 By controlling the gate voltage on the two sides of the junction individually and recording the photocurrent we measure a 6-fold pattern, a clear sign that the photocurrent in our device is governed by the photo-thermoelectric effect20,21,22, 24, 27 (Supplementary Information). The highest responsivity measured when changing the gate voltages is obtained for V g1 = 1.2 V and V g2 = −0.21 V. This corresponds to a pn-configuration with a fairly low doping level of about 0.06 eV (Supplementary Information).

The spectral responsivity (Fig. 2a) is obtained by recording the photovoltage while tuning the wavelength of the quantum cascade laser source from 1000 to 1610 cm−1. A strong photocurrent enhancement for the polarization perpendicular to the gap, peaking at 1515 cm−1, is observed. The peak around 1100 cm−1 is related with the SiO2 surface phonon of the underlying substrate.27

In order to understand the observed behavior we use finite difference time domain (FDTD) simulations to model the scattering process of far-field light into HPPs and the subsequent HPP waveguiding and absorption of the HPPs in the graphene (solid lines in Fig. 2a). A good match with the experimentally observed spectral response is obtained (points in Fig. 2a). The simulated absorption spectrum shows a peak inside the reststrahlen band of h-BN. The spatial distribution of the electric field inside the h-BN layers is shown in Fig. 2d. It is dominated by four rays which are launched at the edges of the split gate and undergo multiple reflections from the top and bottom surfaces. The rays maintain a fixed angle with the c axis. This angle is related to the anisotropy of the permittivity via the analytical formula $${\rm{tan}}\,\theta (\omega ) = i\sqrt {{\epsilon _{{\rm{x,y}}}}(\omega )} {\rm{/}}\sqrt {{\epsilon _{\rm{z}}}(\omega )}$$.1, 3 It predicts that |θ| changes from π/2 to 0 as ω varies across the reststrahlen band. The unusual ray pattern of HPP emission in turn affects the spatial absorption pattern in graphene, which is shown in the simulated spectral-spatial pattern of Fig. 2b. This pattern is dominated by the four families of “hot spots” that correspond to the four HPP rays seen in Fig. 2d. The separation of hot spots within each family is 2d|tan θ|, where d = d t + d b is the total thickness of the h-BN layers (Fig. 1c).

To investigate further the origin of the observed spectral peaks in the photocurrent we carried out scanning near-field photocurrent mapping of our devices.29, 30 In this technique a metallized atomic force microscopy tip is illuminated with an infrared laser and a near-field is generated at the apex of the tip. This enables us to measure the photocurrent with a spatial resolution greatly exceeding the diffraction limit of light. The representative results are shown in Fig. 2c. The device region measured includes the gap of the split gate and one graphene edge localized at the top of the frame. The obtained photocurrent map reveals two series of sinusoidal spatial oscillations (fringes) rather than sharply peaked hot spots seen in Fig. 2b. These smooth oscillations can be explained if we recall that in an h-BN slab of small enough thickness d the HPPs are quantized into discrete eigenmodes with in-plane momenta kl = tan θ(πl + ϕ)/d where l = 0, 1, 2, … is the mode index and ϕ ~ 1 is a phase shift that depends on the boundary conditions (Fig. 3b and e.g., ref. 31). The collimated rays seen in Fig. 2d can be understood as coherent superpositions of many such modes emitted by the split gate. On the other hand, in the photocurrent microscopy the role of the HPP emitter is played by an AFM tip, which apparently couples predominantly to the l = 0 mode.32 The horizontal fringes in Fig. 2c are due to interference of l = 0 polariton waves launched by the tip, which is backreflected at the graphene edge leading to a fringe spacing corresponding to half the wavelength λ p = 2π/k 0 of this mode.32 The vertical fringes are due to interference of the l = 0 partial wave launched at the split gate8 with the tip launched waves. In this case the fringe spacing is λ p. We do not observe HPPs launched by the tip and reflected by the gap as there are no vertical fringes with half the wavelength visible. This interpretation enables us to extract λ p from the fringe spacing in the photocurrent maps. For example, at 1428 cm−1 is λ p = 460 ± 5 nm, which agrees with the calculated wavelength of 455 nm. The observed fringes parallel to the gap on the left of Fig. 2c confirm that phonons are indeed launched by the split gate and are converted into photocurrent.

