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# User talk:Azjps/sandbox

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User talk:Azjps/sandbox/gov school

1) Note that $311_{10} = 100110111_2$, so $2^{311} = 2^{2^8 + 2^5 + 2^4 + 2^2 + 2^1 + 2^0} = 2^{2^8} \times 2^{2^{5}} \times 2^{2^{4}} \times 2^{2^{2}} \times 2^{2^{1}} \times 2^{2^{0}} \quad (*)$. It requires $0$ multiplications to find $2^0, 2^1$, and by repeatedly squaring, we can find $\left(2^1\right)^2 = 2^2,\quad \left(2^2\right)^2 = 2^{2^2},\quad \left(2^{2^2}\right)^2 = 2^{2^2\times 2} = 2^{2^3}$ and so forth. Each squaring requires one multiplication, and so we can achieve $\left\{2^{2^0}, \ldots, 2^{2^8}\right\}$ using $8$ multiplications. Substituting into $(*)$, we find that our expression requires $5$ more multiplications. Thus, $8+5 = \boxed{13}$ multiplications are sufficient.

2)

$\begin{tabular}{|r||r|r|r|r|r|r|r|r|r|r|} \hline (a,n) & a^0 \mod{n} & a^1 \mod{n} & a^2 \mod{n} & a^3 \mod{n}& \cdots&&&&& a^{10} \mod{n} \\ (3,5) & 1 & 3 & 4 & 2 &&&&&& \\ (2,11) & 1&2&4&8&5&10&9&7&3&6 \\ \hline \end{tabular}$ (Error compiling LaTeX. ! Missing $inserted.) Since each of $\left\{a^{0}, a^{1}, \cdots, a^{n-2}\right\}$ are distinct (and consequently is a permutation of $\{1,2,\ldots,n-1\}$), then both are generators with respect to the given modulos. 3) By the Euclidean Algorithm, $\text{gcd}\,(288,84) = \text{gcd}\,(288 - 3\times 84,84)$ $= \text{gcd}\,(36,84) = \text{gcd}\,(36,84-2\times 36)$ $= \text{gcd}\,(36,12) = \boxed{12}$. 4) Let the number of students be $n$. Then the given conditions yield that$\begin{align*}n &\equiv 11 \pmod{13} \\ n &\equiv 5 \pmod{12} \end{align*}(Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.) By trial-and-error, we can easily find that $n \equiv 89 \pmod{156}$ satisfies these conditions, which must be the distinct solution due to the Chinese Remainder Theorem (and since $\text{gcd}\,(12,13)=1$). Since this year we only have $78$ students at Governor's School, we will assume that $\boxed{89}$ is the desired answer, and that we let our counselors pose as students (just kidding). Alternatively, we could note that $12^{-1} \equiv 12 \pmod{13}$ and that $13^{-1} \equiv 1 \pmod{12}$ using the Extended Euclidean Algorithm or by inspection, and thus the answer is $11 \times 12 \times 12 + 5 \times 1 \times 13 \equiv 89 \pmod{156}$. 5) Consider computing $a^m \pmod{n},\ a (this is not really necessary, since we could simply reduce if $a>n$). Assume that it is computationally easy to compute the product of two numbers, both $. Notice then that it is computationally easy to find $a^2$ (which requires just one multiplication, and the numbers are not extremely large as $a). Similarly, it is computationally easy to reduce $a^2 \pmod{n}$. Suppose we express $m = b_kb_{k-1}\cdots b_{1_2}$ in binary form, with $k=\left\lfloor \log_2 m\right\rfloor$ digits (for the given problem, $\left\lfloor \log_2 532452 \right\rfloor = 19$). Then $a^m \equiv a^{2^{b_k} + 2^{b_{k-1}} + \cdots + 2^{b_1}} \equiv a^{2^{b_k}} \times a^{2^{b_{k-1}}} \times \cdots \times a^{2^{b_1}} \pmod{n}$. Each of these terms being multiplied together can be found by squaring $a$, $k$ times (as established above, $k$ is not very large). Additionally, the multiplications are also assumed to be easy from above. Thus, that the exponent yields a huge number does not pose a computational difficulty. 6) The given conditions yield that\begin{align*}n &\equiv 2 \pmod{11} \\ n &\equiv 1 \pmod{13} \\ n &\equiv 3 \pmod{15} \\ n &\equiv 5 \pmod{19} \end{align*}(Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.) Using the Chinese Remainder Theorem repeatedly, the answer comes out to be $\boxed{16953} \pmod{11 \times 13 \times 15 \times 19 = 40755}$, which indeed fits the conditions (of course then $16953$ is the desired answer). Work: solving the first two congruences, and then the second two (by trial-and-error), yields\begin{align*}n &\equiv 79 \pmod{143} \\ n &\equiv 138 \pmod{285}\end{align*}(Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.) Here we notice that $285$ happens to be $2 \times 143 - 1$, which helps us to quickly determine the inverses: $285^{-1} \equiv -1 \pmod{143}$ and $143^{-1} \equiv 2 \pmod{285}$. Then by the Chinese Remainder Theorem, we have $n \equiv 2 \times 143 \times 138 -1 \times 285 \times 79 \equiv 16953 \pmod{40755}$. 7) For a pseudorandom construction, it would not be desirable for an external observer to predict a bit given the previous bits. However, with the given construction, and since $DES$ is a publically known one-way function, an observer could simply carry out $DES$ on the previously generated bits (let $DES_{n}(s) = \underbrace{DES(DES(\cdots(DES(s))\cdots)}_{n\ \text{times}}$, then given $DES_n(s)$, the outsider can easily carry out $DES(DES_n(s)) = DES_{n+1}(s)$), and determine the next bit him/herself. Note that the observer does not need to know the value of the seed $s$, while in the first construction the observer would still need to know $s$. 8) Let Andy want to send a secret message to Blase, but Blase only has a cell-phone. We assume that the limits imposed by Blase's computational restrictions do not apply to Andy. Then Andy computes two large prime numbers $p,q$, and sends $n = p\times q$ to Blase (if desired, Andy can also send the length of his message, $|M|$, to Blase). Blase then generates a one-time pad, $K$, and sends $K^2 \pmod{n}$ back to Andy (squaring is computationally easy). Now Andy can use the Rabin decryption algorithm to find the value of $K$. Let the binary operation $XOR(a,b)$ be denoted as $a \oplus b$. We note some basic properties of $XOR:$ $a \oplus b = b \oplus a$ (commutative), $a \oplus (b \oplus c) = (a \oplus b) \oplus c$ (associative), $a \oplus 0 = a$, $a \oplus a = 0$. Since it is assumed that an outside attacker could not determine the value of $K$ (performing the Rabin decryption without knowing $p,q$ being a difficult problem), Andy now sends $M \oplus K$ back to Blase. By the given, this is also cryptographically secure. To find the message, Blase performs $(M \oplus K) \oplus K = M \oplus (K \oplus K) = M$, as desired. 9) We programmed on Excel because MATLAB was unable to handle the huge exponents. Using $a = 3,5,7$ for simplicity of calculations, we find that\begin{tabular}{|r||r|r|r||r|}

\hline n & 3^{n-1} \pmod{n} & 5^{n-1} \pmod{n} & 7^{n-1} \pmod{n} & \text{Prime\ or\ Composite?} \\ \hline 6173 & 1 & 1 & 1 & P \\ 6179 & & & 4547 & C \\ 8415* & & & & C \\ 4113* & & & & C \\ 4691 & 1 & 1 & 1 & P \\ 5109* & & & & C \\ 7543 & 4466 & & & C \\ 7907 & 1 & 1 & 1 & P \\ \hline

\end{tabular}$(Error compiling LaTeX. ! Missing$ inserted.)

The numbers with asterisks are either trivially divisible by $3$ or $5$. For the numbers that are denoted with $P$, we can be reasonably sure that they are primes, which they indeed turn out to be. However, we cannot be completely sure that they are prime, due to our lack of randomness in choice of bases for the exponents, and because we only carried out a small number of tests.

10) Note that Caleb does not actually need to know Blase's bid $x$, as Caleb only needs to send $E(2x)$. By the statements of RSA, we know that Blase sent $E(x) \equiv x^e \pmod{n}$, where $e,n, E(x)$ are public. Caleb then wants to send $E(2x) \equiv (2x)^e \equiv 2^ex^e \equiv \boxed{2^eE(x) \pmod{n}}$. It is computationally easy for Caleb to perform this exponentiation (just as it was for Blase) and multiplication, and he has sufficient information.

11) $47^{1395} \equiv (-1)^{1395} \equiv -1 \equiv \boxed{47} \pmod{48}$, as $1395$ is odd.

12) $4^{3207} \equiv 4^{5} \times 4^{3202} \equiv 1024 \times 4^{3202} \equiv 0 \times 4^{3202} \equiv \boxed{0} \pmod{1024}$.

13) The following code can factor the product of two primes (MATLAB's factoring function only works for numbers $< 2^{32}$, so we scaled down all of the requested answers by $2\times$)

p = input('Enter prime 1: ')
q = input('Enter prime 2: ')
tic
factor(p * q)
toc

$\begin{tabular}{|r||r|r|r|} \hline \text{bits} & p & q & \text{time} \\ \hline 2 & 3 & 2 & 0.013334 \\ 4 & 11 & 13 & 0.013267 \\ 8 & 129 & 227 & 0.013529 \\ 10 & 803 & 937 & 0.012896 \\ 12 & 3203 & 3701 & 0.012944 \\ 16 & 33893 &47339 & 0.016408 \\ \hline \end{tabular}$

There is not a very strong correlation between the size of the primes because the used numbers are not particularily large. However, the quickest factoring algorithms (elliptic curve, quadratic sieve, and general number field sieve), run in sub-exponential time when it attempts to factor huge numbers.

