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## Bermudan option pricing based on piecewise harmonic interpolation and the reduite

### November 27, 2006

Abstract
We consider an iterative Bermudan option pricing algorithm based on piecewise harmonic interpolation and give an explicit constructive characterisation of the smallest fixed point of the iteration step as the approximate price of the perpetual Bermudan option. The same arguments work for a related iterative algorithm based on the approximation of subharmonic functions via the reduite associated with a given closed ${F}_{\sigma }$  subset of ${\mathbb{R}}^{d}$  .

1 Introduction

We intend to approximate the function that assigns the value of a Bermudan option with payoff function $g$  and no dividends to the logarithmic start prices of the underlying assets by piecewise harmonic functions. In the first step, we will compute a piecewise harmonic approximation to the function that assigns the European option price associated with $g$  and the Bermudan’s maturity $T>0$  to the logarithmic asset prices at the penultimate time $T-t$  where exercise is possible. Then we iteratively compute the expectation of this function after time $t$  , discount, take the maximum with the payoff function $g$  , and perform a reduite-based interpolation (in the one-dimensional setting: a piecewise harmonic interpolation).
Now we would like to answer the following questions: Given the stationarity of perpetual Bermudan option prices, can we prove that there exists a minimal fixed point of the iteration step described above (which would then be an approximation to the perpetual Bermudan price)? If so, can we characterise it explicitly? Is the iteration step monotone?
First, we will discuss these questions in the one-dimensional setting – very little knowledge of potential theory has to be assumed for the proofs in that section.
Second, we shall generalise that approach to higher dimensions; this will entail a few technical subtleties.

2 Piecewise Bermudan option pricing for options on one asset

Consider $\left\{{a}_{0},...,{a}_{m}\right\}\subset \mathbb{R}$  , the set of (mutually distinct) support abscissas, and let $L:{C}^{\infty }\left(\mathbb{R},\mathbb{R}\right)\to {C}^{\infty }\left(\mathbb{R},\mathbb{R}\right)$  be the infinitesimal generator of a Markov semigroup of operators on Lebesgue measurable functions from $\mathbb{R}$  to $\mathbb{R}$  . We call a function $f\in {C}^{\infty }\left(\mathbb{R},\mathbb{R}\right)$  $P$  -harmonic (or shorter: harmonic, if no ambiguity can arise) if and only if $Lf=0$  . Let $\left({P}_{t}{\right)}_{t\ge 0}$  denote the semigroup generated by $L$  .
For the following, assume $L$  to be a second-order differential operator, that is, there are constants ${\beta }_{1},{\beta }_{2}\in \mathbb{R}$  such that $L:f↦{\beta }_{1}{f}^{\prime }+{\beta }_{2}{f}^{\prime \prime }.$  A function $g:\mathbb{R}\to \mathbb{R}$  is said to be subharmonic (superharmonic) if and only if $g$  is rightand left-differentiable (thus, letting $Lg$  become well-defined as a function from $\mathbb{R}$  to $\mathbb{R}\cup \left\{±\infty \right\}$  ) and $Lg\ge 0$  ( $Lg\le 0$  , respectively).
In particular, the supremum (infimum) of countably many harmonic functions is subharmonic (superharmonic).
Lemma 1. Given two support abscissas and ordinates, there is a unique harmonic interpolation, provided $L$  is a second-order differential operator with a non-trivial second-order part (i e ${\beta }_{2}\ne 0$  ) or a non-zero first-order part (i e ${\beta }_{2}\ne 0$  ).
• Proof sketch. The uniqueness is a consequence of the maximum principle for harmonic functions. The existence follows (in our one-dimensional setting) by distinguishing the cases delineated in the statement of the Lemma. If $L$  is a second-order operator and it has only a non-zero term of second order, then the space of solutions are all affine-linear functions from $\mathbb{R}$  to $\mathbb{R}$  . This space is two-dimensional. If there are terms of different order, the space of solutions will have basis elements of the form $exp\left(\alpha \cdot \right)$  and we have to solve a linear or quadratic equation to find the $\alpha$  (or $\alpha$  ’s) satisfying this linear or quadratic equation. Since $L$  is sub-Markovian, there will be at least one real solution to this equation for $\alpha$  .
The Lemma implies
Corollary 1. There cannot be more than two linearly independent harmonic functions: There is a canonical monomorphism from the space of functions to the – two-dimensional – space of pairs of subordinates.
Lemma 2. A subharmonic function from $\mathbb{R}$  to $\mathbb{R}$  is constantly zero if it has three zeros.
• Proof. The leftand right-differentiablility of subharmonic functions entail that for all subharmonic $g$  , $Lg$  will be defined as a function from $\mathbb{R}$  to $\mathbb{R}\cup \left\{±\infty \right\}$  .
If there is only a first order non-zero term, the space of harmonic functions will just coincide with the space of constant functions.
Lemma 3.
• 1. Piecewise harmonic interpolation with respect to the support abscissas $\left\{{a}_{0},...,{a}_{m}\right\}$  preserves subharmonicity on $\left[{a}_{0},{a}_{m}\right]$  : The interpolating function dominates the interpolated function on $A:=\left[{a}_{0},{a}_{m}\right]$  , and if the interpolating function $f$  equals the harmonic function ${f}_{i}$  on $\left[{a}_{i},{a}_{i+1}\right]$  for all $i  , then we have $f=sup\left\{{f}_{0},...,{f}_{m-1}\right\}$  .
• 2. The interpolating function $f$  is strictly dominated by the interpolated function $\mathcal{ℐ}\left(f\right)$  on the intervals $\left(-\infty ,{a}_{0}\right)$  and $\left({a}_{m},+\infty \right)$  .
• Proof sketch.
• 1. The domination part follows from the maximum principle for harmonic functions. From the maximum principle, we also get for all $i  that if ${f}_{i}\ne {f}_{i+1}$  , then $\left\{{f}_{i}={f}_{i+1}\right\}=\left\{{a}_{i+1}\right\}.$  Now there are two possibilities: either ${f}_{i}<{f}_{i+1}$  on $\left(-\infty ,{a}_{i+1}\right)$  and ${f}_{i}>{f}_{i+1}$  on $\left({a}_{i+1},+\infty \right)$  or the other way round ${f}_{i}>{f}_{i+1}$  on $\left(-\infty ,{a}_{i+1}\right)$  and ${f}_{i}<{f}_{i+1}$  on $\left({a}_{i+1},+\infty \right)$  . However, in the former case, the interpolating function would equal ${f}_{i}\wedge {f}_{i+1}$  on $\left[{a}_{i},{a}_{i+2}\right]$  , which is superharmonic, and it would also dominate the subharmonic interpolated function $\eta$  on $\left[{a}_{i},{a}_{i+2}\right]$  .
Then, $\eta -\left({f}_{i}\wedge {f}_{i+1}\right)$  would be nonpositive and subharmonic on $\left[{a}_{i},{a}_{i+2}\right]$  and it would have three zeroes, in ${a}_{i}$  , ${a}_{i+1}$  and ${a}_{i+2}$  . By Lemma  2 , this can only be true if $\eta -\left({f}_{i}\wedge {f}_{i+1}\right)=0$  on $\left[{a}_{i},{a}_{i+2}\right]$  . Thus, $\eta ={f}_{i}\wedge {f}_{i+1}$  on $\left[{a}_{i},{a}_{i+2}\right]$  . Since $\eta$  is subharmonic on $\left[{a}_{i},{a}_{i+2}\right]$  , so must be ${f}_{i}\wedge {f}_{i+1}$  then, and therefore, ${f}_{i}\wedge {f}_{i+1}$  is harmonic on $\left[{a}_{i},{a}_{i+2}\right]$  . This means ${f}_{i}={f}_{i+1}$  (as both ${f}_{i}$  and ${f}_{i+1}$  are harmonic) which contradicts our assumption that ${f}_{i}\ne {f}_{i+1}$  . Therefore, ${f}_{i}>{f}_{i+1}$  on $\left(-\infty ,{a}_{i+1}\right)$  and ${f}_{i}<{f}_{i+1}$  on $\left({a}_{i+1},+\infty \right)$  for all $i  .
Inductively, this yields $f\ge {f}_{i}$  on $\left[{a}_{0},{a}_{m}\right]$  for all $i  , hence $f=sup\left\{{f}_{0},...,{f}_{m-1}\right\}$  on $\left[{a}_{0},{a}_{m}\right]$  .
• 2. The function $f-\mathcal{ℐ}\left(f\right)$  is subharmonic on $\left(-\infty ,{a}_{1}\right]$  and it has two zeroes in ${a}_{0}$  and ${a}_{1}$  . Moreover, it is nonpositive on $\left({a}_{0},{a}_{1}\right)$  . Because of Lemma  2 , then $f-\mathcal{ℐ}\left(f\right)$  has to be positive or negative on $\left(-\infty ,{a}_{0}\right)$  .
In the former case, we are done. In the latter case, due to the maximum principle, $f-\mathcal{ℐ}\left(f\right)$  must be decreasing and therefore in ${a}_{0}$  we would have $L\left(f-\mathcal{ℐ}\left(f\right)\right)\left({a}_{0}\right)<0$  , which is absurd. A symmetric argument works for the proof of the domination of $\mathcal{ℐ}\left(f\right)$  by $f$  on the interval $\left({a}_{m},+\infty \right)$  .
Lemma 4. Piecewise harmonic interpolation to a set of support absicssas $\left\{{a}_{0},...,{a}_{m}\right\}$  is monotone on $\left[{a}_{0},{a}_{m}\right]$  in the sense that if $f\le g$  on $\left[{a}_{0},{a}_{m}\right]$  , then the piecewise harmonic interpolation of $f$  will be dominated by the piecewise harmonic interpolation of $g$  on $\left[{a}_{0},{a}_{m}\right]$  .