In order to better understand which parameters determine the absorption spectrum we also modeled the system analytically.31 In this model we approximate the electric fields at the bottom surface of the h-BN by the electric field inside the gap −a < x < a cut along the y axis in a perfectly conducting plane z = 0 in vacuum:

$${E_x}(x,z = 0) = \frac{{{V_0}}}{\pi }{\rm{Re}}\frac{1}{{\sqrt {{a^2} - {x^2}} }}{\kern 1pt} .$$
(2)

Here V 0 is the voltage across the gap, which is proportional to the field of the incident beam (see inset Fig. 3a).

The Fourier transform of E x is given by (Fig. 3a)

$${\tilde E_x}(k) = {V_0}{J_0}(ka),$$
(3)

where J 0(z) is the Bessel function of the first kind.

We then compute the field inside the h-BN-graphene layered structure using the transfer matrix method (Fig. 3b) by assuming that Eq. (3) represents the field incident on the structure from the bottom. The assumption is not strictly self-consistent because it does not account for the backreaction of h-BN on the split gate, in the form of the HPP rays reflected back to z = 0 plane. A more accurate but also more complicated model that obeys the self-consistency condition is presented in Supplementary. Unfortunately, that latter model can no longer can be solved in a closed form. This is why here we use the simplified analytical model to illustrate the main features of the studied phenomena. We calculate the Fourier transform of the in-plane electric field at the graphene surface as a function of momentum k. The power absorbed in the graphene is then expressed as (Fig. 3d):

$$p = \frac{1}{{4\pi }}{\rm{Re}}\,\sigma (\omega ){\int} {{{\left| {{{\tilde E}_x}(k)} \right|}^2}} dk{\kern 1pt} ,$$
(4)

where σ(ω) is the sheet conductivity of graphene at the laser frequency ω. Here, for simplicity, we neglect the spatial variation of σ near the pn-junction as the hot spots responsible for the absorption are typically found some distance away from the junction (Fig. 3c).

From this model description it becomes clear that the characteristic momentum k ~ 1/a provided by the junction plays a crucial role for the frequency of maximum absorption. By calculating the inverse Fourier transform of $${\tilde E_x}(k)$$ we are able to also calculate the spatial profile of the electric field E x (x) and thus the spatial absorption profile (Fig. 3c). The validity of our analytic model can be seen by the close resemblance between the analytically calculated and FDTD simulated frequency dependent absorption profile (compare Figs. 2b and 3c).

From this model, the origin of the peak in the spectral photoresponse is the competition between the following two processes: the dielectric losses in the h-BN and the (finite) momentum provided by the junction. First, the losses in the h-BN contribute mainly to the low frequency side due to the imaginary part of the permittivity which peaks at the TO phonon frequency (1360 cm−1). The impact of this effect on the device responsivity is enhanced by the obtuse angle with which the HPPs are launched, as the intensity of the HPPs reaching the graphene becomes smaller with traveled distance. Second, the momentum provided by the junction is responsible for the responsivity decay on the high frequency side. Interestingly both of these effects depend on the h-BN thickness and on the gap size. It is important to note that the impact of the h-BN thickness is twofold since it is also changing the HPPs dispersion.1 Thus by choosing the geometrical parameters of the device, the device thickness and gap width, it is possible to tune the frequency as well as amplitude of the photocurrent maximum within the reststrahlen band of h-BN.

In order to show this tunability, and to validate the physical model, we fabricated different device geometries. Experimental responsivity spectra of the different devices are plotted in Fig. 4a. All the spectra were measured using the gate voltage configuration exhibiting the highest responsivity for the respective device. They exhibit different peak frequencies and responsivities and the trend is well captured by the analytically calculated absorption spectra presented in Fig. 4c. The peak frequencies are plotted in Fig. 4b as a function of the relevant geometrical parameters of the system. These are the stack thickness d = d t + d b, where d t and d b are the bottom and top h-BN thicknesses (Fig. 1c), and the split gate gap width 2a. The tunability of the investigated devices spans over 60 cm−1 and the peak frequencies obtained using both the FDTD simulations and the analytic model match the experimental ones. In Fig. 4d we plot the responsivities of the measured devices normalized to the highest one as a function of the peak frequencies. We find that the responsivity follows a bell shaped curve (Fig. 4d) suggesting that the optimal geometry would lead to a peak frequency where there is a trade off between low losses and high launching efficiency. Using the analytic model we obtain a theoretical dependence of the frequency and of the absorbed power as a function of the stack thickness and the gap size (Supplementary Information).

