text:
## Applied Mathematics

•  Guiseppe Caire, TU Berlin
•  The Mathematics of Massive Random Access Communications; zoom link @ https://sites.google.com/view/minds-seminar/home
•  10/22/2020
•  2:30 PM - 3:30 PM
•  Online (virtual meeting) (Virtual Meeting Link)
•  Olga Turanova (turanova@msu.edu)

The Multiple Access Channel (MAC) is one of the most well studied and understood network information theoretic models, describing a scenario where K users wish to deliver their information message to one receiver, sharing the same transmission channel. Beyond the multitude of practical and somwho heuristic MAC protocols (e.g., TDMA, FDMA, CDMA, CSMA, Aloha, and variations thereof), the information theoretic capacity region is well understood under many situations of interest, and in particular in the Gaussian case, modeling the uplink of a wireless system with one access point or base station (receiver) and several users, sharing the same frequency band. More recently, a variant of this model has been proposed for a situation where a very large (virtually unlimited) number of users wish to communicate only very sporadically, such that at any point in time only a finite and relatively small number of users are active''. This scenario is appropriate for machine-type communications and Internet of Things, where a multitude of sensors and objects have only rather sporadic data to send, but they need to send them when they are created, at random times. The identification of the active user set, or the active message set (the list of messages transmitted, irrespectively of who is transmitting them) has some points in common with a compressed sensing problem, where the activity vector (entry 1 if a user/message is active and 0 otherwise) is the key object to be estimated at the receiver side. In this talk we review the basic MAC model and results, a variant for massive random access called unsourced random access'' where all users use the same codebook, and related recent results and algorithms. including some capacity scaling for this model, which remains quite open as far as a full information theoretic characterization is concerned.

## Contact

Department of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824

Phone: (517) 353-0844
Fax: (517) 432-1562

College of Natural Science


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# Schur-Zassenhaus Theorem

## Theorem

Let $G$ be a finite group and $N$ be a normal subgroup in $G$.

Let $N$ be a Hall subgroup of $G$.

Then there exists $H$, a complement of $N$, such that $G$ is the semidirect product of $N$ and $H$.

## Proof

The proof proceeds by induction.

By definition, $N$ is a Hall subgroup if and only if the index and order of $N$ in $G$ are relatively prime numbers.

Let $G$ be a group whose identity is $e$.

We induct on $\order G$, where $\order G$ is the order of $G$.

We may assume that $N \ne \set e$.

Let $p$ be a prime number dividing $\order N$.

Let $\Syl p N$ be the set of Sylow $p$-subgroups of $N$.

By the First Sylow Theorem:

$\Syl p N \ne \O$

Let:

$P \in \Syl p N$
$G_0$ be the normalizer in $G$ of $P$
$N_0 = N \cap G_0$.
$G = G_0 N$

By the Second Isomorphism Theorem for Groups and thence Lagrange's Theorem (Group Theory), it follows that:

$N_0$ is a Hall subgroup of $G_0$
$\index {G_0} {N_0} = \index G H$

Suppose $G_0 < G$.

Then by induction applied to $N_0$ in $G_0$, we find that $G_0$ contains a complement $H \in N_0$.

We have that:

$\order H = \index {G_0} {N_0}$

and so $H$ is also a complement to $N$ in $G$.

So we may assume that $P$ is normal in $G$ (that is: $G_0 < G$).

Let $Z \paren P$ be the center of $P$.

By:

Center is Characteristic Subgroup
$P$ is normal in $G$
Characteristic Subgroup of Normal Subgroup is Normal

$Z \paren P$ is also normal in $G$.

Let $Z \paren P = N$.

Then there exists a long exact sequence of cohomology groups:

$0 \to H^1 \paren {G / N, P^N} \to H^1 \paren {G, P} \to H^1 \paren {N, P} \to H^2 \paren {G / N, P} \to H^2 \paren {G, P}$

which splits as desired.

Otherwise:

$Z \paren P \ne N$

In this case $N / Z \paren P$ is a normal (Hall) subgroup of $G / Z \paren P$.

By induction:

$N / Z \paren P$ has a complement $H / Z \paren P$ in $E // Z \paren P$.

Let $G_1$ be the preimage of $H // Z \paren P$ in $G$ (under the equivalence relation).

Then:

$\order {G_1} = \order {K / Z \paren P} \times \order {Z \paren P} = \order {G / N} \times \order {Z \paren P}$

Therefore, $Z \paren P$ is normal Hall subgroup of $G_1$.

By induction, $Z \paren P$ has a complement in $G_1$ and is also a complement of $N$ in $G$.

$\blacksquare$

## Also known as

Some sources refer to this theorem as Schur's theorem, but that name is also used for an unrelated result in Ramsey theory.

## Source of Name

This entry was named for Issai Schur and Hans Julius Zassenhaus.

format: plain


text:
### A Problem

A major problem in secondary data analysis is that you didn't get to decide what data was collected. Lets say you were interested in how many times a student has read the Twilight books). Specifically, you want to know how effective the ads for the movies and books are. You come up with a model that says $$n_{T} = f(n_{VB}, \text{ads}, \text{gender}, \epsilon)$$ where $$n_{T}$$ is the number of times they read Twilight books and $$n_{VB}$$ is the number of other vampire books they have read in the past year, $$\text{ads}$$ is the number of ads they have seen for the Twilight books or movies and finally gender as Twilight seems to have a differential appeal on the basis of gender. You run off to your favorite data repo and find the prefect data set to answer your question. Someone sat down with a 1,000 high schoolers and asked them about their reading habits as well as how many times they laughed at Twilight trailers. You expect males and females to have difference preferences and therefore respond differently (e.g., different beta coefficients from a linear model) to the ads and the number of other vampire books read in the last year. Instead of creating a model: $n_{T} = \beta_0 + \beta_1 n_{VB} + \beta_2 \text{ads} + \beta_3 \text{gender} + \epsilon$ you decide to use some interactions, getting at a regression model something like: $n_{T} = (\beta_0 + \beta_4 \text{gender}) + (\beta_1 + \beta_5 \text{gender}) n_{VB} + (\beta_2 + \beta_6 \text{gender}) \text{ads} + \beta_3 \text{gender} + \epsilon$ It is a lovely model, it makes sense and neatly answers your question of interest. You fire up R (or your stats package of choice) only to realize when exploring the data that they didn't report if the student was male or female.

Now this is a problem. You already have theorize that gender is part of the true data generating process. To ignore it violates the assumptions of most forms of regression — your error term is now a $$f(\text{gender}, \epsilon)$$ and if gender is part of the data generating process, than Y and your error term are not independent. You can do a regression still but the wonderful properties may not hold. You need a way to account for the observed heterogeneity (e.g., gender of the study subjects).

You read up on the subject matter and discover that, on average, girls read more vampire books and that the ads for Twilight were published in venues with a large teenage female demographic (maybe?). Could you use this information to potentially “discover” the unobserved characteristic using the data you have?

### Expectation–Maximization (EM) Algorithm

EM is an an iterative unsupervised clustering method. To grossly oversimplify, EM takes a set of data ($$\bf X$$) and attempts to find the number of clusters represented in $$\bf X$$ and finds clusters in the data. It finds clusters by first assigning each point a probability of belonging to a each cluster and then re-estimating the probability distribution for each cluster. This process continues until the error has converged at which time the points are all assigned to a given cluster with some degree of uncertainty. The process repeats for various counts of clusters. The quality of each model is accessed using something like the Bayesian information criterion and the model with the number of clusters that maximizes the quality of the model is returned.

We are going to use this with some simulated data to see what risks we run if we ignore the unobserved variables and also how well we can recover that missing information with EM.

### Data Generation and Clustering

Lets put together a little simulation in R. Sticking to the Twilight example, lets have two observed continuous variables ($$x_1$$ and $$x_2$$), one unobserved binary variable ($$z$$) and differences between the two groups along the unobserved variable. Let $$x_1$$ be iid and $x_1 \sim \begin{cases} N(10, 9) &\mbox{if } z = 0 \\ N(16, 9) &\mbox{if } z = 1 \end{cases}$ For simplicity, let $$x_2$$ also iid and distributed in the same fashion as $$x_1$$. We end up having two populations, each with similar variances but unequal means. Lets generate these data in R:

nSample <- 2000  # this is equal to the size of each class (0.5*total)

# x1a and x2a are features x1 and x2 for z=0 (x11 just seemed a tad much)
x1a <- rnorm(nSample, 10, 3)
x2a <- rnorm(nSample, 10, 3)
# x1b and x2b are features x1 and x2 for z=1
x1b <- rnorm(nSample, 16, 3)
x2b <- rnorm(nSample, 16, 3)

One of the best things to do when doing any sort of statistics is to look at the data. I prefer kernel density plots to histograms, so lets take a look at the density plots.

library(ggplot2)
# put the data into a dataframe
clusterData <- data.frame(x1 = c(x1a, x1b), x2 = c(x2a, x2b), class = c(rep("z = 0",
nSample), rep("z = 1", nSample)))
clusterData.gg <- ggplot(clusterData, aes(x1 = x1, x2 = x2, type = class))

# plot the combined density in grey and then overlay with colored density
# for each class
clusterData.gg + stat_density(aes(x = x1), alpha = 0.5) + geom_density(aes(x = x1,
color = class, fill = class), alpha = 0.5) + theme(legend.position = "none") +
scale_x_continuous("Value of x1") + scale_y_continuous("Density")

clusterData.gg + stat_density(aes(x = x2), alpha = 0.5) + geom_density(aes(x = x2,
color = class, fill = class), alpha = 0.5) + theme(legend.position = "none") +
scale_x_continuous("Value of x2") + scale_y_continuous("Density")

The tails are pretty fat but as far as empirical data goes, this is pretty well behaved looking overall. Looking at the composite density, it would be hard to tell that there are actually two different densities represented in the curve. However, our theory/knowledge of the domain says that at least 2 classes exist.