• Proof. Use the maximum principle on each of the intervals $\left[{a}_{i},{a}_{i+1}\right]$  for $i  .
Lemma 5. Let $\mathcal{ℐ}:{\mathbb{R}}^{\left[{a}_{0},{a}_{m}\right]}\to {\mathbb{R}}^{\mathbb{R}}$  denote the operator of piecewise harmonic interpolation with respect to the set of support abscissas $\left\{{a}_{0},...,{a}_{m}\right\}$  . Let $f:\mathbb{R}\to \mathbb{R}$  be subharmonic on $\mathbb{R}$  . Consider a harmonic function $h:\mathbb{R}\to \mathbb{R}$  , assumed to dominate $f$  : $f\le h$  on $\mathbb{R}$  . Then $\mathcal{ℐ}\left(f\right)\le h$  on $\mathbb{R}$  .
• Proof. From the previous Lemma  4 , we already know that $\mathcal{ℐ}\left(f\right)\left(x\right)\le \mathcal{ℐ}\left(h\right)\left(x\right)$  holds for all $x\in \left[{a}_{0},{a}_{m}\right]$  . However, $\mathcal{ℐ}\left(h\right)=h$  , hence $\mathcal{ℐ}\left(f\right)\le h$  on $\left[{a}_{0},{a}_{m}\right]$  and from Lemma  3 , we conclude that $h\ge f\ge \mathcal{ℐ}\left(f\right)$  on the intervals $\left(-\infty ,{a}_{0}\right)$  and $\left({a}_{m},+\infty \right)$  .
Theorem 1. Let $\mathcal{ℐ}:{\mathbb{R}}^{\left[{a}_{0},{a}_{m}\right]}\to {\mathbb{R}}^{\mathbb{R}}$  again denote the operator of piecewise harmonic interpolation with respect to the set of support abscissas $\left\{{a}_{0},...,{a}_{m}\right\}$  . Let $g$  be a subharmonic function, let $c$  be nonnegative and subharmonic, and let $h$  be harmonic. Let $c$  be, moreover, harmonic on each of the intervals $\left[{a}_{i},{a}_{i+1}\right]$  for $i  .
Suppose $c\le g$  on $\left[{a}_{0},{a}_{m}\right]$  and $c,g\le h$  on $\mathbb{R}$  , $r>0$  and let $t>0$  .
Now define $\mathcal{K}:f↦\mathcal{ℐ}\left({e}^{-rt}{P}_{t}\left(\mathcal{ℐ}\left(f\right)\vee c\right)\vee g\right)↾\left[{a}_{0},{a}_{m}\right]$  as well as $Q:=\left\{f↾\left[{a}_{0},{a}_{m}\right]:\begin{array}{c}f:\mathbb{R}\to \mathbb{R}\text{subharmonic},f\ge c\text{on}\left[{a}_{0},{a}_{m}\right],\\ \forall i\in \left\{1,...,m-2\right\}f\text{harmonic on}\left[{a}_{i},{a}_{i+1}\right],\\ f\text{harmonic on}\left(-\infty ,{a}_{1}\right),\left({a}_{m-1},+\infty \right),f\le h\end{array}\right\}.$  Then $\mathcal{K}$  maps the convex and bounded subset $Q$  of ${C}^{0}\left[{a}_{0},{a}_{m}\right]$  continuously to itself. Moreover, due to Lemma  1 , $Q$  is a subset of a finite-dimensional subspace of ${C}^{0}\left[{a}_{0},{a}_{m}\right]$  (this subspace being the space of all functions from $\left[{a}_{0},{a}_{m}\right]$  that are harmonic on each of the intervals $\left[{a}_{i},{a}_{i+1}\right]$  for $i  . By Brouwer’s Fixed Point Theorem, $\mathcal{K}$  has got a fixed point in $Q$  . Finally, $\mathcal{K}$  is a composition of monotone functions on $\left[{a}_{0},{a}_{m}\right]$  and therefore monotone as well.
• Proof sketch. We can divide the proof for $\mathcal{K}\left(Q\right)\subseteq Q$  into three parts:
• 1. The cone of subharmonic functions is closed under $\vee$  , under ${P}_{t}$  , under multiplication by constants and under piecewise harmonic interpolation $\mathcal{ℐ}$  (cf Lemma  3 ), therefore the image of $Q$  under $\mathcal{K}$  can only consist of subharmonic functions.
• 2. The upper bound on the elements of the image $\mathcal{K}\left(Q\right)$  follows from the monotonicity of ${P}_{t}$  and $\mathcal{ℐ}$  (Lemma  4 ), combined with the equations ${P}_{t}h=h$  and $\mathcal{ℐ}\left(h\right)=h$  as well as the Lemma  5 : First, we may state ${e}^{-rt}{P}_{t}\left(\mathcal{ℐ}\left(f\right)\vee c\right)\vee g\le h$  for all $f\le h$  , which by Lemma  5 allows us to deduce $\mathcal{ℐ}\left({e}^{-rt}{P}_{t}\left(\mathcal{ℐ}\left(f\right)\vee c\right)\vee g\right)\le h$  for all $f\in Q$  .
• 3. The lower bound follows again from the monotonicity of $\mathcal{ℐ}$  , but this time only by exploiting $c\le g$  on $\left[{a}_{0},{a}_{m}\right]$  and employing the fact that the space of those functions that are harmonic on each of the intervals $\left[{a}_{i},{a}_{i+1}\right]$  for $i  is invariant under the composition of $\mathcal{ℐ}$  with the restriction to $\left[{a}_{0},{a}_{m}\right]$  (yielding $\mathcal{ℐ}\left(c\right)=c$  on $\left[{a}_{0},{a}_{m}\right]$  ).
Since $c$  is nonnegative, we get that $Q$  is bounded by ${sup}_{\left[{a}_{0},{a}_{m}\right]}h\ge 0$  as a subset of ${C}^{0}\left[{a}_{0},{a}_{m}\right]$  , and because $Q$  is finite-dimensional, we may apply Schauder’s Theorem, provided we are given the continuity of $\mathcal{K}$  . However, this last assertion follows from the maximum principle.
The existence of a minimal fixed point for $\mathcal{K}$  can be proven constructively as well:
Corollary 2. Let us adopt the notation of the previous Theorem. Then the sequence ${\left({\mathcal{K}}^{n}\left(g\vee 0\right)\right)}_{n\in {\mathbb{N}}_{0}}$  is monotone on $\left[{a}_{0},{a}_{m}\right]$  , bounded and dominated by $h$  . Therefore we have the existence of a limit on $\left[{a}_{0},{a}_{m}\right]$  given by $\forall x\in \left[{a}_{0},{a}_{m}\right]q\left(x\right):={lim}_{n\to \infty }{\mathcal{K}}^{n}\left(g\vee 0\right)\left(x\right)={sup}_{n\in {\mathbb{N}}_{0}}{\mathcal{K}}^{n}\left(g\vee 0\right)\left(x\right).$  This limit is an element of $Q$  and therefore can be canonically extended to the whole of $\mathbb{R}$  . By the continuity of $\mathcal{K}$  , $q$  is a fixed point of $\mathcal{K}$  . On $\left[{a}_{0},{a}_{m}\right]$  , the convergence in the last equation will be uniform.
• Proof. The only part of the Corollary that does not follow directly from the preceding Theorem  1 is the uniformity of the convergence and that $q$  will be harmonic on each of the intervals $\left[{a}_{i},{a}_{i+1}\right]$  for $i  . However, monotone convergence on compact sets preserves harmonicity and is always uniform (cf e g Meyer [3– or, more directly, Port and Stone [4,Theorem3.9if $P$  is the Brownian semigroup).
Lemma 6. In the preceding Corollary’s notation, $q$  is the minimal nonnegative fixed point of $\mathcal{K}$  .
• Proof. Any nonnegative fixed point $p$  of $\mathcal{K}$  must be greater or equal $g$  on $\left[{a}_{0},{a}_{m}\right]$  .
Therefore the monotonicity of $\mathcal{K}$  on $\left[{a}_{0},{a}_{m}\right]$  , implies $\forall n\in {\mathbb{N}}_{0}p={\mathcal{K}}^{n}p\ge {\mathcal{K}}^{n}\left(g\vee 0\right)\text{on}\left[{a}_{0},{a}_{m}\right],$  yielding $p\ge {sup}_{n\in {\mathbb{N}}_{0}}{\mathcal{K}}^{n}\left(g\vee 0\right)=q\text{on}\left[{a}_{0},{a}_{m}\right].$
Example 1 (Bermudan vanilla put in a special Black-Scholes model). Assume $P:=\left({P}_{t}{\right)}_{t\ge 0}:={\left({\nu }_{-\mu t,{\sigma }^{2}t}*\cdot \right)}_{t\ge 0},$  where $\sigma >0,\mu :=r-\frac{{\sigma }^{2}}{2},$  thus $P$  can be perceived as the semigroup associated to the logarithmic price process under the risk-neutral measure in the one-dimensional Black-Scholes model). We will assume that (possibly after a linear change of the time scale) $\sigma =1$  . Define $g:=K-exp.$  (the payoff on exercise of a one-dimensional put option with strike price $K$  ). The infinitesimal generator of the Markov semigroup $P$  is $L=\frac{1}{2}\Delta -\mu \nabla .$  Thus, if we assume
 $\begin{array}{c}1-2\mu =1-2r+{\sigma }^{2}\le 0,\end{array}$ (1)
that is, $\mu \ge \frac{1}{2}$  , we obtain $Lg=-\left(\frac{1}{2}-\mu \right)exp\ge 0,$  hence $g$  is, given the condition ( 1 ), $P$  -subharmonic. We can find the $P$  -harmonic functions for $\mu \ne 0$  (otherwise they are simply the affine linear functions) by observing that for all $\alpha \in \mathbb{R}$  ,
 $\begin{array}{ccc}& & 0=Lexp\left(\alpha \cdot \right)=\frac{1}{2}\left({\alpha }^{2}-2\mu \alpha \right)exp\left(\alpha \right)\end{array}$
 $\begin{array}{ccc}& ⇔& \alpha \in \left\{0,2\mu \right\}.\end{array}$
If $\mu \ne 0$  , the functions $1:x↦1$  and $exp\left(2\mu \cdot \right):x↦{e}^{2\mu x}$  are two linearly independent harmonic functions, thus by Corollary  1 , we have already found a basis for the space of harmonic functions. If $\mu =0$  , the harmonic functions are exactly the affine linear functions. In order to obtain the setting of Theorem  1 , we will assume $\mu =\frac{1}{2}$  , thereby making $g=K-exp$  (the payoff on exercise of a put option with strike price $K$  ) as well as $h=K$  harmonic. Then clearly $g\le h$  . In order to satisfy the conditions on $c$  if we set $c=0$  , we could assume in addition ${a}_{m}\le lnK$  (where ${a}_{m}$  is the maximal support abscissa) for instance.

