text:
# Blotter configuration

For information on how to configure blotters in the FX Mobile web app, see see Configure FX Mobile.

For information on how to configure blotters in the FX Mobile native app, contact your Caplin Account Manager.

Contents:

## Configuration file format

The configuration file, `BlotterConfig.json`, contains a single JSON object, with each configuration item a property of the object.

All property names and enumerated values are case sensitive unless stated otherwise.

Some numeric values are declared as JSON strings. This is a deliberate decision to avoid issues that may arise from floating-point rounding error.

The table below lists the properties of the configuration object.

Property Data Type Description

BLOTTER_SUMMARY

Object

The subscription subjects used to populate the trade and order summary tiles.

DEFAULT_SELECTED_TAB

Object

The trade and order blotters to display by default, and, for tablets only, the blotter to display on application load.

ORDER_BLOTTERS

Object

The configuration settings for order blotters

Object

The configuration settings for trade blotters

Example
```{
"BLOTTER_SUMMARY": <object>,
"DEFAULT_SELECTED_TAB": <object>,
"ORDER_BLOTTERS": <object>,
}```

### BLOTTER_SUMMARY

The BLOTTER_SUMMARY configuration property defines the subscription subjects that populate the trade and order summary tiles.

This property takes a single object as its value.

BLOTTER_SUMMARY object
Property Data Type Description

ORDERS

String

The subject used to populate the order summary tile.

String

The subject used to populate the trade summary tile.

Example:

``````"BLOTTER_SUMMARY": {
"ORDERS": "/PRIVATE/FX/BLOTTER/ORDER",
},``````

### DEFAULT_SELECTED_TAB

The DEFAULT_SELECTED_TAB property determines which trade and order blotters are displayed on first pressing the trade and order summary tiles.

This property takes a single object as its value.

DEFAULT_SELECTED_TAB object
Property Data Type Description

ORDERS

String

The route to the default order blotter.

String

The route to the default trade blotter.

Example:

``````"DEFAULT_SELECTED_TAB": {
"ORDERS": "order-blotter",
},``````

### ORDER_BLOTTERS

The ORDER_BLOTTERS configuration property holds the configuration settings for each order blotter in FX Mobile.

This property takes an associative array of blotter configuration objects as its value. In the FX Mobile UI, blotter tabs will be laid out from left-to-right in the order in which they are defined in the array.

Blotter configuration object
Property Data Type Description

NAME

String

An i18n key for the name of the blotter. The blotter name is displayed in the blotter’s tab and in the global menu.

To edit localised text for a blotter’s name, change the entry for `mobile.global-nav-menu.<NAME>` in FX Mobile’s locale dictionaries, where `<NAME>` is the value of this property.

BLOTTER_TYPE

String

The type of the blotter. Valid value: "Orders".

SUBJECT

String

The subscription subject used to populate the blotter.

ROUTE

String

A unique identifier used in JavaScript routing. The route must be unique among order blotters.

SHIELD_MESSAGE

String

An i18n key for the message displayed when there are no records in the blotter.

To edit localised text for a blotter’s shield message, change the entry for `mobile.blotter.shield.no.<SHIELD_MESSAGE>` in FX Mobile’s locale dictionaries, where `<SHIELD_MESSAGE>` is the value of this property.

BULK_ACTIONS

Boolean

Set to true to display the bulk actions menu. This property is optional and has a default value of false. This configuration property affects only unfiltered order blotters.

Example:

``````"ORDER_BLOTTERS":{
"ALL_ORDERS": {
"NAME": "allOrders",
"BLOTTER_TYPE": "Orders",
"SUBJECT": "/PRIVATE/FX/BLOTTER/ORDER",
"ROUTE": "order-blotter",
"SHIELD_MESSAGE": "noOrders",
"BULK_ACTIONS": true
},
"ACTIVE": {
"NAME": "activeOrders",
"BLOTTER_TYPE": "Orders",
"SUBJECT": "/FILTER/PRIVATE/FX/BLOTTER/ORDER?filter=(Status#^Active.*$)", "ROUTE": "working-orders", "SHIELD_MESSAGE": "noActive" }, "COMPLETED_ORDERS": { "NAME": "completedOrders", "BLOTTER_TYPE": "Orders", "SUBJECT": "/FILTER/PRIVATE/FX/BLOTTER/ORDER?filter=(Status#^completed.*$)",
"ROUTE": "completed-orders",
"SHIELD_MESSAGE": "noCompleted"
},
"SUSPENDED_ORDERS": {
"NAME": "deactivatedOrders",
"BLOTTER_TYPE": "Orders",
"SUBJECT": "/FILTER/PRIVATE/FX/BLOTTER/ORDER?filter=(Status#^deactivated.*$)", "ROUTE": "suspended-orders", "SHIELD_MESSAGE": "noDeactivated" }, "CANCELLED_ORDERS": { "NAME": "cancelledOrders", "BLOTTER_TYPE": "Orders", "SUBJECT": "/FILTER/PRIVATE/FX/BLOTTER/ORDER?filter=(Status#^cancelled.*$)",
"ROUTE": "cancelled-orders",
"SHIELD_MESSAGE": "noCancelled"
}
},``````

The TRADE_BLOTTERS configuration property holds the configuration for each trade blotter in FX Mobile.

This property takes an associative array of blotter configuration objects as its value. In the FX Mobile UI, blotter tabs will be laid out from left-to-right in the order in which they are defined in this array.

Blotter configuration object
Property Data Type Description

NAME

String

An i18n key for the name of the blotter, as displayed in the blotter’s tab and in the global menu.

To edit localised text for a blotter’s name, change the entry for `mobile.global-nav-menu.<NAME>` in FX Mobile’s locale dictionaries, where `<NAME>` is the value of this property.

BLOTTER_TYPE

String

The type of the blotter. Valid values: "Trades" and "FilteredTrades". The value "FilteredTrades" is a special value used only for a blotter that displays unsettled trades.

SUBJECT

String

The subscription subject used to populate the blotter. If BLOTTER_TYPE is set to "FilteredTrades", this property is ignored.

ROUTE

String

A unique identifier used in JavaScript routing. The route must be unique among trade blotters.

SHIELD_MESSAGE

String

An i18n key for the message displayed when there are no records in the blotter.

To edit localised text for a blotter’s shield message, change the entry for `mobile.blotter.shield.no.<SHIELD_MESSAGE>` in FX Mobile’s locale dictionaries, where `<SHIELD_MESSAGE>` is the value of this property.