The easiest way I know to do EM in R is to use the mclust package. Lets use that and the data in $$x_1$$ and $$x_2$$ to see what we find:

library(mclust)

## Package 'mclust' version 4.0

# put the data into a matrix just to make using Mclust a bit easier
clusterMatrix <- matrix(NA, nrow = nSample * 2, ncol = 2)
clusterMatrix[, 1] <- c(x1a, x1b)
clusterMatrix[, 2] <- c(x2a, x2b)
# fits the EM model to the data
model <- Mclust(clusterMatrix)

And that is all it takes. Lets see some of the results

summary(model)

## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm
## ----------------------------------------------------
##
## Mclust EII (spherical, equal volume) model with 2 components:
##
##  log.likelihood    n df    BIC
##          -22111 4000  6 -44271
##
## Clustering table:
##    1    2
## 2052 1948

No surprise there, we expected and we found 2 clusters. Lets take a look at those clusters now.

clusterPlots <- data.frame(x1 = clusterData$x1, x2 = clusterData$x2, cluster = factor(model$classification, levels = c(1, 2), labels = c("z = 0", "z = 1")), uncertainity = model$uncertainty,
logUncertainity = log(model$uncertainty)) clusterPlots.gg <- ggplot(clusterPlots) clusterPlots.gg + geom_point(aes(x = x1, y = x2, color = cluster))  The clusters as we defined them were discovered and are centered more or less where we would expect. We can see that they do sit on top of each other a bit in the middle. One of the nice things about EM is it quantifies how uncertain it is in a points class. clusterPlots.gg + geom_point(aes(x = x1, y = x2, color = logUncertainity))  That is somewhat helpful, you can see the general trend (points in the middle have higher $$log(\text{uncerrtainty})$$ values and there are more uncertain) but it might be easier to tell if we split it into groups. clusterPlots$uncertainGroups <- ifelse(clusterPlots$uncertainity <= 0.2, "Highly certain", ifelse(clusterPlots$uncertainity <= 0.4, "Moderately certain", ifelse(clusterPlots$uncertainity <= 0.5, "Weakly certain", ifelse(clusterPlots$uncertainity <= 0.6, "Weakly uncertain",
ifelse(clusterPlots$uncertainity <= 0.8, "Moderately uncertain", "Extremely uncertain"))))) clusterPlots.gg <- ggplot(clusterPlots) clusterPlots.gg + geom_point(aes(x = x1, y = x2, color = uncertainGroups))  ### Application to Regression In the Twilight example, we were using EM to account for some unobserved heterogeneity. Now that we have $$\bf X$$, we might as well simulate some regression models and see how this does. To start with, lets define the data generating process as $y_i = \beta_0 + \beta_{1, z}x_{1, i} + \beta_{2, z}x_{2, i} + \epsilon_i$. For simplicity, we are assuming that the error distribution and the intercept are the same between the two groups. It would not be difficult to allow the intercept to vary with group membership by the inclusion of a dummy variable for group but there is enough going on already with group specific values for $$\beta_1$$ and $$\beta_2$$. Lets generate some data for $$y$$, $$\epsilon$$ and the$\beta$s. # beta estimates for z = 1 are 0.3 b1a <- rnorm(nSample, 1, 0.05) # specific beta1 for z = 0 b1b <- rnorm(nSample, 1.3, 0.05) # specific beta1 for z = 1 b2a <- rnorm(nSample, 1, 0.05) # specific beta2 for z = 0 b2b <- rnorm(nSample, 1.3, 0.05) # specific beta2 for z = 1 intercept <- rnorm(2 * nSample, 3, 0.075) # intercept e <- rnorm(2 * nSample, 0, 0.05) # error term x1 <- c(x1a, x1b) x2 <- c(x1b, x2b) beta1 <- c(b1a, b1b) beta2 <- c(b2a, b2b) y <- intercept + beta1 * x1 + beta2 * x2 + e  And lets test the regression model where we pretend like the $$\beta$$ values are constant $$\forall$$ $$z$$. naiveModel <- lm(y ~ x1 + x2) summary(naiveModel)  ## ## Call: ## lm(formula = y ~ x1 + x2) ## ## Residuals: ## Min 1Q Median 3Q Max ## -14.00 -2.30 0.28 2.45 8.96 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -6.0538 0.3286 -18.4 <2e-16 *** ## x1 1.8902 0.0124 152.8 <2e-16 *** ## x2 1.1386 0.0175 65.1 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 3.32 on 3997 degrees of freedom ## Multiple R-squared: 0.873, Adjusted R-squared: 0.873 ## F-statistic: 1.37e+04 on 2 and 3997 DF, p-value: <2e-16  AIC(naiveModel)  ## [1] 20947  If we just look at the adjusted R-squared (0.873) and p values ($$<2e-16$$, that is, like, really small /sarcasm), it looks like the model did a very good job of explaining the system in question. However, we know what the true data generating process was and all three parameters are significant off. We aren't really getting an accurate idea of what the underlying data generating process is with this model. What happens when we include the EM predicted cluster membership? classOne <- model$classification == 1  # actually z = 1
emModel <- lm(y ~ x1 + I(x1 * classOne) + x2 + I(x2 * classOne))
summary(emModel)

##
## Call:
## lm(formula = y ~ x1 + I(x1 * classOne) + x2 + I(x2 * classOne))
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -13.454  -0.969   0.038   1.037   8.851
##
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)
## (Intercept)        3.5267     0.2780   12.69  < 2e-16 ***
## x1                 1.3463     0.0152   88.80  < 2e-16 ***
## I(x1 * classOne)  -0.1186     0.0190   -6.25  4.6e-10 ***
## x2                 1.1887     0.0153   77.92  < 2e-16 ***
## I(x2 * classOne)  -0.3328     0.0158  -21.04  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.35 on 3995 degrees of freedom
## Multiple R-squared: 0.936,   Adjusted R-squared: 0.936
## F-statistic: 1.47e+04 on 4 and 3995 DF,  p-value: <2e-16

AIC(emModel)

## [1] 18190

anova(naiveModel, emModel)

## Analysis of Variance Table
##
## Model 1: y ~ x1 + x2
## Model 2: y ~ x1 + I(x1 * classOne) + x2 + I(x2 * classOne)
##   Res.Df   RSS Df Sum of Sq    F Pr(>F)
## 1   3997 43958
## 2   3995 22043  2     21915 1986 <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

A significant improvement the simple sweep-it-under-the-rug model but still not all that amazing. We are still not reliably uncovering the true underlying data generating process as well as we might hope. We have to be able to do better than that.

So far we haven't taken advantage of the uncertainty measurements from the EM process. We can use these as weights in the regression to reduce the importance of points which we are most uncertain about group membership. We could also find a way to fix it in with the regression model, perhaps instead of using classZero use something like $$Pr(z_i = 0)$$, but this is the simplest way to bring them into play.

uncertainity <- model\$uncertainty

emModelWeighted <- lm(y ~ x1 + I(x1 * classOne) + x2 + I(x2 * classOne), weights = -log(uncertainity))
summary(emModelWeighted)

##
## Call:
## lm(formula = y ~ x1 + I(x1 * classOne) + x2 + I(x2 * classOne),
##     weights = -log(uncertainity))
##
## Weighted Residuals:
##    Min     1Q Median     3Q    Max
## -22.40  -1.73  -0.14   1.59  12.36
##
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)
## (Intercept)        3.2240     0.2003    16.1   <2e-16 ***
## x1                 1.3107     0.0110   119.7   <2e-16 ***
## I(x1 * classOne)  -0.2015     0.0146   -13.8   <2e-16 ***
## x2                 1.2655     0.0110   115.3   <2e-16 ***
## I(x2 * classOne)  -0.3304     0.0118   -28.1   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.62 on 3995 degrees of freedom
## Multiple R-squared: 0.975,   Adjusted R-squared: 0.975
## F-statistic: 3.96e+04 on 4 and 3995 DF,  p-value: <2e-16

AIC(emModelWeighted)

## [1] 16438

If we weight the estimates by how certain we are in the group membership, we get a much better result. Here I used $$-log(uncertainity)$$ but say similar results using $$1 - uncertainity$$ and $$1/(1 - uncertainity)^{-1}$$. I decided on using $$-log(uncertainity)$$ as I really wanted to weight towards points that I was more certain in against points that I wasn't. The $$-log$$ transformation gives me this desired effect. Using rlm() from MASS gives similar results to any of these weight transformations.

Finally, if just for curiosity's sake, I wanted to see what was “the best I could do.” Since we had assigned the clusters, I wanted to repeat the above regression but using the true assignments instead of the predicted ones.

trueClassOne <- trueClass == 1  # z = 1
cheatingModel <- lm(y ~ x1 + I(x1 * trueClassOne) + x2 + I(x2 * trueClassOne))
summary(cheatingModel)

##
## Call:
## lm(formula = y ~ x1 + I(x1 * trueClassOne) + x2 + I(x2 * trueClassOne))
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -4.258 -0.724 -0.017  0.723  3.945
##
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)
## (Intercept)           2.92893    0.11932    24.6   <2e-16 ***
## x1                    0.99962    0.00769   129.9   <2e-16 ***
## I(x1 * trueClassOne)  0.30467    0.00908    33.6   <2e-16 ***
## x2                    1.00149    0.00686   145.9   <2e-16 ***
## I(x2 * trueClassOne)  0.29878    0.00751    39.8   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.09 on 3995 degrees of freedom
## Multiple R-squared: 0.986,   Adjusted R-squared: 0.986
## F-statistic: 7.24e+04 on 4 and 3995 DF,  p-value: <2e-16

AIC(cheatingModel)

## [1] 12015

Looking at this model and the performance of the EM enhanced models, I have to say while having actual group membership would be best, even in this simple 2D case we readily uncovered the clusters and improved out regression quality drastically compared to ignoring the unobserved variable. What I show here can be readily extended to highly dimensional cases and may detect true clusters more easily. If we had another several $$x$$ variables, we may have been more able to more accurately distinguish between $$z = 0$$ and $$z = 1$$. But that simulation is for another post.

Tags: dataMining EM R statistics regression simulation


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# Copper(I) oxide

(Redirected from Copper(I) hydroxide)
Copper(I) oxide
Identifiers
CAS number 1317-39-1
PubChem 10313194
ChemSpider 8488659
UNII T8BEA5064F
EC number 215-270-7
KEGG C18714
RTECS number GL8050000
Jmol-3D images Image 1
Image 2
Properties
Molecular formula Cu2O
Molar mass 143.09 g/mol
Appearance brownish-red solid
Density 6.0 g/cm3
Melting point 1,232 °C (2,250 °F; 1,505 K)
Boiling point 1,800 °C (3,270 °F; 2,070 K)
Solubility in water Insoluble
Solubility in acid Soluble
Band gap 2.137 eV
Structure
Crystal structure cubic
Thermochemistry
Std molar
entropy
So298
93 J·mol−1·K−1
Std enthalpy of
formation
ΔfHo298
−170 kJ·mol−1
Hazards
MSDS SIRI.org
EU Index 029-002-00-X
EU classification Harmful (Xn)
Dangerous for the environment (N)
R-phrases R22, R50/53
S-phrases (S2), S22, S60, S61
NFPA 704
Related compounds
Other anions Copper(I) sulfide
Copper(II) sulfide
Copper(I) selenide
Other cations Copper(II) oxide
Silver(I) oxide
Nickel(II) oxide
Zinc oxide
Except where noted otherwise, data are given for materials in their standard state (at 25 °C (77 °F), 100 kPa)
(verify) (what is: /?)
Infobox references

Copper(I) oxide or cuprous oxide is the inorganic compound with the formula Cu2O. It is one of the principal oxides of copper. This red-coloured solid is a component of some antifouling paints. The compound can appear either yellow or red, depending on the size of the particles.[1] Copper(I) oxide is found as the reddish mineral cuprite.

## Preparation

Copper(I) oxide may be produced by several methods.[2] Most straightforwardly, it arises via the oxidation of copper metal:

4 Cu + O2 → 2 Cu2O

Additives such as water and acids affect the rate of this process as well as the further oxidation to copper(II) oxides. It is also produced commercially by reduction of copper(II) solutions with sulfur dioxide. Aqueous cuprous chloride solutions react with base to give the same material. In all cases, the color is highly sensitive to the procedural details.