3 Reduite-based approximation of Bermudan option prices

Suppose $P$  is a Markov semigroup on ${\mathbb{R}}^{d}$  ( $d\in \mathbb{N}$  ) and $L$  is the infinitesimal generator of $P$  . We will call a function $f:{\mathbb{R}}^{d}\to \mathbb{R}$  subharmonic if and only if $\forall t>0{P}_{t}f\ge f$  holds pointwise. A function $f:{\mathbb{R}}^{d}\to \mathbb{R}$  will be called superharmonic if and only if $-f$  is subharmonic, and $f:{\mathbb{R}}^{d}\to \mathbb{R}$  will be called harmomic if it is both superand subharmonic.
Let $\mathcal{U}$  denote the operator of upper-semicontinuous regularisation, that is, for all functions $f:{\mathbb{R}}^{d}\to \mathbb{R}$  , $\mathcal{U}f=inf\left\{\ell \ge f:\ell :{\mathbb{R}}^{d}\to \mathbb{R}\text{subharmonic}\right\}$  (of course, this is a priori only defined as a function taking values in $\mathbb{R}\cup \left\{--\infty \right\}$  ).
Consider a harmonic function $h:{\mathbb{R}}^{d}\to \mathbb{R}$  and a closed ${F}_{\sigma }$  set $B$  and define the reduite operator $\mathcal{ℛ}={\mathcal{ℛ}}_{h,B}$  on the set of all subharmonic functions $f:{\mathbb{R}}^{d}\to \mathbb{R}$  dominated by $h$  via $\mathcal{ℛ}f:=\mathcal{U}\left(sup\left\{\ell \le h:\ell :{\mathbb{R}}^{d}\to \mathbb{R}\text{subharmonic},\ell \le f\text{on}B\right\}\right).$  It is a well-known result from potential theory (cf e g the work of Paul-Andre Meyer [3,TheoremeT22) that there will be a greatest subharmonic function dominated by $f$  on $B$  and that this function will be equal to $\mathcal{ℛ}f$  . Moreover, we have that $f=\mathcal{ℛ}f$  on $B$  except on a set of potential zero, in probabilistic/potential-theoretic jargon $f=\mathcal{ℛ}f\text{q.e. on}B,$  where “q.e.” is, as usual, short-hand for “quasi-everywhere”. Now define $Q:=\left\{f\le h:f:{\mathbb{R}}^{d}\to \mathbb{R}\text{subharmonic}\right\}.$  Then our definition of the reduite operator $\mathcal{ℛ}$  implies $\mathcal{ℛ}f\le h$  (as $h$  is dominating the function whose upper-semicntinuous regularisation is, according to our definition, the reduite $\mathcal{ℛ}f$  of $f$  ) and our potential-theoretic characterisation of the reduite – as the greatest subharmonic function dominated by $f$  on $B$  – ensures the subharmonicity of $\mathcal{ℛ}f$  . Therefore, $\mathcal{ℛ}:Q\to Q.$  We also have that $\mathcal{U}$  is monotone (in the sense that for all ${f}_{0}\le {f}_{1}$  , $\mathcal{U}{f}_{0}\le \mathcal{U}{f}_{1}$  ) so that $\mathcal{ℛ}$  must be monotone as well (from the $\subseteq$  -monotonicity of $sup$  and the definition of $\mathcal{ℛ}$  ).
Hence
Lemma 7. Adopting the notation of the preceding paragaph, $\mathcal{ℛ}:Q\to Q$  and whenever ${f}_{0}\le {f}_{1}$  , $\mathcal{ℛ}{f}_{0}\le \mathcal{ℛ}{f}_{1}$  .
Let $g:{\mathbb{R}}^{d}\to \mathbb{R}$  be a subharmonic function such that $g\le h$  and let $r>0$  . The next step is going to be the consideration of the following family of operators:
${\phi }_{t}:f↦{e}^{-rt}{P}_{t}f\vee g$  for $t\ge 0$  . If $f\le h$  , ${P}_{t}f\le {P}_{t}h=h$  for all $t\ge 0$  , since the operators ${P}_{t}$  are positive and linear, and $h$  was assumed to be harmonic. Thus, since $g\le h$  and $r>0$  , one must have ${\phi }_{t}f\le h$  for all $f\le h$  and $t\ge 0$  . Moreover, the operators ${P}_{t}$  preserve subharmonicity and the maximum of two subharmonic functions is subharmonic again, therefore ${\phi }_{t}f$  must be subharmonic for all subharmonic $f$  . Finally, since ${P}_{t}$  is monotone, ${\phi }_{t}$  has to be monotone for all $t\ge 0$  Summarising this, we obtain
Lemma 8. Using the notation introduced previously, ${\phi }_{t}:Q\to Q$  and whenever ${f}_{0}\le {f}_{1}$  , ${\phi }_{t}{f}_{0}\le {\phi }_{t}{f}_{1}$  for all $t\ge 0$  .
As a consequence, we derive from the two Lemmas  7 and  8 the following:
Corollary 3. If we define ${\mathcal{K}}_{t}:=\mathcal{ℛ}\circ {\phi }_{t}$  (adopting the notation of the previous paragraph), we have ${\mathcal{K}}_{t}:Q\to Q$  and whenever ${f}_{0}\le {f}_{1}$  , ${\mathcal{K}}_{t}{f}_{0}\le {\mathcal{K}}_{t}{f}_{1}$  .
This already suffices to prove the following
Theorem 2. Let $t\ge 0$  . Then for all $n\in {\mathbb{N}}_{0}$  ,
 $\begin{array}{c}{{\mathcal{K}}_{t}}^{n+1}\left(g\vee 0\right)\ge {{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right).\end{array}$ (2)
Furthermore, $q:={sup}_{n\in {\mathbb{N}}_{0}}{{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)$  (which a priori is only defined as a function with range in $\mathbb{R}\cup \left\{+\infty \right\}$  ) is an element of $Q$  and indeed is the least nonnegative fixed point of ${\mathcal{K}}_{t}$  .
• Proof.
• 1. The proof of equation ( 2 ) is a straightforward induction on $n$  where we have to use the monotonicity of ${\mathcal{K}}_{t}$  in the induction step.
• 2. Since ${\mathcal{K}}_{t}$  maps $Q$  to itself, the whole sequence ${\left({{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)\right)}_{n\in {\mathbb{N}}_{0}}$  is bounded by $h$  . This entails $q\le h$  as well. Applying Beppo Levi’s Theorem on swapping $sup$  and $\int \cdot d\mu$  – for bounded monotonely increasing sequences of measurable nonnegative functions and an arbitrary measure $\mu$  – to the measures ${P}_{t}\left(\cdot ,x\right)$  , $x\in {\mathbb{R}}^{d}$  and the sequence ${\left({{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)\right)}_{n\in {\mathbb{N}}_{0}}$  , we can exploit the subharmonicity of the functions ${{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)$  , $n\in {\mathbb{N}}_{0}$  , to deduce  $\begin{array}{ccc}\forall x\in {\mathbb{R}}^{d}{P}_{t}q\left(x\right)& =& {sup}_{n\in {\mathbb{N}}_{0}}{P}_{t}\left({{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)\right)\left(x\right)\end{array}$
 $\begin{array}{ccc}& \ge & {sup}_{n\in {\mathbb{N}}_{0}}{{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)\left(x\right)=q\left(x\right),\end{array}$
which is the subharmonocity of $q$  . As we have already seen, $q\le h$  , so $q\in Q$  .
• 3. If we employ Beppo Levi’s Theorem again, we can show that ${\mathcal{K}}_{t}$  and ${sup}_{n\in {\mathbb{N}}_{0}}$  commute for bounded monotonely increasing sequences of functions.
Thereby ${\mathcal{K}}_{t}q={sup}_{n\in {\mathbb{N}}_{0}}{\mathcal{K}}_{t}{{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)={sup}_{n\in \mathbb{N}}{{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)=q.$
• 4. Any nonnegative fixed point $p$  of ${\mathcal{K}}_{t}$  must be greater or equal $g\vee 0$  .
Therefore by the monotonicity of $sup$  and ${\mathcal{K}}_{t}$  , ${sup}_{n\in {\mathbb{N}}_{0}}{{\mathcal{K}}_{t}}^{n}p\ge {sup}_{n\in {\mathbb{N}}_{0}}{{\mathcal{K}}_{t}}^{n}\left(g\vee 0\right)=q.$
Example 2 (Bermudan put option with equidistant exercise times in $t\cdot {\mathbb{N}}_{0}$  on the weighted arithmetic average of a basket in a special Black-Scholes model). Let ${\beta }_{1},...,{\beta }_{d}\in \left[0,1\right]$  be a convex combination and ${\left({P}_{t}\right)}_{t\ge 0}={\left({\nu }_{-t/2,t}\right)}_{t\ge 0}$  . Then one has $L=\frac{1}{2}\Delta -\left(\frac{1}{2},\cdots ,\frac{1}{2}\right)\cdot \nabla$  (cf e g Revuz and Yor’s exposition [5), and both $g:x↦K-{\sum }_{i=1}^{d}{\beta }_{i}exp\left({x}_{i}\right)$  and $h:=K$  (where $K\ge 0$  is the option’s strike price) are harmonic.
Moreover, obviously $g\le h$  . This allows us to deduce that whatever closed ${F}_{\sigma }$  -set $B\subset {\mathbb{R}}^{d}$  we choose (for the sake of computational efficiency one could think of a triangulisation for instance), the (perpetual) Bermudan option pricing algorithm that iteratively applies ${\mathcal{K}}_{t}$  (where $t>0$  is the exercise mesh size of the option) to the payoff function $g\vee 0$  on the $log$  -price space, will monotonely increase and will have a limit which is the smallest nonnegative fixed point of ${\mathcal{K}}_{t}$  .
Acknowledgements. This work originates from research conducted by the author for his doctoral thesis at the University of Oxford. The author is highly indebted to Professor Terry Lyons for numerous helpful discussions. Furthermore, he gratefully acknowledges funding from the German Academic Exchange Service (Doktorandenstipendium des Deutschen Akademischen Austauschdienstes).
References

1. K Ito, H P McKean Jr, Diffusion processes and their sample paths, Grundlehren der mathematischen Wissenschaften 125, Springer, Berlin 1974.
2. T J Lyons, personal communication.
3. P-A Meyer, Probabilites et potentiel, Actualites scientifiqes et industrielles 1318, Hermann, Paris 1966.
4. S C Port, C J Stone, Brownian motion and classical potential theory, Academic Press, New York 1978.
5. D Revuz, M Yor, Continuous martingales and Brownian motion, 3rd ed, Grundlehren der mathematischen Wissenschaften 293, Springer, Berlin 1999.
6. D Stroock, Probability theory – an analytic view, Cambridge University Press, Cambridge 1993.


text:
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 Dokl. Akad. Nauk: Year: Volume: Issue: Page: Find

 Dokl. Akad. Nauk SSSR, 1987, Volume 293, Number 1, Pages 27–32 (Mi dan8226)

MATHEMATICS

On the maximal number of edges for a graph with $n$ vertices in which a given subgraph with $k$ vertices has no more than $l$ edges

A. I. Gol'berg, V. A. Gurvich

The Institute of the Earth Physics, AS USSR, Moscow

Full text: PDF file (768 kB)

Bibliographic databases:
UDC: 519.17
Presented: Ë. Â. Êàíòîðîâè÷

Citation: A. I. Gol'berg, V. A. Gurvich, “On the maximal number of edges for a graph with $n$ vertices in which a given subgraph with $k$ vertices has no more than $l$ edges”, Dokl. Akad. Nauk SSSR, 293:1 (1987), 27–32

Citation in format AMSBIB
\Bibitem{GolGur87} \by A.~I.~Gol'berg, V.~A.~Gurvich \paper On the maximal number of edges for a graph with $n$ vertices in which a given subgraph with $k$ vertices has no more than $l$ edges \jour Dokl. Akad. Nauk SSSR \yr 1987 \vol 293 \issue 1 \pages 27--32 \mathnet{http://mi.mathnet.ru/dan8226} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=882076} \zmath{https://zbmath.org/?q=an:0646.05033}

format: plain


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# Root problem

How to square root a fraction in LaTex?

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The link is invalid. You have to use https not hhttp. · 3 weeks, 6 days ago

Oh, sorry. I guess I miscopied XD · 3 weeks, 6 days ago

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location:  Publications → journals
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Search: MSC category 16D40 ( Free, projective, and flat modules and ideals [See also 19A13] )

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1. CMB Online first

Hou, Ruchen
 On Global Dimensions of Tree Type Finite Dimensional Algebras A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k-$algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum of possible global dimensions is $n-1$. Keywords:global dimension, tree type finite dimensional $k-$algebra, quiverCategories:16D40, 16E10, , 16G20

2. CMB 2009 (vol 52 pp. 145)

Wang, Z.; Chen, J. L.
 $2$-Clean Rings A ring $R$ is said to be $n$-clean if every element can be written as a sum of an idempotent and $n$ units. The class of these rings contains clean rings and $n$-good rings in which each element is a sum of $n$ units. In this paper, we show that for any ring $R$, the endomorphism ring of a free $R$-module of rank at least 2 is $2$-clean and that the ring $B(R)$ of all $\omega\times \omega$ row and column-finite matrices over any ring $R$ is $2$-clean. Finally, the group ring $RC_{n}$ is considered where $R$ is a local ring. Keywords:$2$-clean rings, $2$-good rings, free modules, row and column-finite matrix rings, group ringsCategories:16D70, 16D40, 16S50


text:
A NOTE ON THE VALUE DISTRIBUTION OF f2(f)n FOR n≥2

Title & Authors
A NOTE ON THE VALUE DISTRIBUTION OF f2(f)n FOR n≥2
Jiang, Yan;

Abstract
Let f be a transcendental meromorphic function in the complex plane $\small{\mathbb{C}}$, and a be a nonzero constant. We give a quantitative estimate of the characteristic function T(r, f) in terms of $\small{N(r,1/(f^2(f^{\prime})^n-a))}$, which states as following inequality, for positive integers $\small{n{\geq}2}$, $\small{T(r,f){\leq}$$3+{\frac{6}{n-1}}$$N$$r,{\frac{1}{af^2(f^{\prime})^n-1}}$$+S(r,f)}$.
Keywords
transcendental meromorphic function;deficiency;
Language
English
Cited by
References
1.
A. Alotaibi, On the zeros of $aff(^k)-1$, Complex Var. Theory Appl. 49 (2004), no. 13, 977-989.