14) $\left(x^{-1}\right)x \equiv 1 \pmod{n}$ can be rewritten as $\left(x^{-1}\right)x + kn = 1 \quad (*)$ for some integer $k$. We know that there are solutions to $ax + by = c$ if $c = \text{gcd}\,(a,b)$, which indeed is the case here as $\text{gcd}\,(41,167) = 1$. Thus $x^{-1}$ exists.

To find it, we perform the Extended Euclidean Algorithm on $(*)$:

\begin{align*} 167 &= 4\times 41 + 3 \\ 41 &= 13 \times 3 + 2 \\ 3 &= 2\times 1 + 1 \\ \\ \Longrightarrow 1 &= 3 - 1\times 2 \\ &= 3 - 1 \times (41 - 13 \times 3) \\ &= (14) \times 3 + (-1) \times 41 \\ &= (14)(167-4\times 41) + (-1) \times 41 \\ &= 14 \times 167 - 57 \times 41 \end{align*} (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)

Thus $x^{-1} \equiv -57 \equiv \boxed{110} \pmod{167}$.

15) For RSA, one step involves computing $d$ such that $d \times e \equiv 1 \pmod{\phi(n)}$. The existence of such $d \equiv e^{-1}$, from the previous question, requires that $1 = \text{gcd}\,(\phi(n),e) = \text{gcd}\,((p-1)(q-1),e) = \text{gcd}\, (10 \times 12, 8) = 8$. This is a contradiction, and thus $d$ does not exist.


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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 23rd 2013
• (edited Jan 23rd 2013)

I have briefly recorded the equivalence of FinSet${}^{op}$ with finite Booplean algebras at FinSet – Properties – Opposite category. Then I linked to this from various related entries, such as finite set, power set, Stone duality, opposite category.

(I thought we long had that information on the $n$Lab, but it seems we didn’t)

1. Added to FinSet a remark on the opposite category $FinSet^{op}$ from a constructive perspective:

“In constructive mathematics, for any flavor of finite, $\mathcal{P}$ defines an equivalence of $FinSet$ with the opposite category of that of those complete atomic Heyting algebras whose set of atomic elements is finite (in the same sense as in the definition of $FinSet$).”

I don’t know whether for some values of finite, this characterization can be made more interesting, i.e. whether we can give a condition which does not explicitly mention the set of atomic elements.

• CommentRowNumber3.
• CommentAuthorJohn Baez
• CommentTimeOct 22nd 2020

Typo fix: $FinSet^op$ is freely generated by finite limits (not finite colimits).

format: plain


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# P1.T4.26. Unexpected loss (UL), Ong

Discussion in 'Today's Daily Questions' started by David Harper CFA FRM CIPM, Apr 10, 2012.

1. ### David Harper CFA FRM CIPMDavid Harper

AIMs: Explain the objective for quantifying both expected and unexpected loss. Describe factors contributing to expected and unexpected loss. Define, calculate and interpret the unexpected loss of an asset. Explain the relationship between economic capital, expected loss and unexpected loss.

Questions:

26.1. Of a total original commitment (COM) of $10.0 million, 20.0% is outstanding (OS) such that$8.0 million is unused. The usage given default (UGD) assumption is 50.0% and the loss given default (LGD) assumption is 40.0% where the standard deviation of the LGD is 30.0%. If the probability of default (EDF) is 2.0%, what is the unexpected loss (UL) on the adjusted exposure (AE)?

a. $48,000 b.$421,540
c. $573,495 d.$933,381

26.2. An exposure has a default probability (PD) of 4.0% and loss given default of 50.0%. The standard deviation of the LGD is 25.0%. What is the ratio of the unexpected loss to the expected loss, UL/EL?

a. 1.33
b. 3.72
c. 5.50
d. 9.64

26.3. In the assigned reading on unexpected loss (Ong Chapter 5), unexpected loss (UL) is given as: UL = AE * SQRT[EDF*variance(LGD) + LGD^2*variance(EDF)]. Each of the following is TRUE about this definition of unexpected loss (UL) EXCEPT:

a. It assumes independence (zero default correlation) between the default probability and loss given default (LGD)
b. Economic capital will necessarily equal this value of this unexpected loss, as defined; i.e., EC = UL
c. This unexpected loss (UL) is the standard deviation (volatility) of the unconditional value of the asset at the horizon
d. Whereas expected loss (EL) increases as a linear function of EDF and LGD, unexpected loss (UL) increases as a non-linear function of EDF and LGD

Answers:


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# zbMATH — the first resource for mathematics