Formation of copper(I) oxide is the basis of the Fehling's test and Benedict's test for reducing sugars. These sugars reduce an alkaline solution of a copper(II) salt, giving a bright red precipitate of Cu2O.

It forms on silver-plated copper parts exposed to moisture when the silver layer is porous or damaged. This kind of corrosion is known as red plague.

Little evidence exists for cuprous hydroxide, which is expected to rapidly undergo dehydration. A similar situation applies to the hydroxides of gold(I) and silver(I).

Copper(I) oxide can also be prepared by reacting a copper-ammonia complex with hydrogen peroxide.

## Properties

The solid is diamagnetic. In terms of their coordination spheres, copper centres are 2-coordinated and the oxides are tetrahedral. The structure thus resembles in some sense the main polymorphs of SiO2, and both structures feature interpenetrated lattices.

Copper(I) oxide dissolves in concentrated ammonia solution to form the colourless complex [Cu(NH3)2]+, which is easily oxidized in air to the blue [Cu(NH3)4(H2O)2]2+. It dissolves in hydrochloric acid to give solutions of CuCl2. Dilute sulfuric acid and nitric acid produce copper(II) sulfate and copper(II) nitrate, respectively.[3]

Cu2O degrades to copper(II) oxide in moist air.

## Structure

Cu2O crystallizes in a cubic structure with a lattice constant al=4.2696 Å. The Cu atoms arrange in a fcc sublattice, the O atoms in a bcc sublattice. One sublattice is shifted by a quarter of the body diagonal. The space group is $\scriptstyle Pn\bar{3}m$, which includes the point group with full octahedral symmetry.

## Semiconducting properties

In the history of semiconductor physics, Cu2O is one of the most studied materials, and many experimental observations and semiconductor applications have been demonstrated first in this material:

Today, it is still under investigation in semiconductor optics. In particular, researchers are attempting to create a Bose–Einstein condensate of excitons in Cu2O.[5]

The lowest excitons in Cu2O are extremely long lived; absorption lineshapes have been demonstrated with neV linewidths, which is the narrowest bulk exciton resonance ever observed.[8] The associated quadrupole polaritons have low group velocity approaching the speed of sound. Thus, light moves almost as slowly as sound in this medium, which results in high polariton densities.

Another unusual feature of the ground state excitons is that all primary scattering mechanisms are known quantitatively.[9] Cu2O was the first substance where an entirely parameter-free model of absorption linewidth broadening by temperature could be established, allowing the corresponding absorption coefficient to be deduced. It can be shown using Cu2O that the Kramers–Kronig relations do not apply to polaritons.[5]

## Applications

Cuprous oxide is commonly used as a pigment, a fungicide, and an antifouling agent for marine paints. Rectifier diodes based on this material have been used industrially as early as 1924, long before silicon became the standard. Copper(I) oxide is also responsible for the pink color in a positive Benedict's Test.


text:
# Many-to-many references in Calc

Hi there!

Feel sorry if topic is discussed already, was not able to find similar discussion.

I have a part numbers' compatibility table in Calc, looks like this:

Col. A - Col. B
Item1 - Part1
Item2 - Part1
Item1 - Part2
Item1 - Part3
Item2 - Part3
... - ...

There are many compatible Part cells for each Item, and each Part can match many Items. What I need to achieve is to get all compatible parts for each Item in one row:

Col. A - Col. B
Item1 - Part1, Part2, Part4, Part 10...
Item2 - Part1, Part3, Part4, Part5...
Item3 - Part2, Part7 ...
and so on...

By now, I see 2 ways of doing it:

1). Use Autofilter to go through all Items, Copy filtered Parts and Paste Special with Transpose option to get Parts in one row, then Concatenate them. Bet there are about 40k rows in document, 1.5k unique items - to much routine...

2). Sort unique Items withot dublicate and then for each items use Match to find first occurrence in Parts array and to get this cell address, then in next column use Match again, but for array only after first match, editing this array with Indirect and then calculate cell's address where second match is found, then extend formulae to the next column and match third occurrence, considering previous array address change, calculate cell address for new match and so on... Pretty complicated and unclear formulas - easy to make mistake.

I am pretty sure that there should be some easier way of achieving this goal. Any ideas to guide me the right way?

edit retag close merge delete

Perhaps the solution involves TEXTJOIN as described in many posts by @Lupp lately.

( 2018-01-18 22:20:42 +0100 )edit
1

did you mean, "Item1 - Part1, Part2, Part3, Part 10... "? (you wrote, ..1, Part2, Part4..")

( 2018-01-18 22:27:15 +0100 )edit

You can use a hammer to paint a wall, but it's easier with a brush. You can use a spread sheet to process tables, but it's easier with a database.

( 2018-01-18 22:28:39 +0100 )edit

Thanks for comments, EasyTrieve. I do fully agree on your 'Chose the right tool' metaphor. Unfortunately, I am familiar only with the very basics of Base and SQL, so I would prefer Calc, where my knowledge is a bit better. JohnSUN's and Jim's solutions are elegant and easy to implement in Spreadsheet. Nevertheless, I would be very grateful, if you can past here Base solution for this simple database with one table with two columns - just for educational purposes.

( 2018-01-19 08:08:39 +0100 )edit

@SM_Riga, this might be way over your head right now, but it might provide insight https://stackoverflow.com/questions/2... I suspect you'll tinker with databases more later. But don't start with hard problems, get the basics down first. Good luck.

( 2018-01-21 06:59:19 +0100 )edit

Dear EasyTrieve. Sorry for such late response, was busy last days. You are right, the provided topic is just over my SQL knowledge at this moment, but the link is bookmarked for my to-learn list ;)

( 2018-01-29 17:19:47 +0100 )edit

@SM_Riga, We're all learning, and struggling. Take care.

( 2018-02-14 04:14:57 +0100 )edit

Sort by » oldest newest most voted
1. Copy all distinct values from column A to column C with Data -> More -> Standard Filter, as described at https://ask.libreoffice.org/en/questi....
2. In cell D1, enter the following Array formula (similar to @JohnSUN's answer) by pressing Control+Shift+Enter.

=TEXTJOIN(", ";1;IF(A$1:A$5=C1;B$1:B$5;""))

3. Hold the Control key and drag the formula down column D as far as needed.

more

Many thanks, Jim and JohnSUN. Exactly the way a formula should be - elegant. Was not familiar with array formulas till today, so my gratitude to both of you for not just writing the right formula, but much more for showing new methods to learn and use.

( 2018-01-19 07:34:35 +0100 )edit

What about formula array like as

{=TEXTJOIN(", ";1;IF(A2=A:A;B:B;""))}

more

Dear JonhSUN. As I've mentioned above, many thanks to you for pointing the way to search for solution. I've voted for both answers as they are correct. Jim's reply is just more informative for others searching on similar topic.

( 2018-01-19 07:43:24 +0100 )edit

Yes, you're right - for this reason I voted for Jim's answer too.

( 2018-01-19 09:58:44 +0100 )edit

## Stats

Asked: 2018-01-18 17:00:49 +0100

Seen: 71 times

Last updated: Jan 18 '18


text:
# Question related to g

Gold Member
We know that the force of gravitation is F=GMm/r^2. The acceleration of the body of any mass m is a=GM/r^2 which we call g. So same acceleration regardless of any weight(ignoring air resistance). But when we solve laws of motion problems we take a downward force of -mg. So force depend on mass. Heavier body has more pull towards the earth. So shouldn’t it move faster than the lighter body?

weirdoguy
So force depend on mass.
There is also m in Newton's second law, so it disappears.

Hamiltonian
Gold Member
There is also m in Newton's second law, so it disappears.
So when we do problems related to tension or inclined plane friction problems, we take -mg. The force is always mg. And sometimes we need to find acceleration. But that never is equal to g. Is it because the body is not under free fall and the acceleration will not be g. Only when body is free to move the acceleration will always be g and bodies of any weight will come together.

Lnewqban
weirdoguy
Is it because the body is not under free fall and the acceleration will not be g.

Yes.

Gold Member
Yes.
It’s weird that when in free fall the acceleration doesn’t depend on force but when we take system of bodies like a pulley and attached masses suddenly the acceleration does depend on net force.

weirdoguy
the rotation of the Earth is also included into the vector constant ##\boldsymbol g##

Mentor
It’s weird that when in free fall the acceleration doesn’t depend on force ….
but it does. The gravitational force on the more massive object is greater and that’s how we can get it to accelerate at the same rate as a less massive object subjected to a smaller force.

russ_watters
Gold Member
but it does. The gravitational force on the more massive object is greater and that’s how we can get it to accelerate at the same rate as a less massive object subjected to a smaller force.
Ok. To produce the same acceleration to a more massive body we need more force. Thats Newtons second law F=ma. Gravitational force is F=GMm/r^2 but then the mass is also more for more massive bodies so a=F/m, a=g. No matter the weight all bodies does same acceleration. But when we arrange a pulley and two masses system force on each mass changes. Force on more massive body becomes less than mg so a<g. even though acceleration is downwards but its not g. For lighter body the force of tension is much more than mg. So its direction of acceleration changes. Its not g again.
I made a mistake. Force and acceleration are always connected. Its the g that doesn't depend on mass.

Gold Member
Take a deeper look at the definition of force.
$$F=\frac{d(mv)}{dt}$$
The force is the variation of momentum. Momentum could be seen as the amount of resistance to change in motion, i.e. the principle of inertia. In the case of planets interacting together, we assume mass doesn't vary such that:
$$F=m\frac{dv}{dt}$$
Or:
$$m\frac{GM}{r^2}=m\frac{dv}{dt}$$
So what does it mean? The force ##F## that wants to change the velocity and the inertia that wants to resist that change are both proportional to the mass ##m##. The effect of the mass becomes then effectively irrelevant.

But imagine the mass ##m## would change as time goes on, then:
$$m\frac{GM}{r^2}=\frac{d(mv)}{dt}$$
$$m\frac{GM}{r^2}=m\frac{dv}{dt} + v\frac{dm}{dt}$$
$$\frac{dv}{dt}= \frac{GM}{r^2} - \frac{v}{m}\frac{dm}{dt}$$
Not only the mass is relevant to determine the acceleration, but also the velocity and the rate of change of that mass.

Lnewqban
Homework Helper
Gold Member
It’s weird that when in free fall the acceleration doesn’t depend on force but when we take system of bodies like a pulley and attached masses suddenly the acceleration does depend on net force.

All bodies are weightless while in free fall.


text:
Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01h989r543m
DC FieldValueLanguage
dc.contributor.authorParker, Jeffreyen_US
dc.contributor.otherPlasma Physics Departmenten_US
dc.date.accessioned2014-09-25T22:38:57Z-
dc.date.available2014-09-25T22:38:57Z-
dc.date.issued2014en_US
dc.identifier.urihttp://arks.princeton.edu/ark:/88435/dsp01h989r543m-
dc.description.abstractIn geophysical and plasma contexts, zonal flows are well known to arise out of turbulence. We elucidate the transition from statistically homogeneous turbulence without zonal flows to statistically inhomogeneous turbulence with steady zonal flows. Starting from the Hasegawa--Mima equation, we employ both the quasilinear approximation and a statistical average, which retains a great deal of the qualitative behavior of the full system. Within the resulting framework known as CE2, we extend recent understanding of the symmetry-breaking `zonostrophic instability'. Zonostrophic instability can be understood in a very general way as the instability of some turbulent background spectrum to a zonally symmetric coherent mode. As a special case, the background spectrum can consist of only a single mode. We find that in this case the dispersion relation of zonostrophic instability from the CE2 formalism reduces exactly to that of the 4-mode truncation of generalized modulational instability. We then show that zonal flows constitute pattern formation amid a turbulent bath. Zonostrophic instability is an example of a Type Is instability of pattern-forming systems. The broken symmetry is statistical homogeneity. Near the bifurcation point, the slow dynamics of CE2 are governed by a well-known amplitude equation, the real Ginzburg-Landau equation. The important features of this amplitude equation, and therefore of the CE2 system, are multiple. First, the zonal flow wavelength is not unique. In an idealized, infinite system, there is a continuous band of zonal flow wavelengths that allow a nonlinear equilibrium. Second, of these wavelengths, only those within a smaller subband are stable. Unstable wavelengths must evolve to reach a stable wavelength; this process manifests as merging jets. These behaviors are shown numerically to hold in the CE2 system, and we calculate a stability diagram. The stability diagram is in agreement with direct numerical simulations of the quasilinear system. The use of statistically-averaged equations and the pattern formation methodology provide a path forward for further systematic investigations of zonal flows and their interactions with turbulence.en_US
dc.language.isoenen_US
dc.publisherPrinceton, NJ : Princeton Universityen_US
dc.relation.isformatofThe Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the <a href=http://catalog.princeton.edu> library's main catalog </a>en_US
dc.subjectpattern formationen_US
dc.subjectturbulenceen_US
dc.subjectzonal flowen_US
dc.subject.classificationPlasma physicsen_US
dc.subject.classificationAtmospheric sciencesen_US
dc.titleZonal Flows and Turbulence in Fluids and Plasmasen_US
pu.projectgrantnumber690-2143en_US
Appears in Collections:Plasma Physics

Files in This Item:
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text:
# What’s a Enhance In T?

Home / 未分类 / What’s a Enhance In T?

# What is just really a complement in mathematics?

It is utilised to figure the total amount of the solution of at least two objects. Listed below are few examples

Amount of numbers in the quantity field if it is sort has been inserted into the total amount of their constructive shape. Wherever is multiplied with the item below. An area of a volume is added to some other in an identical thesis help sum it becomes exactly the field of that quantity.

Tri Angle is still a type of multiple group. Wherever there was certainly one of those currencies is broken up from the total amount of this set and then invisibly from your result. Wherever certainly one of this pair is significantly much over one alternative separated by the larger 1 and is multiplied by the smaller 1.

Multiplication of fractions using fractions might be done using the subsequent formula. Even the multiplication of a percentage from its own denominator https://www.njcu.edu/academics/schools-colleges/william-j-maxwell-college-arts-sciences/departments/chemistry (within cases like this fraction) is followed by means of a rest. Wherever the numerator is less compared to the denominator is broken up from the numerator along with the remaining.

Couples are partners who has both partners. Multiplying pair by the half the bunch is currently known as as conjugation. The rest of the fraction is called the fractional point.

The number in which the purpose is really to be situated will be also called the rest. In many scenarios where both numerator and denominator are the exact same that the outcome will be also computed also it is inserted to the sum of their two and also the outcome is a sum. That the denominator expert-writers.net/ is smaller than the numerator could be that your numerator squared and then divided by the numerator.

Complement can be utilized in several aspects of mathematics. It is necessary for computing possibilities where it is employed, and evaluations would be.


text:
# Where can I find Gonthier's Coq code proving the four color theorem?

In a 2008 article in the Notices, Georges Gonthier announced a computer-checked proof of the four color theorem using Coq:

Gonthier, Georges. Formal proof—the four-color theorem. Notices Amer. Math. Soc. 55 (2008), no. 11, 1382–1393. PDF

Unfortunately, the article does not seem to provide a link to the Coq code. Where can I find the code? I'm not having any luck with Google.

-
@darijgrinberg: Thanks, maybe that should be posted as an answer. It's annoying that it's in a proprietary Microsoft packaging format, but given that Gonthier works for Microsoft I guess maybe I shouldn't be surprised... –  Nate Eldredge May 1 '14 at 23:30

## 1 Answer

It is available on Microsoft's site: http://research.microsoft.com/en-us/downloads/5464e7b1-bd58-4f7c-bfe1-5d3b32d42e6d/default.aspx .

Here is a probably temporarily link to a hopefully complete extracted version in .7z format.

The proof depends on the ssreflect library, for which see http://www.msr-inria.fr/projects/mathematical-components-2/ (thankfully as two .tar.gz files).

-
Thanks very much, darij. I will try to build it on Linux, and if I succeed I'll post further details here. –  Nate Eldredge May 1 '14 at 23:51
@NateEldredge - I wonder what the license is on the code. Ideally it could be stored in an online version control repository of some kind, so that people could make use of it. –  David Roberts May 2 '14 at 1:25
@DavidRoberts: Each file says at the top (c) Copyright Microsoft Corporation and Inria. All rights reserved. I didn't see any further license information. But that makes it look like it's probably not allowed. :-( –  Nate Eldredge May 2 '14 at 3:07
@NateEldredge - hmph. :-( indeed. –  David Roberts May 2 '14 at 3:14
You should ask Gonthier -- he and his team seem to usually release with the CeCILL licenses, so this might just be a minor technicality they didn't bother to clarify. –  darij grinberg May 2 '14 at 4:32


text:
# Precisely, what is the polar coordinate system?

We all have seen the polar coordinate system back in high school. Recall that $S^2$ may be parametrized by $(\theta,\phi)$ with the correspondence \begin{align} x(\theta,\phi) &= \sin(\theta)\cos(\phi) \\ y(\theta,\phi) &= \sin(\theta)\sin(\phi) \\ z(\theta,\phi) &= \cos(\theta) \end{align}

where $\theta\in [0,\pi]$ and $\phi\in [0,2\pi)$. There's nothing complicated about this. However, as I began to study some differential geometry and manifold, I find this to be confusing.

Viewing $S^2$ as a $2$-dimensional manifold, we want to think of this parametrization as a coordinate chart $(U,\psi)$, where $\psi$ maps $(x,y,z)$ to the corresponding $(\theta,\phi)$ in $\Bbb R^2$. Then $\psi^{-1}$ would be our parametrization of the sphere.

However, the definition requires that $\psi$ is a diffeomorphism from the open set $U$ onto its image. Clearly $[0,\pi]\times [0,2\pi)$ is not open (this is related to the fact that there's no global chart for $S^2$). Yes, we may restrict to $U\subsetneq S^2$ but that is not my question.

In the language of manifold theory, what does it mean to talk about the global chart or the global parametrization of a sphere?

Even though I know that it's an abuse of languge, strictly speaking, but it appears a lot in the literature. For example, I am reading a book on Riemannian manifold and there's a part that says

We therefore introduce on $\Bbb R^d$ the standard polar coordinates $$(r,\varphi^1,\dots,\varphi^{d-1})$$ where $\varphi=(\varphi^1,\dots,\varphi^{d-1})$ parametrizes the unit sphere $S^{d-1}$. ... Express the (Riemannian) metric in polar coordinates ... in these coordinates at $0\in T_pM$, $0$ corresponds to $p\in M$, $$g_{rr}(0)=1,\ g_{r\varphi}(0)=0.$$

Why are we allowed to use regular rules of taking derivative and such in this "global coordinate"? How do we rigorously justify the use of this global coordinate system?

• You seem to be bothered by the phrase "parameteizes the unit sphere." But this is no more problematic than saying $x=\cos\theta$, $y=\sin\theta$ for $\theta\in[0,2\pi)$ "parametrizes the unit circle." Of course in manifold theory the circle in fact requires two coordinate charts and cannot be parametrized globally, because we must cut out a point for our charts to be defined on open sets. In practical applications (e.g. integration), this doesn't pose a problem; issues at sets of measure zero don't matter. Nov 15, 2017 at 15:21
• @symplectomorphic I decided to make my question more precise. If you wouldn't mind could you take a look at this? mathoverflow.net/questions/286182/… Nov 15, 2017 at 22:28

You should think of this parametrization as a ramified cover of the $2$-sphere $\mathbf{S}^2$ by $\mathbf{R}^2$. Namely, instead of restricting it to get a bijection from $[0,\pi] \times [0,2\pi)$ onto $\mathbf{S}^2$ (which requires a non-canonical choice of endpoints), consider it as a quotient map $$\mathbf{R}^2 \rightarrow \mathbf{S}^2$$ which a calculation of the Jacobian matrix shows is ramified precisely over the points $(0,0,\pm 1) \in \mathbf{S}^2$, when $\theta$ is an integer multiple of $\pi$. Away from these points, the map is a covering projection, and so you obtain charts by taking local sections. You can think of polar coordinates on $\mathbf{R}^n$ in the same way.
Given a metric $g$ on the codomain of a differentiable map between manifolds, you can pull it back to the domain to obtain a metric except on the fibres over the ramified points; this is the formalization of expressing $g$ in polar coordinates. Without knowing precisely what conclusions you wish to draw about $g$ itself it is difficult to say more about how to use this formalism.
• I believe one of the main aim of representing $g$ in polar coordinate in the book is to calculate the volume form and do some integration. So basically you're saying that everything works just fine except for the inverse images of the poles? Would that affect integration? (I'm guessing no since the points have 0 measure, but I'm not sure if there's any blow-up phenomenon that I should be careful of or not.) Nov 15, 2017 at 13:58


text:
# Is there a change of variables formula for integration using a generic measure?

$\int_{\Phi(\Omega)} u d\lambda^n = \int_{\Omega} u\circ \Phi |det J_\Phi| d\lambda^n$
where $\Omega\subset\Re^n$, $\lambda^n$ denotes the n-dimensional Legbesgue measure, and $J_\Phi$ denotes the Jacobian of $\Phi$. Basically, this formula is locked to the Lebesgue measure and I'd like to know if it's possible to replace that measure with a generic measure. Certainly, we could use Radon-Nikodym to generalize it to a few extra measures. I also believe that Federer has generalized this to something similar with the Hausdorff measure. Mostly, I'd like to know if there's something special about the Lebesgue measure that restricts the change of variables formula to this measure. On a similar note, is there an infinite dimensional analogue?


text:
# Accents not rendered correctly \scshape

As mentioned in this unanswered github issue: Wrong accents with scshape,
there is an issue with the rendering of accented letters (Aeéeèuùaà) for the case of the command \descript defined in the same github repository.

Those letters should be displayed as small uppercase, but they are small case.