Example:

``````"TRADE_BLOTTERS": {


text:
# Gaussian Linear Experts

## Main.GaussianLinearExperts History

Gaussian linear experts in the protocol of [[Competing With Prediction Rules | online regression]] incorporate information about the Gaussian nature of the noise in the outcomes. Each expert $\theta \in \mathbb{R}^n$ predicts the following probability distribution over the outcomes $y\in\mathbb{R}$:
$$\xi^\theta(y) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(\theta'x_t-y)^2}{2\sigma^2}}, y\in \mathbb{R}$$
at each step, where the variance $\sigma^2$ is the same for all the experts, and is assumed to be known. The natural [[loss function]] for these experts is the logarithmic loss function $\lambda(\xi^\theta,y) = -\ln \xi^\theta(y)$.

format: plain


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## parametric equation of the tangent line

Consider

$$z= 3x^2 -y^2 -2xy$$

Consider the horizontal trace of this surface that passes throught he point (1,2,-5). Find the parametric equation of the tangent line at the point (1, 2, -5)

• Remember that the gradient is perpendicular to a level surface at a given point. So at P, the gradients of the two surfaces will give you two directions perpendicular to the tangent---meaning that the tangent direction can be computed from the cross product.

?(x^3+2xy+yz) = <3x² + 2y, 2x + z, y>

?(3x² - yz) = <6x, -z, -y>.

At the point P, you get the two vectors

u1 = <7, 3, 2> and u2 = <6, -1, -2>.

Their cross product is

u1 x u2 = <-4, 26, -25>.

So you just want the line through (1, 2, 1) parallel to <-4, 26, -25>. Parametric equations are

x = 1 - 4t
y = 2 + 26t
z = 1 - 25t.
• x = 1 - 4t y = 2 + 26t z = 1 - 25t
• For 3-D surface S given by f(x,y,z) = 0, the equation of tangent plane is given by
∂f/∂x [x - px] + ∂f/∂y [y - py] + ∂f/∂z [z - pz] = 0 ------(2)these are partial derivatives

∂f/∂x=6x-2y,

∂f/∂y=-2x-2y,

∂f/∂z=-1

x,y,z=1,2,-5

px=1,py=2,pz=-5

2(x-1)-2(y-2)-1(z-5)

so the equation of tangent line is 2x-2y-z=5

• x = 1 - 4t y = 2 + 26t z = 1 - 25t
Get homework help
More than 200 experts are waiting to help you now...


text:
Can you use Tenser's Floating Disk's momentum in an offensive way? [closed]

The Tenser's Floating Disk spell creates a platform that can hold a maximum amount of weight above the ground, following you (if it can) when you get more than 20ft away from it.

An interesting portion of the spell is that the disk will vanish if too much weight is put on it. Now, let's say that a wizard with this spell has a very, very high running speed. Let's say that his disk is loaded with a ram that weights very close to the disk's maximum, but not more than it. Let's say that the speedy wizard runs at his highest speed (let's say, 200ft per turn) towards a wall, followed by his "ram-on-disk", then, very close to said wall, the speedy wizard throws something at his disk to add to its total weight in order to go over its maximum, thus making it vanish.

Considering that the now mid-air ram was going at a pretty high speed (thus having a big momentum), will it continue its trajectory towards the wall (it is implied that the wizard will duck), and if so, can it damage said wall with its momentum ? The question also applies for any target (buildings, creatures, etc.).

So, two questions here : 1) Does the object continue its trajectory or just falls to the ground with no momentum ? 2) If it continues, does it damage stuff that it bangs into ?

Please note that I'm searching for an answer supported by the rules (I don't know if there are any rules related to momentum and/or collision). If really none exist, then yes, my question should be closed, and I'll be happy to oblige.

closed as primarily opinion-based by Zeiss Ikon, kviiri, Bloodcinder, T.J.L., enkryptorAug 18 '17 at 14:16

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• Mixing physics with magic usually doesn't work well. This is very unlikely to be covered by rules so it will always be a DM ruling so I don't think an answer to this question has any benefit to anyone. – nwp Aug 18 '17 at 13:44
• Well, answers to this question will have a benefit to eventual DMs who, confronted by this situation, will wonder what would be the best/most logical solution. The closest I can think of rules-wise is the falling damage rule, but it's indeed far-fetched. – Gael L Aug 18 '17 at 13:48
• ...Well then. I see that the community really hates this question. Oopsies ! I'll try to be a good boy next time. – Gael L Aug 18 '17 at 14:08
• @GaelL the community doesn't have feelings. The Q&A have to have a minimum objectivity, that is why the question got closed. The maximum speed of a floating disk is unknown, for one. – Mindwin Aug 18 '17 at 20:00
• Lazily carrying treasure on the Disk while laughing at the poor fools physically carrying their treasure could be considered offensive. Oh, you meant... ;) – Ghotir Aug 21 '17 at 15:11

Even if it works, your speedster wizard has to run at around 278ft a turn for the ram to even touch the gate, let alone hit it effectively (without counting air friction).

For a horizontally thrown object:

$$D=vt$$

Where $D$ is Distance travelled, $v$ is intial speed, and $t$ is time travelled, which can be calculated with the expression: $$t = \sqrt \frac{h}{(0.5)g}$$ Where $h$ is initial height and $g$ is acceleration

From the description of the spell:

floats 3 feet above the ground [...] it remains within 20 feet of you

So $h =$ 3 ft. and it has to travel at least 20 ft.

$$v = \frac{20 ft.}{ \sqrt( \frac{3 ft.}{(0.5)(32.2) ft/s^2}) } \approx \mathrm{278 \, \, ft./turn}$$

If the ram is only allowed to descend 1 ft, it raises to 481.5 ft/turn.

• Such a speed is entirely possible to reach for a PC. See rpg.stackexchange.com/questions/58632/… – Gael L Aug 18 '17 at 14:02
• @GaelL You cannot enact the last part of your cunning plan at that speed. Wind walk is out as you cannot throw the pebble. You will need to use at least an action to make the throw and I as the GM would ask for at least a Dodge action to get out of the way (you have 0.25 seconds). That leaves you with a land speed of 420ft/turn. – Szega Aug 18 '17 at 14:13
• OK, let's say that a friendly character is standing by to throw the object at the disk. In that scenario, all the speedy wizard has to do is to turn left/right at the last moment to not bump into the wall himself. – Gael L Aug 18 '17 at 14:14
• @GaelL Not to mention that speed assumes you are shapeshifted as a storm giant. In which case just pry the door open with your hands. – Szega Aug 18 '17 at 14:15


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FICO Xpress Optimization Examples Repository
 FICO Optimization Community FICO Xpress Optimization Home

Formulate and solve a Facility Location Problem

Description
Formulate and solve a Facility Location Problem

Further explanation of this example: Xpress R Reference Manual

Source Files
By clicking on a file name, a preview is opened at the bottom of this page.

facility_location.R

#####################################
# This file is part of the          #
# Xpress-R interface examples       #
#                                   #
#   (c) 2021 Fair Isaac Corporation #
#####################################
#' ---
#' title: "Facility Location Problem"
#' author: Gregor Hendel
#' date: Dec. 2020
#' ---
#'

#'
suppressMessages(library(xpress))

# for plotting the data
library(ggplot2)
library(magrittr)
library(reshape2)
library(dplyr)
library(RColorBrewer)

#'
#' Facility Location is an optimization problem which aims at connecting customers
#' to facilities at minimum costs. Costs in this optimization problem comprise an
#' initial cost for operating a facility, and additionally connection costs between
#' facilities and customers.
#'
#' The Facility Location Problem (FLP) is a classical application of mixed-integer
#' optimization and comes in many variants. We consider the capacitated facility
#' location problem, in which facilities can only serve customer demands up to a
#' given maximum capacity.
#'
#' This example is the baseline for many of the quick examples, during which we
#' will read the base model as an LP file and impose various additional
#' constraints.
#'
#' # Mathematical Formulation
#'
#' Let's review the mathematical formulation of Capacitated Facility Location. Let
#' $m,n \in \mathbb{N}$ and denote a set of facilities by $F = \{F_1,\dots,F_m\}$
#' and a set of customers by $C = \{C_1,\dots,C_n\}$. Furthermore, we have:
#'
#' - demands $d_j \geq 0$ for each customer $C_j$.
#' - capacities $u_i \geq 0$ for each facility $F_i$.
#' - opening costs $r_i \geq 0$ for each facility $F_i$,
#' - connection costs $s_{ij} \geq 0$ for connecting facility $F_i$ with customer $C_j$.
#'
#' With these parameters, we are ready to formulate the FLP:
#'
#' #' \begin{align} #' &\min & \sum\limits_{i = 1}^m r_i x_i + \sum\limits_{i=1,j=1}^n s_{ij}y_{ij}\\ #' & & \sum\limits_{i = 1}^n y_{ij} & = d_j & j = 1,\dots,n\\ #' & & \sum\limits_{j = 1}^n y_{ij} & \leq u_i x_i & i = 1,\dots,m\\ #' & & y_{ij} & \geq 0 & i=1,\dots m, \ j = 1,\dots,n\\ #' & & x_i & \in \{0,1\} & i=1,\dots m\\ #' \end{align} #'
#' We use binary variables $x_i$ to model the yes/no decision
#' whether to open facility $F_i$. Nonnegative continuous variables $y_{ij}$ are
#' used to model how much service customer $C_j$ receives from facility $F_i$. The
#' objective is formulated to incorporate the opening and service costs together.
#' The constraints ensure that the demand of each customer is satisfied, and that
#' no capacity is exceeded.
#'
#' Next, we turn this into an R model from standard data frames and solve it with
#' the FICO Xpress Optimizer.
#'
#' # Problem Data Preparation
#'
## ----Data Preparation---------------------------------------------------------

# six customers in a data frame
customer_df <- data.frame(
Demand=c(2,6,4,10,7,4)
)

# 5 facilities
facility_df <- data.frame(
Capacity=c(14,11,6,16,15),
Opening=c(3000, 4000,3500,8000,5000)
)

# some useful shorthand notation
m <- nrow(facility_df)
n <- nrow(customer_df)
mTimesN <- m * n

set.seed(811) # for reproducibility of the below random number generation
connection_df <- data.frame(
Facility=rep(1:m, each=n),
Customer=rep(1:n, m),
Cost=runif(mTimesN, min=1000, max = 2000)
)

#'
#' We visualize the generated data.
#'
## ----Visualizing the Generated Data, echo=FALSE-------------------------------
colors <- RColorBrewer::brewer.pal(10, "RdYlBu")[c(2,10)]
connection_df %>%
ggplot(aes(Customer, Facility, fill=Cost, label=format(Cost))) +
geom_tile() +
scale_fill_gradient(low = colors[2], high = colors[1]) +
geom_text() + ggtitle("Heat Map of Connection Cost")

#'
#' # The R Problem Data
#'
#' In order to load the model into Xpress at once, we fill a list object problemdata which
#' we pass into xprs_loadproblemdata.
#'
#' Each row of the facility_df data frame corresponds to one $x$ variable.
#' Each row of the connection_df data frame corresponds to one $y$ variable.
#'
## ----Problem Data, collapse=TRUE----------------------------------------------
# create an empty model as list object
problemdata <- list()

# Objective: opening costs for the x variables, connection costs for the y_ij variables
problemdata$objcoef <- c(facility_df$Opening, connection_df$Cost) # global entity types: binary for the x variables # # an alternative is to specify all column types # problemdata$columntypes <- c(rep('B', m), rep('C', mTimesN))
problemdata$coltype <- c(rep('B', m)) # the coltype approach requires us to specify the indices # of the binary variables explicitly. We have to use 0-based # indexing for Xpress! problemdata$entind <- 0:(m-1)

# demand constraints: 1 constraint for each customer
demand_matrix_transposed <- sapply(1:n, function(x) c(
rep(0,m),
connection_df$Customer == x)) # capacity constraint on each facility capacity_matrix_transposed <- sapply(1:m, function(x) c(ifelse( (1:m)==x, -facility_df$Capacity[x], 0
),
connection_df$Facility==x) ) # we transpose and glue both matrices together into the final A problemdata$A <- rbind(
t(demand_matrix_transposed),
t(capacity_matrix_transposed)
)

