text:
Introduction

The Earth is experiencing a sixth mass extinction event, with current rates of species loss far exceeding the background extinction rate1. More than 25% of known species remaining in the wild are classified as critically endangered, endangered, or vulnerable2. These declines are largely driven by human activities, including destruction or alteration of habitat, overharvest, and the spread of harmful non-native species3.

A strategic plan adopted by parties to the Convention on Biological Diversity (CBD) set global targets for preventing extinction of known imperiled species (Target 12) and expanding protection and effective management of sites with biodiversity conservation significance (Target 11)4. The total annual cost of meeting these targets is estimated at U.S. $76.1 billion5, of which only 12% is currently funded. In the United States (U.S.), the Endangered Species Act (ESA) of 1973 aims to protect threatened and endangered (hereafter ‘imperiled’) species, along with the ecosystems on which they depend. The ESA provides for designation of critical habitat, species management programs, and penalties for knowingly harming or trafficking imperiled species. In 2014, the U.S. Fish and Wildlife Service (USFWS) and other agencies spent over U.S.$1.4 billion on recovery efforts for 1,600 currently listed species6.

Alien invasive species constitute a major environmental stressor7,8 and are considered to be a driving factor in the decline of 3,862 species (16.6%) globally listed as extinct or imperiled2. In the U.S., non-native species are considered the primary cause of decline in 42% of imperiled species via habitat alteration, competitive displacement, predation, spread of pathogens, and hybridization9.

Wild pigs (Sus scrofa), also known as feral hogs, feral swine, or wild boar, are recognized as one of the most widespread and destructive invasive species in the world10,11,12. Wild pigs are native to Eurasia and Northern Africa, but have been widely introduced for centuries, often deliberately, and now occupy every continent except Antarctica13,14. They are extreme generalists in habitat and diet10, reproduce prolifically15, transmit a wide array of parasites and diseases16,17,18,19 and are considered ecosystem engineers20,21,22 due to their habit of rooting soils, resulting in high capacity to negatively impact a wide variety of native species (Table 1).

Wild pigs were introduced to what are now the southeast United States and California by Spanish explorers and settlers in the 16th and 18th centuries, respectively23. They are now the most abundant free-ranging, non-native ungulate in North America24. Recent work predicting the relative probability of wild pig occurrence across the contiguous U.S. based on their physiology and ecology25 suggests that many areas of the country not yet colonized by wild pigs have a high probability of meeting habitat requirements and supporting future population establishment of wild pigs. Furthermore, regions currently occupied by wild pigs and those with the highest invasion potential25,26 are among the most biodiverse in the contiguous U.S. (Fig. 1). The U.S. Department of Agriculture (USDA) invests $20 million annually to manage wild pig damage to agricultural and natural resources, property, animal health, and human health and safety27. Annual costs of controlling invasive species occupying habitats of U.S. endangered species have been estimated to total$29–38 million (1997 $U.S.), and without this continued level of support, 60% of these species could be subject to declines or extinction, even if their habitats are nominally protected28. Given the high costs of managing imperiled and invasive species, along with the potential for alien invasives to necessitate and/or undermine ongoing imperiled species recovery efforts9,29, it is imperative that managers understand the scope of invasives’ potential impacts on at-risk species. Even coarse-scale impact estimates are expected to help managers to identify taxa and regions that may be most at risk and allocate research and management resources effectively. Yet despite their global introduced distribution and highly destructive nature, invasive wild pigs are remarkably understudied and poorly understood in terms of their impacts on native imperiled species. We present estimates of current and future risks of direct wild pig impacts on imperiled species across the contiguous U.S. based on their current and projected species range overlap30,31. To our knowledge, the potential impacts of this incredibly widespread and destructive invasive species on at-risk native species have not yet been estimated anywhere within their introduced range. Our first approximation of the potential scope of wild pig threats to imperiled species - given the current state of knowledge and available data - offers an important starting point for effective allocation of efforts to reduce extinction risks to taxa and regions most heavily impacted by wild pigs in the context of other threats (e.g., habitat conversion, climaate change, other invasives). This assessment can inform targeted management of wild pig spread, as well as efficient allocation of efforts toward more detailed assessment of impact risks. Our simple approach can easily be extended throughout wild pigs’ global range and is readily applicable to estimating potential impacts of many other invasive species. Results Current risk There are a total of 284 imperiled species in the animal taxa we considered (see Methods), of which we identified 141 species expected to be susceptible to wild pig impacts if exposed (Table 1, Supplementary Dataset). Of these species, we estimate that 123 (87.2%) may currently be at risk, with mean wild pig range overlap of 72.7% (29.0% SD) (Fig. 2, Table 2). Crustaceans may be most at risk: eleven of 12 crustacean species co-occur with wild pigs, with a mean of 86.3% (15.5% SD) range overlap. Nearly all mollusks (18 of 19) also co-occur with wild pigs, but mean range overlap is relatively low (58.3 ± 33.9% SD). We estimate that risks are greatest in USFWS Southeast and Pacific Southwest regions (Fig. 1, Table 2). Wild pigs are most pervasive in the Southeast (72.9% of region), co-occurring with 57 imperiled species with an average range overlap of 84.7% (18.8% SD). Wild pigs occupy 41.3% of the Pacific Southwest, co-occurring with 52 imperiled species with an average range overlap of 86.5% (19.6% SD). The Southwest is also highly at risk, with 62.6% wild pig occupancy overlapping the ranges of 36 imperiled species by an average of 83.0% (30.1% SD). Future risk If wild pigs continue their current rate of spread across the contiguous U.S., more imperiled species are likely to be at risk across a greater portion of their range. We estimate that 1,036 (CI: 273–1,800) additional watersheds may be occupied by wild pigs by 2025, an annual increase of 2.17% in the number of occupied watersheds (Fig. 3). In this scenario, 128 (90.8%) (CI: 87.9–95%) of 141 susceptible imperiled species may co-occur with wild pigs in at least 5% of their range, with a mean of 76% (27.1% SD) range overlap (Fig. 2, Table 2). This represents 5 (CI: 1–11) additional imperiled species potentially at risk, and a 6.1% (14.0% SD) mean increase in the extent of species’ pig-occupied ranges. We expect crustaceans to continue to be the most at-risk taxon, with all (12) susceptible species potentially co-occurring with wild pigs by 2025 and a mean increase in range overlap of 11.0%, resulting in an overall mean range overlap of 90.1% (10.1% SD). Birds are expected to experience the greatest increase (10%) in number of species co-occurring with pigs (2 additional species), accompanied by a 9.2% mean increase in range overlap. Although we don’t predict any additional mollusk species to be exposed to pigs by 2025, we expect those currently at risk to experience a mean 10.3% increase in range overlap. Mammals are expected to have the smallest increase in future risk, with one additional species potentially exposed and a mean 1.2% increase in range overlap. We predict wild pigs’ range to expand most in the already at-risk Southeast, where watershed occupancy could increase by 18.3% (CI: 6.8–23.8%) to reach a total occupancy level of 91.2% by 2025 (Fig. 1, Table 2). Mean range overlap is expected to increase by 10.3% and one additional species may be placed at risk, with most imperiled species in the Southeast already co-occurring with wild pigs. We also predict substantial range expansion in the Northeast, Southwest, and Pacific, with 8.6%, 7.4%, and 6.4% increases in wild pig occupancy, respectively. Of these regions, expansion in the Southwest may place the most additional species at risk (4 species; CI: 1–8 species), while expansion in the Pacific region may result in the greatest mean increase in wild pig range overlap with imperiled species (22.4%). Discussion Wild pigs are highly invasive and destructive generalists that occupy much of the contiguous U.S. and have high potential for further spread24,25. We show that wild pigs already co-occur with most imperiled species in the contiguous U.S. that are susceptible to direct impacts, and that wild pigs tend to occupy the majority of those species’ ranges. We predict the risks of wild pig impacts to increase if pigs continue to expand their range as projected over the next 10 years, both in terms of the number of imperiled species at risk and the extent of these species’ ranges in which risks occur. Wild pigs inflict a variety of documented impacts on native animal species (Table 1), with likely impacts on other species occupying similar niches. Their large body size and omnivorous, highly plastic diet provides access to a wide range of animal prey, including all terrestrial vertebrate classes, eggs of reptiles and ground-nesting birds, soil invertebrates, and shallow freshwater invertebrates, with animal matter accounting for up to 30% of wild pigs’ diet10,32 (Table 1). Their rooting and wallowing behavior results in marked ecosystem-level impacts through alteration of soil properties as well as plant community structure and composition10,22, destroying habitat for a wide variety of species, particularly tunneling, burrowing, and litter-dwelling species (Table 1). Our assessment suggests that impacts of wild pigs via these mechanisms on U.S. imperiled species could be extensive. The true magnitude of wild pigs’ impacts may be much greater when other, less-studied impact mechanisms are considered. Pigs are known carriers of diverse parasites and other disease-causing pathogens, with potential to transmit pathogens to both livestock and wildlife18,19. Prevalence varies geographically and among pathogens24, and transmission dynamics are not well understood, but impacts on imperiled species can be serious33. Wild pigs may also compete with native species for resources10. Disease transmission and resource competition may not only exacerbate other impacts on imperiled species, but also increase the total number of species susceptible to wild pig impacts. Potential impacts on medium- to large-bodied mammals may be particularly underestimated due to exclusion of these mechanisms10. In the absence of detailed information about imperiled species’ distributions within their ranges or species-specific interactions with wild pigs, we assumed that species with attributes similar to those for which impacts have been documented may be at risk wherever they co-occur with wild pigs30,31. Due to the coarse scale of the datasets available for our analysis, we recognize that imperiled species and wild pigs may not co-occur at finer spatial scales within all watersheds suggested to be co-inhabited. Certainly, some imperiled species may use different habitats than wild pigs within the watersheds where they do co-occur and therefore may not interact directly with pigs. However, because data to assess co-occurrence at finer scales are unavailable, we suggest that our preliminary assessment based on existing data provides a much-needed first approximation of the potential severity of current and future risks of wild pig impacts on imperiled species in the U.S. Despite some likely error in predictions at the individual species level, the patterns we elucidate in relative potential risk among taxa and regions provide critical initial guidance for effective allocation of management efforts as well as efficient investment in finer-scale assessment of wild pig co-occurrence with and impacts on at-risk species. Furthermore, we suggest that impacts in regions of even broad overlap are likely given pigs’ extreme habitat generalism, diverse diet, and highly destructive rooting habits. Wild pigs can also shift foraging strategies in response to temporal variation in food resources and to opportunistically target highly concentrated prey, and their rooting behavior has been suggested to disproportionately impact specialist species34. This can endanger species that are dependent on highly specialized habitat types or ephemeral conditions despite their overall rarity across the landscape35. We also believe that our projection of wild pigs’ potential distribution by 2025 is likely a conservative one. We project occupancy of 35% of watersheds in the contiguous U.S. by 2025, yet eventual spread of wild pigs to all watersheds where probability of occurrence25 is equal to or greater than that of watersheds where they already occur would result in 93.7% occupancy. Other recent work similarly predicts wild pig expansion throughout most of the contiguous U.S. in the next 3–5 decades26. Furthermore, limitations on pig establishment and density imposed by cold temperatures at northern latitudes and higher elevations25 may be relaxed as climate change results in fewer days with extreme cold temperatures and less snow in these regions, allowing more extensive occupancy and/or occupancy at higher densities26,36. In addition, we have only considered likely patterns of spread by natural dispersal mechanisms, but wild pigs have been introduced to new regions of the U.S. far from their historic range for hunting or other purposes; we expect this practice to continue. Expenditures on management of the 123 imperiled species that currently co-occur with wild pigs totaled$182.2 million in 20146. If wild pigs continue to spread as projected, their impacts could become an additional factor in the management of up to 11 additional species for which 2014 expenditures totaled \$8.56 million. An important goal for the USDA’s wild pig management program is to predict wild pig population expansion and associated risk to animal health37, so that wild pig and imperiled species management activities can be more effectively prioritized and coordinated. The program typically focuses on wild pig damage reduction where populations are well-established, but also partners with state and federal agencies to eliminate wild pigs where their populations are low or where they are just beginning to establish. We show where estimation and reduction of damage from established pigs may benefit the greatest number of imperiled species and where emerging populations may be most critical to control. Prioritizing efforts based on these findings may help minimize future expenditures on both wild pig and imperiled species management. Prioritization of these strategies should also be considered in the context of additional threats to imperiled species, such as habitat conversion, impacts of other invasive species, and climate change.

Our simple and proactive approach to estimating potential risks is modeled after similar invasive species assessments conducted in other systems in which limited information is available30,31, and it can easily be extended to other portions of wild pigs’ global introduced range or used to assess risks from other invasive species. This approach is repeatable and comparable across taxa, and provides a starting point for understanding the relative impacts of invasive species on imperiled species in the context of other stressors (e.g., additional non-native species, overexploitation, habitat loss and degradation, and climate change31). Similar assessments of the spatial distribution of these stressors, combined with information about mechanisms by which they impact imperiled species, could reveal where imperiled species may be most vulnerable to the cumulative effects of multiple stressors and inform coordinated management actions to comprehensively address these threats. Understanding the scope and distribution of alien invasive species’ potential impacts on native imperiled species, given available data, is a crucial first step to managing risks, identifying research priorities, and allocating conservation resources effectively.

Methods

We selected imperiled species expected to be susceptible to wild pig impacts based on published observations of wild pig interactions with other taxa (Table 1). We focused our assessment on animal species (excluding most arthropods) because little is known about the mechanisms by which wild pigs impact plant, fungus, insect, and arachnid species. We focused on risks related to directly observable impacts (i.e., predation, habitat destruction) because the prevalence and severity of other mechanisms (i.e., competition, disease spread) are not well understood. Our assessment thus conservatively estimates the total number of species at risk.

We obtained geographic range data for animal species designated as threatened or endangered in the U.S. from USFWS. Because USFWS manages imperiled species, these range estimates were the most authoritative source available for the U.S. and were more precise than estimates from global sources (e.g., IUCN). However, quality and resolution varied among species and, in some cases, among portions of a species’ range spanning multiple USFWS regions. Most species’ ranges (83.7%) were delineated in whole or in part at the county level. Many (42.2%) were delineated in whole or in part as polygons drawn by USFWS field office biologists. Other less frequently used units of delineation include HUC838 watersheds and USGS 24 K quadrangle or quarter-quadrangle grids39.

We estimated the current distribution of wild pigs in the contiguous U.S. using data from the National Feral Swine Mapping System40, aggregated to watersheds (HUC1038) as described in McClure et al.25. This dataset has been compiled at irregular intervals since 1960 and annually since 2008. Map polygons representing the known geographic extent of established wild pig populations (i.e., present for two or more years and evidence of reproduction) were reported nationally by wildlife professionals in state wildlife resource agencies and the United States Department of Agriculture. Polygon locations were drawn manually using topographical maps to reference areas where pigs have been observed to meet criteria for establishment. These polygons were aggregated to watersheds (HUC10) as described in McClure et al.25 to discretize consistent, comparable, ecologically relevant sampling units.

We estimated wild pigs’ potential future distribution using observed trends in past spread and a previously published model of wild pig occurrence probability25 as follows. We estimated the proportion of wild pig-occupied watersheds at intervals for which data were available from 1960 to 2015, then modeled the trajectory of change in the proportion of occupied watersheds over time. Using an information theoretic approach41, we selected a 3-parameter logistic regression model, which we used to predict the proportion of occupied watersheds in the year 2025 (Supplementary Table S1, Figure S1). Based on this proportion, and assuming that all currently occupied watersheds remain occupied in 2025, we determined the number of additional currently unoccupied watersheds predicted to be occupied by 2025. We then populated this number of unoccupied watersheds to estimate future wild pig distribution.

The spatial distribution of these additional watersheds was selected based on the assumption that watersheds most likely to be occupied in the future were those modeled to have the greatest probability of wild pig occurrence by McClure et al.25 (Fig. 3). This existing model used a logistic discrimination function to estimate the relative probability of wild pig occurrence across the contiguous U.S. given the distribution of habitat covariates at watersheds in which pigs are present relative to randomly sampled ‘background’ watersheds. This approach was selected because it avoids the problematic assumption that background watersheds represent absences or ‘pseudo-absences’ of wild pigs. The inferential model, which was selected within an information-theoretic framework using multi-model inference41, includes environmental covariates customized to represent known physiological and ecological constraints on wild pig distributions. These include mean number of days with temperatures above 35 °C; mean number of days with temperatures below −4 °C; mean snow depth on April 1; mean distance to nearest perennial stream or water body perimeter; percent area with forest cover; forage availability, quantified as percent area with mast-producing or crop cover; and habitat heterogeneity, quantified as the mean number of key habitat elements (water, cover, and forage) available within a radius defined by an average home range size. Cross-validation of model results indicated strong predictive capacity (Pearson’s r = 0.989). Watersheds with the highest estimated probability of occurrence tended to be immediately adjacent to the periphery of wild pigs’ current distribution, and were therefore considered to represent a reasonable prediction of the pattern of future spread by natural dispersal mechanisms; we do not consider the potential for human introductions of wild pigs to novel areas for hunting or other purposes here.

In a geographic information system (GIS), we intersected imperiled species’ ranges with the current and potential range of wild pigs to estimate the number of species currently or potentially at risk of wild pig impacts and the proportion of each species’ range at risk30,31. Only species with at least 5% of their ranges occupied by wild pigs were considered to be potentially at risk, although our results were not sensitive to this threshold as most species had far more extensive range overlap (see Results). Confidence limits on estimates were derived from the confidence interval on the proportion of watersheds projected to be occupied in 2025. Each estimate was further broken down by taxonomic group and USFWS region.

Data availability

The original wild pig occurrence data and threatened and endangered species range data used in this analysis cannot be made available in the manuscript because they were obtained from third party providers. The data providers are not able to share the spatial data publicly due to sensitivities around the location of these species. Threatened and endangered species may be subject to exploitation, and there is a history of hunters using the wild pig data to locate populations then bait, trap, and relocate them for hunting purposes, which greatly interfered with ongoing control and eradication efforts. Threatened and endangered species range data are available by request from the U.S. Fish and Wildlife Service (Mark Saunders, mark_saunders@fws.gov). Wild pig occurrence data are available by request through the Southeastern Cooperative Wildlife Disease Study (SCWDS) National Feral Swine Mapping System (http://swine.vet.uga.edu/nfsms/; Joe Corn, feralpig@uga.edu).


text:
# Modal-set estimation with an application to clustering

13 Jun 2016Heinrich JiangSamory Kpotufe

We present a first procedure that can estimate -- with statistical consistency guarantees -- any local-maxima of a density, under benign distributional conditions. The procedure estimates all such local maxima, or $\textit{modal-sets}$, of any bounded shape or dimension, including usual point-modes... (read more)

PDF Abstract

format: plain


text:
Chapter 9.2, Problem 25E

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
1 views

# Finding a Probability In Exercises 19-26, sketch the graph of the probability density function over the indicated interval and find each probability. See Example 3. f ( t ) = 1 3 e − t / 3 ,        [ 0 , ∞ ] (a) P ( t < 2 ) (b) P ( t ≥ 2 ) (c) P ( 1 < t < 4 ) (d) P ( t > 3 )

(a)

To determine

To calculate: The probability P(t<2), for the probability density function f(t)=13et/3 over the interval [0,)

Explanation

Given Information:

The function, f(t)=13et/3 is a probability density function over the interval [0,).

Formula used:

Consider a function f of a continuous random variables x whose set of values is the interval [a,b], A function is a probability density function when it is non-negative and continuous on the interval [a,b] and when abf(x)dx=1 and the probability P(a<x<b)=abf(x)dx.

Calculation:

Consider the function,

f(t)=13et/3

The function is a probability density function over the interval [0,).

Therefore, f(x) is continuous and non-negative over the interval [0,) and is graphed as shown below:

Now, the probability that t is less than 2 implies that t lies between 0 and 2 is computed as,

P(t<2)=02f(t)dt=0213et/3dt

(b)

To determine

To calculate:

The probability P(t2), for the probability density function f(t)=13et/3 over the interval [0,).

(c)

To determine

To calculate: The probability P(1<t<4), for the probability density function f(t)=13et/3 over the interval [0,).