The template uses XeLaTeX and Raleway font. (You can get it from the git itself)

Relevant code:

\documentclass[a4paper]{deedy-resume-openfont}

\begin{document}

\descript{Accents should be displayed properly: J'ai la tête égarée. Aeéeèuùaà}

\end{document}

% Intro Options
\ProvidesClass{deedy-resume-openfont}[2014/04/30 CV class]

\usepackage{fontspec}
\setmainfont[Color=primary, Path = fonts/lato/,BoldItalicFont=Lato-RegIta,BoldFont=Lato-Reg,ItalicFont=Lato-LigIta]{Lato-Lig}
\setsansfont[Scale=MatchLowercase,Mapping=tex-text, Path = fonts/raleway/]{Raleway-ExtraLight}

\newcommand{\descript}[1]{
\color{subheadings}\raggedright\scshape\fontspec[Path = fonts/raleway/]{Raleway-Medium}\fontsize{11pt}{13pt}\selectfont {#1 \\} \normalfont}

I tried to add French support, without luck:

\usepackage[francais]{babel}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}

• Welcome! Please, add a complete example from \documentclass to \end{document}. Mar 13 '17 at 7:23
• You could see at the link tex.stackexchange.com/questions/269436/…. Mar 13 '17 at 7:41
• @Sebastiano My question is indeed about the same template, but in this case it is not about making text bold. I would like to know how to change the \descript command which includes \scshape not to break accents. Thanks Mar 13 '17 at 7:50
• The Raleway font misses accented small caps. Change font. By the way, the code you show from the deedy-resume-openfont class is not what I'd call clean. Mar 13 '17 at 7:52
• and for any document class, if using xetex do not use \usepackage[T1]{fontenc} or \usepackage[utf8]{inputenc} (The first will force incorrect hyphenation and the second does nothing other than make a warning that it should not be used) Mar 13 '17 at 9:53

I made a few experiments with the Raleway font.

# First attempt

## Input

\documentclass{article}
\usepackage{fontspec}

\setmainfont{Raleway}[
Extension=.otf,
UprightFont=*-Medium,
ItalicFont=*-Medium-Italic,
BoldFont=*-Bold,
BoldItalicFont=*-Bold-Italic,
]

\begin{document}

This is text in Raleway

\textsc{Aeéeèuùaà}

\textsc{Ae\'ee\eu\ua\a}

\end{document}

## Output (with XeLaTeX or LuaLaTeX)

The Raleway font has no small caps glyphs for the accented characters

# Second attempt

\documentclass{article}
\usepackage{fontspec}

\setmainfont{Raleway}[
Extension=.otf,
UprightFont=*-Medium,
ItalicFont=*-Medium-Italic,
BoldFont=*-Bold,
BoldItalicFont=*-Bold-Italic,
]

\def\UndeclareTextComposite#1#2#3{%
\expandafter\let\csname\expandafter\string\csname #1\string#2\endcsname-\string#3\endcsname\relax
}
\UndeclareTextComposite{TU}{\'}{e}

\begin{document}

This is text in Raleway

\textsc{Aeéeèuùaà}

\textsc{Ae\'ee\eu\ua\a}

\end{document}

## Output with XeLaTeX

The same as before

## Output with LuaLaTeX

The mysterious code I added removes the predefined combination \'e, so that the default combining acute character should be used.

The HarfBuzz library used by XeLaTeX seizes the initiative and chooses the predefined combination anyway, so the result is not different. With LuaLaTeX this does not happen but the combining acute accent U+0301 doesn't exist in the font, so we get the message

Missing character: There is no ́ (U+0301) in font [Raleway-Medium.otf]:mode=node;script=latn;language=DFLT;+tlig;+smcp;!

and the “small caps e” does not receive an accent.

## Conclusion

The Raleway font is defective.

Besides, the code you show from the class is not really clean.

• Thank you for your explanation @egreg. I think I reached the limit of the template and I don't intend to focus deeper into it. I am not definitely not fluent in Latex but I like the system so I intend to keep on using it for obtaining professional documents. For this time it seems there is no proper solution. Mar 13 '17 at 10:18
• @RémiDebette When a font doesn't have the needed glyphs, there's not much to do except changing it. Mar 13 '17 at 10:35
• For what it is worth i reported it to the Raleway git Mar 13 '17 at 10:37


text:
Mathematics Department

# Student Seminar Talks

## Welcome to Spring term.

Our student seminars continue this spring term. We will meet on Thursdays in the Math Department - Bailey 207 during common hour unless otherwise noted. We will have lunch at 12:30pm in Bailey 204

Coordinated by Union College Professors Jeff Hatley and Roger Hoerl

## Join us for our final seminar for this term - Note time will be 12:45p-1:45p - Bailey-207

#### It's show time! We will be viewing Secrets of the Surface:

Examines the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, she was both the first woman and the first Iranian to be honored by mathematics' highest prize, the Fields Medal.

Mirzakhani's contributions are explained in the film by leading mathematicians and illustrated by animated sequences. Her mathematical colleagues from around the world, as well as former teachers, classmates, and students in Iran today, convey the deep impact of her achievements. The path of her education, success on Iran's Math Olympiad team, and her brilliant work, make Mirzakhani an ideal role model for girls looking toward careers in science and mathematics.

## Thursday, May 19th, 2022

12:30pm pizza & beverages in Bailey 204

12:55pm Seminar - in Bailey 207

#### Profs. Lucy Oremland and Csilla Szabo, Skidmore College

In mathematics, it is often necessary to build, visualize, and manipulate complex multi-dimensional structures. Instructors and students must become makers - through drawing, simulating, and crafting - in order to engage deeply with key concepts. In this talk we present a solids of revolution making activity for Calculus II (Integral Calculus). As part of the activity, students designed and 3D printed volumes of solids of revolution using our college’s maker space.

We will demonstrate the steps required to create a digital solid of revolution using free online tools, Desmos and Fusion360. If you would like to follow along with the demonstration, please bring a laptop and complete the steps below to get free access to Fusion360.

## What is Actuarial Science, and What Career Opportunities Does It Offer?

John W Robinson, President-Elect, Society of Actuaries

#### Our talk:

The Mathematics Department, Economics Department, Center for Data Analytics, and Becker Career Center are delighted to jointly sponsor a presentation by John W Robinson, recently elected as President-Elect of the Society of Actuaries (SOA). The SOA is the world’s largest actuarial professional organization, with over 32,000 members. In this two-part presentation, John will first explain what actuarial science is and what specific disciplines are involved. After a catered meal break, John will discuss his own career as an actuary, as well as the broader career opportunities that this discipline offers. Feel free to attend either or both parts, depending on interest. This session builds upon the recently approved new minor in Financial and Actuarial Mathematics, and will be organized as follows:

• 4:30 – 5:30pm Part I What is Actuarial Science?
• 5:30 – 6:00pm Catered meal break
• 6:00 – 7:00pm Part II Careers in Actuarial Science

## Thursday, April 21st 2022 - 12:55pm - Bailey 207

Richard Guy’s Strong Law of Small Numbers and How Not to Make Friends

Professor Hanson Smith - University of Connecticut

In this talk we will work through a number of examples from Richard Guy's wonderful paper The Strong Law of Small Numbers. The main theorem is, “You can't tell by looking.” Following Guy, we will prove this by intimidation. Audience participation, questioning, and guessing is highly encouraged. We'll conclude the talk with an exciting application to web comics.

2+1=3 is prime.

2⋅3+1=7 is prime.

2⋅3⋅5+1=31 is prime.

2⋅3⋅5⋅7+1=211 is prime.

2⋅3⋅5⋅7⋅11+1=2311 is prime.

Does it continue?

## Thursday, April 14th 2022 - 12:55pm - Bailey 207

Understanding and Addressing Complexity in Problem-Solving

Professor Roger Hoerl - Union College

Problem-solving is commonly applied in virtually all disciplines. The Hungarian mathematician George Polya wrote a classic text on problem-solving in 1945 that has sold over one million copies. One might argue that statistics as a discipline developed because of unsolved problems in other disciplines, such as conducting the US Census, public health, and agriculture. Unfortunately, much of the literature on problem-solving focuses narrowly on technical complexity and ignores other critical aspects of complexity that must be considered in real problems, such as organizational or political complexity.

In this talk, we will briefly discuss the development of the theory of problem-solving, beginning with Polya. We will then discuss broader aspects of problem-solving complexity that need to be considered for success, noting the critical but often overlooked impact of complexity in properly defining the problem. We will apply these principles to a couple of major problems facing our society today, and offer some thoughts on how to approach complex problems in general.

Pizza lunch at 12:30 in Bailey 204 for those students attending the seminar

## Thursday, April 7th 2022 - 12:55pm

Geometric Constructions in Number Theory

Professor Jeff Hatley - Union College

Thursday, April 7th, 12:55pm, BAIL-207

One of the most satisfying aspects of mathematics is finding bridges between seemingly-disparate fields. These sorts of connections abound between number theory and geometry -- in fact, there is an entire field, called "arithmetic geometry," which studies this sort of thing.

In this talk, we will explore one big (geometric) idea and the insight it provides for two very classical number theory problems. We will begin with a problem of interest to the ancient Greeks, and we will end on a modern, wide-open conjecture whose solution is worth $1 million. The prerequisites for this talk are high school algebra and an affinity for pictures. ## Thursday, November 18th, 2021 ## 12:30pm - 1:45pm pizza & beverages in Bailey 204 ## Study Break - Math Majors/Minors Meet and Greet ## In lieu of a talk this week, please stop by Bailey 204 this Thursday from 12:30-1:45 for a pizza study break. Math majors and minors (and other math students), please come and mingle with each other, and math faculty as well, as we all prepare for finals. ## Thursday, November 11th, 2021 12:30pm pizza & beverages in Bailey 204 12:55pm Seminar - in Bailey 207 ## Some Unusual Dice and their Connections to Voting ## Professor Davide Cervone, Union College We are all familiar with the property that, for numbers, if$a > b$and$b > c$, then$a > c$, and it is reasonable to expect that this property would hold in other situations where one thing "beats" another in some way. It turns out that this is not always the case, however. For example, in voting, it is possible to have candidate$A$be preferred to candidate$B$by a majority of voters, and$B$preferred to$C$by a majority, but also have$C$preferred to$A$by a majority of voters! This was described by the Marquis de Condorcet at the end of the 18th century, and is referred to as a Condorcet cycle, and it makes it hard to decide who should be the winner in an election that contains such a cycle. Similar cycles can be found in other situations as well. For example, it is possible to produce a set of three dice where the first beats the second more than half the time, the second beats the third more often than not, while the third beats the first more than 50% of the time. Such "non-transitive dice" have a close connection to the voting situation, and understanding one can help understand the other. In this talk, we illustrate the connection between voting and non-transitive dice, explore how they work with longer cycles, and generalize these cycles to ones involving the comparison of more than two dice (or candidates) at a time. ## Friday, November 5th 4:00pm pizza & beverages in Bailey 204 4:30pm Seminar - in Bailey 207 ## Solving probability using differential (in)equations ## Professor Joe P. Chen, Colgate University In calculus or analysis we learned the notion of convergence,$\lim_{n\to\infty} x_n =x$. It would be even better if we can quantify the rate of convergence, namely, how fast does$|x_n-x|$decay to$0$as$n \to \infty$? Exponential? Or as a power law? Now translate these questions in the setting of your favorite probabilistic model (random walks, Langevin diffusion, etc. Or you can use last week's talk as an example). All you know is that the model converges to a stationary distribution as time$n\to\infty$. But how fast does it converge? And since the model might be idealized, can you come up with a real-world algorithm that approximately samples this stationary distribution? And how good is the approximation? One answer to this question can be obtained through the analysis of a first-order-in-time differential inequality called the "entropy inequality." After sketching the basic properties of relative entropy, I will explain the structure of the entropy inequality, and illustrate its use in recent research in math and computer science. ## Thursday, October 28th, 2021 12:30pm pizza & beverages in Bailey 204 12:55pm Seminar - in Bailey 207 ## Solving Differential Equations with Probability ## Professor Phanuel De Andrade, Union College In this talk I will discuss how one can solve second order differential equations using the notion of Brownian motion, which is a mathematical model for the random movement of a particle. It was first observed by Robert Brown in 1827 while looking at pollen grains through a microscope. Since then, Brownian motion has been used in many applications to physics, finance and other branches of math. In 1944, Shizuo Kakutani was the first to use a probabilistic approach to solve a differential equation with a boundary condition. We will use this approach to solve a simple boundary value problem that one could encounter in an elementary differential equations class. Thursday, October 21st, 2021 12:30pm pizza & beverages in Bailey 204 12:55pm Seminar - in Bailey 207 The Joy of Abstraction, presented by professor emeritus Kimmo Rosenthal, Union College The imagination is the only genius. It is intrepid and eager and the extreme of its achievement lies in abstraction.” Wallace Stevens. It may seem incongruous for the epigraph to a mathematics talk to be from one of the great American poets. However, while the ubiquity and utility of mathematics is widely acknowledged, its aesthetic appeal is much less so. In this day and age when relevance, applicability, and connections with other disciplines are touted as paramount, is there still a place for purely abstract mathematics viewed more as an intellectual art form? Can the old dictum “art for art’s sake” be replaced by “math for math’s sake”? Abstraction has always appealed to me and indeed guided me. Yet, in some quarters it can provoke outright hostility. We shall take a brief historical tour of mathematical abstraction from the set theory of Georg Cantor (called a “corrupter of youth”) to the mysterious emergence of Nicolas Bourbaki (the famous mathematician who never existed), and finally category theory, which earned the epithet of “abstract nonsense”. There will be a gentle introduction to category theory accessible to anyone taking Math 199, including consideration of the question: can we talk about sets without mentioning elements? ## Thursday, October 14th, 2021 12:30pm pizza & beverages in Bailey 204 12:55pm Seminar - in Bailey 207 ### Connecting cluster algebras to continued fractions and snake graphs Professor Michelle Rabideau - University of Hartford In many areas of mathematics, it is often beneficial to represent objects graphically or numerically. In this talk, we study the class of commutative rings called cluster algebras and their elements called cluster variables. We will focus on one particular characteristic of a cluster variable and study it by considering graphs of square tiles called snake graphs and using numerical tools called continued fractions. The talk will emphasize how mathematicians often change strategies when the current method becomes too abstract, too tedious or too time consuming to be reasonable. ## Thursday, October 7th, 2021 12:30pm pizza & beverages in Bailey 204 12:55pm Seminar - in Bailey 207 ### Ultrametrics, strange geometries, and p-adic numbers Professor Jeffrey Hatley - Union College Different settings call for different ways of measuring distance. I can measure my daily commute by the length of road I travel, or I could measure the distance between my home and Union "as the crow flies." In the city, we might describe distances in terms of blocks -- routes constructed from straight-line paths and right-angle turns. You can even measure distance in more abstract, non-physical settings, such as the distance between you and your second cousin. In this light-hearted talk we'll discuss various notions of distance and explore some of the surprising consequences for "basic" geometric notions, such as what circles and triangles look like. We'll end by explaining why the usual geometry that everyone learns in high school is the strangest of them all. Pre-requisites: If you know what a circle is, this talk is for you! ## Thursday, September 30th 12:30pm pizza & beverages in Bailey 204 12:55pm Seminar - in Bailey 207 ### Come to hear about student research! Josh DaRosa (ME '22), Zekai Hu (MT '22), and Zhebin Yin (MTCS '22) Interested in learning about students' research over the summer / academic year? Josh will present data analysis within PIV fluid dynamics research and PIV measurements of heterogeneous canopies. Zekai (Kai) will present predicting one year survival rate for heart failure patients and COVID-19 patients with heart disease. Zhebin (Irene) will present classification of lung cancer circulating tumor cells in liquid biopsy using deep learning. ## Thursday, September 23rd 12:30pm pizza & beverages in Bailey 204 12:55pm Seminar - Note: due to construction, will be in Bailey 201 ### How To Cut A Cake, presented by Union College professor emeritus Julius Barbanel Suppose that you and I are given a cake that we wish to divide between the two of us. We may value pieces of cake differently. For example, maybe I like the chocolate filling best, but you like the icing best, so a piece of cake that I value as half the cake, you might value as a third of the cake. What would constitute a “good” division of the cake? We shall consider two sorts of criteria, namely “fairness” and “efficiency”. Fairness involves such questions as “Did I get at least half?” or “Did I get at least as much as you?” Efficiency involves questions such as “Is this the best way to divide the cake or is there a way to divide the cake that would make each of us happier?” We will see that there are interesting mathematical ideas and revealing geometric pictures associated with these issues. We will also consider what happens when we wish to divide the cake among more than two people. Greg Malen, Union College, "High-Dimensional Holeyominoes and Hyperbolic Polyforms" A polyomino is a finite rook-connected'' subset of squares on an infinite chessboard, which one can imagine is constructed by taking individual square tiles and gluing them together along their edges. In this talk, I will be considering extremal properties of these structures with respect to the requirement that a fixed number of squares is arranged such that their edges bound a maximal number of holes. These `holeyominoes'' tend towards beautiful symmetries and efficient geometric properties. Building on earlier results with squares in 2-dimensional Euclidean space, I will present new asymptotic bounds for the maximum number of holes which can be bounded by$nd$-dimensional cubes in$d$-dimensional Euclidean space. Furthermore, we will look at on-going efforts to find similar results in the hyperbolic plane for subsets of a$\{p,q\}$-tiling, where$q$regular$p$-gons meet at every vertex. Date/time: Thursday, May 27th at 1:20pm-2:15pm Location: Zoom ID: https://union.zoom.us/j/92591304428 Mary Annese ACAS, CPCU, MAAA, to present “An Introduction to the Actuarial Profession” An introduction to a promising career in what has been touted as one of the nation’s top ranked professions because of its employability, career growth, high-paying salaries, and work-life balance. Date/time: Thursday, May 6th at 1:20pm-2:15pm Location: Zoom ID: https://union.zoom.us/j/92591304428 What you will learn: • What an Actuary is • What Actuaries do • How to become an Actuary • What the Casualty Actuarial Society is • Why this is an attractive career • Your next steps Who should attend? • Undergraduate students majoring in actuarial science, math, statistics, business, or other related fields • Undecided undergraduate students with an interest in math, statistics, business, or other related fields. • Go to: CASstudentcentral.org and register to become a student member. It’s free and provides you access to a wealth of tips, tools and resources. Emily Hoopes-Boyd from Kent State University Date/time: Thursday, April 15th at 1:20pm-2:15pm Location: Zoom ID: https://union.zoom.us/j/92591304428 Title: The Images of Polynomials Evaluated over Matrices Abstract: A version of the L’vov-Kaplansky conjecture states that the image of a multilinear polynomial evaluated over matrices, with entries from the complex numbers, is a vector space. This statement has been proven only for the case of 2×2 matrices, but many partial results have been proven within the last decade. We will consider this problem in a slightly different context; rather than taking the matrix entries to be from the complex numbers, we will consider matrices over an algebraically closed skew field, which we will denote by$K$. We will show that the image of any multilinear polynomial with coefficients from$K$, evaluated over$M_m(K)$, is$M_m(K)$. We will also prove that any matrix in$M_m(K)$may be written as the sum of three or fewer elements from the image of any generalized polynomial. In particular, the image of the polynomial$xy-yx$has some special properties over a variety of matrix rings, including the ring of matrices over the complex numbers. Wednesday, January 29th 12:30pm pizza & beverages in Bailey 204 12:55pm Seminar will be in Bailey 207 Please see flyers posted in Bailey Hall for title and abstract Friday, January 31st 4:00pm light refreshments in Bailey 204 4:15pm Seminar will be in Bailey 207 Please see flyers posted in Bailey Hall for title and abstract Thursday, Jan 23rd Making Our Mark, Women in Mathematics, at Union and Beyond In conjunction with our celebration of the 50th anniversary of co-education at Union College, we invite you all to a special celebration of women in mathematics in our upcoming math seminar. We will hear from Professor Susan Niefield about her experience as a mathematician and as the first woman who joined the math department faculty at Union. We will then briefly introduce some professional organizations that focus on making the mathematical community more inclusive. In particular, members of The Association for Women in Mathematics Chapter at Union will share their stories and talk about their events and resources. Finally, we will watch a short film "Journeys of Women in Mathematics", that shows the difficulties and triumphs of three women mathematicians from three different continents: Neela Nataraj from India, Aminatou Pecha from Cameroon, and Carolina Araujo from Brazil. Thursday, January 16th - Union College welcomes: Mary Annese ACAS, CPCU, MAAA, to present “An Introduction to the Actuarial Profession An introduction to a promising career in what has been touted as one of the nation’s top ranked professions because of its employability, career growth, high-paying salaries, and work-life balance. What you will learn: • What an Actuary is • What Actuaries do • How to become an Actuary • What the Casualty Actuarial Society is • Why this is an attractive career • Your next steps Who should attend? • Undergraduate students majoring in actuarial science, math, statistics, business, or other related fields • Undecided undergraduate students with an interest in math, statistics, business, or other related fields. Go to: CASstudentcentral.org and register to become a student member. It’s free and provides you access to a wealth of tips, tools and resources. ## Past Seminars • Fall 2019 Injectivity of polynomial maps and multistationarity in reaction networks Professor Casian Pantea, West Virginia University Friday, Nov 08th - 1:00pm – BAIL 207 The capacity of biochemical reaction networks to operate at different steady states is crucial in important biological processes like cell division, differentiation, or apoptosis. In this talk we attack the question “when can a certain reaction network admit two or more positive steady states?”, i.e., when can the network be multistationary? This boils down to a difficult question about solutions of some high-dimensional polynomial systems, further complicated by the lack of information on coefficient values. However, it turns out that a lot can be said on multistationarity of reaction networks by studying the injectivity of the corresponding polynomial maps. We will survey some classical and some new results on the topic, and illustrate them using relevant biological examples. The Strange New Universe of Hyperbolic Geometry Professor Ellen Gasparovic, Union College Friday, Nov 1st - 1:00pm – BAIL 201 The fifth postulate in Euclid's Elements states that, in a plane, given a line l and a point P not on l, there is exactly one line through P that is parallel to l. Although this is an axiom in Euclidean geometry, what happens when you don't assume that the so-called "parallel postulate" holds? The answer is that beautiful non-Euclidean geometries emerge, such as that of the hyperbolic plane (what mathematician Janos Bolyai referred to as a "strange new universe"). In this talk, we will learn about what it means for lines to be "parallel" or "ultra-parallel" in this strange new setting, with an eye toward classifying all distance-preserving transformations using the Klein disk model for the hyperbolic plane. The Congruent Number Problem Professor Jeff Hatley, Union College Thursday, Oct 24th - 1:00pm – BAIL 207 The Congruent Number Problem asks the following simple-sounding question: which rational numbers occur as the area of a right triangle with sides of rational length? For example, the familiar (3,4,5) right triangle has area 6, so 6 is a congruent number; but Fermat showed in the year 1640 that 1 is not a congruent number. Our investigation of this question will lead to a surprising and beautiful interaction between algebra and geometry, bringing us to the forefront of modern number theory and a math problem with a$1 Million prize.