## Journal of the American Mathematical Society

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Symmetric spectra. Zbl 0931.55006
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Enumerative tropical algebraic geometry in $$\mathbb{R}^2$$. Zbl 1092.14068
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Well-posedness in Sobolev spaces of the full water wave problem in 3D. Zbl 0921.76017
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Well-posedness of the water-waves equations. Zbl 1069.35056
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Finite dimensional Hopf algebras arising from quantized universal enveloping algebras. Zbl 0695.16006
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The McKay correspondence as an equivalence of derived categories. Zbl 0966.14028
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The unconditional basic sequence problem. Zbl 0827.46008
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Tug-of-war and the infinity Laplacian. Zbl 1206.91002
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Well-posedness of the free-surface incompressible Euler equations with or without surface tension. Zbl 1123.35038
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Quivers, perverse sheaves, and quantized enveloping algebras. Zbl 0738.17011
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A kinetic formulation of multidimensional scalar conservation laws and related equations. Zbl 0820.35094
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Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. Zbl 1078.42013
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Compressed sensing and best $$k$$-term approximation. Zbl 1206.94008
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Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. Zbl 1009.05134
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Local smoothing properties of dispersive equations. Zbl 0667.35061
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The honeycomb model of $$\text{GL}_n(\mathbb C)$$ tensor products. I: Proof of the saturation conjecture. Zbl 0944.05097
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Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. Zbl 0892.03013
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Transcending classical invariant theory. Zbl 0716.22006
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Quiver varieties and finite dimensional representations of quantum affine algebras. Zbl 0981.17016
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The homotopy theory of fusion systems. Zbl 1033.55010
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Entire solutions of semilinear elliptic equations in $$\mathbb{R}^{3}$$ and a conjecture of De Giorgi. Zbl 0968.35041
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Sharp thresholds of graph properties, and the $$k$$-SAT problem (with an appendix by Jean Bourgain). Zbl 0932.05084
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Noetherian hereditary abelian categories satisfying Serre duality. Zbl 0991.18009
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Families of rationally connected varieties. Zbl 1092.14063
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Points on some Shimura varieties over finite fields. Zbl 0796.14014
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Hilbert schemes, polygraphs and the Macdonald positivity conjecture. Zbl 1009.14001
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Asymptotics of Plancherel measures for symmetric groups. Zbl 0938.05061
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2000
Compactifying the space of stable maps. Zbl 0991.14007
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2002
The tame and the wild automorphisms of polynomial rings in three variables. Zbl 1056.14085
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Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. Zbl 0912.14004
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The logarithmic Minkowski problem. Zbl 1272.52012
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Quivers with potentials and their representations. II: Applications to cluster algebras. Zbl 1208.16017
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The Bellman functions and two-weight inequalities for Haar multipliers. Zbl 0951.42007
Nazarov, F.; Treil, S.; Volberg, A.
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Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. Zbl 1312.53096
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Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. Zbl 0958.35126
Bourgain, Jean
1999
Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. Zbl 1202.35146
Frank, Rupert L.; Lieb, Elliott H.; Seiringer, Robert
2008
Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. Zbl 0870.11015
Bergelson, V.; Leibman, A.
1996
The orbifold Chow ring of toric Deligne-Mumford stacks. Zbl 1178.14057
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Conformal restriction: The chordal case. Zbl 1030.60096
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2003
Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $$2\pi$$ and completion of the main proof. Zbl 1311.53059
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2015
Singular Kähler-Einstein metrics. Zbl 1215.32017
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2009
Geometric invariant theory and flips. Zbl 0874.14042
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Heights of projective varieties and positive Green forms. Zbl 0973.14013
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Existence of vortex sheets in dimension two. (Existence de nappes de tourbillon en dimension deux.) Zbl 0780.35073
Delort, Jean-Marc
1991
The regularity of mappings with a convex potential. Zbl 0753.35031
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1992
Harmonic bundles on noncompact curves. Zbl 0713.58012
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1990
Universally optimal distribution of points on spheres. Zbl 1198.52009
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Complexity of Bezout’s theorem. I: Geometric aspects. Zbl 0821.65035
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1993
Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. Zbl 0634.32016
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1988
Knots are determined by their complements. Zbl 0678.57005
Gordon, C. McA.; Luecke, J.
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Torification and factorization of birational maps. Zbl 1032.14003
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Category $${\mathcal O}$$, perverse sheaves and modules over coinvariants of the Weyl group. (Kategorie $${\mathcal O}$$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe.) Zbl 0747.17008
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1990
Double Bruhat cells and total positivity. Zbl 0913.22011
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Navier-Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions. Zbl 0787.34039
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1993
Symmetric decreasing rearrangement is sometimes continuous. Zbl 0688.46014
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1989
Rotation distance, triangulations, and hyperbolic geometry. Zbl 0653.51017
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1988
The method of alternating projections and the method of subspace corrections in Hilbert space. Zbl 0999.47015
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2002
Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than $$2\pi$$. Zbl 1312.53097
Chen, Xiuxiong; Donaldson, Simon; Sun, Song
2015
The honeycomb model of $$\text{GL}_n({\mathbb C})$$ tensor products. II: Puzzles determine facets of the Littlewood-Richardson cone. Zbl 1043.05111
Knutson, Allen; Tao, Terence; Woodward, Christopher
2004
Shrinkwrapping and the taming of hyperbolic 3-manifolds. Zbl 1090.57010
Calegari, Danny; Gabai, David
2006
Global well-posedness of the Benjamin–Ono equation in low-regularity spaces. Zbl 1123.35055
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2007
Sharp effective Nullstellensatz. Zbl 0682.14001
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A variational principle for domino tilings. Zbl 1037.82016
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On the ill-posedness of the Prandtl equation. Zbl 1197.35204
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2010
Canonical bases for cluster algebras. Zbl 1446.13015
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Tensor structures arising from affine Lie algebras. I, II. Zbl 0786.17017
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1993
Complex scaling and the distribution of scattering poles. Zbl 0752.35046
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1991
Flip theorem and the existence of minimal models for 3-folds. Zbl 0649.14023
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Asymptotic distribution of eigenvalues of the Hecke operators $$T_ p$$. (Répartition asymptotique des valeurs propres de l’opérateur de Hecke $$T_ p$$.) Zbl 0871.11032
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1997
A Glimm-Effros dichotomy for Borel equivalence relations. Zbl 0778.28011
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1990
Manifolds with $$1/4$$-pinched curvature are space forms. Zbl 1251.53021
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2009
Beta ensembles, stochastic Airy spectrum, and a diffusion. Zbl 1239.60005
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2011
Independent sets in hypergraphs. Zbl 1310.05154
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2015
Resultants and Chow forms via exterior syzygies. Appendix by Jerzy Weyman: Homomorphisms and extensions between the bundles $$\bigwedge^p U$$ on the Grassmannian. Zbl 1069.14019
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2003
Wiener’s lemma for twisted convolution and Gabor frames. Zbl 1037.22012
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2004
The structure of rational and ruled symplectic 4-manifolds. Zbl 0723.53019
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1990
Analysis of the heterogeneous multiscale method for elliptic homogenization problems. Zbl 1060.65118
E, Weinan; Ming, Pingbing; Zhang, Pingwen
2005
On the equation $$\text{div }Y=f$$ and application to control of phases. Zbl 1075.35006
Bourgain, Jean; Brezis, Haïm
2003
Braid groups are linear. Zbl 0988.20021
Bigelow, Stephen J.
2001
The boundary of negatively curved groups. Zbl 0767.20014
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Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. Zbl 1015.35075
Chen, Gui-Qiang; Feldman, Mikhail
2003
A bilinear approach to the restriction and Kakeya conjectures. Zbl 0924.42008
Tao, Terence; Vargas, Ana; Vega, Luis
1998
Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients. Zbl 0993.16011
Derksen, Harm; Weyman, Jerzy
2000
Lower bounds of rates of decay for solutions to the Navier-Stokes equations. Zbl 0739.35070
Schonbek, Maria E.
1991
Elliptic stable envelopes. Zbl 07304878
Aganagic, Mina; Okounkov, Andrei
2021
Framed motives of algebraic varieties (after V. Voevodsky). Zbl 07304882
Garkusha, Grigory; Panin, Ivan
2021
The test function conjecture for parahoric local models. Zbl 07304879
Haines, Thomas J.; Richarz, Timo
2021
Cartier modules and cyclotomic spectra. Zbl 1467.14058
Antieau, Benjamin; Nikolaus, Thomas
2021
A sequence of polynomials with optimal condition number. Zbl 1458.65164
Beltrán, Carlos; Etayo, Ujué; Marzo, Jordi; Ortega-Cerdà, Joaquim
2021
Small subalgebras of polynomial rings and Stillman’s conjecture. Zbl 1439.13037
Ananyan, Tigran; Hochster, Melvin
2020
Positive entropy actions of countable groups factor onto Bernoulli shifts. Zbl 1437.37015
Seward, Brandon
2020
Residually finite rationally solvable groups and virtual fibring. Zbl 07179135
Kielak, Dawid
2020
Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves. Zbl 1456.11101
Bhargava, M.; Shankar, A.; Taniguchi, T.; Thorne, F.; Tsimerman, J.; Zhao, Y.
2020
Strongly anisotropic type II blow up at an isolated point. Zbl 1436.35253
Collot, Charles; Merle, Frank; Raphaël, Pierre
2020
Invariant metrics on negatively pinched complete Kähler manifolds. Zbl 07171893
Wu, Damin; Yau, Shing-Tung
2020
Effective equidistribution and property $$(\tau )$$. Zbl 1466.11049
Einsiedler, M.; Margulis, G.; Mohammadi, A.; Venkatesh, A.
2020
Tame topology of arithmetic quotients and algebraicity of Hodge loci. Zbl 1460.14027
Bakker, B.; Klingler, B.; Tsimerman, J.
2020
Tsirelson’s problem and an embedding theorem for groups arising from non-local games. Zbl 07171891
Slofstra, William
2020
On the measure of maximal entropy for finite horizon Sinai billiard maps. Zbl 1444.37020
2020
On the remodeling conjecture for toric Calabi-Yau 3-orbifolds. Zbl 1444.14093
Fang, Bohan; Liu, Chiu-Chu Melissa; Zong, Zhengyu
2020
Combinatorial constructions of derived equivalences. Zbl 1454.14045
Halpern-Leistner, Daniel; Sam, Steven V.
2020
Large genus asymptotics for volumes of strata of abelian differentials. Zbl 1452.14025
Aggarwal, Amol
2020
Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. Zbl 1451.05106
Hutchcroft, Tom
2020
$$p$$-adic cohomology of the Drinfeld tower: the 1-dimensional case. (Cohomologie $$p$$-adique de la tour de Drinfeld: Le cas de la dimension 1.) Zbl 1451.11029
Colmez, Pierre; Dospinescu, Gabriel; Nizioł, Wiesława
2020
Weak functoriality of Cohen-Macaulay algebras. Zbl 1458.13015
André, Yves
2020
The 4-dimensional light bulb theorem. Zbl 07225787
Gabai, David
2020
Zero-cycles on homogeneous spaces and inverse Galois problem. (Zéro-cycles sur les espaces homogènes et problème de Galois inverse.) Zbl 07225790
Harpaz, Yonatan; Wittenberg, Olivier
2020
Symplectic topology of $$K3$$ surfaces via mirror symmetry. Zbl 07225792
Sheridan, Nick; Smith, Ivan
2020
Billiards, quadrilaterals, and moduli spaces. Zbl 07268737
Eskin, Alex; Mcmullen, Curtis T.; Mukamel, Ronen E.; Wright, Alex
2020
Virtual homological spectral radii for automorphisms of surfaces. Zbl 1458.57027
Liu, Yi
2020
Exponential self-similar mixing by incompressible flows. Zbl 1456.35158
Alberti, Giovanni; Crippa, Gianluca; Mazzucato, Anna L.
2019
Stably irrational hypersurfaces of small slopes. Zbl 1442.14138
Schreieder, Stefan
2019
Cluster theory of the coherent Satake category. Zbl 1442.22022
Cautis, Sabin; Williams, Harold
2019
A random Schrödinger operator associated with the vertex reinforced jump process on infinite graphs. Zbl 1450.60062
Sabot, Christophe; Zeng, Xiaolin
2019
Khovanov homology from Floer cohomology. Zbl 1401.57005
Abouzaid, Mohammed; Smith, Ivan
2019
Koszul duality for Kac-Moody groups and characters of tilting modules. Zbl 1450.20011
Achar, Pramod N.; Makisumi, Shotaro; Riche, Simon; Williamson, Geordie
2019
Regular supercuspidal representations. Zbl 07121118
Kaletha, Tasho
2019
Absolute continuity of Bernoulli convolutions for algebraic parameters. Zbl 1408.28017
Varjú, Péter P.
2019
A Darboux theorem for derived schemes with shifted symplectic structure. Zbl 1423.14009
Brav, Christopher; Bussi, Vittoria; Joyce, Dominic
2019
Geometric criteria for overtwistedness. Zbl 1408.57026
Casals, Roger; Murphy, Emmy; Presas, Francisco
2019
Multipoint distribution of periodic TASEP. Zbl 07085526
Baik, Jinho; Liu, Zhipeng
2019
Topological noetherianity of polynomial functors. Zbl 1439.13044
Draisma, Jan
2019
A smooth mixing flow on a surface with nondegenerate fixed points. Zbl 1454.37025
Chaika, Jon; Wright, Alex
2019
An improved bound on the Hausdorff dimension of Besicovitch sets in $$\mathbb {R}^3$$. Zbl 1403.42020
Katz, Nets Hawk; Zahl, Joshua
2019
Generating the Fukaya categories of Hamiltonian $$G$$-manifolds. Zbl 1401.53075
Evans, Jonathan David; Lekili, Yankı
2019
No occurrence obstructions in geometric complexity theory. Zbl 1401.68088
Bürgisser, Peter; Ikenmeyer, Christian; Panova, Greta
2019
Existence and uniqueness for anisotropic and crystalline mean curvature flows. Zbl 1420.53073
Chambolle, Antonin; Morini, Massimiliano; Novaga, Matteo; Ponsiglione, Marcello
2019
Contact homology and virtual fundamental cycles. Zbl 1422.53071
Pardon, John
2019
On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow. Zbl 1461.20007
Bader, Uri; Caprace, Pierre-Emmanuel; Lécureux, Jean
2019
Normal subgroups of mapping class groups and the metaconjecture of Ivanov. Zbl 07121117
Brendle, Tara E.; Margalit, Dan
2019
Catalan functions and $$k$$-Schur positivity. Zbl 1423.05192
Blasiak, Jonah; Morse, Jennifer; Pun, Anna; Summers, Daniel
2019
Duality between the pseudoeffective and the movable cone on a projective manifold. Zbl 1429.32031
Nyström, David Witt
2019
Canonical bases for cluster algebras. Zbl 1446.13015
Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim
2018
A proof of the shuffle conjecture. Zbl 1387.05265
Carlsson, Erik; Mellit, Anton
2018
Energy solutions of KPZ are unique. Zbl 1402.60077
Gubinelli, Massimiliano; Perkowski, Nicolas
2018
Monoidal categorification of cluster algebras. Zbl 1460.13039
Kang, Seok-Jin; Kashiwara, Masaki; Kim, Myungho; Oh, Se-Jin
2018
Homological stability for moduli spaces of high dimensional manifolds. I. Zbl 1395.57044
Galatius, Søren; Randal-Williams, Oscar
2018
Shutkas for reductive groups and global Langlands parametrization. (Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale.) Zbl 1395.14017
Lafforgue, Vincent
2018
Definably amenable NIP groups. Zbl 06870167
Chernikov, Artem; Simon, Pierre
2018
Mixed 3-manifolds are virtually special. Zbl 06836097
Przytycki, Piotr; Wise, Daniel T.
2018
Weak null singularities in general relativity. Zbl 1377.83062
Luk, Jonathan
2018
Long gaps between primes. Zbl 1392.11071
Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence
2018
Counterexamples to the Eisenbud-Goto regularity conjecture. Zbl 1390.13043
McCullough, Jason; Peeva, Irena
2018
Quasimodularity and large genus limits of Siegel-Veech constants. Zbl 1404.32025
Chen, Dawei; Möller, Martin; Zagier, Don
2018
Stratification for module categories of finite group schemes. Zbl 06802408
Benson, Dave; Iyengar, Srikanth B.; Krause, Henning; Pevtsova, Julia
2018
Langlands correspondence for isocrystals and the existence of crystalline companions for curves. Zbl 1420.14044
Abe, Tomoyuki
2018
The coarse geometry of Tsirelson’s space and applications. Zbl 1396.46020
Baudier, F.; Lancien, G.; Schlumprecht, Th.
2018
Quantitative null-cobordism. Zbl 1402.53035
Chambers, Gregory R.; Dotterrer, Dominic; Manin, Fedor; Weinberger, Shmuel
2018
The category of singularities as a crystal and global Springer fibers. Zbl 1388.14042
Arinkin, D.; Gaitsgory, D.
2018
On fewnomials, integral points, and a toric version of Bertini’s theorem. Zbl 1439.11091
Fuchs, Clemens; Mantova, Vincenzo; Zannier, Umberto
2018
Quantum unique ergodicity and the number of nodal domains of eigenfunctions. Zbl 1385.58014
Jang, Seung Uk; Jung, Junehyuk
2018
Essential surfaces in graph pairs. Zbl 1452.20038
Wilton, Henry
2018
Simple groups of Morley rank 3 are algebraic. Zbl 06870168
Frécon, Olivier
2018
Decoupling, exponential sums and the Riemann zeta function. Zbl 1352.11065
Bourgain, J.
2017
Gröbner methods for representations of combinatorial categories. Zbl 1347.05010
Sam, Steven V.; Snowden, Andrew
2017
Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics. Zbl 1386.32021
Darvas, Tamás; Rubinstein, Yanir A.
2017
Convexity of the $$K$$-energy on the space of Kähler metrics and uniqueness of extremal metrics. Zbl 1376.32028
Berman, Robert J.; Berndtsson, Bo
2017
Kink dynamics in the $$\phi^4$$ model: asymptotic stability for odd perturbations in the energy space. Zbl 1387.35419
Kowalczyk, Michał; Martel, Yvan; Muñoz, Claudio
2017
Schubert calculus and torsion explosion. Zbl 1380.20015
Williamson, Geordie
2017
The Monge-Ampère equation for $$(n-1)$$-plurisubharmonic functions on a compact Kähler manifold. Zbl 1366.32015
Tosatti, Valentino; Weinkove, Ben
2017
The distribution of sandpile groups of random graphs. Zbl 1366.05098
Matchett Wood, Melanie
2017
Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients. Zbl 1351.15022
Breuer, Jonathan; Duits, Maurice
2017
Lens rigidity for manifolds with hyperbolic trapped sets. Zbl 1377.53098
Guillarmou, Colin
2017
Mod $$p$$ points on Shimura varieties of abelian type. Zbl 1384.11075
Kisin, Mark
2017
Gromov-Witten/pairs correspondence for the quintic 3-fold. Zbl 1360.14134
Pandharipande, R.; Pixton, A.
2017
Higher-order Fourier analysis of multiplicative functions and applications. Zbl 1355.11094
Frantzikinakis, Nikos; Host, Bernard
2017
A classification of irreducible admissible mod $$p$$ representations of $$p$$-adic reductive groups. Zbl 1372.22014
Abe, N.; Henniart, G.; Herzig, F.; Vignéras, M.-F.
2017
Diagonal cycles and Euler systems. II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin $$L$$-functions. Zbl 1397.11090
Darmon, Henri; Rotger, Victor
2017
Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations. Zbl 1406.35377
Serfaty, Sylvia
2017
Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension. Zbl 1393.14034
Kovács, Sándor J.; Patakfalvi, Zsolt
2017
On the Erdős-Szekeres convex polygon problem. Zbl 1370.52032
Suk, Andrew
2017
Harmonic maps and the Schoen conjecture. Zbl 1371.53059
2017
A positive proportion of locally soluble hyperelliptic curves over $$\mathbb {Q}$$ have no point over any odd degree extension. Zbl 1385.11043
Bhargava, Manjul; Gross, Benedict H.; Wang, Xiaoheng
2017
The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions. Zbl 1356.22015
Sun, Binyong
2017
Equivariant properties of symmetric products. Zbl 1367.55005
Schwede, Stefan
2017
Geometric complexity theory. V: Efficient algorithms for Noether normalization. Zbl 1402.14078
Mulmuley, Ketan D.
2017
Measure rigidity for random dynamics on surfaces and related skew products. Zbl 1379.37055
Brown, Aaron; Rodriguez Hertz, Federico
2017
Non-density of small points on divisors on abelian varieties and the Bogomolov conjecture. Zbl 1387.14076
Yamaki, Kazuhiko
2017
A restriction estimate using polynomial partitioning. Zbl 1342.42010
Guth, Larry
2016
$$\mathrm{Pin}(2)$$-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. Zbl 1343.57015
Manolescu, Ciprian
2016
Finite time blowup for an averaged three-dimensional Navier-Stokes equation. Zbl 1342.35227
Tao, Terence
2016
Boundary rigidity with partial data. Zbl 1335.53055
Stefanov, Plamen; Uhlmann, Gunther; Vasy, Andras
2016
A proof of the Howe duality conjecture. Zbl 1342.11051
Gan, Wee Teck; Takeda, Shuichiro
2016
Hypersurfaces that are not stably rational. Zbl 1376.14017
Totaro, Burt
2016
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 70 Tao, Terence 49 Biswas, Indranil 47 Dong, Chongying 45 Borodin, Alexei 45 Kenig, Carlos Eduardo 43 Yang, Dachun 42 Schiffler, Ralf 41 Bourgain, Jean 40 Wei, Juncheng 39 Pandharipande, Rahul 39 Uhlmann, Gunther Alberto 37 Ponce, Gustavo 35 Pillay, Anand 34 Shelah, Saharon 32 Mazorchuk, Volodymyr 32 Tabuada, Gonçalo 31 Linares, Felipe 30 Sogge, Christopher D. 30 Stroppel, Catharina H. 29 Figalli, Alessio 29 Sam, Steven V. 28 Bergelson, Vitaly 28 Chen, Xiuxiong 28 Duong, Xuan Thinh 28 Gee, Toby 28 Guralnick, Robert Michael 28 Tian, Gang 27 Kashiwara, Masaki 27 Kollár, János 27 Lam, Thomas F. 27 Lusztig, George 27 Marsh, Robert James 27 Martel, Yvan 27 Rossi, Julio Daniel 27 Shparlinski, Igor E. 27 Xu, Chenyang 27 Yan, Lixin 26 Brendle, Simon 26 Corwin, Ivan 26 Lee, Sanghyuk 26 Miao, Changxing 26 Tiep Pham Huu 26 Zhu, Bin 25 Argyros, Spiros A. 25 Losev, Ivan V. 25 Shalev, Aner 25 Vega, Luis 25 Volberg, Alexander Lvovich 24 Bismut, Jean-Michel 24 Forstnerič, Franc 24 Hacon, Christopher Derek 24 Markwig, Hannah 24 Raphael, Pierre 24 Song, Liang 24 Stefanov, Plamen D. 24 Van den Bergh, Michel 24 Zhang, Xi 23 Arakawa, Tomoyuki 23 Guo, Boling 23 He, Xuhua 23 Muñoz, Claudio 23 Osękowski, Adam 23 Tataru, Daniel 23 Valdinoci, Enrico 23 Veys, Willem 22 Abdulle, Assyr 22 Abramovich, Dan 22 Balogh, József 22 Damanik, David 22 Frank, Rupert L. 22 Huneke, Craig L. 22 Kang, Seok-Jin 22 Schramm, Oded 22 Shen, Zuowei 22 Sturmfels, Bernd 22 Sudakov, Benny 22 Wang, Weiqiang 22 Yang, Tong 22 Ye, Xiangdong 22 Zhang, Zhifei 21 Assem, Ibrahim 21 Ciubotaru, Dan 21 Fox, Jacob 21 Katzarkov, Ludmil 21 Kukavica, Igor 21 Lu, Guangshi 21 Lück, Wolfgang 21 Merle, Frank 21 Schwede, Karl E. 21 Tang, Xianhua 21 Tien Cuong Dinh 21 Tosatti, Valentino 21 Tseng, Hsian-Hua 21 Xin, Zhouping 21 Zhang, Bingyu 21 Zhang, Ziheng 20 Chen, Gui-Qiang G. 20 Fomin, Sergey Vladimirovich 20 Himonas, A. Alexandrou 20 Li, Hanfeng ...and 21,261 more Authors
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#### Cited in 766 Journals