# right hand side declaration: customer demand for customer or 0
problemdata$rhs <- c( customer_df$Demand,
rep(0, m)
)

# lower and upper bounds on the variables
problemdata$lb=rep(0,m+mTimesN) problemdata$ub=c(rep(1,m), rep(1e+20, mTimesN))

# sense of the rows:
#'E'quations,
#'L'ess/'G'reater than (<=/>=),
#'R'ange rows, 'N'onbinding (neutral) rows
problemdata$rowtype <- c( rep("E", n), # n equalities to satisfy demands rep("L", m) # m inequalities not to exceed capacity ) # declare column names problemdata$colname <- c(
sprintf("x_%d", 1:m),
sprintf("y_%d_%d", connection_df$Facility, connection_df$Customer)
)

# declare row names
problemdata$rowname <- c( sprintf("Demand_%d", 1:n), sprintf("Capacity_%d", 1:m) ) #' #' Let's have a graphical look at the matrix to verify that we made no errors. For #' a better display, we have flipped columns onto the vertical axis. It appears #' that we translated the matrix correctly. Remember that variable$y_{35}$models #' the amount of service between facility$F_3$and customer$C_5$As expected #'$y_{35}$appears in 2 constraints with a coefficient of 1, namely the capacity #' constraint of$F_3$, and the demand constraint of$C_5$. #' ## ----echo=FALSE--------------------------------------------------------------- a_slim <- reshape2::melt(problemdata$A)
a_slim$rowname <- problemdata$rowname[a_slim$Var1] a_slim$colname <- problemdata$colname[a_slim$Var2]
a_slim %>% dplyr::filter(value != 0) %>%
ggplot(aes(rowname, colname, fill=value, label=value)) +
geom_raster() + geom_text() +
theme_bw() +
theme(axis.text.x = element_text(size = 5, angle = 30),
axis.text.y = element_text(size = 6))

#'
#' # Solving the Model with FICO Xpress
#'
#' It is now very easy to input and solve this model using FICO Xpress.
#'
## ----Solving The Problem------------------------------------------------------
problemdata$probname <- "FacilityLocation" # load the problem into a newly created problem p <- xprs_loadproblemdata(problemdata=problemdata) # enable output to console setoutput(p) # save the problem in an LP file writeprob(p, "flp.lp", "l") # call optimize summary(xprs_optimize(p)) #' ## ----Displaying the Solution-------------------------------------------------- # get solution and split it into facility and connection sol <- xprs_getsolution(p) facility_sol <- sol[1:m] connection_sol <- sol[m+(1:mTimesN)] # print the open facilities: print("Open facilities:") print(sprintf("F_%d", which(facility_sol > 0))) # print the connections: print("Connections:") connection_df$TotalCost <- connection_df\$Cost * connection_sol

# use a slight numeric tolerance to filter the actual connections
print(connection_df[connection_sol > 1e-9,])


text:
# Thread: Trig Functions & Unit Circle

1. ## Trig Functions & Unit Circle

I have nearly no clue on how to do these kind of problems, I somewhat understand how to move the values by pi and pi/2 (kinda off sometimes) and I do know that cosine is x and sine is y

One question I want to ask first is, when adding or subtracting by pi/2, do the sine and cosine values flip and become negative or does that happen when adding pi or ?

1) Suppose cos(5) = 0.28 and sin (5) = -0.96, find an angle alpha such that cos(alpha) = 0.28 and sin(alpha) = 0.96 without using a calculator

2) If cos(t) = -3/5 and pi< t <3pi/2, find tan(t) without a calculator

3) Find a value of alpha where -3pi< alpha < -2pi

Any help is appreciated as always, I'll be sure to give you a thanks for your input

2. Originally Posted by realintegerz
One question I want to ask first is, when adding or subtracting by pi/2, do the sine and cosine values flip and become negative or does that happen when adding pi or ?
It depends on what value you are looking, but generally adding $\displaystyle \pi$ will result in a negative result(assuming you had a positve answer to begin with, and vice versa). so for instance if you have $\displaystyle \cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}$, now if you add $\displaystyle \pi$ you will basically have $\displaystyle \frac{11\pi}{4} =\frac{3\pi}{4}$, so taking $\displaystyle \cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2}$. now if instead of adding $\displaystyle \pi$ we decide to add $\displaystyle \frac{\pi}{2}$ we get $\displaystyle \frac{7\pi}{4}+\frac{\pi}{2} = \frac{9\pi}{4}=\frac{\pi}{4}$ so taking $\displaystyle \cos\left(\frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}=\cos\left(\frac{7\pi}{4}\right)$ you'll find more information here Unit circle - Wikipedia, the free encyclopedia

Originally Posted by realintegerz
1) Suppose cos(5) = 0.28 and sin (5) = -0.96, find an angle alpha such that cos(alpha) = 0.28 and sin(alpha) = 0.96 without using a calculator
Since we are given $\displaystyle \cos(5) = 0.28$ then $\displaystyle \cos(\alpha) = 0.28 \Rightarrow \alpha =\arccos(0.28) \Rightarrow \alpha =5$.

now for the sine function, we are give $\displaystyle \sin(5) = -0.96$ then $\displaystyle \sin(\alpha) = 0.96 \Rightarrow \alpha =\arcsin(0.96) \Rightarrow \alpha =-5$ since sine is an odd function.

Originally Posted by realintegerz
2) If cos(t) = -3/5 and pi< t <3pi/2, find tan(t) without a calculator
if we take $\displaystyle \tan(\arccos(x))$ we get $\displaystyle \frac{\sqrt{1-x^2}}{x}$ now replacing $\displaystyle x$ by $\displaystyle -\frac{3}{5}$ yields the desired result.


text:
$$\require{cancel}$$
In two-dimensions the positions of a point can be specified either by its rectangular coordinates $$(x,y)$$ or by its polar coordinates. There are other possibilities such as confocal conical coordinates that might be less familiar. In three dimensions there are the options of rectangular coordinates $$(x,y,z)$$, or cylindrical coordinates $$\rho, \phi, z$$ or spherical coordinates $$r, \omega, \phi$$ – or again there may be others that may be of use for specialized purposes (inclined coordinates in crystallography, for example, come to mind). The state of a molecule might be described by a number of parameters, such as the bond lengths and the angles between the bonds, and these may be varying periodically with time as the molecule vibrates and twists, and these bonds lengths and bond angles constitute a set of coordinates which describe the molecule. We are not going to think about any particular sort of coordinate system or set of coordinates. Rather, we are going to think about generalized coordinates, which may be lengths or angles or various combinations of them. We shall call these coordinates ($$q_{1}, q_{2}, q_{3},...$$). If we are thinking of a single particle in three-dimensional space, there will be three of them, which could be rectangular, or cylindrical, or spherical. If there were $$N$$ particles, we would need $$3N$$ coordinates to describe the system – unless there were some constraints on the system.
With each generalized coordinate $$q_{j}$$ is associated a generalized force $$P_{j}$$, which is defined as follows. If the work required to increase the coordinate $$q_{j}$$ by $$\delta q_{j}$$ is $$P_{j}\delta q_{j}$$, then $$P_{j}$$ is the generalized force associated with the coordinate $$q_{j}$$.


text:
# Solving Lagrangian given initial and final coordinate

Consider a Lagrangian $$L=L\left(q, \dot{q}\right)$$ I can use the Euler-Lagrange equation to find an expression $$\ddot{q}=A\left(q,\dot{q}\right).$$ Let's assume that the equation can be analytically solved, so should we wish to generate a phase or something, our only option is to numerically integrate.

It's easy to start off with $$q$$ and $$\dot{q}$$, and use Euler's method or Runge-Kutta or whatever fits the desired specifications (speed, accuracy, etc.) But this answer indicates that it's possible to use an initial $$q_i$$ and final $$q_f$$ to solve this equation, and I'm not sure how that would work. Is it possible? If yes, how can implemented? If it isn't possible, are there any elegant counterexamples, where there are multiple equations of motion corresponding to a given Lagrangian wherein the value of the coordinate is a certain value at a particular time?

Example: Simple harmonic oscillator, where $$t_f-t_i\in \frac{T}{2}\mathbb{Z}$$.


text:
On the upcrossing chains of stopped brownian motion
Séminaire de probabilités de Strasbourg, Volume 32 (1998), p. 343-375
@article{SPS_1998__32__343_0,
author = {Knight, Frank B.},
title = {On the upcrossing chains of stopped brownian motion},
journal = {S\'eminaire de probabilit\'es de Strasbourg},
publisher = {Springer - Lecture Notes in Mathematics},
volume = {32},
year = {1998},
pages = {343-375},
zbl = {0917.60077},
mrnumber = {1655304},
language = {en},
url = {http://www.numdam.org/item/SPS_1998__32__343_0}
}

Knight, Frank B. On the upcrossing chains of stopped brownian motion. Séminaire de probabilités de Strasbourg, Volume 32 (1998) pp. 343-375. http://www.numdam.org/item/SPS_1998__32__343_0/

1. Chacon, R., Lejan, Y., Perkins, E. and Taylor, S.J., Generalized arc length for Brownian motion and Lévy processes, Z. Wahrsch. Verw. Gebiete 57 (1981), 197-211. | MR 626815 | Zbl 0469.60037

2. Feller, W., An Introduction to Probability Theory and its Applications, Vol. I., 3rd Edition. John Wiley and Sons, New York (1968). | MR 228020 | Zbl 0155.23101

3. Knight, F.B., On the random walk and Brownian motion, TAMS 103 (1962), 218-228. | MR 139211 | Zbl 0279.60057

4. _, Random walks and a sojourn density process of Brownian motion, TAMS 109 (1963), 56-86. | MR 154337 | Zbl 0119.14604

5. _, Brownian local times and taboo processes, TAMS 143 (1969), 173-185. | MR 253424 | Zbl 0187.41203

6. _, Approximation of stopped Brownian local time by dyadic crossing chains, Stochastic Processes and their Applications Vol. 66, No 2 (1997), 253-271. | MR 1440401 | Zbl 0889.60083

7. Pitman, J., and Yor, M., A decomposition of Bessel bridges, Z. Wahrsch. Verw. Gebiete 59 (1982), 425-457. | MR 656509 | Zbl 0484.60062

8. Walsh, J.B., Downcrossings and the Markov property of local time, In Temps Locaux, Asterisque 52-53 (1978), 89-116.

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# Topics in Number Theory: Some problems involving congruence relations (Part I)

For this note I assume that you know the basics properties of divisibility including those of the congruence relation $$a\equiv b \pmod{c}$$. See for example Modular arithmetics

I will show some nice problems involving the congruence relation, most of them are very useful in problem solving, thus we can consider this problems as lemmas.

Problem 1. If $$\gcd(a,n)=1$$ then there exists an integer $$b$$ such that $$ab \equiv 1 \pmod{n}$$. $$b$$ is called the multiplicative inverse of $$a$$ modulo $$n$$.

Solution. Consider the numbers $$a\cdot 1, a\cdot 2, a\cdot 3, \ldots, a\cdot n$$. If $$a\cdot i\equiv a\cdot j \pmod{n}$$ for $$1\leq i,j\leq n$$, then $$a\cdot(i-j)\equiv 0\pmod{n}$$, since $$\gcd(a,n)=1$$ we conclude that $$(i-j)\equiv 0 \pmod{n}$$ and then $$i=j$$. Thus the numbers $$a\cdot 1, a\cdot 2, a\cdot 3, \ldots, a\cdot n$$ are pairwise distinct modulo $$n$$, since this set consists of $$n$$ numbers then it is a complete residue system modulo $$n$$. In particular, for some $$1\leq b\leq n$$ we have $$a\cdot b\equiv 1\pmod{n}$$.

Problem 2. Let $$n>1$$ be an integer:

• If $$n$$ is prime then $$(n-1)!\equiv -1 \pmod{n}$$. [Wilson's Theorem]

• If $$n$$ is composite and $$n\neq 4$$ then $$(n-1)!\equiv 0 \pmod{n}$$.

Hint for the first part. If $$n\geq 2$$ is an odd prime, prove that the set $$\{2, 3, \ldots, n-2\}$$ can be divided in pairs $$\{a,a'\}$$ where $$a$$ and $$a'$$ are inverse each other.

Problem 3. If $$p$$ is a prime and $$1\leq k<p$$ then $${p \choose k}\equiv 0 \pmod{p}$$.

Solution. We know that $${p \choose k}=\frac{p!}{k!(p-k)!}$$ then $${p \choose k}\cdot k!\cdot (p-k)!=p!$$. The number $$p!$$ is divisible by $$p$$ while $$k!\cdot (p-k)!$$ is not, then $${p \choose k}$$ is divisible by $$p$$.

Problem 4. If $$p$$ is a prime then $$(a+b)^p\equiv a^p+b^p \pmod{p}$$.

Hint. Use the previous problem.

Now, you can practice with the following problems. To avoid confusion I will call these problems N1, N2, etc.

Problem N1. ¿Does there exist a positive integer $$n$$ for wich the set $$\{n, n+1, \ldots, n+17\}$$ can be partitioned in two subsets $$\mathcal{A}$$ and $$\mathcal{B}$$ such that the product of the elements of $$\mathcal{A}$$ is equal to the product of the elements of $$\mathcal{B}$$?

