text:
I am about a month into learning Java and I'm just looking for some comments on my code. Specifically, I would like to know if I'm being as efficient as I can be. This is the most complex thing I've come up with so far (that works) and I wanted some feedback on my developing coding style.

public static void main(String[] args) {
Scanner sc = new Scanner(System.in);
double numOne, numTwo, numThree, negativeB,
discriminant, divisor, sqRtDisc, dividendPlus,
dividendMinus;

System.out.println("Enter the value for A: ");
numOne = sc.nextFloat();

System.out.println("Enter the value for B: ");
numTwo = sc.nextFloat();

System.out.println("Enter the value for C: ");
numThree = sc.nextFloat();

negativeB = -numTwo;
discriminant = (numTwo * numTwo) - 4 * numOne * numThree;
divisor = 2 * numOne;
sqRtDisc = Math.sqrt(discriminant);

System.out.println(" ");

if (sqRtDisc > 0) {
dividendPlus = ((negativeB + sqRtDisc) / divisor);
dividendMinus = ((negativeB - sqRtDisc) / divisor);

System.out.println("x = " + dividendPlus);
System.out.println("x = " + dividendMinus);
} else {
System.out.println("x = " + (negativeB / divisor) + " + "
+ (Math.sqrt(-discriminant) / divisor) + "i");
System.out.println("x = " + (negativeB / divisor) + " - "
+ (Math.sqrt(-discriminant) / divisor) + "i");
System.out.println(" ");
}

}

• Thanks for the help, guys. I made a new version of it using what you recommended and I was surprised that I was able to cut it down to 31 lines of code from 43. Works perfectly! – PALADIN 458S Jan 25 '15 at 21:16

## Bugs

You are losing precision by calling sc.nextFloat() instead of sc.nextDouble().

If the equation has complex roots, Math.sqrt(discriminant) will crash.

## Style

You are using too many variables to easily keep track of them mentally. Also, it's better to avoid a giant block of declarations like your

double numOne, numTwo, numThree, negativeB,
discriminant, divisor, sqRtDisc, dividendPlus,
dividendMinus;

We know that we need to store at least a, b, and c, so let's start there. Why not use the conventional terminology instead of numOne, numTwo, numThree?

public static void main(String[] args) {
Scanner sc = new Scanner(System.in);

System.out.println("Enter the value for A: ");
double a = sc.nextDouble();

System.out.println("Enter the value for B: ");
double b = sc.nextDouble();

System.out.println("Enter the value for C: ");
double c = sc.nextDouble();

from there, the next important calculation is the discriminant, and optionally, its square root and the divisor.

    double discriminant = (b * b) - 4 * a * c;
double divisor = 2 * a;

From there, you could go straight to formatting. I suggest System.out.printf() for readability.

    if (discriminant < 0) {         // Complex roots
System.out.printf("x = %d + %d i\n",
-b / divisor, Math.sqrt(-discriminant) / divisor));
System.out.printf("x = %d - %d i\n",
-b / divisor, Math.sqrt(-discriminant) / divisor));
} else {
System.out.println("x = " + (-b + Math.sqrt(discriminant)) / divisor);
System.out.println("x = " + (-b - Math.sqrt(discriminant)) / divisor);
}
}

• One important note not explicitly called out is that you fixed the logic bug whereby the OP was testing if sqRtDisc was greater than 0 (which I would expect it always would be). You have correctly tested discriminant for being positive or negative. – Chris Jan 25 '15 at 18:37
• @Chris As I wrote, "If the equation has complex roots, Math.sqrt(discriminant) will crash." – 200_success Jan 25 '15 at 18:53
• @200_success: Indeed. I actually missed that and edited my comment to take that out (not sure if you saw it pre-edit). My point was a slightly different bug that sqRtDisc would always be positive if it didn't crash because it is the wrong variable to check. Your code does implicitly fix this bug but I thought worth pointing out to the OP in case he hadn't noticed the original bug. Its only a minor point though. – Chris Jan 25 '15 at 19:00

This is too much work for main(), and in Java, it's good to make code more reusable by utilizing methods and classes. Instead, have main() take all the input, then pass them to one or more methods that perform the calculations.

Your main() can look roughly like this:

public static void main(String[] args) {
Scanner sc = new Scanner(System.in);

try {
System.out.println("Enter the value for A: ");
double numOne = sc.nextDouble();

System.out.println("Enter the value for B: ");
double numTwo = sc.nextDouble();

System.out.println("Enter the value for C: ");
double numThree = sc.nextDouble();
} catch (InputMismatchException e) {
System.err.println("Invalid input given");
// handle error here
}

double result = calculate(numOne, numTwo, numThree);

sc.close();
}

You may of course add additional methods or use different names, but this should give you a general idea of how main() should look. Also notice that I've moved around the input variables. It is best to keep variables as close in scope as possible so that it'll be easier to maintain them over time.

I've also added some exception-handling for the input. You can read more about nextDouble()'s functionality in the docs. I've already provided an incomplete example of this, using the most likely exception that it may throw here. You shouldn't allow the program to continue to the calculations until valid input is given. You can either ask the user again for input, or just terminate the program.

Lastly, I've called close() to prevent leaking of resources from the Scanner object. You should always close a Scanner after its use, unless you use try-with-resources, available since Java 7.

• If I put the calculations into another method, do I have to do anything special to call up the calculations into the answer? If a class, do I have to start out with the "extends" command? – PALADIN 458S Jan 25 '15 at 7:39
• @PALADIN458S: You just have to give the methods the input and have them return the final value. You may just need one method for this. In a class, you wouldn't need extends here. That's just needed if you're utilizing inheritance, but you just need helper methods. – Jamal Jan 25 '15 at 7:41


text:
## Electronic Communications in Probability

### Asymptotic Distribution of Coordinates on High Dimensional Spheres

Marcus Spruill

#### Abstract

The coordinates $x_i$ of a point $x = (x_1, x_2, \dots, x_n)$ chosen at random according to a uniform distribution on the $\ell_2(n)$-sphere of radius $n^{1/2}$ have approximately a normal distribution when $n$ is large. The coordinates $x_i$ of points uniformly distributed on the $\ell_1(n)$-sphere of radius $n$ have approximately a double exponential distribution. In these and all the $\ell_p(n),1 \le p \le \infty,$ convergence of the distribution of coordinates as the dimension $n$ increases is at the rate $\sqrt{n}$ and is described precisely in terms of weak convergence of a normalized empirical process to a limiting Gaussian process, the sum of a Brownian bridge and a simple normal process.

#### Article information

Source
Electron. Commun. Probab. Volume 12 (2007), paper no. 23, 234-247.

Dates
Accepted: 15 August 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
http://projecteuclid.org/euclid.ecp/1465224966

Digital Object Identifier
doi:10.1214/ECP.v12-1294

Mathematical Reviews number (MathSciNet)
MR2335894

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

Spruill, Marcus. Asymptotic Distribution of Coordinates on High Dimensional Spheres. Electron. Commun. Probab. 12 (2007), paper no. 23, 234--247. doi:10.1214/ECP.v12-1294. http://projecteuclid.org/euclid.ecp/1465224966.

#### References

• Andrews, George E.; Askey, Richard; Roy, Ranjan. Special functions. Encyclopedia of\n Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. xvi+664\n pp. ISBN: 0-521-62321-9; 0-521-78988-5
• Barthe, Franck; Guédon, Olivier; Mendelson, Shahar; Naor, Assaf. A probabilistic\n approach to the geometry of the $l^n_p$-ball. Ann. Probab. 33 (2005), no. 2, 480–513.\n
• Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons,\n Inc., New York-London-Sydney 1968 xii+253 pp.
• Borel,E. Introduction géométrique á quelques théories physiques. Gauthier-Villars,\n Paris, 1914.
• Borwein, D.; Borwein, J.; Fee, G.; Girgensohn, R. Refined convexity and special cases\n of the Blaschke-Santalo inequality. Math. Inequal. Appl. 4 (2001), no. 4, 631–638.\n
• Busemann, Herbert. Intrinsic area. Ann. of Math. (2) 48, (1947). 234–267. \n
• Busemann, Herbert. The isoperimetric problem for Minkowski area. Amer. J. Math. 71,\n (1949). 743–762.
• Busemann, Herbert. A theorem on convex bodies of the Brunn-Minkowski type. Proc. Nat.\n Acad. Sci. U. S. A. 35, (1949). 27–31.
• Diaconis, Persi; Freedman, David. A dozen de Finetti-style results in search of a\n theory. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 397–423.\n
• Friedman, Avner. Foundations of modern analysis. Holt, Rinehart and Winston, Inc., New\n York-Montreal, Que.-London 1970 vi+250 pp.
• Gardner, R. J. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39\n (2002), no. 3, 355–405 (electronic).
• Poincaré,Henri. Calcul des probabilitiés. Gauthier-Villars, Paris, 1912.
• Rachev, S. T.; Rüschendorf, L. Approximate independence of distributions on spheres\n and their stability properties. Ann. Probab. 19 (1991), no. 3, 1311–1337. \n
• Schechtman, G.; Zinn, J. On the volume of the intersection of two $Lsp nsb p$ balls.\n Proc. Amer. Math. Soc. 110 (1990), no. 1, 217–224.
• Serfling, Robert J. Approximation theorems of mathematical statistics. Wiley Series in\n Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1980.\n xiv+371 pp. ISBN: 0-471-02403-1
• Stam, A. J. Limit theorems for uniform distributions on spheres in high-dimensional\n Euclidean spaces. J. Appl. Probab. 19 (1982), no. 1, 221–228. \n

format: plain


text:
# Schrödinger equation

The Schrödinger equation is an equation that is fundamental to quantum theory. The time independent Schrödinger equation looks like

${\displaystyle {\hat {H}}\psi =E\psi .}$

The ${\displaystyle {\hat {H}}}$ is called the Hamiltonian Hamiltonian mechanics. ${\displaystyle \psi \,\;}$ denote the wavefunctions.