 1,344 Advances in Mathematics 1,055 Transactions of the American Mathematical Society 1,039 Journal of Algebra 719 Communications in Mathematical Physics 670 Duke Mathematical Journal 669 Proceedings of the American Mathematical Society 562 Mathematische Annalen 530 Journal of Functional Analysis 510 Inventiones Mathematicae 497 Journal of Differential Equations 496 Mathematische Zeitschrift 415 Journal of Pure and Applied Algebra 390 Journal of Mathematical Analysis and Applications 371 Journal of the American Mathematical Society 365 Israel Journal of Mathematics 349 Compositio Mathematica 314 Annales de l’Institut Fourier 297 Geometry & Topology 285 Journal of Number Theory 260 Journal für die Reine und Angewandte Mathematik 258 Selecta Mathematica. New Series 242 The Journal of Geometric Analysis 240 Calculus of Variations and Partial Differential Equations 235 Journal of the European Mathematical Society (JEMS) 234 Annals of Mathematics. Second Series 233 Archive for Rational Mechanics and Analysis 226 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 226 Algebraic & Geometric Topology 220 Comptes Rendus. Mathématique. Académie des Sciences, Paris 208 Topology and its Applications 205 Journal of Combinatorial Theory. Series A 202 Journal of Mathematical Physics 191 Geometric and Functional Analysis. GAFA 184 Communications in Algebra 184 Bulletin of the American Mathematical Society. New Series 178 Geometriae Dedicata 175 Journal of Geometry and Physics 175 Manuscripta Mathematica 167 Ergodic Theory and Dynamical Systems 165 Journal of High Energy Physics 163 Transformation Groups 162 Science China. Mathematics 153 Representation Theory 152 Journal of Algebraic Combinatorics 150 Communications in Partial Differential Equations 148 Linear Algebra and its Applications 147 Journal of Algebraic Geometry 146 Journal de Mathématiques Pures et Appliquées. Neuvième Série 145 The Annals of Probability 143 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 140 Journal of Statistical Physics 134 SIAM Journal on Mathematical Analysis 133 Letters in Mathematical Physics 132 Annals of Pure and Applied Logic 132 International Journal of Mathematics 130 Algebras and Representation Theory 129 European Journal of Combinatorics 128 Discrete and Continuous Dynamical Systems 127 Journal of the Institute of Mathematics of Jussieu 124 Discrete & Computational Geometry 120 Probability Theory and Related Fields 116 Mathematical Proceedings of the Cambridge Philosophical Society 114 Discrete Mathematics 114 Journal d’Analyse Mathématique 114 Memoirs of the American Mathematical Society 114 Acta Mathematica Sinica. English Series 114 Forum of Mathematics, Sigma 110 Mathematics of Computation 110 Acta Mathematica 98 Communications in Contemporary Mathematics 95 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 93 Revista Matemática Iberoamericana 91 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 87 Communications on Pure and Applied Mathematics 87 Journal of Mathematical Sciences (New York) 86 The Journal of Symbolic Logic 86 Forum Mathematicum 86 International Journal of Number Theory 85 Nagoya Mathematical Journal 84 Michigan Mathematical Journal 83 Communications on Pure and Applied Analysis 80 Publications Mathématiques 80 Advances in Applied Mathematics 79 Journal de Théorie des Nombres de Bordeaux 78 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 77 Journal of Knot Theory and its Ramifications 73 The Journal of Fourier Analysis and Applications 73 Annales Henri Poincaré 70 Nuclear Physics. B 70 Groups, Geometry, and Dynamics 68 Annals of Global Analysis and Geometry 67 Algebra & Number Theory 66 Publications of the Research Institute for Mathematical Sciences, Kyoto University 66 Random Structures & Algorithms 66 Applied and Computational Harmonic Analysis 65 Theoretical Computer Science 65 Journal of Symbolic Computation 65 Séminaire Lotharingien de Combinatoire 63 Experimental Mathematics 62 Foundations of Computational Mathematics ...and 666 more Journals
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#### Cited in 63 Fields

