The first line of defense against a terrorist attack is oftentimes a vigilant and aware population.
Soft targets are more vulnerable to terrorism because they have lower levels of security, making them easier to attack, whether by a lone actor or a small team of terrorists relying on basic surveillance and simple plans. The past several years have shown that no community is totally immune from this threat. Any public event can easily be a target—a club in Orlando, an office holiday party in San Bernardino, or a marathon in Boston.
Attacks that kill a significant number of people (such as the forty-nine who died more than a year ago in Orlando) or are directed against a symbolic target, meaning it has special significance to a city or to the country (the 2013 Boston Marathon bombing), will likely result in increased media coverage, a primary goal of most terrorist groups.
In the past year, terrorists inspired by the Islamic State have been opportunistic, increasingly tailoring their attacks to relatively simple targets by using vehicles as ramming devices. And while New Jersey–style metal barriers and stanchions can help mitigate this threat, it is impossible to secure all public spaces from vehicular terrorism.
The sheer range of soft targets seems endless—and include shopping malls, sports arenas, museums, restaurants, airports, concert venues and public transportation systems. And not all soft targets are created equal. Because of heightened security measures, it would be more difficult for terrorists to attack a sports stadium or public-transit hub, for instance, than a shopping mall or restaurant.
In other parts of the world, terrorists have targeted hotel chains based in Western countries or those frequented by westerners. Between 2002 and 2015, these hotels have been attacked in Pakistan, Indonesia, Egypt, Jordan and Mali. These attacks, some of which included suicide bombings, resulted in many deaths and widespread property damage, while also having a negative impact on Western tourism. Terrorism expert Peter Bergen said he believes that these kinds of hotels are attractive terrorist targets for two primary reasons: Since they are in the hospitality business, they are unlikely to "turn themselves into fortresses" and they are likely to be frequented by Western tourists, an attractive target.
In the United States, terrorists have frequently targeted transportation, from softer targets like commuter trains and subways to more hardened targets like commercial aviation. The commandeering of airplanes on Sept. 11, 2001, to attack buildings in New York City and Washington, DC is the most high-profile example of the latter. And while American authorities worked assiduously to increase defensive measures designed to protect transportation infrastructure, these types of targets continue to be attractive to terrorists.
After receiving training in Pakistan in 2007, Al Qaeda recruit Bryant Neal Vinas plotted to bomb the Long Island Rail Road, a busy commuter-train service that connects New York City to its outer suburbs. In 2009, Afghan immigrant Najibullah Zazi was arrested before he could carry out his plan to bomb the New York City subway system. Commuter trains and buses have been successfully attacked by terrorists in Madrid, London and Moscow.
Meanwhile, the focus on airplanes remains a high priority for terrorists linked to Al Qaeda and the Islamic State. In June, the New York Times reported that Israeli intelligence had gleaned from ISIS operatives information about the group's quest to make explosives that look exactly like laptop computer batteries in an effort to fool X-ray machines at airports.
As these examples show, soft targets chosen by terrorists share three important characteristics—they are frequented by many people, are easily accessible and can be surveilled without drawing too much attention to the potential attackers. Oftentimes, these targets can be monitored virtually through imagery and details available online. As a whole, soft targets are hard to defend—there are too many of them and defense would require an enormous application of resources.
The United States is forced to make choices regarding where to apply limited resources to the defense of soft targets. Providing better security for federal, state and local government offices and agencies as well as transportation hubs like airports and train stations has, rightly, been the focus of these efforts. However, when one potential target is hardened, other softer targets become more susceptible to attack.
So what, if anything, can the United States do to protect soft targets from terrorist attacks?
The U.S. government should redouble its information-sharing efforts with other governments. It should also expand programs already in place for sharing information with local law enforcement and provide increased resources for sustainment, training and other initiatives dedicated to this kind of threat. Many terrorists or would-be terrorists are increasingly referred to as "known wolves" because they are known to intelligence and police agencies prior to an attack. Broad sharing of information among intelligence agencies and police services coupled with additional training could help identify and track potential attackers before they can execute their plans.
State and local governments should also consider establishing where appropriate heightened security standards—not just for frequently traveled public spaces but also for privately owned businesses and venues. U.S. government agencies like the Department of Homeland Security already provide additional resources for high-profile events like the Super Bowl and on balance, have done an admirable job of providing security without being overly invasive.
Better public-private partnerships are necessary to protect soft targets from attack and to ensure that first responders and emergency medical services are included in planning from the outset. The relationship between the public and private sectors should be mutually beneficial. After all, reliable and consistent security is good for business.
While many businesses already implement a variety of physical security features based on their specific needs—closed-circuit television, perimeter fencing and lighting, automatic door-locking after hours—the government should continue to work with them to define thresholds where additional security assets and procedures are not optional but required. Enhanced standards should be set for lighting, guards, identification and other preventive measures for occasions when hotels, concert halls, or convention sites are hosting events or are expecting large crowds. Furthermore, businesses should be required to coordinate with applicable law-enforcement agencies during and prior to events where larger-than-normal crowds are expected.
In addition, public-information campaigns like "see something, say something" should be advertised prominently in public spaces prior to or during events where large crowds are expected to gather. Because police and guards cannot be deployed everywhere or in sufficient numbers to secure everything and everyone, the first line of defense against a terrorist attack is oftentimes a vigilant and aware population.
Image: A couple (L) and a jogger pass by the Lincoln Memorial Reflecting Pool before sunrise in Washington, September 24, 2013. The nation's capital is enjoying mild temperatures during the first week of autumn. REUTERS/Jason Reed.


text:
HomeHCF ReviewsHAVENHURST: The Hughes Verdict! (aka RESURRECTION OF EVIL)
HAVENHURST: The Hughes Verdict! (aka RESURRECTION OF EVIL)
January 25, 2017 Ross Hughes HCF Reviews, Horror 0
Havenhurst, Resurrection of Evil (2017)
Directed by: Andrew C. Erin.
Written by: Andrew C. Erin., Daniel Farrands
Starring: Belle Shouse, Danielle Harris, Fionnula Flanagan, Jennifer Blanc-Biehn, Julie Benz
Jackie (Benz), a troubled young woman struggling with addiction, is released from rehab and given a second chance with a new job and a furnished apartment at Havenhurst. Guilt-ridden over the loss of her 8-year-old daughter, Jackie is quickly drawn into the mysteries of Havenhurst, in particular the disappearance of apartment 1006's previous lost soul, a young woman (Harris) she befriended in rehab who vanished without a trace.
Aided by a hardened New York police detective (Stamberg) and a lonely foster child (Shouse) who lives under the sadistic shadow of her caretakers, Jackie must not only battle her inner demons… but the very real ones that live within the walls of Havenhurst.
The Trailer:
The Verdict!
Lets start off with the negative and yes I am going to talk about Danielle Harris.
When Bat (our HCF boss) sent me down this screener with the words "You are going to love this as it stars a certain actress", then my horror juices were flowing. You see, I knew straight away what she meant. That this film stared a certain Scream Queen who I have an undying soft spot for. Roughly the same age, I watched Miss Harris way back in the late 80's as she took over from Jamie Lee Curtis to become a target for her Uncle Michael Myers and was the best thing in not only the Halloween sequels, but of Rob Zombie's two remakes as well!
Watching her as a potty mouthed kid in The Last Action Hero only made her stock go up in my eyes and since then she has carved a nice little career in the horror genre, becoming a sort of icon for fans everywhere. From Hatchet 2 to the underrated Blood Mary, there are films out there that show Harris at her best, sadly Havenhurst (also known as Resurrection of Evil in the UK) isn't one of them. Why? Because if you going to hire Harris for a role much similar to Drew Barrymore in the original Scream, then you are going to annoy a lot of people.
If you don't get what I mean, then you will once you sit down and watch the film. But damn you writers- damn you indeed! Its one of the reasons why I have no time for Urban Legend
Luckily another icon takes her place soon after and I am talking about Julie Benz – but wait! She has black hair! Nooooooooooooooooooooooooooo! You see I have this theory that the colour of her hair judges the quality of material on offer. When its black she has starred in Punisher -War Zone and Saw V, not the best of films by a long shot, but when her hair is blonde, well its Buffy. Rambo, Dexter, No Ordinary Family…. *cough* – forget that one…..well you get my point!
Benz and her black hair plays Jackie, a woman struggling with an alcohol addiction and a tragedy from the past, who moves into the huge building block of Havenhurst that with first glance, reminds you of the place that Sharon Stone stepped into in the 90's flick Sliver, only now dark and rundown with a hint of decay in the air. Even when we come to our first sex scene – hey its an horror it has to be one – a camera is filming the woman jumping up and down, her breasts swinging from left to right, faking an orgasm. Who is watching? Is it William Baldwin? Is he still furious that Stone walked out on him all those years ago for being a pervert? While I admit that would have been stupid and dumb (but still cool, I mean who doesn't want a Sliver 2?…anyone?…anyone?),
Jackie is told the rules of her stay by the landlady of the place, Eleanor Mudgett, (The Other's Fionnula Flanagan in her usual creepy panache). Every tenant has a history of moral undoing, you can stay in Havenhurst as long as you don't return to your old life and manage to resist temptation. If you fail and succumb to your vices, then you will be evicted. But being "evicted" doesn't mean being thrown onto the streets, in Havenhurst – it means torture and death.
Obviously Jackie is not told the last part. I mean if she was, then 99% of people would say "Thanks for the offer but I'd rather live in cardboard box!" and then we wouldn't have much of a film, but thankfully for us, but not for her, she signs on the dotted line and moves in, but soon realises that all is not well within the place. Screams in the night, mystery clues left behind by Danielle Harris (damn you writers….damn you……), the longer the film goes on, the move flaws in the plot become apparent, but luckily the film moves with such a brisk pace that it doesn't give the brain a chance to digest what the problems are on screen.
To be fair the film works for most of its running time. This is not a ground breaking horror by any stretch and its much similar to Tobe Hooper's The Toolbox Murders, but what it offers is some effective jump scenes and a cool bad guy that looks like Vin Diesel, after he has gone on a massive diet!
Basically what we have in Havenhurst is a film that starts off like a potential haunted house horror but then makes the viewer fall into numerous trap doors until we settle on a riff of the Saw franchise. Torture porn was the rage back in the noughties and the horror genre really suffered from hundreds of straight to DVD knock offs, but since Found Footage took over, this film surprisingly came across like a breath of fresh air.
While not too gratuitous like many before, it, the film has enough blood to keep the gore hounds happy and while you may sigh at the predictability of it all, I mean why are Cops in horror films so dumb? I was willingly for Jackie and her young female friend Sarah (Belle Shouse) to survive the terror within the walls.
The mystery of Havenhurst won't surprise any horror veteran but while the finale moves towards a big long chase through dark corridors, with the ever dependable Benz always carrying the vital screen presence, you realise that the film is a harmless Friday night entertainment that just does enough to appease those wanting a quick thrill.
I admit and say that the climax left a slight bitter aftertaste and most of you will see it coming, but nevertheless, Havenhurst excels thanks to its strong cast and a mixture of some good jump out of your skin moments, a nice long stalk and chase scene and enough lashes of blood.
If only they didn't do that to Danielle Harris…..damn you writers, damn you indeed!
Brainstorm Media and Twisted Pictures have announced the February 10th limited theatrical and nationwide Cable VOD and Digital HD release of Andrew C. Erin's Havenhurst. Co-written by Erin (Sam's Lake, Simple Things) and Daniel Farrands (Amityville: The Awakening, Haunting in Connecticut, The Id), the film stars genre vet (Julie Benz, "Defiance", "Dexter", "Buffy the Vampire Slayer") as a woman battling her own demons while trying to uncover what happened to her apartment's previous tenant (Danielle Harris, Halloween 4 & 5, Rob Zombie's Halloween, the Hatchet franchise). Havenhurst will debut in Los Angeles at the Laemmle Music Hall.
Havenhurst will also begin weeklong theatrical runs in San Francisco, Denver, Philadelphia, Atlanta, Houston, Kansas City and Phoenix. Nationwide, Havenhurst will be available on Cable VOD and Digital HD platforms, including Charter Spectrum, Comcast, DirecTV Cinema, Dish, iTunes, Amazon Instant Video, Google Play, Vudu and more.
Andrew C. Erin.
Belle Shouse
Daniel Farrands
Jennifer Blanc Biehn
The Hughes Verdict
About Ross Hughes	1819 Articles
Since my mother sat me down at the age of five years of age and watched a little called Halloween, I have been hooked on horror. There is no other genre that gets me excited and takes me to the edge of entertainment. I watch everything from old, new, to cheap and blockbusters, but I promise all my readers that I will always give an honest opinion, and I hope whoever reads this review section, will find a film that they too can love as much as I do! Have fun reading, and please DO HAVE NIGHTMARES!!!!!!
More names join the cast of horror sequel 'See No Evil 2'
A quick bit of casting news here on The Soska Sisters upcoming sequel, See No Evil 2. Greyston Holt ( "Bitten," Lost Boys: The Tribe), Chelan Simmons (pictured above: Tucker & Dale vs. Evil, Final Destination […]
Fiery US theatrical poster for 'Hatchet 3' arrives!
May 23, 2013 Matt Wavish News 0
The official US theatrical poster for Hatchet 3 is here, and man there is a lot going on in the poster. Victor Crowley (Kane Hodder) stands front and centre, and looks as menacing […]
Batman Vs Superman: Dawn Of Justice: The Hughes Verdict!
March 25, 2016 Ross Hughes Action, Fantasy, HCF Reviews, Sci-Fi 0
Its been hard to avoid any spoilers over the last few days for this eagerly awaited smash up between two iconic Superheroes. If I am doing the main review for HCF, then I chose not […]