## How to construct the Schrödinger equation for a system

The operator ${\displaystyle {\hat {H}}}$  is called the Hamiltonian. It contains two parts:

1. The kinetic energy, denoted by the operator ${\displaystyle {\hat {T}}}$
2. The potential energy, denoted by the operator ${\displaystyle {\hat {U}}}$

These two operators represent the principal types of energies in any physical system. Putting these two parts together, we can write this as ${\displaystyle {\hat {H}}={\hat {T}}+{\hat {U}}}$ . We can think of the Hamiltonian as a mathematical object that encodes how the energies in a system can be distributed.

### Kinetic energy operator

In classical mechanics, we know that the kinetic energy of a moving particle is given by

${\displaystyle E={\frac {1}{2}}mv^{2}={\frac {1}{2}}m{\vec {v}}\cdot {\vec {v}}}$

in which ${\displaystyle m}$  corresponds to the mass of the particle, and ${\displaystyle {\vec {v}}}$  is its velocity. Noting that classically the linear momentum ${\displaystyle {\vec {p}}}$  is given by ${\displaystyle {\vec {p}}=m{\vec {v}}}$ , so the kinetic energy can also be written as

${\displaystyle E={\frac {p^{2}}{2m}}}$ .

In order to get the quantum mechanical version of this, we substitute all occurrences of the momentum ${\displaystyle {\vec {p}}}$  with

${\displaystyle {\vec {p}}\to -i\hbar \nabla }$ ,

where ${\displaystyle \nabla }$  is the vector analogue of the partial differential operator ${\displaystyle {\frac {\partial }{\partial x}}}$ .

Hence, the kinetic energy operator is given by

${\displaystyle {\hat {T}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}}$ .

### Total energy operator and the time dependent Schrödinger equation

The total energy operator introduces the time propagation to the wavefunctions so the time dependent Schrödinger equation tells how the quantum system developes is time while ist was found is some state at the original time. It is basically constructed from the condition that its operation on time dependent oscillatory factors characteristic to the plane waves obtained from the classical wave equation describing the mechanical waves like sound or the waves on the water surface should produce the energy quanta ${\displaystyle \hbar \omega }$  as the eigenvalues from the eigen-equation namely

${\displaystyle {\hat {E}}_{TOT}e^{-i\omega t}=\hbar \omega e^{-i\omega t}}$

One therefore may readily guess it as

${\displaystyle {\hat {E}}_{TOT}=i\hbar {\frac {\partial }{\partial t}}}$

While te total energy in the classical mechanics is normally equal to the Hamiltonian the time dependent Schrödinger equation is obtained by equalizing the two:

${\displaystyle {\hat {H}}\psi =i\hbar {\dot {\psi }}.}$

## How to solve the Schrödinger equation

### Example 1: The ground state of the hydrogen atom

The time independent Schrödinger equation for the electron in the Coulomb potential in the spherical coordinates takes the form:

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\left[{1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2} \over \partial \varphi ^{2}}\right]\psi (r,\theta ,\varphi )-{\frac {e^{2}}{4\pi \epsilon _{0}r}}\psi (r,\theta ,\varphi )=E\psi (r,\theta ,\varphi ).}$

For the ground state which is the state with the lowest possible ${\displaystyle E}$  we seek the solution in the simplest possible form not dependent on the spherical angles ${\displaystyle \theta }$  and ${\displaystyle \varphi }$ . The Schrödinger equation greatly simplifies to the second order ordinary partial differential equation of the one radial variable:

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\left[{1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial \over \partial r}\right)\right]\psi _{0}(r)-{\frac {e^{2}}{4\pi \epsilon _{0}r}}\psi _{0}(r)=E_{0}\psi _{0}(r).}$

The part of it contains derivatives multiplied by the various powers of the ${\displaystyle r}$  variable and we guess the solution in the exponential form since the exponential function is the eigenfunction of the differentiation operator while the derivative of the exponential function is proportional to it:

${\displaystyle \psi _{0}(r)=Ne^{-ar}}$

Substituting it to the Schrödinger equation we obtain the following equation:

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\left[-{\frac {2a}{r}}+a^{2}\right]\psi _{0}(r)-{\frac {e^{2}}{4\pi \epsilon _{0}r}}\psi _{0}(r)=E_{0}\psi _{0}(r)}$

The only way this equation is valid for all values of ${\displaystyle r}$  is that the coefficient multiplying the ${\displaystyle 1/r}$  is equal to ${\displaystyle 0}$ :

${\displaystyle {\frac {a\hbar ^{2}}{mr}}={\frac {e^{2}}{4\pi \epsilon _{0}r}}}$

that gives the value of the Bohr radius ${\displaystyle a_{0}}$  (the range how the ground state mostly extents radially):

${\displaystyle a_{0}=1/a={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{me^{2}}}}$

and the ground state energy

${\displaystyle E_{0}=-{\frac {a^{2}\hbar ^{2}}{2m}}=-{\frac {me^{4}}{8{\varepsilon _{0}}^{2}\pi ^{2}\hbar ^{2}}}}$ .

### Example 2: Time dependent perturbation theory of weak field ionization

The first thing one can think to do above finding the ground state or the other allowed energies of the system is to solve the time dependent Schrödinger equation approximately i.e. in the perturbative manner. Let us apply it to the small destruction effects on the ground state of the quantum system and for the simplicity in one spatial dimension which reduces the kinetic energy operator to the second derivative with the proper coefficients. Let the ground state of the atom, for example the atom of Hydrogen will be such that ${\displaystyle E_{0}}$  is the smallest of possible values for the eigen-equation:

${\displaystyle H_{0}\Psi _{0}=E_{0}\Psi _{0}}$

Let us add the external electric field effect to the Hamiltonian ${\displaystyle H_{0}}$  in the simplest linear manner so the Schrödinger equation is

${\displaystyle H_{0}\Psi (t)-exf(t)E_{0}\sin(\omega t)\Psi (t)=i\hbar {\dot {\Psi }}(t)}$ .

We will assume ${\displaystyle \Psi (0)=\Psi _{0}}$  for the initial condition and will check what happens to that state under the influence of the perturbing ${\displaystyle -exf(t)E_{0}\sin(\omega t)}$  term, more specifically where it will go in terms of the expansion in the basis of the unbound states of the system which in very good approximation are simply plane waves. For the convenience of the calculations we added the ${\displaystyle f(\tau )}$  envelope function normalized to ${\displaystyle 1}$  in time so the electric field is turned on and off in the smooth and easy to Fourier transform Gaussian manner:

${\displaystyle f(\tau )=N_{1}e^{-(\tau -\tau _{0})^{2}/T_{0}^{2}}}$ .

it is now convenient to absorb the ${\displaystyle H_{0}}$  term in the equation in the so-called interaction picture introducing the transformed solution:

${\displaystyle \Psi (t)=e^{-iH_{0}t/\hbar }\Phi (t)}$

The original time-dependent equation greatly simplifies to

${\displaystyle -ex(t)f(t)E_{0}\sin(\omega t)\Phi (t)=i\hbar {\dot {\Phi }}(t)}$ .

where the position operator becomes time dependent namely

${\displaystyle x(t)=e^{iH_{0}t/\hbar }xe^{-iH_{0}t/\hbar }}$

With the zero iteration condition ${\displaystyle \Phi (t)=\Psi _{0}}$  we get in the first approximation

${\displaystyle \Phi (t)=\Psi _{0}-{\frac {1}{i\hbar }}\int _{0}^{t}ef(\tau )E_{0}\sin(\omega \tau )e^{iH_{0}\tau /\hbar }xe^{-iH_{0}\tau /\hbar }\Psi _{0}d\tau }$

we are now interested in the unbound continuum state component in the final solution which tells us about the energy spectrum of the ionized electrons. We can get that by projecting the solution onto the continuum state, simply:

${\displaystyle E(k)=|\int _{-\infty }^{\infty }<\Psi (x,+\infty )|e^{ikx}>dx|^{2}}$

where we extended the integration in time from the minus infinity to infinity since the electric field was localized in time with the Gaussian envelope. The only significant term which remains after the integration is:

${\displaystyle E(k)={\frac {e^{2}E_{0}^{2}}{4\hbar ^{2}}}|<\Psi _{0}|x]e^{ikx}>{\tilde {f}}(\hbar ^{2}k^{2}/2m-\hbar \omega -E_{0})|^{2}}$

Where the ${\displaystyle {\tilde {f}}}$  is the Fourier transform of the field envelope function ${\displaystyle f}$  and while ${\displaystyle f}$  is broadly localized the ${\displaystyle {\tilde {f}}}$  is sharply loalized around the point with the condition

${\displaystyle \hbar ^{2}k^{2}/2m-\hbar \omega -E_{0}=0}$

expressing the energy conservation condition

${\displaystyle E(k)=\hbar ^{2}k^{2}/2m=E_{0}+\hbar \omega }$

stating that while the ground state energy is negative there is no ionization below the threshold (for the photon or field frequencies that low that the kinetic energy would be impossibly negative) and the ejected electron kinetic energy is equal the energy of the absorbed photon lowered by the ionization or the binding energy.

