In addition to refinishing kitchen cabinets, Frank's All Surface Refinishing can Refinish your Bathroom Vanities at a fraction of the cost of replacing. Removing the vanities often necessitates removing the countertop and disassembling the plumbing. This can double your already high construction expenses.
Refinishing your Bathroom Vanity is the cost effective solution to your problem. Usually within 3-4 hours we can apply a hard, durable coating, in any color, to your existing Bathroom Vanities.
We use a 2 part resin coating that will stand up to the excessive moisture of normal Bathroom use. We are so confident of your satisfaction and with our satin finish that we provide a Lifetime Warranty against any lifting of the material. This permanent Refinishing process that we have perfected over the last 20 years will last the life of the cabinet. Frank will give you a Free Estimate and we can also suggest color selection.
Many times it is also more economical to refinish your bathroom countertops at the same time as your Vanities. This will allow you to coordinate your color selection, which will enhance your entire Bathroom remodeling project. The results are dramatic. The costs are minimal and we can provide the expertise to bring your project to a speedy completion.


Broaden your understanding of key concepts with these helpful resources.
Explanation: This Methods Map can be used to get a visual representation of key issues and terms that are used in qualitative research.
Explanation: This extensive set of resources, hosted by Huddersfield University, can be used to explore an extensive range of issues, methods and techniques relating to qualitative research in psychology. You can use this to extend your knowledge of the ideas and methods in the book and also to explore beyond this to more techniques.


text:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Net;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Documents;
using System.Windows.Input;
using System.Windows.Media;
using System.Windows.Media.Animation;
using System.Windows.Shapes;

namespace Rawr.ShadowPriest
{
    public partial class SpellPriorityForm : ChildWindow
    {
        public List<string> SpellPriority { get; protected set; }
        public ListBox MirrowList { get; set; }
        private Character character;

        public SpellPriorityForm(List<string> spells, ListBox mirrowList, Character _character)
        {
            SpellPriority = spells;
            MirrowList = mirrowList;
            character = _character;

            InitializeComponent();
            foreach (string spell in Spell.ShadowSpellList)
            {
                if (spells.Contains(spell))
                    lsSpellPriority.Items.Add(spell);
                else
                    cmbSpells.Items.Add(spell);
            }
        }

        private void bAdd_Click(object sender, RoutedEventArgs e)
        {
            if (cmbSpells.SelectedItem == null)
                return;

            lsSpellPriority.Items.Add(cmbSpells.SelectedItem);
            cmbSpells.Items.RemoveAt(cmbSpells.SelectedIndex);
            cmbSpells.SelectedItem = "";
        }

        private void bUp_Click(object sender, RoutedEventArgs e)
        {
            if (lsSpellPriority.SelectedItem == null || lsSpellPriority.SelectedIndex == 0)
                return;

            lsSpellPriority.Items.Insert(lsSpellPriority.SelectedIndex - 1, lsSpellPriority.SelectedItem);
            lsSpellPriority.Items.RemoveAt(lsSpellPriority.SelectedIndex);
        }

        private void bDown_Click(object sender, RoutedEventArgs e)
        {
            if (lsSpellPriority.SelectedItem == null || lsSpellPriority.SelectedIndex == lsSpellPriority.Items.Count - 1)
                return;

            lsSpellPriority.Items.Insert(lsSpellPriority.SelectedIndex + 2, lsSpellPriority.SelectedItem);
            lsSpellPriority.Items.RemoveAt(lsSpellPriority.SelectedIndex);
        }

        private void bRemove_Click(object sender, RoutedEventArgs e)
        {
            if (lsSpellPriority.SelectedItem == null)
                return;

            cmbSpells.Items.Add(lsSpellPriority.SelectedItem);
            lsSpellPriority.Items.RemoveAt(lsSpellPriority.SelectedIndex);
        }

        private void bClear_Click(object sender, RoutedEventArgs e)
        {
            if (lsSpellPriority.Items.Count == 0)
                return;

            foreach (Object o in lsSpellPriority.Items)
                cmbSpells.Items.Add(o);

            lsSpellPriority.Items.Clear();
        }

        private void OKButton_Click(object sender, RoutedEventArgs e)
        {
            this.DialogResult = true;
        }

        private void CancelButton_Click(object sender, RoutedEventArgs e)
        {
            this.DialogResult = false;
        }
    }
}

redpajama_set_name: RedPajamaGithub


My eyes are dimmed by history's words.
I still cannot compare with nature's perfection.
Learning is a passion shared by many of us. There is a great allure to education and a fascination with the accomplishments of civilization. We go to libraries and museums. We go to exhibits showing the diggings from royal tombs. We are enchanted with new inventions. And yet, if we look out our windows and see a tree in its perfection, or gaze into a tide pool, or watch a cat as it strolls its territory, or see the flash of a blue jay, we can see another order of beauty and intelligence in this life.
The works of humanity cannot compare to the works of nature. The works of civilization lack the balance and refinement of nature. Too many times, our accomplishments are tainted by impure motives : profit, hardship, desire for fame, simple greed. We achieve, but we cannot foresee the results because we are unable to place our actions into a greater context.
Nature is a conglomeration of contending forces, of tooth and claw, venom and perfume, mud and excrement, eggs and bones, lightning and lava. It seems chaotic. It seems terrible. And yet, for all its unfathomable workings, it far surpasses the business of our society.


RBW-I is RBW Entertainment's trainee program. Their first batch of trainees will debut as ONEUS and ONEWE in 2018. The second batch, of female trainees, run a channel called 365 Practice on YouTube.
-She speaks Korean, Chinese, Japanese, and English.
-She was on Produce 48 (ranked 80).
-Her nickname is Bambi because of her visuals.
-She is slow at memorizing choreography.
-She was on Produce 48 (ranked 29).
-She went to Hanlim Multi Arts and was in a class with Loona's Chuu and Kim Lip.
-She was on Mixnine (ranked 78).
-Her nickname is "cute blackbean" because she has dark skin.
-She is a current Hanlim Multi Arts student.
-She appeared on Mixnine (ranked 40).
-She appeared on Mixnine (ranked 90).
-She enjoys going shopping by herself.
-She is a fan of Pentagon.
-She appeared on Mixnine (ranked 85).
-She competed in K-pop cover competitions in Japan.
-She was on Mixnine (rank 58).
-She is the "baby face".
-She used to run a YouTube channel.
Read more facts about Seoho, Keonhee, and Hwanung at the ONEUS profile!
Read more facts about Yonghoon, Harin, Kanghyun, Dongmyeong, and Cya at the ONEWE profile!
Who is your RBW-I bias?
Na Goeun was eliminated from produce 48 and ranked 29th.
Author here! If anyone has a photo of new 365 Practice member Eonjin, please let me know where I can find one! She was just introduced and I can't find her on the other social media yet.
jiheun has the same birthday as me, just different years.
When I first saw their YouTube channel, thought their group has many pretty girls to be only doing dance covers. I later came to know that they are actually trainees from RBW (Mamamoo's agency) and will debut later.. ^_^ Fan of their channel..
@ducung_ on Instagram, he passed the audition in Vietnam and in his story you could see the rbw practice room a few days ago. Don't know if he's official trainee.
I sense a future snsd aka national girl group from these girls… I wonder why?
Yuki was born in 2002 so no way she's 15 at this point.
That being said I don't know when her birthday is.


If the top rated skiing, world class fly fishing, two National Parks, and scenic beauty weren't enough reason to live in Jackson Hole, there are significant tax benefits to living in Wyoming as well.
Wyoming has frequently been cited as one of the most tax friendly states in the country. Part of the reason is Wyoming's significant budget surplus (approx. $1 billion). In fact, for the past seven years Bloomberg Wealth Management magazine has named Wyoming "America's Wealth-Friendliest State".
No state income tax: With no state tax on personal or corporate income, "you have more disposable income," Jones says.
Dynasty trusts: In Wyoming, you can shield your real estate from federal estate taxes for up to 1,000 years through a dynasty trust. "You can establish a trust in Wyoming for the benefit of your family or other beneficiaries," Jones says. "You can transfer your real estate into a limited liability company or a family partnership and then put that into the 'dynasty trust.' which can continue for a thousand years." As a result, multiple generations can make use of and enjoy the property, without having to pay estate taxes or worse, having to sell the property in order to pay the taxes. A key point to remember: The trust must be administered in Wyoming.
No inheritance tax or estate tax: "Wyoming repealed its estate tax as of January 1, 2005," Jones says.
No state gift tax: "Somebody who owns property in Wyoming can 'gift' that real estate to their heirs without having to worry about paying a state gift tax," Jones says.
Low property taxes: "Wyoming has very low property taxes compared to other states," Jones says. "The taxes that you do pay here are based on the assessed value of the property." For Teton County, he says, the rate is 1.2 percent of a property's assessed value. The rate for the city of Jackson is 8/10 of one percent.
No excise taxes: When you fill up your car's gas tank or buy a bag of groceries in Wyoming, you won't pay any state tax on your gas or food.
No intangible taxes: Wyoming doesn't make you pay a tax on financial assets like stocks and bonds.
No tax on the sale of real estate.
Please contact me if you are interested in learning more about Wyoming's tax benefits.


text:
OMCA COLLECTIONS
Great Northern Railway | St. Paul
MN | Winold Reiss | New York | L. Langdon Kinn | Glacier National Park
Pikuni and Kainah Tribes of Blackfeet Indians
8.75 in HIGH x 11.5 in WIDE x .25 in DEEP
(22.22 cm HIGH x 29.21 cm WIDE x .63 cm DEEP)
Gift of Eric and Ingrid Woods
blackfeet
kainah
pikuni
waterton lake national park
A: Folder/folio of gray cardboard printed in red and blue with many Natiive American symbols, and "Pikuni and Kainah Tribes of Blackfeet Indians, Glacier National Park, Waterton Lake National Park, America First." The folio fastens with broken string closure. It is lined in red. Inside the portfolio are 12 full color prints of differnt Native American individuals; some of the prints are mounted on blue paper, others on red. B: Lazy Boy--Medicine Man, Glacier National Par, from original portrait by Winold Reiss, New York. Shows man, facing proper left, his hair in braids, wrapped with tan strips of fur; he wears a tan (probably buckskin) shirt, and holds significant symbols in either hand. Mounted on blue paper. C: Mountain Chief, Blackfeet Brave, Glacier National Park, from original portrait by W. Langdon Kinn, New York. Shows man facing forward, wearing buckskin shirt, fringed at shoulders, highly decorated with colorful beadwork; war bonnet on his head of black feathers with white ermine tails suspended from either side. Mounted on red paper. D: Mrs. Devine, Glacier National Park, from original portrait by Winold Reiss, New York. Shows young woman, seated in front of a bright yello backgroun, wearing a red shirt, her arms folded in front of her; she looks down toward the lower proper right corner. Her hair is braided and looped under the blue scarf she is wearing with a large, pale gray "cowboy style" hat on top. Mounted on blue paper. E: Long Time Pipe Woman, Glacier National Park, from original portrait by Winold Reiss, New York. Elderly woman, seat, facing proper left. She wears a blue and brown garment, with her hands folded in her lap. She's wearing several strands of beads. Her braided hair is wrpped with dark brown fur; her white and red headband support a single feather at the back of her head. Mounted on blue paper. F: Mike Short Man, Glacier National Park, from original portrait by Winold Reiss, New York. White background centered with the subject's head. He faces forward, has two braids, wrapped in black. He appears to wear shell earrings; his short necklace has two decorations suspended in the front. He also wears a light brown hat with a red hat band. Mounted on red paper. G: Nightshoots, Glacier National Park, from the original portrait by Winold Reiss, New York. Man faces forward; he's wearing a commercially made yellow shirt with a magenta and yellow scarf around his neck. He also wears a necklace at his neck, over the collar of the shirt. His hair, getting gray at the top, is braided, hangs far down the front of his shirt and is wrapped with white. A tan hat sits squarely on his head with a decorative hat band, and a fluffy feather off to the proper left side. Mounted on red paper. H: Many Mules, Glacier National Park, from original portrait by Winold Reiss, New York. Eldery figure draped in a white blanket, black scarf tied like a bandana over the head and under the chin, with a gray fur toque on top. Mounted on red paper. I: Chief Two Guns White Calf, Glacier National Park, from original portrait by Winold Reiss, New York. Full figure portrait, with the subject seated on a white, black and red blanket, facing the proper left. He wears a blue shirt, green pants or leggings and beaded moccasins. His hair is pulled into two sections, wrapped in red. He has earrings, probably of shell, a beaded necklace trimmed with shells and a single feather over his right ear. Mounted on red paper. J: Alex Eagle Plume, Glacier National Park, from original portrait by Winold Reiss, New York. Child, facing forward, wrapped in a white and light red blanket, with many braids, wrapped in baby blue. Mounted on blue paper. K: Shot on Both Sides, Chief of the Blood Tribe, Waterton Lakes Natiopnal Park, from original portrait by Winold Reiss, New York. The subject faces 3/4 to the proper right. He is wearing white, with a yellow and blue-gray fringed garment over his shoulders. His braided hair is wrapped with white. He wears small shell earrings. Several trimmed feathers are in his hair at the back. Mounted on blue paper. L: Blackfeet Mother, Glacier National Park, from original protrait by W. Langdon Kinn, New York. Woman is seated and faces forward, but looks down slightly to the proper left. She's dressed in red and holds a sleeping child in her arms. The child is wrapped in a multi-colored blanket. Her hair is parted in the middle, and two very long braids hang down the front of her dress, their tips not visible behind the child. Mounted on blue paper. M: Double Steel and Two Cutter, Glacier National Park, from original portrait by Winold Reiss, New York. Two figures, both with looped braids. The figure on the proper right is more decorative, dressed in white fringed garment with a pink scarf over her head, and several pieces of jewelry. The other figure is dress in blue. Mounted on red paper.
This prints were obviously used as home decoration, or classroom information because the all have thumbtack holes at their corners.
Used: decoration | Education
© 2023 Oakland Museum of California
RIGHTS AND REPRODUCTION