Joint Math-CS Student Seminar

DeepFake the Menace?

Dr. Siwei Lyu

University at Albany

Thursday, Oct 17th - 1:00pm – VART 204

The advancements of AI technology, in particular, deep generative models, have enabled the creation of fake images, audios and videos in ways that have not been possible before. Such fake videos, commonly known as the DeepFakes, are eroding our trust to digital media and causing serious ethical, legal, social, and financial consequences. In this talk, I will briefly review the technologies behind the creation of DeepFakes, and then introduce current detection methods of such fake videos and measures that can obstruct the generation of DeepFakes, as well as general technical aspects to combat DeepFakes.

The Interface of Science, Engineering, and Statistics

Professor Roger Hoerl - Union College

Thursday, Oct 10th, 1:00pm, BAIL-207

In essence, science expands our understanding of the fundamental workings of the natural world. Engineering, on the other hand, takes our knowledge of the natural world and attempts to apply it in some way that benefits society. Real engineering applications typically venture beyond the boundaries of our scientific knowledge, resulting in uncertainty, and with some degree of "trial and error" required. Google's experiences with self-driving cars would be an obvious example. At its best, statistics accelerates science and engineering by efficiently filling in the gaps in our scientific theory using empirical data analysis. This seminar will illustrate these concepts using a case study from the author's previous experience as a statistical intern at the DuPont Company.

Connecting STE with M

Professor Robert Rogers - SUNY Fredonia - 2019-2020 "MAA Seaway Section Distinguished Lecturer"

Thursday, Oct 3rd, 1:00pm, VART-204

Theoretic chemistry, internet security, kidney stone treatment, airfoil design; what connects these seemingly diverse science, technology, and engineering topics? This talk will explore the mathematics behind these topics. Furthermore, the mathematical topics required to understand these applications only involve drawing, counting, numbers, some geometry, and some imagination.

Thursday, September 26th, 1:00pm, Bailey 207

Featuring: Student talks

Meichai Chen (Class of 2020), Herschel Norwitz (Class 2021), and

Mushan Zhong (Class of 2020)

**************************************

Meichai Chen ('20)

Weak Orders and Utility Functions
A weak order is a complete and transitive order on a set X, whose elements we refer to as preferences. In this talk, we will explore the connections between weak orders and utility functions. We will then briefly discuss how weak orders and utility functions arise in expected utility theory and prospect theory, and conclude with a short description of how these concepts are used in a study regarding stereotype threat.

Herschel Norwitz ('21)

Distinguishing between Forced and Natural oscillations on the Power Grid

Oscillations are always occurring on the grid. The two forms of oscillations that occur are natural and forced. Observing natural oscillations can tell the health of the system, unfortunately forced oscillations occur that don't effect the health of the system but can make it appear as though the system is unhealthy. In this research a dynamometer and a DC motor were used to inject a forced oscillations into the outlet. Then using Matlab and a python program measurements are taken from an outlet so see if it is not only possible to observe the oscillations but distinguish them.

Mushan Zhong ('20)
Use of Nonlinear Models in Analyzing Experiments with Both Mixture and Process Variables
When conducting statistics experiments; In most cases we can experiment with different combinations of variables without restrictions. While there are some other cases in which the experimental variables are ingredients, and must sum to 100%. These are called mixture variables. Our research focused on how to approach problems with both process and also mixture variables. We tested different models to see how well they could fit a given set of data, as well as how well they predicted new data. The objective was to develop models that can be applied to smaller data sets, which would allow researchers to run smaller, cheaper, and faster experiments.

Featuring: Student summer research

Dan Resnick (class 2021)

Sam Kemp (class 2021)

Friday, September 20th, 1:00pm, Bailey 207

Dan Resnick (class 2021)

Stellar Wind Collisions: Thin Shell Geometry
I will be talking about the theory behind the stellar wind problem and the assumptions/setup to where we started working. I will discuss the methods we used and where we stopped on the problem, and where we plan to go forward with the project.

Sam Kemp (class 2021)
Simulation of Forces Between Inclusions on a Lipid Bilayer
A program was written in C++ to simulate the forces between two inclusions on a lipid bilayer using the finite element method. This was first done using Dirichlet boundary conditions and then with mixed boundary conditions at the edge of the inclusions.

• Spring 2019

The Historical Roots of Gödel’s Theorems

Professor Andrea Pedeferri - Philosophy Department @ Union College

Tuesday, May 28th, 1:00pm, Bailey 207

At the beginning of “On formally undecidable propositions of Principia Mathematica and related systems” Gödel writes: “The development of mathematics toward greater precision has led [...] to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules.” In just few sentences Gödel summarizes a century of key developments in mathematics that made that period one of the most exciting and optimistic for the discipline. As Hilbert wrote in 1925 and in 1930 “we are all convinced that [...] in mathematics there is no ignorabimus”, “We must know. We will know”.

Then comes 1931. Gödel writes: “One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question [...]. It will be shown below that this is not the case [...]. The precise analysis of this curious situation leads to surprising results concerning consistency proofs for formal systems.” The impact of Gödel’s results was immense.

In my talk I will follow the main developments of mathematics highlighted by Gödel in order to show how they are crucial to understand the impact and the reach of Gödel’s theorems and to fully appreciate their dramatic but revolutionary nature.

Rationalizable Voting Rules and the S-Correspondence

Professor Ashley Piggins, Department of Economics, National University of Ireland, Galway (joint work with Conal Duddy, Department of Economics, University College, Cork)

Tuesday, May 21st, 4:00pm, Bailey 207

Perhaps the most important feature in Kenneth Arrow’s famous “Impossibility Theorem” (which helped him earn a Nobel prize in Economics) is IIA – the Independence of Irrelevant Alternatives condition on voting rules, aka social choice. We introduce a related but new social choice property, the S-independence condition. We characterize the “S-correspondences” – those social choice rules that satisfy S-independence along with three more standard axioms: strong Pareto optimality, neutrality and anonymity. This class is closely related to the S-rules of Bossert and Suzumura (2008a, J. Econ. Theory 138, 311-320). S-independence can be justified by a new rationalizability argument that shows it is equivalent to assuming IIA in a somewhat different framework.

The History of Fermat's Last Theorem

Professor Ravi Ramakrishna, Cornell University

Thursday, May 16th, 1:00pm, Bailey 207

In 1994, Wiles, assisted by Taylor, finally settled the 340-year-old question of Fermat's Last Theorem. In this talk I will give some of the history of this problem, with particular focus on events of 1847 and the 1980-90s. I'll then talk a little about developments off the last 25 years in the subject.

This talk is NOT aimed at faculty in number theory. It is intended for undergrads that like math, but have not taken many upper level courses.

Mathematical Modeling in Computational Neuroscience

Paulina Volosov, Rensselaer Polytechnic Institute

Thursday, May 2nd, 1:00pm, Bailey 207, with lunch at 12:30pm in Bailey 204

Computational neuroscience is a field of active research for applied mathematicians. The brain has been actively studied since the second half of the 19th century, but there are still an endless number of unanswered questions about how and why the brain works the way it does.

This presentation will give a basic introduction to some of the questions asked in neuroscience and show how a mathematical model can be built and used to shed light on this complicated biological system. Namely, how can a mathematician attempt to learn about the structure of the brain and networks of neurons? What is it that a mathematical model tries to accomplish? And why is this useful?

Who Discovered Integral Calculus?

Professor Emeritus Julius Barbanel, Union College

Thursday, April 25th, 1:00pm, Bailey 207, with lunch at 12:30pm in Bailey 204

Isaac Newton and Gottfried Leibniz (both working in the late 17th and early 18th centuries CE) are generally considered to be the inventors of calculus. We will argue in this talk that part of the credit for this discovery should be given to the ancient Greek mathematician Archimedes (who worked in the 3rd century BCE). Archimedes pursued two lines of research involving areas and volumes. These approaches, known as the Method of Exhaustion and the Mechanical Method, can be viewed as early examples of ideas that we think of as being part of integral calculus. We will focus on Archimedes’ Method of Exhaustion.

Presenting Mathematics - an open discussion

Professor Leila Khatami and Professor Jue Wang, Union College

Thursday, April 18th, 1:00pm, Bailey 207, with lunch at 12:30pm in Bailey 204

As Steinmetz symposium approaches, many students are contemplating their presentations and are thinking about ways to make an engaging presentations to effectively share their work with fellow students, family members, and professors. Even those who are not presenting this year, will most probably present their work in the future. In this seminar, we will talk about some of the characteristics of a good mathematics presentation, as well as some of common missteps to avoid. As part of the discussion, we will also watch and analyze a short sample presentation.

Math, ECBE, Physics and Astronomy joint seminar

Thursday April 11, 2019, during common hour in Karp 005

Towards Cyber-Physical Electrical Power Systems: where the laws of nature and the rules of algorithms collide!

Luigi Vanfretti
Electrical, Computer, and Systems Engineering, RPI

Electrical power networks are undergoing unprecedented changes. On one hand, the adoption of distributed energy resources (DER) and renewable energy sources (RES), both of which have a large degree of variability in small time-scales, puts challenges to the traditional, historical-and-experience-based design and operation of electrical power networks. On the other hand, digitization and automation, opens opportunities for a more carbon neutral electrical energy system by helping to harmonize these new energy sources with the rest of the power grid, not without also bringing along the potential threats of the cyber world. This talk aims to give an overview of these challenges, and to present different research efforts conducted by the presenter to address how to transform today’s electrical grid into a cyber-physical power system. This includes the development of an experimental facility to conduct, real-time hardware-in-the-loop simulation experiments of power networks with “cyber” assets. This approach allows to characterize how the interaction of systems governed by the laws of nature will interact with engineered systems governed by rules of algorithms. Finally, with the rise of electrification in transport, and in particular aircraft, and the rise of more autonomous machines, the talk will also discuss the need for development of a new course on modeling and simulation for cyber-physical systems (CPS) and the teaching approach adopted which brings a “digital” toolbox and know-how to the next generation of electrical engineers that will have to increasingly deal with complex CPS.