 5,933 Algebraic geometry (14-XX) 5,298 Partial differential equations (35-XX) 3,365 Number theory (11-XX) 2,785 Group theory and generalizations (20-XX) 2,750 Differential geometry (53-XX) 2,636 Combinatorics (05-XX) 2,017 Dynamical systems and ergodic theory (37-XX) 1,943 Manifolds and cell complexes (57-XX) 1,916 Several complex variables and analytic spaces (32-XX) 1,892 Nonassociative rings and algebras (17-XX) 1,718 Associative rings and algebras (16-XX) 1,546 Probability theory and stochastic processes (60-XX) 1,481 Commutative algebra (13-XX) 1,384 Topological groups, Lie groups (22-XX) 1,360 Quantum theory (81-XX) 1,325 Global analysis, analysis on manifolds (58-XX) 1,210 Functional analysis (46-XX) 1,169 Fluid mechanics (76-XX) 1,088 Category theory; homological algebra (18-XX) 1,003 Harmonic analysis on Euclidean spaces (42-XX) 979 Mathematical logic and foundations (03-XX) 955 Operator theory (47-XX) 951 Statistical mechanics, structure of matter (82-XX) 941 Algebraic topology (55-XX) 853 Convex and discrete geometry (52-XX) 795 Functions of a complex variable (30-XX) 718 Numerical analysis (65-XX) 656 Computer science (68-XX) 640 Linear and multilinear algebra; matrix theory (15-XX) 530 $$K$$-theory (19-XX) 499 Calculus of variations and optimal control; optimization (49-XX) 399 Ordinary differential equations (34-XX) 393 Measure and integration (28-XX) 335 Information and communication theory, circuits (94-XX) 331 Special functions (33-XX) 307 Real functions (26-XX) 297 General topology (54-XX) 259 Field theory and polynomials (12-XX) 248 Relativity and gravitational theory (83-XX) 248 Operations research, mathematical programming (90-XX) 231 Abstract harmonic analysis (43-XX) 225 Order, lattices, ordered algebraic structures (06-XX) 199 Potential theory (31-XX) 198 Geometry (51-XX) 147 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 134 Biology and other natural sciences (92-XX) 133 Mechanics of deformable solids (74-XX) 131 Mechanics of particles and systems (70-XX) 130 Statistics (62-XX) 126 Approximations and expansions (41-XX) 120 Systems theory; control (93-XX) 119 Integral transforms, operational calculus (44-XX) 93 Difference and functional equations (39-XX) 92 Optics, electromagnetic theory (78-XX) 85 History and biography (01-XX) 79 Geophysics (86-XX) 76 Integral equations (45-XX) 71 General and overarching topics; collections (00-XX) 28 Classical thermodynamics, heat transfer (80-XX) 26 General algebraic systems (08-XX) 21 Astronomy and astrophysics (85-XX) 9 Sequences, series, summability (40-XX) 6 Mathematics education (97-XX)


text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# How do you determine ground state electron configuration?

An atom in its ground state has the same number of electrons as protons. It is also know as a neutral atom or an uncharged atom.

So, looking at , an atom of Iron ("Fe") in its ground state would have 26 electrons.

The electrons fill the orbitals closest to the nucleus first and then move out until all of the electrons are accounted for.

The first orbital filled is the 1s. It can hold a maximum of 2 electrons.

Next, the 2s, which can also hold a maximum of 2 electrons. Then the 2p, which can hold up to 6 electrons.

Then 3s (max 2 electrons), 3p (max 6 electrons), 4s (max 2), 3d (can hold a maximum of 10 electrons), then 4p (max 6), 5s (max 2), 4d (max 10), 5p (max 6), 6s (max 2), 4f (can hold a maximum of 14 electrons), 5d (max 10), 6p (max 6), 7s (max 2), 5f (max 14), 6d (max 10), 7p (max 6), 8s (max 2) and so on.

So, the for Iron would then be:

1s^2, 2s^2, 2p^6, 3s^2, 3p^6, 4s^2, 3d^6.

If you add up the electron numbers (the numbers following the letters s , p, and d, you will see that they add up to 26, which is the number of electrons that iron has in its ground state. You now know where each of the 26 electrons are located.

I hope this helps.