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to write data in parquet format I have some data in my ArrayList> can we write this data  in parquet file or not, if yes then how,please recommend and Java program,thanks in advance.
Please Note I am not using any Spark,Kafka...

A: Previous link mostly contains Avro schema examples (I'm wondering, why people don't read the question before an answer). I did not manage to find any reliable examples so far. What you can use is this schema file and this article. Also, there is a repo with more examples.

A: Have you googled it?
Try this, for example.

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction} \label{sec:introduction}
In the last decade, the rapid developments in machine learning and artificial intelligence have captured the imagination of many scientists, across a full spectrum of disciplines. Mathematics has not been immune to this phenomenon. In fact, quite the opposite has happened and many mathematicians have been at the forefront of this fundamental research effort. Their contributions ranged from statistics, to optimization, to approximation theory, and last but not least, to harmonic analysis and related representation theory.
It is this last aspect that we want to focus on in this paper, as it has lead to many intriguing developments associated with the general theory of deep learning, and more specifically, with convolutional neural networks, see \cite{mallat2016}.

Convolutional neural nets (CNNs) are a popular type of architecture in deep learning, which has shown an outstanding performance in various applications, e.g., in image and video recognition, in image classification \cite{ciregan2012multi}, \cite{ciresan2011flexible}, or in natural speech processing \cite{collobert}. CNNs can be effectively modeled by multiscale contractions with wavelet-like operations, applied interchangeably with pointwise nonlinearities \cite{mallat2016}. This results in a wealth of network parameters, which can negatively impact the numerical performance of the network. Thus, a form of dimensionality reduction or data compression is needed in order to efficiently process the information through the artificial neural network.
For these purposes many examples of CNNs use \emph{pooling} as a type of layer in their networks. Pooling is a dimension reduction technique that divides the input data into subregions and returns only one value as the representative of each subregion. Many examples of such compression strategies have been proposed to-date. Max pooling and average pooling are the two most widely used traditional pooling strategies and they have demonstrated good performance in application tasks \cite{goodfellow2016deep}. In addition to controlling the computational cost associated with using the network, pooling also helps to reduce overfitting of the training data \cite{graham}, which is a common problem in many applications.

In addition to the classical examples of maximal and average pooling, many more pooling methods have been proposed and were implemented in neural net architectures. Among those constructions, a significant role has been played by ideas from harmonic analysis, due to their role in providing effective models for data compression and dimension reduction.
Spectral pooling was proposed in \cite{spectral} to perform the reduction step by truncating the representation in the frequency domain, rather than in the original coordinates. This approach claims to preserve more information per parameter than other pooling strategies and to increase flexibility for the size of pooling output.
Hartley pooling was introduced in \cite{Hartley} to address the loss of information that happens in the dimensionality reduction process. Inspired by the Fourier spectral pooling, the Hartley transform was proposed as the base, thus avoiding the use of complex arithmetic for frequency representations, while increasing the computational effectiveness and network's discriminability.
Transformation invariant pooling based on the discrete Fourier transform was introduced in \cite{tip} to achieve translation invariance and shape preservation thanks to the properties of the Fourier transform.
Wavelet pooling \cite{waveletp} is another alternative to the traditional pooling procedures. This method decomposes features into a two level decomposition and discards the first-level subbands to reduce the dimension, thus addressing the overfitting, while reducing features in a structurally conscious manner.
Multiple wavelet pooling \cite{mwaveletp} builds upon the wavelet pooling idea, while introducing more sophisticated wavelet transforms such as Coiflets and Daubechies wavelets, into the process.
An even more general approach was proposed in \cite{elpp}, where $\ell_p$ pooling was defined based on the concept of a representation in terms of general frames for $\mathbb{R}^d$, to provide invariance to the system.

In this paper we follow in the footsteps of the aforementioned constructions and propose a novel method for reducing the dimension in CNNs, which is based on a fundamental concept from harmonic analysis. Inspired by the \textit{Hardy--Littlewood maximal function}, \cite {HL}, cf., \cite{coifman1974weighted}, \cite{muckenhoupt1972weighted} for its modern treatment, we introduce a novel pooling strategy, called \textit{maxfun pooling}, which can be viewed as a natural alternative to both max pooling and average pooing. In particular, max pooling takes the maximum value in each pooling region as the scalar output, and average pooling takes the average of all entries in each pooling region as the scalar output. As such, maxfun pooling can be interpreted as a novel and creative way of interpolating between max and average pooling algorithms.

In what follows, we introduce a discrete and computationally feasible analogue of the Hardy-Littlewood maximal function. The resulting operator depends on two integer parameters $b$ and $s$, corresponding to the size of the pooling region and the stride. We limit the support of this operator to be finite, we discretize it and define the maximal function pooling operation, denoted throughout this paper by \textit{maxfun pooling}. The maxfun pooling computes averages of subregions of different sizes in each pooling region, and it selects the largest average among all. To demonstrate the features of maxfun pooling, we present two different applications. First, we study its properties in the realm of convolutional sparse coding. It has been shown that feedforward convolutional neural networks can be viewed as convolutional sparse coding \cite{papyan2016convolutional}. Moreover, under the point of view presented by the convolutional sparse coding, stable recovery of the signal contaminated with noise can be achieved, given simple sparsity conditions \cite{papyan2016convolutional}. Equivalently, it implies that feedforward neural networks maintain stability under noisy conditions. The case of pooling function analyzed via convolutional sparse coding is studied in \cite{kabkab2017spatial}, where the two common pooling functions, max pooling and average pooling are analyzed. We follow the framework presented in \cite{kabkab2017spatial} and we show that stability of the neural network under the presence of noise is also preserved with maxfun pooling. We close this paper with a different application, presenting illustrative numerical experiments utilizing maxfun pooling for image classification.

\section{Preliminaries}  \label{sec:prelim}

In this section, we elaborate upon the role of pooling in neural networks, and discuss two traditional strategies, max and average pooling. We focus on image data as our main application domain, but we mention that the subsequent definitions can be readily generalized.

We view images as functions on a finite lattice $X\colon [M]\times [N]\to\mathbb{R}$, where $[K]:=\{0,1,\dots,K-1\}$, or $X\in\mathbb{R}^{M\times N}$ in short. Its $(i,j)-th$ coordinate is denoted $X_{i,j}$. In practice, it is convenient to fold images into vectors. Slightly abusing notation, for $X\in\mathbb{R}^{M\times N}$, we also let $X\in\mathbb{R}^{MN}$ denote its corresponding vectorization,
\begin{equation}
\label{eq:vectorization}
X
=(X_{1,1}, X_{2,1},\dots, X_{M,1}, X_{1,2},\dots, X_{M,2}, \dots, X_{M,N})^T.
\end{equation}
Throughout this chapter, we will not make distinctions between vectors and images.
As a consequence, we shall assume without any loss of generality that pooling layer's input is always nonnegative.

\subsection{Max and average pooling}

Both the maximum and average pooling operators depend on a collection of sets $\mathcal{Q}\subset [M]\times[N]$, which we refer to {\it pooling regions}. There is a standard choice of $\mathcal{Q}$, which we describe below. Fix an odd integer $s$ and set $m=\lfloor M/s\rfloor$ and $n=\lfloor N/s \rfloor$. For each pair of integers $(k,\ell)$, we define the square $Q_{k,\ell}$ of size $s\times s$ by
\begin{equation}
\label{eq:Q}
Q_{k,\ell }
:= \{(i,j)\colon (k-1)s \leq i < ks, \, (\ell -1)s \leq j < \ell s\}.
\end{equation}
It is common to refer to $s$ as the {\it stride}. The stride determines the size of each pooling region $Q_{k,\ell}$ and the total number of squares in $\mathcal{Q}$. The use of a stride implicitly reduces the input data dimension since an image of size $M\times N$ is then reduced to one of size $m\times n$.

For this collection $\mathcal{Q}$, the {\it average pooling} operator $P_{\text{avg}}\colon\mathbb{R}^{M\times N}\to\mathbb{R}^{m\times n}$ is given by
\[
(P_{\text{avg}} X)_{k,\ell}:=\frac{1}{s^2} \sum_{(i,j)\in Q_{k,\ell}} X_{i,j}.
\]
For the same collection, the {\it max pooling} operator $P_{\text{max}}\colon \mathbb{R}^{M\times N}\to\mathbb{R}^{m\times n}$ is defined as
\[
(P_{\text{max}} X)_{k,\ell}:=\max_{(i,j)\in Q_{k,\ell}} X_{i,j}.
\]
In other words, $P_{\text{avg}}$ simply averages the input image on each $Q_{k,\ell}$, while $P_{\text{max}}$ is the supremum of the values of the image restricted to $Q_{k,\ell}$.