### Example 3: Theory of Above Threshold Ionization (ATI)

Above Threshold Ionization may be explained by solving the time dependent Schrödinger equation in the approximate manner. The Schrödinger equation for the free electron in the field of the electromagnetic wave in one dimension and in the radiation gauge is given by

${\displaystyle {\frac {1}{2m}}\left({\frac {\hbar }{i}}{\frac {\partial }{\partial x}}-eA(t)\right)^{2}\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}}$ ,

where

${\displaystyle A(t)={\frac {E_{0}}{\omega }}\cos(\omega t)}$

so the electric field is given by

${\displaystyle E(t)=-{\frac {\partial A}{\partial t}}=E_{0}\sin(\omega t)}$

Substituting

${\displaystyle {\frac {}{}}\Psi (t)=e^{iC(t)+ikx}}$

we are obtaining the equation for ${\displaystyle C(t)}$

${\displaystyle {\dot {C}}(t)=\left(\hbar k-eA(t)\right)^{2}/2m}$

with the solution

${\displaystyle C(t)=-{\frac {\hbar ^{2}k^{2}}{2m}}t+\alpha k\sin(\omega t)-\beta \sin(2\omega t)-\gamma t}$ ,

where

${\displaystyle \alpha ={\frac {(eE_{0})\hbar }{m\omega ^{2}}}}$
${\displaystyle \beta ={\frac {(eE_{0})^{2}}{8m\omega ^{3}}}}$
${\displaystyle \gamma ={\frac {(eE_{0})^{2}}{4m\omega ^{2}}}}$ .

The Schrödinger equation for the electron in the wave field and in the atomic potential will be given by

${\displaystyle \left[H_{0}(t)+V(x)\right]\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}}$ ,

where, ${\displaystyle H_{0}}$  is the Hamiltonian of the free electron in the field. Adding and subtracting the energy of the ground state from which the electron is to be ionized we obtain

${\displaystyle \left[H_{0}(t)+V(x)+E_{0}-E_{0}\right]\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}}$

Because in the ground state the electron kinetic energy is equall the total energy but with the opposite sign (virial theorem) and only this energy will be left after the fast removal of the electron, we neglect in this equation the sum ${\displaystyle v(x)+E_{0}}$  for all ${\displaystyle x}$  and obtain the approximate equation

${\displaystyle \left[H_{0}(t)-E_{0}\right]\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}}$ ,

where the only remaining of the atomic potential is the constant.

This equation may now be solved using the previous result for the free electron in the field and expanding the ground state into its Fourier components

${\displaystyle \Psi (0)=Ne^{-|x|/a_{0}}=\int c_{k}e^{ikx}dk}$ ,

with

${\displaystyle c_{k}={\frac {N}{a_{0}}}{\frac {1}{k^{2}a_{0}^{2}+1}}}$

This equation has therefore the solution

${\displaystyle \Psi (t)=\int c_{k}e^{ikx}e^{iE_{0}t}e^{iC(t)}dk}$

We obtain the ionization spectrum from the formula

${\displaystyle E(k)=|\int _{-\infty }^{\infty }<\Psi (\tau )|e^{ikx}>g(\tau )d\tau |^{2}}$ ,

telling how much of the plane wave of the free electron with the given kinetic energy is at the and of the ionization process, where ${\displaystyle g(\tau )}$  is the averaging function of the measuring detector for example

${\displaystyle g(\tau )=N_{1}e^{-(\tau -\tau _{0})^{2}/T_{0}^{2}}}$ .

Expanding the factor

${\displaystyle {\frac {}{}}e^{i\alpha k\sin(\omega t)-i\beta \sin(2\omega t)}=\sum _{n}{\mathcal {J}}_{n}(-\alpha k,\beta )e^{-in\omega t}}$

with the generalized Bessel functions ${\displaystyle {\mathcal {J}}_{n}}$  defined by the inverse transform, we obtain

${\displaystyle E(k)=|\sum _{n}c_{k}{\mathcal {J}}_{n}(-\alpha k,\beta ){\tilde {g}}(\hbar ^{2}k^{2}/2m-n\hbar \omega -E_{0}+\gamma )|^{2}}$

(${\displaystyle {\tilde {g}}(\omega )}$  is the Fourier transform of the detector function) therefore the sum of sharp or fuzzy maxima localized around the energy condition of emitted electrons ${\displaystyle \hbar ^{2}k^{2}/2m=n\hbar \omega +E_{0}-\gamma }$  depending on the speed i.e the averaging parameter of the detector ${\displaystyle T_{0}}$ .

### Example 4: Basic theory of the Trojan Wave Packet

Because the Trojan wave packet is an approximate eigenstate of the Hamiltonian its search starts from the Schrödinger´s equation for the Hydrogen atom in the field of the circularly polarized wave in the coordinate system rotating with the frequency of the field

${\displaystyle H\Psi =E{\tfrac {}{}}\Psi }$ ,

where

${\displaystyle H={\frac {\mathbf {p} }{2}}^{2}-{\frac {1}{r}}-{\epsilon }x-\omega L_{z}}$ ,

and ${\displaystyle L_{z}}$  is the operator of z-component of the angular momentum.

For the simplification one can restrict himself to only two spacial dimensions. This equation after neglecting the curvature terms of the Laplacian and assumption that some terms are slowly varying or even constant takes in the polar coordinate system the simplified and separated form

${\displaystyle \left[-{\frac {\partial ^{2}}{2\partial r^{2}}}+{\frac {l^{2}}{2r^{2}}}-{\frac {1}{r}}-{\epsilon }r-{\frac {\partial ^{2}}{2r_{0}^{2}\partial \phi ^{2}}}-{\epsilon }r_{0}\cos \phi +\epsilon r_{0}\right]\chi =E\chi }$ ,
${\displaystyle {\frac {}{}}\chi =e^{il\phi }\Psi ,l=\omega ^{2}r_{0}}$

where ${\displaystyle \hbar =1,m_{e}=1}$ , and ${\displaystyle r_{0}}$  is the point of the potential of the part of the Hamiltonian dependent only on the radial coordinate and therefore also the radius of the classical orbit. Because the radial potential has the minimum the Trojan wavepacket is in this theory the product of a well localized radial Gaussian function and a well localized excited state of the quantum mathematical pendulum, corresponding to the pendulum totally reversed upside down which is also the Mathieu function with the characteristic value (and the energy) equal with a good approximation to ${\displaystyle r_{0}\epsilon }$ . The more precise theory shows that it is the pendulum of a negative and fractional electron mass equal -1/3 (of the electron) and that is why it is on the contrary the ground state with approximately the same energy but in the reversed spectrum.

To notice that let us more keep the correction from the expansion of the centrifugal term

${\displaystyle \Delta H=-2i{\frac {l}{r_{0}^{3}}}{\frac {\partial }{\partial \phi }}(r-r_{0})}$

and let us consider its action on the Fourier component of ${\displaystyle \chi }$ , e.g. ${\displaystyle e^{ik}}$ .

The additional centrifugal potential will be

${\displaystyle \Delta V=2i{\frac {l}{r_{0}^{3}}}k(r-r_{0})}$

It will cause the shift of the minimum and the energy lowering of the radial harmonic oscillator equal to

${\displaystyle \Delta E=-{\frac {2}{r_{0}^{2}}}k^{2}}$

Replacing back ${\displaystyle k^{2}}$  with the operator of the z component of the angular momentum we get

${\displaystyle \Delta H={\frac {2\partial ^{2}}{r_{0}^{2}\partial \phi ^{2}}}}$

Adding this correction in the Schródinger equation we get the equation with the negative and fractional electron mass -1/3 (${\displaystyle m_{e}}$ ):

${\displaystyle \left[-{\frac {\partial ^{2}}{2\partial r^{2}}}+{\frac {l^{2}}{2r^{2}}}-{\frac {1}{r}}-{\epsilon }r+{\frac {3\partial ^{2}}{2r_{0}^{2}\partial \phi ^{2}}}-{\epsilon }r_{0}\cos \phi +\epsilon r_{0}\right]\chi =E\chi }$ ,

which allows the Gaussian localization of the electron in the maximum but not in the minimum of the potential in the angular direction.

## References

[1] Qi-Chang Su and J. H. Eberly, Numerical simulations of multiphoton ionization and above-threshold electron spectra, Phys. Rev. A 38, 3430 (1988)

[2] H. R. Reiss and V. P. Krainov, Approximation for a Coulomb-Volkov solution in strong fields, Phys. Rev. A 50, R910 (1994)


text:
Palindromes are common topic when it comes to questions or exercises in computer science. For instance, they can be found in several exercises of Project Euler.

In the English language, palindromes are used to define both symmetrical words and numbers. However, this post will focus on numbers only.

## Using stringified numbers

String manipulation is very convenient when checking properties such as this. So, the most common approach is to stringify the number first.