redpajama_set_name: RedPajamaCommonCrawl


text:
Concrete's Contribution to the Rise and Decline of Civilization
As the world becomes more encased in concrete, the detrimental effects of its widespread use are increasing in scale and severity.
Generating Thermal Energy Requires Water, Lots of It
September 5, 2018, 12pm PDT
Water is an important part of the thermal energy process – one that we may not think as much about.
As Drought Returns to Texas, Report Recommends Stricter Rules on Outdoor Water Use
May 5, 2018, 5am PDT
A report from a coalition of nonprofits recommends, among other restrictions, allowing outdoor watering only twice per week.
The Texas Tribune
Let's Talk Water
January 12, 2016, 2pm PST
Water has been an integral part of the designed landscape for centuries, but is it time to reevaluate its relevance?
The Passaic River: A Postindustrial River Reimagined
Writer Jeff Byles chronicles the Passaic River's relationship to Paterson, New Jersey and how its revival can prove beneficial for the city in this second installment of the Paterson series.
What Droughts Say About Planners and Water Officials
When water policy and land use planning operate in separate spheres, it's more difficult to design for efficient resource use. Better communication is needed in the drought-stricken southwest.
Next City
An Interactive Map of California's Urban Water Use
January 23, 2015, 11am PST
California has commenced new water use reporting requirements, which the Pacific Institute has compiled into an interactive map and database to provide accessible evidence about where the state's water goes.
Pacific Institute
U.S. Water Use Hits 45-Year Low
November 6, 2014, 9am PST
Recent data from the United States Geological Survey (USGS) shows that the country is using less water even as its population grows.
United States Geological Survey
'Droughtshaming' and the Death of Fun
September 22, 2014, 1pm PDT
Laura Bliss provides a summary of the controversy surrounding plans to bring the "Slide the City" event to Temple Avenue in Downtown Los Angeles.
Deep Flaws Found in California's Water Use Reporting
The special status of "senior rights holders" among California water interests is well documented. Not as well documented is the system which allows errors in water use reports to protect the interests of a fortunate few.
Historic Drought Offers Historic Opportunity
August 19, 2012, 7am PDT
Charles Fishman pens an opinion piece for The New York Times arguing that America's worst drought since the 1950s offers "an opportunity to tackle long-ignored water problems and to reimagine how we manage, use and even think about water."


text:
Q: How to arrange Bootstrap grid with one column vertically centered next to a 4x4 grid? I am still learning how to use Bootstrap, and I was wondering how you can arrange the grid more freely.
I am trying to have a single div centered vertically on the left, and a 4x4 grid on the right. I've provided an example image of how I want to arrange my divs in the format I want. Please see the
Example.
How would you accomplish something like this? Thank you!

A: I would use grid system instead of bootstrap in this case. Something like this:
HTML:

Image

Text 1

Text 2

Text 3

Text 4

CSS:
.grid-container {
            display: grid;
            height: 50vh;
            padding: 10px;
            background-color: coral;
            border: 1px solid;
            grid-template-columns: 1fr 1fr 1fr 1fr;
            grid-template-rows: 1fr 1fr;
            grid-row-gap: 20px;
            grid-column-gap: 20px;
        }
        .child {
            background-color: cornflowerblue;
            border: 1px solid;
        }
        .a {
            grid-area: 1/1/3/3;
        }

A: With bootstrap I recommend you to use flex properties
Example

<link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.0-beta3/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-eOJMYsd53ii+scO/bJGFsiCZc+5NDVN2yr8+0RDqr0Ql0h+rP48ckxlpbzKgwra6" crossorigin="anonymous">

IMAGE

1

2

3

4

*

*d-flex: display flex

*align-items-center: this centers all your items vertically

*align-content-around: is how items are spaced horizontally, around takes the free space and as you can see distributes it equally

redpajama_set_name: RedPajamaStackExchange


Irony abounds at Zouch Manor.
When Mr. Zouch reached into his jacket pocket, though, to retrieve is own pair of spectacles, he found they were missing. He immediately commanded a full search of the house, but we turned up little more than three buttons, a canvas bag of dimmicks, the bones of a long-forgotten rodent, and an uncomfortably explicit love letter that seemed to confirm Mr. Zouch's periodic boasts about his youthful conquests.
The next morning, as we listened to Mr. Zouch mutter about his myopia, the editors found several lists of suitable replacements.
More than one of us is hoping he'll choose the Cobain frame with Calvin Harris lenses.


text:
Strong 40 gives Addin plenty of confidence
Wayne Cousins
Tue 3 Apr 2018, 12:09 PM
Addin Fonua-Blake has taken plenty of confidence from his strong first half performance against the Canberra Raiders.
The hard running prop punched out a quality 40 minutes after the side lost interchange forward Kelepi Tanginoa with a season ending knee injury.
"I felt pretty happy that the coaching staff trusted me to play that first 40, '' Fonua-Blake said.
"It just came as a result of the injuries. I was pretty stoked to get it out and it gives me more confidence going forward.
"At some stages there in the first half, I felt a bit tired but it is all a mental thing.
"I kept telling myself that I could do it, so to do it in the end was pretty good."
Fonua-Blake's efforts were not lost on Coach Trent Barrett who praised the Tongan and Kiwi international post match for his tireless efforts. All up, the 22-year-old played 66 minutes, made 18 runs for 189 metres, and completed 27 tackles.
His work-rate was not lost on his team-mates either.
"Addin did very well to get those (first) 40 minutes out for such a big lad and with how fast the game is now. It was pretty impressive,'' said forward Shaun Lane.
"The amount of work that he has put in over the last few years has been really impressive.
"He is just getting better and better with each game, and the results are really starting to show now.
"To be putting out the minutes that he is and all the work he does, he doesn't even look tired now when he is out there."
Centre Brian Kelly said Fonua-Blake has taken his game to another level alongside fellow Kiwi international Martin Taupau.
"This year Addin has shown a lot of energy. On the weekend, he was carrying two to three players a good 20 metres,'' Kelly said.
"It is a good way to start our sets and at the back end, it really helps us out.
"Addin and Marty have got their little plays they do and have got a few little weapons under their belt. They are working together really well, so it is a plus for us."
Did you enjoy this piece of content from Manly Media?
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NRL History
Legend Q&A: Luke Williamson
Sea Eagles: 2020 round 1 predicted team
Sea Eagles extend Addin Fonua-Blake
Fonua-Blake and Taupau in Kiwis squad
Addin Fonua-Blake to make his NRL debut


River City Carpet Cleaning in Forest Lake, QLD, has experience in all areas of Carpet Cleaners and can help you find the right Carpet Cleaners solutions. stain removal, rugs, and guarantees and many other Carpet Cleaners products/services are available. Talk to us at 0422 17#### for professional advice on any aspect of Carpet Cleaners.


text:
Q: Why is minifyEnabled is false in release builds by default? In build.gradle (app) file we have this by default,
buildTypes {
    release {
        minifyEnabled false
        proguardFiles getDefaultProguardFile('proguard-android-optimize.txt'), 'proguard-rules.pro'
    }
}

According to https://developer.android.com/studio/build/shrink-code,
minifyEnabled true make the code more secure (making it hard to reverse engineer) and also shrink the size in release build, which we use for publishing the app.
I know that using minifyEnabled true makes compile time longer, but usually debug builds are used for development and testing, which is not affected anyways.
What I'm looking for is that, what are the disadvantages (such as performance impacts) cause by using minifyEnabled true at runtime. I'm not worried about the build time of a release build.
Google Play Console also recommending us to enable it therefore I'm wondering why is minifyEnabled is disabled by default.

A: By adding minifyEnabled true in your release build you can obfuscate code but it is set to false by default in release builds because if set true it shall require proguard rules to be written to tell the compiler which classes are to be ignored while obfuscating the code. If minifyEnabled is set to true by default and the developer forgets to add proguard rules, it can lead to runtime crashes.
However, setting minifyEnabled to true and shrinkResources to true can reduce your apk size.

A: As NIKHIL AGGARWAL said, setting minifyEnabled to true can lead to runtime crashes.
You should research each used library so that include their ProGuard rules into your project. Also sometimes after Gradle or libraries upgrade you should again check ProGuard rules and accordingly update them in your project.
You should add data classes of your network model to ProGuard.
You should check your release build on several devices/emulators with different Android versions to understand where it can crash. Debugging in release build is difficult and with obfuscation it becomes a nightmare.
However after you set minifyEnabled = true and avoid crashes you will decrease apk size and impede to cracking attempts.


MRI (magnetic resonance imaging) of the brain is a diagnostic technique that uses magnetic field and radio waves to produce images of the brain and brain stem. It is a safe and painless method. An MRI can detect cysts, tumors, bleeding, swelling, infections, inflammation, aneurysms, and developmental deformities of the brain.
I had an MRI of brain for tinnitus. Please explain the cause.
Is there a possibility to perform MRI of brain when on ventilator?
My father has severe headaches and nausea. I want a second opinion on his MRI of brain.
My USG of 36 weeks showed big head with mild prominence. What to do?
Why does my body vibrate?
Is eye floaters an optical problem or neurological or a combination of both?
Due to cervical spondylosis, will I have problem during pregnancy?
Do you have a question on Mri Of Brain? We have medical experts to help you right away.


text:
\section{Introduction}

The purpose of the regression of a univariate response $y$ on a $p$-dimensional
predictor vector ${\mathbf{x}}$ is to make inference on the conditional distribution of $y|{\mathbf{x}}$.
Following \citet{cook1998b}, ${\mathbf{x}}$ can be replaced by its standardized version
\begin{eqnarray}
{\mathbf{z}}= [\Sigma_{{\mathbf{x}}}]^{-{1}/{2}} ({\mathbf{x}}-\mu_{{\mathbf{x}}})\ ,
\label{standardize}
\end{eqnarray}
where $\mu_{{\mathbf{x}}}$ and $\Sigma_{{\mathbf{x}}}$ denote
the mean and covariance matrix of ${\mathbf{x}}$ respectively assuming non-singularity
of $\Sigma_{{\mathbf{x}}}$.

The goal of {\it dimension reduction in regression} is to
find out a $p\times d$ matrix $\gamma$ such that
\begin{eqnarray}
y \;\, \rule[0em]{.03em}{.67em} \hspace{-.25em {\mathbf{z}} | \gamma' {\mathbf{z}}\ ,
\label{modelcdrs}
\end{eqnarray}
where ``$\;\, \rule[0em]{.03em}{.67em} \hspace{-.25em$" indicates independence.
Then the $p$-dimensional {${\mathbf{z}}$} can be replaced
by the $d$-dimensional vector $\gamma' {\mathbf{z}}$
without specifying any parametric model
and without losing any information on predicting $y$.
The column space ${\rm Span}\{\gamma\}$ is called a
{\it{dimension reduction subspace}}.
The smallest applicable $d$ is called the {\it dimension of the regression}.

Based on the inverse mean ${\rm E}({\mathbf{z}}|y)$, \citet{li1991a} proposed Sliced Inverse
Regression (SIR) for dimension reduction in regression.
It is realized that SIR can not recover the symmetric dependency \citep*{li1991b,cook1991}.
After SIR, many dimension reduction methods have been introduced.
Sliced Average Variance Estimate (SAVE) proposed by \citet*{cook1991}
and Principle Hessian Directions (pHd) proposed by \citet*{li1992} are another two popular ones.
Both pHd and SAVE refer to the second inverse moment, centered or non-centered.
Compared with SAVE, pHd can not detect certain dependency hidden in the second moment
\citep*{yin2002,ye2003} and the linear dependency \citep*{li1992,cook1998a}.
Among those dimension reduction methods using only the first two inverse moments,
SAVE seems to be the preferred one.  Nevertheless, SAVE is not always the winner.
For example, \citet*{ye2003} implied that
a linear combination of SIR and pHd may perform better than SAVE
in some cases.  It is not surprising since \citet{li1991b} already suggested that
a suitable combination of two different methods might sharpen the dimension reduction results.
\citet*{ye2003}  further proposed that
a bootstrap method could be used to pick up the ``best" linear combination of two
known methods, as well as the dimension of the regression,
in the sense of the variability of the estimators,
although lower variability under the bootstrap procedure does not
necessarily lead to a better estimator. \citet*{liwang2007} pointed out
that linear combinations of two known methods selected by the bootstrap criterion
may not perform as well as a single new method, their Directional
Regression method (DR), even though the bootstrap one is computationally intensive.