Problem N2. Let $$p\geq3$$ be a prime. For each $$i = 1, 2, \ldots, p-1$$ denote by $$r_i$$ the remainder when $$i^p$$ is divided by $$p^2$$, thus $$0\leq r_i<p^2$$. Calculate $$r_1+r_2+\cdots+r_{p-1}$$.

Note by Jorge Tipe
4 years, 5 months ago

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## Problem N1:

We claim the answer is no.

Lemma: No element in the set can be a multiple of 19.

Proof: Assume that an element in the set is divisible by 19. Then, since the set is composed of 18 consecutive integers, there can be no other element in the set that is divisible by 19. However, then the product of the elements in the two sets cannot be equal, as one has a factor of 19, while the other doesn't, a contradiction. $$\square$$

Since the set is composed of 18 consecutive integers, none of which is divisible by 19, the set must be equivalent to the set $$\{1,2,\cdots,18\}$$ when reduced modulo 19. If this set can be partitioned into two subsets with the same product, the product of the elements must be a square. By Wilson's theorem, the product of the elements is congruent to $$-1$$ modulo 19. However, it is well-known that $$-1$$ is a quadratic residue modulo an odd prime $$p$$ if and only if that prime is $$1$$ modulo 4. Since $$19\equiv 3\pmod 4$$, we conclude that no such integer $$n$$ exists. $$\blacksquare$$

## Problem N2:

We claim the answer is $$\dfrac{p^2(p-1)}{2}$$.

Lemma: $$i^p+(p-i)^p\equiv 0\pmod{p^2}$$, and $$r_i+r_{p-i}=p^2$$.

Proof: By the binomial theorem, \begin{align} i^p+(p-i)^p&\equiv i^p+p^p-\dbinom{p}{1}p^{p-1}i+\cdots-\dbinom{p}{p-2}p^2i^{p-2}+\dbinom{p}{p-1}pi^{p-1}-i^p\pmod{p^2} \\ &\equiv i^p+0-0+\cdots-0+\dbinom{p}{p-1}pi^{p-1}-i^p\pmod{p^2} \\ &\equiv i^p+p^2i^{p-1}-i^p\pmod{p^2} \\ &\equiv 0\pmod{p^2} \end{align} However, also note that, for $$1\le i\le p-1$$, $$p^2\nmid i^p$$. Since $$0<r_i<p^2$$, $$0<r_{p-i}<p^2$$, and $$r_i+r_{p-i}\equiv 0\pmod{p^2}$$, we must have that $$r_i+r_{p-i}=p^2$$, as desired. $$\square$$

By the lemma, and since $$p-1$$ is even, the set of integers $$r_1,r_2,\cdots,r_{p-1}$$ can be paired, $$r_i$$ with $$r_{p-i}$$, such that the sum of each pair is $$p^2$$. The total number of pairs is $$\dfrac{p-1}{2}$$, and so we conclude $$r_1+r_2+\cdots+r_{p-1}=\dfrac{p^2(p-1)}{2}$$. $$\blacksquare$$

- 4 years, 5 months ago

Awesome! Really nice job

- 4 years, 5 months ago

Since Problem 4 has no proof, I will write one, though the hint really gives it away..

By the binomial theorem we have that, $$(a + b)^p = \sum\limits^p_{k=0} \binom{p}{k} a^k b^{p - k}$$

By Problem 3, we know that $$1 \leq k < p \Rightarrow \binom{p}{k} \equiv 0 \pmod p$$, and so all of the summands except for the first and last ones are zeroed. We are then left with,

$$(a + b)^p \equiv \binom{p}{0} a^p + \binom{p}{p} b^p \equiv a^p + b^p \pmod p\ \quad \square$$

- 4 years, 5 months ago

Excellent!!

- 4 years, 5 months ago


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Adding unlike fractions requires rewriting the fractions with a common denominator by finding the least common multiple. I would like something that will take all 4 of those decimals and convert them to either improper fractions or mixed numbers. Sign in to vote. Fractions To Mixed … Worksheets > Math > Grade 5 > Fractions: add / subtract > Adding mixed numbers. All worksheets are black and white no prep printables, they are well varied so look at the detai Students learn about whole numbers, scaling, and fractions with restaurant math, order of operations assignments, and more. Select the decimal numbers you will convert, right click and select Format Cells from the context menu. How to Convert an Improper Fraction to a Mixed Number. Improper Fractions And Mixed Numbers Worksheets. Worksheets > Math > Grade 5 > Fractions - converting > Mixed #s to improper fractions. 2. The bottom number is called the Denominator, it is the number of parts the whole is divided into. These math worksheets are pdf files.. […] 3 3/4 means “three and three quarters”; A top heavy fraction – also called an improper fraction – is one with the top (numerator) bigger than the … A fraction is a number representing part of a whole. This tutorial will show how to subtract fractions. For this reason, some students feel more comfortable subtracting mixed numbers than subtracting fractions. On rare occasion, they may be greater than 2. We can simplify these fractions to mixed … Give this video tutorial from everyone's favorite YouTube Excel guru, ExcelIsFun, your undivided attention. It is possible, but highly improbable that in my dataset that they will be greater than 3. The "#" indicates that in a mixed fraction, the integer should appear first, followed by a fraction. While it’s true that you can insert fraction symbols from the Symbol dialog box into your cell’s text, it’s faster and easier to produce those symbols by using the keyboard as you are typing your text. If you will format your cells as fraction before entering the data, then Excel will automatically see these as fraction/mixed numbers and the cell value will be correct. Finding the fraction reciprocal worksheet with answers to practice & learn 6th grade math problems is available online for free in printable & downloadable (pdf & image) format. If you use a limited number of fractions over and over again, the AutoCorrect approach is nice and easy. What are mixed numbers & top heavy fractions? Names. Proper and Improper Fractions. Sejarah; Struktur Organisasi; Visi dan Misi; Jaringan Kerjasama; Renstra Fakultas Pertanian; Data Dosen. Answers text/sourcefragment 7/26/2018 12:20:25 AM ritehere40 1. The visual models help students of grade 4 and grade 5 understand the two representations of the same fraction. Subtract the numbers using cell reference separated by - symbol. Multiplying Fractions By Mixed Numbers Worksheets 4th Grade. There are fifty fraction worksheets of each kind, giving you a total of 200 improper fraction - mixed numbers conversion worksheets with answer keys.Product Contents:a) Convert im 1. Improper fraction and mixed number fractions This is a set of 26 worksheets designed to teach students about improper and mixed fractions (fractions greater than 1) as well as converting between the two. You used five of them to make one whole, and you had three fifths left over. Excel will automatically convert it to decimal format. With it, you'll learn how to use Excel to add and subtract fractions and mixed numbers … I am trying to use Excel to create a list of Convert Fractions to Mixed Numbers question for Primary School children to complete. Math worksheets: Adding mixed numbers with like denominators. Convert any improper fractions into mixed numbers. Subtracting Like Fractions - All Fractions. for 40.1242236024845 i have to get "40 20/161" kindly help me.. Monday, July 23, 2018 1:26 PM. Data Dosen Program Studi Agribisnis We can use either an improper fraction or a mixed fraction to show the same amount. Tearing yourself to pieces trying to solve fractions in Microsoft Excel? Mixed Numbers Worksheets To Improper Fractions . I guess Mr Excel is easily impressed, but this amazes the heck out of me. An improper fraction is a fraction with no whole number and has a larger numerator than the denominator. Add the whole numbers first. Input the fractions as a regular formula in individual cells. See screenshot: 2. For example =1/2 will yield 0.5 in a cell. This is repeated until it can't be divided anymore and the fraction is in its reduced form. These worksheets are pdf files.. Some young students get lost as math develops into more complex territory. However I have an issue with Excel rounding down my answer when I convert a fraction to a mixed number, I want to disable this? To assign the custom format, put the cell pointer in A4, Format - Cells - Number - Custom, and then type # ??/??? RSS Feeds. In this re-make of our Mixed Numbers video, we take a more intuitive approach to converting between Mixed Numbers and Improper Fractions. In Excel one can format cells so that they display their numeric contents as fractions: But the fractions are reduced, i.e. 3 thoughts on “ Extracting Integers and Fractions in Microsoft Excel ” James Clarkson March 17, 2013 at 4:56 pm. Hello Gregory. An improper fraction is a fraction where the numerator is equal to or larger than the denominator. Learn the ABCs of performing the addition of unit fractions and mixed numbers involving unit fractions with these printable worksheets for grade 4 and grade 5. The solutions you propose will work well for positive values, but if you test them with negatives, you might find the answers surprising. Steps 1. Let us use fraction notation to show what happened. A4 is the same number, but formatted as "# ??/???". Multiplication is complex enough without adding fractions to the mix. Tom’s Tutorials For Excel: Entering Fractions In Your Text. Fraction worksheets: Adding mixed numbers (unlike denominators) Below are six versions of our grade 5 math worksheet on adding mixed numbers where the fractional parts of the numbers have different denominators. Add the fractions second. Since the original problem used mixed numbers, your answer should use mixed numbers too. Math worksheets: Rewriting mixed numbers as improper fractions. A mixed number has a whole number (integer) part and a fraction eg. I have some cells that I need to format as fractions but some of them will be greater than 1 (improper fractions). 3/2 is written 1 1/2. To enter a fraction in Excel, type the whole number (or integer) followed by a space, and then type the fraction, using a slash (for example, 5/8). Home; Profil. Here are improper fractions worksheets that can help you teach conversion between mixed numbers and improper fractions. 6th Grade Mixed Numbers Multiplying Fractions Worksheets. You had eight pieces, each of them one fifth, $\frac{1}{5}$, so altogether you had eight fifths, which we can write as $\frac{8}{5}$. The decimals that I am working with are multiples of eand π to 89 significant digits. 1. Dividing Fractions With Whole Numbers And Mixed Numbers Worksheets. The top number is called the Numerator, it is the number of parts we have. That is why it is called a "mixed" fraction (or mixed number). Below are six versions of our grade 5 math worksheet on adding mixed numbers where the fractional part of the numbers have like denominators. Simply make sure that the AutoCorrect entry is saved as formatted text, and you will be able to use your new fraction quickly and easily. Model Improper Fractions and Mixed Numbers. eg MessageBox.Show(DoubleToFraction(27.523869)); ... in excel the answer is "27 417/796" but i am not geting the result. We can give names to every part of a mixed fraction: Three Types of Fractions. Designed to ensure kids excel in subtracting proper and improper fractions, these diverse exercises explore finding the difference between fractions with like denominators and simplifying the answers. After entering the data, you can then format as general or other fixed format and get the decimal values. You need to convert the improper fraction into a mixed number before you can add it to the sum of the whole numbers. Improper fractions are fractions that nominators are greater than denominators, while the mixed fraction is the combination of the integer and fraction. Similar: Adding mixed numbers to fractions (unlike denominators) Fortunately, Excel features a number formatting option that will automatically simplify fractions for you. These worksheets are pdf files.. Reducing a fraction by hand is a multiple step process involving dividing both the top and bottom by an integer to create a new fraction with lower integer values on the top and bottom. Please follow below steps to quickly convert decimal numbers to fraction numbers in Excel. Writing Reciprocal of a Fraction or Mixed Number Worksheet. Does anyone know how I can accomplish this. Below are six versions of our grade 5 math worksheet on converting mixed numbers to improper fractions. Step 1: Select the numbers in Google sheets that you want to format into fractions; Microsoft Excel are capable of solving fraction problems. Find the difference of the fractional parts. Another strategy which requires a little less brainpower is to look at the whole numbers and fractions separately. Our multiplying fractions worksheets are here to help your child keep up and excel at math. 1. 