\subsection{Maximal function}

For a locally integrable function $f \in L^1_\text{\rm loc}(\mathbb{R}^d)$, we can define its {\it Hardy--Littlewood maximal function} $M(f)$ as
\begin{equation}
\label{e.8.1}
\forall\; x \in \mathbb{R}^d, \quad M(f)(x) = \sup_{B(x)} \frac{1}{m^d(B(x))} \int_{B(x)} |f|\; dm^d,
\end{equation}
where the supremum is taken over all open balls $B(x)$ centered at $x$ and the integral is taken with respect to the Lebesgue measure $m^d$. Because of this last property, sometimes we talk about the {\it centered} maximal function, as opposed to the non-centered analogue. The function $M$ was introduced by Godfrey H. Hardy and John E. Littlewood in 1930 \cite{HL}. Maximal functions had a profound influence on the development of classical harmonic analysis in the 20th century, see, e.g., \cite{Grafakos} and \cite{Stein}. Among other things, they play a fundamental role in our understanding of the differentiability properties of functions, in the evaluation of singular integrals, and in applications of harmonic analysis to partial differential equations. One of the key tools in this theory is the following Hardy--Littlewood lemma.

\begin{theorem}
\label{theorem}
{\bf Hardy--Littlewood lemma}\\
Let $(\mathbb{R}^d, \mathcal{M}(\mathbb{R}^d), m^d)$ be the Lebesgue measure space. Then, for any $f \in L^1(\mathbb{R}^d)$,
\[
\forall\; \alpha>0, \quad m^d\left( \{ x\in \mathbb{R}^d: M(f)(x) >\alpha \} \right)
\le \frac{3^d}{\alpha} \int_{\mathbb{R}^d}|f|\; dm^d.
\]
\end{theorem}
In view of Theorem \ref{theorem}, the maximal function is sometimes interpreted as encoding the worst possible behavior of the signal $f$. This point of view is further exploited through the {\it Calder\'on-Zygmund decomposition theorem}.

\begin{theorem}
\label{theorem2}
{\bf Calder\'on-Zygmund decomposition}\\
Let $(\mathbb{R}^d, \mathcal{M}(\mathbb{R}^d), m^d)$ be the Lebesgue measure space. Then, for any $f \in L^1(\mathbb{R}^d)$ and any $\alpha >0$, there exists $F, \Omega \subset \mathbb{R}^d$ such that $\mathbb{R}^d = F \cup \Omega$, $F \cap \Omega = \emptyset$, and
\begin{enumerate}
    \item $f(x) \le \alpha$, $m^d$-$a.e.$ in $F$
    \item $\Omega$ is the union of cubes $Q_k$, $k=1, \ldots$, whose interiors are pairwise disjoint, edges are parallel to the coordinate axes, and for each $k=1, \ldots$, we have
    \[
    \alpha < \frac{1}{m^d(Q_k)} \int_{Q_k} |f|\; dm^d \le 2^d \alpha.
    \]
\end{enumerate}
\end{theorem}
The Calder\'on-Zygmund decomposition allows us to split an arbitrary integrable function into its ``good" (i.e., small) and ``bad" (i.e., large) parts, which is a standard technique in addressing problems in the theory of Calder\'on-Zygmund operators. In this approach, the maximal function helps us control the undesirable behaviour of the signal by constraining the regions (represented by balls or cubes) on which this happens. It is this aspect of the maximal functions that we also aim to exploit in our construction of the {\it maxfun pooling}, which will be defined in the next section.

\section{Maxfun Pooling}  \label{sec:maxfun-def}

For maxfun pooling, we first fix an odd stride $s$ and parameter $b$ such that $2b+1\leq s$. Consider the same collection $\mathcal{Q}$ with $Q_{k,\ell}$ defined in \eqref{eq:Q}, and let $q_{k,\ell}$ denote the center of $Q_{k,\ell}$, which is well defined since $s$ is assumed odd. For each $r\leq b$, we define the sub-squares,
\begin{equation}
\label{eq:suppB}
B_{k,\ell,r}
\subset Q_{k,\ell},
\end{equation}
where $B_{k,\ell,r}$ denotes the square of side length $2r+1$ whose center is also $q_{k,\ell}$.

The {\it maxfun pooling} operator $M_{b,s}\colon \mathbb{R}^{M\times N}\to \mathbb{R}^{m\times n}$ is given by
\begin{equation}
\label{eq:maxfun}
(M_{b,s} X)_{k,\ell}
:=\max_{1\leq r\leq b} \, \left( \frac{1}{(2r+1)^2} \sum_{(i,j)\in B_{k,\ell,r}} X_{i,j} \right).
\end{equation}
The quantity $(M_{b,s} X)_{k,\ell}$ is the average of $X$ on $B_{k,\ell,r}$, and the optimal choice of $r$ depends on the restriction of $X$ to $Q_{k,\ell}$.

\par
Let us briefly justify the definition of maxfun pooling. Similar to the maximal function given in \eqref{e.8.1} where the supremum is taken over all balls $B$ centered at a fixed $x$, each $B_{k,\ell,r}$ is centered at $q_{k,\ell}$. Our convention regarding the nonnegative inputs of the pooling layer aligns with the use of the absolute value in the definition of the maximal function. Finally, the computational complexity of evaluating maxfun can be further reduced by choosing the $b$ parameter to be strictly smaller than $(s-1)/2$.

\par
For each region $Q_{k,\ell}$ and any image $X$, clearly we have the inequalities
\[
(P_{\text{avg}} X)_{k,\ell}
\leq (M_{b,s}X)_{k,\ell}
\leq (P_{\text{max}} X)_{k,\ell}.
\]
From this point of view, the output of maxfun on each pooling region is quantitatively between that of average and max pooling. It is not difficult to construct examples of $X$ and $Y$ for which $P_{\text{avg}} X =M_{b,s}X$ and $P_{\text{max}} Y= M_{b,s}Y$.

Another main conceptual difference between these three pooling operators is the amount of information of $X$ that determines their outputs. Average pooling depends on the values of $X$ on the entire region $Q_{k,\ell}$, whereas max pooling selects the largest value of $X$ on $Q_{k,\ell}$, which is not necessarily representative of $X$. From this perspective, average pooling is more robust to perturbations. Maxfun pooling is adaptive in the sense that the optimal region $B_{k,\ell,r}$ depends on $X$, which is a different procedure for selecting a representative value.

\section{Convolutional Sparse Coding} \label{sec:sparse-coding}
The sparse coding problem is an important problem in signal processing, where one aims at finding a low dimensional representation using few atoms for high dimensional data \cite{papyan2016working}. Following standard conventions, we treat an image $X$ as a vector $X\in\mathbb{R}^N$ using the vectorization operator defined in \eqref{eq:vectorization}. Given a vector $X \in \mathbb{R}^N$, and a dictionary $D \in \mathbb{R}^{N\times M}$, the sparse coding problem attempts to find the sparsest vector $\Gamma \in \mathbb{R}^M$ such that $X= D\Gamma$. In other words, for a fixed dictionary $D\in \mathbb{R}^{N\times M}$, the sparse coding problem attempts to solve:
\begin{equation}
\min_{\Gamma}\|\Gamma\|_0 \quad s.t. \quad D\Gamma=X,
\label{sc1}
\end{equation}
where $\|\Gamma\|_0$ is the $l_0$ pseudo norm, which provides the number of non-zero elements in vector $\Gamma$.
Each column in $D$ represents one element in the dictionary, and finding the dictionary to represent data $X$, so that minimal number of dictionary elements are used, solves the sparse coding problem.

Restriction on the sparsity of $\Gamma$ with respect to the mutual coherence of the dictionary $D$ can guarantee uniqueness of the solution to (\ref{sc1}). The mutual coherence of a matrix $D$, see \cite{donoho2006stable}, is defined as:
\begin{equation}
\mu(D)=\max_{i \neq j}\frac{|di^Td_j|}{\|d_i\|_2\cdot \|d_j\|_2},
\end{equation}
where $d_i$'s are the columns of matrix $D$.
However, finding the solution remains NP hard. Relaxation of the model to allow noise and form error bounds leads to the following formulation:
\begin{equation}
\min_{\Gamma} \|\Gamma\|_0 \quad s.t. \quad \|D\Gamma-X\|<\epsilon.
\end{equation}
When high dimensional signals are present, an alternative method called the convolutional sparse coding model (CSC) was proposed. One attempts to represent the whole signal $X \in \mathbb{R}^N$ as a multiplication of a global convolutional dictionary $D \in \mathbb{R}^{N\times Nm_1}$ and a sparse vector $\Gamma \in \mathbb{R}^{Nm_1}$. $D$ is constructed by shifting a local matrix of size $n_0 \times m_1$ in all possible positions, as shown in Figure \ref{Dconvolution}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{sparsecoding.pdf}
\caption{Convolutional Sparse Coding, level 1 \cite{papyan2016convolutional}}
\label{Dconvolution}
\end{figure}
We define the $j$th stripe $\gamma_j$ of the sparse vector $\Gamma$ as a group of $2n_0-1$ adjacent sparse vectors of length $m_1$, starting at the $j$th vector of length $m_1$. See Figure \ref{Dconvolution} for an illustration.

The stripe $\gamma_j$ gives the representation of a patch of $X$, $x_j$ of length $n_0$ by $x_j=\Omega_j \gamma_j$. $\Omega_j \in \mathbb{R}^{n_0\times (2n_0-1)m_1}$ is a submatrix of $D$, called a stripe dictionary consisting of $n_0$ consecutive rows of $D$ and the columns of zeros removed. The $l_{0,\infty}$ norm of the global sparse vector $\Gamma_1$ is defined by the maximum number of non-zeros in any stripe of length $(2n_0-1)m_1$ extracted from it, i.e.,
\begin{equation}
\|\Gamma\|_{0,\infty}=\max_{i \in \{1, ..., N\}} \|\gamma_i\|_0.
\end{equation}
Here $\|\cdot\|_0$ is the zero norm that gives the number of nonzero elements of a vector.

A multi-layer convolutional sparse coding model is defined so that the output sparse vector $\Gamma$ from the previous layer serves as the input vector in the next layer, and we aim at finding a new representation $\Gamma_2$ for a new set of dictionary $D_2$. Formally, the problem of finding solutions to multi-layer convolutional sparse coding problem is defined as the deep coding problem $DCP_{\lambda}$ in \cite{papyan2016convolutional}:
\begin{align}
\label{problem5}
\begin{aligned}
&Find \quad \{\Gamma_i\}_{i=1}^L \\
&X=D_1\Gamma_1, \\
&\Gamma_1 = D_2\Gamma_2, \\
&\vdots\\
&\Gamma_{L-1} = D_L\Gamma_L, \\
\end{aligned}
&&
\begin{aligned}
 &s.t.\\
 & \|\Gamma_1\|_{0,\infty} &\leq \lambda_1\\
 & \|\Gamma_2\|_{0,\infty} &\leq \lambda_2\\
 \\
 &\|\Gamma_L\|_{0,\infty} &\leq \lambda_L
\end{aligned}
\end{align}
where $\lambda_i$ are bounds on sparsity of the output vector $\Gamma_i$ at each level, and $L$ is the number of layers. Note that we want to find representations of the input vectors at each layer that are sparse in terms of its stripe sparsity, defined by $\|\cdot\|_{0,\infty}$.\par
In practice, the input signal $X$ can be contaminated with noise, and we have $Y=X+E$ as the input signal instead of $X$, where $E$ represents noise. In this case, we relax the constraint and allow the representation to vary within some error bounds of the input signal. The deep coding problem when noise is present ($DCP_{\lambda}^{\epsilon}$) \cite{papyan2016convolutional} is defined as:
\begin{align}
\label{DCSPe}
\begin{aligned}
&Find \quad \{\Gamma_i,P_i\}_{i=1}^L \\
&\|Y-D_1\Gamma_1\|_2\leq \epsilon_1, \\
&\|\Gamma_1-D_2\Gamma_2\|_2\leq \epsilon_2, \\
&\vdots\\
&\|\Gamma_{L-1}-D_L\Gamma_L\|_2\leq \epsilon_L, \\
\end{aligned}
&&
\begin{aligned}
&s.t.\\
 & \|\Gamma_1\|_{0,\infty} &\leq \lambda_1\\
 & \|\Gamma_2\|_{0,\infty} &\leq \lambda_2\\
 \\
 &\|\Gamma_L\|_{0,\infty} &\leq \lambda_L.
 \end{aligned}
\end{align}
Here $\epsilon_i$ is the error bound that are allowed in the $i$th layer. \par
Uniqueness of the solution to the $DCP_{\lambda}$ model, and the stability of the solution to the $DCP_{\lambda}^{\epsilon}$ problem have been shown in \cite{papyan2016convolutional}. The connection between deep convolutional sparse coding problem and feed forward neural network has also been demonstrated in \cite{papyan2016convolutional}, where it is proven that one can view the output vector $\Gamma_i$ from each layer of the $DCP_{\lambda}$ problem as the output from one layer of feed forward simplified convolutional neural network (CNN), and thus the deep convolutional sparse coding problem can be viewed as a signal reconstruction problem for simplified CNN models.