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# Large Eddy Simulation Review¶

## Introduction¶

In M-Star CFD, turbulence is modeled using large eddy simulation (LES) with a Smagorinsky sub-grid closure model. Within this approach, the velocity field is decomposed into (i) a filtered component and (ii) a sub-grid component. The filtered component is represented in the visual and numerical output. The sub-grid component is not shown explicitly, but the effects on the filtered flow field are captured by replacing the molecular viscosity with an effective viscosity within the momentum transport equations. In this sense, the effective viscosity of the fluid, $$\nu^*$$, varies as a function of space and time according to:

$\nu^* = \nu_0+\nu_t$

where $$\nu_0$$ is the molecular viscosity and $$\nu_t$$ is the sub-grid eddy viscosity calculated from:

$\nu_t=\left(C_s \Delta_x \right)^2 \overline{ {S}},$

where $$C_s$$ is an empirically-determined Smagorinsky coefficient, $$\Delta_x$$ is the local lattice spacing, and $$\overline{S}$$ is the norm of the filtered strain rate tensor defined by:

$\overline{\textrm{S}}_{ij}=\frac{1}{2}\left( \partial_j \overline{u}_i+ \partial_i \overline{u}_j \right),$

where $$\overline{u}$$ is the filtered velocity and $$\overline{\textrm{S}}_{ij}$$ is the characteristic filtered rate of strain. The value of $$C_s$$ is set by comparing LES simulation predictions to those obtained from DNS simulation. The default value used in M-Star CFD is 0.1, a selection based on results from the literature. [1]

## Energy Dissipation¶

The energy dissipation rate (EDR) is calculated from:

$\epsilon = 2 \nu \displaystyle \sum_{i,j} \overline{\textrm{S}}_{ij} \overline{\textrm{S}}_{ij},$

where $$\epsilon$$ is measured in W/kg and $$\nu$$ is the kinematic viscosity (momentum diffusivity) measured in m^2/s.

For systems with an LES filter, this expression becomes:

$\epsilon = 2 \left(\nu_0+\nu_t\right) \displaystyle \sum_{i,j} \overline{\textrm{S}}_{ij} \overline{\textrm{S}}_{ij},$

such that $$\epsilon$$ can be decomposed into resolved and unresolved components:

$\epsilon_{\textrm{res}} = 2 \nu_0 \displaystyle \sum_{i,j} \overline{\textrm{S}}_{ij} \overline{\textrm{S}}_{ij}$

and

$\epsilon_{\textrm{unres}} = 2 \nu_t \displaystyle \sum_{i,j} \overline{\textrm{S}}_{ij} \overline{\textrm{S}}_{ij}.$

Note that $$\epsilon_{\textrm{unres}}$$ is related to the third power of $$\overline{\textrm{S}}_{ij}$$ through the additional terms defining $$\nu_t$$

In most LES models, the resolved energy dissipation is order-of-magnitude smaller than the resolved energy dissipation. That is, a majority of the energy that is thermalized by the fluid exits through unresolved, sub-grid fluid eddies. Although the magnitude of this energy dissipation is known, the nature of the eddies within the sub-grid model is unspecified. [2]

Note that, for laminar and transitional simulations, where the LES filter is not necessary, the unresolved energy dissipation will be zero. For turbulent systems, if the LES filter is deactivated by manually setting $$C_s$$ to zero, the unresolved energy dissipation will be zero but the resolved energy will likely fall short of the reported energy input. Within this latter case, however, the reported energy dissipation will approach the input power as the simulation resolution approaches that required to perform direct numerical simulations.

Also note that, within a lattice-Boltzmann simulation, the local strain rate can be computed directly from the non-equilibrium components of the probability density function at each point. Within the context of an LES filter, this approach provides a more self-consistent description of strain than explicit finite differencing across the filtered velocity field.

The resolved and unresolved EDR computed from finite difference approaches can be extracted from the filtered velocity field and compared to those reported directly from the solver. Although the results are likely to be order-of-magnitude consistent, those reported directly from the solver are more accurate.

## References¶

[1] DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann method.Y. Huidan, S.. Girimaji, L. Luo, Journal of Computational Physics 2005, 209:2, 599-616

[2] Turbulent Flows, S. Pope, Cambridge University Press 2000, Cambridge


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# Seminars & Colloquia Calendar

Geometric Analysis Seminar

## Construction of the moduli space of Higgs bundles using analytic methods.

#### Yue Fan, University of Maryland

Location:  via Zoom, https://rutgers.zoom.us/j/93880280324
Date & time: Tuesday, 27 October 2020 at 2:50PM - 3:50PM

Abstract: Introduced by Hitchin, a Higgs bundle $$(E,\Phi)$$ on a complex manifold $$X$$ is a holomorphic vector bundle $$E$$ together with an $$\End E$$-valued holomorphic 1-form $$\Phi$$. The moduli space of Higgs bundles was constructed by Nitsure where $$X$$ is a smooth projective curve and by Simpson where $$X$$ is a smooth projective variety. They both used Geometric Invariant Theory, and the moduli space is a quasi-projective variety. It is a folklore theorem that the Kuranishi slice method can be used to construct this moduli space as a complex space where $$X$$ is a closed Riemann surface. I will present a proof of this folklore theorem and show that the resulting complex space is biholomorphic to the one in the category of schemes. Moreover, I will briefly talk about some applications of this new construction.


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# What is “velocity distance” in astrophysics?

I was reading some papers on astrophysics, and in several of them, I've encountered velocity being used as distance. Or more precisely, distance being in dimensions of distance over time. For example, a paper referred to a group of galaxies at "roughly $2000\:\mathrm{km\:s^{-1}}$ in distance." Another used the specific phrase "velocity distance." It said there was a super-cluster "an observed concentration of galaxies at a velocity distance of ${\sim}2500$ to $4000\:\mathrm{km\:s^{-1}}$." I searched for what this could mean for quite a bit, and couldn't find it. If anyone has any idea what this means, some insight would be greatly appreciated.

It is used because of uncertainty in the Hubble constant. The relationship between recession velocity and distance is given by

$v=H_{0}d$

Where $H_{0}$ is the Hubble constant. Since that isn't known precisely, distances aren't known precisely, so talking about distances doesn't necessarily make sense.

Taking about the distance velocity on the other hand is using something they can be measured pretty precisely from the redshift. And you know that if object B had twice the velocity of object A then it is twice as far away even if you don't know the actual distances.

Since distances depend on the Hubble parameter then, especially back when it was only known to be in the range 50-100 km/s/Mpc there wasn't a great deal of point in reporting distances: to compare distances between two papers you'd have to convert them to velocities using whatever value of the Hubble parameter each paper used, since there is no guarantee they'd use the same value.

• So to be clear, is the velocity distance essentially the velocity a galaxy would be moving if no other forces besides the Hubble expansion acted on it? – RothX Apr 27 '17 at 14:19
• That's a better answer than mine - a very obvious (but never thought about by me) reason as to why it's done this way. – WetSavannaAnimal Apr 27 '17 at 14:22
• @RothX Phill's answer is also, of course, the reason why the same data are often talked about as $z$ (redshift) values; without thinking about it too much, I guess I'd always just assumed that it was just easier to have an everyday sized number instead of some godzillion number of parsecs. – WetSavannaAnimal Apr 27 '17 at 14:28
• @RothX kind of. For the great majority of things in the universe, the expansion velocity is much much greater than any extra velocity they might have. For nearby objects (such as in the local group of galaxies the local gravitational motion within the cluster is bigger, but then we are measuring the distance by other means (such as supernovae) target than from the redshift. So for galaxies where it is worth measuring the redshift, out is just assumed (quite reasonably) that all of the redshift velocity is due to cosmic expansion – PhillS Apr 27 '17 at 17:26

Hubble's law will convert the velocity to a proper (as opposed to co-moving) distance for you; indeed Wikipedia states one form of Hubble's law as:

"... the observation ... that ... Objects observed in deep space (extragalactic space, 10 megaparsecs (Mpc) or more) are found to have a Doppler shift interpretable as relative velocity away from Earth"

This law comes from the FLRW metric in the special case when the the standard $\Lambda{\rm CDM}$ cosmological equation of state prevails, and is also, of course, confirmed by observations first made by Edwin Hubble (but formerly theoretically predicted by Georges Lemâitre and contemporary theoreticians).

• So is the velocity distance essentially the velocity a galaxy would be moving if no other forces besides the Hubble expansion acted on it? – RothX Apr 27 '17 at 14:17
• @RothX yes, that's right – WetSavannaAnimal Apr 27 '17 at 14:20


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# How do you solve 2 + (3/(x-1)) = 10?

Mar 10, 2016

You must start by putting on an equal denominator.

#### Explanation:

The LCD (least common denominator) is $x - 1$

$\frac{2 \left(x - 1\right)}{x - 1} + \frac{3}{x - 1} = \frac{10 \left(x - 1\right)}{x - 1}$

We can now eliminate the denominators and solve.

$2 x - 2 + 3 = 10 x - 10$

$11 = 8 x$

$\frac{11}{8} = x$

Practice exercises:

1. Solve for x.

$\frac{2}{2 x + 3} + \frac{3}{x - 1} = \frac{5}{x - 1}$

Challenge problem:

Solve for x: $\frac{1}{x} + \frac{1}{x y} = \frac{2}{y + x}$

Good luck!


text:
### IIT JAM MATHS

DOUBLE INTEGRAL
BETA AND GAMMA FUNCTION
VOLUME
SURFACE AREA
GRADIENT, DIVERGENCE AND CURL
LINE INTEGRAL
GREENS THEOREM
SURFACE INTEGRAL
GAUSS DIVERGENCE THEOREM
STOKES THEOREM
CONSERVATIVE VECTOR FIELD
LIMITS
CONTINUITY
Continuity
DIFFERENTIABILITY
APPLICATION OF DERIVATIVES
MEAN VALUE THEOREM
Part B: FUNCTION OF TWO VARIABLES
INTRODUCTION
DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE
ORTHOGONAL TRAJECTORY
DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS
CAUCHY EULER EQUATIONS
DIFFERENTIAL EQUATION OF SECOND ORDER
Section 5: LINEAR ALGEBRA
VECTOR SPACE AND LINEAR EQUATIONS
ORTHOGONALITY
EIGENVALUES AND EIGENVECTORS
LINEAR TRANSFORMATION
Section 6: GROUP THEORY
Unit Group
Subgroup
CYCLIC GROUP
PERMUTATION GROUP
ISOMORPHISM OF GROUPS
EXTRNAL DIRECT PRODUCT
HOMOMORPHISM OF GROUP
SET THEORY
Set Theory
REAL NUMBERS
TOPOLOGY ON REAL LINE
Open Sets
REAL SEQUENCES
INFINITE SERIES
POWER SERIES

# Change of Order-1

Lesson Progress
0% Complete

Let us take the following example :

Theorem 1 :

Change the order of integration in $\int\limits_0^{a/2} {\int\limits_{{x^2}/a}^{x - {x^2}/a} {fdydx} }$ where f is a function of x and y.