2.
A. Alotaibi, On the zeros of $af(f^{(k)})^n-1$ for $n{\geq}2$, Comput. Methods Funct. Theory 4 (2004), no. 1, 227-235.

3.
A. A. Goldberg and V. I. Ostrovskii, Value distribution of meromorphic functions, vol. 236, American Mathematical Society, Providence, RI, 2008.

4.
W. K. Hayman, Meromorphic Functions, Oxford University Press, 1964.

5.
X. J. Huang and Y. X. Gu, On the value distribution of $f^2f(^k)$, J. Aust. Math. Soc. 78 (2005), no. 1, 17-26.

6.
Y. Jiang and B. Huang, A note on the value distribution of $f^l(f^{(k)})^n$, in submission.

7.
I. Lahiri and S. Dewan, Inequalities arising out of the value distribution of a differential monomial, J. Inequal. Pure Appl. Math. 4 (2003), no. 2, Article 27, 6 pp. (electronic).

8.
P. Li and C. C. Yang, On the value distribution of a certain type of differential polyno-mials, Monatsh. Math. 125 (1998), no. 1, 15-24.

9.
C. K. Tse and C. C. Yang, On the value distribution of $f^l(f^{(k)})^n$, Kodai Math. J. 17 (1994), no. 1, 163-169.

10.
J. P. Wang, On the zeros of $f^n(z)f^{(k)}(z)-c(z)$, Complex Var. Theory Appl. 48 (2003), no. 8, 695-703.

11.
C. C. Yang and P. C. Hu, On the value distribution of $ff^{(k)}$, Kodai Math. J. 19 (1996), no. 2, 157-167.

12.
C. C. Yang and H. X. Yi, A unicity theorem for meromorphic functions with deficient values, Acta Math. Sinica 37 (1994), no. 1, 62-72.

13.
Q. D. Zhang, A growth theorem for meromorphic function, J. Chengdu Inst. Meteor. 20 (1992), 12-20.


text:
Outlook: EDINBURGH INVESTMENT TRUST PLC is assigned short-term Ba1 & long-term Ba1 estimated rating.
Dominant Strategy : Hold
Time series to forecast n: 01 Mar 2023 for (n+6 month)
Methodology : Inductive Learning (ML)

## Abstract

EDINBURGH INVESTMENT TRUST PLC prediction model is evaluated with Inductive Learning (ML) and Factor1,2,3,4 and it is concluded that the LON:EDIN stock is predictable in the short/long term. According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Hold

## Key Points

1. Stock Forecast Based On a Predictive Algorithm
2. What is a prediction confidence?
3. Can neural networks predict stock market?

## LON:EDIN Target Price Prediction Modeling Methodology

We consider EDINBURGH INVESTMENT TRUST PLC Decision Process with Inductive Learning (ML) where A is the set of discrete actions of LON:EDIN stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4

F(Factor)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Inductive Learning (ML)) X S(n):→ (n+6 month) $R=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

n:Time series to forecast

p:Price signals of LON:EDIN stock

j:Nash equilibria (Neural Network)

k:Dominated move

a:Best response for target price

For further technical information as per how our model work we invite you to visit the article below:

How do AC Investment Research machine learning (predictive) algorithms actually work?

## LON:EDIN Stock Forecast (Buy or Sell) for (n+6 month)

Sample Set: Neural Network
Stock/Index: LON:EDIN EDINBURGH INVESTMENT TRUST PLC
Time series to forecast n: 01 Mar 2023 for (n+6 month)

According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Hold

X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)

Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)

Z axis (Grey to Black): *Technical Analysis%

## IFRS Reconciliation Adjustments for EDINBURGH INVESTMENT TRUST PLC

1. When measuring the fair values of the part that continues to be recognised and the part that is derecognised for the purposes of applying paragraph 3.2.13, an entity applies the fair value measurement requirements in IFRS 13 Fair Value Measurement in addition to paragraph 3.2.14.
2. Historical information is an important anchor or base from which to measure expected credit losses. However, an entity shall adjust historical data, such as credit loss experience, on the basis of current observable data to reflect the effects of the current conditions and its forecasts of future conditions that did not affect the period on which the historical data is based, and to remove the effects of the conditions in the historical period that are not relevant to the future contractual cash flows. In some cases, the best reasonable and supportable information could be the unadjusted historical information, depending on the nature of the historical information and when it was calculated, compared to circumstances at the reporting date and the characteristics of the financial instrument being considered. Estimates of changes in expected credit losses should reflect, and be directionally consistent with, changes in related observable data from period to period
3. A net position is eligible for hedge accounting only if an entity hedges on a net basis for risk management purposes. Whether an entity hedges in this way is a matter of fact (not merely of assertion or documentation). Hence, an entity cannot apply hedge accounting on a net basis solely to achieve a particular accounting outcome if that would not reflect its risk management approach. Net position hedging must form part of an established risk management strategy. Normally this would be approved by key management personnel as defined in IAS 24.
4. Annual Improvements to IFRSs 2010–2012 Cycle, issued in December 2013, amended paragraphs 4.2.1 and 5.7.5 as a consequential amendment derived from the amendment to IFRS 3. An entity shall apply that amendment prospectively to business combinations to which the amendment to IFRS 3 applies.

*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.

## Conclusions

EDINBURGH INVESTMENT TRUST PLC is assigned short-term Ba1 & long-term Ba1 estimated rating. EDINBURGH INVESTMENT TRUST PLC prediction model is evaluated with Inductive Learning (ML) and Factor1,2,3,4 and it is concluded that the LON:EDIN stock is predictable in the short/long term. According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Hold

### LON:EDIN EDINBURGH INVESTMENT TRUST PLC Financial Analysis*

Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementCBaa2
Balance SheetBaa2B3
Leverage RatiosCCaa2
Cash FlowCBaa2
Rates of Return and ProfitabilityB2Baa2

*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?

### Prediction Confidence Score

Trust metric by Neural Network: 93 out of 100 with 669 signals.

## References

1. A. Tamar, D. Di Castro, and S. Mannor. Policy gradients with variance related risk criteria. In Proceedings of the Twenty-Ninth International Conference on Machine Learning, pages 387–396, 2012.
2. Mnih A, Hinton GE. 2007. Three new graphical models for statistical language modelling. In International Conference on Machine Learning, pp. 641–48. La Jolla, CA: Int. Mach. Learn. Soc.
3. Imai K, Ratkovic M. 2013. Estimating treatment effect heterogeneity in randomized program evaluation. Ann. Appl. Stat. 7:443–70
4. Burgess, D. F. (1975), "Duality theory and pitfalls in the specification of technologies," Journal of Econometrics, 3, 105–121.
5. Andrews, D. W. K. (1993), "Tests for parameter instability and structural change with unknown change point," Econometrica, 61, 821–856.
6. S. Devlin, L. Yliniemi, D. Kudenko, and K. Tumer. Potential-based difference rewards for multiagent reinforcement learning. In Proceedings of the Thirteenth International Joint Conference on Autonomous Agents and Multiagent Systems, May 2014
7. Çetinkaya, A., Zhang, Y.Z., Hao, Y.M. and Ma, X.Y., Is TPL a Buy?. AC Investment Research Journal, 101(3).
Frequently Asked QuestionsQ: What is the prediction methodology for LON:EDIN stock?
A: LON:EDIN stock prediction methodology: We evaluate the prediction models Inductive Learning (ML) and Factor
Q: Is LON:EDIN stock a buy or sell?
A: The dominant strategy among neural network is to Hold LON:EDIN Stock.
Q: Is EDINBURGH INVESTMENT TRUST PLC stock a good investment?
A: The consensus rating for EDINBURGH INVESTMENT TRUST PLC is Hold and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of LON:EDIN stock?
A: The consensus rating for LON:EDIN is Hold.
Q: What is the prediction period for LON:EDIN stock?
A: The prediction period for LON:EDIN is (n+6 month)

## People also ask

What are the top stocks to invest in right now?


text:
# HTML and LaTeX code conversion using TeX4HT

All the display equations/centered equations are no need to convert to MathML or PNG and need to retain in LaTeX coding only.

If it is possible convert to HTML tag in-between text part only convert to HTML tag.

But no need to convert to same positions of Superscript and Subscript, Same positions of Limits Superscript and Subscript and so on in the text part.

\documentclass{book}

\usepackage{amssymb,amsmath}

\begin{document}

Schematic representation of the model, including the following populations: $e$ vaue is $e=2$ and %e and e=2 convert to html tag
thalamic reticular nucleus; $s^2_3$ %s^2_3 should be in latex as its tag
primary and secondary thalamic relay nuclei themselves with gains $\pi + 2\pi n (n=0,1,\ldots)$ %\pi + 2\pi n (n=0,1,\ldots) convert to html tag
and $G_{ii}$ %G_{ii} convert to html tag. Schematic to and $G_{ii}$ is a text part position

\end{document}

-
may be it is just me, but I am having a hard time understanding the question. Could please try to ask it again? What is it you are asking for? Are you getting an error from htlatex? What command did you use? What is the error? –  Nasser Feb 20 '13 at 7:35
aren't you looking for tex.stackexchange.com/q/27510/2891 and tex.stackexchange.com/q/33708/2891 ? –  michal.h21 Feb 20 '13 at 7:56
@Balakrishnan I think your needs are covered with questions linked in my previous comment –  michal.h21 Feb 20 '13 at 9:37

After my research i have found the link http://www.tug.org/applications/tex4ht/mn5.html#QQ1-5-23. Because i have need to maintain centered/display/SUB SUP equations in LaTeX format only and no need PNG/MathML format. After that i will convert LaTeX Equations to Math Type OLE format. htlatex Config file:

\Preamble{html}
\newtoks\eqtoks

\def\EquationAlt{\afterassignment\setimg\eqtoks}
\def\setimg{\Picture*[\HCode{\the\eqtoks}]{}\the\eqtoks\EndPicture}

\def\SubSupOp#1{%
\edef\temp{\expandafter\gobble\string#1}%
\expandafter\let\csname old\temp\endcsname=#1
\edef\temp{\noexpand\SUBSUPOP{\expandafter\noexpand
\csname old\temp\endcsname}}%
\HLet#1=\temp
}
\def\gobble#1{}
\def\SUBSUPOP#1{\let\curOP=#1%
\let\next=\putOP \let\OPsub=\empty \let\OPsup=\empty
\futurelet\nextop\getOP}
\def\getOP{%
\ifx _\nextop \let\next=\getsub
\else\ifx ^\nextop \let\next=\getsup\fi\fi \next}
\def\getsub#1#2{\def\OPsub{#2}\let\next=\putOP
\futurelet\nextop\getOP}
\def\getsup#1#2{\def\OPsup{#2}\let\next=\putOP
\futurelet\nextop\getOP}
\def\putOP{\Picture+{  align="middle"}{\curOP_{\OPsub}^{\OPsup}}\EndPicture}

\SubSupOp\sum
\SubSupOp\int

\begin{document}

\Configure{graphics*}
{png}
\Picture[pict]{\csname Gin@base\endcsname .png
\space width="40px" height="40px" }%

htlatex myfile.tex test.cfg


text:
Local area network & Metropolitan area network

Local area network (LAN)
The term "local area network" is commonly used to describe a network of devices in a limited area (a house, office, building...). This type of network is usually capable of achieving high data transfer rate (up to 10 Gbps!) at low cost.