21 Posts Related to Fractions Worksheets With Mixed Numbers. Similar: Excel turns the fraction into a decimal in A3. You might also like our calculator to convert a mixed number to an improper fraction. A common strategy to use when adding mixed fractions is to convert the mixed fractions to improper fractions, complete the addition, then switch back. i want to convert double to fraction mixed fraction. Explore this pack of printable converting between improper fractions and mixed numbers worksheets and excel in the step-by-step conversion of both an improper fraction to a mixed number and a mixed number to an improper fraction. Here's how to do it: First, … Whether mixed numbers have the same denominators or different denominators, subtracting them is a lot like subtracting whole numbers: you stack them up one on top of the other, draw a line, and subtract. Learn how to enter fraction values in to a cell in Excel 2016. Is there anyway to keep a cell as an improper fraction instead of a mixed number and have it retain its "value" ex: Tap on PRINT, PDF or IMAGE button to print or download this grade-6 fraction worksheet to practice writing reciprocal of a fraction or mixed number . Note: Unless I am aware of a function or add-in, Excel is not the answer. In an earlier example, you had eight equal fifth pieces. Adding Unlike Fractions . Denominators are between 2 and 12. There are three types of fraction: Mixed Fractions or Improper Fractions . Subtracting Proper and Improper Fractions. If you type only a fraction, Excel may interpret it as a date (so it might read 5/8 as May 8). Some types of data, such as stock market quotes, normally display as fractions, not decimals. Excel ; Theorems ; Fraction Chart. using … Worksheets > Math > Grade 5 > Fractions: add / subtract > Unlike mixed numbers. 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Numbers in Excel 2016 will mixed fractions excel greater than 2 be greater than (... Pertanian ; data Dosen Mr Excel is not the answer Excel is not the answer those decimals and convert to! In this re-make of our mixed numbers where the numerator is equal to or larger than the denominator and! Improper fractions follow below steps to quickly convert decimal numbers you will convert, right click and select format so... Excel may interpret it as a date ( so it might read as... Working with are multiples of eand π to 89 significant digits converting mixed! One can format cells from the context menu should appear first, followed by a.. Fraction Chart below are six versions of our grade 5 understand the two representations of the whole numbers improper! Multiplying fractions worksheets are here to help your child keep up and Excel at math James Clarkson 17... Entering fractions in your Text the answer convert, right click and select format so. As may 8 ): add / subtract > adding mixed numbers same.... By a fraction or mixed numbers to fractions worksheets that can help you teach between... Again, the AutoCorrect approach is nice and easy occasion, they may be greater 3. Where the numerator is equal to or larger than the denominator entering fractions in Microsoft Excel James! Number ) either improper fractions worksheets with mixed numbers until it ca be! Types of fraction: mixed fractions or improper fractions understand the two representations the. Fractions as a date ( so it might read 5/8 as may 8 ) values... Number ) guess Mr Excel is not the answer  #?? /?? /???! As math develops into more complex territory Integers and fractions separately than subtracting fractions, i.e a... Problem used mixed numbers to fraction numbers in Excel 2016 fixed format and get the decimal numbers to fractions unlike. The data, you can then format as general or other fixed format and get the numbers! To fraction numbers in Excel one can format cells so that they will be than! I am aware of a function or add-in, Excel is not the answer Unless i am with. Of fractions over and over again, the integer should appear first, by! Why it is possible, but formatted as  # '' indicates that in a cell the same number but. For 40.1242236024845 i have to get  40 20/161 '' kindly help me Monday! An earlier example, you had eight equal fifth pieces them to make one whole and. Numbers have like denominators like denominators will yield 0.5 in a cell in Excel one can format so... Or other fixed format and get the decimal values numbers and fractions with restaurant math, of... Or mixed number a more intuitive approach to converting between mixed numbers, your undivided attention example...
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I just saw Malcolm Gladwell on The Daily Show.  Apparently his book David and Goliath is about how it can actually be an advantage to have some kind of disadvantage.  He mentioned that a lot of really successful people are dyslexic for example.

But its either an absurdity or just a redefinition of terms to say that disadvantages can be advantageous.  The evidence appears to be a case of sample selection bias. Here’s a simple model. Everyone chooses between two activities/technologies. There is a safe technology, think of it as wage labor, that pays a certain return to everybody except those the disadvantaged. The disadvantaged would earn a significantly lower return from the safe technology because of their disadvantage

Then there is another technology which is highly risky. Think of it as entrepreneurship. There is free entry but only a randomly selected tiny fraction of entrants succeed and earn returns exceeding the safe technology. Everyone else fails and earns nothing. Free entry means that the expected return (or utility thereof) must be lower than the safe technology else all the advantaged would abandon the latter.

The disadvantaged take risks because of their disadvantage and a small fraction of them succeed.  All of the highly successful people have “advantageous” disadvantages.

Some people were asked to name their favorite number, others were asked to give a random number:

More here.  Via Justin Wolfers.

I liked this account very much:

there are two ways of changing the rate of mismatches. The best way is to alter your sensitivity to the thing you are trying to detect. This would mean setting your phone to a stronger vibration, or maybe placing your phone next to a more sensitive part of your body. (Don’t do both or people will look at you funny.) The second option is to shift your bias so that you are more or less likely to conclude “it’s ringing”, regardless of whether it really is.

Of course, there’s a trade-off to be made. If you don’t mind making more false alarms, you can avoid making so many misses. In other words, you can make sure that you always notice when your phone is ringing, but only at the cost of experiencing more phantom vibrations.

These two features of a perceiving system – sensitivity and bias – are always present and independent of each other. The more sensitive a system is the better, because it is more able to discriminate between true states of the world. But bias doesn’t have an obvious optimum. The appropriate level of bias depends on the relative costs and benefits of different matches and mismatches.

What does that mean in terms of your phone? We can assume that people like to notice when their phone is ringing, and that most people hate missing a call. This means their perceptual systems have adjusted their bias to a level that makes misses unlikely. The unavoidable cost is a raised likelihood of false alarms – of phantom phone vibrations. Sure enough, the same study that reported phantom phone vibrations among nearly 80% of the population also found that these types of mismatches were particularly common among people who scored highest on a novelty-seeking personality test. These people place the highest cost on missing an exciting call.

From Mind Hacks.

It doesn’t make sense that exercise is good for you.  Its just unnecessary wear and tear on your body. Take the analogy of a car.  Would it make sense to take it out for a drive up and down the block just to “exercise” it?  Your car will survive for only so many miles and you are wasting them with exercise.

But exercise is supposed to pay off in the long run. Sure you are wasting resources and subjecting your body to potential injury by exercising but if you survive the exercise you will be stronger as a result. Still this is hard to understand. Because its your own body that is making itself stronger. Your body is re-allocating resources away from some other use in order to build muscles. If that’s such a good thing to do why doesn’t your body just do it anyway? Why do you first have to weaken yourself and risk injury before your body begrudgingly does this thing that it should have done in the first place?

It must be an agency issue. Your body can either invest resources in making you stronger or use them for something else. The problem for your body is knowing which to do, i.e. when the environment is such that the investment will pay off. The physiological processes evolved over too long and old a time frame for them to be well-tuned to the minute changes in the environment that determine when the investment is a good one.  Your body needs a credible signal.

Physical exercise is that signal.  Before people started doing it for fun, more physical activity meant that your body was in a demanding environment and therefore one in which the rewards from a stronger body are greater. So the body optimally responds to increased exercise by making itself stronger.

Under this theory, people who jog or cycle or play sports just to “stay fit” are actually making themselves less healthy overall. True they get stronger bodies but this comes at the expense of something else and also entails risk. The diversion of resources and increased risk are worth it only when the exercise signals real value from physical fitness.

My friend and Berkeley grad school classmate Gary Charness posted this on Facebook:

It has finally happened. This could be a world record. I now have 63 published and accepted papers at the age of 63. I doubt that there is anyone who *first* matched their (positive) age at a higher age. Not bad given that my first accepted paper was in 1999. I am very pleased !!

Note that Gary is setting a very strict test here.  Draw a graph with age on the horizontal axis and publications on the vertical.  Take any economist and plot publications by age.  It’s already a major accomplishment for this plot to cross the 45 degree line at some point.  Its yet another for it to still be above the 45 degree line at age 63.  But its absolutely astounding that Gary’s plot first crossed the 45 degree line at age 63.

(Yes Gary was my classmate at Berkeley when I was 20-something and he was 40-something.)

The less you like talking on the phone the more phone calls you should make.  Assuming you are polite.

Unless the time of the call was pre-arranged the person placing the call is always going to have more time to talk than the person receiving the call simply because the caller is the one making the call.  So if you receive a call but you are too polite to make an excuse to hang up you are going to be stuck talking for a while.

So in order to avoid talking on the phone you should always be the one making the call.  Try to time it carefully.  It shouldn’t be at a time when your friend is completely unavailable to take your call because then you will have to leave a voicemail and he will eventually call you back when he has plenty of time to have a nice long conversation.

Ideally you want to catch your friend when they are just flexible enough to answer the phone but too busy to talk for very long.  That way you meet your weekly quota of phone calls at minimum cost in terms of time actually spent on the phone.  What could be more polite?