This article aims to develop a new class of, instead of a single one,
dimension reduction methods using only the first two inverse moments,
as well as the corresponding large sample tests for the dimension of the regression
and an efficient criterion for selecting a suitable candidate from the class.
Theoretically, it can cover SIR, pHd, SAVE and their linear combinations.
Practically, it can achieve higher power in recovering the dimension reduction subspace.
In Section~2, we review the necessary dimension reduction context.
In Section~3, we introduce a simple candidate matrix $M_{{\mathbf{z}}{\mathbf{z}}'|y}$
which targets the entire inverse second moment subspace.
It is indeed the candidate matrix of an intermediate method
between pHd and SAVE.
In Section~4, we propose a new class of dimension reduction methods
called Sliced Inverse Moment Regression (SIMR),
along with weighted chi-squared tests for
the dimension of the regression.
In Section~5, we use SIMR to analyze a simulated example
and illustrate how to select a good candidate of SIMR.
Simulation study shows that SIMR may have
consistently greater power than SIR, pHd, and SAVE, as well as DR
and another new method Inverse Regression Estimator \citep*{cookni2005}.
In Section~\ref{sectionozonedata}, a real example is used to
illustrate how the proposed method works.
It is implied that a class of dimension reduction methods, along with
a suitable criterion for choosing
a good one among them, may be preferable in practice to any single method.
We conclude this article with discussion and proofs of the results presented.

\section{Dimension Reduction Context}

\subsection{Central Dimension Reduction Subspace (CDRS)}

\citet*{cook1994b,cook1996} introduced the notion of
{\it{central dimension reduction subspace}} (CDRS),
denoted by $S_{y|{\mathbf{z}}}$, which is the intersection of all dimension reduction subspaces.
Under fairly weak restrictions, the CDRS $S_{y|{\mathbf{z}}}$ is still a dimension reduction subspace.

In this article, we always assume that $S_{y|{\mathbf{z}}}$ is a dimension reduction subspace
and that the columns of $\gamma$ is an orthonormal basis of $S_{y|{\mathbf{z}}}$.
In practice,
we usually first transform the original data $\{{\mathbf{x}}_i\}$ into their standardized version
$\{{\mathbf{z}}_i\}$ by replacing $\Sigma_{\mathbf{x}}$ and $\mu_{\mathbf{x}}$  in (\ref{standardize})
with their usual sample estimates $\hat{\Sigma}_{{\mathbf{x}}}$ and $\hat{\mu}_{{\mathbf{x}}}$.
Then we can estimate $S_{y|{\mathbf{x}}}$ by
\begin{eqnarray*}
\hat{S}_{y|{\mathbf{x}}}=[\hat{\Sigma}_{{\mathbf{x}}}]^{-{1}/{2}} \hat{S}_{y|{\mathbf{z}}}\ ,
\end{eqnarray*}
where $\hat{S}_{y|{\mathbf{z}}}$ is an estimate of ${S}_{y|{\mathbf{z}}}$.
Therefore, the goal of dimension reduction in regression is to
find out the dimension of the regression $d$ and the CDRS
${S}_{y|{\mathbf{z}}}={\rm Span}\{\gamma\}$.

Following \citet*{li1991a} and \citet*{cook1998b}, we also assume:
(1) $E({\mathbf{z}}|\gamma' {\mathbf{z}}) = P_{\gamma}{\mathbf{z}}$,
where $P_{\gamma}=\gamma \gamma'$, known as the {\it linearity condition};
(2) ${\rm Var} ({\mathbf{z}}|\gamma' {\mathbf{z}}) = Q_{\gamma}$,
where $Q_{\gamma}={\rm I} - P_{\gamma}$, known as the
{\it constant covariance condition}.
These two conditions hold if ${\mathbf{z}}$ is normally distributed,
although the normality is not necessary.

\subsection{Candidate Matrix}
\label{sectioncandidatematrix}

\citet*{ye2003} introduced the concept of {\it candidate matrix}, which is a
$p\times p$ matrix $A$ satisfying $A = P_{\gamma} A P_{\gamma}$.
They showed that any eigenvector corresponding to any nonzero eigenvalue of $A$ belongs to
the CDRS ${\rm Span}\{\gamma\}$.
Besides, the set of all candidate matrices, denoted by $\mathcal{M}$,
is closed under
scalar multiplication, transpose, addition, multiplication, and
thus under linear combination and expectation.

They also showed that the matrices $[\mu_1(y) \mu_1(y)']$ and $[\mu_2(y)-{\rm I}]$ belong to
$\mathcal{M}$ for all $y$, where $\mu_1(y)={\rm E}({\mathbf{z}}|y)$ and $\mu_2(y)={\rm E}({\mathbf{z}}{\mathbf{z}}'|y)$.
They proved that the symmetric matrices that SIR, SAVE, and $y$-pHd estimate all belong to $\mathcal{M}$:
\begin{eqnarray*}
M_{{\rm SIR}} &=& {\rm Var} ({\rm E}({\mathbf{z}}|y)) = {\rm E} [\mu_1(y) \mu_1(y)']\ , \\
M_{{\rm SAVE}} &=& {\rm E}[(I-{\rm Var} ({\mathbf{z}}|y))^2] \\
 &=& {\rm E} ([\mu_1(y) \mu_1(y)']^2 + [\mu_2(y) - {\rm I}]^2  \\
 & &  -  [\mu_1(y) \mu_1(y)'][\mu_2(y) - {\rm I}] -  [\mu_2(y) - {\rm I}][\mu_1(y) \mu_1(y)'] )\ , \\
M_{y{\rm -pHd}} &=& {\rm E}[(y-{\rm E}(y)){\mathbf{z}}{\mathbf{z}}'] = {\rm E}[y (\mu_2(y)-{\rm I})]\ .
\end{eqnarray*}

\section{Candidate Matrix $M_{{\mathbf{z}}{\mathbf{z}}'|y}$}

\subsection{A Simple Candidate Matrix}

The matrices $[\mu_1(y)\mu_1(y)']$ and $[\mu_2(y)-{\rm I}]$ are actually two fundamental components of
$M_{{\rm SIR}}$, $M_{{\rm SAVE}}$, and $M_{y{\rm -pHd}}$ (see Section~\ref{sectioncandidatematrix}).
$M_{{\rm SIR}}$ only involves the first component $[\mu_1(y)\mu_1(y)']$,
while both $M_{{\rm SAVE}}$ and $M_{y{\rm -pHd}}$ share the second component $[\mu_2(y)-{\rm I}]$.
Realizing that this common feature may lead to the connection between SAVE and pHd,
we investigate the behavior of the matrix $[\mu_2(y)-{\rm I}]$.
To avoid the inconvenience due to ${\rm E} ([\mu_2(y)-{\rm I}])=0$, we define
\begin{eqnarray*}
M_{{\mathbf{z}}{\mathbf{z}}'|y} &=& {\rm E} ( [ {\rm E}({\mathbf{z}}{\mathbf{z}}'-{\rm I}|y) ]^2 ) = {\rm E} ([\mu_2(y)-{\rm I}]^2).
\end{eqnarray*}
Note that $M_{{\mathbf{z}}{\mathbf{z}}'|y}$ takes a simpler form
than the rescaled version of $\mbox{sirII}$ \citep[Remark R.3]{li1991b}
while still keeping the theoretical comprehensiveness.
It also appears as a component in one expression of the {\it directional regression} matrix $G$
\citep*[eq.(4)]{liwang2007}.
We choose its form as simple as possible for less complicated large sample
test and potentially greater test power.
To establish the relationship between $M_{y{\rm -pHd}}$ and $M_{{\mathbf{z}}{\mathbf{z}}'|y}$,
we need:
\begin{lemma}
Let $M$ be a $p\times q$ random matrix defined on a probability space
$(\Omega,{\cal F},P)$, then there exists an event $\Omega_0 \in {\cal F}$ with probability $1$,
such that,
$${\rm Span} \{ E ( MM' ) \}
    = {\rm Span} \{ M(\omega), \omega \in \Omega_0 \}.$$
\label{asaveone}
\end{lemma}
A similar result can also be found in \citet*[Proposition 2(i)]{yin2003}.
The lemma here is more general.  By
the definition of $M_{{\mathbf{z}}{\mathbf{z}}'|y}$,
\begin{corollary}
${\rm Span} \{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \} = {\rm Span} \{ [\mu_2(y)-{\rm I}], y \in \Omega(y)\}$,
where $\Omega(y)$ is the support of $y$.
\label{asavetwo}
\end{corollary}
Based on Corollary \ref{asavetwo}, \citet*[Lemma 3]{ye2003},
and the fact that $[\mu_2(y)-{\rm I}] \in \mathcal{M}$ for all $y$,
matrix $M_{{\mathbf{z}}{\mathbf{z}}'|y}$ is in fact a candidate matrix too.
Corollary \ref{asavetwo} also implies a strong connection between $M_{y{\rm -pHd}}$
and $M_{{\mathbf{z}}{\mathbf{z}}'|y}$:
\begin{corollary}
$ {\rm Span}\{ M_{y{\rm -pHd}} \} \subseteq {\rm Span}\{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \} $.
\label{asavethree}
\end{corollary}

To further understand the relationship between
$M_{y{\rm -pHd}}$ and $M_{{\mathbf{z}}{\mathbf{z}}'|y}$,
recall the {\it{central $k$-th moment dimension reduction subspace}} \citep*{yin2003},
$S_{y|{\mathbf{z}}}^{(k)}={\rm Span}\{\eta^{(k)}\}$.
The corresponding random vector $(\eta^{(k)})' {\mathbf{z}}$
contains all the available information about $y$ from the first $k$
conditional moments of $y|{\mathbf{z}}$.  In other words,
$\left.y \;\, \rule[0em]{.03em}{.67em} \hspace{-.25em \{{\rm E}(y|{\mathbf{z}}),\ldots,{\rm E}(y^k|{\mathbf{z}})\} \right| (\eta^{(k)})' {\mathbf{z}}$.
Similar to
\[
{\rm Span}\{{\rm E}(y{\mathbf{z}}),\ldots,{\rm E}(y^k{\mathbf{z}})\}
= {\rm Span}\{{\rm E}(y \mu_1(y)),\ldots,{\rm E}(y^k \mu_1(y))\}
\subseteq S_{y|{\mathbf{z}}}^{(k)} \subseteq S_{y|{\mathbf{z}}},
\]
the subspace ${\rm Span}\{ {\rm E}(y [\mu_2(y)-{\rm I}]),\ldots,{\rm E}(y^k [\mu_2(y)-{\rm I}])\}$ is also contained in
$S_{y|{\mathbf{z}}}^{(k)}$.
Parallel to \citet*[Proposition 4]{yin2002}, the result on $M_{{\mathbf{z}}{\mathbf{z}}'|y}$ is:
\begin{proposition}
(a) If $y$ has finite support $\Omega(y)=\{a_0,\ldots,a_k\}$, then
\begin{eqnarray*}
{\rm Span} \{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \}  = {\rm Span} \{ {\rm E}[y^i (\mu_2(y)-{\rm I})],\ i=1,\ldots,k \}.
\end{eqnarray*}
(b) If $y$ is continuous and $\mu_2(y)$ is continuous on $y$'s support $\Omega(y)$,
then
\begin{eqnarray*}
{\rm Span} \{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \}  = {\rm Span} \{ {\rm E}[y^i (\mu_2(y)-{\rm I})],\ i=1,2,\ldots \}.
\end{eqnarray*}
\label{asavefour}
\end{proposition}
According to Proposition~\ref{asavefour} and \citet*[Proposition 4]{yin2002},
the relationship between
${\rm E}[y (\mu_2(y)-{\rm I})]=M_{y{\rm -pHd}}$ and $M_{{\mathbf{z}}{\mathbf{z}}'|y}$
is fairly comparable with the relationship between
${\rm E}(y \mu_1(y) ) = {\rm E}(y{\mathbf{z}})$ and $M_{{\rm SIR}}$.
Both ${\rm E}(y{\mathbf{z}})$ and $M_{y{\rm -pHd}}$ actually target
the central mean (first moment) dimension reduction subspace \citep*{cook2002},
while $M_{{\rm SIR}}$ and $M_{{\mathbf{z}}{\mathbf{z}}'|y}$ target the central $k$-th moment dimension
reduction subspace given any $k$, or equivalently the CDRS $S_{y|{\mathbf{z}}}$ as $k$ goes to infinite.
In order to understand the similarity from another perspective,
recall the {\it inverse mean subspace} of $S_{y|{\mathbf{z}}}$ \citep*{yin2002}:
$$S_{{\rm E}({\mathbf{z}}|y)}={\rm Span}\{{\rm E}({\mathbf{z}}|y), y\in \Omega(y)\}.$$
Similarly, we define the {\it inverse second moment subspace} of $S_{y|{\mathbf{z}}}$:
$${\rm Span}\{{\rm E}({\mathbf{z}}{\mathbf{z}}'|y)-{\rm I}, y\in \Omega(y)\}.$$
By definition, matrices $M_{{\rm SIR}}$ and $M_{{\mathbf{z}}{\mathbf{z}}'|y}$ are designed to recover
the entire inverse mean subspace and the entire inverse second moment subspace respectively,
while ${\rm E}(y{\mathbf{z}})$ and $M_{y{\rm -pHd}}$ are only able to recover portions of those subspaces.
We are therefore interested in combining matrices $M_{{\rm SIR}}$ and $M_{{\mathbf{z}}{\mathbf{z}}'|y}$
because they are both comprehensive.