Solution :

Step 1 : The first step involves getting the bounding curves of domain of integration.

Let us compare the limits of given integral to its standard form $\int\limits_0^{a/2} {\int\limits_{{x^2}/a}^{x - {x^2}/a} {fdydx}} \equiv \int\limits_{x = a}^{x = b} {\int\limits_{y = f(x)}^{y = {f_2}(x)} {fdydx}}$ comparing the limit, we get the bounding curves

The bounding curves are

$C_1 : y = \dfrac{{{x^2}}}{a}$

$C_2 : y = x - \dfrac{{{x^2}}}{a}$

$C_4 : x = a/2$

The first two curves are the parabolas

$C_1 : x^2 = ay$

and last two are straight lines

$C_2 : (x - a/2)^2 = -a(y - a/4)$ $C_3 : x = 0$ $C_4 : x = a/2$

Step 2 : The second step involves the plotting of the bounding curves and getting the domain of integration.
If the equation of curve is quadratic in one variable and linear in other variable, it will be a parabolas.

The equation of parabola is usually given in any of the two forms
(a) $(y - \beta)^2 = 4a(x - \alpha)$

(b) $(x - \alpha)^2 = 4b(y - \beta)$

The curves $C_1, C_2, C_3, C_4$ are plotted as

The region bounded by all the four curves is the domain of integration.

Step 3 : Third step involves determining the limits after changing the order.
On changing the order, the integral converts to

$\int\limits_{y = c}^{y = d} {\int\limits_{x = {g_1}(y)}^{y = {g_2}(y)} {fdxdy} }$

For finding the limit of x, let us draw an arrow parallel to x-axis and intersecting the region of integration arbitrarily

The arrow first intersects curve $C_2$ at point A, write the equation of curve $C_2$ in the form of $x = g_1(y)$

$x = \dfrac{a}{2} \pm \dfrac{1}{2}\sqrt {{a^2} - 4ay}$

Since, point A lies towards positive x

So, $x = \dfrac{a}{2} + \dfrac{1}{2}\sqrt {{a^2} - 4ay}$

This is the lower limit of x

The arrow then intersects curve $C_1$ at point B, write the equation of $C_1$ in the form of $x = g_2(y)$

$x = \sqrt {ay}$

This is the upper limit of x.

In order to transverse the whole region of integration the arrow has to be moved from y = 0 to $y = \dfrac{a}{4}$ so, these are the limit of y.
Hence, the integral after changing the order becomes

$I = \int\limits_0^{a/4} {\int\limits_{\dfrac{a}{2} + \dfrac{1}{2}\sqrt {{a^2} - 4ay} }^{\sqrt {ay} } {fdxdy} }$


text:
Multi-armed quantum bandits: Exploration versus exploitation when learning properties of quantum states

Josep Lumbreras1, Erkka Haapasalo1, and Marco Tomamichel1,2

1Centre for Quantum Technologies, National University of Singapore, Singapore
2Department of Electrical and Computer Engineering, Faculty of Engineering, National University of Singapore, Singapore

Abstract

We initiate the study of tradeoffs between exploration and exploitation in online learning of properties of quantum states. Given sequential oracle access to an unknown quantum state, in each round, we are tasked to choose an observable from a set of actions aiming to maximize its expectation value on the state (the reward). Information gained about the unknown state from previous rounds can be used to gradually improve the choice of action, thus reducing the gap between the reward and the maximal reward attainable with the given action set (the regret). We provide various information-theoretic lower bounds on the cumulative regret that an optimal learner must incur, and show that it scales at least as the square root of the number of rounds played. We also investigate the dependence of the cumulative regret on the number of available actions and the dimension of the underlying space. Moreover, we exhibit strategies that are optimal for bandits with a finite number of arms and general mixed states.

► References

[1] T. Lattimore and C. Szepesvári. Bandit algorithms''. Cambridge University Press. (2020).
https://doi.org/10.1017/9781108571401

[2] A. Slivkins. Introduction to multi-armed bandits''. Foundations and Trends in Machine Learning 12, 1–286 (2019).
https://doi.org/10.1561/2200000068

[3] S. Bubeck and N. Cesa-Bianchi. Regret analysis of stochastic and nonstochastic multi-armed bandit problems''. Foundations and Trends in Machine Learning 5, 1–122 (2012).
https://doi.org/10.1561/2200000024

[4] D. Bouneffouf, I. Rish, and C. Aggarwal. Survey on applications of multi-armed and contextual bandits''. In 2020 IEEE Congress on Evolutionary Computation (CEC). Pages 1–8. (2020).
https://doi.org/10.1109/CEC48606.2020.9185782

L. Tang, R. Rosales, A. Singh, and D. Agarwal. Automatic ad format selection via contextual bandits''. In Proceedings of the 22nd ACM International Conference on Information and Knowledge Management. Page 1587–1594. Association for Computing Machinery (2013).
https://doi.org/10.1145/2505515.2514700

[6] M. Cohen, I. Lobel, and R. Paes Leme. Feature-based dynamic pricing''. Management Science 66, 4921–4943 (2020).
https://doi.org/10.1287/mnsc.2019.3485

[7] W. Thompson. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples''. Biometrika 25, 285–294 (1933).
https://doi.org/10.1093/biomet/25.3-4.285

[8] H. Robbins. Some Aspects of the Sequential Design of Experiments''. Bulletin of the American Mathematical Society 58, 527–535 (1952).
https://doi.org/10.1090/S0002-9904-1952-09620-8

[9] T.L Lai and H. Robbins. Asymptotically efficient adaptive allocation rules''. Advances in Applied Mathematics 6, 4–22 (1985).
https://doi.org/10.1016/0196-8858(85)90002-8

[10] P. Auer, N. Cesa-Bianchi, and P. Fischer. Finite-time analysis of the multiarmed bandit problem''. Mach. Learn. 47, 235–256 (2002).
https://doi.org/10.1023/A:1013689704352

[11] B. Casalé, G. Di Molfetta, H. Kadri, , and L. Ralaivola. Quantum bandits''. Quantum Mach. Intell. 2 (2020).
https://doi.org/10.1007/s42484-020-00024-8

[12] D. Wang, X. You, T. Li, and A. Childs. Quantum exploration algorithms for multi-armed bandits''. In Proceedings of the AAAI Conference on Artificial Intelligence. Volume 35, pages 10102–10110. (2021).

[13] P. Rebentrost, Y. Hamoudi, M. Ray, X. Wang, S. Yang, and M. Santha. Quantum algorithms for hedging and the learning of ising models''. Phys. Rev. A 103, 012418 (2021).
https://doi.org/10.1103/PhysRevA.103.012418

[14] O. Shamir. On the complexity of bandit linear optimization''. In Proceedings of The 28th Conference on Learning Theory. Volume 40 of Proceedings of Machine Learning Research, pages 1523–1551. PMLR (2015).

[15] P. Rusmevichientong and J. Tsitsiklis. Linearly parameterized bandits''. Mathematics of Operations Research 35 (2008).
https://doi.org/10.1287/moor.1100.0446

[16] J. Barry, D.T. Barry, and S. Aaronson. Quantum partially observable markov decision processes''. Phys. Rev. A 90, 032311 (2014).
https://doi.org/10.1103/PhysRevA.90.032311

[17] M. Ying, Y. Feng, and S. Ying. Optimal policies for quantum markov decision processes''. International Journal of Automation and Computing 18, 410–421 (2021).
https://doi.org/10.1007/s11633-021-1278-z

[18] M. Paris and J. Rehacek. Quantum state estimation''. Springer Publishing Company, Incorporated. (2010). 1st edition.
https://doi.org/10.1007/b98673

[19] S. Aaronson. Shadow tomography of quantum states''. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Page 325–338. STOC 2018. Association for Computing Machinery (2018).
https://doi.org/10.1145/3188745.3188802

[20] S. Aaronson, X. Chen, E. Hazan, S. Kale, and A. Nayak. Online learning of quantum states''. Journal of Statistical Mechanics: Theory and Experiment 2019 (2018).
https://doi.org/10.1088/1742-5468/ab3988

[21] J. Bretagnolle and C. Huber. Estimation des densités: risque minimax''. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 47, 119–137 (1979).
https://doi.org/10.1007/BF00535278

[22] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel. On Quantum Rényi Entropies: A New Generalization and Some Properties''. Journal of Mathematical Physics 54, 122203 (2013).
https://doi.org/10.1063/1.4838856

[23] M. Wilde, A. Winter, and D. Yang. Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy''. Communications in Mathematical Physics 331, 593–622 (2014).
https://doi.org/10.1007/s00220-014-2122-x

[24] W. Hoeffding. Probability inequalities for sums of bounded random variables''. Journal of the American Statistical Association 58, 13–30 (1963).
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[25] P. Auer. Using confidence bounds for exploitation-exploration trade-offs''. J. Mach. Learn. Res. 3, 397–422 (2003).
https://doi.org/10.5555/944919.944941

[26] D. Varsha, T. Hayes, and S.Kakade. Stochastic linear optimization under bandit feedback.''. In Proceedings of the 21st Conference on Learning Theory. Pages 355–366. (2008).

[27] P. Rusmevichientong and J. N. Tsitsiklis. Linearly parameterized bandits''. Mathematics of Operations Research 35, 395–411 (2010).
https://doi.org/10.1287/moor.1100.0446

[28] Y. Abbasi-Yadkori, D. Pál, and Cs. Szepesvári. Improved algorithms for linear stochastic bandits''. In Advances in Neural Information Processing Systems. Volume 24. Curran Associates, Inc. (2011).

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Cited by

[1] Zongqi Wan, Zhijie Zhang, Tongyang Li, Jialin Zhang, and Xiaoming Sun, "Quantum Multi-Armed Bandits and Stochastic Linear Bandits Enjoy Logarithmic Regrets", arXiv:2205.14988.

[2] Xinyi Chen, Elad Hazan, Tongyang Li, Zhou Lu, Xinzhao Wang, and Rui Yang, "Adaptive Online Learning of Quantum States", arXiv:2206.00220.

The above citations are from SAO/NASA ADS (last updated successfully 2022-08-13 14:15:28). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-08-13 14:15:27).


text:
Moscow Mathematical Journal

Volume 20, Issue 1, January–March 2020  pp. 27–42.

Matrix Polar Decomposition and Generalisations of the Blaschke–Petkantschin Formula in Integral Geometry

Authors:  Peter J. Forrester (1)
Author institution:(1) Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia

Summary:

In the work [Bull. Austr. Math. Soc. 85 (2012), 315–234], S. R. Moghadasi has shown how the decomposition of the $N$-fold product of Lebesgue measure on $\mathbb{R}^n$ implied by matrix polar decomposition can be used to derive the Blaschke–Petkantschin decomposition of measure formula from integral geometry. We use known formulas from random matrix theory to give a simplified derivation of the decomposition of Lebesgue product measure implied by matrix polar decomposition, applying too to the cases of complex and real quaternion entries, and we give corresponding generalisations of the Blaschke–Petkantschin formula. A number of applications to random matrix theory and integral geometry are given, including to the calculation of the moments of the volume content of the convex hull of $k \le N+1$ points in $\mathbb{R}^N$, $\mathbb{C}^N$ or $\mathbb{H}^N$ with a Gaussian or uniform distribution.

2010 Math. Subj. Class. 15B52; 52A22.

Keywords: Blaschke–Petkantschin formula, matrix polar decomposition, integral geometry


text:
## Welcome to Cross Validated

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It is built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we are working together to build a library of detailed answers to every question about statistics. Our Help Center provides further guidance to help in posting good questions and answers.

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# Can two random variables have the same distribution, yet be almost surely different?

Is it possible that two random variables have the same distribution and yet they are almost surely different?

Let $X\sim N(0,1)$ and define $Y=-X$. It is easy to prove that $Y\sim N(0,1)$.

But $$P\{\omega : X(\omega)=Y(\omega)\} = P\{\omega : X(\omega)=0,Y(\omega)=0\} \leq P\{\omega : X(\omega)=0\} = 0 \, .$$

Hence, $X$ and $Y$ are different with probability one.

Any pair of independent random variables $X$ and $Y$ having the same continuous distribution provides a counterexample.

In fact, two random variables having the same distribution are not even necessarily defined on the same probability space, hence the question makes no sense in general.

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# Can two random variables have the same distribution, yet be almost surely different?

Is it possible that two random variables have the same distribution and yet they are almost surely different?

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Let $X\sim N(0,1)$ and define $Y=-X$. It is easy to prove that $Y\sim N(0,1)$.

But $$P\{\omega : X(\omega)=Y(\omega)\} = P\{\omega : X(\omega)=0,Y(\omega)=0\} \leq P\{\omega : X(\omega)=0\} = 0 \, .$$

Hence, $X$ and $Y$ are different with probability one.

This same trick works much more generally and even in cases that might "appear" simpler to someone first encountering the subject. For example, consider $X$ and $1-X$ where $X$ is a Bernoulli random variable with probability of success being $1/2$. - cardinal Mar 20 '12 at 17:07

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text:
Location of Repository

## Discovery of the Binary Pulsar PSR B1259-63 in Very-High-Energy Gamma Rays around Periastron with H.E.S.S

### Abstract

10 pages, 8 figures, accepted in Astronomy and Astrophysics on 2 June 2005, replace: document unchanged, replaced author field in astro-ph entry - authors are all members of the H.E.S.S. collaboration and three additional authors (99+3, see document)We report the discovery of very-high-energy (VHE) $\gamma$-ray emission of the binary system PSR B1259-63/SS 2883 of a radio pulsar orbiting a massive, luminous Be star in a highly eccentric orbit. The observations around the 2004 periastron passage of the pulsar were performed with the four 13 m Cherenkov telescopes of the H.E.S.S. experiment, recently installed in Namibia and in full operation since December 2003. Between February and June 2004, a $\gamma$-ray signal from the binary system was detected with a total significance above 13 $\sigma$. The flux was found to vary significantly on timescales of days which makes PSR B1259-63 the first variable galactic source of VHE $\gamma$-rays observed so far. Strong emission signals were observed in pre- and post-periastron phases with a flux minimum around periastron, followed by a gradual flux decrease in the months after. The measured time-averaged energy spectrum above a mean threshold energy of 380 GeV can be fitted by a simple power law $F_0(E/1 TeV)^{-\Gamma}$ with a photon index $\Gamma = 2.7\pm0.2_{stat}\pm0.2_{sys}$ and flux normalisation $F_0 = (1.3\0.1_{stat}\pm0.3_{sys})\times10^{-12} TeV^{-1} cm^{-2} s^{-1}$. This detection of VHE $\gamma$-rays provides unambiguous evidence for particle acceleration to multi-TeV energies in the binary system. In combination with coeval observations of the X-ray synchrotron emission by the RXTE and INTEGRAL instruments, and assuming the VHE \$\gamma¤-ray emission to be produced by the inverse Compton mechanism, the magnetic field strength can be directly estimated to be of the order of 1 G

Topics: [SDU.ASTR] Sciences of the Universe [physics]/Astrophysics [astro-ph], [PHYS.ASTR.CO] Physics [physics]/Astrophysics [astro-ph]/Cosmology and Extra-Galactic Astrophysics [astro-ph.CO]
Publisher: EDP Sciences
Year: 2005
DOI identifier: 10.1051/0004-6361:20052983
OAI identifier: oai:HAL:in2p3-00024875v1
Provided by: Hal-Diderot


text:
What units are represented by pr - micrometer?

In studying column abundances, I often encounter the units $$pr-\mu m$$. What does this stand for? Part per micrometer? how can this be an abundance unit?

Incredibility I cannot figure out what it means after quite a bit of online searching.

• Likely per micrometer. The pr is not standard. I imagine it would be commonly used in display technology and micofabrication for density. In molecular fields you have the related ppm for parts per million. – Paul Childs Jun 6 '19 at 1:03


text:
## Elementary Algebra

Each piece of fuel line will be $\frac{23}{8}$ feet long.
The piece of fuel line is 5$\frac{3}{4}$ feet long, or $\frac{23}{4}$ feet long. He needs to cut it into 2 equal pieces. So, we divide $\frac{23}{4}$ by 2: $\frac{23}{4}$ $\div$ 2 = $\frac{23}{4}$ $\times$ $\frac{1}{2}$ = $\frac{23}{8}$ Each piece of fuel line will be $\frac{23}{8}$ feet long.


text:
# Tag Info

Let $X,Y$ be metric spaces and denote their distance functions by $d_X$ and $d_Y$. A mapping $f: X\to Y$ is said to be Hölder continuous with exponent $\alpha \in (0,1)$ (sometimes $\alpha$-Hölder continuous), and denoted $f\in C^\alpha(X,Y)$ or $f\in C^{0,\alpha}(X,Y)$ if

$$\sup_{x_1,x_2\in X; x_1\neq x_2} \frac{d_Y\left( f(x_1),f(x_2)\right)}{d_X\left(x_1,x_2\right)^\alpha} < \infty$$

The value of the supremum is sometimes denoted $[f]_\alpha$ and is called the Hölder coefficient.

The case $\alpha = 1$ (which is always denoted $C^{0,1}$ and not $C^1$ so as not to be confused with the space of continuously differentiable functions) corresponds to Lipschitz continuity.

The notion of Hölder continuity is used to quantify how continuous (and how close to differentiable) a function is. It can be extended also to higher derivatives: letting $X$ and $Y$ be subsets of Euclidean spaces, we can define the space $C^{k,\alpha}(X,Y)$ to be subspace of $k$-times continuously differentiable functions all of whose $k$th partial derivatives are $\alpha$-Hölder continuous. This can be made in to a Banach space with the norm

$$\|f\|_{k,\alpha} = \|f\|_{C^k} + \sum_{|\gamma| = k} [D^\gamma f]_\alpha$$

The various Hölder spaces are frequently used to study quantitative estimates of differentiability in harmonic analysis and analysis of elliptic and parabolic partial differential equations.