(d)

To determine

To calculate: The probability P(t>3), for the probability density function f(t)=13et/3 over the interval [0,).

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started


text:
### Artificial IntelligenceAIMA Exercises

Show that the three forms of independence in Equation (independence-equation) are equivalent.

Show that the three forms of independence in Equation (independence-equation) are equivalent.

Submit Solution


text:
# Mean occurrences of letters in complete strings given by a Bernoulli scheme

Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous drawn letters and that $\sum_{n=1}^K p_n =1$; this model should be known in the literature as Bernoulli scheme.

Suppose moreover that the letters are not chosen uniformly at random, i.e. there exists a letter (without loss of generality the first one) whose corresponding probability $p_1$ satisfies the property $p_1 \geq p_n$ for all $2 \leq n \leq K$ and this inequality is strict for at least one letter.

We keep sampling letters according to this scheme until we create a string $s$ where all $K$ letters appear at least once.

How many occurrences of the first letter do one have on average in such string? If an exact result is impossible, can one bound this mean or say something its magnitude?

Any hint, partial solution or reference for this problem would be really appreciated.

-
this question belongs on math stackexchange - it's more a homework question than a research question – Anthony Quas Mar 6 '12 at 17:54
Actually it is a difficult question that might not have been answered in the literature. – Brendan McKay Mar 7 '12 at 4:35

## 1 Answer

The magic words are "Coupon collector problem". For some generalizations, you can check out

The moment zeta function and applications Authors:
Rivin, Igor Publication:
eprint arXiv:math/0201109 Publication Date:
01/2002

-
Thanks Igor for your suggestions. Do you know whether my specific question was already answered somewhere in the literature? The reformulation of my question using the CCP terms is: what is the mean of the largest number of cupons of the same type, once that the collection is complete? – alezok Mar 6 '12 at 20:27
@Ale: Yes, I understand your question: I believe that something very close to this is covered in "the coupon-collector problem revisited", by A Boneh et al (google finds the PDF). – Igor Rivin Mar 6 '12 at 21:25
See the list on page 7 in the paper... – Igor Rivin Mar 6 '12 at 21:27


text:
Product rule for differentiation is different from the sum and difference rule for derivatives. In sum and difference rules the derivative of a sum or difference is the sum or difference of the derivatives. In other words we find the derivatives separate and add or subtract as required.
But the derivative of the product of a function is not the product of derivatives.
For example, consider f(x) = x5. By using the power rule for derivative of a function we get f'(x) = 5x4.  If we treat the function f(x) as product as f(x) = x3.x2, then  the derivative of x3 = 3x2 and the derivative of x2 = 2x. If we multiply the two derivatives we get (3x2).(2x) = 6x3 which is not the derivative of x5 .Hence the differentiation of a product is analyzed using the definition of the derivative and a rule is arrived at for finding the derivative of a function which is expressed as a product of two or more functions.
Many rules exist under the name Product Rule related to various concepts like the product rule for exponents, product rule for logarithms, product rule for limits and product rule for derivatives. Let us first view few instances where the product rules are defined other than for derivatives.

## Product Rule Definition

The Product Rule for derivatives can be defined as follows.

If two functions are differentiable, so is their product and the derivative of their product is the sum of the products obtained by multiplying a function with the derivative of the other function.

Using notation we can write this as:

$\frac{d}{dx}$ $[f(x)g(x)]$ = $f(x)g'(x)+g(x)f'(x)$

## Product Rule for Exponents

Product rule for exponents is a basic algebraic rule that is used to simplify when products of same variable with different exponents occur.


text:
1. ## algebra question (squares?)

1 Write down the first ten square numbers (perfect squares) [can do this]
2 Show that when each of these squares are divided by 3, the remainder is either 0 or 1
3 Write down three consecutive numbers the first of which is divisible by three[can do up to this part]
4 Prove that the square of any number is of the form of 3n or 3n+1
5 Use modular arithmetic(what's that) to prove the same result

2. Originally Posted by requal
1 Write down the first ten square numbers (perfect squares) [can do this]
2 Show that when each of these squares are divided by 3, the remainder is either 0 or 1
3 Write down three consecutive numbers the first of which is divisible by three[can do up to this part]
4 Prove that the square of any number is of the form of 3n or 3n+1
5 Use modular arithmetic(what's that) to prove the same result
In part 2 you showed that when a square is divided by 3, the remainder is either 0 or 1.

This means that the number can be written in the form:

k = 3n+1
or
k = 3n.

3. Okay i got the worked solutions of the question, but I don't understand it, someone please clarify it for me.

iii, 3n, 3n+1, 3n+2
iv Any number must be of the form of 3k, 3k+1, 3k+2

Hence, $(3k)^2=9k^2=3(3k^2)=3n$
And, $(3k+1)^2=9k^2+6k+1=3(3k+2)+1=3n+1$
And, $(3k+2)^2=9k^2+12k+4=3(3k^2+4k+1)+1=3n+1$

Can someone explain to me how they got the solution?? I mean how can n is equal to three randomly different terms

4. Originally Posted by requal
Can someone explain to me how they got the solution?? I mean how can n is equal to three randomly different terms
Well Its another way of saying every number when divided by 3 leaves a remainder of either 0,1 or 2.


text:
## Engineering Mechanics: Statics & Dynamics (14th Edition)

$C_x=4.97KN$ $D_x=2.23KN$ $D_y=7.2KN$
According to the principle of impulse and momentum $\Sigma M_A=\frac{dm}{dt}(d_{DB}v_B-d_{DA}v_A)$ We plug in the known values to obtain: $-C_x\times 2=600(0-(1.5-0.12)\times 12)$ $\implies C_x=4968N=4.97KN$ We know that the resultant force in the $x$-direction is given as $\Sigma F_x=\frac{dm}{dt}(v_B)_x-(v_A)_x$ $\implies D_x+C_x=600(12-0)$ This simplifies to: $D_x=2232N=2.23KN$ Now the resultant force in the $y$-direction is given as $\Sigma F_y=\frac{dm}{dt}[(v_B)_y-(v_A)_y]$ We plug in the known values to obtain: $D_y=600(0-(-12))$ This simplifies to: $D_y=7200N=7.2KN$


text:
# inverse of Vandermonde's Matrix without using determinants

I want to show, that the Vandermonde's Matrix $A:==\begin{pmatrix}1&x_0&\cdots&x_0^d\\1&x_1&\cdots&x_1^d\\\vdots&\vdots&\ddots&\vdots\\1&x_d&\cdots&x_d^d\end{pmatrix} \in \mathbb{R}^{(d+1)\times (d+1)}$ is invertible , when $x_0, ..., x_d \in \mathbb{R}$ are pairwise distinct. Determinants are unknown for us so far.

My idea was to use Gaussian elimination:

\begin{align*}\begin{pmatrix}1&x_0& x_0^2 &\cdots&x_0^d\\1&x_1& x_1^2 &\cdots&x_1^d\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&x_d& x_d^2 &\cdots&x_d^d\end{pmatrix} &\longmapsto \begin{pmatrix}1&x_0& x_0^2 &\cdots&x_0^d\\0&x_1-x_0& x_1^2-x_0^2 &\cdots&x_1^d-x_0^d\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&x_d-x_0& x_d^2-x_0^2 &\cdots&x_d^d-x_0^d\end{pmatrix} \\ &\longmapsto \begin{pmatrix}1&x_0& x_0^2 &\cdots&x_0^d\\0&x_1-x_0& x_1^2-x_0^2 &\cdots&x_1^d-x_0^d\\ 0 & 0 & x_2^2-x_1^2 & \dots & x_2^d-x_1^d\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&0& x_d^2-x_1^2 &\cdots&x_d^d-x_1^d\end{pmatrix} \\ &\longmapsto \ldots \end{align*}

After the first transformation, we can see that $x_1,..x_d$ must be different from $x_0$ otherwise the row echelon form wouldn't have full rank $\Rightarrow$ not invertible. Analog for the others.

Is my proof correct?

• probably the simplest argument: if a vector is send by this matrix to zero then it gives us a degree-$\leq d$ polynomial $p$ such that $p(x_i)=0$ $\forall i$ – user8268 May 18 '14 at 20:18
• – Martin Sleziak May 20 '14 at 7:40


text:
Question 6

# In the following question, select the related letters from the given alternatives.MOT : PRW :: TAG : ?

Solution

The logic here is

M + 3 = P

O + 3 = R

T + 3 = W

Similarly,

T + 3 = W

A + 3 = D

G + 3 = J

$$\therefore\$$TAG is related to WDJ in the same way MOT is related to PRW.

Hence, the correct answer is Option D


text:
## Solution of problem 11306 in AMM

Let $a,b,$ and $c$ be the lengths of the sides of a nondegenerate triangle, let $p=(a+b+c)/2$, and let $r$ and $R$ be the inradius and circumradius of the triangle, respectively. Show that $\dfrac{a}{2}\cdot \dfrac{4r-R}{R}\leq\sqrt{(p-b)(p-c)}\leq \dfrac{a}{2},$

and determine the cases of equality.

My solution.

## A famous functional equation

That is following problem: Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$f(x^2 + y + f(y)) = (f(x))^2 + 2\cdot y\; \forall x,y\in\mathbb{R}.(*)$
It is famous! Why? Because, it is from AMM(problem 10908, posted by Wu Wei Chao) and it is one in problems from Bulgarian TST 2003, Vietnam TST 2004 and Iran TST 2007. However, in Vietnam TST 2004 it has form:


text:
# Derivation of Faraday's Law from Lorentz's

1. Aug 7, 2014

### iScience

$$\vec{F}=q\vec{v}\times\vec{B}$$

$$\frac{d\vec{F}}{dq}=\vec{v}\times\vec{B}$$

$$\int\frac{d\vec{F}}{dq} \cdot ds=\int(\frac{d\vec{s}}{dt}\times\vec{B}) \cdot ds$$

from here, I went about it two different ways:

1.) Here I assumed everything was at right angles and got rid of all the vectors and vector products

$$\varepsilon=\int \frac{ds}{dt}B ds=\int \frac{ds}{dt}B \frac{ds}{dt}dt$$

By u substitution

$$u=\frac{ds}{dt}, du=dt$$
$$\varepsilon=\int B(u^2)du=\frac{Bv^3}{3}$$

where v = ds/dt

That was the first way i went about it, but i didn't feel any closer to Faraday's law.

2.) Here I left the vectors alone on the RHS; I figured since $\hat{v}$ and d$\hat{s}$ were perpendicular, the quantity ($\vec{v}$s) would be a time derivative of the area formed

$$\varepsilon=\int\frac{ds}{dt}B ds=\int(\vec{v}\times\vec{B}) \cdot d\vec{s}=\dot{A}B$$

$$\varepsilon=\frac{BA}{dt}$$

don't know where the minus sign is; probably was supposed to do something with the cross product, but didn't know what.

Well I got alot further with the second "method," but is this a valid derivation? and what went wrong with the first method?

Last edited: Aug 7, 2014
2. Aug 9, 2014

### iScience

Oh! PS, well, more like pre-script... my goal is to derive faraday's induction law from lorentz force law


text:
Is Iaijutsu Focus bonus damage subject to double damage? Or triple?

I am a fan of the Iajutsu Master. Though I've never really been a fan of charging builds, I recently saw the weapon enhancement Valorous from Unapproachable East which does double damage on a charge and it got me thinking.

Is there any game mechanic that would double Iajutsu Focus damage? Remember, it is not sneak attack damage, it is not precision based damage or skirmish damage. It is pure bonus damage based on the uncanny speed of the strike.

I have not played since 2010 but I used to play Iaijutsu Masters as a favorite class and I never once made any attempt to try to double the IF bonus damage. Perhaps there is a ruling out there somewhere which prohibits it (and I simply forgot)?

The rules for multiplying damage are pretty clear:

Multiplying Damage

Sometimes you multiply damage by some factor, such as on a critical hit. Roll the damage (with all modifiers) multiple times and total the results. Note: When you multiply damage more than once, each multiplier works off the original, unmultiplied damage.

Exception: Extra damage dice over and above a weapon’s normal damage are never multiplied.

These rules cover any damage multiplication, critical hit or otherwise, and exempt any form of bonus damage dice, sneak attack or otherwise.

So no, Iaijutsu Focus damage dice are not multiplied in any circumstance. However, the iaijutsu master prestige class includes the strike from the void feature at 5th level, which adds his Charisma bonus for each die Iaijutsu Focus adds to his damage. That flat damage would be multiplied on a critical hit or in other circumstances that result in damage multiplication. Also, notably, since the bonus from Charisma, even applied multiple times, is a bonus to the damage, not the damage itself, it would be multiplied as in normal mathematics, not by adding multipliers.

That is, if a 5th-level iaijutsu master makes an Iaijutsu Focus check to add three damage dice and has Strength 20 and Charisma 16, for example, his damage bonus would be

$$+5_{Str} +3\text{d}6_{\text{Iaijutsu Focus}} +3_{Cha} +3_{Cha} +3_{Cha} \\ = 14+3\text{d}6$$

If this damage is doubled, it would be

$$\left[+5_{Str} +3\text{d}6_{\text{Iaijutsu Focus}} +3_{Cha} +3_{Cha} +3_{Cha}\right]\times2 \\ = +10_{2\times Str} +3\text{d}6_{\text{Iaijutsu Focus}} +6_{2\times Cha} +6_{2\times Cha} +6_{2\times Cha} \\ = 28 + 3\text{d}6$$

Rather than simply taking the $3 \times Cha$ of the original an adding another multiplier for $4 \times Cha$.

• Ah, there you have it! That's why I never tried to multiply IF bonus damage. However, on the multiplying of the extra damage, I can see where you are coming from, but I believe you are mistaken. The 5th level Iaijutsu Master class ability: Strike from the Void (Ex): An iaijutsu master of 5th level or higher adds her Charisma modifier to each extra damage die gained from using the Iaijutsu Focus skill. – Aceon Sep 20 '16 at 2:33
• Therefore if you add Cha to each extra bonus damage dice, then it can NOT be considered base damage and thus not subject to any multiplications, as it is added to each bonus damage dice and as you pointed out - bonus damage dice are never multiplied. – Aceon Sep 20 '16 at 2:36
• @Aceon Hmm. That’s a very good point, but I think it could be read either way. That would have to be a DM ruling, I think. I’ll update my answer when I get a chance. – KRyan Sep 20 '16 at 13:30


text:
• F F Wu

Articles written in Bulletin of Materials Science

• Formation of ternary Mg–Cu–Dy bulk metallic glasses

The glass-forming ability (GFA) of ternary Mg–Cu–Dy alloys was systematically investigated by using differential scanning calorimetry (DSC) and X-ray diffractometry (XRD) techniques. The results showed that a series of ternary Mg–Cu–Dy bulk metallic glasses (BGMs) with a diameter of 4–8 mm were successfully fabricated in the system with conventional Cu-mold casting method. Mg55Cu32Dy13, Mg60Cu27Dy13, Mg65Cu25Dy10 and Mg70Cu17Dy13 BMGs exhibit a clear glass transition, a broad supercooled liquid region and different crystallization and melting behaviours. They have supercooled liquid region ($\Delta T_{x}$) from 41 K to 65 K, reduced glass transition temperature ($T_{rg}$) from 0.5363 to 0.5974 and 𝛾 parameter from 0.4038 to 0.4136. The 𝛾 shows a relatively good agreement with the GFA of the BGMs. On the other hand, a high fracture compressive strength of 624 MPa was obtained for Mg60Cu27Dy13 BMG.

• Microstructural evolution and mechanical properties of hypereutectic Al–Si alloy processed by liquid die forging

The microstructural evolution and mechanical properties of a hypereutectic Al–Si alloy processed by liquid die forging were investigated. It is found that the grain size of the primary Si was significantly reduced by liquid die forging with increased pressure. The volume fraction of eutectic silicon was decreased with increased pressure. By liquid die forging with pressure up to 180 MPa, the average size of the primary Si was reduced to about 18 𝜇m, which results in the remarkable increase in the fracture strength and hardness of the hypereutectic Al–Si alloy.

• # Bulletin of Materials Science

Volume 43, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
# To negate a statement, why do you have to invert each quantifier and negate each sub-expression?

To negate a statement, you have to invert each quantifier and negate each sub-expression. Example: \begin{align} \DeclareMathOperator{\epsilon}{\varepsilon} f \text{ is continuous at a} &\iff \forall\epsilon>0\exists\delta>0\forall x\bigl(|x-a|<\delta\implies |f(x)-f(a)|<\varepsilon\bigr)\\[4pt] f \text{ is discontinuous at a} &\iff \exists\epsilon>0\forall\delta>0\exists x\bigl(|x-a|<\delta\implies |f(x)-f(a)|\ge\varepsilon\bigr) \end{align} I'm trying to justify why this is the case. It seems intuitively clear that $$\neg\forall xP(x)\iff \exists x\bigl(\neg P(x)\bigr)$$: if a proposition is not true for all $$x$$, then there just needs to be one counterexample. Moreover, $$\neg \exists xP(x)\iff \forall x\bigl(\neg P(x)\bigr)$$ because both statements say that there aren't any examples of a proposition being true.

My idea is to use these two "rules" recursively: \begin{align} f \text{ is discontinuous at a} &\iff \neg\Bigl(\forall\epsilon>0\exists\delta>0\forall x\bigl(|x-a|<\delta\implies |f(x)-f(a)|<\varepsilon\bigr)\Bigr)\\[4pt] &\iff \exists\epsilon>0\neg\Bigl(\exists\delta>0\forall x\bigl(|x-a|<\delta\implies |f(x)-f(a)|<\varepsilon\bigr)\Bigr)\\[4pt] &\iff \exists\epsilon>0\forall\delta>0\neg\Bigl(\forall x\bigl(|x-a|<\delta\implies |f(x)-f(a)|<\varepsilon\bigr)\Bigr)\\[4pt] &\iff \exists\epsilon>0\forall\delta>0\exists x\neg\bigl(|x-a|<\delta\implies |f(x)-f(a)|<\varepsilon\bigr)\\[4pt] &\iff \exists\epsilon>0\forall\delta>0\exists x\Bigl(|x-a|<\delta\implies\neg\bigl(|f(x)-f(a)|<\varepsilon\bigr)\Bigr)\\[4pt] &\iff \exists\epsilon>0\forall\delta>0\exists x\bigl(|x-a|<\delta\implies|f(x)-f(a)|\ge\varepsilon\bigr)\\[4pt] \end{align} Does this line of reasoning make sense? Moreover, can the "invert each quantifier and negate each sub-expression" rule be applied in any situation?

• Yes, because $\lnot \forall$ is the sane as $\exists \lnot$ and similar with the negated existential quantifier. Intuition: "it is not true that all cats are black" means that "there are some cat that is not black" Jun 29 at 11:57
• Could the downvoted please justify it?
– Joe
Jun 29 at 12:01
• Negation of continuity is wrong: You should take negation of last sentence too. Jun 29 at 12:36
• @zkutch: Is that because the negation of $p\rightarrow q$ is $p \wedge \neg q$? The negation of $p \rightarrow q$ is not $p \rightarrow \neg q$.
– Joe
Jun 29 at 12:57
• @MauroALLEGRANZA: Thanks for clarifying. Is it possible if post your comment as an answer so that we can remove this question from the "unanswered questions" list?
– Joe
Jun 30 at 12:30

Correct, because $$¬∀x$$ is the sane as $$∃x¬$$ and similar for the case with the negated existential quantifier.
$$\lnot \forall \varepsilon \ \exists \delta \ \forall x \ \mathcal A \ \$$ will be : $$\ \ \exists \varepsilon \ \forall \delta \ \exists x \ \lnot \mathcal A$$.


text:
# Tick the correct answer in the following : Area of a sector of angle p (in degrees) of a circle with radius R is : (a) $$\frac{p}{180} \ × \ 2\pi R$$ (b) $$\frac{p}{180} \ × \ \pi R^2$$ (c) $$\frac{p}{360} \ × \ 2\pi R$$ (d) $$\frac{p}{720} \ × \ 2\pi R^2$$

We know that area A of a sector of angle $$\theta$$ in a circle of radius r is given by

$$A \ = \ \frac{ \theta }{360} \ × \ \pi r^2$$

But here , $$r \ = \ R$$ and $$\theta \ = \ p$$

$$\therefore$$ $$A \ = \ \frac{p}{360} \ × \ \pi R^2 \ = \ \frac{p}{720} \ × \ 2\pi R^2$$

$$\therefore$$(D) is the correct answer.


text:
# Showing the ergodicity of a rotation on the unit sphere

Consider the rotation $R_\alpha(z) = \alpha z, R_\alpha : S^1 \to S^1$. Show that

$R_\alpha$is ergodic with respect to Haar measure on $S^1$ $\iff$ $\alpha$ is not a root of unity.

I don't know how to show the $\Leftarrow$ implication. Can someone please help me? I know I have to consider some Fourier series, but I fail to solve this exercise.

Thank you!

Show that the Fourier coefficients $c_k(\psi)$ of any $L^2$ function $\psi$ satisfy $c_k(\psi\circ R_\alpha)=\alpha^kc_k(\psi)$ for $k\in\mathbb Z$. After that use the fact that $\alpha^k\ne1$ (notice that if $\psi$ is invariant, the Fourier coefficients of $\psi$ and $\psi\circ R_\alpha$ are the same for all $k$).


text:
# How do you solve a/5> 10?

Jan 21, 2016

$a > 50$

#### Explanation:

One of the permitted operations for inequalities (that doesn't effect the validity or orientation of the inequality) is multiplication of both sides by the same constant provided that constant is greater than zero.

Multiplying both sides of $\textcolor{b l u e}{\frac{a}{5} > 10}$ by $\textcolor{b l u e}{5}$ gives:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{a > 50}$


text:
## Questions

1. Consider the utility function given by U x = ln x and

the set of gambles with the possible outcomes $10,$20

and $30. For each exercise, it may be useful to use a spreadsheet or other numerical tools. (a) Graph the indifference curves implied by expected utility in the MarschakMachina triangle. What is the slope of the indifference curves? (b) Now suppose that the decision maker maximizes probability weighted utility, with theweights given by π p = p0.7 i p0.7 i + 1 pi 0.7 1 0.7 . Graph a few examples of indifference curves. What does the probability weighting do to the shape of these curves relative to those in part a? (c) Repeat the exercise in b, now assuming the decision maker maximizes rank-dependent expected utility. Thus, now the probability weighting function is applied to cumulative probabilities. (d) Finally, consider the regret utility function given by U x, y = x y 2 if xy x y 2 if x < y Plot the implied indifference curves supposing the alternative choice would yield$19 with certainty and

compare the shape of these curves to those of the

other models considered. How do these curves

change when the foregone gamble is altered?

2. Suppose that there were three possible states of nature

as represented in the table below:

States of Nature 1 2 3

Probability 0.35 0.4 0.25

Gamble 1 $1,000$2,000 $3,000 Gamble 2$1,800 $1,800$1,800

Gamble 3 $2,500$1,500 $1,500 (a) What conditions would be required for the regret theory utility function to predict preference cycling when choosing between pairs of the three possible gambles? Graph an example of a function that would satisfy these conditions. (b) Is it possible for preference cycling to occur given the same choices under expected utility maximization with probability weights? Show why or why not. (c) Would it be possible for preference cycling to occur given the same choices under rank-dependent expected- utility maximization? Show why or why not. 3. Confirm that the common ratio effect as found in Example 9.4 could be explained either by probability weighting or by regret theory. To do this, find utility and weighting functions that satisfy the models and that lead to the choices found in Example 9.4. 4. Consider two gambles, each with outcomes$10, $20,$30,

and \$40. Gamble 1 has probabilities p10, p20, p30,

1 p10 p20 p30 for these outcomes and gamble 2

has probabilities q10, q20, q30, and 1q10 q20 q30.

Suppose that Gamble 2 stochastically dominates

Gamble 1. Thus, p10 >q10, p10 +p20 > q10 + q20, and

p10 +p20 +p30 >q10 +q20 +q30. Show that anyone who

maximizes rank-dependent expected utility must prefer

Gamble 2 to Gamble 1.

5. Suppose that you are a policymaker considering

instituting a tax to combat climate change. The current

climate research is conflicting as to the probability that

carbon emissions will lead to catastrophic climate

change. Suppose that the probability of a catastrophic

climate change is equal to 0.3 × cϕ, where c represents

carbon dioxide emissions and, depending on which

scientists you listen to, ϕ can be as low as 0.1 or as

high as 0.9. As a policymaker you wish to maximize

expected social welfare. If a catastrophic climate

change occurs, social welfare will be equal to 0, no

matter how much other production occurs. If a

catastrophic climate does not occur, then social welfare

is given by the profit of the emitting industry,

π = p t y k y = 3 t y 0.15y2, where y is

output, p is the output price, k . is the cost of production,

and t is the tax you impose. The firm chooses

y to maximize profits, according to y= 3 t 0.3.

Carbon dioxide emissions are given by c =y.

(a) Suppose you display α-maxmin expected-utility

preferences, with α=1 (fully ambiguity averse).

What tax will you choose? What level of social

welfare will be realized if a catastrophic climate

change is not realized? You may use a spreadsheet

to determine the answer if needed.

(b) Suppose α=0 (fully ambiguity loving). What tax

will you choose? What level of social welfare will

be realized if a catastrophic climate change is not

realized?

(c) Suppose that a definitive study shows that ϕ=0.2.

What is the expected social welfaremaximizing

tax? What is the resulting expected social welfare

and the social welfare resulting if no catastrophic

climate change is realized?


text:
# Math Help - Integral and infinity: Can somebody check this solution ?

1. ## Integral and infinity: Can somebody check this solution ?

It seems to be strange answer to me.

2. Originally Posted by totalnewbie
It seems to be strange answer to me.
Hello,

I've attached an image.

EB


text:
## Thursday, March 17, 2005

### The Strong Coupling Constant

First off, before I get to the substance of the post, let me welcome Mark Trodden to the physics blog world. He's part of the even more elite "Central New York physics" blog world.

This post is going to be an outline of how one computes a value for the strong coupling constant ($\alpha_{s}$) using lattice QCD. Now the strong coupling constant isn't really constant at all. It depends on two things, the regulation scheme you use, and the energy you determine it at. The Particle Data Group always quotes the values in the MSbar scheme (a particular type of dimensional regulation, don't worry if you don't know what it is) at the energy scale $\mu=M_{Z}=91$ GeV. So that's our goal, to get a number for $\alpha$ from lattice QCD. The world average, which includes an older lattice calculation, is $\alpha_{s}(M_{Z}) = 0.1187(20)$, where the number in brackets is the error on the last two digits.

How do we get this number from lattice QCD? Well, let's think about the inputs to a lattice QCD simulation, they are 5 quark masses, and the lattice spacing (the finite volume doesn't matter in this calculation). Now, in the language of quantum field theory, these are bare parameters, quantum effects will renormalize them, so the actual physical values that the simulation predicts will be different. For example, you might put 75 MeV in for the strange quark mass, but if you turn the crank, and then look at a physical quantity which tells you m_{s} you might find that it's 72.3 GeV. The same is true of the lattice spacing $a$. The spacing you put in, say $a=0.09$ fm ($1$ fm = $10^{-15}$m), will get renormalized, we'll call the renormalized spacing $a'$.

Unlike quark masses, which are very hard to extract, it's pretty easy to extract the renormalized spacing. What you do is compute some mass differences of heavy mesons. The ideal case, that our group uses, is the bound state of a b and an anti-b quark, called an Upsilon. Like the hydrogen atom (or better, positronium) the Upsilon has a ground state, and a whole spectrum of excited states. In addition, because the b quark is so heavy, the dynamics of the system is basically non-relativistic. The latter fact means that we can compute masses of all the excited states fairly cheaply. Even better, we can compute the mass differences between the excited states. This is better because many systematic errors cancel in the differences.

The general procedure then is to start with gauge configurations generated by the MILC collaboration. These are configurations of gauge fields which have the non-perturbative effects of the light quarks "folded in" (or in lattice QCD jargon, they're unquenched). With these configurations we can determine the mass splittings in the Upsilon system, using our non-relativistic quark formalism. Of course, we don't get the actual mass differences out, we're working on a computer, which will only give us dimensionless numbers, so what we get is DM * a', that is the mass difference, multiplied by the renormalized lattice spacing.

Hopefully, you can see where this is going, since we know DM * a', and Upsilon mesons are things you can actually make in the lab, we can use our calculation, and a measurement of DM, to extract a'. We're really fortunate, because several groups (including CLEO here at Cornell) have measured DM very precisely. With that, we can get very precise values for a'.

Now what does this have to do with the strong coupling constant? Remember, I said that the coupling constant depends on the scale at which we measure it, well a' sets the scale for everything on the lattice. And now we know the scale.

The next step is to actually get a value for \alpha, which takes some doing. The first thing you do is take your non-perturbative gauge field configuurations and compute something that you expect to be perturbative. A popular choice is the average value of the gauge fields around 1x1 square on the lattice (which we'll call P). This is a very short distance thing, around 1 fm per side. Now remember, in QCD if something is short distance, we ought to be be able to compute in in perturbation theory. So we fire up our perturbation theory codes and compute P.

My part in this calculation was computing P (and some other quantities) to second order, that is computing P1 and P2 in the expansion,
$$P = 1 + P1 \alpha_{V} + P2 \alpha_{V}^{2} + P3 \alpha_{V}^{3} + ...$$
In this expression $\alpha_{V}$ is the QCD coupling constant, evaluated in a lattice regulated theory (not the MS bar thing we want, but close) and at the scale a' (which we know). Unfortunately, for really high precision second order calculations are not enough, and through heroic efforts two coworkers (Howard Trottier and Quentin Mason) computed the third order coefficient P3.

Now we have everything we need, we've determined P from our simulation, and we also know what it is in perturbation theory. With that we can solve for $\alpha_{V}$ at the scale a'! There are a couple of extra things we have to do after than, first we convert from the lattice regulated coupling to the MSbar one we want, and then we run the scale up to M_{Z}. These steps require more perturbation theory at third order (Howard and Quentin had to do the former calculation, the latter (scale running) was already known), so they're also non-trivial.

What's more, if you just do this using one lattice spacing a', and one short distance quantity (P) you don't get a very precise answer. One reason is that the perturbation theory is not very convergent. That is, contributions from the P4 term (which we haven't computed) are not very small. There are two things we did to get around this problem. One is to run at three different lattice spacings. This allows one to estimate P4 (and P5) from fits, which helps control the error. The other is to use more than one short distance quantity. In the paper there are 28 different short distance quantities used. It turns out that the first trick, using multiple lattice spacings, is the one that really cuts down the error.

So what's the final result? We find
$$\alpha_{s}(M_{Z}) = 0.1177(13)$$
which is more accurate than the world average, and is the single most accurate determination of $\alpha_{s}$.

Of all lattice QCD calculations, I think this one is the most elegant. It nicely mixes perturbative and non-perturbative physics in a non-trivial way. To quote the paper, this calculation "demonstrates that the QCD of confinement is the same theory as the QCD of jets; lattice QCD is full QCD, encompassing both its perturbative and nonperturbative aspects."

## Tuesday, March 15, 2005

### Physics at the south pole

I learned about two really cool experiments today, which I wanted to highlight. I don't follow experimental physics nearly as well as I should, given that I agree with what David Polizter said in his Nobel lecture: "I must say that I do regard theoretical physics as a fundamentally parasitic profession, living off the labors of the real physicists." In general, experiments are what drives the field. Certainly areas with active experimental programs are usually healthier than those without. Condensed matter physics is one example, another is neutrino physics, which is what I'm going to post about.

The colloquim speaker today was Buford Price from Berkley. Dr. Price has many interests, it would be impossible to sum up his whole talk here, even if I could have taken notes at the pace he was going. Instead, I'll say a few words about the amazing neutrino experiment that he was involved in.

Neutrinos are very weakly interacting particles, they can travel vast distances without interacting with anything. The are copiously produced in many astronomical objects that we are interested in studying, from the "mundane" (supernovae) to the more exotic (gamma ray bursters). Many groups are now engaged in studying neutrinos, produced in all sorts of situations. But traditional detectors have trouble studying the highest energy neutrinos. Enter AMANDA.

The AMANDA-II Telescope consists of 19 strings of optical modules buried between 1 and 1.5 miles beneath the snow surface of the geographic south pole. The total number of OMs in the array is 680, although only about 600 are fully operational at any given moment. Cherenkov light is converted to electrical signals by the photomultiplier tubes within each OM.

Buried under the south pole, experimentalists have turned ultra-pure ice into the worlds largest telescope. Because of its size, AMANDA is able to see neutrinos that have much higher energies than more modest, man-made detectors. So far, they've only seen an isotropic distribution of high energy neutrinos, no point source can be resolved.

This of course raises the question: What do you do when your giant telescope made of ice isn't sensitive enough? Yup, you guessed it, you build a bigger, more sensitive, telescope. This one is called ICECUBE. The idea is basically the same as AMANDA, you drill holes in the ice, two or three kilometers down. Then you lower your detectors down on strings. What's different is the scale, ICECUBE occupies a cubic kilometer of space. To put that in perspective, if you stacked up every human being living or dead (all of them) they'd only take up about a third of it. With its size ICECUBE will be sensitive to neutrinos with PeV energies.

## Monday, March 07, 2005

### Hans Bethe, 1906-2005

Hans Bethe, one of the Twentieth century's greatest theoretical physicists passed away peacefully last night, at the age of ninety-eight. For Cornell, this is a very sad day. Bethe was very a very important part of the Cornell community. He basically put Cornell on the map as far as physics goes. Today, thanks to his work, Cornell is a world leading physics school, with strong programs in virtually all areas of physics.

The strength and diversity of the department here is a direct reflection of Hans Bethe's strength and diversity as a theorist. John Bacall's quote is appropriate: "If you know his work, you're inclined to think that he is many different people, all of whom have gotten together and formed a conspiracy to sign their papers under the same name."

In an era of increasing specialization, Bethe worked in many fields. For example, he won the Nobel Prize for his work in stellar astrophysics. Basically, he worked out why stars shine. Many quantum field theorists will know about another important piece of work, his back of the envelope calculation of the Lamb shift. This was the first calculation which showed that renormalization might work, and spurred the development of renormalization theory by Schwinger and Feynman. He did pioneeing work in condensed matter physics, quantum mechanics, and astrophysics, staying active well into his nineties.

It would be totally impossible to summarize his career here, Bethe's contributions to modern physics have been too wide. Indeed his biographer, Sam Schweber has been working for years on a multi-volume biography. For those who are interested in learning a bit more about Hans Bethe, Cornell has a webpage with some biographical information, and videos. The video "I can do that", linked from the "reading" page, offers a nice overview.

## Thursday, March 03, 2005

### A crisis in particle theory?

Peter Woit is worried that particle theory is in trouble, since very few theory papers have made the top 50 most cited list since 1999. According to Peter,

Even more so than last year, this data shows that particle theory and string theory flat-lined around 1999, with a historically unprecedented lack of much in the way of new ideas ever since. Among the top 50 papers, the only particle theory ones written since 1999 are a paper about pentaquarks by Jaffe and Wilczek from 2003 at number 20, the KKLT flux vacua paper at number 29 and a 2002 paper on pp waves at number 32.

How many more years of this will it take before leaders of the particle theory community are willing to publicly admit that there's a problem and start a diiscussion about what can be done about it?

It would seem Peter is living in a different world than me. The problem isn't new ideas, it's the ability to test them which is lacking. Particle theory, particularly of the phenomenological bent, is actually pretty active. Just off the top of my head, here are three very new, very active ideas people have had in particle physics in the last couple of years

1) Split supersymmetry
2) Little Higgs models
3) Warped fermions

Search the arxiv for them, you'll get lots of results. Of course, without an experiment to tell us which new idea is right (or, more likely, that they're all wrong) none of them will get hundreds of citations, but that doesn't imply a lack of new ideas. In fact, it's quite the opposite, there are too many new ideas!

Now assume the LHC turns on and finds solid evidence for option number 2, a little Higgs model, then you can be sure that the original little Higgs papers will rocket up the citition list. Until the LHC turns on, though, we're stuck in a "propose a model, explore some of its consequences, move on" sort of mode. That's precisely the sort of thing that leads to lots of new ideas, and not a lot of citations.

My advice to those who think particle theory is in trouble, read hep-ph on a regular basis :) Of course, particle theory could well be in trouble shortly, it all depends on what the LHC sees, but for now, I'd say it's really quite active. Certainly more active than 5/10 years ago, when there was pretty much SUSY, in the form of the MSSM, and nothing else.


text:
Representations of $p$-adic symplectic groups
Compositio Mathematica, Tome 90 (1994) no. 2, pp. 123-181.
@article{CM_1994__90_2_123_0,
title = {Representations of $p$-adic symplectic groups},
journal = {Compositio Mathematica},
pages = {123--181},
volume = {90},
number = {2},
year = {1994},
zbl = {0797.22008},
mrnumber = {1266251},
language = {en},
url = {http://archive.numdam.org/item/CM_1994__90_2_123_0/}
}
TY  - JOUR
TI  - Representations of $p$-adic symplectic groups
JO  - Compositio Mathematica
PY  - 1994
DA  - 1994///
SP  - 123
EP  - 181
VL  - 90
IS  - 2
UR  - http://archive.numdam.org/item/CM_1994__90_2_123_0/
UR  - https://zbmath.org/?q=an%3A0797.22008
UR  - https://www.ams.org/mathscinet-getitem?mr=1266251
LA  - en
ID  - CM_1994__90_2_123_0
ER  -
Tadić, Marko. Representations of $p$-adic symplectic groups. Compositio Mathematica, Tome 90 (1994) no. 2, pp. 123-181. http://archive.numdam.org/item/CM_1994__90_2_123_0/

[Ba] Barbasch, D., The unitary dual for complex classical groups, Invent. Math. 96 (1989), 103-176. | MR 981739 | Zbl 0692.22006

[BnDKa] Bernstein, J., Deligne, P. and Kazhdan, D., Trace Paley-Wiener theorem for reductive p-adic groups, J. Analyse Math. 42 (1986), 180-192. | MR 874050 | Zbl 0634.22011

[BnZ1] Bernstein, I.N. and Zelevinsky, A.V., Representations of the group GL(n, F), where F is a local non-Archimedean field, Uspekhi Mat. Nauk. 31 (1976), 5-70. | MR 425030 | Zbl 0348.43007

[BnZ2] Bernstein, J. and Zelevinsky, A.V., Induced representations of reductive p-adic groups I, Ann. Sci. École Norm Sup. 10 (1977), 441-472. | Numdam | MR 579172 | Zbl 0412.22015

[BlWh] Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, Princeton University Press, Princeton, 1980. | MR 554917 | Zbl 0443.22010

[Bu] Bourbaki, N., Algébre, Hermann, Paris, 1947-59.

[Cr] Cartier, P., Representations of p-adic groups; a survey, Symp. Pure Math. 33, part 1, pp. 111-155, Amer. Math. Soc., Providence, Rhode Island, 1979. | MR 546593 | Zbl 0421.22010

[Cs] Casselman, W., Introduction to the theory of admissible representations of p-adic reductive groups, preprint.

[F] Faddeev, D.K., On multiplication of representations of classical groups over finite field with representations of the full linear group (in Russian), Vestnik Leningradskogo Universiteta 13 (1976), 35-40. | Zbl 0436.20025

[GbKn] Gelbart, S.S. and Knapp, A.W., L-indistinguishability and R groups for the special linear group, Advan. in Math. 43 (1982), 101-121. | MR 644669 | Zbl 0493.22005

[GfKa] Gelfand, I.M. and Kazhdan, D.A., Representations of GL(n, k), Lie Groups and their Representations, Halstead Press, Budapest, 1974, pp. 95-118. | Zbl 0348.22011

[Go] Goldberg, D., Reducibility of induced representations for Sp(2n) and SO(n), preprint. | Zbl 0851.22021

[Gu] Gustafson, R., The degenerate principal series for Sp(2n), Mem. of the Amer. Math. Society 248 (1981), 1-81. | MR 631958 | Zbl 0482.22013

[Jc] Jacquet, H., Generic representations, Non-Commutative Harmonic Analysis, Lecture Notes in Math. 587, Springer Verlag, Berlin, 1977, pp. 91-101. | MR 499005 | Zbl 0357.22010

[Jn] Jantzen, C., Degenerate principal series for symplectic groups, Mem. Amer. Math. Society 488 (1993), 1-110. | MR 1134591 | Zbl 0814.22004

[KaL] Kazhdan, D. and Lusztig, G., Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153-215. | MR 862716 | Zbl 0613.22004

[Ke] Keys, D., On the decomposition of reducible principal series representations of p-adic Chevalley groups, Pacific J. Math. 101 (1982), 351-388. | MR 675406 | Zbl 0438.22010

[KuRa] Kudla, S.S. and Rallis, S., Ramified degenerate principal series representations for Sp(n), Israel Y. Math. 78 (1992), 209-256. | MR 1194967 | Zbl 0787.22019

[M] Moy, A., Representations of GSp(4) over a p-adic field: parts 1 and 2, Compositio Math. 66 (1988), 237-328. | Numdam | MR 948308 | Zbl 0662.22012

[R1] Rodier, F., Décomposition de la série principale des groupes réductifs p-adiques, Non-Commutative Harmonic Analysis, Lecture Notes in Math. 880, Springer-Verlag, Berlin, 1981. | MR 644842 | Zbl 0465.22009

[R2] Rodier, F., Sur les représentations non ramifiées des groupes réductifs p-adiques; l'example de GSp(4), Bull. Soc. Math. France 116 (1988), 15-42. | Numdam | MR 946277 | Zbl 0662.22011

[SaT] Sally, P.J. and Tadić, M., Induced representations and classifications for GSp(2, F) and Sp(2, F), Memoires Soc. Math. France. 52 (1993), 75-133. | Numdam | MR 1212952 | Zbl 0784.22008

[Sh] Shahidi, F., A proof of Langlands conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. 132 (1990), 273-330. | MR 1070599 | Zbl 0780.22005

[Si1] Silberger, A., Isogeny restrictions of irreducible representations are finite direct sums of irreducible admissible representations, Proceedings Amer. Math. Soc. 73 (1979), 263-264. | MR 516475 | Zbl 0368.22009

[Si2] Silberger, A., Discrete series and classifications for p-adic groups I, Amer. J. Math. 103 (1981), 1231-1321. | MR 636960 | Zbl 0484.22026

[T1] Tadić, M., Classification of unitary representations in irreducible representations of general linear group (non-archimedean case), Ann. Sci. École Norm. Sup. 19 (1986), 335-382. | Numdam | MR 870688 | Zbl 0614.22005

[T2] Tadić, M., Induced representations of GL(n, A) for p-adic division algebras A, J. reine angew. Math. 405 (1990), 48-77. | MR 1040995 | Zbl 0684.22008

[T3] Tadić, M., Notes on representations of non-archimedian SL(n), Pacific J. Math. 152 (1992), 375-396. | MR 1141803 | Zbl 0724.22017

[T4] Tadić, M., On Jacquet modules of induced representations of p-adic symplectic groups, Harmonic Analysis on Reductive Groups, Proceedings, Bowdoin College 1989, Progress in Mathematics 101, Birkhauser, Boston, 1991, pp. 305-314. | MR 1168490 | Zbl 0760.22008

[T5] Tadić, M., A structure arising from induction and restriction of representations of classical p-adic groups, preprint. | Zbl 0874.22014

[T6] Tadić, M., On regular square integrable representations of p-adic groups, preprint. | MR 1600280 | Zbl 0903.22008

[Wd] Waldspurger, J.-L., Un exercice sur GSp(4, F) et les représeniations de Weil, Bull. Soc. Math. France 115 (1987), 35-69. | Numdam | MR 897614 | Zbl 0635.22016

[Wi] Winarsky, N., Reducibility of principal series representations of p-adic Chevalley groups, Amer. J. Math. 100 (1978), 941-956. | MR 517138 | Zbl 0475.43005

[Z1] Zelevinsky, A.V., Induced representations of reductive p-adic group II, On irreducible representations of GL(n), Ann. Sci. École Norm Sup. 13 (1980), 165-210. | Numdam | MR 584084 | Zbl 0441.22014

[Z2] Zelevinsky, A.V., Representations of Finite Classical Groups, A Hopf Algebra Approach, Lecture Notes in Math 869, Springer-Verlag, Berlin, 1981. | MR 643482 | Zbl 0465.20009


text:
# Is it possible to cook meat using starlight and multiple magnifying instruments

So you can burn a piece of paper with magnifying glass and sunlight.

Light from stars are similar to that emitted by the sun, just much, much weaker.

Theoretically, is it possible to cook a thinly sliced piece of meat using powerful magnifying instruments such as gigantic converging lens or dish or combination of those to converge light (and thus, energy) from stars?

This does not have to be on earth, it could be carried out in space for example so there is less interference from external factors such as clouds etc.

If it can not be done, what is the limiting factor?

• Well, if you go far enough into space, and close enough to another star, you can just do the same thing as you do with the sun :) – milo Dec 23 '16 at 10:20
• May be you can, its just a matter of size of your lenses, but the time its takes will be crucial and meat may not be eatable – Digvijay Yadav Dec 23 '16 at 10:21
• Also, take a look at this: what-if.xkcd.com/145 . It's a different question because unlike the moon, the stars ARE hot, but still interesting. – milo Dec 23 '16 at 10:22

Purely in theory, yes you could.

You don't really need the light to be of a specific wavelength to generate heat. You mostly just need a high enough concentration of photons.

The problem is that the number of photons falling on earth from a star is many times lower than that from the sun, so you'd have to concentrate the energy from an immense area to get the same intensity.

Ignoring, for the moment, the filtering from the earth's atmosphere (and such), we receive about 1021 photons per square meter per second from the sun. The brightest star (other than the sun) is Sirius. We receive about 109 photons per square meter per second from Sirius, so we'd need to magnify by a factor of about 1012 just to match the intensity we reach from the sun without any magnification1.

Doing a bit of quick math, that works out to a single round magnifier about 10 kilometers in diameter, or equal area in other forms.

Then figure that cooking meat with sunlight does require at least some sort of magnifier, so we can probably multiply that by a factor of at least 3, and probably more like 7-10 or so.

Of course, for now I'm ignoring a such minor details as how you even design (not to mention building) a system even close to that size. There would be decidedly non-trivial challenges involved. For example, the largest mirrors we've built yet have diameters around 10 meters (or non-round mirrors of roughly equivalent area). These are already built in sections, with a computer to control movement of the individual pieces to maintain the illusion of a single mirror acting as a unit. Trying to multiply that out to multiple kilometers--well, I don't think anybody's even contemplated what that would take yet.

1. side note: both these numbers are really for the number of photons hitting the top of the atmosphere, not what we see on earth after being filtered by the atmosphere. We don't really care a whole lot about that though--the intent is just to get at least some idea of what it takes to concentrate starlight to approximately match sunlight.

Back in the days of Kepler, the theory was, that if sum of the combined light from all visible to the naked eye stars, approximately 5-10,000 max, was equal to or greater than, that received from the Sun. Then we would live in permanent daylight. This theory is still true today!

It would be possible to cook some meat in sunlight. If you live in a desert, just pop a black skillet in the sun and wait couple of hours, this would store enough energy from the sun to fry a egg, albeit slowly, because you need about 160°C to start frying anything, although 55°C is about enough for a rare steak. Now try the same thing under a full moon, it won't work, why? Because the moon only reflects the shorter wavelengths of light and not the infra red wavelengths, which it absorbs, and it is the infra red energy that you need to cook. By the time starlight actually reaches us, all the infra red radiation has been absorbed by interstellar dust or hydrogen molecules.

Therefore, the answer to you question would a be definite NO! No matter how many or how big your optical lenses were!

Another point is, that if infra red radiation from stars were actually reaching us on Earth, there would be a lot of amateur and professional astronomers with burnt retinas.

• I am sure you are right about how the moon filters sunlight, but starlight is not subject to the same filtration and contains plenty of IR, ionising radiation. To restate, the energy of each photon is the same in starlight, there are just many less of them. Do you need IR to cook, or is it just the most efficient absorption frequency for organic matter? – JMLCarter Dec 23 '16 at 12:46
• I would have thought shorter wavelength part of the light carries more energy. What are you basing the statement: 'it is the infra red energy that you need to cook' on? – TelKitty Dec 23 '16 at 12:51

In fact weaker light does not contain photons with less energy, it contains less photons. Which means that the heating and the ionising effect (aka burning) will be the same, just slower. So you could just leave your meat in space for a really long time and eventually it would be burnt. Or if you are not very hungry you can have a really really tiny "piece of meat", like a single atom, it might not take so long.

Cosmic ray ionisation is a real problem/effect both in space and on the surface of the earth, especially for semi-conductor electronics in which it can cause data corruption, a bit-flip. Satellites have to be built with measures to protect against such corruption, either through shielding or detecting it.

• So this is why Tyrannosaurus Rex never throve in space, no? Your answer is correct in many ways, but I'm not sure about the "just slower" bit: cooking of meat involves change of proteins that in turn requires higher temperature, so I don't think the meat won't get there unless the flux of light is high enough to be equal to $\sigma\,T^4$, where $T$ is high enough to cause this change and $\sigma\,T^4$ is the rate of radiation emission per unit surface area that balances the incoming radiation at steady state. You would need to arrange your magnifying instruments for this to be so. – WetSavannaAnimal Dec 23 '16 at 11:54
• I was thinking it might slowly burn but not cook, since ionisation can still occur. – JMLCarter Dec 23 '16 at 12:04
• I'm no protein chemist, so I don't really know, but I have a hunch that meat does need to get to a specific temperature to "cook" as we'd normally understand that word. But I think in principle you could gather enough light to do this - there's no principle barring this and in particular it won't worry the second law as there's no possibility of the meat's rising to a higher temperature than the source starlight - it's just going to be really hard to do it with starlight! – WetSavannaAnimal Dec 23 '16 at 12:09


text:
# What's the relationship between Variational Autoencoders and the Expectation Maximization Algorithm?

I know that the EM algorithm is used in latent variable models, specifically to do maximum likelihood estimation iteratively. Similarly, the VAE can be used for latent variable models and, although they are usually used for generative modelling or posterior inference, they can also be used for parameter inference. So I was wondering what's the relationship between them and when it's better to use one or the other.

What is the relationship between VAE and EM?

# Notation

Observed data: $$\mathcal{D} = \{\vx_1, \vx_2, \ldots, \vx_N\}$$

Latent variables denoted by $$\vz$$.

# Expectation Maximization Algorithm (Standard Version)

The EM algorithm is often (e.g. see Wikipedia) described as follows.

Start with a guess $$\vtheta^{(0)}$$, then until convergence:

• Compute expectations $$\Ebb_{p(\vz \mid \vx, \vtheta^{(t)})}[\log p_{\vtheta}(\vx, \vz)]$$ for every data point $$\vx\in \mathcal{D}$$.
• Choose parameter value $$\vtheta^{(t+1)}$$ to maximize expectations $$\vtheta^{(t+1)} = \arg\max_{\vtheta} \sum_{\vx\in\mathcal{D}}\Ebb_{p(\vz \mid \vx, \vtheta^{(t)})}[\log p_{\vtheta}(\vx, \vz)]$$

# Expectation-Maximization Algorithm (Rewritten)

One can rewrite the algorithm above in a slightly different way. Rather than computing expectations in the first step, we compute the distributions $$p(\vz\mid, \vx, \vtheta^{(t)})$$. The EM algorithm then looks as follows:

Start with a guess $$\vtheta^{(0)}$$, until convergence:

• Compute distributions $$\left\{p(\vz\mid, \vx, \vtheta^{(t)}) \, : \, \vx \in \mathcal{D}\right\}$$
• Choosen new parameter value in the same way as before $$\vtheta^{(t+1)} = \arg\max_{\vtheta} \sum_{\vx\in\mathcal{D}}\Ebb_{p(\vz \mid \vx, \vtheta^{(t)})}[\log p_{\vtheta}(\vx, \vz)]$$

# Variational Autoencoders

Why did I rewrite it like that? Because one can write the ELBO, which is usually considered as a function of $$\vx$$ parametrized by $$\vtheta$$ and $$\vphi$$ ($$\vphi$$ are the parameters of the encoder $$q_{\vphi}$$), as a functional of $$q_{\vphi} and a function of$$\vtheta$$that is parameterized by$$\vx\$ (indeed the data is fixed). This means the ELBO can be written as:

$$\begin{equation*} \mathcal{L}_{\boldsymbol{\mathbf{x}}}(\boldsymbol{\mathbf{\theta}}, q_{\boldsymbol{\mathbf{\phi}}})= \begin{cases} \displaystyle \log p_{\boldsymbol{\mathbf{\theta}}}(\boldsymbol{\mathbf{x}})- \text{KL}(q_{\boldsymbol{\mathbf{\phi}}}\,\,||\,\,p_{\boldsymbol{\mathbf{\theta}}}(\boldsymbol{\mathbf{z}}\mid \boldsymbol{\mathbf{x}})) \qquad \qquad &(1)\\ \qquad \\ \displaystyle \mathbb{E}_{q_{\boldsymbol{\mathbf{\phi}}}}[\log p_{\boldsymbol{\mathbf{\theta}}}(\boldsymbol{\mathbf{x}}, \boldsymbol{\mathbf{z}})] - \mathbb{E}_{q_{\boldsymbol{\mathbf{\phi}}}}[\log q_{\boldsymbol{\mathbf{\phi}}}] \qquad \qquad &(2) \end{cases} \end{equation*}$$

Now we can find two identical steps as those of the EM algorithm by performing maximization of the ELBO with respect to $$q_{\vphi}$$ first, and then with respect to $$\vtheta$$

• E-step: Maximize $$(1)$$ with respect to $$q_{\vphi}$$ (this makes the KL-divergence zero and the bound is tight) $$\left\{p_{\boldsymbol{\mathbf{\theta}}^{(t)}}(\boldsymbol{\mathbf{z}}\mid \boldsymbol{\mathbf{x}})= \arg\max_{q_{\boldsymbol{\mathbf{\phi}}}} \mathcal{L}_{\boldsymbol{\mathbf{x}}}(\boldsymbol{\mathbf{\theta}}^{(t)}, q_{\boldsymbol{\mathbf{\phi}}})\,\, : \,\, \boldsymbol{\mathbf{x}}\in\mathcal{D}\right\}$$
• M-step: Maximize $$(2)$$ with respect to $$\vtheta$$ $$\boldsymbol{\mathbf{\theta}}^{(t+1)} = \arg\max_{\boldsymbol{\mathbf{\theta}}} \sum_{\boldsymbol{\mathbf{x}}\in\mathcal{D}} \mathcal{L}_{\boldsymbol{\mathbf{x}}}(\boldsymbol{\mathbf{\theta}}, p_{\boldsymbol{\mathbf{\theta}}^{(t)}}(\boldsymbol{\mathbf{z}}\mid \boldsymbol{\mathbf{x}}))$$

The relationship between the Expectation Maximization algorithm and Variational Auto-Encoders can therefore be summarized as follows:

• EM algorithm and VAE optimize the same objective function.

• When expectations are in closed-form, one should use the EM algorithm which uses coordinate ascent.

• When expectations are intractable, VAE uses stochastic gradient ascent on an unbiased estimator of the objective function.

As you stated, both EM and VAE are machine learning techniques/algorithms to find the latent variables z. However, despite the overall goal and even the objective function being the same, there are differences because of the complexities of the model.

There are 2 issues at hand where EM (and its variants) have limitations. These are mentioned in the original VAE paper by Kingma. I am pasting the relevant part instead of rewriting the already well-written problems/challenges in the paper (https://arxiv.org/abs/1312.6114)

From section 2.1 of the paper -


text:
# Difference between revisions of "2015 AMC 10A Problems/Problem 13"

## Problem 13

Ashraful has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Ashraful have?

$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

## Solution 1

Let Ashraful have $x$ 5-cent coins and $\left( 12 - x \right)$ 10-cent coins. It is easily observed that any multiple of $5$ between $5$ and $5x + 10(12 - x) = 120 - 5x$ inclusive can be obtained by a combination of coins. Thus, $24 - x = 17$ combinations can be made, so $x = 7$. But the answer is not $7,$ because we are asked for the number of 10-cent coins, which is $12 - 7 = \boxed{\textbf{(C) } 5}$

## Solution 2

Since the coins are 5-cent and 10-cent, all possible values that can be made will be multiples of $5.$ To have exactly $17$ different multiples of $5,$ we will need to make up to $85$ cents. If all twelve coins were 5-cent coins, we will have $60$ cents possible. Each trade of a 5-cent coin for a 10-cent coin will gain $5$ cents, and as we need to gain $25$ cents, the answer is $\boxed{\textbf{(C) } 5}$

For small scale substitutions of 5 cent for 10 cent, like in this problem, it is quite fast, however, if the problem is a little bit more complex, knowing that you need 85 cents, explained above, it is also possible to use a system of equations. It would be $$5x+10y=85$$ $$x+y=12$$ Solving this $x$ (5-cent coins) $= 7$ and $y$ (10-cent coins) $= 5$, so again the answer is $\boxed{\textbf{(C) } 5}$

~savannahsolver


text:
# Projective varieties: Reduction to the affine case

I ran across a certain type of argument for the second time now. Assume that $f$ is some rational map between projective varieties $X$ and $Y$, then supposedly I may replace $X$ with an open affine subset. It follows that the target space is also affine.

This might be a dumb question, but how exactly do I construct those open sets? I initially just took the standard affine covering of $X$, but I don't see how the image of such an open set under $f$ is affine. Wouldn't the image have to have empty intersection with some hyperplane?

-
what you can do is take an affine open subset of $Y$ then take a preimage under the map, then restrict to an open affine subset of the preimage. –  Dima Sustretov Mar 10 '12 at 13:22
Thanks; I know that the function field in the target space will then be isomorphic to the function field of Y, but is the same true for the domain? –  Guest Mar 12 '12 at 12:26
function field of a dense open subset is of course the same as the function field of the whole variety –  Dima Sustretov Mar 12 '12 at 13:11
I see, thank you. That essentially answers my question, so if you want to reformulate your comment as an answer, I'll gladly accept it. –  Guest Mar 12 '12 at 14:31
glad to be of help. –  Dima Sustretov Mar 13 '12 at 23:44


text: In this paper, we investigate the Ulam-Hyers-Rassias stability for the Drygas functional equation $f( x + y ) + f( x - y ) = 2f (x) + f (y)+ f(-y)$ in the setting quasi-Banach spaces using the approach of Brzdek’s fixed point theorem. Also, we gave a general result on the hyperstability of Drygas functional equation. The results obtained in this paper extend various previously known results to the setting of quasi-Banach space.


text:
# MCO16203 - Editorial

Author: Zi Song Yeoh

EASY

+1, -1 Trick

### PROBLEM:

You have a n \times n binary grid. There are q operations, which involves flipping all the bits in a subrectangle. Output the final grid.

### EXPLANATION:

For subtask 1, iterating through all bits in the subrectangle and flipping them naively gives a simple O(qn^{2}) solution.

To solve the full task, we need to make the update process faster. First, let’s consider the problem in 1D. Suppose we only have a 1 \times n grid instead. Let a_{1}, a_{2}, ..., a_{n} be the original bits. We define a new sequence d_{1}, d_{2}, ..., d_{n + 1}, so that d_{i} = a_{i} - a_{i - 1} for 2 \le i \le n, d_{n + 1} = a_{n}, d_{1} = a_{1}. Then, to flip a range [l, r], we just flip d_{l} and d_{r + 1}, and the resulting values of a_{i} can be restored with a simple loop at the end.

The 2D case is similar. First, we define the sequence d for each row, and now we note that when we update a subrectangle, we’re effectively performing range [l, r] updates of a subsegment of d_{i} in a column. We can use the same trick again to reduce each query to flipping 4 corners of a rectangle. Finally, we can restore the grid by taking the prefix sums on rows and columns. You can see the code for the details.

The complexity of the solution is O(q + n^2).

### AUTHOR’S AND TESTER’S SOLUTIONS:

Author’s solution can be found here.

Tester’s solution can be found here.

//