\subsection{ SAVE versus SIR and pHd}

\citet*{ye2003} showed that
\begin{eqnarray}
{\rm Span}\{  M_{{\rm SIR}} \} \subseteq {\rm Span}\{  M_{{\rm SAVE}} \},
\label{yeweiss}
\end{eqnarray}
We then prove further the following proposition:
\begin{proposition}
${\rm Span}\{ M_{{\rm SAVE}} \} = {\rm Span}\{ M_{{\rm SIR}} \} + {\rm Span}\{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \}.$
\label{asavefive}
\end{proposition}
A straightforward result following Proposition \ref{asavefive} and Corollary \ref{asavethree} is:
\begin{corollary}
$ {\rm Span}\{ M_{y{\rm -pHd}} \}, {\rm Span}\{ M_{{\rm SIR}} \}, {\rm Span}\{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \}
    \subseteq {\rm Span} \{ M_{{\rm SAVE}} \}. $
\label{asavesix}
\end{corollary}
Corollary \ref{asavesix}
explains why SAVE is able to provide better estimates of the CDRS
than SIR and $y$-pHd in many cases.

\section{Sliced Inverse Moment Regression Using
    Weighted Chi-Squared Tests}

\subsection{Sliced Inverse Moment Regression}

In order to simplify the candidate matrices using the first two inverse moments
and still keep the comprehensiveness of SAVE,
a natural idea is to combine $M_{{\mathbf{z}}{\mathbf{z}}'|y}$ with $M_{{\rm SIR}}$ as follows:
\[
\alpha M_{{\rm SIR}} + (1-\alpha) M_{{\mathbf{z}}{\mathbf{z}}'|y}
= {\rm E} (\alpha[\mu_1(y) \mu_1(y)'] +  (1-\alpha)[\mu_2(y) - {\rm I}]^2 ),
\]
where $\alpha\in (0,1)$. We call this matrix $M^{(\alpha)}_{{\rm SIMR}}$
and the corresponding dimension reduction method {\it Sliced Inverse Moment Regression} (SIMR or SIMR$_{\alpha}$).
Note that the combination here is simpler than the $\mbox{SIR}_\alpha$ method \citep*{li1991b,gannoun2003}
while retaining the least requirement on comprehensiveness.
Actually, for any $\alpha\in (0,1)$, SIMR$_\alpha$ is as comprehensive as SAVE at a theoretical level based on
the following proposition:
\begin{proposition}
${\rm Span}\{M^{(\alpha)}_{{\rm SIMR}}\}={\rm Span}\{M_{{\rm SAVE}}\},\ \forall \alpha \in (0,1).$
\label{asaveseven}
\end{proposition}
Combined with Corollary~\ref{asavesix}, we know that any linear combination of SIR, pHd and
SAVE can be covered by SIMR$_\alpha$:
\begin{corollary}
${\rm Span}\{a M_{{\rm SIR}}+ b M_{y{\rm -pHd}} + c M_{{\rm SAVE}}\}\subseteq {\rm Span}\{M^{(\alpha)}_{{\rm SIMR}}\}$,
where $a, b$, and $c$ are arbitrary real numbers.
\label{simr8}
\end{corollary}
Note that the way of constructing SIMR$_\alpha$ makes it easier to develop a
corresponding large sample test for the dimension of the regression (Section~4.3).

From now on, we assume that the data $\{(y_i,{\mathbf{x}}_i)\}_{i=1,\ldots,n}$ are i.i.d.~from
a population which has finite first four moments and conditional moments.

\subsection{Algorithm for SIMR$_\alpha$}

Given i.i.d.~sample $(y_1,{\mathbf{x}}_1)$,...,$(y_n,{\mathbf{x}}_n)$,
first standardize ${\mathbf{x}}_i$ into $\hat{\mathbf{z}}_i$,
sort the data by $y$,
and divide the data into $H$ slices with intraslice sample sizes
$n_h$, $h=1,\ldots,H$.
Secondly construct the intraslice sample means
$\overline{({\mathbf{z}}{\mathbf{z}}')}_h$ and $\bar{{\mathbf{z}}}_h$:
\begin{eqnarray*}
\overline{({\mathbf{z}}{\mathbf{z}}')}_h &=& \frac{1}{n_h}\sum_{i=1}^{n_h} \hat{\mathbf{z}}_{ih} \hat{\mathbf{z}}_{ih}'\ ,
\\
\bar{{\mathbf{z}}}_h &=& \frac{1}{n_h}\sum_{i=1}^{n_h} \hat{\mathbf{z}}_{ih}\ ,
\end{eqnarray*}
where $\hat{\mathbf{z}}_{ih}$'s are predictors falling into slice $h$.
Thirdly calculate
\begin{eqnarray*}
\hat{M}^{(\alpha)}_{{\rm SIMR}} &=& \sum_{h=1}^H  \hat{f}_h
 \left( (1-\alpha)[\overline{({\mathbf{z}}{\mathbf{z}}')}_h - {\rm I}_p] [\overline{({\mathbf{z}}{\mathbf{z}}')}_h - {\rm I}_p]'
 + \alpha [\bar{{\mathbf{z}}}_h] [\bar{{\mathbf{z}}}_h]'\right)
\\
&=& \hat{U}_{n} \hat{U}_{n}'\ ,
\end{eqnarray*}
where $\hat{f}_h = n_h/n$ and
\begin{eqnarray*}
\hat{U}_{n} =   \left(
    \ldots, \sqrt{1-\alpha}\ [\overline{({\mathbf{z}}{\mathbf{z}}')}_h - {\rm I}_p]\sqrt{\hat{f}_h}, \ldots ,
\,
        \ldots,\sqrt{\alpha}\ \bar{{\mathbf{z}}}_h \sqrt{\hat{f}_h} ,\ldots
        \right)_{p \times (pH+H)}.
\end{eqnarray*}
Finally calculate the eigenvalues
$\hat{\lambda}_{1} \geq \cdots \geq \hat{\lambda}_{p}$ of $\hat{M}^{(\alpha)}_{{\rm SIMR}}$
and the corresponding eigenvectors $\hat{\gamma}_1, \ldots, \hat{\gamma}_p$~.
Then ${\rm Span}\{ \hat{\gamma}_{1},\ldots,\hat{\gamma}_{d} \}$ is an estimate of the
CDRS ${\rm Span}\{\gamma\}$, where $d$ is determined by the weighted chi-squared
test described in the next section.

\subsection{A Weighted Chi-Squared Test for SIMR$_\alpha$}
\label{weightedtest}

Define the population version of $\hat{U}_{n}$:
\begin{eqnarray}
& & B \nonumber\\
&=& \left(
    \ldots ,
    \sqrt{1-\alpha}\ [{\rm E}({\mathbf{z}}{\mathbf{z}}'|\tilde{y}=h) - {\rm I}_p]\sqrt{f_h},
    \ldots,
    \sqrt{\alpha}{\rm E}({\mathbf{z}}|\tilde{y}=h)\sqrt{f_h},
    \ldots
      \right)
\nonumber\\
&=&\left( (\Gamma_{11})_{p \times d} , (\Gamma_{12})_{p \times (p-d)} \right)
   \left( \begin{array}{cc} D_{d \times d} & 0 \\ 0 & 0 \end{array} \right)
   \left( \begin{array}{l} (\Gamma_{21}')_{d \times (pH+H)}
        \\ (\Gamma_{22}')_{(pH+H-d) \times (pH+H)} \end{array} \right)
\label{svd}
\end{eqnarray}
where $\tilde{y}$ is a slice indicator with
$\tilde{y} \equiv h$ for all observations falling into slice $h$,
${f_h} = P( \tilde{y} = h )$ is the population version of $\hat{f}_h$,
and (\ref{svd}) is the {\it singular value decomposition} of $B$.

Denote $\tilde{U}_{n}=\sqrt{n}(\hat{U}_{n} - B)$.
By the multivariate central limit theorem and the multivariate version of Slutsky's theorem,
$\tilde{U}_n$ converges in distribution to a certain random $p \times (pH+H)$ matrix $U$
as $n$ goes to infinity \citep*{gannoun2003}.
Note that the singular values are invariant under right and left multiplication by orthogonal matrices.
Based on \citet*[Theorem 4.1 and 4.2]{eaton1994},
the asymptotic distribution of
the smallest $(p-d)$ singular values of $\sqrt{n} \hat{U}_n$ is
the same as the asymptotic distribution of
the corresponding singular values of the following $(p-d) \times (pH+H-d)$ matrix:
\begin{eqnarray}
\sqrt{n} \Gamma_{12}' \hat{U}_n \Gamma_{22}.
\label{eatontylerone}
\end{eqnarray}
Construct statistic
\begin{eqnarray*}
\hat{\Lambda}_{d} = n  \sum_{h=d+1}^p \hat{\lambda}_h,
\end{eqnarray*}
which is the sum of the squared smallest $(p-d)$
singular values of $\sqrt{n} \hat{U}_n$.
Then the asymptotic distribution of $\hat{\Lambda}_{d}$ is the same as
that of the sum of the squared singular values of (\ref{eatontylerone}).  That is
\[
n {\rm Trace}(  [\Gamma_{12}' \hat{U}_n \Gamma_{22}] [\Gamma_{12}' \hat{U}_n \Gamma_{22}]' )
= n [{\rm Vec} (\Gamma_{12}' \hat{U}_n \Gamma_{22})]'
    [{\rm Vec} (\Gamma_{12}' \hat{U}_n \Gamma_{22})],
\]
where ${\rm Vec}(A_{r \times c})$ denotes
$({{a_1}}',\ldots,{{a_c}}')'_{rc \times 1}$ for
any matrix $A=({{a_1}},\ldots,{{a_c}})$.
By central limit theorem and Slutsky's theorem again,
\begin{eqnarray*}
{\rm Vec}(\tilde{U}_n) \stackrel{\mathcal{L}}{\rightarrow} N_{(p^2H+pH)}(0,V)
\end{eqnarray*}
for some nonrandom $(p^2H+pH) \times (p^2H+pH)$ matrix $V$.
Thus,
\begin{eqnarray*}
\sqrt{n} [{\rm Vec} (\Gamma_{12}' \hat{U}_n \Gamma_{22})]  \stackrel{\mathcal{L}}{\rightarrow}
 N_{(p-d)(pH+H-d)} (0, W),
\end{eqnarray*}
where $W=[\Gamma_{22}' \otimes \Gamma_{12}'] V [\Gamma_{22}' \otimes \Gamma_{12}']'$
is a $(p-d)(pH+H-d) \times (p-d)(pH+H-d)$ matrix.
Combined with Slutsky's theorem, it yields the following theorem:
\begin{theorem}
The asymptotic distribution of $\hat{\Lambda}_{d}$ is the same as that of
\begin{eqnarray*}
\sum_{i=1}^{(p-d)(pH+H-d)} \alpha_i K_i
\end{eqnarray*}
where the $K_i$'s are independent $\chi_1^2$ random variables,
and $\alpha_i$'s are the eigenvalues of the matrix $W$.
\label{theoremasaveone}
\end{theorem}

Clearly,
a consistent estimate of $W$ is needed
for testing the dimension of the regression based on
Theorem \ref{theoremasaveone}.
The way we define $M^{(\alpha)}_{{\rm SIMR}}$
allows us to partition $\hat{U}_n$ into
\begin{eqnarray*}
\hat{U}_{n,1} &=&   \left(
    \ldots, \sqrt{1-\alpha}\ [\overline{({\mathbf{z}}{\mathbf{z}}')}_h - {\rm I}_p]\sqrt{\hat{f}_h}, \ldots ,
            \right)_{p \times pH}\ ,
\\
\hat{U}_{n,2} &=&   \left(
    \ldots,\sqrt{\alpha}\ \bar{{\mathbf{z}}}_h \sqrt{\hat{f}_h} ,\ldots
        \right)_{p \times H}\ .
\end{eqnarray*}
The asymptotic distribution of the matrix $\hat{U}_{n,2}$ has been fully
explored by \citet*{bura2001}, resulting in a weighted chi-squared test
for SIR.  The similar techniques can also be applied on the matrix
$\hat{U}_{n,1}$, and therefore the matrix  $\hat{U}_n$ as a whole, although
the details are much more complicated.

Define the population versions of $\hat{U}_{n,1}$ and $\hat{U}_{n,2}$,
\begin{eqnarray*}
B_{1} &=& \left(
    \ldots ,
   \sqrt{1-\alpha}\ [{\rm E}({\mathbf{z}}{\mathbf{z}}'|\tilde{y}=h) - {\rm I}_p]\sqrt{f_h},
    \ldots
      \right)_{p \times pH},
\\
B_{2} &=& \left(
    \ldots,
    \sqrt{\alpha}\ {\rm E}({\mathbf{z}}|\tilde{y}=h)\sqrt{f_h},
    \ldots
      \right)_{p \times H}.
\end{eqnarray*}
Then
$\hat{U}_n = \left(
    \hat{U}_{n,1} , \hat{U}_{n,2}
      \right)$,
and $B = \left(
    B_1,B_2
      \right)$.