Some languages, such as Python, are particularly flexible and allow the check to be as short as:

def is_string_palindrome(input_string):
return input_string == input_string[::-1]

def is_number_palindrome(number):
return is_string_palindrome(str(number))

Other languages, like C++, require a bit more of manual work. A solution would be to traverse the strings from both ends until meeting at the middle.

bool IsPalindrome(const String& s)
{
for (int i = 0; i < s.length() / 2; ++i)
{
if (s[i] != s[s.length() - 1 - i])
return false;
}
return true;
}

While using strings certainly eases the problem, allocating them is an expensive operation. There is a cheaper way to compute this.

It is possible, using math, to come to the very same conclusion in a CPU-friendly way. The approach is the same: we need to compare the nth-first and nth-last digits of the number. If they are different, the number is not a palindrome. If we run out of digits to check, we can guarantee that it is.

## Defining numeric palindromes mathematically

We can define the function $$IsPalindrome(x)$$ as:

$IsPalindrome(x) = \begin{cases} IsPalindrome(\lvert x \rvert) & \text{if x < 0}\\ True & \text{if x < 10}\\ IsPalindrome(Middle(x)) & \text{if FirstDigit(x) = LastDigit(x)}\\ False & \text{otherwise} \end{cases}$

The first case is to allow negative numbers to be taken into account. The second is the base case of this recursive function: single-digit numbers are, by nature, symmetric.

We still need to define the other functions used by the recursive case. For reference, $$FirstDigit(x)$$ and $$LastDigit(x)$$ refer to the left-most and right-most digit of $$x$$, respectively.

### First digit of a number

Getting the left-most digit of a number is just a matter of either:

• Looping, dividing the number by 10 each time, until the result is less than 10.
• Dividing the number by the closest power of 10 below the number.

This is possible because dividing a number by its base results in its radix point (decimal point for base 10) moving one digit to the left. This can be generalised for any number of digits: moving it $$N$$ number of places to the left is equal to dividing the number by $$10^N$$.

For instance, $$\frac{999}{10} = 99.9 \approx 9$$ and $$\frac{238}{100} = 2.38 \approx 2$$ .

In the case of the second approach, we can use the logarithm base 10 to determine the number of digits the number is made up of. In other words, its length.

$Length(x) = \lfloor \log_{10}(x)\rfloor + 1$

The reason why we use $$\lfloor \log_{10}(x)\rfloor + 1$$ instead of $$\lceil \log_{10}(x)\rceil$$ is because $$\log_{10}(1) = 0$$. If we try to calculate the ceiling of it, it'll stay as $$0$$ instead of being rounded-up to $$1$$.

With that in mind, it is possible to get the first digit with:

$FirstDigit(x) = \biggl\lfloor \frac{x}{10^{Length(x) - 1}} \biggr\rfloor$

## Last digit of a number

The process is analogue to the left-most digit. In this case, however, we use the remainder of the division instead of the division itself. This is a job for $$\bmod$$.

Since we only care about the last digit and not the last $$N$$ digits, is suffices to do this:

$LastDigit(x) = x \bmod 10$

## Middle part of a number

The middle is the number without its first and last digits.

Unfortunately, this works if the middle doesn't contain any leading zeroes. If it does, then it'll be malformed for our purposes. For example: the middle of $$10101$$ should be $$010$$ but it is $$10$$ instead because leading zeroes are ignored.

This problem is solved by removing any trailing digits as well. In the case of a palindrome, if those trailing digits aren't zeroes then we can rest assured that it isn't really a palindrome.

For simplicity purposes, let's focus on numbers that don't contain zeroes. We can get rid of the first digit by subtraction and then the last digit by division.

$Middle(x) = \frac{x - (FirstDigit(x) * 10^{Length(x) - 1})}{10}$

For example, the middle segment of $$321123$$ is:

\begin{aligned} Middle(321123)&=\biggl\lfloor \frac{321123 - (FirstDigit(321123) * 10^{Length(321123) - 1})}{10}\biggr\rfloor \\ &=\biggl\lfloor \frac{321123 - (3 * 10^{5})}{10}\biggr\rfloor \\ &=\biggl\lfloor \frac{321123 - 300000}{10}\biggr\rfloor \\ &=\biggl\lfloor \frac{21123}{10}\biggr\rfloor \\ &=\lfloor 2112.3 \rfloor \\ &= 2112 \end{aligned}

## Putting it all together

With all these in mind, it is possible to create an algorithm that detects if a number is a palindrome.


text:
Non-differentiable function

1. Sep 18, 2012

damabo

I know that $e^{rx}$ is an infinitely differentiable function.
However, say you have f= x. this is clearly one time differentiable, giving 1. a second time it can be derived as well, giving 0. is 0 differentiable, indeed differentiation gives zero.
So when is a function non-differentiable. i'm thinking of cusps, such as on x=0 with the function f=|x|, or other functions where the derivative is undefined. correct? are there other cases? it would also be interesting to see a function that is only finitely differentiable.

Last edited: Sep 18, 2012
2. Sep 18, 2012

pwsnafu

3. Sep 18, 2012

damabo

that's a nice one!

4. Sep 18, 2012

chiro

5. Sep 18, 2012

damabo

thanks, will apply that formula.


text:
# Verifying mgf for expectation, variance, and correlation

I have the following joint MGF of A and B. t1 corresponds to A and t2 corresponds to B.

I tried to find the expectation, variance, and correlation for A and B. I was wondering if the following values were correct:

E(A) = 1 Var (A) = 1

E(B) = 2 Var(B) = 2

Corr (A,B) = -1

If anyone could verify these values with me, then I would really appreciate it.

Thanks!

-

## 1 Answer

In the answer to the prior question of regarding this same distribution, I gave an interpretation of $(A,B)$ as $(Z_1, Z_1+Z_2)$, where $Z_1$ and $Z_2$ were independent Poisson variable with unit mean.

Using this $\mathbb{E}(A) =\mathbb{E}(Z_1) = 1$, $\mathrm{Var}(A) =\mathrm{Var}(Z_1) = 1$. Similarly $\mathbb{E}(B) = \mathbb{E}(Z_1) + \mathbb{E}(Z_2) = 2$, and $\mathrm{Var}(B) =\mathrm{Var}(Z_1) + \mathrm{Var}(Z_2) = 2$.

Covariance $\mathrm{Cov}(A, B) = \mathrm{Cov}(Z_1, Z_1+Z_2) = \mathrm{Cov}(Z_1, Z_1) + \mathrm{Cov}(Z_1,Z_2) = \mathrm{Var}(Z_1) + 0 = 1$.

Correlation follows $\mathrm{Cor}(A, B) = \frac{1}{\sqrt{2}}$.

-
Thanks! If Cor (A,B) = Cov(A,B) / sigma (A) * sigma (B), then how do you get 1? The denominator must be root 2 and I don't believe the covariance is also root 2... –  icobes Sep 29 '11 at 5:18
@icobes Yes, my bad. I have corrected the post. I was meaning to write covariance, not correlation. –  Sasha Sep 29 '11 at 5:37


text:
hey everyone,
im not here to just get answers because im lazy, im here because i got a 10 page notebook and ive done 90 pct of the questions and just a few is left. This question really confuses me, i just need help to solve it so i can be done
thnx if u help me(im new here)

A student designs a special container as a part of an egg drop experiment. she believes the container can withstand a fall as long as the speed of the container does not exceed 80 ft/s. shes uses this equation
v=sqrt(v0)2+2ad (the whole thing is square root)
to model the velocity v, in feet per second, as a function of constant acceleration, a, in feet per second squared and the drop distance d in feet. the acceleration due to gravity is 32ft/s2

a) will the egg break if the student drops the egg from shoulder height(5 ft) off a building 80 ft high?

i did so far
v=sqrt(v0)2+2(32)(85)

b) what is the maximum height the egg can be dropped from?

no help guys?

You are on the right track...now if the egg is "dropped" what is $v_0$?

Hello, EASYPEAAZY!

A student designs a special container as a part of an egg-drop experiment.
She believes the container can withstand a fall as long as the speed of the container does not exceed 80 ft/s.
She uses this equation: $v\:=\:\sqrt{v_o^2 +2ad}$ to model the velocity $v$ in ft/sec,
as a function of constant acceleration, $a = 32 \text{ft/sec}^2$ and the drop distance $d$ in feet.

a) Will the egg break if the student drops the egg from shoulder height (5 ft) off a building 80 ft high?

I did so far: $v\:=\:\sqrt{v_o^2+2(32)(85)}$

Note that the egg is dropped.
That is: . $v_o \,=\,0$

b) What is the maximum height the egg can be dropped from?

We have: . $\sqrt{0^2+64d} \:\le\:80$

Square: n . . . . . $64d \:\le\:6400$

Therefore: . . . . . . $d \:\le\:100$

ah i see velocity is 0, i just remembered that thnx

a) so i solved it and got v=sqrt(64)(85).....-> v=sqrt(5440)----->v=73.76 ft/s...so it will not break (thnx mark and soroban)

^ tell me if its right

b) i did 80=sqrt(0^2+64d)---> 80^2=sqrt(64d)^2--> 6400=64d--->d=100... 100 is the maximum height that it will not break(its a lil different than what you did soroban)

tell me if those are right please(thnx to the 2 posters above me for giving me help)

$v_f=8\sqrt{85}\text{ }\frac{\text{ft}}{\text{s}}$


text:
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# Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2018: 0.38

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Volume 10, Issue 2

# Groups whose word problems are not semilinear

Robert H. Gilman
/ Robert P. Kropholler
/ Saul Schleimer
Published Online: 2018-10-30 | DOI: https://doi.org/10.1515/gcc-2018-0010

## Abstract

Suppose that G is a finitely generated group and $\mathrm{WP}\left(G\right)$ is the formal language of words defining the identity in G. We prove that if G is a virtually nilpotent group that is not virtually abelian, the fundamental group of a finite volume hyperbolic three-manifold, or a right-angled Artin group whose graph lies in a certain infinite class, then $\mathrm{WP}\left(G\right)$ is not a multiple context-free language.

