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Home » Label » BIS » Brahms - Piano Works Vol. 3
Studio quality download available at eClassical.com
Brahms - Piano Works Vol. 3
Composer Johannes Brahms
Performer Jonathan Plowright
Period Romantic
Catalogue Number BIS-2127 SACD
Format SACD Hybrid
This hybrid disc plays on both CD and SACD players
SACD Surround - SACD Stereo - CD Stereo
Release date Jan 2016
Total time 81'19
Editor's Choice, Gramophone
Critical acclaim for Jonathan Plowright's two previous discs in his series of Brahms's works for solo piano includes distinctions such as '10/10' on the website Classics Today and top marks in Diapason as well as 'Instrumental Choice of the Month' in BBC Music Magazine. On his third disc Plowright presents the listener with two seemingly contradictory aspects of Johannes Brahms: the deeply serious, emotionally charged, but also meticulously constructed late Piano Pieces of Op. 118 and their precursor Op. 76 appear side by side with the unapologetic extroversion and scintillating verve of most (if not all) of the 16 Waltzes Op. 39, and the flamboyant Variations on a Hungarian Melody. Brahms's fascination with Hungarian gypsy music stemmed from his friendship with the violinist Eduard Reményi, and found an outlet in several works, among which the Hungarian Dances are probably the most famous. The composer was also a friend and admirer of Johann Strauss II, and in his Op. 39 trespassed – with great success – into the domains of Vienna's 'Waltz King'. It is tempting to think that Brahms early experience as a pianist in taverns and restaurants induced him to compose this sophisticated light music, but there were also commercial reasons: he composed the waltzes for piano four hands and subsequently made no less than two arrangements for solo piano of the entire set.
Buy selected tracks
Variations on a Hungarian Melody, Op.21, No.2 7'34
01 Thema. Allegro 0'15 $ 0.00 EUR
02 Variation I 0'17 $ 0.00 EUR
03 Variation II 0'16 $ 0.00 EUR
04 Variation III 0'14 $ 0.00 EUR
05 Variation IV 0'17 $ 0.00 EUR
06 Variation V. Con espressione 0'22 $ 0.00 EUR
07 Variation VI 0'20 $ 0.00 EUR
08 Variation VII. Poco più lento, dolce espressivo. Quasi pizzicato 0'31 $ 0.00 EUR
09 Variation VIII. Ancora un poco più lento. Espressivo, dolce 0'41 $ 0.00 EUR
10 Variation IX. Dolce 0'29 $ 0.00 EUR
11 Variation X. Legato, dolce 0'18 $ 0.00 EUR
12 Variation XI. Dolce 0'18 $ 0.00 EUR
13 Variation XII. Espressivo 0'17 $ 0.00 EUR
14 Variation XIII. Con forza 0'22 $ 0.00 EUR
15 Allegro (il doppio movimento)—Tempo I più animato 2'37 $ 0.00 EUR
Eight Pieces, Op.76 26'59
16 1. Capriccio in F sharp minor. Un poco agitato 3'43 $ 0.00 EUR
17 2. Capriccio in B minor. Allegretto non troppo 3'47 $ 0.00 EUR
18 3. Intermezzo in A flat major. Grazioso 2'43 $ 0.00 EUR
19 4. Intermezzo in B flat major. Allegretto grazioso 1'50 $ 0.00 EUR
20 5. Capriccio in C sharp minor. Agitato, ma non troppo presto 2'55 $ 0.00 EUR
21 6. Intermezzo in A major. Andante con moto 4'05 $ 0.00 EUR
22 7. Intermezzo in A minor. Moderato semplice 4'09 $ 0.00 EUR
23 8. Capriccio in C major. Grazioso ed un poco vivace 3'47 $ 0.00 EUR
Waltzes, Op.39 20'03
24 No.1 in B major. Tempo giusto 0'45 $ 0.00 EUR
25 No.2 in E major. Dolce 1'15 $ 0.00 EUR
26 No.3 in G sharp minor. Dolce 0'46 $ 0.00 EUR
27 No.4 in E minor. Poco sostenuto 1'14 $ 0.00 EUR
28 No.5 in E major. Grazioso 1'37 $ 0.00 EUR
29 No.6 in C sharp major. Vivace 0'56 $ 0.00 EUR
30 No.7 in C sharp minor. Poco più andante 2'26 $ 0.00 EUR
31 No.8 in B flat major. Dolce, sotto voce 1'24 $ 0.00 EUR
32 No.9 in D minor. Espressivo 1'15 $ 0.00 EUR
33 No.10 in G major. Poco scherzando 0'32 $ 0.00 EUR
34 No.11 in B minor 1'17 $ 0.00 EUR
35 No.12 in E major 2'30 $ 0.00 EUR
36 No.13 in B major 0'37 $ 0.00 EUR
37 No.14 in G sharp minor 1'20 $ 0.00 EUR
38 No.15 in A flat major 1'15 $ 0.00 EUR
39 No.16 in C sharp minor 0'54 $ 0.00 EUR
Piano Pieces, Op.118 24'17
40 1. Intermezzo in A minor. Allegro non assai, ma molto appassionato 1'46 $ 0.00 EUR
41 2. Intermezzo in A major. Andante teneramente 6'35 $ 0.00 EUR
42 3. Ballade in G minor. Allegro energico 3'27 $ 0.00 EUR
43 4. Intermezzo in F minor. Allegretto un poco agitato 2'25 $ 0.00 EUR
44 5. Romance in F major. Andante - Allegretto grazioso - Tempo I 4'26 $ 0.00 EUR
45 6. Intermezzo in E flat minor. Andante, largo e mesto 5'38 $ 0.00 EUR
Album total 81'19
Composer Brahms, Johannes
Piano Plowright, Jonathan


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Car Rental Charleston
Hertz car rental in Charleston
Rent a car in Charleston to discover the small-town charm and history of West Virginia's capital city. Home to a range of museums, restaurants and the imposing Governor's Mansion, Charleston is located on the River Kanawha and boasts scenic views of the surrounding green valleys.