Let $f$, $\hat{f}$ and $1_H$ be $H \times 1$ vectors with
elements $f_h$, $\hat{f}_h$ and $1$ respectively;
let $G$ and $\hat{G}$ be $H \times H$ diagonal matrices with
diagonal entries $\sqrt{{f}_h}$ and $\sqrt{\hat{f}_h}$ respectively;
and let
\[
\hat{F} = ({\rm I}_H-\hat{f} 1_H'),\>\>
{F} = ({\rm I}_H-{f} 1_H'),
\]
\[
\left( \begin{array}{l} (\Gamma_{21}') \\ (\Gamma_{22}') \end{array} \right)
=
\left( \begin{array}{ll}   (\Gamma_{211}')_{d \times pH}        & (\Gamma_{212}')_{d \times H}
            \\ (\Gamma_{221}')_{(pH+H-d) \times pH}     & (\Gamma_{222}')_{(pH+H-d) \times H}
\end{array} \right).
\]
Finally, define four matrices
\begin{eqnarray*}
M &=& (\dots, {\rm E}({\mathbf{x}}|\tilde{y}=h),\ldots)_{p \times H},
\\
N &=& (\dots, {\rm E}({\mathbf{x}}'|\tilde{y}=h),\ldots)_{1 \times pH} = {\rm Vec}(M)',
\\
O &=& (\dots, {\rm E}({\mathbf{x}}{\mathbf{x}}'|\tilde{y}=h),\ldots)_{p \times pH},
\\
C &=& [ O - M ({\rm I}_H \otimes {\mu}_{{\mathbf{x}}}') - {\mu}_{{\mathbf{x}}} N]_{p \times pH},
\end{eqnarray*}
and their corresponding sample versions $M_n$, $N_n$, $O_n$, and $C_n$.
By the central limit theorem,
\begin{eqnarray*}
\sqrt{n}  {\rm Vec} ( [(C_n,M_n) - (C,M)])
    \stackrel{\mathcal{L}}{\rightarrow}
N_{(p^2H+pH)}(0,\Delta)
\end{eqnarray*}
for a nonrandom $(p^2H+pH) \times (p^2H+pH)$ matrix $\Delta$.
As a result,
\begin{theorem}The covariance matrix in Theorem~\ref{theoremasaveone} is
\[
W =   (K
    \Gamma_{22}
    )'
    \otimes
    (\Gamma_{12}' {\Sigma}_{{\mathbf{x}}}^{-1/2})
\Delta
 (K    \Gamma_{22}
    )
    \otimes
    (\Gamma_{12}' {\Sigma}_{{\mathbf{x}}}^{-1/2} )'\ ,
\]
where
\[
K=    \left( \begin{array}{cc}
        \sqrt{1-\alpha} ({F} {G})  \otimes {\Sigma}_{{\mathbf{x}}}^{-1/2} & 0
        \\
        0 & \sqrt{\alpha}\ F G
    \end{array}
    \right)
\]
\label{theoremasavetwo}
\end{theorem}

The only difficulty left now is to obtain a consistent estimate of $\Delta$.
By the central limit theorem,
\begin{eqnarray*}
\sqrt{n}  {\rm Vec} ( [(O_n,M_n,\hat{\mu}_{{\mathbf{x}}}) - (O,M,\mu_{{\mathbf{x}}})] )
    \stackrel{\mathcal{L}}{\rightarrow}
N_{(p^2H+pH+p)}(0,\Delta_0)
\end{eqnarray*}
where $\Delta_0$ is a nonrandom $(p^2H+pH+p) \times (p^2H+pH+p)$ matrix,
with details shown in the Appendix.  On the other hand,
\begin{eqnarray*}
{\rm Vec} (C_n,M_n) &=&
    \left( \begin{array}{ccc}
        {\rm I}_{p^2H} &     -{\rm I}_H \otimes \hat{\mu}_{{\mathbf{x}}} \otimes {\rm I}_p-{\rm I}_{pH}\otimes\hat{\mu}_{{\mathbf{x}}}
                        & 0
        \\
        0 &         {\rm I}_{pH}                     & 0
    \end{array}
    \right){\rm Vec} (O_n,M_n,\hat{\mu}_{{\mathbf{x}}})
\\
&=& g ( [{\rm Vec} (O_n,M_n,\hat{\mu}_{{\mathbf{x}}})] )
\end{eqnarray*}
for a certain mapping $g: \mathcal{R}^{(p^2H+pH+p)} \rightarrow \mathcal{R}^{(p^2H+pH)}$
such that
$${\rm Vec}(C,M)=g( [{\rm Vec}(O,M,\mu_{{\mathbf{x}}})] ).$$
Thus the close form of $\Delta$ can be obtained by Cram\'{e}r's theorem \citep{cramer1946}:
\begin{eqnarray}
\Delta= [ \dot{g} ( [{\rm Vec}(O,M,\mu_{{\mathbf{x}}})] ) ]
    \Delta_0
    [ \dot{g} ( [{\rm Vec}(O,M,\mu_{{\mathbf{x}}})] ) ]',\label{delta}
\end{eqnarray}
where the $(p^2H+pH) \times (p^2H+pH+p)$ derivative matrix
\begin{eqnarray}
\dot{g}\left[{\rm Vec}(O,M,\mu_{{\mathbf{x}}})\right]=
\left(\begin{array}{ccc}
{\rm I}_{p^2H} & -{\rm I}_H\otimes\mu_{{\mathbf{x}}}\otimes {\rm I}_p-{\rm I}_{pH}\otimes \mu_{{\mathbf{x}}} & \dot{g}_{13}\\
0 & {\rm I}_{pH} & 0
\end{array}\right) \label{gdot}
\end{eqnarray}
with $
\dot{g}_{13} =-\left(\ldots,{\rm I}_p\otimes{\rm E}({\mathbf{x}}'|\tilde{y}=h),\ldots\right)'-{\rm Vec}(M)\otimes {\rm I}_p
$ .

In summary, to compose a consistent estimate of matrix $W$,
one can (i) substitute the usual sample moments to get the sample estimate of $\Delta_0$;
(ii) estimate $\Delta$ by substituting the usual
sample estimates for ${\rm E}({\mathbf{x}}'|\tilde{y}=h)$, $\mu_{{\mathbf{x}}}$ and $M$
in (\ref{delta}) and (\ref{gdot}); (iii) obtain the usual sample estimates of
$\Gamma_{12}$ and $\Gamma_{22}$ from the singular value decomposition of $\hat U_n$;
(iv) substitute the usual sample estimates for $F$, $G$, ${\Sigma}_{{\mathbf{x}}}$, $\Gamma_{12}$
and $\Gamma_{22}$ in Theorem \ref{theoremasavetwo} to form an estimate of $W$.
Note that both $\Delta$ and $\Delta_0$ do not rely on $\alpha$.
This fact can save a lot of computational time when multiple $\alpha$'s need to be checked.

To approximate a linear combination of chi-squared random variables,
one may use the statistic proposed by \citet*{satterthwaite1941},
\citet*{wood1989}, \citet*{satorra1994}, or \citet*{bentler2000}.
In the next applications, we will present tests based on Satterthwaite's statistic
 for illustration purpose.

\subsection{Choosing Optimal $\alpha$}
\label{sectionchoosealpha}

\citet*{ye2003} proposed a bootstrap method to pick up the ``best" linear combination of two known methods
in terms of variability of the estimated CDRS $\hat{S}_{y|{\mathbf{z}}}$.
The bootstrap method works reasonably well with known dimension $d$
of the regression,
although less variability may occur with a wrong $d$
(see Section~\ref{simulationstudy} for an example).
Another drawback is its computational intensity \citep*{liwang2007}.

Alternative criterion for ``optimal" $\alpha$ is based on the weighted chi-squared tests
developed for ${\rm SIMR}$. When multiple tests with
different $\alpha$ report the same dimension $d$, we simply pick up the $\alpha$ with
the smallest $p$-value. Given that the true dimension $d$ is detected,
the last eigenvector $\hat{\gamma}_d$ added into the estimated CDRS
with such an $\alpha$ is the most significant
one among the candidates based on different $\alpha$.
In the mean time, the other eigenvectors $\hat{\gamma}_1,\ldots, \hat{\gamma}_{d-1}$
with selected $\alpha$ tend to be more significant than other candidates too.
Based on simulation studies (Section~\ref{simulationstudy}), the performance
of the $p$-value criterion is comparable with the bootstrap one with known $d$.
The advantages of the former include that it is compatible with
the weighted chi-squared tests and it requires much less computation.

When a model or an algorithm is specified for the data analysis, cross-validation
could be used for choosing optimal $\alpha$ too, just like how people did for
model selection. For example, see \citet[chap. 7]{hastie2001}.
It will not be covered in this paper since we aim at model-free dimension reduction.

\section {Simulation Study}
\label{simulationstudy}

\subsection{A Simulated Example}
\label{simulatedexample}

Let the response $y=2z_1\epsilon + z_2^2 + z_3$,
where $({\mathbf{z}}',\epsilon)'=(z_1,z_2,z_3,z_4,\epsilon)'$
are i.i.d sample from the $N_5(0,{\rm I}_5)$ distribution.
Then the true dimension of the regression is $3$ and the true CDRS is
spanned by $(1,0,0,0)'$, $(0,1,0,0)'$, and $(0,0,1,0)'$, that is, $z_1$,
$z_2$ and $z_3$.

Theoretically,
$M_{{\rm SIR}}={\rm Diag}\{0,0,$ ${\rm Var}({\rm E}(z_3|y)),$ $0\}$,
$M_{y{{\rm -pHd}}} = {\rm Diag}\{0,2,$ $0,$ $0\}$,
and $M_{r{{\rm -pHd}}} = {\rm Diag}\left\{0,2,0,0\right\}$
have rank one and therefore are only able to
find a one-dimensional proper subspace of the CDRS.
The linear combination of any two of them suggested by \citet*{ye2003}
can at most find a two-dimensional proper subspace of the CDRS.
On the contrary,
both SAVE and SIMR are able to recover the complete CDRS at a theoretical level.

\subsection{A Single Simulation}
\label{sectionsinglesimulation}

We begin with a single simulation with sample size $n=400$.
SIR, $r$-pHd, SAVE and SIMR are applied to the data.
Number of slices $H=10$ are used for SIR, SAVE, and SIMR.
The {\tt R} package {\tt dr} \citep[version 3.0.3]{weisberg2002,weisberg2009}
is used for SIR, $r$-pHd, SAVE, as well as their corresponding marginal dimension tests.
SIMR$_\alpha$ with $\alpha=0, 0.01, 0.05$,
$0.1\sim 0.9$ paced by $0.1$, $0.95, 0.99, 1$ are applied.

For this typical simulation, SIR identifies only the direction $(.018, .000,$
$-{\mathbf{.999}},$ $-.035)'$.
It is roughly $z_3$, the linear trend.
$r$-pHd identifies only the direction $(.011,\mathbf{.999},$ $-.038,-.020)'$,
which is roughly $z_2$, the quadratic component.
As expected, SAVE works better.
It identifies $z_2$ and $z_1$.
However, the marginal dimension tests for SAVE \citep{shao2007} fail to
detect the third predictor, $z_3$.  The $p$-value of the corresponding test
is $0.331$.

Roughly speaking, SAVE with its marginal dimension test is comparable with SIMR$_{0.1}$
in this case.
The comparison between SAVE and
SIMR$_\alpha$ suggests that the failure of SAVE might due to its weights
combining the first and second inverse moments.  As $\alpha$ increases,
SIMR$_\alpha$ with $\alpha$ between $0.3$ and $0.8$ all succeed in detecting all the three
effective predictors $z_1$, $z_2$ and $z_3$.  The CDRS estimated by those candidate
matrices are similar to each other, which implies that the results with different $\alpha$
are fairly consistent.
The major difference among SIMR$_\alpha$
is that the order of the detected predictors changes roughly from
$\{z_2, z_1, z_3\}$ to $\{z_3, z_2, z_1\}$ as $\alpha$ increases from $0.3$
to $0.8$.
As expected, SIMR$_\alpha$ is comparable with SIR if $\alpha$ is close to $1$.

For this particular simulation, SIMR$_\alpha$ with $\alpha$ between $0.3$ and
$0.8$ are first selected.
If we know the true CDRS, the optimal $\alpha$
is the one minimizing the distance between the estimated CDRS and the true CDRS.
Following \citet*[p.~974]{ye2003}, the three distance
measures {\tt arccos($q$)}, $1-q$, $1-r$ behave similarly
and imply the same $\alpha=0.6$ for this particular
simulation.
Since the true CDRS is unknown, bootstrap criterion and $p$-value criterion
(Section~\ref{sectionchoosealpha}) are applied instead.

The left panel of Figure~\ref{figure12} shows the variability of bootstrapped estimated CDRS.
Distance $1-r$ is used because it is comparable across different dimensions.
The minimum variability is attained at $d=3$ and $\alpha=0.6$, which happens to
the optimal one based on the truth.
Another 200 simulations reveal that
about $75\%$ ``optimal" $\alpha$ based on bootstrap fall in $0.5\sim 0.6$.
SIMR with $\alpha$ chosen by bootstrap
criterion attains $1-r=0.0086$ away from the true CDRS on average.
Note that low variability not necessarily implies that
the estimated CDRS is accurate.
For example, SIMR$_1$ or SIR can only detect one direction $z_3$.
However the estimated one-dimensional CDRS is fairly stable
under bootstrapping (see Figure~\ref{figure12}).