Matthew Rabin was here last week presenting his work with Erik Eyster about social learning. The most memorable theme of their their papers is what they call “anti-imitation.” It’s the subtle incentive to do the opposite of someone in your social network even if you have the same preferences and there are no direct strategic effects.

You are probably familiar with the usual herding logic. People in your social network have private information about the relative payoff of various actions. You see their actions but not their information. If their action reveals they have strong information in favor of it you should copy them even if you have private information that suggests doing the opposite.

Most people who know this logic probably equate social learning with imitation and eventual herding. But Eyster and Rabin show that the same social learning logic very often prescribes doing the opposite of people in your social network. Here is a simple intuition. Start with a different, but simpler problem.  Suppose that your friend makes an investment and his level of investment reveals how optimistic he is. His level of optimism is determined by two things, his prior belief and any private information he received.

You don’t care about his prior, it doesn’t convey any information that’s useful to you but you do want to know what information he got. The problem is the prior and the information are entangled together and just by observing his investment you can’t tease out whether he is optimistic because he was optimistic a priori or because he got some bullish information.

Notice that if somebody comes and tells you that his prior was very bullish this will lead you to downgrade your own level of optimism. Because holding his final beliefs fixed, the more optimistic was his prior the less optimistic must have been his new information and its that new information that matters for your beliefs. You want to do the opposite of his prior.

This is the basic force behind anti-imitation. (By the way I found it interesting that the English language doesn’t seem to have a handy non-prefixed word that means “doing the opposite of.”) Suppose now your friend got his prior beliefs from observing his friend. And now you see not only your friend’s investment level but his friend’s too. You have an incentive to do the opposite of his friend for exactly the same reason as above.

This assumes his friend’s action conveys no information of direct relevance for your own decision. And that leads to the prelim question. Consider a standard herding model where agents move in sequence first observing a private signal and then acting.  But add the following twist. Each agent’s signal is relevant only for his action and the action of the very next agent in line.  Agent 3 is like you in the example above.  He wants to anti-imitate agent 1. But what about agents 4,5,6, etc?

If you are like me and you believe that thinking is better path to success than not thinking, its hard not to take it personally when an athlete or other performer who is choking is said to be “overthinking it.” He needs to get “untracked.” And if he does and reaches peak performance he is said to be “unconscious.”

There are experiments that seem to confirm the idea that too much thinking harms performance.  But here’s a model in which thinking always improves performance and which is still consistent with the empirical observation that thinking is negatively correlated with performance.

In any activity we rely on two systems:  one which is conscious, deliberative and requires “thinking.”  The other is instinctive.  Using the deliberative system always gives better results but the deliberation requires the scarce resource of our moment-to-moment attention.  So for any sufficiently complex activity we have to ration the limited capacity of the deliberative system and offload many aspects of performance to pre-programmed instincts.

But for most activities we are not born with an instinctive knowledge how to do it.  What we call “training” is endless rehearsal of an activity which establishes that instinct.  With enough training, when circumstances demand we can offload the activity to the instinctive system in order to conserve precious deliberation for whatever novelties we are facing which truly require original thinking.

An athlete or performer who has been unsettled, unnerved, or otherwise knocked out of his rhythm finds that his instinctive system is failing him.  The wind is playing tricks with his toss and so his serve is falling apart.  Fortunately for him he can start focusing his attention on his toss and his serve and this will help.  He will serve better as a result of overthinking his serve.

But there is no free lunch.  The shock to his performance has required him to allocate more than usual of his deliberative resources to his serve and therefore he has less available for other things.  He is overthinking his serve and as a result his overall performance must suffer.

(Conversation with Scott Ogawa.)

I coach my daughter’s U12 travel soccer team. An important skill that a player of this age should be picking up is the instinct to keep her head up when receiving a pass, survey the landscape and plan what to do with the ball before it gets to her feet.  The game has just gotten fast enough that if she tries to do all that after the ball has already arrived she will be smothered before there is a chance.

Many drills are designed to train this instinct and today I invented a little drill that we worked on in the warmups before our game against our rivals from Deerfield, Illinois. The drill makes novel use of a trick from game theory called a jointly controlled lottery.

Imagine I am standing at midfield with a bunch of soccer balls and the players are in a single-file line facing me just outside of the penatly area.  I want to feed them the ball and have them decide as the ball approaches whether they are going to clear it to my left or to my right. In a game situation, that decision is going to be dictated by the position of their teammates and opponents on the field. But since this is just a pre-game warmup we don’t have that.  I could try to emulate it if I had some kind of signaling device on either flank and a system for randomly illuminating one of the signals just after I delivered the feed.  The player would clear to the side with the signal on.

But I don’t have that either and anyway that’s too easy and quick to read to be a good simulation of the kind of decision a player makes in a game.  So here’s where the jointly controlled lottery comes in.  I have two players volunteer to stand on either side of me to receive the clearing pass.  Just as I deliver the ball to the player in line the two girls simultaneously and randomly raise either one hand or two.  The player receiving the feed must add up the total number of hands raised and if that number is odd clear the ball to the player on my left and if it is even clear to the player on my right.

The two girls are jointly controlling a randomization device.  The parity of the number of hands is not under the control of either player.  And if each player knows that the other is choosing one or two hands with 50-50 probability, then each player knows that the parity of the total will be uniformly distributed no matter how that individual player decides to randomize her own hands.

And the nice thing about the jointly controlled lottery in this application is that the player receiving the feed must look left, look right, and think before the ball reaches her in order to be able to make the right decision as soon as it reaches her feet.

We beat Deerfield 3-0.

I loved that show. She died last month. Here are 30 selected episodes. Definitely check out the Keith Jarett one.

1. Facebook’s business problem is that it is the social network of people you see in real life.  All the really interesting stuff you want to do and say on the internet is stuff you’d rather not share with those people or even let them know you are doing/saying.
2. What is the rationale for offsides in soccer that doesn’t also apply to basketball?
3. If the editors of all the journals were somehow agreeing to publish each others’ papers what patterns would we look for in the data to detect that?
4. I need to know in advance the topic of the next 3 Gerzensee conferences so that I can start now writing papers on those topics in hopes of getting invited.

Suppose you are writing a referee report and you are recommending that the paper be rejected. You have a long list of reasons. How many should you put in your report? If you put only your few strongest arguments you run the risk that the author (or editor) finds a response to those and accepts the paper.

You will have lost the chance to use your next few strongest arguments to their full effect, even if there is a second round. The reason has to do with a basic friction of rhetoric.  Nobody really knows what’s true or false, but the more you’ve thought about it the better informed you are. So there is always a signaling aspect to rhetoric. Even if the opponent can’t find a counterargument, when it is known that you rank your argument low in terms of persuasiveness, your argument will as a result be in fact less persuasive.  Your ranking reveals that you believe that the probability is high that a counterargument could be found, even if by chance this time it wasn’t.

On the other hand you also don’t want to put all of your arguments down. The risk here is that the author refutes all but your strongest one or two arguments. Then the editor may conclude that your decision to reject was made on the basis of that long list of considerations and now that a large percentage of them have been refuted this seals the case in favor. Had you left out all the weak arguments your case would look stronger.

It may even be optimal to pick a non-interval subset of arguments. That is you might give your strongest argument, leave out the second strongest but include the third strongest. The reason is that you care not just about the probability that any single one of your arguments is refuted but the probability that a large subset of your arguments survive. And here correlation matters. It may be that a refutation of the strongest argument is likely also to partially weaken the second-strongest. You pick the third because it is orthogonal to the first.

Grocery chain Trader Joe’s has opened up a legal can of whup ass on its self-professed “best customer,” Pirate Joe’s.

Vancouver, British Columbia shopkeeperMichael Hallatt, claims to have spent more than $350,000 at Trader Joe’s in the past two years. Trader Joe’s would like him to stop shopping there. What gives? Hallatt, makes frequent drives across the border to shop the U.S. stores, then resells popular Trader Joe’s branded products in his own store, cannily called Pirate Joe’s. Various commentators are at a loss to explain why Trader Joe’s would cut off its best customer. But isn’t it obvious? Trader Joe’s always had the option of opening a store in Vancouver. Because it never did, it must be that it would not be profitable. Now the joint profits between Trader Joe’s and Pirate Joe’s cannot be higher than the profit that Trader Joe’s would have earned if it opened its own store. At worst Trader Joe’s could just replicate what Pirate Joe’s is doing, but probably it could do it more efficiently. So Trader Joe’s which earns only a share of the joint TJ/PJ profit must be less profitable than it would be if it opened its own store in Vancouver which it has already calculated to be unprofitable. Here is a nice essay on the idea that “over thinking” causes choking. It begins with this study: A classic study by Timothy Wilson and Jonathan Schooler is frequently cited in support of the notion that experts, when performing at their best, act intuitively and automatically and don’t think about what they are doing as they are doing it, but just do it. The study divided subjects, who were college students, into two groups. In both groups, participants were asked to rank five brands of jam from best to worst. In one group they were asked to also explain their reasons for their rankings. The group whose sole task was to rank the jams ended up with fairly consistent judgments both among themselves and in comparison with the judgments of expert food tasters, as recorded in Consumer Reports. The rankings of the other group, however, went haywire, with subjects’ preferences neither in line with one another’s nor in line with the preferences of the experts. Why should this be? The researchers posit that when subjects explained their choices, they thought more about them. The upcoming Northwestern home game versus Ohio State has now sold out. Danny Ecker at Crain’s Chicago Business has the post-mortem: Sales so far show the school was effective in its experimental“Purple Pricing” offer for about 5,000 single-game seats for the game. The modified Dutch auction system, which guarantees that buyers don’t pay any more for tickets than anyone else in their section, ended up selling out at$195, $151 and$126 for seats on the sideline, corner and end zones, respectively.