Pooling is a common operation included in CNNs that serves as feature extraction method to reduce redundancy of representation of signal and save computational resources. It has been shown that adding max pooling and average pooling in the feed forward path preserves the stability of the neural network \cite{kabkab2017spatial}. We demonstrate that the maxfun pooling, preserves the stability of a convolutional neural network when added in between layers of convolutions.

Given a input signal $X$, the deep convolutional sparse coding problem with pooling ($DCPP$) is defined by \cite{kabkab2017spatial}
\begin{align}
\label{DCPP}
\begin{aligned}
&Find \quad \{\Gamma_i,P_i\}_{i=1}^L \\
&X=D_1\Gamma_1, \\
&P_1 = D_2\Gamma_2, \\
&\vdots\\
&P_{L-1} = D_L\Gamma_L, \\
\end{aligned}
&&
 \begin{aligned}
 &s.t.\\
& \|\Gamma_1\|_{0,\infty} &\leq \lambda_1\\
 & \|\Gamma_2\|_{0,\infty} &\leq \lambda_2\\
 \\
 &\|\Gamma_L\|_{0,\infty} &\leq \lambda_L,
 \end{aligned}
 &&
\begin{aligned}
\\
 &P_1=Pool_{b_1,s_1}(\Gamma_1),\\
 &P_2=Pool_{b_2,s_2}{}(\Gamma_2),\\
 &\vdots\\
 & P_L=Pool_{b_L,s_L}(\Gamma_L),
 \end{aligned}
\end{align}
where $Pool_{_ib,s}$ denotes the pooling operation at the step $i$. We take $Pool_{b_i,s}$ to be the maxfun pooling, i.e., $Pool_{b_i,s} =M_{b,s}$, as defined in (\ref{eq:maxfun}).

Problem (\ref{DCPP}) intends to find a stable sparse representation $\Gamma_1$ of $X$ with dictionary elements in $D_1$, given restriction on the stripe-sparsity of $\Gamma_1$. Then pooling operation is performed on $\Gamma_1$ to get $P_1$. In second layer, we attempt to find the sparse representation $\Gamma_2$ of $P_1$ with dictionary elements in $D_2$. The stripe-sparsity of $\Gamma_2$ is restricted to be no greater than $\lambda_2$. We repeat the process $L$ times.

If our input signal $X$ is contaminated by noise $E$, we are still interested in finding a sparse representation that is stable. Define the deep convolutional sparse coding problem with pooling when noise is present ($DCPP^{\epsilon}$) by \cite{kabkab2017spatial}
\begin{align}
\label{DCPPe}
\begin{aligned}
&Find \quad \{\Gamma_i,P_i\}_{i=1}^L \\
&\|Y-D_1\Gamma_1\| \leq \epsilon_1, \\
&\|P_1 - D_2\Gamma_2\| \leq \epsilon_2, \\
&\vdots\\
&\|P_{L-1} - D_L\Gamma_L\|\leq \epsilon_L, \\
\end{aligned}
&&
 \begin{aligned}
 &s.t.\\
& \|\Gamma_1\|_{0,\infty} &\leq \lambda_1\\
 & \|\Gamma_2\|_{0,\infty} &\leq \lambda_2\\
 \\
 &\|\Gamma_L\|_{0,\infty} &\leq \lambda_L,
 \end{aligned}
 &&
\begin{aligned}
\\
 &P_1=Pool_{b_1,s_1}(\Gamma_1),\\
 &P_2=Pool_{b_2,s_2}{}(\Gamma_2),\\
 &\vdots\\
 & P_L=Pool_{b_L,s_L}(\Gamma_L),
 \end{aligned}
\end{align}
It has been shown in \cite{kabkab2017spatial} that when max pooling and average pooling are used, the stability of solution to the $DCPP^{\epsilon}$ problem is preserved. We show that when we use maxfun pooling, the stability result also holds. We prove the following theorem:
\begin{theorem}
\label{theorem1}
Suppose a vector $X$ satisfies the $DCPP$ model in (\ref{DCPP}), but is contaminated with noise $E$, where $\|E\|_2 \leq \epsilon$, resulting in $Y=X+E$. Suppose $\{\Gamma^*_i,P_i^*\}_{i=1}^L$ solves the problem in (\ref{DCPP}) and $\{\hat{\Gamma}_i,\hat{P}_i\}_{i=1}^L$ solves the problem in (\ref{DCPPe}). If
\begin{equation}
\begin{split}
&\|\Gamma^*_i\|_{0,\infty} \leq \lambda_i <\frac{1}{2} \Big (1+\frac{1}{\mu(D_i)} \Big), \quad \forall 0 \leq i \leq L,\\
&\epsilon_0 = \epsilon, \quad \epsilon_i^2 =\frac{4\epsilon_{i-1}^2}{1-(2\|\Gamma_i^*\|_{0,\infty}-1)\mu(D_1)} \quad \forall i \geq 1,\\
&\text{then for all}\quad 1\leq i \leq L,\\
&\|P_i^*-\hat{P}_i\|^2_2 \leq \|\Gamma_i^*-\hat{\Gamma}_i\|_2^2 \leq \epsilon_i^2.
\end{split}
\label{assumption}
\end{equation}
Here maxfun pooling is used as the pooling operation and we assume that the minimum pooling region size $b \geq 1$.
\label{T1}
\end{theorem}
In order to prove Theorem \ref{T1}, we first prove the following Lemma for maxfun pooling.
\begin{lemma}
Let $X$ and $\hat{X}$ be two functions in $\mathbb{R}^{N\times N}$, and let $P = M_{b,s}(X)$, $\hat{P}=M_{b,s}(\hat{X})$ be the outcome of maxfun pooling of $X$ and $\hat{X}$,respectively, and assume that $s \geq b$. Then $\|P-\hat{P}\|_2 \leq \|X-\hat{X}\|_2$, where $\|\cdot\|_2$ denotes the Frobenius norm.
\label{lemma4.2}
\end{lemma}
\begin{proof}
Let $B_{k,\ell,r} $ be the set of indices that represents points in the centered sub-square of side length $2r + 1$ in the $(k,\ell)$-th pooling region. Let
\[
\gamma_{k,\ell,r}=\frac{1}{(2r + 1)^2}\sum_{(i,j) \in B_{k,\ell,r}}X_{i,j}
\quad\text{and}\quad
\hat{\gamma}_{k,\ell,r} =\frac{1}{(2r + 1)^2}\sum_{(i,j) \in B_{k,\ell,r}}\hat{X}_{i,j}.
\]
We take $r_{k,\ell}^*=\underset{1\leq r\leq b}{\text{argmax}}\gamma_{k,\ell,r}$ to indicate the $r$ parameter of the sub-square with the maximum of $\gamma_{k,\ell,r}$ over all $r$ for each $(k,\ell)$-th region; and we take $\hat{r}_{k,\ell}^*=\underset{1\leq r\leq b}{\text{argmax}}\hat{\gamma}_{k,\ell,r}$ to indicate the $r$ parameter of the sub-square with maximum of $\hat{\gamma}_{k,\ell,r}$ over all $r$ for each $(k,\ell)$-th region.
We also take the $r_{min}^*$ to be minimum of all $r_{k,\ell}^*$, and $\hat{r}_{min}^*$ to be minimum of all $\hat{r}_{k,\ell}^*$. Furthermore, we let $K_1$ be the set of indices of $(k,\ell)$ so that $\gamma_{k,\ell,r_{k,\ell}^*} \geq \hat{\gamma}_{k,\ell,\hat{r}_{k,\ell}^*}$. Let $K_2$ be the set of indices of $(k,\ell)$ so that $\gamma_{k,\ell,r_{k,\ell}^*} < \hat{\gamma}_{k,\ell,\hat{r}_{k,\ell}^*}$. Then we have
\begin{align}
&\|P-\hat{P}\|^2_2 \nonumber \\
 &=\sum_{(k,l)} \left(\underset{1\leq r \leq b}{\max} \Big(\frac{1}{(2r + 1)^2}\sum_{(i,j) \in B_{k,\ell,r}}X_{i,j} \Big)- \underset{1\leq r \leq b}{\max} \Big(\frac{1}{(2r + 1)^2}\sum_{(i,j) \in B_{k,\ell,r}}\hat{X}_{i,j}\Big)\right)^2\\
&= \sum_{(k,\ell)\in K_1} (\gamma_{k,\ell,r_{k,\ell}^*}-\hat{\gamma}_{k,\ell,\hat{r}_{k,\ell}^*})^2+\sum_{(k,\ell)\in K_2} (\hat{\gamma}_{k,\ell,\hat{r}_{k,\ell}^*}-\gamma_{k,\ell,r_{k,\ell}^*})^2\\
\label{equation7}
&\leq \sum_{(k,\ell)\in K_1} (\gamma_{k,\ell,r_{k,\ell}^*}-\hat{\gamma}_{k,\ell,r_{k,\ell}^*})^2 +\sum_{(k,\ell)\in K_2} (\hat{\gamma}_{k,\ell,\hat{r}_{k,\ell}^*}-\gamma_{k,l,\hat{r}_{k,\ell}^*})^2\\
\label{equation8}
&=\sum_{(k,\ell) \in K_1}\left(\frac{1}{(2r_{k,\ell}^*+1)^2}\sum_{(i,j) \in B_{k,l,r_{k,\ell}^*}}X_{i,j}-\frac{1}{(2r_{k,\ell}^*+1)^2}\sum_{(i,j) \in B_{k,\ell,r_{k,\ell}^*}}\hat{X}_{i,j}\right)^2 \nonumber \\
&\quad  + \sum_{(k,\ell) \in K_2}\left(\frac{1}{(2\hat{r}_{k,\ell}^*+1)^2}\sum_{(i,j) \in B_{k,\ell,\hat{r}_{k,\ell}^*}}\hat{X}_{i,j}-\frac{1}{(2\hat{r}_{k,\ell}^*+1)^2}\sum_{(i,j) \in B_{k,\ell,\hat{r}_{k,\ell}^*}}X_{i,j}\right)^2\\
\label{equation10}
&\leq \sum_{(k,\ell) \in K_1}\frac{1}{(2r_{k,\ell}^*+1)^2}\sum_{(i,j) \in B_{k,\ell,r_{k,\ell}^*}}(X_{i,j}-\hat{X}_{i,j})^2 \nonumber \\
& \quad +\sum_{(k,\ell) \in K_2}\frac{1}{(2\hat{r}_{k,\ell}^*+1)^2}\sum_{(i,j) \in B_{k,\ell,\hat{r}_{k,\ell}^*}}(\hat{X}_{i,j}-X_{i,j})^2\\
\label{equation11}
&\leq \frac{1}{(2r_{min}^*+1)^2}\sum_{(k,\ell) \in K_1,(i,j) \in B_{k,\ell,r_{k,\ell}^*}}(X_{i,j}-\hat{X}_{i,j})^2  \nonumber \\
&\quad +
\frac{1}{(2\hat{r}_{min}^*+1)^2}\sum_{ (k,\ell)\in K_2, (i,j)\in B_{k,\ell,\hat{r}_{k,\ell}^*}}(\hat{X}_{i,j}-X_{i,j})^2\\
\label{equation12}
&\leq \sum_{(k,\ell) \in K_1, (i,j) \in B_{k,\ell,b} }(X_{i,j}-\hat{X}_{i,j})^2+\sum_{(k,\ell) \in K_2, (i,j) \in B_{k,\ell,b} }(\hat{X}_{i,j}-X_{i,j})^2\\
\label{equation13}
&\leq \|X-\hat{X}\|_2^2.
\end{align}
The inequality (\ref{equation7}) comes from the fact that $\hat{\gamma}_{k,l,\hat{r}_{k,\ell}^*}$ is the maximum over all $r$ and thus $\hat{\gamma}_{k,\ell,\hat{r}_{k,\ell}^*}\geq \hat{\gamma}_{k,\ell,r_{k,\ell}^*}$, and similarly $\gamma_{k,\ell,r_{k,\ell}^*}\geq\gamma_{k,\ell,\hat{r}_{k,\ell}^*}$. In (\ref{equation8}), $B_{k,\ell, r_k^*}$ and $B_{k,\ell, \hat{r}_k^*}$ are the corresponding set of indices for which $\gamma_{k,\ell,r_{k,\ell}^*}$ and $\hat{\gamma}_{k,\ell,\hat{r}_{k,\ell}^*}$ are maximums across all $r$'s, respectively. The inequality (\ref{equation10}) holds based on the inequality $(\sum_{i=1}^n a_i)^2 \leq n\sum a_i^2$.
 Inequality (\ref{equation12}) follows from the fact that $r\leq b$ and stride size $s \geq 2b + 1$. $B_{k,\ell,b}$ represents the indices of the sub-square of length $2b+1$ at the initial position in the $(k,l)$th pooling region.
\end{proof}
We now prove Theorem \ref{T1}.
\begin{proof}
By Theorem 3 in \cite{papyan2016working}, we know that for a signal $Y=X+E$, if
\begin{equation*}
\begin{split}
&1.\|\Gamma_1^*\|_{0,\infty} < \frac{1}{2}(1+\frac{1}{\mu(D_1)}) \text{ and } \|E\|_2=\|Y-D_1\Gamma^*_1\|_2 \leq \epsilon_0,\\
&2.\|\hat{\Gamma}_1\|_{0,\infty} < \frac{1}{2}(1+\frac{1}{\mu(D_1)}) \text{ and } \|Y-D_1\hat{\Gamma}_1\|_2 \leq \epsilon_0,
\end{split}
\end{equation*}
then
\begin{equation}
\|\Delta\|_1^2=\|\Gamma^*_1-\hat{\Gamma}_1\|_2^2 \leq \frac{4\epsilon_0^2}{1-(2\|\Gamma_1\|_{0,\infty}-1)\mu (D_1)}=\epsilon_1^2.
\end{equation}
Since $\|\Gamma_1^*\|_{0,\infty} \leq \lambda_1$ and $\|\hat{\Gamma}_1^*\|_{0,\infty} \leq \lambda_1$ by assumption in problem \ref{DCPP} and \ref{DCPPe}, and $\lambda$ is bounded by assumption \ref{assumption} in theorem \ref{T1}, the first parts of 1 and 2 hold. Since $\Gamma_1^*$ is the solution to the $DCPP$ problem, it must be true that $\|Y-D_1\Gamma_1^*\| \leq \epsilon_0$. $\|Y-D_1\hat{\Gamma}_1\| \leq \epsilon_0$ by assumption in problem \ref{DCPPe}. Therefore, we have $\|\Delta\|_2^2 \leq \epsilon_1^2$. And hence by Lemma \ref{lemma4.2}, we have
\begin{equation}
\|P^*_1-\hat{P}_1\|_2^2 \leq \|\Delta_1\|_2^2 \leq \epsilon_1^2.
\end{equation}
At second level, the same argument holds so that $\|\Gamma_2^*\| <\lambda_2 <\frac{1}{2}(1+\frac{1}{\mu(D_1)})$ and $\|\hat{\Gamma}_2\|<\lambda_2 <\frac{1}{2}(1+\frac{1}{\mu(D_1)})$. $\|P^*_1-D_2\Gamma_2^*\|_2<\epsilon_2$ by assumption in problem \ref{DCPP}. $\|\hat{P}_1-D_2\hat{\Gamma}_2\|_2<\epsilon_2$ by assumption in problem \ref{DCPPe}. Therefore, by theorem 3 in \cite{papyan2016working} we have
\begin{equation}
\|\Gamma_2^*-\hat{\Gamma}_2\|_2^2 \leq \epsilon_2^2,
\end{equation}
and by Lemma 1, we get
\begin{equation}
\|P_2^*-\hat{P}_2\|_2^2 \leq \|\Gamma_2^*-\hat{\Gamma}_2\|_2^2 \leq \epsilon_2^2.
\end{equation}
Following this argument for $1\leq i \leq L$, we complete the proof and showed that
\begin{equation}
\|P_i^*-\hat{P}_i\|_2^2 \leq \|\Gamma_i^*-\hat{\Gamma}_i\|_2^2 \leq \epsilon_i^2, \quad \forall \quad 1 \leq i \leq L.
\end{equation}
\end{proof}