In this simple analytic model the frequency dependence of the gap voltage V 0 (Eq. (2)) is neglected. Thus, the coupling between the far-field light and the split gate is not taken fully into account.33 This leads to some discrepancy between simple theory and experiment. However, the mentioned above more sophisticated model based on the Galerkin method (Supplementary Information) is in much better agreement with the experimental results and FDTD simulations (Fig. 4b).

## Discussion

Finally, we will address the photodetection device performance. We remark that the device operates at zero bias, leading to an extremely low noise level (~4 nV/$$\sqrt {{\rm{Hz}}}$$) from which we estimate a noise equivalent power (NEP) of 26 pW/$$\sqrt {{\rm{Hz}}}$$ (Methods section). From our simulations we found that the active area is about 2.5 μm2, i.e., only 2.5% of the device area. Thus, the device can be easily scaled to smaller dimensions, with the potential to enhance the performance by another factor of 40 because the total device resistance would be decreased and thus the Johnson-Nyquist noise would decrease as well leading to a lower NEP. Current state-of-the art detectors based on other technologies are described in refs 34, 35. At room temperature typically silicon bolometers are used. Our detectors can be further optimized to have similar detectivity as silicon bolometers, but offer several distinct advantages: it allows a smaller pixel size, higher operation speed and simpler fabrication as no suspension of the device is necessary.

Our novel nano-optoelectronic infrared detectors operate at room temperature, are highly efficient, and can be used for a wide range of on-chip sensing applications.

## Methods

### Sample fabrication

All the stack elements (top and bottom h-BN and graphene) are mechanically cleaved and exfoliated onto freshly cleaned Si/SiO2 substrates. First the selected top h-BN is detached from the substrate using a PPC (poly-propylene carbonate) film and is then used to lift by Van der Waals forces the graphene and the bottom h-BN consecutively. The as-completed stack is released onto the split gate. The split gate electrodes are prepared by lithography, titanium (5 nm)/gold (30 nm) evaporation and focus ion beam irradiation to create the gap. The source and drain electrodes mask is designed in a AZ-5214 photoresist film by laser lithography and is exposed to a plasma of CHF3/O2 gases to partially etch the stack. The graphene is finally contacted by the edges by evaporating titanium (2 nm)/gold (30 nm) and lift off in acetone. The recipe used for making those contacts is detailed in ref. 10.

### Measurements

The device is illuminated by a linearly polarized quantum cascade laser with a frequency tunable from 1000 to 1610 cm−1. The device position is scanned using a motorized xyz-stage. The laser is modulated at 128 Hz using a chopper and the current at the junction is measured using a current pre-amplifier and lock-in amplifier. The polarization of the light is controlled using a ZnSe wire grid polarizer. The light is focused using ZnSe lenses with a numerical aperture of ~0.5. The power for each frequency is measured using a thermal power meter and the photocurrent spectra are normalized by this power to calculate the responsivity.

### Noise equivalent power estimation

We calculate a noise equivalent power (NEP) given by NEP = S noise/R internal = 26pW/$$\sqrt {{\rm{Hz}}}$$ where S noise is the voltage noise and R internal is the internal responsivity. Because graphene pn-junction photodetectors operate at zero bias the electrical noise is of thermal Johnson-Nyquist type given by $${S_{{\rm{noise}}}} = \sqrt {4{k_{\rm{B}}}TR} = 4{\kern 1pt} {\rm{nV/}}\sqrt {{\rm{Hz}}}$$. Where k B is the Boltzmann constant, T = 300 K and R = 1 kΩ is the resistance for which the calculated NEP is minimum corresponding to a carrier concentration of n = 0.2 × 1012 cm−2. The internal responsivity is given by R internal = R external/η = 150 V/W where R external = 1 V/W is the experimental responsivity and η = A abs/A spot = 0.5% the percentage of absorbed light. A spot = 491 μm2 is the laser spot area and A abs = σW = 2.5 μm2 is the active area where σ = 250 nm is the absorption cross section obtained by FDTD simulations and W = 10 μm is the width of the device.

### Simulations

The full wave simulations were performed using Lumerical FDTD. The frequency dependent permittivity of the h-BN was taken from ref. 1. The optical conductivity of the graphene was calculated using the local random phase approximation at T = 300 K with a scattering time of 500 fs. For each device the appropriate Fermi energy was simulated (Supplementary Information), however this did not influence the results significantly. In the simulations the Fermi energy of the graphene is spatially constant (see the comment after Eq. (4)) but frequency dependent. A plane wave source was used and the absorption cross section was calculated by normalizing to the incident power. For simplicity the calculated absorption does not take into account the cooling length of the graphene nor the carrier density profile.

### Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

It is a great pleasure to thank Klaas-Jan Tielrooij for many fruitful discussions. This work used open source software (www.matplotlib.org, www.python.org, www.povray.org). F.H.L.K. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0522), support by Fundacio Cellex Barcelona, the Mineco grants Ramón y Cajal (RYC-2012-12281) and Plan Nacional (FIS2013-47161-P and FIS2014-59639-JIN), and support from the Government of Catalonia trough the SGR grant (2014-SGR-1535). Furthermore, the research leading to these results has received funding from the European Union Seventh Framework Programme under grant agreement no.696656 Graphene Flagship, and the ERC starting grant (307806, CarbonLight). Y.G. and J.H. acknowledge support from the US Office of Naval Research N00014-13-1-0662. P.A.-G. acknowledges funding from the Spanish Ministry of Economy and Competitiveness through the national projects FIS2014-60195-JIN.

## Author information

Authors

### Contributions

A.W. and F.H.L.K. conceived the experiment. A.W. and R.P. performed the far-field experiments, analyzed the data and wrote the manuscript. A.W. performed the FDTD simulations. A.W. and P.A.-G. performed the near-field experiments. D.D. and Y.G. fabricated the devices. M.B.L. and S.N. assisted with measurements, interpretation and discussion of the results. J.-S.W. and M.M.F. developed the analytic model. K.W. and T.T. synthesized the h-BN. J.H., R.H. and F.H.L.K. supervised the work and discussed the results. All authors contributed to the scientific discussion and manuscript revisions.

### Corresponding author

Correspondence to Frank H. L. Koppens.

## Ethics declarations

### Competing interests

R.H. is co-founder of Neaspec GmbH, a company producing scattering-type scanning near-field optical microscope systems such as the ones used in this study. All other authors declare no competing financial interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Woessner, A., Parret, R., Davydovskaya, D. et al. Electrical detection of hyperbolic phonon-polaritons in heterostructures of graphene and boron nitride. npj 2D Mater Appl 1, 25 (2017). https://doi.org/10.1038/s41699-017-0031-5

• Revised:

• Accepted:

• Published:

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text:
# Cardinality of Image of Mapping not greater than Cardinality of Domain

## Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Let $\card S$ denote the cardinal number of $S$.

Let $S \sim \card S$.

Then:

$\card {\Img f} \le \card S$

## Proof

$f: S \to \Img f$

is a surjection.

Let $h$ be a mapping such that:

$h: \card S \to S$

is a bijection.

$f \circ h: \card S \to \Img f$

is a surjection.

Construct a set $R$ such that:

$R = \set {x \in \card S: \forall y \in x: \map f {\map h x} \ne \map f {\map h y} }$

It follows that:

$R \subseteq \card S$
$\card R \le \card S$

Suppose:

$x \in R$
$y \in R$
$\map f {\map h x} = \map f {\map h y}$

Then, $x$ and $y$ are ordinals and by Ordinal Membership is Trichotomy:

$x < y \lor x = y \lor y < x$

If $x < y$, then:

$\map f {\map h x} \ne \map f {\map h y}$ by the definition of $R$.

Similarly, $y < x$ implies that:

$\map f {\map h x} \ne \map f {\map h y}$

Therefore, $x = y$.

It follows that the restriction $f \circ h \restriction_R : R \to \Img f$ is an injection.

$\Box$

Finally, by the definition of surjection, for all $x \in \Img f$, there is some $y \in \card S$ such that:

$\map f {\map h y} = x$

But since this is true for some $y$, the set:

$\set {\map y \in \card S: \map f {\map h y} = x}$

has a minimal element.

For this minimal element $y$, it follows that:

$\forall z \in y: \map f {\map h z} \ne \map f {\map h y}$

since if it were equal, this would contradict the fact that $y$ is a $\in$-minimal element.

It follows that the restriction $f \circ h \restriction_R : R \to \Img f$ is a bijection.

By the definition of set equivalence, $R \sim \Img f$.