The right panel in Figure~\ref{figure12} shows that the $p$-value
criterion also picks up $\alpha=0.6$ for this single simulation
(check the line $d=3$, which is the highest one that still goes below
the significance level $0.05$).  Based on the same $200$ simulations,
about $80\%$ of the ``best" $\alpha$ selected by $p$-value criterion
fall between $0.4$ and $0.7$.  On average,
SIMR with $\alpha$ selected by $p$-values attains $1-r=0.0082$,
which is comparable with the bootstrap ones.

\subsection{Power Analysis}

We conduct 1000 independent simulations
and summarize in Table~\ref{empiricalpower05} the empirical powers and sizes of the marginal dimension tests with significance level 0.05 for
SIR, SAVE, $r$-pHd, and SIMR$_{\alpha}$ with $\alpha$ chosen by the $p$-value criterion.
For illustration purpose, we omit the simulation results of $y$-pHd
because there is little difference between $y$-pHd and $r$-pHd in this case.
The empirical powers and sizes with significance level 0.01 are omitted too
since their pattern is similar to Table~\ref{empiricalpower05}.

In Table~\ref{empiricalpower05}, the rows
$d\leq 0$, $d\leq 1$, $d\leq 2$ and $d\leq 3$ indicate different null hypotheses.
Following \citet*{bura2001}, the numerical entries in the rows $d\leq 0$,
$d\leq 1$, and $d\leq 2$ are empirical estimates of the powers of the corresponding tests,
while the entries in the row $d\leq 3$ are empirical estimates of the sizes of the tests.

As expected, SIR claims $d=1$ in most cases.
$r$-pHd works a little better.  At the significance level 0.05,
$r$-pHd has about $30\%$ chance to find out $d\geq 2$ (Table~\ref{empiricalpower05}).
At level 0.01, the chance shrinks to about $15\%$.
Both SAVE and SIMR perform much better than
SIR and pHd.   Compared with SAVE,
SIMR has consistently greater powers for the null hypotheses $d\leq 0$,
$d\leq 1$ and $d\leq 2$
across different choices of sample size, number of slices and significant level.
For example, under the null hypothesis $d\leq 2$ with sample size $400$,
the empirical powers of SIMR at
level 0.05 are $0.939$ under 5 slices and $0.943$ under 10 slices,
while the corresponding powers of SAVE are only $0.399$ and $0.213$ respectively
(Table~\ref{empiricalpower05}).
Those differences become even bigger at level 0.01.
The empirical sizes of SIMR are roughly under the nominal size 0.05
although they tend to be larger than the others.

For comparison purpose, the methods
{\it inverse regression estimator} (IRE) \citep*{cookni2005,wen2007,weisberg2009})
and {\it directional regression} (DR) \citep*{liwang2007}
are also applied. Roughly speaking, IRE performs
similar to SIR in this example.  Given that the truth dimension $d=3$ is known,
both DR and SIMR are among the best in terms of mean($1-r$).
For example, at $n=600$, DR achieves mean($1-r$) $=0.0050$ with $H=5$, $0.0053$ with $H=10$
and $0.0059$ with $H=15$, while SIMR's are $0.0048$, $0.0046$, and $0.0053$.
Nevertheless, the powers of the marginal tests for DR are between
SAVE and SIMR in this case. Roughly speaking, DR's power tests are comparable
with SIMR$_\alpha$'s with $\alpha$ between $0.2$ and $0.3$.
For example, at $H=10$ and level $0.05$,
the empirical powers of DR against $d\leq 2$
are $0.247$ with $n=200$, $0.800$ with $n=400$, and $0.974$ with $n=600$.

Among the six dimension reduction methods applied, SIMR is the most reliable one.
Besides, the chi-squared tests for SIMR do not seem to be very sensitive to the numbers of slices.
Nevertheless, we suggest that the number of slices should not be greater than 3\%-5\% of the sample size
based on the simulation results.

\section{A Real Example: Ozone Data}
\label{sectionozonedata}

To examine how SIMR works in practice, we consider a data set taken from \citet*{breiman1985}.
The response {\tt Ozone} is the daily ozone concentration in parts per million,
measured in Los Angeles basin, for 330 days in 1976.
For illustration purpose, the dependence of {\tt Ozone} on the following four predictors
is studied next:
{\tt Height}, Vandenburg 500 millibar height in meters;
{\tt Humidity} in percents;
{\tt ITemp}, Inverse base temperature in degrees Fahrenheit;
and {\tt STemp}, Sandburg Air Force Base temperature in degrees Fahrenheit.

To meet
both the linearity condition
and the constant covariance condition,
simultaneously power transformations on the predictors are estimated
to improve the normality of their joint distribution.
After replacing {\tt Humidity}, {\tt ITemp}, and {\tt STemp} with
${\tt Humidity}^{1.68}$, ${\tt ITemp}^{1.25}$, and ${\tt STemp}^{1.11}$ respectively,
SIR, $r$-pHd, SAVE and SIMR are applied to the data.
For SIR, SAVE, and SIMR, various numbers of slices are applied, and the results are
fairly consistent. Here we only present the outputs based on $H=8$.

At significance level $0.05$, SIR suggests the dimension of the regression $d=1$,
while $r$-pHd claims $d=2$.
Using the visualization tools described by \citet*{cook1994} and \citet{cook1998b},
the first pHd predictor appears to be somewhat symmetric about the response $Ozone$,
and the second pHd predictor seems to be similar to the first SIR predictor,
which are not shown in this article.
The symmetric dependency explains why SIR is not able to find the first pHd predictor.
The resulting inference based on pHd is therefore
more reliable than the inference based on SIR.

When checking the predictors of SAVE,
visual tools show a clear quadratic or even higher order polynomial dependency
between the response and the first SAVE predictor.
The second SAVE predictor is similar to the second pHd predictor,
and the third SAVE predictor is similar to the first pHd predictor.
Both SIR's and pHd's tests miss the first SAVE predictor.

Now apply SIMR to the ozone data. Bootstrap criterion picks up $\alpha=0.2$
while $p$-value criterion suggests $\alpha=0$. Nevertheless, both
SIMR$_{0.2}$ and SIMR$_0$ lead to very similar estimated CDRS in this case
(see Table~\ref{ozonetabletwo}).
As expected , they recovers all the three SAVE predictors.
Actually, those three estimated CDRS appear to be almost identical.

\section{Discussion}

SIMR$_\alpha$ and SAVE are theoretically equivalent since that the subspaces spanned by their
underlying matrices are identical.  Nevertheless, simulation study shows that
SIMR$_\alpha$ with some chosen $\alpha$ may perform better than SAVE.
The main reason is that SAVE is only a fixed combination of the first two
inverse moments.
The simulation example in Section~\ref{simulationstudy} implies that any fixed combination can not always be
the winner.
Apparently, SIMR$_{0.6}$ can not always be the winner either.  For example, if the simulation
example is changed to $y=2 z_1\varepsilon+z_2^2+0.1 z_3$, SIMR$_\alpha$ with $\alpha$ closer to $1$ will
perform better.
For practical use, multiple methods, as well as their combinations,
should be tried and unified.
SIMR$_\alpha$ with $\alpha\in (0,1)$ provide a simple solution to it.

As a conclusion, we propose SIMR using weighted chi-squared tests
as an important class of dimension reduction methods,
which should be routinely considered during the search for the central dimension reduction
subspace and its dimension.

\section*{Appendix}

{\bf Proof of Lemma \ref{asaveone}:}
By definition,
${\rm Span} \{ {\rm E} ( MM' ) \} \subseteq {\rm Span} \{ M(\omega), \omega \in \Omega_0 \} $,
if $P(\Omega_0)=1$.
On the other hand, for any $v_{p \times 1} \neq 0$,
\begin{eqnarray*}
& &v' {\rm E} ( M(\omega)M'(\omega) ) =0
\>\Rightarrow\> v' {\rm E} ( M(\omega)M'(\omega) )v =0
\\
&\Rightarrow& {\rm E} ( [v'M(\omega)] [v'M(\omega)]' ) =0
\>\Rightarrow\> [v'M(\omega)] \equiv 0, \mbox{ with probability }1
\end{eqnarray*}
Since $\{v: v'{\rm E}(MM')=0\}$ only has finite dimension, there exists an $\Omega_0$ with probability 1, such that,
$$\dim( {\rm Span} \{ {\rm E} ( M(\omega)M'(\omega) ) \} )
        \geq \dim( {\rm Span} \{ M(\omega), \omega \in \Omega_0 \} ).$$
Thus,
${\rm Span} \{ {\rm E} ( M(\omega)M'(\omega) ) \} = {\rm Span} \{ M(\omega), \omega \in \Omega_0 \}$

\vspace{0.2cm}
\noindent
{\bf Proof of Corollary \ref{asavethree}:}
\begin{eqnarray*}
{\rm Span}\{ M_{y{\rm -pHd}} = {\rm E}[ y (\mu_2(y)-{\rm I}) ] \} \subseteq {\rm Span} \{ [\mu_2(y)-{\rm I}], \forall y \} = {\rm Span} \{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \}.
\end{eqnarray*}

\vspace{0.2cm}
\noindent
{\bf Proof Proposition \ref{asavefour}: }\
Define $\mu_{i}= {\rm E} [ ({\mathbf{z}}{\mathbf{z}}'-{\rm I})|y=a_i ]={\rm E} ({\mathbf{z}}{\mathbf{z}}'|y=a_i)-{\rm I}$
and $f_i={\rm Pr}(y=a_i)$ for $i=0,...k$,
then $\Sigma_{i=0}^k f_i = 1$ and $\Sigma_{i=0}^k f_i \mu_{i} = {\rm E} ( ({\mathbf{z}}{\mathbf{z}}'-{\rm I}))=0$.
The rest of the steps follow the exactly same proof as
in \citet*[A.3. Proposition 4]{yin2002}.

\vspace{0.2cm}
\noindent
{\bf Proof of Proposition \ref{asavefive}:}\
By Lemma \ref{asaveone},
\begin{eqnarray*}
{\rm Span}\{ M_{{\rm SAVE}} \} &=& {\rm Span}\{ [ \mu_1(y)\mu_1(y)' + (\mu_2(y)-{\rm I})], \forall y \}
\\
&\subseteq& {\rm Span}\{ \mu_1(y), \forall y \} + {\rm Span}\{ (\mu_2(y)-{\rm I}), \forall y \}
\\
&=& {\rm Span}\{ M_{{\rm SIR}} \} + {\rm Span}\{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \}
\\
&\subseteq& {\rm Span}\{ M_{{\rm SIR}} \} + [ {\rm Span}\{ \mu_1(y)\mu_1(y)' + (\mu_2(y)-{\rm I}), \forall y \}
\\
&&   + {\rm Span}\{ \mu_1(y), \forall y \} ]
\\
&\subseteq& {\rm Span}\{  M_{{\rm SIR}} \} + {\rm Span}\{  M_{{\rm SAVE}} \} + {\rm Span}\{  M_{{\rm SIR}} \}
\\
&=& {\rm Span}\{  M_{{\rm SAVE}} \}.
\end{eqnarray*}

\vspace{0.2cm}
\noindent
{\bf Proof of Proposition \ref{asaveseven}:}\
By Lemma \ref{asaveone},
\begin{eqnarray*}
{\rm Span}\{ M^{(\alpha)}_{{\rm SIMR}} \}
&=& {\rm Span}\{ (\mu_1(y), [\mu_2(y)-{\rm I}]), \forall y \}
\\
&=& {\rm Span}\{ \mu_1(y), \forall y \} + {\rm Span}\{ [\mu_2(y)-{\rm I}], \forall y \}
\\
&=& {\rm Span}\{ M_{{\rm SIR}} \} + {\rm Span}\{ M_{{\mathbf{z}}{\mathbf{z}}'|y} \}
\\
&=& {\rm Span}\{  M_{{\rm SAVE}} \}.
\end{eqnarray*}