\section{Classification Experiments} \label{sec:classification}

As described in Section~\ref{sec:introduction}, pooling as a technique for dimension reduction arises in a variety of signal processing contexts.
Accordingly, the sparse coding problem described in Section~\ref{sec:sparse-coding} is not the only possible application for maxfun pooling.
In this section we consider maxfun for supervised classification problems.

Prior work has examined properties of pooling operators in the context of supervised classification.
For example,~\cite{boureau2010theoretical} performed theoretical analyses and conducted empirical experiments comparing maximum and average pooling.  Their findings indicated that, depending upon the data and features at hand, either maximum or average pooling may be preferable.
The authors also identified potential benefits of pooling methodologies that ``interpolate" between maximum and average pooling.  As demonstrated in Section~\ref{sec:maxfun-def}, maxfun pooling also has the property that it resides between maximum and average pooling in a precise sense.
Thus it is of interest to explore how maxfun relates not only to average and maximum pooling, but also to other intermediate pooling techniques.

\subsection{Approach}

Our goal is to highlight differences between different pooling strategies; therefore we construct a classification problem where pooling plays a significant role in developing the feature representation.
Instead of the gradual dimension reduction one might observe throughout a standard feedforward deep neural network (where 2x2 or 3x3 pooling regions might be used at any given layer) here we consider an approach akin to very shallow networks followed by a single, aggressive pooling step.
The preliminary experiments presented in this section are in the same spirit as those of~\cite{boureau2010theoretical}, albeit with a different implementation.

Since natural images often manifest meaningful correlations in the spatial dimension we choose to focus upon image classification problems.
We generate raw features by extracting the outputs from the first layer of a standard CNN (prior to pooling) that has been pre-trained on natural images.
We then apply various pooling techniques along the spatial dimensions of these feature maps.
Finally, the vectorized feature maps are used in conjunction with a traditional multi-class support vector machine (SVM)~\cite{hearst1998support} to compare the utility of the different pooling methods.

For our experiments we use a subset of the Caltech-101 data set~\cite{fei2007learning}. To mitigate the impact of class imbalance, we limit our study to classes that have between 80 and 130 instances.  The result is an 18-class classification problem with mild class imbalance.
To standardize the spatial dimensions all images are first zero padded to make them square; e.g. a 100x120 pixel image is padded to produce a 120x120 pixel image.
Images are kept centered when padding (in the previous example 10 rows would be added to the top and 10 to the bottom of the image).
Finally, we resize all images to 128x128 pixels.
This provides a collection of natural images with the same dimensions where some attempt has been made to preserve the aspect ratio.

For raw features we use the outputs from the first layer of the Inception-v3 network~\cite{szegedy2016rethinking}.  This deviates from~\cite{boureau2010theoretical}, which used SIFT to construct feature maps (note their study predated a number of key developments in convolutional neural networks).
We then spatially decompose the resulting feature tensor into multiple (possibly overlapping) pooling regions.
This decomposition is applied independently along each channel, and thus preserves the cardinality of the channel dimension.
After identifying the pooling regions, we apply various pooling operators.
These pooled tensors are then vectorized and constitute the feature maps used to solve the multiclass classification problem.

\subsubsection{Baselines}
In addition to maximum and average pooling, we also compare maxfun to  other pooling strategies that sit in between these two extremes.
In particular, we consider the ``stochastic pooling" method of~\cite{zeiler2013stochastic} and the ``mixed pooling" method of~\cite{lee2016generalizing}. For stochastic pooling, the authors recommend a particular weighted average motivated by their original stochastic formulation:
\begin{equation} \label{eq:stochastic-pooling}
	v_{k,\ell} = c_{k,\ell} \sum_{(i,j) \in Q_{k,\ell}} X_{i,j}^2, \qquad c_{k,\ell} = \frac{1}{\sum_{(i,j) \in Q_{k,\ell}} X_{i,j}} ,
\end{equation}
where $v_{k,\ell}$ denotes the pooled value and $X$ is a (non-negative) two-dimensional feature map, following the notation of Section~\ref{sec:prelim}.
Note $X$ could represent a raw image or some higher-level representation thereof.
The non-negativity assumption is consistent with image pixel intensities or with features extracted from the output of a RELU layer within a CNN.  See~\cite{zeiler2013stochastic} for full details on the stochastic origins of~\eqref{eq:stochastic-pooling}.

Mixed pooling is defined as a convex combination of maximum and average pooling; that is
\begin{equation} \label{eq:mixedpooling}
	v_{k,\ell} = \alpha \, (P_{\text{max}}X)_{k,\ell} + (1-\alpha)\, (P_{\text{avg}} X)_{k,\ell} ,
\end{equation}
where $\alpha \in (0,1)$.

The maxfun and the mixed pooling strategies both introduce hyperparameters to our experiments; in the case of the maximal function we elected to add a minimum radius $r_{min} > 1$ as a lower bound on $r$ in \eqref{eq:maxfun} while for the mixed pooling strategy we must select the scalar $\alpha$ in~\eqref{eq:mixedpooling}.
In both cases we use a $k$-fold cross-validation procedure with $k=3$ to select these hyperparameters.
Our training and testing data sets are of size 975 and 649 with the partition chosen uniformly at random.

\subsection{Preliminary Results and Discussion}
Results for our pooling experiments are summarized in table \ref{tbl:empirical}.
Across all pooling methods we consider two scenarios: one where the pooling regions partition the spatial domain and another where the pooling windows overlap.
For this study we implemented both centered and non-centered versions of the maxfun pooling.
The centered variant is a realization of \eqref{eq:maxfun} while the non-centered variant allows the squares to be centered at points other than the center of pooling region.

\begin{table}
	\centering
	\begin{tabular}{l|cc}
		pooling  & \multicolumn{2}{c}{SVM accuracy} \\
		strategy & $(b = 21, s=21)$ & $(b=21, s=11)$\\ \hline\hline
		average & 0.5763 & 0.6102\\
		maximum & 0.5932 & 0.5932 \\
		mixed & 0.6287 & 0.6240\\
		stochastic & 0.6502 & 0.6641\\
		maxfun & 0.6225 & 0.6102\\
		centered maxfun  & 0.6626 & 0.6579
	\end{tabular}
	\caption{Empirical results for our Caltech-101 classification experiments. Pooling regions have dimensions $b \times b$ and the pooling stride is $s$. Thus, when $s=b$ (column 1), the pooling regions partition the spatial domain.  When when $s<b$ (column 2) the pooling regions overlap.}
	\label{tbl:empirical}
\end{table}

Table~\ref{tbl:empirical} suggests that the centered maxfun pooling strategy provides relatively good results, on par with those of the stochastic pooling method.
Note that these results are not state-of-the-art for this problem.
The simplified network topology and aggressive pooling we utilize here are sub-optimal from a pure performance standpoint.
Recall our goal was to highlight relative differences among pooling strategies; at this stage of this work we are yet not focused on absolute performance.

It is interesting to observe the the more flexible non-centered maxfun does not appear as effective as the centered variant.
We might speculate that edge effects are causing some issues with the non-centered variant.
Alternatively, it is possible that the location of the pooling window may be a more significant driver in feature importance than the spatial extent of the pooling subregion.

Obviously the experiments we describe here are highly preliminary.  One direction for future work is to expand the scope these studies and to further inquire into differences between the centered vs non-centered maxfun variants.
Furthermore, there could be value in a more detailed comparison, both theoretically and empirically, of maxfun with other pooling variants that ``interpolate'' between average and maximum pooling.
Ultimately, there is an open question as to whether maxfun can benefit modern CNN architectures if it were to be included as a pooling layer.
This would entail defining a suitable notion of a gradient for discrete maxfun pooling and conducting appropriate numerical studies.
These studies would also need to address the added computational expense of maxfun over simpler methods, such as stochastic pooling~\eqref{eq:stochastic-pooling}.

\section{Conclusions}
In this paper we introduced a novel pooling method, maxfun pooling, inspired by the Hardy--Littlewood maximal function. We proved that this new pooling strategy maintains the stability of a convolutional neural network and we demonstrated its experimental performance by comparing it with state-of-the-art pooling algorithms in classification tasks.\par
Many functions and transformations originating from harmonic analysis have demonstrated the ability to extract useful features from the input data sets for classification or segmentation purposes. Maxfun pooling is another example of such a strategy that effectively extracts features from outputs of layers of neural networks and produces faithful representation of input data.
\bibliographystyle{cjj}

redpajama_set_name: RedPajamaArXiv


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If you would like to see something different in your life and if you like cooking or want to learn how to cook, we offer you very special itinerary to learn very good Turkish cuisines with smiley and friendly Turkish people. You can cook, taste and eat. First class service and guided transportation through the historical and picturesque sites of Istanbul.
Points of Interest: Istanbul - Sultanahmet Imperial Mosque, Hippodrome, Topkapi Palace, Welcome Traditional Turkish Dinner, Turkish Cooking Lesson Istanbul, Egyptian (Spice) Bazaar and Shopping Possibility in Grand Bazaar.
All entrance fees mentioned in the itinerary; Transportation in a fully air-conditioned, non-smoking coach; Professional English-speaking tour guides; Return airport transfer to Istanbul Ataturk Airport.
Day 1: Istanbul - Arrival Day (Welcome dinner included).
Welcome to Turkey, arrival into Istanbul. Transfer from airport and check in to your hotel, you will have a welcome drink and a chat with gourmet experts in Turkish cuisine. Tonight you will have a welcome dinner, be introduced to our friendly crew and other travelers while enjoying traditional Turkish food and enjoy the mystical atmosphere of Istanbul at night.
Day 2: Turkish Cooking Lesson Istanbul (Breakfast included).
After a morning orientation sightseeing tour of Istanbul, you participate in your first Turkish cooking lesson: Preparing lunch which you will enjoy! After the cooking lesson, you will eat what you cooked. After lunch visit to TOPKAPI PALACE. The great palace of the Ottoman sultans from the 15th to the 19th centuries housing an exquisite collection of crystal, silver, and Chinese porcelain, robes worn by the sultans and their families, the famous jewels of the Imperial Treasury, miniatures, the Holy Mantle; enshrining relics of the Prophet Mohammed.
HIPPODROME: Ancient Hippodrome, the scene of chariot races, with the three monuments; the Obelisk of Theodosius, the bronze Serpentine Column and the Column of Constantine. Overnight in Istanbul.
Day 3: Istanbul Cooking Lesson (Breakfast included).
After breakfast, you will visit Istanbul Bosphorus Cruise by boat along the waterway seperating Europe and Asia, then visit the famous Egyptian (Spice) Bazaar, with elaborate displays of herbs, spices, dried fruits, nuts and other snacks and ingredients. Then we will go to one of the city's most famous baklava shops to watch the masters produce this favorite Turkish sweet. This will be a good experience for you. In the afternoon you will go back to the kitchen for another cooking lesson followed by tea, coffee and goodies.
This tour require minimum 2 people to book.


Edward Everett Hale (1822–1909) was an American author, historian and Unitarian minister. Hale was born on April 3, 1822, in Boston, Massachusetts, the son of Nathan Hale, proprietor and editor of the Boston Daily Advertiser, and the brother of Lucretia Peabody Hale, Susan Hale, and Charles Hale. Edward Hale was the nephew of Edward Everett, the orator and statesman, while his father was the nephew of Nathan Hale, the war hero who was executed by the British for espionage during the Revolutionary War. He was also a descendant of Richard Everett and related to Helen Keller.
Look up and not down;Look forward and not back;Look out and not in;Lend a Hand.
Do you pray for the Senators, Dr. Hale? someone asked the chaplain. No, I look at the Senators and pray for the country.


Commerce. Daily Life. Natural world. Power. Religion/Spirituality.
This lion-head figurine was discovered in 1965. It is considered one of the most important finds of the city of Ur's treasure. It was discovered hidden in a pottery jar located in one of the columned rooms of the royal palace of King Zimri Lim in Mari, the founder of Mari and the initiator of the first dynasty. Mari is an ancient city located in north east Syria, on the Euphrates River.
The treasure was a gift from the king of Ur, located in the Lower Euphrates region, to the king of Mari on the Middle Euphrates. Such gifts of treasures indicates the high status and dignity of the kingdom of Mari vis-à-vis the other city states of the region. Mari dates back to the 3rd millennium BC, when it was the capital of the Middle Euphrates.
This statuette is made of lapis lazuli and gold. It features a mythological creature with the face of a lion, wings of a bird, and tail of a fish. The lapis lazuli body is ornamented with diagonally striated lines and a pair of gently arching vertical lines, indicating the wings and the body respectively. The head and the tail are executed with sheets of gold fixed above a core of tar. This mythological creature is named Anzu. He is analogous to the god Ningurso from Sumerian mythology and is the sacred deity in southern Mesopotamia.
The statuette is carefully and symmetrically pierced, twice at the top of the shoulders of the creature and one at the tail, indicating that is was originally meant to be sown into a fabric, as a jewel to be fixed on the chest of the wearer, above the breast-bone.


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Ab inbev stock
MichaelByrd November 13, 2021, 4:36pm #1
When does Anheuser Busch InBev release its earnings? AnheuserBusch InBev SA / NV is expected to publish its next quarterly results on Thursday, November 4, 2021. View the earnings forecast for AnheuserBusch InBev SA / NV. What was AnheuserBusch InBev SA / NV profit in the last quarter? AnheuserBusch InBev SA / NV (NYSE: BUD) announced its results on Wednesday, July 28.
When did Anheuser Busch InBev acquire SABMiller?
Following the SABMiller acquisition, the company's portfolio now includes five of the top ten best-selling beer brands and 18 brands with more than $1 billion in retail sales. AB InBev was created in 2008 as a result of the merger of the Belgian companies InBev and AnheuserBusch. The company owns a 62% economic interest in Ambev and acquired SABMiller in 2016.
Where is Anheuser Busch InBev located in the world?
It operates in the following geographic segments: North America, Central America, South America, Europe, Middle East, Africa, Asia-Pacific, and global export and portfolio companies. The company was founded on August 2, 1977 and is located in Leuven, Belgium.
What's the estimated earnings of Anheuser-Busch InBev?
AnheuserBusch InBev SA / NV earnings are expected to increase from one dollar per share to one dollar per share in the coming year, which is an increase.
When does AB InBev report third quarter earnings?
AnheuserBusch InBev SA / NV BUD, also known as AB InBev, is expected to announce Q3 2020 results on October 29. The company posted surprising gains in its most recent quarter, but profits lagged the Sachs consensus average over the past four quarters.
What was the consensus estimate for AB InBev?
However, averaged over the past four quarters, gains have not been in line with Sachs' consensus estimate. Zacks' consensus forecast for AB InBev's third quarter earnings is 73 cents per share, down from the same quarter last year.
What is the Zacks rank of AB InBev?
AB InBev is ranked 4th by Zacks (Sell) and ESP for Profit. Here are some companies to watch out for as their model shows they have the right mix of elements to generate more profit. Nu Skin Enterprises, Inc. NUS currently has ESP+ earnings and a #1 Zacks ranking.
How much is Anheuser Busch InBev worth?
AnheuserBusch InBev has agreed to purchase the remaining non-treasury shares of the Craft Brew Alliance, bringing the Portland-based company to an estimated $321 million. Today's best supply chain and logistics news from WSJ.
Is the Mexican arm of Anheuser Busch InBev suing constellation?
BRUSSELS (Reuters) Mexican subsidiary AnheuserBusch InBev has filed a lawsuit against Constellation Brands, citing a second breach of their agreement allowing Constellation to sell Mexican beer in the United States.
What is the stock price of Anheuser Busch?
AnheuserBusch InBev SA/NV was listed on March 11, 2020, when the coronavirus (COVID19) reached pandemic status according to the World Health Organization. Since then, BUD stocks have risen and are now trading at USD.
What is the payout ratio of Anheuser Busch?
This payout percentage is at a stable and healthy level, below 75%. According to earnings estimates, AnheuserBusch InBev SA/NV will have a dividend payment percentage next year. This indicates that AnheuserBusch InBev SA / NV can maintain or increase its dividend. View the dividend history of AnheuserBusch InBev SA / NV.
What are the analyst ratings for Anheuser Busch?
There are currently 2 sell ratings, 10 hold ratings, and 7 buy ratings for stocks. Wall Street analysts agree that investors should own shares in AnheuserBusch InBev SA/NV.
Who owns Anheuser Busch now?
AnheuserBusch is owned by AnheuserBusch InBev. It is located in St. Louis, Missouri. AnheuserBusch is owned by AnheuserBusch InBev. It is located in St. Louis, Missouri.
Who bought Anheuser Busch brewery?
AB InBev was created after the acquisition of the American brewery AnheuserBusch by the Belgian-Brazilian brewery InBev, a merger of AmBev and Interbrew.
What are the Anheuser-Busch brands?
AnheuserBusch InBev (abbreviated to AB InBev) is a Belgian-Brazilian multinational beverage company based in Leuven, Belgium. This is the largest brewery in the world. AB InBev owns 14 brands. They make over a billion dollars a year. International brands include Becks, Corona, Budweiser, Stella Artois and Brahma.
What beers does Anheuser-Busch make?
AnheuserBusch's best-known beers include brands like Budweiser, das Busch (originally known as Busch Bavarian Beer), and Michelob, as well as Natural Light.
What kind of company is Anheuser Busch InBev?
AnheuserBusch InBev SA / NV is a holding company engaged in the manufacture and sale of alcoholic and non-alcoholic beverages. It operates in the following geographic segments: North America, Central America, South America, Europe, Middle East, Africa, Asia-Pacific, and global export and portfolio companies.
Is the sea world Anheuser Busch Park for sale?
Monday's announcement of the sale of Saint Louis-based AnheuserBusch to Belgian beer giant InBev questions the future of 10 Anheuser Busch theme parks, including SeaWorld.
How many Anheuser Busch breweries are there in the world?
Anheuser Busch International, Inc. was founded in 1981 as a subsidiary responsible for the company's international activities and investments. Prior to the acquisition in 2008, AnheuserBusch operated 15 breweries worldwide: 14 in China and one in the UK.
Why are the Anheuser Busch parks for sale?
Since the parks, as well as packaging company AnheuserBusch, are considered non-core assets, InBev will likely sell them to pay some of the cost of the $52 billion King of Beers Brewer acquisition.
Who is the CEO of Anheuser Busch InBev?
Michelle Dukeris becomes CEO of AnheuserBusch InBev Vestibulum luctus, mauris ullamcorper volutpat ullamcorper, quam felis mollis sem and fermentum felis nisi ut arcu. Open Share Share: Share to Facebook Share to Twitter Share via Email Close Share PRESS RELEASE.
Where is the headquarters of Anheuser Busch?
NS. Opened in 1852, LOUIS, Missouri, St. Louis is the flagship brewery, headquarters of AnheuserBusch, and home to Budweiser. The brewery has a beautiful interior and has been offering tours for guests since the 1890s.
When did the Anheuser Busch brewery in open?
Opened in the 1850s, St. Louis is the flagship brewery, headquarters of AnheuserBusch, and home to Budweiser. The brewery has a beautiful interior and has offered tours to the public since the 1890s. This is the center of innovation, producing more than 30 of its brands, including the popular Budweiser and Michelob ULTRA.
When was InBev acquired by Anheuser Busch?
In 2008, InBev completed its acquisition of AnheuserBusch, creating AnheuserBusch InBev, further extending InBev's previous status as the world's largest beer producer into one of the world's five largest producers of consumer goods.
What did SABMiller make before it was acquired by AB InBev?
AB InBev has been responsible for brands such as Budweiser, Corona, Stella Artois and many more. Prior to the acquisition, SABMiller was based in England and was also a multinational brewery known for its beverages such as Miller, Fosters and Tyskie beers.
Who was the largest brewer in the world before SABMiller?
Before the SABMiller acquisition, it was the largest brewery in the world, accounting for approximately 27% of the world's volume. AB InBev has been responsible for brands such as Budweiser, Corona, Stella Artois and many more.
What was the market value of AB InBev after the merger?
Following the merger, the share price of trader AB InBev soared to an all-time high, with a market value of more than $200 billion. The target company's stock price also rose to new all-time highs. This makes the merger more beneficial for everyone.
When did anheuser busch inbev acquire sabmiller stock
In October 2015, AnheuserBusch InBev announced a successful cash offer to acquire its UK multinational rival SABMiller.
Who is the parent company of SABMiller South Africa?
SABMiller. Companies such as South African Breweries and Fosters Group, which were subsidiaries of SABMiller and were not sold following the acquisition of SABMiller by AnheuserBusch InBev SA/NV, are now subsidiaries of AB InBev.
When did anheuser busch inbev acquire sabmiller oil
AnheuserBusch InBev to acquire SABMiller in a $106 billion deal on .
How does the acquisition of SABMiller affect Anheuser Busch InBev?
The acquisition of SABMiller by Anheuser will significantly expand the beverage company's business, but will continue to have an impact on its finances going forward. Even after the sale of SABMiller's stake in MillerCoors, AB InBev's net debt-to-EBITDA ratio could be roughly reached to fund the merger.
How big is the AB InBev SABMiller deal?
Following the third-largest acquisition in history by the Financial Times and the largest in the UK, where SABMiller's headquarters will remain until its core functions are combined with AB InBev in Belgium, the new company will generate annual sales of 55,000 have million dollars. That's $44 billion before closing.
Why was the acquisition of SABMiller so important?
The merger with SABMiller will be important to AB InBev, primarily due to its leadership position in the high-growth markets of Africa and South America.
What's the current price of Anheuser Busch InBev?
They have an AnheuserBusch InBev valuation of $113, which is roughly the current market price. According to the Wall Street Journal, AB InBev is in talks with banks to fund a potential $122 billion acquisition of SABMiller.
How did the SABMiller deal leave AB InBev with a lasting hangover?
How the SABMiller AB deal left InBev with a lingering hangover. It was a deal designed to round out a wave of two-decade acquisitions and make AnheuserBusch InBev the undisputed king of beer. Instead, Megabrew's takeover of rival SABMiller left the Belgian-Brazilian owner a long hangover.
How much would Anheuser Busch InBev pay for SABMiller?
But with cash flow increasing and significant reductions in production costs, Anheuser may now seek to acquire SABMiller and consolidate its share of certain markets in South America and Africa, where beer is expected to grow rapidly. They have an AnheuserBusch InBev valuation of $113, which is roughly the current market price.
Is the AB InBev and SABMiller merger a horizontal merger?
Because AB InBev and SABMiller are a horizontal merger of two companies operating in similar industries, they used Bloomberg's relative price function (as shown in Figure 3 in the Appendix) to determine other similar companies and prices relative to its performance.
When did Busch Gardens take over SeaWorld Orlando?
In 1999, Busch Gardens Tampa purchased nearly 200 acres of land to transport animals when the new hotel was never built. In early 2001, the company sold Aurora SeaWorld to Six Flags and the first wildlife sanctuary, Discovery Cove, opened adjacent to Orlando SeaWorld.
When did anheuser busch inbev acquire sabmiller products
AnheuserBusch InBev (AB InBev) is the world's largest brewer, a position it further strengthened when it acquired rival SABMiller (of South Africa) in October 2016 for more than $100 billion. The deal gave AB InBev access to SABMiller's massive African presence. Here are four plans for the continent.
Who owns Anheuser Busch company?
AnheuserBusch is a wholly owned subsidiary of AnheuserBusch InBev, the world's largest brewery and one of the five largest consumer goods companies.
What company owns Busch Beer?
AnheuserBusch was the last major American brewery mostly owned and operated by the United States. He has been working with InBev ever since. AnheuserBusch's best-known beers include brands like Budweiser, das Busch (originally known as Busch-Bayerisches Bier), and Michelob, as well as Natural Light.
Who bought Budweiser beer?
Budweiser, introduced by AnheuserBusch in Saint Louis in 1876, is the best-known American brand of beer. In 2008, AnheuserBusch was acquired by the Belgian-Brazilian conglomerate InBev.
Where is Budweiser corporate office?
Budweiser Brewery, also known as Anheuser Busch Brewery, is located in St. Louis, Missouri and is the global headquarters of the Anheuser Busch Brewing Company.
When did Anheuser Busch and AmBev merge?
AnheuserBusch was founded in 1852 in St. Louis, Missouri, USA as Anheuser & Co. In 2004, Interbrew and AmBev merged to form the largest brewery in the world, InBev. The $1 billion deal brought together the third largest brewery (Interbrew) and the fifth largest brewery (Ambev) in the world.
Where are the headquarters of Anheuser Busch InBev?
AnheuserBusch InBev SA/NV, commonly known as AB InBev, is a multinational brewing and beverage company based in Leuven, Belgium. AB InBev has a functional global management office in New York and regional headquarters in Sao Paulo, London, St. Louis, Mexico City, Bremen, Johannesburg and other countries.
How much does Anheuser Busch InBev make a year?
Even before the SABMiller acquisition, it was the world's largest brewery and one of the world's largest consumer goods companies. The company had billions of dollars in annual sales before the merger in 2019, while ABInBev had billions of dollars in 2016.
How many beer brands does AB InBev have?
There are about 630 beer brands in 150 countries in the world. AB InBev was created by InBev (a merger of Belgium-based Interbrew and Brazil-based AmBev) through the acquisition of AnheuserBusch in the United States.
What kind of rating does Anheuser Busch InBev have?
Wall Street analysts have given AnheuserBusch InBev SA/NV a Hold rating, but there may be better buying opportunities in the stock market. Some of MarketBeats' previous successful trading ideas have resulted in weekly earnings of 515%. MarketBeat had just presented five new stock ideas, but AnheuserBusch InBev SA/NV was not among them.
Is the AB InBev stock at a resistance level?
AB INBEV has a high resistance. Technical data points to a potential breakout of resistance levels after achieving significant daily returns. However, the RSI seems to suggest that the stock is on the brink of collapse. Also, fundamentals are not yet strong enough to support a breakout.
Is there a long entry in AB InBev?
Long entry for ABI as predicted by its AI algorithm. The expected time horizon is approximately 5 days. AB INBEV has a high resistance. Technical data points to a potential breakout of resistance levels after achieving significant daily returns.
When did Abi newbelco SA share price drop?
Since the incorporation of ABI Newbelco SA / NV AnheuserBusch InBev SA / NV in 2016, the share price appears to have fallen significantly, > 70%. Do you have any idea why the stock price has fallen so much?
Who are the major shareholders of Molson Coors?
Insiders who own Molson Coors Beverage stock include Andrew Thomas Molson, Brian Tabolt, James A. Winnefeld, Jr., Louis Vashon, Peter H. Course and Simon ■■■. See Institutional Property Trends at Molson Coors Beverage. Which Major Investors Are Selling Molson Coors Beverage Shares?
How much money does Molson Coors beverage make?
TAP shares can currently be bought for about $. How Much Money Does the Molson Coors Drink Make? Molson Coors Beverage has a market cap of one billion dollars and generates billions of dollars in revenue every year.
When did Molson Coors release their quarterly results?
Molson Coors Beverage (NYSE: TAP) announced its results on Wednesday, July 28. The company reported earnings per share (EPS) for the quarter, beating the Zacks consensus.
What are the analysts ratings on Molson Coors?
Eight Wall Street analysts issued buy, hold and sell recommendations from Molson Coors Beverage last year. There are currently 2 Sales Ratings, 3 Retention Ratings, and 3 Stock Buy Ratings. Wall Street analysts agree that investors should own shares in Molson Coors Beverage.
Who is the Mexican arm of Anheuser Busch InBev?
Mexican subsidiary AnheuserBusch InBev has filed a lawsuit against Constellation Brands for a second breach of their agreement, allowing Constellation to sell Mexican beer in the United States. Ireland-based C&C Group PLC (CCGGY, Financial) (LSE: CCR, Financial) has taken control of the pandemic shutdowns.
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Community and chorus join their voices in harmony with Handel's great masterwork. It is presented with a 25-piece orchestra, excellent soloists and projections of beautiful calligraphy by Timothy Botts. The McHenry County College Chorus will supplement the on-stage choir. This is one of our area's most meaningful holiday traditions. Let the Raue resound with the glory of Handel's music telling the story that reflects the greatest desires of mankind throughout history. Find more info here.


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Stories » Bryce Dessner & Jonny Greenwood - St Carolyn by the Sea | There Will Be Blood / Silver Platters review
Bryce Dessner & Jonny Greenwood - St Carolyn by the Sea | There Will Be Blood / Silver Platters review
Posted: April 21, 2014 12:00 AM | By: Admin
In mid 2013, Nonesuch released an album featuring compositions by Bryce Dessner. A little research revealed that Dessner was the guitar player from the rock band The National, but that he had initially been interested in classical and found that more mainstream music paid the bills. The disc featured none other than the Kronos Quartet, an ensemble known for their adventurousness and they gave the young composer a premier worthy of praise. Now, not to slight the Kronos' adventurousness, but I didn't think much would come of young Dessner beyond that recording. It wasn't because it wasn't good, it was impressive even (more so to the fact that the Kronos Quartet will often play compositions by fringe composers who have little access to the mainstream than any reflection on the music). So when a poster came into work from Deutsche Grammaphone featuring Bryce Dessner mere months later, a firm mental note was taken. Furthermore, Dessner, an upstart in the classical music world from as far as I know, was top billed above Jonny Greenwood, who has come into fame both as lead guitarist for Radiohead and for his striking films scores, including music to There Will Be Blood and The Master. The idea that Dessner got top billing said something to me, that the yellow label clearly saw something in his pieces that they believed in; this was clear to me before I even listened to St Carolyn by the Sea / There Will Be Blood. READ THE FULL Silver Platters REVIEW.
Crossover Media Projects with Bryce Dessner | Johnny Greenwood
Bryce Dessner | Johnny Greenwood
St. Carolyn by the Sea | There Will Be Blood
Never before have classical and rock converged in so organic, compelling and sensual a way as they do in the three short orchestral works by New York composer and guitarist Bryce Dessner on his new release St. Carolyn by the Sea, performed with the Copenhagen Phil and conducted by André de Ridder. The same is true of Jonny Greenwood, the composer of a suite of six miniatures for the soundtrack of There Will Be Blood which rounds out the album. The pairing of Dessner and Greenwood on the disc didn't come about by chance, but because André de Ridder likes programming their music together, for live concerts too. The defining strand of the collaboration is not so much the similar musical backgrounds of Dessner and Greenwood, but more their mutual preoccupation with quintessentially American themes such as the vast expanses of the country's landscape, or a sense of nostalgic longing; themes which, previously, were largely the domain of other musical genres.
12 NEW ON / 62 TOTAL
SYND: Performance Today, C24
Direct: SiriusXM, Music Choice, MOOD
Markets include: New York, Los Angeles, San Francisco, Boston, Philadelphia, Minneapolis, Denver, Portland, Cincinnati, New Orleans, Milwaukee, Detroit, Austin, Kansas City, Long Island NY, Berkeley CA
Online: Tainradio, Jazzweekly, RadioIO, Greenarrow, Live 365, WGOE, HuffPost


4k00:14Sunset over a calm lake in suburbian Stockholm. Aspudden, Stockholm, Sweden.
hd00:10Silhouettes of a family of two people sitting on a pier at sunset. Mother and daughter together practicing yoga next to a city water body on a summer evening.