• Winter 2019

### The Spectacular Spectral Theory

Thursday, March 7th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Do you know the reason for the collapse of the Tacoma Narrows Bridge in 1940? Have you ever wondered about the mathematical idea behind Google’s page ranking and Netflix’s movie recommendation? Do you know why scientists believe that distant stars are largely composed of hydrogen?

This talk is an invitation to the spectacular spectral theory, which is the key to answering all of these questions.

### Using linear algebra to understand knots

Cynthia Curtis, Professor of Mathematics at The College of New Jersey – Union College Class of 1987

Thursday, February 14th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Knots are prevalent in nature, and the study of knotting is important in diverse areas such as DNA, bonding of molecules, and statistical mechanics. Understanding knots has been fundamental within mathematics to our ability to understand three-dimensional spaces. In this talk we use linear algebra to generate polynomials, which help decide whether two given knots are different. This is a surprisingly hard question! The polynomials can also help us know when to look for hidden symmetries in the knots. The first knot polynomial we introduce was found by James Waddell Alexander II in 1923. We then discuss new polynomials arising from research with undergraduates Vincent Longo, Alyssa Springstead, and Hoang Cao at The College of New Jersey.

### Generalizing composition of functions and Operads

Peter Bonventre, Union College Class of 2011

Friday, February 8th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Given two single-variable functions, we are allowed to take their composite to produce a new function again of a single variable. In this talk, we will ask: “In what other contexts does ‘composition of functions’ make sense?”. We will slowly broaden our definitions of “function” and “composition”, starting with the types of functions that appear in the Calculus sequence, and moving to include well-behaved geometric figures. This will lead us to the abstract concept of an operad. We will give several examples, as well as an interpretation of what these new objects can do for us.

### Kimmo Rosenthal, Union College

Thursday, January 31st, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

The imagination is the only genius. It is intrepid and eager and the extreme of its achievement lies in abstraction Wallace Stevens.

It may seem incongruous for the epigraph to a math talk to be from one of the great American poets. However, while the ubiquity and utility of mathematics is widely acknowledged, its burnish of aestheticism is much less so. Can the old dictum “art for art’s sake” be replaced by “math for math’s sake”? In this day and age, when relevance, applicability, and connections with other disciplines are touted as paramount, is there still a place for purely abstract mathematics viewed more as an intellectual art form? Abstraction has always appealed to me and indeed guided me. Why does it often provoke outright hostility? We shall follow the path of abstraction from the set theory of Cantor (called a “corrupter of youth”) to point-set topology, followed by the mysterious emergence of Bourbaki (the mathematician who never existed), and finally category theory, which earned the epithet of “abstract nonsense”. Of course, there will be some mathematics along the way, reasonably modest in scope.

### Alan Taylor, Union College

Thursday, January 24th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Suppose we have a collection of women and a collection of men, and each woman finds some of the men acceptable (and the rest not). When is it possible to match each woman with a man she considers acceptable, subject to the obvious constraint that the matching be one to one? The answer to this metaphorical question is a beautiful result in finite combinatorics known as Hall’s marriage theorem. We will discuss Hall’s theorem, sketch a proof of it, and consider a couple of natural questions it suggests, all with the hope of providing an illustration of how research gets done in mathematics.

### Summer Opportunities for Math Students

Thursday, January 17th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

This week’s seminar will focus on ways in which you can put your mathematical skills to use over the summer.

Julia Greene ’19 will speak about the Teaching Experiences for Undergraduates Program, Professor Jeff Hatley will speak about Research Experiences for Undergraduates, and Keri Willis of the Becker Career Center will speak about summer internships.

• Fall 2018

### Turning the lights out, mathematician style

Leila Khatami, Union College

Thursday, November 1st, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

“LIGHTS OUT!” is a single player game played on a 5 by 5 grid where each cell has a button that can be turned on or off. Pressing a button toggles the light in the cell and its neighboring cells. The game starts with some cells turned on and some turned off. The goal of the game is to turn all cells off. The game was originally introduced in 1995 as a handheld electronic came. Nowadays, the original game, as well as many of its variants, are readily accessible in app stores and elsewhere. It is not obvious (or even true!) that all starting configurations of the game are “solvable”. In this talk, we use mathematical tools to see if a game is “solvable”. We also briefly discuss ways to find the most efficient solutions for solvable games.

### Solving the General Cubic Equation

ax3ax3+bx2bx2+cx+d=0cx+d=0

### Paul Friedman, Union College

Thursday, October 25th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

The solution to the general quadratic equation,

ax2+bx+c=0ax2+bx+c=0,

is well-known to most high school students:

It was also known to many “ancient” cultures … some dating back to 2000 BC! However, the solution to the general cubic equation, ax3+bx2+cx+d=0ax3+bx2+cx+d=0, is not as well known, and it was not found until the 1500s.

In this talk, we will look at how the Renaissance mathematicians Scipione del Ferro, Tartaglia, and Cardano, solved the cubic equation, though we will do so using modern language and notation. As a cute consequence, we will be able to derive some remarkable identities, such as:

### William Zwicker, Union College

Thursday, Oct 18th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Countries A and B are dividing up a disputed island with several cities, linked by roads:

• Each city must go entirely to A, or entirely to B
• A city may be worth more to one country than to the other
• You must be able to drive among A’s cities without going through B’s

Can cities be allocated in a way that leaves neither country jealous of the other’s share?

No – not in general. But with certain road networks one can always get within one city of this ideal. Which networks are these? With more than 2 countries, the question gets harder . . . in interesting ways.

In its “classical” setting Fair Division concerns sharing a single, continuously divisible resource. Some solutions can be adapted to this new setting, but a lot needs to change.

### Volker Remmert, , of the Bergische Universitat Wuppertal

Tuesday, Oct 9th, 4:45pm, Bailey 207, with reception at 4:15pm in Bailey 204

Over the last twenty years or so there has been a steady flow of historical studies on remigration into Germany in the immediate post-war period. These studies have described three main forms of academic remigration to Germany after World War II:

1) returning to universities in Germany on a permanent basis as university professors;

2) returning as visiting professors, assessing Germany without any obligation to stay;

3) returning for guest lectures and academic visits.

In this context my interest is in Jewish émigré mathematicians and their stance to Germany in the immediate post-war period.

### Danielle Gregg ’19 and Robert Righi ’19, Union College Undergraduates

Thursday, Oct 4th, 1:00pm, Bailey 207

Much recent research has focused on discerning topological and geometric features of data. For example, by observing the “birth” and “death” of holes via an algebraic method known as persistent homology, we can distinguish noise from significant features in data. In analyzing the “shape” of data our research diverges into two separate fields: music and health science. How can one use geometric and topological methods to classify a variety of degenerative diseases of the eye or compare songs within an artist’s discography? Come learn about what two Union students researched over the past summer as well as the often non-linear research process.

### Julie Bergner, University of Virginia and Cornell University

Thursday, Sept 27th, 1:00pm, Bailey 207

The Catalan numbers are given by a recursively-defined sequence and arise from over 200 different kinds of combinatorial objects. In 2013, two of my undergraduate research students, Gerardo Alvarez and Ruben Lopez, showed that a family of directed graphs called action graphs gives a new way to obtain this sequence. Since these graphs are defined inductively, one might ask what sequences we can get by using a different initial graph but the same induction process. Last year, three more students, Cedric Harper, Ryan Keller, and Mathilde Rosi-Marshall, looked into this question. They found new families, called kk-initial action graphs, which produce self-convolutions of the Catalan sequence. In this talk we’ll introduce the sequences and graphs involved and talk about how these comparisons were made.

### Jonathan Campbell, Vanderbilt University

Friday, Sept 21st, 1:00pm, Bailey 207

Give two polyhedra of equal volume, can you cut up one into a finite number of pieces, and reassemble it into the other? This was a problem posed by Hilbert in a famous address. I’ll go through the two dimensional analogue of this problem, and present Dehn’s beautiful solution to Hilbert’s question. Time permitting, I’ll give some hint of how this easily stated problem shows up in my own research.

### Professor Brenda Johnson, Union College

Thursday, Sept 13th, 1:00pm, Bailey 207

Sudoku is a popular puzzle involving a 9×9 grid in which one has to arrange the numbers 1 through 9 so that each row, column, and block contains all nine numbers. There are many interesting mathematical questions involving sudoku puzzles. In this talk, we’ll focus on a couple of questions related to counting sudokus. After discussing how many possible solutions there are for 9×9 and 4×4 sudokus, we’ll look at ways in which one can generate new sudokus from old ones, and whether or not these techniques can be used to generate all sudokus of a given size.


text:
## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$690,000$
To convert the given number to decimal notation, move the decimal point 5 places to the right to obtain: $6.9 \times 10^5=690,000$


text:
# What is the covariance matrix analogue of testing for variance > 0?

I have a mathematical context--a computer program I am writing--in which, if I am dealing with scalars, I have a variance, and at a certain point, I (or rather the program) take its reciprocal, i.e. the variance's multiplicative inverse.

When instead I am dealing with vectors, I use a covariance matrix, and instead of the reciprocal, I take the inverse matrix--again, the (matrix-)multiplicative inverse.

The variance, or the covariance matrix changes over time as I run the same procedures repeatedly.

The procedure can result in a "variance" that's negative, which of course makes no sense. It can also result in a variance that's zero, which might be OK, except that then the program will divide by zero. This is not because of a bug in the code. It's just that the code wasn't intended to be used under certain conditions, but what those conditions are, isn't clear. (Except that it's clear that if the code generates a "variance" $< 0$, those weren't the right conditions!)

One way to avoid "variances" $\leq 0$ would be to force the procedure to return a value just above 0 when it would have otherwise returned a value $\leq 0$. Something like: if sigma <= 0, then assign 0.000001 to sigma. This is easy to implement.

My question: What would the covariance matrix analogue (or analogues) of the test for variance $\leq 0$ be, and why? I have a hypothesis that it's a test for whether the covariance matrix's determinant is greater than 0, but I'm not quite sure why that would make sense.

(I'm not sure whether I have given enough information to make the question answerable. I'm not sure whether I can give enough information, if not, so I hope that I have.)

## 1 Answer

A covariance matrix should be at least positive semidefinite and typically positive definite.

If the matrix is positive semidefinite than some of your individual RV's are perfectly correlated, which can cause lots of complications. Generally you will want to avoid these and insist on a positive definite covariance matrix.

In case you're not familiar with the definition, a symmetric matrix $A$ is positive semidefinite if $x^{T}Ax \geq 0$ for all $x$ and positive definite if $x^{T}Ax > 0$ for all non-zero $x$. In terms of the eigenvalues of $A$, $A$ is positive semidefinite if the eigenvalues are all greater than or equal to 0 and positive definite if the eigenvalues are strictly greater than 0.

• Thanks--that sounds like it might be just what I need. I'll investigate. – Mars Sep 13 '16 at 3:58