Some of the most popular LAN technologies are Ethernet, Token Ring and FDDI. Most LAN networks use TCP/IP to communicate. Twisted-pair cabling is usually used in a LAN.

Examples of this type of network are a small office network inside a single building or your home network.

Metropolitan area network (MAN)
The term „metropolitan area network“ is used to describe a network in a single metropolitan area, hence the name. This type of network is usually bigger than a LAN and smaller than a WAN. An example of this type of network would be a network that connects two company offices inside the same city.


text:
# Astrophysics and Astronomy Commons™

## All Articles in Astrophysics and Astronomy

2,705 full-text articles. Page 1 of 57.

Rotation Lightcurves Of Small Jovian Trojan Asteroids, 2015 Illinois Wesleyan University

#### Rotation Lightcurves Of Small Jovian Trojan Asteroids, Linda French, Robert D. Stephens, Daniel Coley, Lawrence H. Wasserman, Jennifer Sieben

##### Linda French

Several lines of evidence support a common origin for, and possible hereditary link between, cometary nuclei and Jovian Trojan asteroids. Due to their distance and low albedos, few comet-sized Trojans have been studied. We present new lightcurve information for 19 Trojans ≲ 30 km in diameter, more than doubling the number of objects in this size range for which some rotation information is known. The minimum densities for objects with complete lightcurves are estimated and are found to be comparable to those measured for cometary nuclei. A significant fraction (~40%) of this observed small Trojan population rotates slowly (P > 24 hours ...

2015 Bates College

#### Analysis And Comparison Of Physical Properties And Morphology Of Impact Melt Flows On Venus And The Moon To Determine Atmospheric Controls, Christine Elizabeth Dong

##### Geology Theses

Impact melt flows form when target material melted by an impact breaches the rim of the impact crater. The melt flows out and away from the crater, forming a lava flow. Such flows have been identified on Venus and the Moon by previous studies. This study aims to document impact melt flows on these two bodies to determine the differences, if any, between them, and to ascertain whether these differences are due to atmospheric conditions. Previously recorded morphological data from Robert Herrick of the Lunar and Planetary Institute and Neish et al. (2014) are used in conjunction with S-band (12 ...

A Photometric Study Of The Eclipsing Binary Star Uv Lyncis, 2015 Stephen F Austin State University, Department of Physics and Astronomy

#### A Photometric Study Of The Eclipsing Binary Star Uv Lyncis, Edward J. Michaels

##### Bright Ideas Conference

UV Lyncis is a short period, over contact eclipsing binary star. These stars are physically in contact with each other, sharing a common atmosphere. Photometric observations were made on 10 nights in February and March 2014 using the 0.3 meter robotic telescope and a CCD science imager at the Waffelow Creek Observatory. 3051 images were acquired, calibrated and then reduced to record the changing brightness as the two stars orbited and eclipsed each other. Data collected resulted in complete light curves in 5 band passes and several times of minima. A period analysis confirmed previous studies that the orbital ...

Stellar Spectroscopy: New Methods And Insights, 2015 The University of Western Ontario

#### Stellar Spectroscopy: New Methods And Insights, Sanaz S. Golriz

##### University of Western Ontario - Electronic Thesis and Dissertation Repository

The study of the chemical evolution of stars is of crucial importance since they play a major role in the enrichment of the chemistry of the universe. Throughout their lifetime, stars undergo several processes that can alter their chemistry. Gradually, the nucleosynthesis products from the interior of the star are radiatively and convectively levitated and mixed with the upper layers of the atmosphere. In the later stages of their evolution, low to intermediate mass stars (0.8-8.0~M☉) eject a significant fraction of these nucleosynthesis products, resulting in a circumstellar envelope of gas and dust around the central star ...

Supernatural Cosmic Origins: Challenging The Reigning Paradigm, 2015 Liberty University

#### Supernatural Cosmic Origins: Challenging The Reigning Paradigm, Rachel Blattner

##### Senior Honors Theses

Contemporary scientific study primarily uses a paradigm based upon naturalism, materialism, and empiricism on which to base research. The widely accepted cosmological model the big bang theory adheres to this paradigm. Despite many weaknesses in this model and in the paradigm itself, researchers continue to favor the modification of the accepted model over the adoption of other more comprehensive models. The paradigm from which the models proposed by Russell Humphries, John Hartnett, and Jason Lisle come justifies the six-day creation young-earth biblical account and better fits observational evidence with fewer arbitrary assumptions than the paradigm from which the big bang ...

The Ptolemaic System: A Detailed Synopsis, 2015 Kennesaw State University

#### The Ptolemaic System: A Detailed Synopsis, John Cramer Dr.

##### Oglethorpe Journal of Undergraduate Research

The Ptolemaic System, constructed by Claudius Ptolemeus (the Latin form of his name), was the most influential of all Earth centered cosmological systems. His ingenious and creative work is primarily recorded in his book The Mathematical Systematic Treatise which the Arabs characterized as “the greatest” and, in so doing, gave the book its most used name, Almagest.

The Copernican System: A Detailed Synopsis, 2015 Kennesaw State University

#### The Copernican System: A Detailed Synopsis, John Cramer Dr.

##### Oglethorpe Journal of Undergraduate Research

Dissatisfied with the problems of the geocentric system inherited from Claudius Ptolemy, Nicholas Copernicus began the change from geocentrism to heliocentrism. His eponymous system was expounded first in the Commentariolus (written about 1508 and circulated privately in manuscript form) and then more fully and finally in his book, De Revolutionibus Orbium Coelestium (On the Revolutions of the Celestial Orbs) published as he lay dying in 1543.

2015 The University of Western Ontario

#### Methods And Results Toward Measuring Magnetic Fields In Star-Forming Regions, Scott C. Jones

##### University of Western Ontario - Electronic Thesis and Dissertation Repository

Protolunar Disk Evolution And The Depletion Of Volatile Elements In The Moon, 2015 Southwest Research Institute

#### Protolunar Disk Evolution And The Depletion Of Volatile Elements In The Moon, Robin M. Canup, Channon Visscher, Julien Salmon, Bruce Fegley Jr.

##### Faculty Work: Comprehensive List

We explore how the evolution of the protolunar disk could lead to a depletion in K, Na, and Zn in the Moon relative to Earth even in the absence of escape.

2015 Utah State University

#### The Riemann Curvature Tensor, Its Invariants, And Their Use In The Classification Of Spacetimes, Jesse Hicks

##### Presentations and Publications

The equivalence problem in general relativity is to determine whether two solutions of the Einstein field equations are isometric. Petrov has given a classification of metrics according to their isometry algebras. This talk discusses the use of the Petrov classification scheme, together with the use of scalar curvature invariants, to address the equivalence problem. These are the slides for a presentation at the Mathematics Association of America Spring 2015 conference at Brigham Young University.

Geometrization Conditions For Perfect Fluids, Scalar Fields, And Electromagnetic Fields, 2015 Utah State University

#### Geometrization Conditions For Perfect Fluids, Scalar Fields, And Electromagnetic Fields, Charles G. Torre, Dionisios Krongos

##### Presentations and Publications

Rainich-type conditions giving a spacetime “geometrization” of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and elec- tromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equa- tions are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Ge- ometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and for- mulas for constructing ...

2015 Brigham Young University - Provo

#### Radio Emission Toward Regions Of Massive Star Formation In The Large Magellanic Cloud, Adam Johanson

##### All Theses and Dissertations

Four regions of massive star formation in the Large Magellanic Cloud (LMC) were observed for water and methanol maser emission and radio continuum emission. A total of 42 radio detections were made including 27 new radio sources, four water masers, and eight compact HII regions. The lobes of a radio galaxy were resolved for the first time, and the host galaxy identified. Seven sources were associated with known massive young stellar objects (YSOs). A multi-wavelength analysis using both the infrared and radio spectrum was used to characterize the sources. Mid-infrared color-magnitude selection criteria for ultracompact HII (UCHII) regions in the ...

The Jcmt Gould Belt Survey: Evidence For Radiative Heating In Serpens Mwc 297 And Its Influence On Local Star Formation, 2015 University of Massachusetts - Amherst

#### The Jcmt Gould Belt Survey: Evidence For Radiative Heating In Serpens Mwc 297 And Its Influence On Local Star Formation, D. Rumble, J. Hatchell, R. A. Gutermuth, H. Kirk, J. Buckle, S.F. Beaulieu, D.S. Berry, H. Broekhoven-Fiene, M.J. Currie, M. Fich, T. Jenness, D. Johnstone, J.C. Mottram, D. Nutter, K. Pattle, J.E. Pineda, C. Quinn, C. Salji, S. Tisi, S. Walker-Smith, J. Di Francesco, M.R. Hogerheijde, D. Ward-Thompson, L.E. Allen, L.A. Cieza, M.M. Dunham, P.M. Harvey, K.R. Stapelfeldt, P. Bastien, H. Bunter, M. Chen, A. Chrysostomou, S. Coude, C.J. Davis, E. Drabek-Maunder, A. Duarte-Cabral, J. Greaves, J. Gregson, W. Holland, G. Joncas, J.M. Kirk, L.B.G. Knee, S. Mairs, K. Marsh, B.C. Matthews, G. Moriarty-Schieven, J. Rawlings, J. Richer, D. Robertson, E. Rosolowsky, S. Sadavoy, H. Thomas, N. Tothill, S. Viti, G.J. White, C.D. Wilson, J. Wouterloot, J. Yates, M. Zhu

##### Robert A. Gutermuth

We present SCUBA-2 450 and 850 μm observations of the Serpens MWC 297 region, part of the James Clerk Maxwell Telescope (JCMT) Gould Belt Survey of nearby star-forming regions. Simulations suggest that radiative feedback influences the star formation process and we investigate observational evidence for this by constructing temperature maps. Maps are derived from the ratio of SCUBA-2 fluxes and a two-component model of the JCMT beam for a fixed dust opacity spectral index of β = 1.8. Within 40 arcsec of the B1.5Ve Herbig star MWC 297, the submillimetre fluxes are contaminated by free–free emission with a ...

2015 Otterbein University

#### Empyreal Radiance: An Application Of Sonification In The Field Of Astrophysics, Ryan Loth

##### Distinction Papers

Broadly, this paper discusses the application of sonification and its potential for increasing knowledge. The paper is broken up into three sections: the theory of sonification, sonification for artistic purposes, and lastly an extensive look at one process of sonification dealing with solar winds in space. Concerning the theory of sonification, the paper will divulge into the process of sonification and ask questions about the limitations of it as well. The second section discusses how sonification is a way to build the curiosity of not just scientists, but also the general public. The final section addresses my composition Empyreal Radiance ...

2015 The University of Western Ontario

#### Rationality Of The Spectral Action For Robertson-Walker Metrics And The Geometry Of The Determinant Line Bundle For The Noncommutative Two Torus, Asghar Ghorbanpour

##### University of Western Ontario - Electronic Thesis and Dissertation Repository

In noncommutative geometry, the geometry of a space is given via a spectral triple $(\mathcal{A,H},D)$. Geometric information, in this approach, is encoded in the spectrum of $D$ and to extract them, one should study spectral functions such as the heat trace $\Tr (e^{-tD^2})$, the spectral zeta function $\Tr(|D|^{-s})$ and the spectral action functional, $\Tr f(D/\Lambda)$.

The main focus of this thesis is on the methods and tools that can be used to extract the spectral information. Applying the pseudodifferential calculus and the heat trace techniques, in addition to computing the newer ...