Keywords: Group theory; formal languages; word problem

MSC 2010: 20F10; 68Q45

## References

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R. H. Gilman, Formal languages and infinite groups, Geometric and Computational Perspectives on Infinite Groups (Minneapolis/New Brunswick 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 25, American Mathematical Society, Providence (1996), 27–51. Google Scholar

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M.-C. Ho, The word problem of ${ℤ}^{n}$ is a multiple context-free language, Groups Complex. Cryptol. 10 (2018), no. 1, 9–15. Google Scholar

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D. F. Holt, S. Rees and C. E. Röver, Groups with context-free conjugacy problems, Internat. J. Algebra Comput. 21 (2011), no. 1–2, 193–216.

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R. P. Kropholler and D. Spiriano, The class of groups with MCF word problem is closed under free products, preprint (2017).

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J. Lehnert and P. Schweitzer, The co-word problem for the Higman–Thompson group is context-free, Bull. Lond. Math. Soc. 39 (2007), no. 2, 235–241.

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A. Mateescu and A. Salomaa, Formal languages: An introduction and a synopsis, Handbook of Formal Languages. Vol. 1, Springer, Berlin (1997), 1–39. Google Scholar

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D. E. Muller and P. E. Schupp, Groups, the theory of ends, and context-free languages, J. Comput. System Sci. 26 (1983), no. 3, 295–310.

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M. Nivat, Transductions des langages de Chomsky, Ann. Inst. Fourier (Grenoble) 18 (1968), no. 1, 339–455.

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A. Y. Ol’shanskii and M. V. Sapir, Length and area functions on groups and quasi-isometric Higman embeddings, Internat. J. Algebra Comput. 11 (2001), no. 2, 137–170.

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D. W. Parkes and R. M. Thomas, Groups with context-free reduced word problem, Comm. Algebra 30 (2002), no. 7, 3143–3156.

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A. Piggott, On groups presented by monadic rewriting systems with generators of finite order, Bull. Aust. Math. Soc. 91 (2015), no. 3, 426–434.

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S. Salvati, MIX is a 2-MCFL and the word problem in ${ℤ}^{2}$ is captured by the IO and the OI hierarchies, J. Comput. System Sci. 81 (2015), no. 7, 1252–1277. Google Scholar

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H. Seki, T. Matsumura, M. Fujii and T. Kasami, On multiple context-free grammars, Theoret. Comput. Sci. 88 (1991), no. 2, 191–229.

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Published Online: 2018-10-30

Published in Print: 2018-11-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1440140

This material is based upon work supported by the National Science Foundation grant DMS-1440140 while the authors were in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the Fall 2016 Semester.

Citation Information: Groups Complexity Cryptology, Volume 10, Issue 2, Pages 53–62, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144,

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© 2018 Walter de Gruyter GmbH, Berlin/Boston.


text:
# Squeeze Theorem/Limits Question. Answers don't match up.

1. Jun 17, 2013

### fire9132

1. The problem statement, all variables and given/known data

Evaluate using the Squeeze Theorem.
$\lim_{x\rightarrow0} (2^{x} - 1) cos\frac{1}{x}$

2. Relevant equations

3. The attempt at a solution

$-1 ≤ cos\frac{1}{x} ≤ 1 \\ 1 - 2^{x} ≤ (2^{x} -1)cos\frac{1}{x} ≤ 2^{x} - 1 \\ 0 ≤ (2^{x} -1)cos\frac{1}{x} ≤ 0 \\ ∴ \lim_{x\rightarrow0} (2^{x} - 1) cos\frac{1}{x} = 0$

That's the solution I got and the solution that's in my textbook. But when I checked on wolframalpha, it says the limit does not exist.

Did I do something wrong and is my textbook wrong? or is it something conceptually that I don't understand about this? Is wolframalpha wrong?
Thanks :)

2. Jun 17, 2013

### Staff: Mentor

3. Jun 18, 2013

### jbunniii

You are correct that the limit is zero, but your reasoning has some flaws. The last line is clearly not correct, for it is equivalent to
$$(2^x - 1)\cos\left(\frac{1}{x}\right) = 0$$
which is not true for all $x$. The previous line is correct, but only if $2^x - 1 \geq 0$.

It's somewhat easier to work with absolute values:
$$\left|\cos\left(\frac{1}{x}\right)\right| \leq 1$$
Now multiply both sides by $|2^x - 1|$ and see what you can conclude.

4. Jun 19, 2013

### shortydeb

for x < 0, this statement is false.


text:
Article | Open | Published:

# Influence of Tibetan Plateau snow cover on East Asian atmospheric circulation at medium-range time scales

## Abstract

The responses of atmospheric variability to Tibetan Plateau (TP) snow cover (TPSC) at seasonal, interannual and decadal time scales have been extensively investigated. However, the atmospheric response to faster subseasonal variability of TPSC has been largely ignored. Here, we show that the subseasonal variability of TPSC, as revealed by daily data, is closely related to the subsequent East Asian atmospheric circulation at medium-range time scales (approximately 3–8 days later) during wintertime. TPSC acts as an elevated cooling source in the middle troposphere during wintertime and rapidly modulates the land surface thermal conditions over the TP. When TPSC is high, the upper-level geopotential height is lower, and the East Asia upper-level westerly jet stream is stronger. This finding improves our understanding of the influence of TPSC at multiple time scales. Furthermore, our work highlights the need to understand how atmospheric variability is rapidly modulated by fast snow cover changes.

## Introduction

The lower boundary conditions of the atmosphere, including land surface and sea surface conditions, force the atmosphere and can serve as indicators of atmospheric circulation. One of the most important lower boundary conditions is snow cover. Snow cover, a crucial component of the cryosphere, affects the global climate system via the sensitivity of the radiation balance to the high-albedo characteristics of snow and the energy allocation involved in the melting of snowpack1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16. The Tibetan Plateau (TP) is the highest plateau in the world and is known as the third pole17. Thermal forcing by the TP influences weather and climate18,19,20,21,22,23. Due to its high elevation, the TP is colder than other surface regions at the same latitude and has much more snow cover. The variability of TP snow cover (TPSC) influences the land surface thermal conditions and thus influences the weather and climate in and around the TP1,7,9,11,12,13,15,24,25,26,27,28,29.

The response of atmospheric variability to TPSC at seasonal, interannual and decadal time scales has been extensively investigated7,9,11,12,13,15,24,25,26,27,28,29. TPSC is regarded as an important indicator for climate prediction. Generally, climate prediction methods use the seasonal average TPSC conditions. For example, the winter or spring mean state of TPSC has been used as an indicator of summer monsoon precipitation anomalies over China7,13,15,26,27. However, the faster subseasonal variability of TPSC revealed by daily snow cover data and its effects on the atmosphere have largely been ignored. Recent studies have shown that accurate snow cover initialization can improve subseasonal and seasonal forecasts30,31,32,33, implying that snow cover is a potential indicator for forecasts at shorter time scales. The internal atmospheric variabilities with high-frequency are significant over the TP34; and TPSC may serve as part of the source of these variabilities. A better understanding of the atmospheric effects of TPSC at multiple time scales, including shorter time scales, allows us to understand all aspects of atmospheric variability.

This study demonstrates that the subseasonal variability of TPSC, which has previously received little attention, is a dominant variability of TPSC and is nonnegligible. We further investigate the atmospheric effects caused by such fast subseasonal variability of TPSC. The anomalous TPSC rapidly influences land surface sensible heat flux over the TP, which directly affects subsequent local and downstream upper-level atmospheric circulation. This process of response of downstream atmospheric circulation are at an approximate 3–8 days lag, which falls in the medium range. Hopefully, this work will bring attention to the fast subseasonal variability of TPSC and its effects on the atmosphere.

## Results

### The subseasonal variability of Tibetan Plateau snow cover

We first investigated whether the subseasonal component of TPSC is significant. Here, the subseasonal variability means the variability with a period shorter than a season (e.g., 120 days). We analyzed the subseasonal variation of TPSC during wintertime by using daily snow cover data (Fig. 1a). Most areas have standard deviation greater than 20%. Meanwhile, some areas have values greater than 28% over the central and eastern TP. We also note that some parts of the western TP (marked by black oval in Fig. 1a) and the Himalayas have smaller standard deviations than areas over the central and eastern TP. The western TP and Himalayas are covered by snow during more than 90% of wintertime (Fig. 1b). The high climatological mean of the wintertime snow cover in these areas leads to smaller standard deviations because they are covered by snow almost all winter; however, these areas only represent a small percentage of the whole TP. For regions across almost the entire TP, especially the central and eastern TP, strong subseasonal variability generally occurs.