So:

 $\displaystyle \Img f$ $=$ $\displaystyle \card R$ $\quad$ Equivalent Sets have Equal Cardinal Numbers $\quad$ $\displaystyle$ $\le$ $\displaystyle \card S$ $\quad$ Subset implies Cardinal Inequality $\quad$

$\blacksquare$


text:
# Toronto Math Forum

## APM346-2012 => APM346 Math => Final Exam => Topic started by: Djirar on December 20, 2012, 01:30:05 PM

Title: Problem 1
Post by: Djirar on December 20, 2012, 01:30:05 PM
Solve the first-order equation $2 u_t + 3 u_x =0$ with the auxiliary condition $u = \sin x$ when $t=0$.

Title: Re: Problem 1
Post by: Chen Ge Qu on December 20, 2012, 01:35:12 PM
I thought we were supposed to wait until Prof. Ivrii posted the problems...?

In any case, my solution to Problem 1 is attached.
Title: Re: Problem 1
Post by: Ian Kivlichan on December 20, 2012, 02:20:06 PM
I'm adding my solution.. the others do not mention why the solution works for all of $(x, t)$ (though it does not change the final result).
Title: Re: Problem 1
Post by: Pei Zhou on December 20, 2012, 04:49:02 PM
$$\frac{dx}{3}=\frac{dt}{2}\implies x-\frac{3}{2}t =C \implies u=f(x-\frac{3}{2}t)$$
Step 2 Initial condition: $u(x,0)=f(x)=\sin(x)$ and therefore
$$\boxed{u(x,t)=\sin(x-\frac{3}{2}t)}.$$
Several students put the wrong sign $u=\sin(x+\frac{3}{2}t)$, several  made mistakes on Step 2 and got marks halved. Few made really grave mistakes like trying method of separation, but majority did well and got all 20 (correct but ugly solutions/answers are not punished).


text:
# Does there exist a supersmooth non-polynomial function?

Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ almost everywhere. Similarly, let's call $f:\mathbb{R}\rightarrow\mathbb{R}$ Baire supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ except on a meager set.

• Is every Lebesgue supersmooth function a polynomial? $\mathbf{No}$. Answered by Pietro Majer in the comments by referring to this question.

• Is every Baire supersmooth function a polynomial? $\mathbf{No}$. Answered by Pietro Majer in the comments by referring to this question.

• Do we get the same solutions if we replace "meager" and "measure zero" with "countable"?

• What happens in higher dimensions?

I am asking this question mainly out of curiosity and because it seems like such a function would be a good example to have in mind if one exists.

• @AlexandreEremenko: I don't think it's that obvious, because we get to choose $x$ (or a conull/comeager set of $x$) depending on the sequence $a_n$ – Nate Eldredge Oct 2 '14 at 3:20
• @Noam D. Elkies: How single $x_0$ and $a_n=1$ implies that $f$ is analytic? – Alexandre Eremenko Oct 2 '14 at 3:30
• As to the first two questions: no: see the examples here, where the exceptional set is the Cantor set mathoverflow.net/questions/94038/… – Pietro Majer Oct 2 '14 at 6:47
• @Joseph: a further question would be: is there a Lebesgue (resp., Baire) supersmooth function which is not locally polynomial a.e. (resp., on a residual set)? – Pietro Majer Oct 2 '14 at 7:30
• @Pietro Majer. I asked the further question you suggested here mathoverflow.net/q/182423/22277. – Joseph Van Name Oct 3 '14 at 0:13


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Journal article

### A classical model for derived critical loci

Abstract:

Let $f:U\to{\mathbb A}^1$ be a regular function on a smooth scheme $U$ over a field $\mathbb K$. Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the "derived critical locus" Crit$(f)$, an example of a new class of spaces in derived algebraic geometry, which they call "$-1$-shifted symplectic derived schemes". They show that intersections of algebraic Lagrangians in a smooth symplectic $\mathbb K$-scheme, and stable moduli schemes of coherent sheaves on a Calabi-Ya...

Publication status:
Published
Peer review status:
Peer reviewed
Version:
Accepted Manuscript

### Authors

More by this author
Institution:
University of Oxford
Department:
Oxford, MPLS, Mathematical Inst
Publisher:
International Press Publisher's website
Journal:
Journal of Differential Geometry Journal website
Publication date:
2013-04-16
EISSN:
1945-743X
ISSN:
0022-040X
URN:
uuid:267b8993-772c-46b0-87de-81cc8a736cd2
Source identifiers:
396010
Local pid:
pubs:396010
Keywords: