text:
From H^{1} to L^{2}

# Characteristic boundary value problems: estimates from H1 to L2

## Abstract.

Motivated by the study of certain non linear free-boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the solution to certain boundary value problems, in the conormal Sobolev space , can be transformed into an a priori estimate of the same problem.

###### Key words and phrases:
Boundary value problem, characteristic boundary, pseudo-differential operators, anisotropic and conormal Sobolev spaces, Magneto-Hydrodynamics.
###### 2000 Mathematics Subject Classification:
35L40, 35L50, 35L45.

## 1. Introduction and main results

The present paper is motivated by the study of certain non linear free boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics (MHD).

The well-posedness of initial boundary value problems for hyperbolic PDEs was studied by Kreiss [15] for systems and Sakamoto [28, 29] for wave equations. The theory was extended to free-boundary problems for a discontinuity by Majda [17, 18]. He related the discontinuity problem to a half-space problem by adding a new variable that describes the displacement of the discontinuity, and making a change of independent variables that “flattens”the discontinuity front. The result is a system of hyperbolic PDEs that is coupled with an equation for the displacement of the discontinuity. Majda formulated analogs of the Lopatinskiĭ and uniform Lopatinskiĭ conditions for discontinuity problems, and proved a short-time, nonlinear existence and stability result for Lax shocks in solutions of hyperbolic conservation laws that satisfy the uniform Lopatinskiĭ condition (see [3, 19] for further discussion).

Interesting and challenging problems arise when the discontinuity is weakly but not strongly stable, i.e. the Lopatinskiĭ condition only holds in weak form, because surface waves propagate along the discontinuity, see [11, 14]. A general theory for the evolution of such weakly stable discontinuities is lacking.

A typical difficulty in the analysis of weakly stable problems is the loss of regularity in the a priori estimates of solutions. Short-time existence results have been obtained for various weakly stable nonlinear problems, typically by the use of a Nash-Moser scheme to compensate for the loss of derivatives in the linearized energy estimates, see [2, 7, 12, 32, 34].

A fundamental part of the general approach described above is given by the proof of the well-posedness of the linear boundary value problems (shortly written BVPs in the sequel) obtained from linearizing the nonlinear problem (in the new independent variables with “flattened”boundary) around a suitable basic state. This requires two things: the proof of a linearized energy estimate, and the existence of the solution to the linearized problem.

In case of certain problems arising in MHD, a spectral analysis of the linearized equations, as required by the Kreiss-Lopatinskiĭ theory, seems very hard to be obtained because of big algebraic difficulties. An alternative approach for the proof of the linearized a priori estimate is the energy method. This method has been applied successfully to the linearized MHD problems by Trakhinin (see [33, 35] and other references); typically the method gives an a priori estimate for the solution in the conormal Sobolev space (see Section 2 for the definition of this space) bounded by the norm of the source term in the same function space (or a space of higher order in case of loss of regularity).

Once given the a priori estimate, the next point requires the proof of the existence of the solution to the linearized problem. Here one finds a new difficulty. The classical duality method for the existence of a weak solution requires an a priori estimate for the dual problem (usually similar to the given linearized problem) of the form (from the data in the interior to the solution, disregarding for simplicity the boundary regularity). In case of loss of derivatives, when for the problem it is given an estimate of the form , one would need an estimate of the form for the dual problem, see [10].

The existence of a solution directly in would require an a priori estimate for the dual problem in the dual spaces (possibly of the form in case of loss of regularity), but it is not clear how to get it.

This difficulty motivates the present paper. We show that an a priori estimate of the solution to certain BVPs in the conormal Sobolev space can be transformed into an a priori estimate, with the consequence that the existence of a weak solution can be obtained by the classical duality argument.

The most of the paper is devoted to the proof of this result. In the Appendix we present some examples of free-boundary problems in MHD that fit in the general formulation described below.

For a given integer , let denote the dimensional positive half-space

 Rn+:={x=(x1,…,xn)∈Rn:x1>0}.

We also use the notation . The boundary of will be sistematically identified with .
We are interested in a boundary value problem of the following form

 Lγu+ρ♯u=F, inRn+, (1a) bγψ+Mγu+b♯ψ+ℓ♯u=g, onRn−1. (1b)

In (1a), is the first-order linear partial differential operator

 Lγ=Lγ(x,D):=γIN+n∑j=1Aj(x)∂j+B(x), (2)

where the shortcut , for , is used hereafter and denotes the identity matrix. The coefficients , () are real matrix-valued functions in , the space of restrictions to of functions of 1.
In (1b),

 bγ=bγ(x′,D′):=γb0(x′)+n∑j=2bj(x′)∂j+β(x′), (3a) Mγ=Mγ(x′,D′):=γM0(x′)+n∑j=2Mj(x′)∂j+M(x′) (3b)

are first-order linear partial differential operators, acting on the tangential variables ; for a given integer , the coefficients , and , (for ) are functions in taking values in the spaces and respectively. Finally, in (1a) and , in (1b) stand for “lower order operators” of pseudo-differential type, acting “tangentially” on , whose symbols belong to suitable symbol classes introduced in Section 3.1. The operators , , must be understood as some “lower order perturbations” of the leading operators , and in the equations (1); in the following we assume that the problem (1), with given operators , , , obeys a suitable a priori estimate which has to be “stable” under the addition of arbitrary lower order terms , , in the interior equation (1a) and the boundary condition (1b) (see the assumptions , below). The structure of the operators (3) and , , in (1) will be better described later on.
The unknown , as well as the source term , are valued functions of , the unknown is a scalar function of 2 and the boundary datum is an valued function of . We may assume that and are compactly supported in the unitary dimensional positive half-cylinder . Analogously, we assume that and are compactly supported in the unitary dimensional ball . For an arbitrary , we also set and .
The BVP (1) has characteristic boundary of constant multiplicity in the following sense: the coefficient of the normal derivative in displays the block-wise structure

 A1(x)=(AI,I1AI,II1AII,I1AII,II1), (4)

where , , , are respectively , , , sub-matrices, such that

 AI,II1|x1=0=0,AII,I1|x1=0=0,AII,II1|x1=0=0, (5)

and is invertible over . According to the representation above, we split the unknown as ; and are said to be respectively the noncharacteristic and the characteristic components of .
Concerning the boundary condition (1b), we firstly assume that the number of scalar boundary conditions obeys the assumption . As regards to the structure of the boundary operator in (3b), we require that actually it acts nontrivially only on the noncharacteristic component of ; moreover we assume that the first-order leading part of only applies to a subset of components of the non characteristic vector , namely there exists an integer , with , such that the coefficients , of take the form

 Mj=(Msj0),M=(MI0),j=0,2,…,n, (6)

where the matrices () and belong respectively to and . If we set , then the operator in (3b) may be rewritten, according to (6), as

 Mγu=MsγuI,s+MIuI, (7)

where is the first-order leading operator

 Msγ(x′,D′):=γMs0+n∑j=2Msj∂j. (8)

As we just said, the operator must be understood as a lower order perturbation of the leading part of the boundary operator in (7); hence, according to the form of , we assume that only acts on the component of the unknown vector , that is

 ℓ♯(x′,D′,γ)u=ℓ♯(x′,D′,γ)uI,s. (9)

A BVP of the form (1), under the structural assumptions (4)-(7), comes from the study of certain non linear free boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-hydrodynamics. Such problems model the motion of a compressible inviscid fluid, under the action of a magnetic field, when the fluid may develop discontinuities along a moving unknown characteristic hypersurface. As we already said, to show the local-in-time existence of such a kind of piecewise discontinuous flows, the classical approach consists, firstly, of reducing the original free boundary problem to a BVP set on a fixed domain, performing a nonlinear change of coordinates that sends the front of the physical discontinuities into a fixed hyperplane of the space-time domain. Then, one starts to consider the well posedness of the linear BVP obtained from linearizing the found nonlinear BVP around a basic state provided by a particular solution. The resulting linear problem displays the structure of the problem (1), where the unknown represents the set of physical variables involved in the model, while the unknown encodes the moving discontinuity front. The solvability of the linear BVP firstly requires that a suitable a priori estimate can be attached to the problem.
Let the operators , , be given, with structure described by formulas (2), (3a), (4)-(8) above. We assume that the two alternative hypotheses are satisfied:

• A priori estimate with loss of regularity in the interior term. For all symbols , and , taking values respectively in , and , there exist constants , , depending only on the matrices , , , , , in (2), (3), (7), (8) and a finite number of semi-norms of , , , such that for all functions , compactly supported on , , compactly supported on , and all the following a priori energy estimate is satisfied

 γ(||u||2H1tan,γ(Rn+)+||uI|x1=0||2H1/2γ(Rn−1))+γ2||ψ||2H1γ(Rn−1)≤C0(1γ3||F||2H2tan,γ(Rn+)+1γ||g||2H3/2γ(Rn−1)), (10)

where , and , , are respectively the pseudo-differential operators with symbols , , .

• A priori estimate without loss of regularity in the interior term. For all symbols and , taking values respectively in and , there exist constants , , depending only on the matrices , , , , , in (2), (3), (7), (8) and a finite number of semi-norms of , , such that for all functions , compactly supported on , , compactly supported on , and all the following a priori energy estimate is satisfied

 γ(||u||2H1tan,γ(Rn+)+||uI|x1=0||2H1/2γ(Rn−1))+γ2||ψ||2H1γ(Rn−1)≤C0γ(||F||2H1tan,γ(Rn+)+||g||2H3/2γ(Rn−1)), (11)

where , and , are respectively the pseudo-differential operators with symbols , .

The symbol class and the related pseudo-differential operators will be introduced in Section 3.1. The function spaces and the norms involved in the estimates (10), (11) will be described in Section 2.
By the hypotheses and , we require that an a priori estimate in the tangential Sobolev space (see the next Section 2 and Definition 3 below) is enjoyed by the BVP (1). The structure of the estimate is justified by the physical models that we plan to cover (see the Appendix B). The inserting of the zeroth order terms , , in the interior source term and the boundary datum is a property of stability of the estimates (10), (11), under lower order operators. We notice, in particular, that the addition of in the boundary condition (1b) only modifies the zeroth order term for the part that applies to the components of the noncharacteristic unknown vector , see (7), (9). This behavior of the boundary condition, under lower order perturbations, is inspired by the physical problems to which we address. It happens sometimes that the specific structure of some coefficients involved in the zeroth order part of the original ”unperturbed” boundary operator (7) is needed in order to derive an a priori estimate of the type (10) or (11) for the corresponding BVP (1); hence these coefficients of the boundary operator must be kept unchanged by the addition of some lower order perturbations.
Note also that the two a priori estimates in (10), (11) exhibit a different behavior with respect to the interior data: in (10) a loss of one tangential derivative from the interior data occurs, whereas in (11) no loss of interior regularity is assumed. According to this different behavior, a stability assumption under lower order perturbations of the interior operator is only required in .
Both the estimates exhibit the same loss of regularity from the boundary data.

The aim of this paper is to prove the following result.

###### Theorem 1.

Assume that the operators , , have the structure described in (2), (3a), (4)-(8). Let .

• If the assumption holds true, then for all symbols , there exist constants , , depending only on the matrices , , , , , in (2), (3), (7), (8), and a finite number of semi-norms of , , , such that for all functions , compactly supported on , , compactly supported on , and all the following a priori energy estimate is satisfied

 γ(||u||2L2(Rn+)+||uI|x1=0||2H−1/2γ(Rn−1))+γ2||ψ||2L2(Rn−1)≤˘C0(1γ3||F||2H1tan,γ(Rn+)+1γ||g||2H1/2γ(Rn−1)), (12)

where and .

• If the assumption holds true, then for every pair of symbols there exist constants , , depending only on the matrices , , , , , in (2), (3), (7), (8), and a finite number of semi-norms of , , such that for all functions , compactly supported on , , compactly supported on , and all the following a priori energy estimate is satisfied

 (13)

where and .

The paper is organized as follows. In Section 2 we introduce the function spaces to be used in the following and the main related notations. In Section 3 we collect some technical tools, and the basic concerned results, that will be useful for the proof of Theorem 1, given in Section 4.
The Appendix A contains the proof of the most of the technical results used in Section 4. The Appendix B is devoted to present some free boundary problems in MHD, that can be stated within the general framework developed in the paper.

## 2. Function Spaces

The purpose of this Section is to introduce the main function spaces to be used in the following and collect their basic properties. For and , we set

 λs,γ(ξ):=(γ2+|ξ|2)s/2 (14)

and, in particular, .
The Sobolev space of order in is defined to be the set of all tempered distributions such that , being the Fourier transform of . For , the Sobolev space of order reduces to the set of all functions such that , for all multi-indices with , where we have set

 ∂α:=∂α11…∂αnn,α=(α1,…,αn),

and , as it is usual.
Throughout the paper, for real , will denote the Sobolev space of order , equipped with the depending norm defined by

 ||u||2s,γ:=(2π)−n∫Rnλ2s,γ(ξ)|ˆu(ξ)|2dξ, (15)

( are the dual Fourier variables of ). The norms defined by (15), with different values of the parameter , are equivalent each other. For we set for brevity (and, accordingly, ).
It is clear that, for , the norm in (15) turns out to be equivalent, uniformly with respect to , to the norm defined by

 ||u||2Hsγ(Rn):=∑|α|≤sγ2(s−|α|)||∂αu||2L2(Rn). (16)

Another useful remark about the parameter depending norms defined in (15) is provided by the following counterpart of the usual Sobolev imbedding inequality

 ||u||s,γ≤γs−r||u||r,γ, (17)

for arbitrary and .

###### Remark 2.

In Section 4, the ordinary Sobolev spaces, endowed with the weighted norms above, will be considered in (interpreted as the boundary of the half-space ) and used to measure the smoothness of functions on the boundary; regardless of the different dimension, the same notations and conventions as before will be used there.

The appropriate functional setting where one measures the internal smoothness of solutions to characteristic problems is provided by the anisotropic Sobolev spaces introduced by Shuxing Chen [8] and Yanagisawa, Matsumura [36], see also [30] . Indeed these spaces take account of the loss of normal regularity with respect to the boundary that usually occurs for characteristic problems.
Let be a monotone increasing function such that in a neighborhood of the origin and for any large enough.
For , we set

 Z1:=σ(x1)∂1,Zj:=∂j,forj≥2.

Then, for every multi-index , the differential operator in the tangential direction (conormal derivative) of order is defined by

 Zα:=Zα11…Zαnn.

Given an integer the anisotropic Sobolev space of order is defined as the set of functions such that , for all multi-indices and with , see [22] and the references therein. Agreeing with the notations set for the usual Sobolev spaces, for , will denote the anisotropic space of order equipped with the depending norm

 ||u||2Hm∗,γ(Rn+):=∑|α|+2k≤mγ2(m−|α|−2k)||Zα∂k1u||2L2(Rn+). (18)

Similarly, the conormal Sobolev space of order is defined to be the set of functions such that , for all multi-indices with . For , denotes the conormal space of order equipped with the depending norm

 ||u||2Hmtan,γ(Rn+):=∑|α|≤mγ2(m−|α|)||Zαu||2L2(Rn+). (19)

In the end, we remark that the following identity holds true. However, for a Sobolev order the continuous imbedding is fulfilled with the strict inclusion relation.
Since the functions we are dealing with, throughout the paper, vanish for large (as they are compactly supported on ), without the loss of generality we assume the conormal derivative to coincide with the differential operator from now on 3. This reduction will make easier to implement on conormal spaces the technical machinery that will be introduced in the next Section.

## 3. Preliminaries and technical tools

We start by recalling the definition of two operators and , introduced by Nishitani and Takayama in [26], with the main property of mapping isometrically square integrable (resp. essentially bounded) functions over the half-space onto square integrable (resp. essentially bounded) functions over the full space .
The mappings and are respectively defined by

 w♯(x):=w(ex1,x′)ex1/2,a♮(x)=a(ex1,x′),∀x=(x1,x′)∈Rn. (20)

They are both norm preserving bijections.
It is also useful to notice that the above operators can be extended to the set of Schwartz distributions in . It is easily seen that both and are topological isomorphisms of the space of test functions in (resp. ) onto the space of test functions in (resp. ). Therefore, a standard duality argument leads to define and on , by setting for every

 ⟨u♯,φ⟩:=⟨u,φ♯−1⟩, (21) ⟨u♮,φ⟩:=⟨u,φ♭⟩ (22)

( is used to denote the duality pairing between distributions and test functions either in the half-space or the full space ). In the right-hand sides of (21), (22), is just the inverse operator of , that is

 φ♯−1(x)=1√x1φ(logx1,x′),∀x1>0,x′∈Rn−1, (23)

while the operator is defined by

 φ♭(x)=1x1φ(logx1,x′),∀x1>0,x′∈Rn−1, (24)

for functions . The operators and arise by explicitly calculating the formal adjoints of and respectively.
Of course, one has that ; moreover the following relations can be easily verified (cf. [26])

 (ψu)♯=ψ♮u♯, (25) ∂j(u♮)=(Zju)♮,j=1,…,n, (26) ∂1(u♯)=(Z1u)♯+12u♯, (27) ∂j(u♯)=(Zju)♯,j=2,…,n, (28)

whenever and (in (25) and are even allowed).
From formulas (27), (28) and the boundedness of , it also follows that

 ♯:Hmtan,γ(Rn+)→Hmγ(Rn) (29)

is a topological isomorphism, for each integer and real .
The previous remarks give a natural way to extend the definition of the conormal spaces on to an arbitrary real order . More precisely we give the following

###### Definition 3.

For and , the space is defined as

 Hstan,γ(Rn+):={u∈D′(Rn+):u♯∈Hsγ(Rn)}

and is provided with the norm

 ||u||2s,tan,Rn+,γ:=||u♯||2s,γ=(2π)−n∫Rnλ2s,γ(ξ)|ˆu♯(ξ)|2dξ. (30)

It is obvious that, like for the real order usual Sobolev spaces, is a Banach space for every real ; furthermore, the above definition reduces to the one given in Section 2 when is a positive integer. Finally, for all , the operator becomes a topological isomorphism of onto .
In the end, we observe that the following

 ♯:C∞(0)(Rn+)→S(Rn),♮:C∞(0)(Rn+)→C∞b(Rn)

are linear continuous operators, where denotes the Schwartz space of rapidly decreasing functions in and the space of infinitely smooth functions in , with bounded derivatives of all orders; notice also that the last maps are not onto. Finally, we remark that

 ♯−1:S(Rn)→C∞(Rn+) (31)

is a bounded operator.

### 3.1. A class of conormal operators

The operator, defined at the beginning of Section 3, can be used to allow pseudo-differential operators in acting conormally on functions only defined over the positive half-space . Then the standard machinery of pseudo-differential calculus (in the parameter depending version introduced in [1], [6]) can be re-arranged into a functional calculus properly behaved on conormal Sobolev spaces described in Section 2. In Section 4, this calculus will be usefully applied to derive from the estimate (10) or (11) associated to the BVP (1) the corresponding estimate (12) or (13) of Theorem 1.
Let us introduce the pseudo-differential symbols, with a parameter, to be used later; here we closely follow the terminology and notations of [9].

###### Definition 4.

A parameter-depending pseudo-differential symbol of order is a real (or complex)-valued measurable function on , such that is with respect to and and for all multi-indices there exists a positive constant satisfying:

 |∂αξ∂βxa(x,ξ,γ)|≤Cα,βλm−|α|,γ(ξ), (32)

for all and .

The same definition as above extends to functions taking values in the space (resp. ) of real (resp. complex)-valued matrices, for all integers (where the module is replaced in (32) by any equivalent norm in (resp. )). We denote by the set of depending symbols of order (the same notation being used for both scalar or matrix-valued symbols). is equipped with the obvious norms

 |a|m,k:=max|α|+|β|≤ksup(x,ξ)∈Rn×Rn,γ≥1λ−m+|α|,γ(ξ)|∂αξ∂βxa(x,ξ,γ)|,∀k∈N, (33)

which turn it into a Fréchet space. For all , with , the continuous imbedding can be easily proven.
For all , the function is of course a (scalar-valued) symbol in .
Any symbol defines a pseudo-differential operator on the Schwartz space , by the standard formula

 ∀u∈S(Rn),∀x∈Rn,Opγ(a)u(x)=a(x,D,γ)u(x)


text:
# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

All Issues

## Discrete & Continuous Dynamical Systems - A

January 1997 , Volume 3 , Issue 1

Select all articles

Export/Reference:

1997, 3(1): 1-24 doi: 10.3934/dcds.1997.3.1 +[Abstract](1379) +[PDF](3712.6KB)
Abstract:
We study the global attractor of semilinear parabolic equations of the form

$u_t=u_{x x}+f(u,u_x),\ x\in\mathbb{R}$/$\mathbb{Z}, \ t>0.$

Under suitable conditions on $f$, the equation generates a global semiflow on a suitable function space. The general theory of inertial manifolds does not apply to this equation due to lack of the so-called spectral gap condition. Using a totally different method, we show that the global attractor is the graph of a continuous mapping of finite dimension. We also show that this dimension is equal to $2[N$/$2]+1$, where $N$ is the maximal value of the generalized Morse index of equilibria and periodic solutions. Note that we do not make any assumption regarding the hyperbolicity of those solutions. We further prove that there exists no homoclinic orbit nor heteroclinic cycle.

1997, 3(1): 25-34 doi: 10.3934/dcds.1997.3.25 +[Abstract](1029) +[PDF](1750.1KB)
Abstract:
The problem of the existence and multiplicity of periodic (harmonic and subharmonic) solutions to the parameter-dependent second order equation $x' '+g(x)=s+w(t)$ is investigated for $|s|$ large under suitable "jumping" conditions on $g'(\pm\infty)$. The results which are obtained complete and complement some recent theorems on the periodic Ambrosetti-Prodi problem.
1997, 3(1): 35-58 doi: 10.3934/dcds.1997.3.35 +[Abstract](1104) +[PDF](3038.5KB)
Abstract:
The paper introduces a notion of "shift-differentials" for maps with values in the space BV. These differentials describe first order variations of a given function $u$, obtained by horizontal shifts of the points of its graph. The flow generated by a scalar conservation law is proved to be generically shift-differentiable, according to the new definition.
1997, 3(1): 59-78 doi: 10.3934/dcds.1997.3.59 +[Abstract](1145) +[PDF](2748.4KB)
Abstract:
It is well known that an Euler-Bernoulli beam may be exactly controlled with a single control acting on an end of the beam. In this article we show that for certain boundary conditions, the same result holds for a beam that is surrounded by an incompressible, inviscid fluid with a sufficiently small density. The proof involves reducing the control problem to a moment problem and using compactness properties of the Neumann to Dirichlet map for the Laplacian operator to obtain the needed estimates.
1997, 3(1): 79-90 doi: 10.3934/dcds.1997.3.79 +[Abstract](986) +[PDF](2231.1KB)
Abstract:
In a previous paper [2] we made a classification of generic binary differential equations (BDE's)

$a(x,y)dy^2+2b(x,y)dxdy+c(x,y)dx^2=0$

near points at which the discriminant function $b^2-ac$ has a Morse singularity. Such points occur naturally in families of BDE's and here we describe the manner in which the configuration of solution curves change in their natural 1-parameter versal deformations.
The results in this paper can be used to describe, for instance, the changes in the structure of the asymptotic curves on a 1-parameter family of smooth surfaces acquiring a flat umbilic and on integral curves determined by eigenvectors of 1-parameter families of $2\times 2$ matrices. It also sheds light on the structure of the rarefraction curves associated to a $2\times 2$ system of conservation laws in 1 space variable.

1997, 3(1): 91-106 doi: 10.3934/dcds.1997.3.91 +[Abstract](927) +[PDF](1994.2KB)
Abstract:
We shall prove here the global solvability for small initial data for symmetric hyperbolic systems with integro-differential coefficients. In this way, we will extend some results obtained in [5], [6], [8], [11] for the classic Kirchhoff equation and in [3] for regularly hyperbolic systems.
M. Bauer and
1997, 3(1): 107-116 doi: 10.3934/dcds.1997.3.107 +[Abstract](1138) +[PDF](1797.8KB)
Abstract:
We consider billiard trajectories on the geodesic triangle $D$ in the hyperbolic half-plane with internal angles $0,0$ and $\pi /2$ at the vertices $\infty,1$, and $i$. The billiard map of $D$ sends a given trajectory $\gamma$ with the boundary $\delta D$ of $D$ to the next intersection point of $\gamma$ with $\delta D$.
By choosing an appropriate cross section for the billiard map, we show that the decay of correlation of the first return map is slower than $n^-2$. As a by-product, we enumerate the billiard trajectories in terms of their cutting sequences and relate the boundary expansion of the billiard map to continued fractions with even partial quotients.
1997, 3(1): 117-133 doi: 10.3934/dcds.1997.3.117 +[Abstract](1255) +[PDF](2590.2KB)
Abstract:
We construct a two-parameter family of self-similar solutions to both the compressible and incompressible two-dimensional Euler equations with axisymmetry. The equations can be reduced under the situation to two systems of ordinary differential equations. In the compressible and polytropic case, the system in autonomous form consists of four ordinary differential equations with a two-dimensional set of stationary points, one of which is degenerate up to order four. Through asymptotic analysis and computations of numerical solutions, we are fortunate to be able to recognize a one-parameter family of exact solutions in explicit form. All the solutions (exact or numerical) are globally bounded and continuous, have finite local energy and vorticity, and have well-defined initial and boundary values at time zero and spatial infinity respectively. Particle trajectories of some of these solutions are spiral-like. In the incompressible case, we also find explicit self-similar axisymmetric spiral solutions, which are, however, somewhat less physical due to unbounded pressures or infinite local energy near their swirling centers.
1997, 3(1): 135-151 doi: 10.3934/dcds.1997.3.135 +[Abstract](1320) +[PDF](3162.0KB)
Abstract:
We generalize the Aubry-Mather theorem on the existence of quasi-periodic solutions of one dimensional difference equations to situations in which the independent variable ranges over more complicated lattices. This is a natural generalization of Frenkel-Kontorova models to physical situations in a higher dimensional space. We also consider generalizations in which the interactions among the particles are not just nearest neighbor, and indeed do not have finite range.

2018  Impact Factor: 1.143

format: plain


text:
# sandwich

0th

Percentile

##### Making Sandwiches with Bread and Meat

Constructing sandwich covariance matrix estimators by multiplying bread and meat matrices.

Keywords
regression
##### Arguments
x

a fitted model object.

either a bread matrix or a function for computing this via bread.(x).

meat.

either a bread matrix or a function for computing this via meat.(x, …).

arguments passed to the meat function.

##### Details

sandwich is a simple convenience function that takes a bread matrix (i.e., estimator of the expectation of the negative derivative of the estimating functions) and a meat matrix (i.e., estimator of the variance of the estimating functions) and multiplies them to a sandwich with meat between two slices of bread. By default bread and meat are called.

Some theoretical background along with implementation details is given in Zeileis (2006).

##### Value

A matrix containing the sandwich covariance matrix estimate. Typically, this should be an $k \times k$ matrix corresponding to $k$ parameters.

##### References

Zeileis A (2006), Object-Oriented Computation of Sandwich Estimators. Journal of Statistical Software, 16(9), 1--16. URL http://www.jstatsoft.org/v16/i09/.


text:
Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 28 Aug 2016, 14:49

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# 780 (Q50, V48) debrief and thank you

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Intern
Joined: 18 Oct 2009
Posts: 32
Schools: Kellogg, Ross, MIT
Followers: 1

Kudos [?]: 11 [8] , given: 13

780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

02 Dec 2009, 08:23
8
This post received
KUDOS
1
This post was
BOOKMARKED
Summary: If I had one tip for the GMAT taker, it would be USE GMAT CLUB. Period. So much information is available and if you use it critically you have all the external tools for success. I have no doubt someone could just use the questions discussions and rock it for $0 in prep materials. Also gives you faith in the human spirit that people are willing to help. My second tip would be, work smart, not hard. After reviewing a million GMAT club question discussions, I saw that the hardest Qs REQUIRE working smart. Hacking through a problem (ie writing out every iteration in a combination problem by hand, or using a formula when doing it by hand reveals a shortcut), even if you get it right, should be recorded as an error because you are likely working more slowly than you could. Likewise, when you see an answer explanation using an excel spreadsheet, just skip it. Either the answerer is missing the point, or the question is not well written and isn't worth your time. (Or the answerer is trying to put to rest an endless discussion, good but just not useful to you). Prep materials: All of OG12, Manhattan GMAT Number Properties, Kaplan Prime, Kaplan Math, and Kaplan Advanced. I used MGMAT SC for idioms and the question bank. I felt that the reviews of these materials here were accurate. Manhattan GMAT CAT tests (they come with any book) were the best single resource because the questions can get wicked and the analysis is super helpful. Kaplan Advanced is relatively brief to work through but you get good bang for your$ and I think it's kinda funny (yes, I have cracked).
I also worked through the free GMAT Club materials including several of the lists of questions. Some of these questions are great, and some are just not well written. Similarly, with the verbal (not just GMAT club, in general), I think some of those answers are just made up, they are so ridiculous, and personally I will never understand them so I didn't waste sleep over it.. well OK sometimes I did, but I shouldn't have.

Prep method: In a quirk of fate I took 3 months leave from my job in September so I had a lot of time to study. I got my math skills basically straight, then hit the practice tests, using the other questions for warmup. On my first timed practice set, I panicked and bombed. On my first whole practice test I got really bored and bombed verbal. This is why I did at least 8 practice tests with AWA to build up my mental stamina and get used to the timing. I did 14 practice tests total (mostly Kaplan, MGMAT). In addition to getting my timing straight, doing CAT tests gives you a shot at seeing harder questions. My battle was finding enough truly hard questions to practice in a timed setting and this helped.

Timing: My timing ended up OK but I still had the FEAR so I kept the timer visible at all times and recalculated time remaining every 3rd question or so. It's theoretically a waste of time but it kept me calm.

Pre-test hours: I just thought about anything BUT GMAT and spent pre-test time looking at Cyber Monday deals online.

Test time: No big shocker about the location or setup, except fiddling with the headset, pen, laminate board, etc. Left handed people like myself can expect to smear ink all over the darn place. I felt like the keyboard was weird and it threw off my writing but whatever. Also there was a guy with Tourette's in my test room which paradoxically kept things mellow (it wasn't like a cussing thing, more of a grunting thing).
Math: Seemed a little easier than I had faced before, which gave me doubt, but I just kept telling myself "you are rocking it... the easy questions must be experimental". Question mix was what I expected from the practice tests.
Verbal: Usually my strong point, but about 5 minutes in, my mind went AWOL as I tried to pin down how math had went. From the practice tests I knew to expect this feeling so I buckled down and told myself this is where I can pick up the relatively easy points. Verbal seemed fairly similar to my prep work, meaning there were more than a few questions that were just picky and stupid and pointless. Yes, I hate verbal.

The Aftermath: Grinned like an idiot. Bragged about myself to my parents. Etc., etc. It was 2 days ago and it still hasn't sunk in yet. I guess my dreams of Bschool could be a reality...or at least the GMAT is no longer standing in the way. I have nothing but thanks for everyone in the GMAT Club community, the generosity that people have shown with their time and efforts really, truly has improved this experience for many.
_________________

"It never hurts to help!"

 Kaplan GMAT Prep Discount Codes e-GMAT Discount Codes Jamboree Discount Codes
Senior Manager
Joined: 25 Jul 2007
Posts: 378
Location: Times Square
Schools: Baruch / Zicklin
Followers: 4

Kudos [?]: 38 [0], given: 148

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

02 Dec 2009, 08:50
amazing score! congrats!!!
Intern
Joined: 08 Jun 2009
Posts: 34
Followers: 0

Kudos [?]: 5 [0], given: 1

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

02 Dec 2009, 19:19
very good score man .

how did you manage to take 3 months off ? , did u take vacation specifically for gmat ?

thanks,
Intern
Joined: 18 Oct 2009
Posts: 32
Schools: Kellogg, Ross, MIT
Followers: 1

Kudos [?]: 11 [1] , given: 13

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

02 Dec 2009, 23:12
1
This post received
KUDOS
Klusner, nah, that would be way too smart for me.... I saved a bunch of money working a "tough" job and got six months leave without pay to save my sanity (wanted to resign but realized I might need a salary eventually and was volunteering to job search in the worst economy in my lifetime). Then I decided to take the GMAT and apply to b-school since I had the free time (and obviously I also want to change jobs). Awesome career plan eh?
_________________

"It never hurts to help!"

Intern
Joined: 12 Nov 2009
Posts: 46
Followers: 0

Kudos [?]: 14 [0], given: 2

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

03 Dec 2009, 01:21
Wow..congrats!
How much were you scoring during the prac tests?
Intern
Joined: 08 Jun 2009
Posts: 34
Followers: 0

Kudos [?]: 5 [0], given: 1

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

03 Dec 2009, 08:22
eekthecat , i am in a similar position of doing a stressful job . But i have learnt to handle the stress and it doesnt bother me at all.

I was also thinking of taking a sabatical ) , lets see how that goes.

thanks,
Manager
Joined: 29 Jul 2009
Posts: 224
Followers: 3

Kudos [?]: 73 [0], given: 6

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

03 Dec 2009, 08:43
Congrats eekthecat. Can you post how you did on the MGMAT tests before the real exam? Thanks.
Intern
Joined: 18 Oct 2009
Posts: 32
Schools: Kellogg, Ross, MIT
Followers: 1

Kudos [?]: 11 [0], given: 13

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

03 Dec 2009, 10:55
Klusner, I'm with ya, what was killing me was the social part, the job was overseas and I hadn't seen my friends+family for too long. Now I've spent too much time with them and I'm VERY ready to go back to work in January. Sabbatical or equivalent is a fantastic idea, I think go-getters (likely describing most of us) can't relax on a 3 day weekend. It took me two weeks to stop being afraid that I had to go back.

As for the scores:
MGMAT 1, 5 weeks before: 680 (Q45, V37)
MGMAT 2, 3 weeks before: 710 (Q47, V40)
MGMAT 3, 2 weeks before: 720 (Q47, V41)
MGMAT 4, 2.5 weeks before: 740 (Q47, V45)
MGMAT 5, 1 week before: 770 (Q50, V45)
MGMAT 6, 3 days before: 740 (Q47, V45) (this was my last practice test but my internet went out in the middle of the quant and messed up the time so I didn't stress too much about this score)

Kaplan CATs (don't have the dates here, but took interspersed with the MGMAT at about equal intervals):
750 (Q45, V41); 800 (Q53, V55); 770 (Q50, V54); 740 (Q50, V46)

GMAT Prep, I messed up the software and took questions 2x, so I don't consider my scores representative, more like practice. What I'm noting on the scores is 1. my verbal really did improve, I didn't do more studying so I think this was just mental stamina gains and 2. The Kaplan scoring is a little less consistent.
_________________

"It never hurts to help!"

Intern
Joined: 08 Jun 2009
Posts: 34
Followers: 0

Kudos [?]: 5 [0], given: 1

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

03 Dec 2009, 15:48
ooooooooooooooooohh , what practise scores man

Keep it up bro
Intern
Joined: 06 Aug 2009
Posts: 8
Followers: 0

Kudos [?]: 4 [0], given: 0

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

08 Dec 2009, 17:55
I had read your profile a couple of days before the test and it really motivated me! thank you for sharing your story. I found the quant pretty easy as well, but it ended up screwing me over, since so many of the questions were just straight forward and did not have the usual GMAT type nuance in it so it completely threw me off!

Anyways, great score...you should definitely apply to Stanford GSB..by far one of the best programs in the Country.
Intern
Joined: 18 Oct 2009
Posts: 32
Schools: Kellogg, Ross, MIT
Followers: 1

Kudos [?]: 11 [0], given: 13

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

09 Dec 2009, 16:33
MXB908 I know what you mean, I was expecting the crazy tricks to come out (a la GMAT Club or Manhattan GMAT) and they just didn't!

As for Stanford... I like what I see there. Not to mention the campus is just beautiful. I will be putting all my energy into that app starting tomorrow!
_________________

"It never hurts to help!"

Current Student
Joined: 24 Nov 2009
Posts: 67
Location: Azerbaijan
Concentration: General Management, Entrepreneurship
GMAT 1: 680 Q44 V39
GRE 1: 311 Q158 V153
GPA: 3.74
WE: Project Management (Computer Software)
Followers: 2

Kudos [?]: 9 [0], given: 11

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

10 Dec 2009, 11:50
Awesome!!!

congrats!!!!!
Intern
Joined: 23 Nov 2009
Posts: 3
Location: Nairobi
Followers: 0

Kudos [?]: 0 [0], given: 0

adfasfasf [#permalink]

### Show Tags

10 Dec 2009, 14:31
nice score
Manager
Joined: 08 Jul 2009
Posts: 171
Followers: 0

Kudos [?]: 25 [0], given: 26

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

10 Dec 2009, 14:59
Thanks for sharing your experience and practice scores. You did an amazing job!
Intern
Joined: 03 Dec 2009
Posts: 9
Followers: 0

Kudos [?]: 0 [0], given: 0

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

17 Dec 2009, 12:02
Contratulations, Amazing
Intern
Joined: 17 Dec 2009
Posts: 28
Followers: 0

Kudos [?]: 0 [0], given: 0

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

17 Dec 2009, 16:36
Hey shadow,

For the quant section can you breakdown how many questions you blindly guessed on, how many you guessed by eliminating 2 or 3, and how many questions your sure you got right?

The reason I ask is because I don't understand how when I get 5 or 10 wrong on the GMAT Prep Exam I still have a score from 45-50 but I haven't received that same score on the real GMAT.

How many questions does one have to actually get right? Can some experts shed some light?
This exam is confusing the heck out of me.
Intern
Joined: 18 Oct 2009
Posts: 32
Schools: Kellogg, Ross, MIT
Followers: 1

Kudos [?]: 11 [0], given: 13

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

17 Dec 2009, 20:16
Not sure if this was meant for me but I will give my own experience. I don't think I blindly guessed/eliminated 2 for any questions. Full disclosure: I never did that in practice either, I just made common errors, get a wrong result that GMAT predicted, and move forward thinking I'm right when I'm wrong. So I'm sure I did this during the test as well.

As for the GMAT Prep; # of wrong/right answers is not the predictor for score, it is the difficulty of the questions that you get right (try this link for more info). If you are getting them right, the questions should get harder, and if you're getting them wrong, the questions would get easier. So, if you get 27 Q47-level difficulty questions right, and 10 Q51-level questions wrong, it might average out to Q50. If that makes any sense.

So, what is more important than # right/wrong is your overall approach, like having the skills solid, getting your timing right (moving on, not digging yourself into a hole), and keeping your wits about you during testing. If your GMAT score is not reflective of your GMAT prep score, maybe it is nerves, maybe you've seen the GMAT prep questions one too many times?
_________________

"It never hurts to help!"

Intern
Joined: 17 Dec 2009
Posts: 28
Followers: 0

Kudos [?]: 0 [0], given: 0

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

17 Dec 2009, 20:43
Hey Eek, that was for you sorry about that. Thanks for your reply.

Yes, I have taken the Gmat Prep many times so that could have played a role. I've also taken the MGMAT exams and scored in the same range. And nerves definitely played a role especially during the Quant section. I just can't for the life of me figure out how to do well in Quant on the REAL exam which is why I was asking how many I would need to get right versus how many I could guess.

But if you're saying that you didn't even guess on any questions in the Quant then that's amazing. I'm certainly not there.
Manager
Joined: 27 May 2008
Posts: 126
Followers: 2

Kudos [?]: 63 [0], given: 4

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

21 Dec 2009, 01:20
Great Score,, congrats !!
Intern
Joined: 24 Jan 2010
Posts: 13
Followers: 0

Kudos [?]: 0 [0], given: 0

Re: 780 (Q50, V48) debrief and thank you [#permalink]

### Show Tags

24 Jan 2010, 11:27
Congrats !!!
Re: 780 (Q50, V48) debrief and thank you   [#permalink] 24 Jan 2010, 11:27

Go to page    1   2    Next  [ 25 posts ]

Similar topics Replies Last post
Similar
Topics:
6 Debrief: 780 (Q50, V47) 0 16 Mar 2015, 13:18
21 780 GMAT Q50/V47 Debriefing. Retake? 12 01 Jun 2014, 13:36
1 Finished with the GMAT 780 (Q: 50, V: 48) 2 31 Aug 2013, 09:45
I just got a 780 (Q50;V48) - here are some tips! 11 15 Aug 2009, 14:43
2 Final debrief - 780 (Q50,V46) 21 13 Aug 2008, 08:10
Display posts from previous: Sort by

# 780 (Q50, V48) debrief and thank you

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.


text:
Introduction

I recently read the “Gaussian Blue Noise” paper by Ahmed et al. and was very impressed by the quality of their results and the rigor of their method.

They provide a theoretical framework to analyze the quality of blue noise from a frequency analysis perspective and propose an improved technique for generating blue noise patterns. This use of blue noise is very different from “blue noise masks” and regular dithering – about which I have a whole series of posts if you’re interested.

Authors of the GBN paper show some gorgeous blue noise stippling patterns:

Their work is based on an offline optimization procedure and has inspired me to solve a different problem – progressive stippling. I thought it’d look super cool and visually pleasing to have some “progressive” blue-noise stippling animations from an image.

By progressive, we mean creating a set of N sample sets, where:

• each one has 1…N samples,
• a set K is a subset of the set K+1 (set K+1 adds one more point to the set K),
• each set K fulfills the blue noise criterion.

By fulfilling all three criteria (especially the second), the “quality” (according to the third) of the last complete set cannot be as high as in, for example, the “Gaussian Blue Noise” paper. In optimization, every new constraint and additional requirement sacrifices other requirements.

We won’t have as excellent blue noise properties as in the cited GBN paper, but as we’ll see, we can get “good enough” and visually pleasing results.

Progressive sampling is a commonly used technique in Monte Carlo rendering for reconstruction that improves over time (critical in real-time/interactive applications) but less commonly in stippling/dithering – I haven’t found a single reference to progressive stippling.

Here, I propose a straightforward method to achieve this goal.

Method

As a method, I augment my “simplified” variation of the greedy void and cluster algorithm with a trivial modification to the initial “energy” potential field.

Instead of uniform initialization, I initialize it with the image intensity scaled by some factor “a.”

(Depending on the type of stippling, with black “ink” dots or with the bright “light splats,” you might want to “negate” this initialization)

Factor “a” will balance the image content’s importance compared to filling the whole image plane with the blue noise distribution property.

In my original implementation, the Gaussian kernel used for energy modification was unnormalized and didn’t include a correction for small sigma. I fixed those to allow for easier experimentation with different spatial sigmas.

One should consider the boundary conditions for applying the energy modification function (zero, mirror, clamp) for the desired behavior. For example, spherically-wrapped images like environment maps should use “wrap” while displayable ones might want to use “clamp.” I ignore it here and use the original “wrap” formulation, suitable for use cases like spherical images.

We stop the iteration after we reach the desired point count – for example, equal to the average brightness of the original image.

This is the whole method!

Here is the complete code:

def gauss_small_sigma(x: np.ndarray, sigma: float):
p1 = scipy.special.erf((x - 0.5) / sigma * np.sqrt(0.5))
p2 = scipy.special.erf((x + 0.5) / sigma * np.sqrt(0.5))
return (p2 - p1) / 2.0

def void_and_cluster(
input_img: np.ndarray,
percentage: float = 0.33,
sigma: float = 0.9,
content_bias: float = 0.5,
negate: bool = False,
) -> Tuple[np.ndarray, np.ndarray]:
size = input_img.shape[0]

wrapped_pattern = np.hstack((np.linspace(0, size / 2 - 1, size // 2), np.linspace(size / 2, 1, size // 2)))
wrapped_pattern = gauss_small_sigma(wrapped_pattern, sigma)
wrapped_pattern = np.outer(wrapped_pattern, wrapped_pattern)
wrapped_pattern[0, 0] = np.inf

lut = jnp.array(wrapped_pattern)
jax.device_put(lut)

def energy(pos_xy_source):
return jnp.roll(lut, shift=(pos_xy_source[0], pos_xy_source[1]), axis=(0, 1))

points_set = []

energy_current = jnp.array(input_img) * content_bias * (-1.0 if negate else 1.0)
jax.device_put(energy_current)

@jax.jit
def update_step(energy_current):
pos_flat = energy_current.argmin()
pos_x, pos_y = pos_flat // size, pos_flat % size
e = energy_current[pos_x, pos_y]
return energy_current + energy((pos_x, pos_y)), pos_x, pos_y, e

final_stipple = np.zeros_like(lut) if negate else np.ones_like(lut)

samples = []
for x in range(int(input_img.size * percentage)):
energy_current, pos_x, pos_y, e = update_step(energy_current)
samples.append((pos_x, pos_y, input_img[pos_x, pos_y]))
final_stipple[pos_x, pos_y] = 1.0 if negate else 0.0
return final_stipple, np.array(samples)

And it works pretty well:

With different factors “a,” we get the different balancing of filling the overall image space more as contrasted to a higher degree of importance sampling with the equal sample count:

When the parameter goes towards infinity, we get the pure progressive sampling of the original image. The other extreme will produce pure blue noise, filling space and ignoring the image content:

The behavior of low values is interesting. With the increasing point count, they first reveal the shape, then give a ghostly “hint” of the shape underneath, and finally turn into almost uniform spatial filling:

Why does this work?

Results look pretty remarkable, given the straightforward and almost “minimal” method, so why does it work?

When we initialize the energy field with the image, we find the brightest/darkest point (depending on whether we negated the image content) as the one with minimum energy. After the first update, we add energy to this position. The next iteration finds and updates a different point. Here is a visualization (slightly exaggerated) for 1, 2, 11, and 71 points:

Each iteration both excludes the point from being reprocessed (sets it as “infinite energy”) as well as produces a Gaussian “splat” that makes the points around it less desirable to be “picked” in the following close iterations. The algorithm will choose something further away from it. The “a” constant and the Gaussian sigma balance the repulsion of points with the priority of the pixel intensity.

Edge sampling

When stippling, classic literature (ref1 ref2) emphasizes how the stippling of the edges is more critical than stippling the constant areas – both for “artistic” reasons and because the human visual system has strong edge detection mechanisms.

The above example worked very well for two reasons.

The most important secret was that Kodak images – like most images on the internet – are sharpened for display and printing. If you look at the boat paddle in the picture, you will notice a bright halo around it. Almost all images have sharpening, tiny halos, and gradient reversals.

The second reason relates to the method: looking at single points in combination with blue noise like “repulsion” will push many points to the edges.

But if we look at the simplest analytical unsharpened example – a circle, we find that the shape is somewhat visible, but not overly so:

The solution is straightforward, and along the lines of observation of sharpened internet images, we can pre-sharpen the image arbitrarily (just before generating the initialization energy).

The amount of sharpening determines how strong we are going to reproduce the edges:

All of the above are very usable, and I like it the most and find it the most useful when our constant “a” is low (biasing the results towards preserving space-filling blue-noise properties more than contents brightness). Visible edges can make a huge difference in how we perceive low-contrast shapes.

Improving the final state further

The suboptimality of the described method at the final state can be improved slightly (at the cost of breaking the progressive results) by “fine-tuning” the positions of points based on the last point set and according to the energy defined like in the GBN paper. However, one would have to allow it not to go too far from the original positions to preserve the progressive property.

This approach is not unlike standard methods in deep learning, where one fine-tunes the training/optimization results on specific, more relevant examples – typically using fewer iterations and a slower learning rate to avoid over-fitting.

Use for importance sampling

The use-case I have described so far is… cute and produces fun and pretty animations, but can it be useful? Probably the concept of “progressive stippling” – not so much beyond artistic expression.

But if we squint our eyes, we make the following observations:

1. Thanks to the “progressive” property, points are sorted according to “importance,”
2. Importance consists of two components:
1. Blue noise spatial distribution,
2. The value of the input pixel.
3. We don’t have to stop at the average brightness and can continue until we fill all the points.

Note, however, that the sorting/importance is not according to just the brightness or good spatial distribution; it’s a hybrid metric that combines both and depends on the spatial relationships and distribution.

Could we use it for blue noise distributed importance sampling of images?

Importance sampling of images is commonly used for evaluating image-based lighting or sampling 2D look-up tables for terms we lack analytical solutions. Most approaches rely on complicated spatial division structures built offline and traversed during the sampling process.

I propose to use the sorted progressive stippling points (and their pixel values) in the importance sampling process.

Let’s use a different Kodak image, a crop of the lighthouse (and inverting for the sampling of the brights):

If we plot the brightness of the consecutive points in the sequence, we observe it is a very “noisy” version and distorted version of the image histogram:

The distortion is non-obvious, hard to predict, and depends on the spatial co-locality of the points in the image. Different images with identical histograms will get different results based on where the similarly valued pixels are.

To use it in practice for importance sampling, I propose to use a smoothened, normalized version of this histogram as a PDF. Based on the inverse-sampled value, pick a point from the sequence. The sample stores information about its intensity and the original location in the image.

For simplicity’s sake and a quick demonstration, I will use an ad-hoc inverse CDF of exponentiating a random number to the power of 3 (equivalent to \$\frac{1}{3x^{\frac{2}{3}}} PDF).

For example, with the uniform random variable value of 0.1, we will pick 0.1**3 == 0.001 percentile point from the progressive stippling sequence.

The following figure presents what the four different batches of 1000 random samples from this distribution look like:

And by comparison, the same sample counts, but with pure importance sampling based on the image brightness:

The latter has significantly more “clumping” and worse space coverage outside bright regions. This difference might matter for the cases of significant mismatch of the complete integrand (often a product of multiple terms) with the image content.

Many small details are necessary to make this method practical for Monte Carlo integration. A pixel’s fixed (x, y) position is not enough to sample other terms appearing in an actual Monte Carlo integral, like a continuous PDF of a BRDF. One obvious solution would be to perform jittering within pixel extents based on additional random variables.

I don’t test the effectiveness of this method. I believe it would give less variance at low sample counts but not improve the convergence (this is typical with blue noise sampling).

Finally, this method can be expanded to more than just two dimensions – as long as we have a discrete domain and can splat samples into it.

But I leave all of the above – and evaluation of the efficacy of this method to the curious reader.

Conclusions

I have presented an original problem – progressive stippling of images and a method to compute it with a few lines of code of modification of the “void and cluster” blue noise generation method.

It worked surprisingly well, and I consider it a fun, successful experiment, just look again at those animations:

It seems to have some applications beyond fun animations, like very fast blue-noise preserving importance sampling of textures, environment maps, and look-up tables. I hope it inspires some readers to experiment with that use case.

This entry was posted in Code / Graphics and tagged , , , , , , , , , . Bookmark the permalink.


text:
# Chapter 2, Functions - Section 2.8 - One-to-One Functions and their Inverses - 2.8 Exercises: 55

$f^{-1}(x) = \frac{4x}{1 - x}$

#### Work Step by Step

$f(x) = \frac{x}{x + 4}$ $y = \frac{x}{x + 4}$ y(x + 4) = x yx + 4y = x 4y = x - yx 4y = x(1 - y) $x = \frac{4y}{1 - y}$ $f^{-1}(x) = \frac{4x}{1 - x}$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
## Maximum Likelihood Estimator

Suppose X1...Xn are iid and have PDF $\displaystyle f(x; \theta) = \frac{1}{\theta} e^{\frac{-x}{\theta}} \ \ \ 0<x<\infty$

Find the MLE of P(X<2).

I know the MLE of theta is $\displaystyle \overline{X}$

so would $\displaystyle P(X<2) = 1 - \frac{1}{\overline{X}} e^{\frac{-2}{\overline{X}}}$?


text:
# Tag Info

11

The PCA analysis does not really tell you what the bonds do but it tells you how the rates move together. The variations of $n$ rates (i.e. 1 y, 2y, ...) are split up in (at first) abstract factors like $$\Delta R_i = \sum_{j=1}^n e_{i,j} f_j$$ where $\Delta R_i$ is the change in the rate $i$ and $f_j$ is factor $j$ and $e_{i,j}$ is the (factor loading=) ...

3

The main problem is that you cannot achieve Libor in the markets. So the old-fashioned method of discounting at Libor doesn't work any more. As an example, if you compound up the 3m Libor with today's price on a 3x6 FRA, you won't get 6m Libor. Traditionally, that would mean arbitrage, but these days it's just a fact of life. You cannot achieve 3m Libor for ...

1

If a bank lends 6m Libor and finances it by borrowing 3m Libor and borrowing forward 3x6 libor, this is not arbitrage, as the bank is assuming 6m credit risk whilst his financing is 3m credit risk. (There are also other factors like regulatory capital, tying up balance sheet for 6m, etc.) So the text book case where the 3x6 FRA (or front Eurodollar) is equal ...

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


text:
Chin. Phys., 2007, Vol. 16(9): 2671-2675    DOI: 10.1088/1009-1963/16/9/030
 NUCLEAR PHYSICS Prev   Next

# Effect of strong magnetic field on electron capture of iron group nuclei in crusts of neutron stars

Liu Jing-Jing, Luo Zhi-Quan, Liu Hong-Lin, Lai Xiang-Jun
Institute of Theoretical Physics, China West Normal University, Nanchong 637002, China
Abstract  In this paper electron capture on iron group nuclei in crusts of neutron stars in a strong magnetic field is investigated. The results show that the magnetic fields have only a slight effect on electron capture rates in a range of 10$^{8}-10^{13}$G on surfaces of most neutron stars, whereas for some magnetars the magnetic fields range from 10$^{13}$ to 10$^{18}$~G. The electron capture rates of most iron group nuclei are greatly decreased, reduced by even four orders of magnitude due to the strong magnetic field.
Received:  12 December 2006      Published:  20 September 2007
 PACS: 23.40.-s (β-decay;double β-decay; electron and muon capture) 26.30.-k (Nucleosynthesis in novae, supernovae, and other explosive environments) 95.30.Cq (Elementary particle processes) 97.10.Cv (Stellar structure, interiors, evolution, nucleosynthesis, ages) 97.60.Jd (Neutron stars)
Fund: Project supported by the National Natural Science Foundation of China (Grant No 10347008).


text:
# Integral problem. Do I need a u sub?

$$\int_0^1 e^x - xe^{x^{2}}dx$$

I think I can do this:

$$\int_0^1 e^x dx- \int_0^1xe^{x^{2}}dx$$

$$(e - 1) - \int_0^1xe^{x^{2}}dx$$

I'm not sure how to handle the right side?

If I set $u = x^2$, then $\frac{du}{dx} = 2x$ and then $du = 2x\cdot dx$

Where do I go from here?

EDIT

Is this right?

$\frac{du}{2} = x \cdot dx$

so $$\frac{1}{2} \int_0^1 e^u \cdot du$$

$$\frac{1}{2} e^{x^{2}} ] _0^1$$ $$= \frac{e}{2} - \frac{1}{2}$$

so...

$$\int_0^1 e^x - x \cdot e^{x^2} = (e - 1 ) - \frac{e}{2} + \frac{1}{2}$$

• All you did so far is correct. Just perform the substitution (let $x$ disappear)... – Yves Daoust May 12 '18 at 14:03

One of the most frequently applied trick(especially in evaluating electromagnetism integrals) is: $$xdx=\frac12d(x^2)$$


text:
# The Future Space Launch Market and Stratospheric Ozone

Traffic within (and through) the stratosphere was perhaps the first recognized anthropogenic force for stratospheric perturbation. In the early 1970’s, a proposed fleet of 400 Supersonic Stratospheric Transports (SSTs) — or equivalently, High-Speed Civil Transports (HSCTs) — was severely scrutinized to address concerns which included the impact of SST HO$_\text{x}$ and NO$_\text{x}$ emissions on stratospheric ozone. Initial 1-D photochemical model estimates of steady state ozone loss arising from such a fleet frequently exceeded 50% or more and contributed to a moratorium on US development of SSTs.

Enhancements in stratospheric water and nitrogen oxides produce enhancements in the cycling of ozone via the general ozone-loss processing motif depicted in figure 1. In brief, precursor species Y may be activated to form active species X. Once formed, X reacts quickly with ozone to form XO. XO then reacts quickly with atomic oxygen to reform species X. The cycle produces a net loss of stratospheric ozone. Chain reaction termination steps are idiosyncratic to the physicochemical environment. In some cases, as depicted in figure 1,  X and XO will recombine to reform the long-lived precursor species.

Figure 1: general motif for ozone loss processing in the middle stratosphere. Klobas Thesis 2018

These initial model results featured exaggerated estimates of ozone loss for several reasons. The kinetic rates of the HO$_\text{x}$– and NO$_\text{x}$– mediated reactions were highly uncertain – and most importantly, a previously unknown source of atmospheric chlorine significantly reduced the efficiency by which atmospheric NO$_\text{x}$ might process ozone. Elevated concentrations of chlorine from chlorofluorocarbons served to lock away nitrogen dioxide (produced from SST emission of NO) and produce chlorine nitrate, per the following reactions.

$\text{NO} + \text{O}_3 \rightarrow \text{NO}_2 + \text{O}_2$

$\text{Cl} + \text{O}_3 \rightarrow \text{ClO} + \text{O}_2$

$\text{ClO} + \text{NO}_2 + \text{M} \rightarrow \text{ClONO}_2 + \text{M}^*$

Chlorine nitrate is a long-lived reservoir of both odd chlorine and odd nitrogen, and the discovery of this reaction essentially resolved the major threat to ozone previously predicted from high-altitude SSTs.

Space launch vehicles also perturb stratospheric inventories of trace gases. During the early 1970’s, and contemporaneous with the SST debates in congress, were initial studies on the implications of increased use of solid rocket motors (SRMs).  SRMs frequently employ ammonium perchlorate ($\text{NH}_4\text{ClO}_4$) oxidizers with alumina fuel, producing large local enhancements in chlorine and stratospheric surface area following a launch. The effects are quantifiable and a series of in situ rocket plume encounters in the late 1990’s revealed large, transient ozone holes spanning several hundred kilometers after a launch.

Contemporary commercial space launch vehicles tend to shy away from SRMs for many reasons including: low specific impulse, inability to prematurely terminate/restart burns, and environmental concerns. Liquid rocket motors (LRMs) are more gentle to the ozone layer than SRMs; however, they still produce trace gas emissions as they transit the stratosphere. HO$_\text{x}$ and NO$_\text{x}$ are the primary reactive families produced in a LRM plume, depending on the fuel source utilized. LO$_2$/RP-1, a kerosene fuel/liquid oxygen oxidizer, will emit water, nitrogen oxides, contributing to ozone loss processing. LO$_2$/LH$_2$ is cleaner burning,  but still produces large enhancements in HO$_\text{x}$.

This leads to an interesting scenario: as anthropogenic halogens decline throughout the century as a result of the highly-successful Montreal protocol (and subsequent amendments), the efficiency of the odd-nitrogen and odd-hydrogen catalytic cycles will increase. Also projected to increase is the rate of launches and the mass of material placed into orbit as the private space launch market grows exponentially.  Today’s launch of the Falcon Heavy by SpaceX is an impressive milestone in monitoring this trend.

The news surrounding this event reminded me of a paper from a while back — an econometric analysis of the space launch market and the impact its  expansion might have on the stratospheric ozone layer.

Ross et al. (2009) provide constraints on the mass which may be placed into orbit depending on future choices in fuel mix.  They provide the following figure (figure 2), which I reproduce for fair use academic purposes. In the figure, losses of ozone are presented as a function of payload rate, and fractional use of SRMs vs LRMs.

Figure 2, reproduced from Ross et al. (2009)

It should be noted that the projected losses are parameterized from model studies of 20th century atmospheres. Effects from the changing climate, such as stratospheric cooling, changes in the strength of the Brewer Dobson Circulation, and declining halogen burdens may significantly change the calculus. Unfortunately these studies have yet to be performed.

The SpaceX Falcon Heavy launch window opens in several hours and, if successful, will be a transformational moment in commercial spaceflight, opening up deep space to private venture. Tune in to the launch livestream here.

# Impacts from Comets and Asteroids

The evolution of the Earth, and life on Earth, has been punctuated by periodic impacts with extraterrestrial bodies — some leading to mass extinction events. Most famous among these is the 65 Mya Cretaceous-Tertiary (K-T) extinction, during which three quarters of all surface species were eliminated following the impact of a 15 km diameter asteroid travelling at 20 km/s at the Chicxulub site (the cenote ring — sinkholes in the underlying standstone —  is traced on figure 1).

Figure 1: approximate location of cenote ring (and extrapolation of the semicircle to the marine environment) at the Chicxulub site in the Yucatan penninsula

Asteroid densities range between 3000 kg / m$^3$ (rocky) to 8000 kg / m$^3$ (metallic), depending on their composition. A back-of-the-envelope calculation provides an estimate of the impactor’s energy:

$\text{E}_\text{kinetic} = \frac{1}{2}\text{m}{v}^2$

where m ranges between $4\times 10^{16} - 1\times 10^{17}$ kg and v = 20,000 m / s and the kinetic energy ranges between $8\times 10^{24} - 2\times 10^{25}$ J [one should note that the impact velocity of an asteroid averages 18 km /s but may range between the Earth’s escape velocity (~10 km/s)  and the solar system’s escape velocity (~ 70 km/s]. Another back-of-the-envelope conversion (one ton of TNT = $4\times 10^9$ J) reveals that this impact released an energy equivalent to $2\times 10^9 - 5\times 10^9$ megatons of TNT (not a typo).

An interesting thing happens when explosions produce large fireballs. If the fireball is greater than the scale height of the atmosphere (~ 8 km), the fireball will backfire, producing a vacuum straw which draws material from within the fireball directly to the stratosphere and mesosphere. This was certainly the case with the K-T impactor.

The chemical and climate effects of this event were evaluated by a team from the NCAR and the University of Colorado in 2017. Bardeen and coworkers found catastrophic decadal reductions in surface and sea-surface temperatures following the massive injection of aciniform carbon, black carbon, and charcoal into the mesosphere and stratosphere by the event. They determined that this layer of carbon reduced surface insolation by one-billionfold for several months, and this impact winter didn’t recover to even one percent of the present downwelling flux until two years later. Meanwhile, the carbon layer in the atmosphere produced stratospheric heating between 50 — 100 K, cooking away the ozone layer: enhanced rates of HOx-mediated ozone destruction and reduced rates of ozone production resulted in surface UV flux increases of up to 300% for a period of about six years following.

Such impacts of large, monolithic asteroids or comets are infrequent. The time between expected impacts can be expressed as a power law function of impactor diameter in which D is given in meters and alpha is the fractional composition of ocean (0.7) or land (0.3), depending on the type of impact for which one wishes to compute the recurrence interval. If the impactor is greater than the average depth of the ocean (3.6 km), alpha is unity.

$\tau_\text{impact} = \frac{3.71\text{x}10^{-2}\text{D}^{2.377}}{\alpha}$

The threat from an asteroid or comet impact is ranked between 0 — 10 on the Torino Scale, in which an increasing metric conveys both the increasing probability of impact and a higher potential destructiveness. Though only known Earth-crossing bodies are ranked on the Torino scale, from the recurrence interval relation, it is evident that the catastrophic Chicxulub event is not a major concern ($\tau = 3.13\times 10^8$ years); however, smaller events are expected to pose a threat especially to the ozone layer.

Impacts of smaller bodies with the ocean will vaporize large amounts of ocean water and the trace species within it. Pierazzo et al. (2010) quantify the risk to ozone following such an event, finding that impactors greater than 500 m in diameter ($\tau$ = 97 ky, ~ 60,000 MT TNT) would increases the stratospheric burden of NOx, ClOx, BrOx, and HOx to such an extent that years-long decreases in total column ozone of up to 30% at midlatitudes would occur, resulting in twofold increases in erythemal radiation.

Similar results are found by Birks et al. (2007), who explore impactors between 150 — 1200 m in diameter. They find that asteroids of 450 m diameter ($\tau$ = 75 ky, ~ 40,000 MT TNT) are likely to cause significant years-long perturbations in total column ozone, and that smaller impactors are likely to only have continental or regional effects, though they do not model the effect of stratospheric soot injections, and these effects may be very significant, depending on the area over which biomass burning occurs.

This leads me to two very informative articles which were published a few days ago in the Journal of Geology. Wolbach and coworkers produced two comprehensive papers (paper one & paper two) which provide an explanation for the Younger Dryas cooling event (13 kya), which is characterized by a 1300 year disruption in climate, the collapse of the thermohaline circulation, the disappearance of the Clovis culture (note: not the Klobas culture), and the extinction of large animals across the world. The authors provide trace element and isotopic data from ~200 sedimentary and ice core sites around the world indicating that an abrupt and widespread input of carbon to the atmosphere coincided with an enhancement in platinum resultant from comet impact.

They posit that an Earth-crossing comet on the order of 100 km in size fragmented and produced a debris swarm several hundred Earth radii in length, estimating that the recurrence time between an encounter of the Earth and such a debris swarm is on the order of 50 ky. For each encounter, some $10^{13} - 10^{14}$ g of comet bits will rain down at 30 km/s (comets have an average velocity around 30 km/s at 1 AU) and produce a series of air bursts and land impacts. The authors find from ice core and sedimentary record that up to 10 million square kilometers of the Earth burned following the impact 13 kya (about 5% of the total land surface area of the planet at the time), producing a long-lived impact winter with extreme effect on the climate. Though the direct effects of the comet on the atmosphere dissipated within a decade, feedbacks were perturbed in such a way that the climate system was disrupted for a millennium afterward (causing iceberg calving, meltwater flooding, and subsequent thermohaline compensation).

The authors also briefly discuss the ozone impact of the comet encounter, noting that while soot falls out of the atmosphere after a few months the recovery of ozone lags behind for up to a decade — and that ozone depletion may play a primary role in the extinctions observed during this period.

So, I guess it’s time to add another entry to the stratospheric threat matrix. More on that later, when the work is complete.

# Climate Feedbacks from Stratospheric Ozone Loss Following Energetic Particle Precipitation

Incoming energetic particles (EPs), primarily protons and electrons, mediate some chemical and physical processes in the Earth’s atmosphere. Variations in the incoming flux of EPs can thus result in perturbations to the chemical partitioning of the stratosphere.

Though EPs arise from multiple origins, each source process produces a characteristic energy spectrum. Galactic cosmic rays (GCRs) are a highly energetic source of protons originating from outside the solar system, with characteristic energies between $10^{-12}$ J — $10^2$ J per nucleon. Solar cosmic rays (SCRs) originating from coronal mass ejections have energy spectra which typically peak around $10^{-7}$ J per nucleon. Fluxes of GCRs and SCRs are inversely correlated with 11-year periodicity. When the 11-year sunspot cycle is at a maximum, so too are SCR fluxes, while GCR fluxes are at a minimum due to pressure from solar winds.

Regardless of their origin, sufficiently energetic EPs ($> 10^{-11}$ J) will collide with atmospheric gases and produce charged secondary products, classified by their soft (electrons, positrons, and photons) and hard (muons, pions, etc.) components. These secondary products will then interact with atmospheric species in a variety of ways. One such scheme resulting in the production of NO$_\text{x}$ is presented below.

The reaction pathway is initiated following interaction with energetic secondary electrons, denoted $\text{e}^*$.

$\text{e}^* + \text{N}_2 \rightarrow \text{N}^+ + \text{N} + 2\text{e}^-$

$\text{e}^* + \text{N}_2 \rightarrow \text{N} + \text{N} + \text{e}^-$

$\text{e}^* + \text{N}_2 \rightarrow \text{N}_2^+ + 2\text{e}^-$

$\text{e}^* + \text{O}_2 \rightarrow \text{O}_2^+ + 2\text{e}^-$

$\text{e}^* + \text{O}_2 \rightarrow \text{O} + \text{O}^+ + 2\text{e}^-$
Subsequent recombination/exchange chemistry further enhances atomic nitrogen.

$\text{N}_2^+ + \text{O} \rightarrow \text{NO}^+ + \text{N}$

$\text{N}_2 + \text{N}^+ \rightarrow \text{N}_2^+ + \text{N}$
$\text{N}_2^+ + \text{e}^- \rightarrow \text{N} + \text{N}$
$\text{NO}^+ + \text{e}^- \rightarrow \text{N} + \text{O}$

$\text{N}^+ + \text{O} \rightarrow \text{N} + \text{O}^+$

Finally, reaction of atomic nitrogen with molecular oxygen produces nitric oxide.

$\text{N} + \text{O}_2 \rightarrow \text{NO} + \text{O}$

Similarly, EPP events may produce HO$_\text{x}$. For more details, Mironova et al. (2015) provide a very thorough review on the interaction of EPPs with the atmosphere.

Once formed, HO$_\text{x}$ and NO$_\text{x}$ will engage mesospheric ozone loss processes. A schematic for the HO$_\text{x}$-mediated destruction of ozone in the middle atmosphere is presented in figure 1 — and NO$_\text{x}$-mediated processes will be quite similar. Ozone loss processes produced in the immediate environment of EPP interactions are frequently called direct effects. Direct effects from NO$_\text{x}$ and HO$_\text{x}$ produced by EPPs are a significant source of variability in mesospheric ozone (upwards of 25% of mesospheric ozone).

Figure 1: A graph theory representation of the HOx-mediated catalytic destruction of ozone in the middle stratosphere. Klobas thesis 2018.

Mesospheric HO$_\text{x}$ from EPPs is not long-lived and does not have effects beyond the immediate environment in which it is formed; however, NO$_\text{x}$ has a long lifetime and will transport to the middle stratosphere, where it will further engage ozone loss processing. Ozone loss following transport from the region in which the NO$_\text{x}$ was formed is referred to as an indirect effect of EPPs. Ozone losses from indirect effects are estimated to be as large as 15% in the middle and upper stratosphere.

The action of these events is largely confined to the polar regions because (1) EPs are entrained to geomagnetic fields which direct particle fluxes to the magnetic poles, and (2) mesospheric and stratospheric subsidence of NO$_\text{x}$ occurs at the poles.

One of the most fascinating papers I’ve ever read on the potentially disastrous effects of EPPs concerns a seemingly unlikely situation: a magnetic field reversal of the Earth concurrent with the transit of the Solar System through a region of high interstellar density.  The story goes like this:

The density of the interstellar medium is heterogeneous. If the solar system were to transit a region with higher density, enhanced rates of EP precipitation would be expected as the heliosphere contracts. The coincidence of both situations seems highly improbable at first glance; however, a review of the expected frequencies and durations of these events provides the evidence to the contrary. The solar system has transited regions of enhanced density ($\geq 100$ H atoms/cm$^3$ vs 0.3 H atoms/cm$^3$ currently) $\approx$ 135 times in the previous 4.5 Gy and the average duration of a transit was $\approx$ 1 My. The Earth’s magnetic field reverses stochastically, however on average once every 300 ky, the event itself lasting several thousand years. A statistical treatment thus indicates that there may have been as many as 7 instances in the prior 250 My of a magnetic field reversal coinciding with the transit of the solar system through a region of enhanced density — and that EP fluxes would remain elevated for up to a period of thousands of years.

Such 1000-year periods of enhanced GCR precipitation are projected to lead to widespread, long-term depletions of the stratospheric ozone layer. Pavlov et al. (2005) use a 2-D dynamical/chemical model to evaluate the new steady-state ozone solution following a 300-fold enhancement of GCR, finding that total column ozone decreases by 40\% globally, and by up to 80% at high latitudes after a 5 — 10 year adjustment period.

It’s a completely insane story with extraordinarily dire consequences. The fact that it’s not only possible, but expected to occur on a basis as frequent as the collision of a 5 km asteroid (one third of a Chicxulub impactor), with 1000-year implications is alarming; while Bruce Willis might be able to save us from an incoming asteroid, there’s nothing we can do to restart the geomagnetic field or redirect the velocity of the Solar System through less dense space.

To put this all in context, the Chicxulub event is estimated to have perturbed the ozone layer, the climate, and pretty much everything on Earth for about a decade or so, but I’ll write a post about that another time..

Two papers recently appeared on my google scholar alerts on the topic of EPPs.

The first, by Andersson et al. (2018), provides an accounting of total chemical forcing from EPP processes on ozone, while exploring the effect of medium-energy electrons (MEE, 300 — 1000 keV), whose effect has only recently been incorporated into chemical models [WACCM, this paper, SOCOL in Arsenovic et al. (2016)]. Andersson and workers find that the inclusion of MEE results in a strong enhancement of indirect ozone processing rates as low as 30 km — accounting for up to 8% of the ozone depending on the pressure level.

The second, by Meraner and Schmidt (2018), explores the climate impact of the radiative forcing effects from EPP-induced ozone loss. The authors find that the winter stratosphere increases in temperature as a result of ozone loss and gradient-driven transport is subsequently reduced.This effect could slightly reduce the strength of the polar vortex, though they state that the modeled response is small and that the effect of EPPs on climate may be minimal. This finding was surprising to me as previous works (though with less-sophisticated models) predicted stratospheric cooling  with resultant strengthening of the polar vortex and a much larger surface warming effect.

# Perturbation of Ozone by Enhancements in Meteoric Smoke Following Major Meteor Showers

Between 5 – 300 tons of meteoric material is deposited in the Earth’s atmosphere each
day. The uncertainty in that estimate is large due to limitations in the different instrumental and analytical techniques used to obtain meteoric mass fluxes — and the fact that no single technique can provide an integrated estimate over the entire size distribution. Recent estimates of mass fluxes seem to be biased toward the lower quarter of this quantity (e.g., 40 — 50 tons per day) — and this estimate contains both contributions from meteorite and cosmic dust infall.  In the case of cometary dust, the average velocity of an incoming meteorite is less than 15 km/s (though greater than 11 km /s — Earth’s escape velocity), and some 20% of the infalling mass is converted to meteoric smoke (nanometer-scale metallic particles of meteoric origin) via ablative processes (whose efficiency increases as a function of velocity). This accounts for roughly 10 tons meteoric smoke per day.

These particles subsequently sediment toward the poles during their 4-year lifetime, serving as mesospheric and stratospheric cloud nuclei and possibly participating directly in ozone chemistry.

Meteor showers are periodic events that occur when the Earth’s orbit sweeps past the path of a cometary debris trail and can present orders-of-magnitude enhancements in the daily meteoric flux rate over zonally localized regions. This morning I woke up to see a paper on correlations between large meteor showers and changes in total ozone: Gorbanev et al. (2017) explore TOMS measurements over several decades, using autocorrelation techniques. The decreases they find are significant — on the order of 5 DU. Figure 1, below, demonstrates the autocorrelation peaks of total ozone with the radar-returned meteor infall rate.

Figure 1: The autocorrelation functions from the total
ozone measured during annual meteor showers as a function of Time lag (days). (From Gorbanev et al. [2017])

The authors then demonstrate the disruption of the seasonal Autumnal enhancement in northern hemispheric ozone by the occurrence of the Leonid showers (figure 2, below). Following peak meteor activity, total ozone declines by about 5 DU over a period of 14 days (November 18 — December 2). Following the disturbance, total ozone resumes its seasonal trend.

Figure 2: 1999 northern hemispheric total ozone.  The Leonid meteor shower disrupts the seasonal increase in ozone by about 5 DU over a period of two weeks. The seasonal trend recovers and resumes afterward. (From Gorbanev et al. [2017])

The authors conclude that this signal can be used to identify interactions between meteoric material and the atmosphere. Their story is open access and available at this link.

# Partitioning of Halogens in Volcanic Gas Emissions

Volcanic eruption columns are complex and dynamic chemical environments. Along with sulfur dioxide and water, volcanic eruption columns may contain large quantities of the halogens chlorine and bromine, and to a lesser extent iodine. Plume temperatures range from very hot at the crater rim to ambient within the umbrella cloud. Entrainment of environmental air during convection is likely to produce enhancements in water, rapid parcel cooling and resultant kinetic freeze-out of chemical species.  Until recently, it had been assumed that volcanic halogen gases primarily existed in their hydrogen halide forms, i.e., hydrogen chloride, hydrogen bromide, and hydrogen iodide.

This is important because hydrogen halides are extraordinarily soluble in water and water-soluble species are effectively screened from the stratosphere during eruption column ascent. Indeed, conventional wisdom held that volcanic eruptions posed an insignificant source of stratospheric halogen gases until Textor et al. (2003) demonstrated that, under some conditions, significant quantities of hydrogen chloride might survive transport in the eruption column to the stratosphere.

Bobrowski et al. (2003) first detected an oxidized halogen — the bromine monoxide radical, BrO — in a volcanic plume using DOAS. Subsequently, Bobrowski et al. (2007) explained the presence of BrO by invoking a bromine explosion mechanism similar to the one described by Barrie and Platt (1997) in regard to the very different phenomenon of polar sea-salt processing.

The mechanism goes like this:

Some quantity of volcanic HBr is thermally dissociated to produce the Br radical.

$\text{HBr} \xrightarrow[]{\Delta} \text{H} + \text{Br}$

This radical then reacts with entrained ozone to form BrO.

$\text{Br} + \text{O}_3 \rightarrow \text{BrO} + \text{O}_2$

The BrO may react with the hydroperoxyl radical, which may also form due to thermal chemistry in the volcanic plume, producing hypobromous acid.

$\text{BrO} + \text{HO}_2 \rightarrow \text{HOBr} + \text{O}_2$

HOBr can then react with dissolved bromine (e.g., in an acidic droplet on a particle surface) to produce molecular bromine.

$\text{HOBr}_\text{(g)} + \text{Br}^-_{\text{(aq)}} + \text{H}^+_{\text{(aq)}} \rightarrow \text{Br}_{2\text{(g)}} + \text{H}_2\text{O}_\text{(l)}$

Molecular bromine is rapidly photolyzed to produce two more bromine radicals, which react with ozone.

$\text{Br}_2 + \text{h}\nu \rightarrow 2\text{Br}$

$2\text{Br} + 2\text{O}_3 \rightarrow 2\text{BrO} + 2\text{O}_2$

Thus, for every initial bromine atom, two bromine monoxide molecules are produced, provided there is enough ozone in the entrained air volume.

Finally, bromine atoms regenerate themselves to perpetuate the autocatalytic cycle:

$\text{BrO} + \text{h}\nu \rightarrow \text{Br} + \text{O}$

This O atom may produce more ozone.

$\text{Net}: \text{Br} + \text{Br}^-_\text{(aq)} + \text{O}_3 + \text{H}^+_\text{(aq)} \xrightarrow[]{surface} 2\text{BrO} + \text{products}$

If significant quantities of volcanic halogen species were to partition to the stratosphere, they would contribute to ozone loss processes, as demonstrated by Klobas et al. (2017). Figure 1, below, demonstrates how the co-injection of volcanic halogens may significantly perturb column ozone following a Pinatubo-sized volcanic eruption in a RCP 2.6 climate change scenario. Panels a — b relate the effect from sulfate aerosol alone in the year 2018 (a) and 2100 (b). Notably, as the stratospheric burden of chlorine declines due to the decay of anthropogenic chlorofluorocarbons (CFCs), sulfate-only volcanic eruptions will have less and less impact on total column ozone — but, if the volcanic eruption also injects halogens, as in panels c — f, ozone depletion is predicted even when background halogen levels are low.

Figure 1: Ozone response to stratospheric halogen injection in contemporary and future RCP 2.6 atmospheres. (a and b) Pinatubolike eruptions (SO2 injection only), (c and d) coinjection of 0.014 HCl:SO2 (EESC effectively increased by ~0.3 ppbv), and (e and f) coinjection of 0.14 HCl:SO2 (EESC effectively increased by ~3 ppbv). Note that the color scale is nonlinear in order to encompass the broad range of ozone changes. Global averages (90°S–90°N) of total column ozone perturbation are traced atop each panel as a function of time. Temporal average ozone anomalies are traced right. Global-temporal averages are enumerated in the top right. Black triangles indicate injection latitude and time. (Klobas thesis 2018)

The existence of oxidized halogens in a volcanic plume is a game-changer when it comes to stratospheric injection efficiency. While hydrogen halides possess Henry’s law constants in the 1e13 — 1e15 mol/atm neighborhood, halogen monoxides are closer to 0.5 — 0.8 mol/atm. Compare this to the Henry’s law constant for sulfur dioxide, which we know partitions effectively to the stratosphere, of about 1e5 mol/atm. IF — and it’s a big if — volcanic halogen species can be found in great quantity in their non-hydride forms, the stratospheric injection efficiency for these halogen gases may be much higher than previously considered.  Indeed, in the recent past, groups have reported finding UTLS BrO, OClO, and IO in aged volcanic plumes with SCIAMCHY and GOME-2.

So, an interesting paper showed up in a google scholar alert in my email inbox this morning: Bobrowski and colleagues measured the evolution of halogen gases using both in situ and remote sensing techniques AND quantified the oxidized fractions. What they found confirmed my suspicions: while chlorine exists mainly in the form of hydrogen chloride, up to 35% of total bromine and 18% of total iodine are bound in oxidized species. Mind you this wasn’t an explosively erupting volcano with stratospheric significance — they measured emissions from a lava lake — so this isn’t direct evidence of oxidized halogen species in a stratospheric eruption column (and eruption columns are dark and ozone-poor, so photochemistry will be suppressed everywhere except the boundaries), but it’s another step in that direction.

The stratospheric injection efficiency of volcanic halogens is a developing story and more research is needed to really constrain the magnitude of the threat to stratospheric ozone. The fact that oxidized halogen species are significantly present in volcanic emissions is another indication that prior estimates of the very low halogen injection efficiency may have been too conservative.

# Interactions between Climate Change and Volcanism

An interesting review detailing the inter-linkage between climate change and volcanic activity hit my google scholar alert inbox.  Cooper et al. (2018) discuss not only how large explosive volcanic eruptions might impact surface radiative forcing via sulfate aerosols scattering incoming sunlight, but also how symptoms of the changing climate might influence the frequency of volcanic eruptions.

The story goes like this: as the cryosphere is destabilized, landlocked ice sheets will melt. In places with very thick ice sheets, this will produce a very abrupt (on a geological timescale) shift in lithospheric pressure loading. As one might infer from figure 1, below, such a shift might significantly enhance the fraction of melt partitioning in the vicinity. More melt, of course, translates to more frequent volcanism, though there isn’t an instantaneous response — it takes centuries for the new magma to reach the surface.

Figure 1: Example of decompression melting. An abrupt change in surface loading, (e.g., isothermal decompression) provokes a phase transition.  (adapted from Arham Bahar / Universiti Malaysia Kelantan)

As I was reading this review I was reminded of a work relating another lesser-known link between climate change and volcanism. Aubry et al. (2016) explore how climate change might influence eruption column height.

Initially, a volcanic plume may be considered in loose analogy to a molecular beam — the expansion of a high pressure gas into a vacuum through an aperture. The expanding volcanic gas carries with it some momentum, thrusting itself upward. At some point, friction with (and entrainment of) the surrounding atmosphere counters this momentum. Depending on the strength of the eruption, the gas thrust region may extend from the crater rim to several km above the crater rim. Above the gas thrust region, convective thrust drives vertical transport within the eruption column. Air parcels within the eruption column are considerably buoyant and will convect to their point of neutral buoyancy. Finally, at this point, as in a cloud top, the parcels will diffuse horizontally, forming the umbrella cloud.

The maximum height of an eruption column is limited by the amount of heat transferred to the atmosphere per unit time (power, Watts), as it is convection which ultimately drives this process. Ultimate column height can be expressed according to the following equation, in which N is the the Brunt–Väisälä frequency, M is the mass ejection rate, and horizontal transport coefficients k1 + k2 = 1.

$\text{H}\propto\text{N}^{-\text{k}_1}\text{M}_0^{\text{k}_2}$

The Brunt–Väisälä frequency is dependent on the lapse rate ($\Gamma = -\frac{\text{dT}}{\text{dz}}$), the specific heat capacity $\text{c}_\text{p}$, the gravitational constant g, and the zonal temperature T.
$\text{N}^2 = \frac{\text{g}}{\text{T}}\left(\frac{\text{g}}{\text{c}_\text{p}}-\Gamma \right)$
In order for a volcanic eruption to influence global climate (and stratospheric chemistry), its eruption column must penetrate the tropopause (defined as the point of inflection in the lapse rate) — and climate change is expected to produce changes in both the tropopause height and the lapse rate. In the future, in most regions, the Brunt–Väisälä frequency will decrease. In turn, the plume height for any given eruption mass ejection rate will be shorter. Aubry et al. (2016) conclude that stratospheric input of volcanic gases will be less frequent in a future climate change scenario.

Figure 2: Lapse rate ($-\frac{dT}{dz}$) as a function of year within the RCP 6.0 emissions storyline. Note that lapse rate increases as the climate changes. (Klobas thesis 2018)

Finally, since we’re on the topic of unexpected climate-change/volcanism interactions, I’d like to bring attention to a work by Laakso et al. (2016) in which they simulate what might happen if a large volcanic eruption were to occur during albedo modification geoengineering by stratospheric aerosol injection –aka solar radiation management (SRM).

Figure 3 below provides the total aerosol burden of sulfate in Tg. For controls, they simulate a volcano-only scenario (black solid), a SRM-only scenario (black dashes), the sum of the SRM-only and volcano-only scenarios (magenta dashes). There are two experimental perturbations: SRM is immediately stopped when the volcano erupts (blue solid) and SRM continues as normal despite the volcanic eruption (red solid).

Surprisingly, the time required to recover to the pre-eruption sulfate burden during SRM is halved relative to the volcano-only case, even if the SRM program continues as normal after the eruption. Why? Because coagulation processes will produce larger aerosols more quickly, reducing the lifetime of sulfate aerosol via gravitational sedimentation. They go on to show that the cessation of SRM following a volcanic eruption could cause very rapid surface warming as a result of reduced aerosol lifetimes. The authors obviously explore more than this — and it’s open access — so I encourage you to read it yourself for more info.


text:
# Solution to this limit

Can anyone tell me how to calculate this limit. It is a puzzle so I think there must be some trick.

$lim_{x \to \dfrac{\pi}{20}} \left( {(\sec x)}^{\cos x} + {(\csc x)}^{\sin x} \right)$

• I suspect that it is $\pi/2$, which makes it a bit more interesting. – Aravind Oct 18 '15 at 19:00

$\sin$ and $\cos$ are analytic, so they're continuous everywhere, and in particular,

$$\lim_{x \to \frac{\pi}{20}} \sin x = \sin \frac{\pi}{20}$$

and

$$\lim_{x \to \frac{\pi}{20}} \cos x = \cos \frac{\pi}{20}$$

Since $\sin \frac{\pi}{20} \neq 0$ and $\cos \frac{\pi}{20} \neq 0$, the same applies to $\sec$ and $\csc$, so you can just evaluate this limit by substitution: replace all the $x$ with $\frac{\pi}{20}$ and you're done.

Notice, both the limits exist separately so we can break the given limit as follows $$\lim_{x\to \frac{\pi}{20}}\left(\sec^{\cos x}(x)+\csc^{\sin x}(x)\right)$$ $$=\lim_{x\to \frac{\pi}{20}}\left(\sec x\right)^{\cos x}+\lim_{x\to \frac{\pi}{20}}\left(\csc x\right)^{\sin x}$$ $$=\left(\sec \frac{\pi}{20}\right)^{\cos \frac{\pi}{20}}+\left(\csc \frac{\pi}{20}\right)^{\sin \frac{\pi}{20}}$$

$$\lim_{x \to \frac{\pi}{20}} \left( {(\sec(x))}^{\cos(x)} + {(\csc(x))}^{\sin(x)} \right)=$$ $$\lim_{x \to \frac{\pi}{20}} \sec^{\cos(x)}(x)+\lim_{x \to \frac{\pi}{20}} \csc^{\sin(x)}(x)=$$ $$\sec^{\cos\left(\frac{\pi}{20}\right)}\left(\frac{\pi}{20}\right)+\csc^{\sin\left(\frac{\pi}{20}\right)}\left(\frac{\pi}{20}\right)\approx 2.34901$$


text:
# CAT Quant Practice Problems

Question: In $\Delta ABC$, the internal bisector of $\angle A$ meets $BC$ at $D$. If$AB{\rm{ }} = {\rm{ }}4, {\rm{ }}AC{\rm{ }} = {\rm{ }}3{\rm{ }} \;and \;\angle A = 60^\circ$, then the length of $AD$ is
1. $2\sqrt 3$
2. $\frac{{12\sqrt 3 }}{7}$
3. $\frac{{15\sqrt 3 }}{8}$
4. $\frac{{6\sqrt 3 }}{7}$

Correct Option:2

## CAT Quant Questions with Video Solutions

CAT Quant Practice Problems
4.5 (89.41%) 17 vote[s]