2015 Cleveland State University

#### On The Detection Of Exoplanets Via Radial Velocity Doppler Spectroscopy, Joseph P. Glaser

##### The Downtown Review

Since the discovery of the first exoplanet occurred in the late 1980s, physicists and astronomers have confirmed the existence of over 1783 planets orbiting other stars. Over the years, several methods have been developed for detecting these celestial systems, but none has thus far been more successful than the use of Radial Velocity Doppler Spectroscopy. As such, this paper attempts to break down the barriers placed before new entries into the field by detailing the theoretical premise and experimental process of this most successful technique. The content enclosed is geared for the senior undergraduate student with some knowledge of vector-based ...

A 24Μm Point Source Catalog Of The Galactic Plane From Spitzer/Mipsgal, 2015 University of Massachusetts - Amherst

#### A 24Μm Point Source Catalog Of The Galactic Plane From Spitzer/Mipsgal, R. A. Gutermuth, Mark Heyer

##### Robert A. Gutermuth

In this contribution, we describe the applied methods to construct a 24 μm based point source catalog derived from the image data of the MIPSGAL 24 μm Galactic Plane Survey and the corresponding data products. The high quality catalog product contains 933,818 sources, with a total of 1,353,228 in the full archive catalog. The source tables include positional and photometric information derived from the 24 μm images, source quality and confusion flags, and counterpart photometry from matched 2MASS, GLIMPSE, and WISE point sources. Completeness decay data cubes are constructed at 1' angular resolution that describe the varying ...

The Formation Of The Trajectories Of The Planets Of The Solar System, 2015 Yanbikov Vildyan Shavkyatovich

#### The Formation Of The Trajectories Of The Planets Of The Solar System, Vildyan Yanbikov

##### Vildyan Yanbikov

Процесс превращения круговой орбите планеты по эллиптической. Процесс превращения эллиптической орбите вокруг Солнца (прецессия перигелия планеты).

Geometrization Conditions For Perfect Fluids, Scalar Fields, And Electromagnetic Fields, 2015 Department of Physics, Utah State University

#### Geometrization Conditions For Perfect Fluids, Scalar Fields, And Electromagnetic Fields, Charles G. Torre, Dionisios Krongos

##### Charles G. Torre

Rainich-type conditions giving a spacetime “geometrization” of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and elec- tromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equa- tions are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Ge- ometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and for- mulas for constructing ...

Alignment Of The Western And Eastern Passage Tombs At Knowth Tomb 1, 2015 Dublin Institute of Technology

#### Alignment Of The Western And Eastern Passage Tombs At Knowth Tomb 1, Frank Prendergast, Tom Ray

##### Book/Book Chapter

This report presents a summary account of two surveys undertaken by the authors to determine and interpret the alignments of the western and eastern passage tombs at Knowth, Co. Meath. The findings indicate that contrary to earlier suggestions, the eastern passage and the western passage (inner and outer) are not aligned towards sunrise and sunset respectively at the period of the vernal and autumnal equinoxes.


text:
# D meson decay

1. Nov 30, 2011

### KBriggs

I am trying to work out the Feynman diagram for the decay

$$D^0\to K^++\pi^-$$

But I can't seem to get it unless the D meson is its own antiparticle. Could someone tell me if this is the case? Ie, is $$|\bar{u}c\rangle = |u\bar{c}\rangle$$?

Thanks

2. Nov 30, 2011

### Bill_K

That's because the decay is actually D0 → K- + π+. Maybe you want to start with an anti-D0

3. Nov 30, 2011

### KBriggs

No, the question gives three decays, one of which is the one you posted and another is the one I posted.

Could I change D to anti-D by having the two constituent quarks exchange a gluon, so that the decay I posted would then proceed from the antiparticle?

See for example decay modes 42 and 213 here: http://pdg.lbl.gov/2008/listings/s032.pdf

Last edited: Nov 30, 2011
4. Nov 30, 2011

Staff Emeritus
Do you know what Cabibbo suppression is? That decay is doubly Cabibbo suppressed. That should help you.

5. Nov 30, 2011

### KBriggs

Update: seem to work if the c decays into d and a W^+ meson, which then decays into u and anti-s

any problems there?

this decay is apparently the much less likely than the one you posted. Why is that?

EDIT: I don't yet know what cabibbo suppression is. Do you know of a resource that explains it? Google turns up a lot of scholarly stuff that assumes I know it already.


text:
# How does combat with weapon and shield work?

In Ars Magic 5th I am not able to find how combat works for a character with weapon and shield.

Attack total is:

Dexterity + Combat Ability + Weapon Attack Modifier + Stress Dice

Defense total is:

Speed + Combat Ability + Weapon Defense Modifier + Stress Dice

If a character fights with a sword and a shield, does it add both Defense Modifiers? Is the same done with Attack Modifiers?

• I have spanish edition, so I needed to translate it back to english. Probably it is a fault on my side. – Envite Sep 9 '15 at 17:25
• I'm removing the tag again, and removing the instructions you wrote in the tag wiki to use both tags, because it's unnecessary to overtag questions. If people want to find all questions about Ars Magica, they can search/favourite ars-magica*. – SevenSidedDie Sep 9 '15 at 18:01
• Should you, then, remove the double tag from all ars-magica-5 questions? – Envite Sep 9 '15 at 18:05
• @Envite Probably most of those questions aren't about every edition and so the tag doesn't belong, but digging through old questions just to fix one tag is a waste of people's time and eyeballs on the front page (since editing bumps questions). They'll get removed organically as they return to the front page or as people trip over them while looking for other things. – SevenSidedDie Sep 9 '15 at 18:06

## 1 Answer

In the English language edition, on page 171, under "Combat Scores" it says "If the character is using a weapon and a shield, add together the modifiers of the weapon and the shield to get the final modifier."

So, yes, add both modifiers to both totals.

• As far as I can see the rules do not, however, consider the case where a combatant is fighting with two Single Weapons; or worse fighting Sword and Dagger, although I think that style may be anachronistic for the setting. – Senji Sep 11 '15 at 0:52
• Thanks! I'll double check the possibly corresponding paragraph in spanish. – Envite Sep 11 '15 at 8:58
• Found it. Correct. – Envite Sep 11 '15 at 15:55


text:
## Year 2017-2018

Time: Tuesdays 3-4pm
Location: MW 154

## Schedule of talks:

 TIME SPEAKER TITLE August 29   Tue, 3pm Dave Anderson  (OSU) Determinantal formulas for degeneracy loci and Brill-Noether theory September 5   Tue, 3pm Hsian-Hua Tseng  (OSU) On the quantum K-rings of flag varieties of type A September 12   Tue, 3pm Angelica Cueto  (OSU) Tropical geometry of genus 2 curves September 19   Tue, 3pm Herb Clemens  (OSU) Eliptically-fibered Calabi-Yau (CY) threefolds and fourfolds in string theory October 3   Tue, 3pm Seth Baldwin  (UNC) Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups October 17   Tue, 3pm Roi Docampo  (Oklahoma) Differentials on the arc space October 26   Thu, 4:10pm CH 240 Renzo Cavalieri  (Colorado State U.) Colloquium talk: From classical to tropical enumerative geometry October 31   Tue, 3pm Christin Bibby  (U. Michigan) Arrangements with group actions and representation stability November 7   Tue, 3pm Chris Manon  (U. Kentucky) Problems with Khovanskii bases November 14   Tue, 3pm Daniel Litt  (Columbia U.) Arithmetic Representations of Fundamental Groups November 21   Tue, 3pm TBA  () TBA November 28   Tue, 3pm Jose Gonzalez  (UC Riverside) Negative curves on blown-up weighted projective planes December 5   Tue, 3pm Matthew Satriano  (U. Waterloo) TBA January 9   Tue, 3pm TBA  ( ) TBA January 16   Tue, 3pm Gabriel Khan  (OSU) A conjectural picture of complex structures on tori (and other spaces) January 23   Tue, 3pm TBA  () TBA January 30   Tue, 3pm TBA  () TBA February 6   Tue, 3pm TBA  () TBA February 13   Tue, 3pm TBA  () TBA February 20   Tue, 3pm Nicola Tarasca  (Rutgers) TBA February 27   Tue, 3pm TBA  () TBA March 1   Thu, 4:10pm CH 240 William Fulton  (Michigan) Colloquium - TBA March 6   Tue, 3pm TBA  () TBA March 13   Tue, 3pm (break) March 20   Tue, 3pm Valery Lunts  (Indiana) TBA March 27   Tue, 3pm TBA  () TBA April 3   Tue, 3pm TBA  () TBA April 10   Tue, 3pm TBA  () TBA April 16-18   Tue, 4:10pm CH 240 Sergey Fomin  (U. Michigan) Zassenhaus Lecture Series 2018 April 24   Tue, 3pm TBA  () TBA

## Abstracts

(Anderson): Relations between degeneracy loci and divisors on curves go back over a hundred years: Giambelli's formula for the class of the locus where a matrix drops rank is a cornerstone of Schubert calculus; Brill and Noether's estimate for the dimension of the space of special divisors on a curve also comes from considerations of degeneracy loci. In this talk, I will describe work with Linda Chen and Nicola Tarasca which connects more recent developments in Schubert calculus with formulas by Eisenbud-Harris, Pirola, and Chan-Lopez-Pflueger-Teixidor for the genus of a Brill-Noether curve.

(Tseng): The quantum K-ring of a smooth projective variety X, introduced by Givental and YP Lee, is a deformation of the Grothendieck ring of coherent sheaves on X defined using holomorphic Euler characteristics on moduli spaces of stable maps to X (these are K-theoretic version of Gromov-Witten invariants). In this talk we discuss some properties of the quantum K-rings of X=Flr+1, the variety of complete flags in Cr+1, including:

(1) a presentation of the quantum K-ring;

(2) finiteness of quantum product;

(3) canonical polynomial representatives of Schubert classes.

This is based on joint work in progress with David Anderson and Linda Chen.

(Cueto): In this talk, I will discuss the structure of tropical and non-Archimedean analytic genus 2 curves and their moduli from three perspectives:

(1) as 2-to-1 covers of P1 branched at 6 points;

(2) as solutions to the hyperelliptic equation y2=f(x), where f has degree 5; and

(3) as metric graphs dual to genus 2 nodal algebraic curves over a valued field.

Our first description allows us to give an explicit combinatorial rule to characterize each such metric graph together with a harmonic map to a metric tree on 6 leaves, in terms of the valuations of 6 branch points in P1. Even though the tropicalization of a plane hyperelliptic curve shows no genus, we provide explicit re-embeddings of the input planar hyperelliptic curve that reveals the correct metric graphs.

From our third viewpoint, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates on the tropical moduli space, we propose an alternative set of invariants that provides new length data. This is joint work with Hannah Markwig.

(Clemens): The talk will construct one example of the equivalence' between an elliptically-fibered CY-threefold endowed with a semi-stable E8 + E8 bundle and a semi-stable degeneration of an elliptically-fibered CY-fourfold. I will then loosely describe how this equivalence is used to break' the SU(5)-symmetry of grand unified theory (GUT) to the SU(3)xSU(2)xU(1)-symmetry of what physicists call the Standard Model of the physics of our cosmos.

(Baldwin): The cohomology ring of flag varieties has long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

(Docampo): We study the sheaf of Kähler differentials on the arc space of an algebraic variety. We obtain explicit formulas that can be used effectively to understand the local structure of the arc space. The approach leads to new results as well as simpler and more direct proofs of some of the fundamental theorems in the literature. The main applications include: an interpretation of Mather discrepancies as embedding dimensions of certain points in the arc space, a new proof of a version of the birational transformation rule in motivic integration, and a new proof of the curve selection lemma for arc spaces. This is joint work with Tommaso de Fernex.