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NYC loses appeal to ban large sugary drinks
Haley Draznin, CNN
Updated 5:10 PM EDT, Thu June 26, 2014
New York sugary drinks ban ruled illegal
New York appeals courts rules that city's ban on large sugary drinks is illegal
Decision says Board of Health "exceeded scope of its regulatory authority"
Industry group praises the decision, says ban was unfair to small businesses
Mayor Bill de Blasio: "We cannot turn our backs on the high rates of obesity and diabetes"
New York CNN —
Generic image of a large soda filling up at fountain
Freedom to be fat
New York's Court of Appeals ruled Thursday that New York City's ban on large sugary drinks, which was previously blocked by lower courts, is illegal.
"We hold that the New York City Board of Health, in adopting the 'Sugary Drinks Portion Cap Rule,' exceeded the scope of its regulatory authority," the ruling said.
Christopher Gindlesperger, a spokesman for the American Beverage Association, praised the ruling.
"We are pleased that the lower courts' decisions were upheld," he said in a statement. "It would have created an uneven playing field for thousands of small businesses in the city and limited New Yorkers' freedom of choice. With this ruling behind us, we look forward to collaborating with city leaders on solutions that will have a meaningful and lasting impact on New Yorkers and families across the country."
lklv roth ny drinks ban explainer_00001904.jpg
New York's big drink debate
Former New York City Mayor Michael Bloomberg had pushed for the ban on sugary drinks larger than 16 ounces as a way to fight obesity and other health problems.
In a statement, Bloomberg's successor, Mayor Bill de Blasio, expressed disappointment with the decision.
"The negative effects of sugary drink over-consumption on New Yorkers' health, particularly among low-income communities, are irrefutable," the statement said. "We cannot turn our backs on the high rates of obesity and diabetes that adversely impact the lives of so many of our residents. While we are still examining the Court's decision, it is our responsibility to address the causes of this epidemic, and the City is actively reviewing all of its options to protect the health and well-being of our communities."
Sugary drinks linked to 180,000 deaths worldwide


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French, 10 results 10
Secretariat Records, 41 results 41
Archives and Documentation of International Organizations, 3 results 3
UNESCO. Funds-in-Trust, 8 results 8
Ishii, Kikujirō, 1866-1945, 1 results 1
UNESCO Office Islamabad, 1 results 1
International Centre for Synchrotron Light for Experimental Science and Applications in the Middle East (Jordan), 1 results 1
UNDP, 1 results 1
European Organization for Nuclear Research, 1 results 1
Japan, 49 results 49
Asia and the Pacific, 49 results 49
Cambodia, 8 results 8
Afghanistan, 3 results 3
Congo DR, 2 results 2
South Sudan, 1 results 1
Pakistan, 1 results 1
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Mali, 1 results 1
File, 49 results 49
Japan File
Relations avec le Japon
Japanese Trust Fund for Preservation of the World Cultural Heritage: Project ...
Japanese Trust Fund for Preservation of the World Cultural Heritage: Project on Conservation and Restoration of Royal Plaza, Bayon and Angkor Vat, Cambodia
Japanese Trust-Fund for Preservation of the World Cultural Heritage: project ...
Japanese Trust-Fund for Preservation of the World Cultural Heritage: project on "Selection of Khmer Inscriptions in Angkor" - Publication of epigraphic reference book
Plan of Operation for a Funds-in-Trust Project on Programme for Enhancement o...
Plan of Operation for a Funds-in-Trust Project on Programme for Enhancement of Literacy in Afghanistan Phase III (ELA 3)
Credentials for Satoru Takahashi,... Alternate Delegate of Japan to the Inter...
Credentials for Satoru Takahashi,... Alternate Delegate of Japan to the International Conference for the creation of an International Computation Center
Instrument of Acceptance of the Government of Japan regarding the acceptance ...
Instrument of Acceptance of the Government of Japan regarding the acceptance of the convention for the establishment of the International Computation Centre
Full Powers of the Minister for Foreign Affairs of Japan delivered to Mr. Sue...
Full Powers of the Minister for Foreign Affairs of Japan delivered to Mr. Suemasa Okamoto, Ambassador Extraordinary and Minister Plenipotentiary of Japan to the Netherlands with authority to sign the Convention...
Grant Agreement and Plan of Operation for Strategic Strengthening of Flood Wa...
Grant Agreement and Plan of Operation for Strategic Strengthening of Flood Warning and Management Capacity (Phase 2) in the Islamic Republic of Pakistan
Public Opinion Survey in Japan on UNESCO & United Nations Part I
Public Opinion Survey in Japan on UNESCO & United Nations Part II
Statement of the Government of Japan declaring its willingness to participate...
Statement of the Government of Japan declaring its willingness to participate in the UNESCO Book Coupon scheme as purchaser of Book Coupons and supplier of Books
Statement of the Government of Japan declaring its willingness to participate in the Unesco Coupon Scheme as purchaser of film coupons and supplier of films
Full Powers for Toru Hagiwara,... Plenipotentiary of Japan to attend the Inte...
Full Powers for Toru Hagiwara,... Plenipotentiary of Japan to attend the International Conference for the creation of an International Computation Centre and to sign and seal any and all international instruments...
Full Powers of the Emperor of Japan for Toru Hagiwara and Masa-Akira Katsumot...
Full Powers of the Emperor of Japan for Toru Hagiwara and Masa-Akira Katsumoto to attend the Conference for the Universal Copyright Convention and to sign
Pleins pouvoirs pour la conférence de révision de la Convention Universelle s...
Pleins pouvoirs pour la conférence de révision de la Convention Universelle sur le Droit d'Auteur, Paris, 5-24 juillet 1971
Statistiques intellectuelles - Japon
Note of the Vice-Minister for Foreign Affairs to the Director-General of UNES...
Note of the Vice-Minister for Foreign Affairs to the Director-General of UNESCO informing that following persons have been designated by the Government as composing the Japanese Delegation to the Intergovernment Conference...
Documentation - Japon
Japan Grant for the Project Humanitarian Assistance for African Countries
Japan - Convention concerning the Protection of the World Cultural and Natura...
Japan - Convention concerning the Protection of the World Cultural and Natural Heritage
Japan - Instrument of Acceptance - Convention for the Safeguarding of the Int...
Japan - Instrument of Acceptance - Convention for the Safeguarding of the Intangible Cultural Heritage, adopted in Paris on 17 October 2003
Japanese Trust Fund for Preservation or the World Cultural Heritage: Project ...
Japanese Trust Fund for Preservation or the World Cultural Heritage: Project Preparation Mission to Angkor
Japanese Trust Fund for the Preservation of the World Cultural Heritage: Trai...
Japanese Trust Fund for the Preservation of the World Cultural Heritage: Training for National Capacity Building for the Conservation of Cultural Monuments, Cambodia
Japanese Trust Fund for Preservation of the World Heritage: Project on Traini...
Japanese Trust Fund for Preservation of the World Heritage: Project on Training for National Capacity Bulding for the Conservation of Cultural Monuments, Cambodia
Guide des Echanges - Japon
Relations avec le Gouvernement du Japon
Ratifications of the Emperor of Japan regarding the Universal Copyright Conve...
Ratifications of the Emperor of Japan regarding the Universal Copyright Convention and the Protocol No. I, and the Protocol No. II, and the Protocol No. III annexed to the Convention
Participation Programme - Project agreement No. 4926 - Mutual Appreciation of...
Participation Programme - Project agreement No. 4926 - Mutual Appreciation of Eastern and Western Cultural Values
Plan d'Opération - International Institute of Seismology and Earthquake ...
Plan d'Opération - International Institute of Seismology and Earthquake Engineering (JPN.1) (Phase I)
Japan - Instrument of Acceptance - International Convention against Doping in...
Japan - Instrument of Acceptance - International Convention against Doping in Sport
UNESCO-Japan Project Agreement - Humanitarian Assistance for African Countrie...
UNESCO-Japan Project Agreement - Humanitarian Assistance for African Countries - Democratic Republic of Congo
Japon - Convention relative aux zones humides d'importance international...
Japon - Convention relative aux zones humides d'importance internationale particulièrement comme habitats de la sauvagine, Ramsar
Statement of the Government of Japan declaring its willingness to participate in the Unesco Coupon Scheme as purchaser of Coupons for scientific material and supplier of scientific material
Note de l'agence du gouvernement japonais à Génève au M. le Professeur P...
Note de l'agence du gouvernement japonais à Génève au M. le Professeur Paul Scherrer, Président de la Commission suisse de l'énergie atomique, désignant M. Keiichi Tatsuke...et M. Bunshichi Hoshi...comme observateurs du Japon à la Conférence...
Note of the Minister for Foreign Affairs of Japan certifying that the followi...
Note of the Minister for Foreign Affairs of Japan certifying that the following persons have been designated as composing the Japanese delegation to the Intergovernmental Conference for the Protection of Cultural Property in the Event of Armed Conflict...
Convention Universelle sur le Droit d'auteur révisée, Paris, le 24 Juill...
Convention Universelle sur le Droit d'auteur révisée, Paris, le 24 Juillet 1971
1er Congrès international des Arts populaires - Japon
Conférence internationale des fouilles - Documentation - Japon
Plan of Operations: Japanese Trust Fund for the Preservation of the World Cul...
Plan of Operations: Japanese Trust Fund for the Preservation of the World Cultural Heritage - Project on Preservation and Restoration of the Royal Plaza, the Bayon and Angkor Wat, Phase II
Japanese Trust Fund for Preservation of the World Cultural Heritage: Project on Conservation and Restoration of the Royal Plaza, the Bayon and Angkor Wat, Phase II (556-CMB-74)
Japan - Notification of Acceptance - Centre for Synchrotron-Light for Experim...
Japan - Notification of Acceptance - Centre for Synchrotron-Light for Experimental Science and Applications in the Middle East (SESAME)
Japon - Protocol to Amend the Convention on Wetlands of International Importa...
Japon - Protocol to Amend the Convention on Wetlands of International Importance especially as Waterfowl Habitat
UNESCO-Japan Grant Agreement - Humanitarian Assistance for African Countries ...
UNESCO-Japan Grant Agreement - Humanitarian Assistance for African Countries - Democratic Republic of Congo and Mali
Note from the Embassy of Japan, Afghanistan, on the Project on Literacy for E...
Note from the Embassy of Japan, Afghanistan, on the Project on Literacy for Empowering Afghan Police II (LEAP II)
Plan of Operation between UNESCO and the Government of the Islamic Republic o...
Plan of Operation between UNESCO and the Government of the Islamic Republic of Afghanistan for the project on Literacy for Empowering Afghan Police II (LEAP II)