\vspace{0.2cm}
\noindent
{\bf Proof of Theorem~\ref{theoremasavetwo}:}\
Actually, $B= {\Sigma}_{{\mathbf{x}}}^{-1/2} (C, M) K$,
\[
\hat{U}_n
= \hat{\Sigma}_{{\mathbf{x}}}^{-1/2} (C_n, M_n)
    \left( \begin{array}{cc}
        \sqrt{1-\alpha}\ (\hat{F} \hat{G})  \otimes \hat{\Sigma}_{{\mathbf{x}}}^{-1/2} & 0
        \\
        0 & \sqrt{\alpha}\ \hat{F} \hat{G}
    \end{array}
    \right).
\]
Note that
$\left( \Gamma_{12}'B_1, \Gamma_{12}'B_2 \right)  = 0_{(p-d) \times (pH+H)}$,
$B_1 \Gamma_{221} + B_2 \Gamma_{222} = 0_{p \times (pH+H-d)}$,
${\rm Span}\{C' {\Sigma}_{{\mathbf{x}}}^{-1/2} \Gamma_{12} \} \subseteq {\rm Span}\{1_H \otimes {\rm I}_p \}$,
${\rm Span}\{M' {\Sigma}_{{\mathbf{x}}}^{-1/2} \Gamma_{12} \} \subseteq {\rm Span}\{1_H \}$,
$1'_H \hat{F}$ $=$ $0$,
$1'_H F = 0$.
Writing $\hat{{\rm I}}_p=\hat{\Sigma}_{{\mathbf{x}}}^{-1/2} \Sigma_{{\mathbf{x}}}^{1/2}$,
\begin{eqnarray*}
&&  \sqrt{n} \Gamma_{12}' \hat{U}_{n} \Gamma_{22}
\\
&=& \sqrt{n} \Gamma_{12}' \hat{U}_{n,1} \Gamma_{221}
    +\sqrt{n} \Gamma_{12}' \hat{U}_{n,2} \Gamma_{222}
\\
&=& \sqrt{1-\alpha}\ \sqrt{n} \Gamma_{12}'
        (\hat{{\rm I}}_p -{\rm I}_p +{\rm I}_p) \Sigma_{{\mathbf{x}}}^{-1/2}
        (C_n-C+C)
        [(\hat{F} \hat{G} - FG + FG) \otimes {\rm I}_p]
\\
&&
        ({\rm I}_H \otimes \Sigma_{{\mathbf{x}}}^{-1/2}) [{\rm I}_H \otimes (\hat{{\rm I}}'_p -{\rm I}_p +{\rm I}_p)]
        \Gamma_{221}\ +\ \sqrt{\alpha}\ \sqrt{n} \Gamma_{12}'
        (\hat{{\rm I}}_p -{\rm I}_p +{\rm I}_p)
\\
&&
    \Sigma_{{\mathbf{x}}}^{-1/2}
        (M_n-M+M)
        (\hat{F} \hat{G} - FG + FG)
        \Gamma_{222}
\\
&=& \sqrt{1-\alpha}\ \sqrt{n} \Gamma_{12}'
        \Sigma_{{\mathbf{x}}}^{-1/2}
        (C_n-C)
        [FG \otimes {\rm I}_p]
        ({\rm I}_H \otimes \Sigma_{{\mathbf{x}}}^{-1/2})
        \Gamma_{221}
\\
&&  +\sqrt{\alpha}\ \sqrt{n} \Gamma_{12}'
        \Sigma_{{\mathbf{x}}}^{-1/2}
        (M_n-M)
        FG
        \Gamma_{222}
   \ +\ O_p(n^{-1/2})
\\
&=& \sqrt{n} \Gamma_{12}' {\Sigma}_{{\mathbf{x}}}^{-1/2}
    [(C_n,M_n) - (C,M)] K \Gamma_{22}
   \ +\ O_p(n^{-1/2}).
\end{eqnarray*}

Therefore, the asymptotic distribution of
$\Gamma_{12}' \hat{U}_{n} \Gamma_{22}$
is determined only by
the asymptotic distribution of $(C_n,M_n)$.

\vspace{0.3cm}
\noindent
{\bf The detail of $\Delta_0$, $(p^2H+pH+p)\times (p^2H+pH+p)$:}
\[
\Delta_0=\left(
\begin{array}{ccc}
\Delta_0^{1,1} & \Delta_0^{1,2} & \Delta_0^{1,3}\\
\Delta_0^{2,1} & \Delta_0^{2,2} & \Delta_0^{2,3}\\
\Delta_0^{3,1} & \Delta_0^{3,2} & \Delta_0^{3,3}
\end{array}
\right),
\]
where
$\Delta_0^{1,1}={\rm diag}\left\{\ldots,{\rm Cov}\left({\rm Vec}({\mathbf{x}}{\mathbf{x}}')|\tilde{y}=h\right)/f_h,\ldots\right\},\> p^2H\times p^2H$;
$\Delta_0^{2,1}$ $=$ ${\rm diag}\left\{\ldots,{\rm Cov}\left({\mathbf{x}},{\rm Vec}({\mathbf{x}}{\mathbf{x}}')|\tilde{y}=h\right)/f_h,\ldots\right\},\> pH\times p^2H$;
$\Delta_0^{2,2}={\rm diag}\{$ $\ldots,$ ${\rm Cov}({\mathbf{x}}|\tilde{y}=h)/f_h,$ $\ldots\},\> pH\times pH$;
$\Delta_0^{3,1}=\left[\ldots,{\rm Cov}\left({\mathbf{x}},{\rm Vec}({\mathbf{x}}{\mathbf{x}}')|\tilde{y}=h\right),\ldots\right],$ $p\times p^2H$;
$\Delta_0^{3,2}=\left[\ldots,{\rm Cov}\left({\mathbf{x}}|\tilde{y}=h\right),\ldots\right],\> p\times pH$;
$\Delta_0^{3,3}=\Sigma_{{\mathbf{x}}},\> p\times p$;
$\Delta_0^{1,2}=\left(\Delta_0^{2,1}\right)'$;
$\Delta_0^{1,3}=\left(\Delta_0^{3,1}\right)'$;
$\Delta_0^{2,3}=\left(\Delta_0^{3,2}\right)'$.

redpajama_set_name: RedPajamaArXiv


text:
Q: json to human readable date date shows one less than actual date I convert json date to human readable date but it shows less one then actual date. I
used this code to convert it:
SimpleDateFormat df = new SimpleDateFormat("yyyy-MM-dd");
            Calendar calendar = Calendar.getInstance();
            Long timeInMillis =    Long.valueOf(AttendanceModelList.get(position).getEmpdate());
            calendar.setTimeInMillis(timeInMillis);
            Date date=new Date(timeInMillis);
            viewHolder.textemployeedate.setText(df.format(date));

Please help

A: You say as summary:

Your calendar date is one day less than expected when you try to
  interprete a global timestamp of type java.util.Date as calendar
  date.

This phenomenon can happen due to timezone effects or midnight change. Before viewing the technical solution, you have to ask yourself:

*

*What is your default (system) timezone using TimeZone.getDefault()?

*Do you run your code on a server which has not the expected timezone?

*In which timezone do you wish to view the calendar date? (the timezone associated with your expected "actual" date)

How to specify the timezone?
java.util.Date d = ...; // from your JSON-timeInMillis?
SimpleDateFormat sdf = new SimpleDateFormat("yyyy-MM-dd");
String tz = "Asia/Kolkata"; // or any other valid tz id
sdf.setTimeZone(TimeZone.getTimeZone(tz));
System.out.println(sdf.format(d));


Yesterday I managed to finish and block my father's christmas gift. Today I managed to finish and block my mother's christmas gift. They are both fantastic, and quite frankly my dad's gift may get a part two. They are actually going to be quite difficult to give away I love them so much. But, I can't show you until after christmas.
Instead, I figured I'd ball up some yarn for a project of my own. I got a new winter coat this year, and I realized that I have yet to knit any accompanying accessories for it. I also realized I have yet to knit anything from loop bumps I've spun. So, I looked in my stash, found a lovely two ply, and wound it into cakes.
Let's talk about winding yarn for a second here. Recently most of my winding is done by hand, with a nostepinne I picked up at the Taos Wool Festival. It is from KCL Woods. But even before the nostepinne purchase, I'd been winding by hand for quite some time. I think this has something to do with working predominately in handspun, and loving the handspun so much that even winding it is a pleasure.
However, I have close to 900 yards of this loop handspun. So, I figured it would be a little overwhelming to wind it by hand when I was eager to start immediately. I wound the first smaller ball with a regular royal ball winder, but for the second skein, which was HUGE, I pulled out my strauch ball winder. This one comes in handy when I need something for a particularly large skein. I can winder faster with the cheap royal, but it was worth it to pull out the Strauch. So I figured I'd grab a photograph to show the difference. Added bonus, I grabbed the nostepinne and threw that in the shot as well.
I could not finish the knitting project with the Jacob Fleece yarn I plied earlier this week, so I was forced to spin some more. I totally forgot what a pain this particular fleece is to spin, and how it is quite dirty (my fault for giving up on a fleece that was less than amazing) and how it makes my entire house dirty when I spin it. I forgot how I only spin this in the warm months, out on the porch, to save myself from that. It took two days, but I do have another skein of light gray yarn. This time I have 108 yards.
I also decided to work on a new spinning project. I've had this superwash merino from Enchanted Knoll Farms in my stash for years. I think I purchased it online, and when I got it, because all that was showing were the more muted colors, I put it away. Well, once I opened it up, I realized why I purchased it in the first place. I love the blue and green against the brown, and I wouldn't really call it all that muted at all.
Now with two spinning projects on the spindles, I should have plenty to keep me occupied when I am absolutely sick of Christmas gift knitting.
While I took this week off to get crafting done, or more specifically, get some fleece washing done, that hasn't happened at all. Instead, I've been doing deep cleaning around the house. Seeing as that feels wonderful and productive, I am not too upset. And I have plenty of fleece to process before I truly need to wash fleece again. No big loss.
One of my big jobs was the "toss the stash" job. I needed to reorganize my yarn cubes and I needed to do some dusting as well. I've been selling off yarn regularly, due to just not using commercial yarn very often and due to financial circumstances. I also have a lot more handspun than I used to, and it was all over the place. So, I reorganized, handspun in two cubes and the rest separated by weight. It looks much better now, and it is completely inspiring to study the cubes and consider what I can do with the yarn.
I also organized my knitting/crafting book cabinet and entered all the books and magazines in my ravelry library. This is very convenient when pattern searching, and I am excited to have it done.
Happy Thanksgiving everyone! Have a wonderful day.
Due to circumstances beyond my control, this Christmas will be a frugal one. So, much knitting is happening in this household. This is great for me, it has renewed my knitting interest a bit, and I am enjoying my progress. However, it is bad for the blog as my intention is to take pictures and then save them for after Christmas.
But, I did do some spinning. I have been working on that jacob fleece for a good long time now, I have spun through the darker colors and had 3 balls of gray in varying shades to ply. I finally got around to that yesterday. I am consistently amazed at the yarn this fleece makes. I basically start with some very very rustic looking singles, spin them long draw, ply, give it a very good thwack, and end up with this really bouncy yarn. Is it rustic? Heck yeah, but it also looks amazing.
This spin yielded 82 yards of worsted weight 3 ply. It really is wonderful tonal stuff and I can't wait to use it, but since it is for a christmas gift, I won't even be able to show you for a month!
I am afraid I've put myself in a bit of a bind with this gift though, I keep running out of yarn and needing to spin more. Not quite how I intended it to be, but I suspect the end result will be worth it indeed.
Day #1 of vacation was yesterday. And I spent the entire day cleaning. I got started in the morning and was just too inspired to quit. I did the kind of cleaning we all need to do sometimes, but is completely uninspiring. The kind where we take the pillows off the couch and vacuum the couch, the kind where you dust and wash all the baseboards. That kind of cleaning. It felt really really good to get done, and I think it will make me feel much more relaxed during the week.
That being said, I didn't have any time for crafting at all. So, I decided I'd take a picture of a project that has been on the needles for a bit now. I love it. It is a bias knit cowl I am knitting in the handspun singles I recently blended and spun.
The pattern calls for using two different size needles, and this makes for a waffle like soft squishy fabric that I am completely in love with. And it lends itself very well to the thick/thin nature of handspun. I am having trouble working on other stuff simply because this one is so addicting.
Here's another one I've taken quite a bit of time to finish. I started this well before I left on my New Mexico trip. I decided that I could spin this on my spindles, and once I got started, I realized it was spinning quite thin very naturally. So, I went with it, even though that means a long project and I was already in the middle of another thin and long project.
This is dyeabolical's colorway flowershop inferno. I did this one in a 3 ply fingeringweight. This is superwash merino, but I have this colorway in another fiber as well, so I am considering spinning that and chain plying just to compare the two. But, that will have to wait for a few other fantastic projects I have hanging around.
I have a total of 500 yards on this one. The colors are all crazy and so I have no idea what kind of project this should be. Maybe something in linen stitch!
I am fortunate enough to be on vacation this week. Just staying home and doing all the things that need to get done around the house. I am quite excited about it. I decided that the first thing I should do is clean, since that will allow me to enjoy my vacation more.
And then put away the drum carder. I love this batt, and should have 4 oz. or so by the time I am done. Again, this is dyed locks paired with undyed locks from a CVM Romeldale fleece I have in my stash. I think it is quite fabulous!


James Franco is in talks to star in Gus Van Stats upcoming biopic based on the life of gay activist turned ex-gay nutsy religious zealot wacko devotee, Michael Glatze.
Glatz who was at one time the was an award winning gay rights activist and editor of Young Gay America, Glatze found "God" and then claimed that his homosexuality came to him "because I was already weak". He attributes his sexual "confusion" to the death of his father when he was 13, and his mother when he was 19. He was 14 when he decided he was gay, and 20 when he came out publicly.
In 2004, he launched YGA as a "virtuous counterpart" to adult-oriented gay magazines.
Gus Van Sant (Good Will Hunting, Milk) is on board to produce the $3.5 million movie with shooting scheduled to begin in New York this July, reports the New York Post.
Why Van Stat picked this particular self-loathing ex-gay racist douchebag to make a movie about is unknown.
Note to Glatze: You are still repulsive.


text:
Q: how to finalize the spoiler There is a code http://jsfiddle.net/VWCnd/5/. How to add the indicators ↓ ↑.
Example image http://i.stack.imgur.com/OxpaP.png
Js:
$(document).ready(function(){
 $('.spoiler_title').click(function(){
  $(this).parent().children('div.spoiler_toggle').toggle();
  return false;
 });
});

html:

    Title

Body

CSS:
.spoiler_title {
    display:block;
    background-color: #551A8B;
    color: #fff;
    padding: 10px 10px;
}
.spoiler_toggle {
    display:none;
}
.spoiler_bg {
    background: #9C6AC9;
    padding: 5px 5px;
}

Thanks.

A: HERE IS THE DEMO
CODE -
JS:
$(document).ready(function(){
 $('.spoiler_title').click(function(){
     $(this).find(".sp_but").text(($(this).find(".sp_but").text() == '↓' ? '↑' : '↓'))
  $(this).parent().children('div.spoiler_toggle').toggle();
  return false;
 });
});

HTML:

    Title ↓

Body

CSS:
.spoiler_title {
    display:block;
    background-color: #551A8B;
    color: #fff;
    padding: 10px 10px;
}
.spoiler_toggle {
    display:none;
}
.sp_but{
    float: left;
    margin-right:20px;
}
.spoiler_bg {
    background: #9C6AC9;
    padding: 5px 5px;
}

A: You could always do something like this:
http://jsfiddle.net/VWCnd/7/
 $('.spoiler_title').click(function(){
     var $this = $(this),
         visible  = $this.parent().children('div.spoiler_toggle').toggle().is(':visible');
     $this.find('.spoiler_indicator').html(visible? '↑' : '↓');
     return false;
 });


Last week Throwback Thursday did not happen because we were busy celebrating Thanksgiving. Before I feature anymore of our retired collections, I want to share some of the throwback photos I have received for readers in past few weeks.
From Marjorie: Sonali (2009) in Molly's aviator outfit.
Also from Marjorie – Lindsey (Girl of the Year 2001) in the cargo outfit from 2000 and Lanie (GOTY 2010) in the Ready for Fun outfit from 2004.
Madelon's Kanani is modeling the 1996 Pleasant Company ice skating outfit.
This one is from my archive. It is a display that was in the Doubletree Hotel in Times Square when we visited in 2006. The black and red jacket and hat as well as Coconut's Statue of Liberty headband were AGPNY store exclusives at the time. We have that headband for Coconut.
If you have photos of retired American Girl (or other brands) dolls, clothing or furniture and want to help us take a look into the past, email your photos and descriptions to us at share@dolldiaries.com.
I won the November reader photo contest and I forgot to put that I would like a 18 inch doll shirt! I am sorry it was that I was so happy! PS mine was Fun with leaves by Sabrina!
Cute pictures! By the way, does anyone know which of the Doll Diaries posts has a picture of Mia and McKenna together with some stuff that their owner got in a giveaway? One of them (Mia?) is wearing a white sweater with a cat face. I know this sounds weird but I have looked all over the site and cannot find it! I need it for a doll project I am doing.
So cute! I WANT Kanani's sweater- it would look adorable on me!
did you get my Thanksgiving photos?
Thank you so much Katie, that would be perfect!
I only have a bitty baby outfit a kanai outfit and a lanie outfit does that count?
Thanks for sharing my pictures! I really like that ice skating picture. I think the dress is a Chanukah dress, and one of the first outfits from the Girl of Today line. Very pretty. On a side note, I was at Target today and saw two gorgeous Hearts for Hearts Girls: Nahji from India and Lillian from Belarus. I had to take them home–I was immediately in love.
Molly's Aviator outfit and recital dress are the 2 I am missing! Sonali looks great! She pulls it off with style = I had the khaki outfit with the flannel shirt – it was called A+B+C+1,2,3, in the 2001 Speing catalog = My jLY wore Ready For Fun – wise advice: Don't EVER sell part of your doll collections. These were real P.C clothes, way better than a lot of the Mattel ones and are rare and hard to find. The figure skating ouutfit was 1st releasedin (1998)?? as a Hanukkah dress , but came with a silver headband and small star of David in sterling silver. I have the necklace and sold the outfit to a collector. (quietly banging my head against the wall) AG recently had a very simlar re-do of the outfit with a silver tafetta skirt and the sweater was thin blue with sequins. Cyber Monday. It's home and on Rebecca because I was stupid enough to sell my Lindsey too. My Kit and Molly collections are really extensive. I need to send in Molly wearing her Dude Ranch outfit and Kit playing baseball. OOOOOOOOOH – I just got the Cecile and MG book series and blated through them yesterday. The illustrations ARE awful, but here's what AG needs to make: Cecile's beautilful rosary beads, she uses them in "Troubles For Cecile" and there's a small picture, Cecile's picnic lunch of baguette and figs, brown and white checked dress, plus cover outfits for Cecile and the white lace delicate dress for MG, her quill pen and inkpot, French book, and slate – AG are you listening??????
iluvdolls – please, please send in some of your photos for Throwback Thursday. I would love to see Molly in her Dude Ranch outfit and Kit's baseball outfit is one that I would like to add to our Kit collection.
Char,I have a quick question I won the November reader photo contest and I emailed you my address address and I was so happy! I forgot to say that I would like a shirt for a 18 inch doll.Can you send me a shirt for a 18 inch doll? Mine was fun with leaves by Sabrina!
Thank you so much Char! I was getting worried! I just play with American girl dolls more than Barbies!
I really like the ready for fun outfit!
Some one else has a Lindsey!! I bet a lot of peoples Lindsey doll is in storage or their attics because they "grew out of it"! And must not know how much she is worth!!!
I have a Lindsey too, I bought her off eBay during the summer. She had really loose legs that were sent to get fixed in the Dublin doll hospital, some marks on the vinyl and bad hair I manages to fix the marks and her legs are fine but her hair is still bad.
Char- Do you mist your doll's hair? I was wondering if you should mist it or not. I really want my dolls to be in great condition, so I just wanted to know what you reccomend.
Vivienne – only when they need it. The ladies in the AG salon mist the dolls' hair – but they are usually dealing with hair that needs it.
Think you so much! I sent an email to you. But it might be a few minutes before you get it.
Oops. I meant *thank instead of think.
Vivienne- my Kailey desperately needed misting (I was going to Downy Spray it but I heard from a hairstylist that it deteriorates the doll's hair) so I misted it and now her hair is in the best condition it's ever been.
Thanks Katie – you are awesome!
Yes, I have that noted.
How do you make the icon by your name char??
Go to Gravatar.com and sign up there – you can pick our picture.
Thank you Katie! I adore McKenna and Mia, and their juxtaposition was so cute!
Thanks Char I think I got it!
Wait let me try one more time……….
Where did you get the mini shopping bags from?
Erica – you can get them at the American Girl stores – the Creativi-tees area has them and I know they have them at the Bistro or Sweet Treats bar, too.
Char– Do you have any AMERICAN GIRL DOLLS?
Sarah – I have Ivy, Felicity and will be getting Molly, too. My daughters have quite a few of the others.


Mempho Music Festival, brought to you by Big River Presents, has announced the full lineup for it's second edition, which will return to Memphis, Tennessee, on October 6th and 7th, 2018.
Nestled in the heart of Shelby Farms Park, Mempho Fest blends today's top performers with Memphis' rich musical history, creating a festival experience filled with music, food, and nature for people of all ages. This year, for the first time ever at Shelby Farms Park, camping and "glamping" options will be available.
New to 2018, Mempho Music Festival is partnering with the Memphis Area Women's Council to promote the Memphis Says NOMORE campaign—aimed at raising awareness for domestic violence and sexual assault—by providing a safe, inclusive, andwelcoming environment for all attendees. Mempho has also jumped on board the #BringYourSoul movement, celebrating theoriginality, soul, and change that Memphis is known for. Additionally, Mempho Music Festival has teamed up with the OceanicGlobal Foundation—a non-profit that educates individuals on issues impacting our ocean through art, music, and emergingtechnologies—and is proud to take the pledge to go completely straw-free!
This year, Mempho Music Festival has partnered with CID Entertainment to provide VIP and Super VIP experiences with on-site and off-site lodging options. For more information on packages, click HERE.A limited supply of GA, VIP, and Super VIP pre-sale tickets and packages are available on Monday, June 11th, for returning fans,starting at $79 for Single Day and $139 for 2-Day tickets. General on-sale begins on Friday, June 15th, at 10 A.M. CT, starting at$89 for Single Day and $159 for 2-Day tickets. Prices will increase on July 13th and September 28th, so reserve your tickets while supplies last.


text:
Meet Your ASUR President, Kamal Bilal
Evan Furgurson
I met Kamal Bilal in the Fall of 2014 when we were both incoming freshman and roommates. Shortly after, I found out about his aspirations to become ASUR president. He has recently achieved his goal, as Kamal was elected as ASUR president and was sworn into the position on Wednesday. The athlete, artist, rapper, Johnston student, and now student body president of The University of Redlands is humble and thoughtful. Often he gives the appearance of being a quiet person, but when the time comes, he is quite vocal and eloquent. With such a multi-talented and passionate person, the best place to start is at the beginning.
Learning a bit about Kamal's past sheds some light on his current ventures. Born and raised on the Southside of Chicago, Kamal has always been very involved. When he was in 8th grade, he won The Money Smart Kid award, making him the ambassador of the Federal Reserve in Chicago for a year. He comments: "I was hella busy in high school with soccer, basketball, homework and a social life all at the same time–dude forget it."
In college, Kamal has maintained this high level of dedication and involvement. During his freshman year at U of R, Kamal began plans for a summer camp to put on in Chicago. The camp, Eye Love You, is an inner-city camp where local youth get a chance to practice art, sports and learn about healthy living. Eye Love You has had one successful session so far in Chicago and is now gearing up for this coming summer. Last year Kamal threw an all campus party to raise money for it, combining his passions to help people and promote student artists. Kamal will be teaching a class in the fall of 2017 involving the camp, that aims to motivate and guide projects that empower communities. In the past few years Kamal has been involved in many organizations around campus like BLACC (Brilliant Leaders Advocating Color Consciousness), Enactus (a green business club), and student government. He has also written a few articles for the Bulldog Weekly. Kamal has also been a forward for the Men's soccer team for the last two years, most recently winning the SCIAC championships and making it to the NCAA Elite Eight this past December.
He spends his summers not only running the camp he created, but also collaborating and making art with his friends. He is a very passionate and talented rapper who performs under the pseudonym Rick Ph7air. Also known as Ricky, he has made his own albums, videos and cover art. Kamal is also a talented photographer and graphic designer. Kamal is passionate about many forms of art but especially sharing and creating with others. Rick Ph7air recently performed at the event Green Slam, put on by ASUR social affairs and SEA (Students for Environmental Action). I am excited for more performances by Ricky at great events the entire campus can enjoy.
I sat with the tall, athletic, and balanced Chicago-native as he held a cold Mexican import between his hands and asked him a few questions in anticipation of being sworn in as ASUR president.
When did you first want to run for president and why?
"I wanted to be president my freshmen year, but I couldn't haha, so I became vice-president for BLACC. After that I wanted more experience in student government so I joined Senate, then Cabinet…I want to have influence on this campus, use my privilege of being in many different groups to hear different student voices…. I just know I like to serve, to put people in a position to succeed, and I think being president is a great way to do that. I just want to have a positive impact on a lot of people."
What movie always makes you cry?
Marley and Me and King Kong are two movies that I cried hella hard in.
What will be your main priorities when you take office?
"My main priorities are bon appetite reform, compost, Greek life, specifically the party policy, and transparency. But I'm already working on these right things now, so next semester we will have different stuff to work on"
What message would you like to send to the student body?
"I want people to know that if they want something to happen, let's try to make it happen. If you get a few people together with an idea or project in mind for the University, we can do it. I want to make myself very accessible so I can not only support with new ideas but also make communications more clear between the school and students…My cabinet and I will be available around campus, in SLIC and online, so contact us." Kamal_Bilal@redlands.edu
Today's political climate is dividing and deep-seated. It effects everyone to a different extent, but is palpable on this campus. Kamal Bilal is a multi-talented leader who brings a new energy, perspective and mission. It will be exciting to see how Kamal handles all that is to come in the upcoming year.
About Evan Furgurson
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text:
Question of Elementary Education in the Third Russian State Duma (1907-1912)
Author: SantaMaria, Phillip
Includes an overview and perspective of the Third Duma, chapters on "The Committee on Public Education," "Legislative Work For Elementary Education," "The Church and Primary Education," and "The Nationalities and Education." Supplemented by numerous tables, including tables on "Literacy in the Russian Empire in 1897," "Growth of Education in Russia at the End of the Eighteenth Century," "Total Expenditures of the Russian Government and Expenditure of the Ministry of Public Instruction (1795-1917)," and others.
Chapter I. The Third Duma, Overview and Perspective
Chapter II. The Committee on Public Education
Chapter III. Legislative Work for Elementary Education
Chapter IV. The Church and Primary Education
Chapter V. The Nationalities and Education
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