To measure the regional variability of TPSC, a TPSC index (TPSCI) was defined (see Methods). The TPSCI is directly proportionate to the snow-covered area over the TP. We decomposed the TPSCI (with the annual cycle removed; Supplementary Fig. 1a) into a subseasonal component (Supplementary Fig. 1b) and an interannual-to-decadal component (Supplementary Fig. 1c). The subseasonal component has a greater variability than the interannual-to-decadal component, especially during wintertime. The subseasonal component explains 65.8% of the total non-seasonal TPSCI variability, while the interannual-to-decadal component explains the remaining part. We also selected a representative case to illustrate rapid snow cover changes (Supplementary Figs. 2, 3). Supplementary Fig. 2 shows the TPSCI (red line) and its subseasonal anomalies (blue line) during the cold season of 2012/2013, while Supplementary Fig. 3 shows the spatial snow cover on selected dates. Although the climatology of the TPSCI (i.e., annual cycle) increases during this period (gray line), the TPSCI shows both increasing and decreasing changes in this season. For example, the TPSCI is 53.1% on 2012-12-29 (Supplementary Fig. 3a), which is higher than its climatology. Then, the TPSCI decreases over the following 7 days. The TPSCI decreases to 24.2% on 2013-01-06 (Supplementary Fig. 3b). Most of the TP area is without snow cover, except for some parts of the western TP and the Himalayas. Then, the TPSCI increases over the month and reaches 63.9% on 2013-01-20 (Supplementary Fig. 3c). Subsequently, the TPSCI decreases again and is close to its climatology on 2013-01-27 (Supplementary Fig. 3d). During the period from 2013-01-06 to 2013-01-27 (about half a month), the change in the TPSCI was up to 39.7%. The obvious subseasonal variation, such as the 39.7% change in the snow-covered area (~1.1 × 106 km2) encompasses a broad area. The fast variability of snow cover over such a broad area may induce the atmospheric variability at a rapid time scale.

The above results demonstrate that the subseasonal variability of TPSC, which has previously received little attention, is dominant and nonnegligible, while seasonal averages may neglect this significant variability. It is thus valuable to further investigate the atmospheric effects caused by such fast variability.

### Influence of Tibetan Plateau snow cover on atmosphere

We found that the subseasonal variability of TPSC influences subsequent upper-level regional atmospheric circulation, especially the East Asia upper-level westerly jet stream (EAJS), during wintertime. The lead-lag relationship between TPSC and EAJS falls into medium-range time scales (~3–8 days). Both a statistical analysis based on ERA-interim reanalysis data and numerical experiments were performed to reveal this relationship.

The EAJS is one of the most distinct sources of atmospheric variability over East Asia35. The wintertime EAJS over subtropical regions varies across a wide range of time scales and is closely associated with many features of East Asia weather and climate36,37,38,39. Therefore, understanding variations in the of EAJS is important for accurate forecasting. The climatological mean of the wintertime upper-level (300-hPa) zonal wind reflects strong westerly winds over East Asia (Contours in Fig. 2). There is a strong westerly wind belt over the northwestern Pacific and Japan, with a zonal wind greater than 50 m s−1. This upper-level westerly wind has the highest of global upper-level zonal wind values and is known as the jet core35.

Composites were performed with respect to the TPSCI to assess the difference between positive and negative extreme cases of snow cover (see Methods). Figure 2 shows the composite evolution of the difference in 300-hPa zonal wind (U300). The composites show a lead–lag relationship between TPSC and U300. These composites represent the responses of the atmosphere to increased TPSC. First, there is a significantly anomalous U300 with a north–south dipole structure around the TP at lag 1–2 days (Fig. 2a). Then, the persisting anomalous dipole structure of the U300 moves eastward (Fig. 2b). The positive anomalous westerlies reach the EAJS core region at lag 5–6 days (Fig. 2c). The composite of the anomalous U300 near the south of the Japanese archipelago (near 20–30°N, 120–150°E) is significantly positive, with values of ~5 m s−1 (anomalous westerlies). Meanwhile, the anomalies still show a north-south dipole structure. There are negative anomalies near 40–50°N and 120–150°E (anomalous easterlies). Then, the U300 anomalies keep propagating eastward and weaken (Fig. 2d).

The above lead–lag composites indicate that the U300 varies with TPSC. Note that the composites contain signals not only from the lower boundary conditions (i.e., snow cover) but also from the internal atmospheric variability. In other words, both the TPSC-forced atmospheric response and the initial internal atmospheric variability may contribute to the TPSC variability and the downstream jet variability. To isolate the TPSC-forced atmospheric response (feedbacks exerted by snow) and to demonstrate that the composites contain causality of relationship and to further reveal the mechanism of interest, we performed a series of numerical experiments (see Methods). We carried out positive anomalous TPSC experiments (ExpPOS) and negative anomalous TPSC experiments (ExpNEG). The difference between the ExpPOS and ExpNEG represents the atmospheric response to the increased TPSC, while the response to the decreased TPSC has the opposite sign of this difference.

The response of the U300 in the numerical experiments is similar to that in the composite. This result clearly shows that a dipole structure of the U300 response originates from the TP (Fig. 3a). Then, the anomalous U300 moves eastward (Fig. 3b). The most evident response of the U300 over the EAJS core occurs after a 5–6 days lag (i.e., 5–6 days after the model initial time; Fig. 3c), which is consistent with the observed response (Fig. 2c). The U300 response is significantly positive near the EAJS core region. Then, the U300 anomalies continue to propagate eastward and weaken (Fig. 3d). The numerical experiments reproduce the dipole structure of the U300 anomalies in response to TPSC variability found in the reanalysis. However, there is still a slight difference between the reanalysis composites and the numerical experiments response signal over the TP and the Indian subcontinent. The intensity of the zonal wind response in the numerical experiments is not as strong as that in the reanalysis composites. The slight difference between the reanalysis and model experiments could be caused by many factors, including model biases, model lateral boundary forcing constraints, which limit the model response to TPSC changes, and the limited effect of the initial change in snow cover in the models (only over TP) compared to the real cases (different everywhere). Despite these biases, the numerical experiments could reproduce the major patterns and evolution of the U300 anomalies.

The response of upper-level atmospheric circulation extends over East Asia, especially over the EAJS region. An EAJS index (EAJSI) was defined to measure the variability of the EAJS. The EAJSI was calculated by averaging the U300 speed over the EAJS core region (magenta box in Fig. 2a). This index is directly proportional to the intensity of the EAJS. The composite of the EAJSI with respect to TPSC is shown in Fig. 4 (black line in Fig. 4a). The EAJSI shows a delayed variation with respect to TPSC (EAJSI lags TPSC). The EAJSI increases after a lag of 0–5 days and peaks after a 6-day lag. Then, the EAJSI anomaly decreases. The EAJSI from the ExpPOS numerical experiments relative to that from the ExpNEG experiments isolates the response of the EAJSI to TPSC (blue line and light blue shading in Fig. 4a). The EAJSI in the numerical experiments also increases in the first 5 days after the initial time of the model. Specifically, the largest EAJSI response also occurs after a 6-day lag. Then, the response weakens. Thus, the EAJSI has a delayed response to TPSC, which makes TPSC can acts as an indicator of EAJS at medium-range time scales.

### Land surface energy budget and its atmospheric effects

The numerical experiments suggest that the observed relationships at medium-range time scales can be reproduced in the model. Snow cover influences the land surface albedo40. By checking the land surface energy fluxes over the TP between ExpPOS and ExpNEG, we found that TPSC influences both the albedo and surface energy (Table 1). The snow cover condition over the TP in ExpPOS and ExpNEG is the only difference at the initial time step. The difference in albedo between ExpPOS and ExpNEG is 0.37 on the first day in the numerical experiments. The increased albedo strongly increases the upward-reflected shortwave radiation. This difference in solar surface energy leads to a decrease in the absorbed solar radiation. Thus, the net shortwave radiation is decreased, while the response of the net longwave radiation is much smaller than that of the net shortwave radiation. The decreased absorbed solar radiation is mainly emitted by the land surface as sensible heat flux. Apart from the snow-albedo effect, snow cover insulates the overlying air from the soil layer below (snow-thermodynamic effect). The snow-thermodynamic also reduces the sensible heat flux at land surface. In contrast, the latent heat flux response is low. The overall responses of surface energy to the increased/decreased TPSC lead to anomalous cooling/heating effects, due to the snow-albedo effect and snow-thermodynamic effect.

The persistence of the snow anomaly signals in the observational and numerical experiments are shown in Fig. 4b. The composite at a 0-day lag of the TPSCI in the Interactive Multi-Sensor Snow and Ice Mapping System (IMS) analysis is 24.0%. In addition, the difference between the TPSCI of the ExpPOS and ExpNEG at the first day of simulation is 24.5%, which is almost equal to that of the composite because the experimental design is based on the composite of TPSC. Then, the TPSCI anomaly in both the observational and numerical experiment composites decreases with lag days (Fig. 4b). The results show that the observed snow anomaly signal persists up to approximately 1 week and shows a decreasing tendency. The numerical experiments can reproduce both the persistence of the snow anomaly signal and the decreasing tendency within 6 days. The observational TPSCI anomalies fall sharply after 7 days, while the model that simulated TPSCI anomalies still shows a relatively slow decreasing tendency. The persistences of the TPSCI in ExpPOS and ExpNEG show similar results (Supplementary Note 1 and Supplementary Fig. 4). Although there is some bias 7 days after the initial date, the numerical experiments can generally reproduce both the persistence of the snow anomaly signal and the decreasing tendency at a medium range. A persistent surface energy response with a decreasing tendency is expected. Here, we take the sensible heat flux as a representative component of land surface energy to show the persistence of the land surface energy response (Supplementary Fig. 5). A strong negative response of sensible heat flux is caused by the snow-albedo effect for the first day, as discussed above. Then, the sensible heat flux increases (absolute value decreases) with lag days. At a 6-day lag, the sensible heat flux response is –22.8 W m−2, implying that the surface energy can persist to a lag of approximately 1 week.