(Bibby): We are interested in certain arrangements of subvarieties on which a wreath product group acts. We give a combinatorial description of its poset of layers (connected components of intersections) as a generalization of Dowling and partition lattices. This combinatorial structure is an aid in understanding the cohomology of the complement as a representation of the group. In particular, when one considers a sequence of these representations, its stability can be viewed as a consequence of combinatorial stability. Joint work with Nir Gadish.

(Manon): I'll give an introduction with examples to the notion of Khovanskii basis. Khovanskii bases are generating sets for algebras which are distinguished by their computational properties as well as their relationships with tropical geometry and toric geometry. After a few examples, I'll survey some recent results on their existence and classification.

(Litt): Let X be an algebraic variety over a field k. Which representations of π1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X. As a sample application of our techniques, we show that if X is a normal variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) satisfying the following: any irreducible, non-trivial p-adic representation of the fundamental group of X, which arises from geometry, is non-trivial mod pN.

(Gonzalez): The Cox rings of weighted projective planes blown-up at a general point have been studied in connection to the noetherianity of symbolic Rees algebras and to the finite generation of the Cox ring of the moduli space of stable rational pointed curves. We will present some sufficient conditions for the finite and nonfinite generation of the Cox ring of such blowups. This is joint work with Javier Gonzalez and Kalle Karu.

(Khan): One of the central questions in complex geometry is to understand the moduli space of complex structures on a given manifold. In general, this is a mysterious object. Locally, we can study deformations of complex structures using Kodaira-Spencer theory. However, the global geometry can be very complicated and have singularities. In this talk, we consider the case where we have a Riemannian-flat metric and show how this can be used to understand the geometry of the moduli space. From here, we lay out some conjectures about how this can be done more generally, and give some heuristic evidence for these conjectures.

## Past Seminars

Ohio State University Algebraic Geometry Seminar-Year 2016-2017

Ohio State University Algebraic Geometry Seminar-Year 2015-2016

Ohio State University Algebraic Geometry Seminar-Year 2014-2015

Ohio State University Algebraic Geometry Seminar-Year 2013-2014


text:
# Introduction to Earth Data ScienceEarth Lab CU Boulder

## About the Introduction to Earth Data Science Textbook

Introduction to Earth Data Science is an online textbook for anyone new to open reproducible science and the Python programming language. There are no prerequisites for this material, and no prior programming knowledge is assumed.

This textbook is designed for the Earth Analytics Bootcamp for the Earth Data Analytics Professional Certificate taught by instructors at CU Boulder.

### Overview

In this textbook, you will learn how to analyze and visualize earth and environmental science data using the Python programming language. You will also get familiar with a suite of open source tools that are often used in open reproducible science workflows including bash, git and Github.com, and Jupyter Notebook.

This textbook is highly technical, and each chapter covers some aspect of scientific programming and open reproducible science workflows. Additional sections and chapters will continue to be added.

Updated:


text:
MathSciNet bibliographic data MR1988329 (2004e:35060) 35J60 (35B05 35B40 53C21 58J05) Cheung, Ka Luen; Leung, Man Chun Asymptotic behavior of positive solutions of the equation $\Delta u+Ku\sp {\frac{n+2}{n-2}}=0$$\Delta u+Ku\sp {\frac{n+2}{n-2}}=0$ in $\Bbb R\sp n$$\Bbb R\sp n$ and positive scalar curvature. Dynamical systems and differential equations (Kennesaw, GA, 2000). Discrete Contin. Dynam. Systems 2001, Added Volume, 109–120. Journal

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# Fall 2019 Exam 1 Solution

Here are the final numerical results for each section of the exam. You can use them to check your work if you do the exam for practice. If you have trouble with the problems, or don't get the answers shown here, stop by during office hours or make and appointment and we can go over them.

### Question 1

(a) P=$160, Q=48,000; Q_{Ai}=480; Q_{Ci}=120; graph omitted. (b) Market results: P^d=$156; P^s=$168; Q=46,400. Individual results: Q_{Ai}=464, Q_{Bi}=232, \Delta CS_{Ai}=-$3776. Percent of tax borne by each group: buyers=8/12=67%, sellers=4/12=33%. Effective tax rates: ETR_{Ai}=4.6%, ETR_{Bi}=6.2%; tax is regressive since the effective tax rate is higher for the group with lower income.

### Question 2

Tariff needed=$100; new P=$300; new total Q consumed=6000; new Q_L=2000; new Q_F=4000; \Delta CS=-$800k; Delta PS=+$150k; \Delta Rev=+$400k; \Delta SS=-$250k. The policy is very expensive for what it delivers: the cost in lost CS is more than five times the PS it delivers to domestic producers, the intended beneficiaries. Alternatively, it creates $250k in overall losses to deliver$150k of gain to producers.

### Question 3

Initial equilbrium: P=$700, Q=300; efficient Q=340; efficient P^d=$580, P^s=$780; \Delta CS=+$38.4k, \Delta PS=+$25.6k, \Delta Rev=-$68k, \Delta Ext=+$7.2k. Overall: \Delta SS=+$3.2k.

### Question 4

Policy 1: Q_X=8k; \Delta Rev=$160k; DWL=$20k. Policy 2: in each market Q will be 9k, \Delta Rev will be $90k, and DWL will be$5k; combining the two markets gives the overall impact: \Delta Rev=$180k and DWL=$10k. Policy 2 is better: the broader but lower tax raises more revenue and produces half the DWL. Alternatively, the DWL per dollar of revenue for Policy 1 is 12.5% while for Policy 2 it is only 5.5%.

URL: https://wilcoxen.maxwell.insightworks.com/pages/4904.html
Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 10/13/2019


text: The symmetric square of a genus $2$ curve is a blow up of a 2-torus in one point (the canoncal divisor in the Jacobian). Nice example for Hilbert schemes.


text:
## alli14344 Help me please if you want :D 10 months ago 10 months ago

1. alli14344

2. alli14344

28 4 20 5.3

3. alli14344

10 17.8 625 5

4. rajee_sam

a^2 + b^2 = c^2 12^2 + 16 ^2 = c^2 144 + 256 = c^2 400 = C ^2 c=20

5. luckythebest

c = 20. By pythagoras theorem, hypotenuse = sqrt. of sum of squares of sides. c = $\sqrt{256+144}$ => $\sqrt{400}$ => + or - 20. However sides cannot be negative; hence c = 20 option C

6. alli14344

:) thank you


text:
Instantiates one new measure per protected attribute group in a task. Each metric is then evaluated only on predictions made for the given specific subgroup.

## Usage

groupwise_metrics(base_measure, task, intersect = TRUE)

## Arguments

base_measure

(Measure())
The base metric evaluated within each subgroup.

Task
mlr3::Task() to instantiate measures for.

intersect

logical
Should multiple pta groups be intersected? Defaults to TRUE. Only relevant if more than one pta columns are provided.

## Value

list

List of mlr3::Measures.

  t = tsk("compas")
l$train(t)$predict(t)\$score(m, t)


text:
# Chapter 3 - Reading Check Questions (Comprehension): 9

$$10\frac{km}{h \cdot s}$$

#### Work Step by Step

Use the equation on page 44. $$\frac{100 km/h - 0}{10 s}=10\frac{km}{h \cdot s}$$

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:
ansi-terminal-0.8.0.4: Simple ANSI terminal support, with Windows compatibility

System.Console.ANSI

Description

Through this module, this library provides platform-independent support for control character sequences following the 'ANSI' standards (see further below) for terminal software that supports those sequences, running on a Unix-like operating system or Windows.

The sequences of control characters (also referred to as 'escape' sequences or codes) provide a rich range of functionality for terminal control, which includes:

• Colored text output, with control over both foreground and background colors
• Clearing parts of a line or the screen
• Hiding or showing the cursor
• Moving the cursor around
• Reporting the position of the cursor
• Scrolling the screen up or down
• Changing the title of the terminal

The native terminal software on Windows is 'Command Prompt' or PowerShell. Before Windows 10 version 1511 (known as the 'November [2015] Update' or 'Threshold 2') that software did not support such control sequences. For that software, this library also provides support for such sequences by using emulation.

Terminal software other than the native software exists for Windows. One example is the 'mintty' terminal emulator for 'Cygwin', 'MSYS' or 'MSYS2', and dervied projects, and for 'WSL' (Windows Subsystem for Linux).

The 'ANSI' standards refer to (1) standard ECMA-48 Control Functions for Coded Character Sets' (5th edition, 1991); (2) extensions in ITU-T Recommendation (previously CCITT Recommendation) T.416 (03/93) 'Information Technology – Open Document Architecture (ODA) and Interchange Format: Character Content Architectures (also published as ISO/IEC International Standard 8613-6); and (3) further extensions used by 'XTerm', a terminal emulator for the X Window System. The escape codes are described in a Wikipedia article at http://en.wikipedia.org/wiki/ANSI_escape_code and those codes supported on current versions of Windows at https://docs.microsoft.com/en-us/windows/console/console-virtual-terminal-sequences.

The whole of the 'ANSI' standards are not supported by this library but most (if not all) of the parts that are popular and well-supported by terminal software are supported. Every function exported by this module comes in three variants, namely:

• A variant that has an IO () type and doesn't take a Handle (for example, clearScreen :: IO ()). This variant just outputs the ANSI command directly to the standard output channel (stdout) and any terminal corresponding to it. Commands issued like this should work as you expect on both Unix-like operating systems and Windows.
• An 'h...' variant that has an IO () type but takes a Handle (for example, hClearScreen :: Handle -> IO ()). This variant outputs the ANSI command to the supplied handle and any terminal corresponding to it. Commands issued like this should also work as you expect on both Unix-like operating systems and Windows.
• A '...Code' variant that has a String type (for example, clearScreenCode :: String). This variant outputs the sequence of control characters as a String, which can be added to any other bit of text before being output. The use of these codes is generally discouraged because they will not work on legacy versions of Windows where the terminal in use is not ANSI-enabled (see further above). On Windows, where emulation has been necessary, these variants will always output the empty string. That is done so that it is possible to use them portably; for example, coloring console output on the understanding that you will see colors only if you are running on a Unix-like operating system or a version of Windows where emulation has not been necessary. If the control characters are always required, see module System.Console.ANSI.Codes.

Example:

module Main where

import System.Console.ANSI

-- Set colors and write some text in those colors.
main = do
setSGR [SetColor Foreground Vivid Red]
setSGR [SetColor Background Vivid Blue]
putStrLn "Red-On-Blue"
setSGR [Reset]  -- Reset to default colour scheme
putStrLn "Default colors."

For many more examples, see the project's extensive Example.hs file.

Synopsis

# Cursor movement by character

Arguments

 :: Int Number of lines or characters to move -> IO ()

Arguments

 :: Int Number of lines or characters to move -> IO ()

Arguments

 :: Int Number of lines or characters to move -> IO ()

Arguments

 :: Int Number of lines or characters to move -> IO ()

Arguments

 :: Handle -> Int Number of lines or characters to move -> IO ()

Arguments

 :: Handle -> Int Number of lines or characters to move -> IO ()

Arguments

 :: Handle -> Int Number of lines or characters to move -> IO ()

Arguments

 :: Handle -> Int Number of lines or characters to move -> IO ()

Arguments

 :: Int Number of lines or characters to move -> String

Arguments

 :: Int Number of lines or characters to move -> String

Arguments

 :: Int Number of lines or characters to move -> String

Arguments

 :: Int Number of lines or characters to move -> String

# Cursor movement by line

The difference between movements "by character" and "by line" is that *Line functions additionally move the cursor to the start of the line, while functions like cursorUp and cursorDown keep the column the same.

Also keep in mind that *Line functions are not as portable. See https://github.com/feuerbach/ansi-terminal/issues/10 for the details.

Arguments

 :: Int Number of lines to move -> IO ()

Arguments

 :: Int Number of lines to move -> IO ()

Arguments

 :: Handle -> Int Number of lines to move -> IO ()

Arguments

 :: Handle -> Int Number of lines to move -> IO ()

Arguments

 :: Int Number of lines to move -> String

Arguments

 :: Int Number of lines to move -> String

# Directly changing cursor position

Arguments

 :: Int 0-based column to move to -> IO ()

Arguments

 :: Handle -> Int 0-based column to move to -> IO ()

Arguments

 :: Int 0-based column to move to -> String

Arguments

 :: Int 0-based row to move to -> Int 0-based column to move to -> IO ()

Arguments

 :: Handle -> Int 0-based row to move to -> Int 0-based column to move to -> IO ()

Arguments

 :: Int 0-based row to move to -> Int 0-based column to move to -> String

# Saving, restoring and reporting cursor position

Save the cursor position in memory. The only way to access the saved value is with the restoreCursor command.

Restore the cursor position from memory. There will be no value saved in memory until the first use of the saveCursor command.

Looking for a way to get the cursors position? See getCursorPosition.

Emit the cursor position into the console input stream, immediately after being recognised on the output stream, as: ESC [ <cursor row> ; <cursor column> R

In isolation of getReportedCursorPosition or getCursorPosition, this function may be of limited use on Windows operating systems because of difficulties in obtaining the data emitted into the console input stream. The function hGetBufNonBlocking in module System.IO does not work on Windows. This has been attributed to the lack of non-blocking primatives in the operating system (see the GHC bug report #806 at https://ghc.haskell.org/trac/ghc/ticket/806).

# Clearing parts of the screen

Note that these functions only clear parts of the screen. They do not move the cursor.

# Scrolling the screen

Arguments

 :: Int Number of lines to scroll by -> IO ()

Scroll the displayed information up or down the terminal: not widely supported

Arguments

 :: Int Number of lines to scroll by -> IO ()

Scroll the displayed information up or down the terminal: not widely supported

Arguments

 :: Handle -> Int Number of lines to scroll by -> IO ()

Scroll the displayed information up or down the terminal: not widely supported

Arguments

 :: Handle -> Int Number of lines to scroll by -> IO ()

Scroll the displayed information up or down the terminal: not widely supported

Arguments

 :: Int Number of lines to scroll by -> String

Arguments

 :: Int Number of lines to scroll by -> String

# Select Graphic Rendition mode: colors and other whizzy stuff

Arguments

 :: [SGR] Commands: these will typically be applied on top of the current console SGR mode. An empty list of commands is equivalent to the list [Reset]. Commands are applied left to right. -> IO ()

Set the Select Graphic Rendition mode

Arguments

 :: Handle -> [SGR] Commands: these will typically be applied on top of the current console SGR mode. An empty list of commands is equivalent to the list [Reset]. Commands are applied left to right. -> IO ()

Set the Select Graphic Rendition mode

Arguments

 :: [SGR] Commands: these will typically be applied on top of the current console SGR mode. An empty list of commands is equivalent to the list [Reset]. Commands are applied left to right. -> String

# Changing the title

Arguments

 :: String New title -> IO ()

Set the terminal window title

Arguments

 :: Handle -> String New title -> IO ()

Set the terminal window title

Arguments

 :: String New Icon Name and Window Title -> String

XTerm control sequence to set the Icon Name and Window Title.

# Checking if handle supports ANSI (not portable: GHC only)

Use heuristics to determine whether the functions defined in this package will work with a given handle.

For Unix-like operating systems, the current implementation checks that: (1) the handle is a terminal; and (2) a TERM environment variable is not set to dumb (which is what the GNU Emacs text editor sets for its integrated terminal).

For Windows, the current implementation performs the same checks as for Unix-like operating systems and, as an alternative, checks whether the handle is connected to a 'mintty' terminal. (That is because the function hIsTerminalDevice is used to check if the handle is a terminal. However, where a non-native Windows terminal (such as 'mintty') is implemented using redirection, that function will not identify a handle to the terminal as a terminal.)

# Getting the cursor position

Attempts to get the reported cursor position, combining the functions reportCursorPosition, getReportedCursorPosition and cursorPosition. Returns Nothing if any data emitted by reportCursorPosition, obtained by getReportedCursorPosition, cannot be parsed by cursorPosition.

On Windows operating systems, the function is not supported on consoles, such as mintty, that are not based on the Win32 console of the Windows API. (Command Prompt and PowerShell are based on the Win32 console.)

Attempts to get the reported cursor position data from the console input stream. The function is intended to be called immediately after reportCursorPosition (or related functions) have caused characters to be emitted into the stream.

For example, on a Unix-like operating system:

hSetBuffering stdin NoBuffering -- set no buffering (the contents of the
-- buffer will be discarded, so this needs
-- to be done before the cursor positon is
-- emitted)
reportCursorPosition
hFlush stdout -- ensure the report cursor position code is sent to the
-- operating system
input <- getReportedCursorPosition

On Windows operating systems, the function is not supported on consoles, such as mintty, that are not based on the Win32 console of the Windows API. (Command Prompt and PowerShell are based on the Win32 console.)

Parses the characters emitted by reportCursorPosition into the console input stream. Returns the cursor row and column as a tuple.

For example, if the characters emitted by reportCursorPosition are in String input then the parser could be applied like this:

let result = readP_to_S cursorPosition input
case result of
[] -> putStrLn $"Error: could not parse " ++ show input [((row, column), _)] -> putStrLn$ "The cursor was at row " ++ show row
++ " and column" ++ show column ++ "."
(_:_) -> putStrLn \$ "Error: parse not unique"


text:
# The Mass Flow Conveyor Behavior Panel

The Mass Flow Conveyor Behavior panel defines the behavior properties for a mass flow conveyor.

The following properties are on this panel:

### Speed

Defines the speed of the conveyor.

### Generative Rate

This is used if you want the conveyor to be a source that generates mass flow units. You define a rate, in units per model time unit, and the flow unit that is to be generated. For example, if you are simulating a bottling line, and your model is defined in minutes, then a generative rate of 100 means the conveyor will generate 100 bottles per minute.

### Upstream Output Order

Defines the ordering by which this conveyor should be chosen by an upstream conveyor in distributing its output flow. This is only used when the conveyor is one of multiple downstream conveyors that is connected to a single upstream conveyor, in other words, when the conveyor upstream of this conveyor must choose between multiple downstream conveyors in sending flow units. See Distributing Flow Downstream for more information.

### Excess

Defines how the conveyor should handle excess flow that cannot be sent downstream. If Accumulate is chosen, then the excess will accumulate on the conveyor. If Track as Loss is chosen, then any excess flow will be tracked as loss, as if the product were to fall off the end of the conveyor. See Distributing Flow Downstream for more information.

### Width Rule

Defines a rule for determining the simulated width of the conveyor. This is used in combination with the Width Value.

For detailed information on how the simulated width affects conveyor behavior, see The Math Behind Mass Flow Conveyors. The width rule lets you customize how the $W$ value in those math equations is determined. Here we use the same notation, defining $W$ as the simulated width of the conveyor, and introducting $V$ as the user-defined width value.

• Conveyor Width - Uses the standard Width property.

• $W = \text{conveyor width}$

• Virtual Width - The simulated width is the Width Value.

• $W = V$

• Number of Lanes - The simulated width is calculated so as to simulate a user-defined number of unit lanes traveling down the conveyor. Enter the desired number of lanes as the Width Value.

• $W = V \cdot \Large{\frac{U_y}{R}}$

For intuition on this equation, note that the value $\frac{U_y}{R}$ is essentially the width associated with a single lane of units flowing down the conveyor: $U_y$ is the width of a single unit, and $R$ is the round-bottle density factor. $V$ therefore serves to scale the conveyor width to be the number of lanes you want to simulate.

It's important to remember that, for this and other rules, fractional values are perfectly valid. For example, if you were to enter 0.5 as the width value here, it would simulate 'one half' of a single lane of flow. This is essentially the same as a single lane of units with an 'air gap' of $U_x$ between each unit/carton/bottle/etc. This is similar to the Single Lane Flow Unit Distance rule described below, except that, in that rule, the inter-unit distance is fixed, while in this rule, the inter-unit distance scales with the unit's length. Bigger bottles/cartons/etc. would have larger distances between each other, and so forth. See Why Does the Math Matter? for more information.

• Single Lane Flow Unit Distance - The simulated width is calculated so as to simulate a single lane of units, where you can define the distance from the center of one unit/bottle/carton/etc. to the center of the next unit. Enter this distance as the Width Value.

• $W = \Large{\frac{U_x}{V} \cdot \frac{U_y}{R}}$

Again for intuition here, note that, if you were to enter the width value $V$ as the same as the unit length $U_x$, i.e. an inter-unit distance of $U_x$, the left term would cancel out and you would be left with $\frac{U_y}{R}$, which again is the width associated with a single lane of back-to-back bottles. Increasing $V$ decreases $W$, meaning it extends the distance between bottles in the single lane.

• Single Lane Gap - The simulated width is calculated so as to simulate a single lane of units with a user-defined 'air gap' between units, i.e. the distance from the back of one unit to the front of the next. Enter this 'air gap' distance as the Width Value.

• $W = \Large{\frac{U_x}{U_x + V} \cdot \frac{U_y}{R}}$

### Width Value

Defines a value associated with the Width Rule.


text:
# Change enumeration style in exercises using xsim

Forgive me if this has already been asked, but I couldn't find this question, exactly. I'm using the xsim package (documentclass book) and I'd like my "top"-level enumeration to be labeled with letters in my exercises, but with Arabic numerals outside of those. So, for instance, if I type

\begin{enumerate}
\item Hello
\end{enumerate}

outside of the exercises, I want it to read

1. Hello

but if I type it within an exercise environment I want it to read

a. Hello

Obviously I could hack this by changing my enumeration labeling style each time I enumerate within an exercise, but there has to be a better way. Right? Keep in mind I'm using the default exercise environment supplied by xsim.

• please post MWE of the same Jun 16, 2017 at 5:39

You can either add the re-definition with the pre-hook or begin-hook options to the exercise environments

\documentclass{article}
\usepackage{xsim}

\xsimsetup{
exercise/begin-hook = \renewcommand\theenumi{\alph{enumi}} ,
solution/begin-hook = \renewcommand\theenumi{\alph{enumi}}
}

\begin{document}

\begin{exercise}
\begin{enumerate}
\item bla bla
\end{enumerate}
\end{exercise}

Outside:
\begin{enumerate}
\item bla bla
\end{enumerate}

\end{document}

or just add the redefinition inside the code of a custom environment template:

\documentclass{article}
\usepackage{xsim}

\DeclareExerciseEnvironmentTemplate{custom}
{%
\UseExerciseTemplate{begin}{default}%
\renewcommand\theenumi{\alph{enumi}}%
}
{\UseExerciseTemplate{end}{default}}

\xsimsetup{
exercise/template=custom ,
solution/template=custom
}

\begin{document}

\begin{exercise}
\begin{enumerate}
\item bla bla
\end{enumerate}
\end{exercise}

Outside:
\begin{enumerate}
\item bla bla
\end{enumerate}

\end{document}

• That was exactly what I was looking for, thank you! Jun 17, 2017 at 7:08