On the secondary market, sideline seats have sold for an average of $190, corner seats for$135 and end-zone seats for $127. That suggests that fans haven’t been able to flip them for a profit — at least, not yet. Suppose you and a friend of the opposite sex are recruited for an experiment. You are brought into separate rooms and told that you will be asked some questions and, unless you give consent, all of your answers will be kept secret. First you are asked whether you would like to hook up with your friend. Then you are asked whether you believe your friend would like to hook up with you. These are just setup questions. Now come the important ones. Assuming your friend would like to hook up with you, would you like to know that? Assuming your friend is not interested, would you like to know that? And would you like your friend to know that you know? Assuming your friend is interested, would you like your friend to know whether you are interested? Assuming your friend is not interested, same question. And the higher-order question as well. These questions are eliciting your preferences over you and your friend’s beliefs about (beliefs about…) you and your friend’s preferences. This is one context where the value of information is not just instrumental (i.e. it helps you make better decisions) but truly intrinsic. For example I would guess that for most people, if they are interested and they know that the other is not that they would strictly prefer that the other not know that they are interested. Because that would be embarrassing. And I bet that if you are not interested and you know that the other is interested you would not like the other to know that you know that she is interested. Because that would be awkward. Notice in fact that there is often a strict preference for less information. And that’s what makes the design of a matching mechanism complicated. Because in order to find matches (i.e. discover and reveal mutual interest) you must commit to reveal the good news. In other words, if you and your friend both inform the experimenters that you are interested and that you want the other to know that, then in order to capitalize on the opportunity the information must be revealed. But any mechanism which reveals the good news unavoidably reveals some bad news precisely when the good news is not forthcoming. If you are interested and you want to know when she is interested and you expect that whenever she is indeed interested you will get your wish, then when you don’t get your wish you find out that she is not interested. Fortunately though there is a way to minimize the embarrassment. The following simple mechanism does pretty well. Both friends tell the mediator whether they are interested. If, and only if, both are interested the mediator informs both that there is a mutual interest. Now when you get the bad news you know that she has learned nothing about your interest. So you are not embarrassed. However it doesn’t completely get rid of the awkwardness. When she is not interested she knows that *if* you are interested you have learned that she is not interested. Now she doesn’t know that this state of affairs has occurred for sure. She thinks it has occurred if and only if you are interested so she thinks it has occurred with some moderate probability. So it is moderately awkward. And indeed you know that she is not interested and therefore feels moderately awkward. The theoretical questions are these: under what specification of preferences over higher-order beliefs over preferences is the above mechanism optimal? Is there some natural specification of those preferences in which some other mechanism does better? Update: Ran Spiegler points me to this related paper. A firm has a basic goal: maximize profits. And then it has day-to-day decisions. It is far too complicated to every day try to trace through the consequences of those basic decisions on the fundamental objective of maximizing profits. A manager who tried to do that would spend so much time thinking that by the time he figured it out the day would be over and he’d have to start thinking again about tomorrow’s decision. So firms don’t hire managers like that. Managers cling to intermediate goals, like say maximize market share. The best intermediate goals are the ones that are easy to monitor and which do a pretty good job of proxying for the underlying goal. These intermediate goals eventually become part of the culture of the firm and knowledge of their connection to the underlying goal can get lost. The manager can’t distinguish between intermediate goals and fundamental goals. Now a consultant comes in to advise the manager. A consultant’s job is to show the manager how best to pursue his goals. So the very first thing a consultant should do is find out what the manager’s goals are. And here’s where the dilemma arises. The consultant might actually be smart enough to figure out that the manager’s goals are just intermediate goals. Does he say “De Gustibus” and advise the manager on how to pursue his goals even if he can see that in this particular instance it works against what the manager should really be maximizing? Or does he have enough ambition in his job as advisor to try to convince the manager that his goals are all wrong, that he should really be maximizing something else? I honestly wonder what the smart consultant does in these situations. More generally, in everyday life we have arguments about what’s the right thing to do. A lot of the time these arguments are confounded by the inability to distinguish whether we are arguing about the right course of action given our common goals (an argument that can be settled) or whether we have really just chosen different intermediate goals (loggerheads.) Presh Talwalker tells us about this study of parking strategies: They observed two distinct strategies: “cycling” and “pick a row, closest space.” They compared the results. “What was interesting,” [Professor Andrew Velkey found], “was although the individual cycling were spending more time driving looking for a parking space, on average they were no closer to the door, time-wise or distance-wise, than people using ‘pick a row, closest space.’” And commenters are inferring that hunting for the best spot is a sub-optimal strategy. But those that are searching for the best parking spot are not interested in reducing their expected parking time, rather they care about the second moment. When we have an appointment there is a deadline effect: our payoff drops precipitously if we arrive past the deadline. Faced with such a payoff function we are typically wiling to increase our expected parking time if in return we can at least increase the probability of getting lucky with a really good spot. “Pick a row, closest space” guarantees we will be a bit late. “Cycling” may increase the average searching time but at least gives us a chance of being on time. Dr. Doom did not get approval for the tub from the Department of Buildings, which resulted in the violation. He’s been ordered to remove not only the tub, but also the deck and party room (replete with a bar and bathroom) which he had constructed on the roof of his$5.5 million East First Street pad. The ruling apparently came about after a complaint levied in February.

If you’re still grappling to get a sense of what will be lost now that these parties—at least, in the form they previously inhabited—will cease to exist, here’s a wonderful quote Roubini gave toNew York which paints quite the picture of both his allure and the nature of the shindigs.

“[The models who attend my parties] love my beautiful mind. I am ugly, but they’re attracted to the brains. I’m a rock star among geeks, wonks and nerds,” he said. “[What makes the parties so great are] fun people and beautiful girls. I look for 10 girls to one guy.”

1. Exercise: find a name such that when you sing The Name Game (“banana fana fo …”) all three words you get are insults.
2. The Northwestern Women’s Lacrosse team has won the NCAA championship like every year but two in the past decade.  The two losses make the overall dynasty more impressive.  Discuss.
3. Why do fat people slide farther when they reach the bottom of a water slide?
4. When hiking in a group, if an accurate measure of (changes in) elevation is unavailable but you have a watch and a GPS it’s better to share the work of carrying the backpack by dividing the time rather than the distance.

You are walking back to your office in the rain and your path is lined by a row of trees. You could walk under the trees or you could walk in the open. Which will keep you drier?

If it just started raining you can stay dry by walking under the trees. On the other hand, when the rain stops you will be drier walking in the open. Because water will be falling off the leaves of the tree even though it has stopped raining. Indeed when the rain is tapering off you are better off out in the open. And when the rain is increasing you are better off under the tree.

What about in steady state? Suppose it has been raining steadily for some time, neither increasing nor tapering off. The rain that falls onto the top of the tree gets trapped by leaves. But the leaves can hold only so much water. When they reach capacity water begins to fall off the leaves onto you below. In equilibrium the rate at which water falls onto the top of the tree, which is the same rate it would fall on you if you were out in the open, equals the rate at which water falls off the leaves onto you.

Still you are not indifferent: you will stay drier out in the open. Under the tree the water that falls onto you, while constituting an equal total volume as the water that would hit you out in the open, is concentrated in larger drops. (The water pools as it sits on the leaves waiting to be pushed off onto you.) Your clothes will be dotted with fewer but larger water deposits and an equal volume of water spread over a smaller surface area will dry slower.

It is important in all this that you are walking along a line of trees and not just standing in one place. Because although the rain lands uniformly across the top of the tree, it is probably channeled outward away from the trunk as it falls from leaf to leaf and eventually below. (I have heard that this is true of Louisiana Oaks.) So the rainfall is uniform out in the open but not uniform under the tree. This means that no matter where you stand out in the open you will be equally wet, but there will be spots under the tree in which the rainfall will be greater than and less than that average. You can stand at the local minimum and be drier than you would out in the open.

Why are conditional probabilities so rarely used in court, and sometimes even prohibited?  Here’s one more good reason:  prosecution bias.

Suppose that a piece of evidence X is correlated with guilt.  The prosecutor might say, “Conditional on evidence X, the likelihood ratio for guilt versus innoncence is Y, update your priors accordingly.”  Even if the prosecutor is correct in his statistics his claim is dubious.

Because the prosecutor sees the evidence for all suspects before deciding which ones to bring to trial.  And the jurors know this.  So the fact that evidence like X exists against this defendant is already partially reflected in the fact that it was this guy they brought charges against and not someone else.

If jurors were truly Bayesian (a necessary presumption if we are to consider using probabiilties in court at all) then they would already have accounted for this and updated their priors accordingly before even learning that evidence X exists.  When they are actually told it would necessarily move their priors less than what the statistics imply, perhaps hardly at all, maybe even in the opposite direction.

A balanced take in the New Yorker.  Here is an excerpt.

A core objection is that neuroscientific “explanations” of behavior often simply re-state what’s already obvious. Neuro-enthusiasts are always declaring that an MRI of the brain in action demonstrates that some mental state is not just happening but is really, truly, is-so happening. We’ll be informed, say, that when a teen-age boy leafs through the Sports Illustrated swimsuit issue areas in his brain associated with sexual desire light up. Yet asserting that an emotion is really real because you can somehow see it happening in the brain adds nothing to our understanding. Any thought, from Kiss the baby! to Kill the Jews!, must havesome route around the brain. If you couldn’t locate the emotion, or watch it light up in your brain, you’d still be feeling it. Just because you can’t see it doesn’t mean you don’t have it. Satel and Lilienfeld like the term “neuroredundancy” to “denote things we already knew without brain scanning,” mockingly citing a researcher who insists that “brain imaging tells us that post-traumatic stress disorder (PTSD) is a ‘real disorder.’ ”

And

It’s perfectly possible, in other words, to have an explanation that is at once trivial and profound, depending on what kind of question you’re asking. The strength of neuroscience, Churchland suggests, lies not so much in what it explains as in the older explanations it dissolves. She gives a lovely example of the panic that we feel in dreams when our legs refuse to move as we flee the monster. This turns out to be a straightforward neurological phenomenon: when we’re asleep, we turn off our motor controls, but when we dream we still send out signals to them. We really are trying to run, and can’t.

He died earlier this week.  If you grew up in Southern California, and you watched TV, you may have forgotten Cal Worthington but his dog Spot, the acres and acres of cars, the “Go See Cal”, the giant selection of cars and trucks on sale, the “open every day til midnight” and the music in the way “nineteen” springboarded the cars vintage out of your set and into your ears are all still stored away in some synapses somewhere in there and they’re all gonna come flowing out when you watch this video and probably bring with them a whole bunch of other stuff lost in there that you are gonna be pretty tickled to find again.  RIP Cal Worthington.

Should restaurants put salt shakers on the table?  A variety of food writers weigh in on the question here.

The naive argument is that salt shakers give diners more control. They know their own tastes and can fine tune the salt to their liking. The problem with this argument is that salt shaken over prepared food is not the same as salt added to food as it is cooked.  A chef adds salt numerous times through the cooking process to different items on the plate because some need more salt than others.

So the benefit of control comes at the cost of excess uniformity in the flavor. But beyond that, there is an interesting strategic issue. When there is no salt shaker on the table the chef chooses the level of saltiness to meet some median or average diner’s taste for salt. All diners get equally salty food independent of their taste. Diners to the left of the median find their dish too salty and diners to the right wish they had a salt shaker.

A reduction in the level of saltiness benefits those just to the left of the median at the expense of those far to the right and at an optimum those costs outweigh the benefits.

But when there is a salt shaker, the chef can reduce the level of saltiness at a lower cost because those to the right can compensate (albeit imperfectly) by adding back the salt. So in fact the optimal level of salt added by a chef whose restaurant puts salt shakers on the table is lower.

So the interesting observation is that salt shakers on the table benefit diners who like less salt (and also those that like a lot of salt) at the expense of the average diner (who would otherwise be getting his salt bliss point but is now getting too little).

Imagine that the President convenes his top economic advisors to get a recommendation on a pressing policy issue. They say unequivocally “do X.” The President asks why and they say “its complicated. Do X.” The President, not happy with that, decides he is going to read the economic literature on the pros and cons of doing X. After a thorough study he comes back to his advisors and says “You economists don’t understand your own science. I read the literature and I should do Y.”

I think we would agree that’s a bad outcome. For probably exactly the same reason that Doctors don’t seem to be happy with economist Emily Oster’s apparent advice to pregnant women to drink alcohol “like a European adult.”

But let’s assume that Emily truly can interpret the published statistical literature better than her Obstetrician. There is another reason to question her recommendations.

An advisor’s job is to advise on the risks of an activity. Because the advisor is the expert on that. The decision-maker is the expert on her own preferences. The correct decision is based on weighing both of these.

A recommendation to have up to a glass of wine per day while pregnant confounds the two sides. What it really means is “I like wine a lot.  I also read about the risks and decided that my taste for wine was strong enough that I am willing to live with the risks.” Thus her recommendation amounts to “If you like wine as much as I do you should drink up to a glass per day when you are pregnant.”

When I asked my doctor about drinking wine, she said that one or two glasses a week was “probably fine.” But “probably fine” isn’t a number.

The problem is that there is no way to quantify how much she likes wine and so no way for her readers to know whether they like wine as much as she does. Likewise it is too much for Emily to demand her doctors to say much more than “probably fine.”

The doctors’ advice is based on some assumption about the patient’s taste for wine weighed against the risks. Emily’s advice is based on a different assumption. As for the risks, when Emily reads the literature and concludes that the evidence is weak of the danger of drinking alcohol she then jumps to the conclusion that it is weaker than what the doctors thought. She makes the identifying assumption that their recommendation was conservative because they overestimated the risks and not because they underestimated her taste for wine. But there does not seem to be any basis for that assumption because her doctors never told her what they believed the risks to be and they never asked her how much she likes wine.

One of my favorite subjects is woven through the highly entertaining This American Life segment about Emir Kamenica.  I highly recommend listening.  Unrelated to the main story, at the very beginning Michael Lewis details a very nice defense of plagiarism:

The concept was alien to me, I mean I just thought it was an odd concept because you repeat what other people say all the time, I was just repeating what someone else said, it was a very intelligent thing to repeat. And I thought I was saving us a lot of trouble… save him the trouble of having to read something really awful and I wouldn’t have to write a boring book report or even read the boring book and so I was doing both of us a favor.  And it seemed sortof counterintuitive to have to generate a thing that had already been done.

Roy is coming to plant flowers in Zoe’s garden. Zoe loves flowers, her utility for a garden with $x$ flowers is

$z(x) = x.$

Roy plants a unit mass of seeds and the fraction of these that will bloom into flowers depends on how attentive Roy is as a gardener. Roy’s attentiveness is his type $\theta$. In particular when Roy’s type is $\theta$, absent any sabotage by Zoe, there will be $\theta$ flowers in Zoe’s garden in Spring. Roy’s attentiveness is unknown to everyone and it is believed by all to be uniformly distributed on the unit interval.

Jane, Zoe’s neighbor, is looking for a gardener for the following Spring. Jane has high standards, she will hire Roy if and only if he is sufficiently attentive. In particular, Jane’s utility for hiring Roy when his true type is $\theta$ is given by

$j(\theta) = \theta - 2/3.$

(Her utility is zero if she does not hire Roy.)

Roy tends to one and only one garden per year. Therefore Roy will continue to plant flowers in Zoe’s garden for a second year if and only if Jane does not hire him away from her.

Consequently, Zoe is contemplating sabotaging Roy’s flowers this year. If Zoe destroys a fraction $1 - \alpha$ of Roy’s seeds then the total number of flowers in Zoe’s garden when Spring arrives will be $x = \alpha\theta$. Of course sabotage is costly for Zoe because she loves flowers.

There will be no sabotage in the second year because after two years of gardening Roy goes into retirement. Therefore, if Zoe destroys $1-\alpha$ in the first year and Roy continues to work for Zoe in the second year, Zoe’s total payoff will be

$z(\alpha\theta) + z(\theta)$

whereas if Roy is hired away by Jane, then Zoe’s total payoff is just $z(\alpha\theta)$.

This is a two-player (Zoe and Jane) extensive-form game with incomplete information. The timing is as follows. First, Roy’s type is realized. Nobody observes Roy’s type. Zoe moves first and chooses $\alpha \in [0,1]$. Then Spring arrives and the flowers bloom.  Jane does not observe $\alpha$ but does observe the number of flowers in Zoe’s garden. Then Jane chooses whether or not to hire Roy away from Zoe. Then the game ends.

Describe the set of all Perfect Bayesian Equilibria.

I haven’t decided yet and I can’t figure out which side this is evidence for:

Me:  Oh I have to remember to set up your desk today because I promised that I would and that if I didn’t I would give you $2. 7 year old: I was hoping you would forget. Me: Are you saying you would rather have$2 than your desk?

7yo:  No, I am saying I would rather have $2 today and my desk tomorrow. Me: Hold on, what would you rather have:$2 today and your desk tomorrow or $2 today, another$2 tomorrow and then your desk the next day?

7yo:   $2 today, another$2 tomorrow and then my desk the next day.

Me:  $2 today,$2 tomorrow, $2 the next day, and then your desk the day after that? 7yo: Yep. Me: And what is the number of days you would like to have$2 before you finally get your desk?

7yo:  Infinity.

Here’s the preview:

Students all over are starting college this month, and some of them still have a nagging question: what, exactly, got me in? An admissions officer tells us the most wrongheaded things applicants try. And Michael Lewis has the incredible story of how a stolen library book got one man — Emir Kamenica — into his dream school. (Photo: Emir as a Harvard undergrad. Credit Terri Wang.)


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fixed points and the cycle index

Let $C_n$ be the cyclic group of order $n$ acting on a finite set $X$ and let $Z(C_n, X; p_1,p_2,\dots)$ be the cycle index of the corresponding permutation group.

I wonder whether the knowledge of the cycle index alone is enough to determine the number of fixed points of the action of a given element $g\in C_n$ on $X$?

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Yes; fixed points are cycles of length 1. –  Qiaochu Yuan Feb 16 '11 at 14:12
Thank's, I missed the obvious. –  Martin Rubey Feb 16 '11 at 14:57