The TP acts as a cooling source during wintertime19. Decreases in the sensible heat flux strengthen the cooling effect. Due to the high elevation of the TP, the cooling effect directly affects the upper-level atmosphere (above a pressure level of 500 hPa). The decrease in sensible heat flux for ExpPOS relative to ExpNEG is reflected by the upper-level atmospheric temperature. A substantial upper-level cooling response occurs over the TP. The temperature over the TP decreases rapidly after a 1–2 days lag (after the model initial time; Fig. 5a). The composite reanalysis of temperature also represents the upper-level cooling response (Supplementary Fig. 6). Such a cool response of TPSC is similar to that found in a previous study31, but at seasonal time scales. The geopotential height changes accordingly (Fig. 5b). Geopotential height demonstrates a negative response over the TP. The maximum response of geopotential height occurs near the 500- to 200-hPa pressure level over the TP. The negative response of the geopotential height over the TP moves eastwards with climatic westerly winds. The negative response of the geopotential height reaches the Japanese archipelago after an approximately 5–6 day lag (Fig. 5c). The zonal wind is in geostrophic balance with geopotential heights. Adaptive cyclone anomalies lead to the dipole structure of the zonal wind response, namely, the easterlies/westerlies to the north/south of the center of the geopotential height response (Fig. 5d).

Apart from the TP, snow cover is also widely distributed over Eurasia during boreal winter. Future studies on the possible rapid modulation by fast changes in snow cover over other parts of Eurasia are potentially valuable. This work also validates the sensitivity of the large-scale circulation simulated by numerical model to TPSC initialization. A more realistic TPSC initialization should help to decrease uncertainty in predictions or simulations. Although the model reproduces both the persistence of the snow anomaly signal and its tendency at the medium range, the model has some biases in simulating the sharp variations in TPSC after 7 days of observations. More works are necessary to further improve the snow and snow-atmosphere interaction processes in land surface models.

## Methods

### Data

Two publicly available datasets are used in this study. (1) Daily snow cover data at a 24-km resolution are obtained from the Interactive Multi-Sensor Snow and Ice Mapping System (IMS) snow cover analysis41. (2) Daily averaged large-scale atmospheric circulation data are obtained from ERA-Interim42. The data range from November 1997 to February 2017. Each winter comprises 120 days from November 1 to February 28 of the following year. This study spans 20 extended winters (from 1 November 1997 to 28 February 2017).

### The definitions of snow cover and jet stream indices

To measure the regional variability of TPSC, a TPSC index (TPSCI) is defined, which represents the percentage of the snow-covered area over the TP. The unit of the TPSCI is %. The TPSCI is calculated from the IMS snow cover analysis. The TPSCI is defined as TPSCI =$$\frac{1}{n}\mathop {\sum }\nolimits_{i = 1}^n x_i \times 100\,\%$$, where x is the IMS snow cover analysis over the TP. If one grid point is covered by snow, x = 1; otherwise, x = 0. The grid points over the TP are defined as points at an altitude of >3000 m and within 25–43°N and 65–105°E. In total, according to these criteria, there are 7658 grid points over the TP. The TPSCI for the numerical experiments is also calculated. The TPSCI definition is the same as that used for the IMS analysis, but x is the snow cover fraction in the model grid.

The EAJS index (EAJSI) is defined by averaging the 300-hPa zonal wind over the jet core region. The jet core region is the area in which the zonal wind is greater than 50 m s−1 over the northwestern Pacific Ocean (magenta box in Fig. 2a).

### Validation of the snow cover analysis

The IMS snow cover analysis, which is provided by the National Oceanic and Atmospheric Administration (NOAA), is an interactive system that is used to examine satellite images and other sources of data on snow cover and to generate maps of snow cover distribution41,43,44. The visible and infrared spectral data from the Polar Operational Environmental Satellites (POES) and Geostationary Orbiting Environmental Satellites were primarily used to generate snow cover data. Moderate Resolution Imaging Spectrometer (MODIS) imagery was used as well. In addition, ground weather observations from many countries were used. Since the visible and infrared data suffer from persistent cloud cover, which makes observations difficult, microwave products from SSM/I (Special Sensor Microwave Imager) and AMSR-E (Advanced Microwave Scanning Radiometer for EOS) are being used in the IMS product. The IMS system also includes the model output from Snow Data Assimilation System (SNODAS) and station-mapped products. The spatial resolutions of visible, infrared, microwave and SNODAS products used in the IMS System vary from 1 to 40 km.

The IMS product was manually created by NOAA NESDIS (The National Environmental Satellite, Data, and Information Service) satellite-product group analysts looking at all available satellite imagery, automated snow mapping algorithms, and other ancillary data. The IMS analysts use these multiple sources with different spatial resolutions within the interactive multisensor snow mapping system and re-gridded it to map snow at a 24-km spatial resolution. The analyst begins charting using the map from the previous day, then uses the satellite inputs accordingly. The IMS system allows for faster processing time to produce snow cover maps from satellite remote sensing data.

Reference44 validated the IMS snow cover analysis by a comparison with ground-based measurements over the continental United States. They found that the IMS maps demonstrate a good correspondence with the ground-based measurements. The daily rate of agreement between the products mostly ranges between 80 and 90% in the Northern Hemisphere during the winter season, when about a quarter to one-third of the continental US territory is covered with snow. Furthermore, they suggested that, when mapping snow cover, IMS analysts use the same technique and similar sources of data (e.g., satellite imagery, in situ data, and automated snow remote sensing products) over the whole Northern Hemisphere. Therefore, it is reasonable to assume that the accuracy of snow cover mapping over the mid-latitude region of Eurasia is similar to that over North America. Reference 45 found that the overall accuracy of IMS snow cover analysis is higher than 91% compared to station observations over the TP. Based on refs 44,45, the IMS snow cover analysis should be reliable to measure the overall variability of the TPSC. To test this hypothesis, we also evaluated the reliability of the IMS snow cover analysis over the TP. Here, we evaluated not only its overall accuracy but also the subseasonal variability of the TPSC in the IMS dataset.

To assess the quality of the IMS snow cover analysis over the TP, we used daily snow depth data collected from national meteorological stations of the China Meteorological Administration (CMA). The station data cover the period from 2000 to 2010, and we studied 55 stations over the TP region (Supplementary Fig. 7). The stations are mainly spread over the eastern TP. In the central and western TP, the weather observations are particularly lacking due to the bitter natural geographical environment and sparse population. To our knowledge, these station data are the only long-term operational observations including snow cover over the TP. It should be noted that CMA meteorological stations tend to be located in inhabited valleys. This may lead to differences between the station observations and satellite observations.

We evaluated the overall accuracy of the IMS snow cover analysis over the TP. We used an evaluation method similar that of refs 44,45. The IMS pixel values (snow or no snow) were compared to the snow depth values of the station data. For a given station location that coincides with an IMS pixel, a “match” was tagged if the IMS pixel value was “snow” and the snow depth value from station data was “trace” or greater. Similarly, if the IMS pixel was classified as “land” and the station data recorded a snow depth of “0”, a “match” occurred. A “mismatch” occurred if either the IMS pixel was “snow” and the station data showed a snow depth of ‘0’ or the IMS pixel was classified as “land” and the station data showed a “trace” or greater snow depth. If the station value was missing for that day, it was tagged as “missing”. Comparisons to the IMS data were not performed for those stations tagged as a “missing”. The daily agreement between the IMS snow pixels and station data was calculated as the number of matching pixels divided by the sum of matching pixels (NMatching) and mismatching pixels (NMismatching) multiplied by 100%:

$${\mathrm {OA}} = N_{{\mathrm{Matching}}}{\mathrm{/}}\left( {N_{{\mathrm{Matching}}} + N_{{\mathrm{Mismatching}}}} \right) \times 100\%$$
(1)

where OA is the overall accuracy of IMS snow cover analysis. The overall accuracy was calculated for each day during the period of 2000–2010 (sample N = 3650). The overall accuracy over the TP based on our validation evaluation is 92%. Reference 45 found that the overall accuracy of IMS snow cover analysis is higher than 91% over the TP. Our evaluation is consistent with that of ref. 45. Both our validation and ref. 45 suggest that the overall accuracy of IMS snow cover analysis over the TP demonstrate a good correspondence with ground-based measurements.

Our study focuses on the subseasonal variability of the TPSC. It is necessary to evaluate whether the IMS snow cover analysis can capture the subseasonal variability of the TPSC. Thus, the ‘TPSCI’ of both the station observations and IMS analysis for each winter during the period of 2000–2010 are compared. Note that here, the “TPSCI” is not the same as that defined previously. Here, the “TPSCI” only covers the 55 stations shown in Supplementary Fig. 7.

Also note that the snow condition observed at the stations is snow depth. Adopted from Biosphere-atmosphere Transfer Scheme (BATS)46, the snow cover fraction (fsno) is parameterized as a nonlinear function of snow depth (hsno):

$$f_{{\mathrm{sno}}} = h_{{\mathrm{sno}}}{\mathrm{/}}\left( {10z_{g,0} + h_{{\mathrm{sno}}}} \right)$$
(2)

where zg, 0 (= 0.01 m) is the ground roughness length. We estimate fsno for each station by using this parameterized function. The average of fsno for all 55 stations is the “TPSCI” of station observations.

The capture of the fsno for one station by IMS analysis is estimated using the following steps. First, the five IMS snow cover grids nearest to one station are selected. Then, the average of these five grids is regarded as the capture of one station by IMS analysis. Finally, the average of the fsno by IMS analysis for all 55 stations is regarded as the “TPSCI” for IMS analysis. Similar to the method in the main text, we apply a 120-day high-pass filter to remove the interannual and decadal components for these two “TPSCI” time series and focus on the subseasonal time scale.

The “TPSCI” of both stations and IMS for each winter during the period of 2000–2010 are compared (Supplementary Fig. 8). Overall, the IMS data capture the peaks of the “TPSCI” anomaly well for each winter during the period of 2000–2010. For example, the above-anomaly to below-anomaly transitions are captured correctly by IMS analysis (Supplementary Fig. 8i and Fig. 8j). The correlation coefficient (CC) between the two time series for each winter (the sample size N = 120) ranges from 0.39 to 0.79. The CC for the two time series for all 10 winters is 0.56 (the sample size N = 1200), indicating that the IMS snow cover analysis captures the subseasonal variability of the TPSC well.

The above evaluation shows that the IMS snow cover analysis can capture the general subseasonal variability of the TPSC. This dataset is valuable, especially for the TP, where observations are particularly lacking due to the bitter natural geographical environment and sparse population.

### Composite

This study investigates the influence of TPSC on East Asia upper-level circulation using a composite analysis. Extreme anomalous TPSC events are selected according to the TPSCI from the IMS snow cover analysis. We first extract the subseasonal component of the TPSCI by using a 120-day high-pass filter to remove the interannual and decadal components. We then standardize the index. The extreme anomalous TPSC events are selected based on the following criteria. The TPSCI must be greater than 0.5 (less than −0.5) for 6 or more successive days. The largest (smallest) day among these 6 or more successive days is regarded as the “start day” of this one extreme positive (negative) anomalous TPSC event. The number of events for the composite are 45 and 64 for positive and negative events, respectively. We averaged the 45 (64) samples of positive (negative) anomalous event samples at the “start day”. The difference between the average positive and negative anomalous events is the composite that lags the TPSCI by 0 days. The difference of the average atmospheric fields values for the following 1–8 days after the “start day” are the composites that lag TPSCI by 1 to 8 days. To focus on the subseasonal time scale, we applied a 120-day high-pass filter to remove the interannual and decadal components of the atmospheric fields.

### Significance tests

The statistically significance test for the composites is based on the two-sided Student’s t test.

### Numerical model

To confirm the conclusions from the composites and to further reveal the mechanism of interest, numerical experiments are performed. The numerical model used here is the Advanced Weather Research and Forecasting Model (WRF-ARW, version 3.9.1), which was developed by the National Center for Atmospheric Research (NCAR). The WRF-ARW has been applied to climate research, including studies of land–air interactions. The model domain is centered at 35°N and 115°E, covering an area of 9250 km (west–east) × 6350 km (south–north) with a horizontal resolution of 50 km. There are 28 levels in the vertical direction. The land surface parameterization scheme used in this study is the Noah-MP land surface model47. Noah-MP contains a multi-layer snow pack with liquid water storage and melt/refreeze capability and a snow-interception model describing loading/unloading, melt/refreeze, and the sublimation of the canopy-intercepted snow. Important physics options include the WRF single-moment 6-class microphysics scheme48, the NCAR Community Atmosphere Model (CAM 3.0) spectral-band shortwave and longwave radiation schemes49, the Yonsei University planetary boundary layer scheme50, and the Kain–Fritsch convective parameterization scheme51. The WRF is driven by atmospheric and surface forcing data extracted from the National Centers for Environmental Prediction (NCEP) FNL Operational Model Global Tropospheric Analyses.

### Design of numerical experiments

To test the response of atmospheric circulation to TPSC, two ensemble experiments are performed, namely, positive anomalous TPSC experiments (ExpPOS) and negative anomalous TPSC experiments (ExpNEG). The initial lower boundary condition TPSC is modified based on the IMS snow cover analysis. The target of the experimental design is to generate increased/decreased TPSC in ExpPOS/ExpNEG that are consistent with the IMS composite. The simultaneous composites of the daily TPSC from the IMS snow cover analysis were treated as “anomalies” (Supplementary Fig. 9a). The climatological TPSC plus half of the “anomalies” obtain the initial forcing for ExpPOS, whereas the initial forcing for ExpNEG is obtained by the climatological TPSC minus half of the “anomalies”. Hence, the difference in TPSC between ExpPOS and ExpNEG are the ‘anomalies’ shown in Supplementary Fig. 9b. In the land surface scheme, the physical snow depth (hsno) and snow water equivalent (msno) are the direct variables related to the snow cover fraction (fsno). We still transformed the fsno to the hsno by using Eq. (2). The hsno in grids of IMS analysis was interpolated into model grid. The snow density (ρsno) is estimated as 100 kg m−3 for ExpPOS to emulate the fresh snow increase in positive anomalous TPSC events, and the ρsno is estimated as 350 kg m−3 for ExpNEG to emulate the aged snow remaining in negative anomalous TPSC events. The msno is then calculated from the modified hsno and the estimated ρsno. Except for hsno and msno, other variables at the initial time step remain unchanged. As a result, the initial fsno between ExpPOS and ExpNEG is the “anomalies” (Supplementary Fig. 9b).

The only difference at the initial time step of the model are the snow cover conditions over the TP for ExpPOS and ExpNEG. All other variables at initial time step remain unchanged. Then, the model is integrated freely. To eliminate the impacts of atmospheric variability embedded in the initial conditions, we carried out several runs for each experiment with different initial dates and conducted an ensemble mean. Both numerical experiments contain 12 members with different initial times. Here, we use the 1st, 11th, and 21st day in each month from November 2013 to February 2014 as the initial dates. Each member is run continuously for 15 days. The ensemble means of each member for different experiments are used for analysis. Since the initial date can be considered as random, the internal atmospheric variability in the initial time step could be largely smoothed out after the ensemble average is applied. Comparisons of ensemble mean results between ExpPOS and ExpNEG could efficiently reveal the TPSC-forced atmospheric responses. The numerical experiments can reproduce the persistence of the snow anomaly signal (Fig. 4b). Using the ensemble means of each member with different initiation times can remove the synoptic variability of the atmosphere. The difference between ExpPOS and ExpNEG is considered to represent the response or the sensitivity of the atmosphere to TPSC at medium-range time scales.

### Code availability

All figures were produced using NCAR Command Language (NCL) version 6.4.0, an open source software free to the public, by UCAR/NCAR/CISL/TDD, https://doi.org/10.5065/d6wd3xh5. All the NCL scripts used in this study are available from the corresponding author upon request.

## Data availability

The IMS snow cover data are available at http://nsidc.org/data/docs/noaa/g02156_ims_snow_ice_analysis/. The ERA-interim data are available at http://apps.ecmwf.int/datasets/. The NCEP FNL data are available at https://rda.ucar.edu/datasets/ds083.2/. The WRF source codes can be obtained at http://www2.mmm.ucar.edu/wrf/users/download/get_sources.html. The topographic data from Global Relief Model data of ETOPO1 can be obtained at https://data.nodc.noaa.gov/cgi-bin/iso?id=gov.noaa.ngdc.mgg.dem:316. Other data that support the findings of this study are available from the corresponding author upon request.

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

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## Acknowledgements

This research is supported by the Strategic Priority Research Program of Chinese Academy of Sciences, Grant No. XDA2006010104, Natural Science Foundation of China (41705056, 41475063), the Natural Science Foundation of Jiangsu Province (BK20170638), the Startup Foundation for Introducing Talent of NUIST, and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). This work is also supported by Jiangsu Collaborative Innovation Center for Climate Change.

## Author information

### Affiliations

1. #### Key Laboratory of Meteorological Disaster, Ministry of Education (KLME)/Joint International Research Laboratory of Climate and Environment Change (ILCEC)/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters (CIC-FEMD), Nanjing University of Information Science & Technology, 210044, Nanjing, China

• Wenkai Li
• , Pang-Chi Hsu
•  & Jiangfeng Wei
2. #### Institute for Climate and Global Change Research, School of Atmospheric Sciences, Nanjing University, 210023, Nanjing, China

• Wenkai Li
• , Weidong Guo
•  & Bo Qiu
3. #### Joint International Research Laboratory of Atmospheric and Earth System Sciences, Nanjing University, 210023, Nanjing, China

• Wenkai Li
• , Weidong Guo
•  & Bo Qiu
4. #### Department of Geography and Department of Atmospheric and Oceanic Sciences, University of California, 90095, Los Angeles, CA, USA

• Yongkang Xue

### Contributions

W.G. led the overall scientific questions and designed the research. W.L. analyzed the data and drafted the manuscript. B.Q., Y.X., P.H., and J.W. gave conceptual advice. All authors contributed to the discussion of the results and to revising the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Weidong Guo.

## Electronic supplementary material

### DOI

https://doi.org/10.1038/s41467-018-06762-5