With a rental car from Hertz you'll be able to reach the Hawks Nest State Park in just under an hour, and New River Gorge National Park in just over an hour. You can also drive three hours to explore Monongahela National Forest. If you prefer city vibes to a countryside hike, Lexington and Columbus are both under three hours away by car. Downtown Charleston is a mere 10 minutes' drive from West Virginia International Yeager Airport.
So, whether you'd like to hire a sophisticated sedan for visiting the capitol building or a spacious SUV for taking the family on a nature expedition, we have the perfect rental car for you.
Start planning your riverside city trip today with Hertz car rental in Charleston.
Pickup Locations Charleston
Town Center Marriott
Opening hours: Mon-Sat 8:00AM-5:00PM, Sun closed
Address: 1100A Highway 72 West
Driving in and around Charleston
Parking in Charleston
Washington Trust Lot (25301) provides convenient parking, close to the Clay Center, Capitol Market and the Charleston Convention & Visitors Bureau.
Chase Tower – Douglas Parking (25301) can be found in Downtown Charleston, offering more than 500 spaces near Haddad Riverfront Park and Heritage Towers.
Charleston Town Center Washington Street Garage (25301) has 550 spaces and is conveniently located for the shops and restaurants in Charleston's center.
Key roads in Charleston
Interstate 64 (I-64) runs east-west through West Virginia, starting in outer St. Louis and running through Louisville, Charleston and Richmond to end in coastal Norfolk.
US Route 60 (US-60) is a major US highway running east-west from Brenda, near Phoenix, to Virginia Beach. In Charleston, it is the main road over Patrick Street Bridge and through the town center.
Interstate 77 (I-77) is a south-north highway connecting Columbia in South Carolina to Cleveland in Ohio, via the metropolitan outskirts of Charlotte and other cities in West Virginia, including Charleston.
How to get from Charleston to Hawks Nest State Park
It takes an hour to drive the 50 miles from Charleston to Hawks Nest State Park.
Leave the car rental car park on MacCorkle Avenue and head east towards Maple Street.
Merge onto I-64 E via the slip road to Charleston and follow the road.
Take Exit 96 for US-60 E towards Belle and continue to US-60 E.
Turn left onto US-60 E/Midland Trail.
Follow Midland Trail to Hawks Nest State Park.
How to get from Charleston to Lexington, Kentucky
Lexington is just under a three-hour drive from Charleston. Take I-64 W/I-77 N from Lee Street and follow the I-64 W until you reach Exit 81, then take the exit and merge onto I-75 S towards Richmond/Knoxville. Take Exit 110 for US-60 towards Lexington.
A quick guide to Charleston
Best hotels in Charleston
Charleston Marriott Town Center
200 Lee Street E, Charleston, West Virginia, 25301
Located in Downtown Charleston, a few minutes from the State Capitol building, this modern Marriott hotel provides an indoor pool, fitness center and 16 event rooms. The on-site Brick Salt Bar+Kitchen offers a seasonal menu for guests, and you can even bring your pets to stay.
Hampton Inn Charleston-Southridge
1 Preferred Place, Charleston, West Virginia 25309
The Hampton Inn is ideally placed off US-119, just 15 minutes from Yeager Airport. Guests can take advantage of complimentary parking, a delicious hot breakfast, plus a fitness center that's perfect for cramming a quick workout into a packed schedule.
Embassy Suites by Hilton Charleston
300 Court Street, Charleston, West Virginia, 25301
Indulge yourself at Embassy Suites with free made-to-order breakfast, complimentary evening reception and an indoor pool. Other amenities at this Downtown hotel include free airport transportation, room service, laundry service, a business center and pet friendly rooms.
Best restaurants in Charleston
Bricks & Barrels
1214 Smith Street, Charleston, West Virginia, 25301
Bricks & Barrels is a popular restaurant among the locals thanks to its unique ambience and fresh food. With dining tables made from wooden wine and whiskey barrels, rustic brick walls and private seating alcoves made to create the impression of dining inside of a barrel, it's obvious where the restaurant name comes from. Expect a premium steak and seafood menu.
350 MacCorkle Avenue SE, Charleston, West Virginia, 25314
Treat yourself to a fine dining experience in the decadent rooms of this historic Railroad Depot building. Established in 1976, Laury's offers indulgent French-inspired dishes such as veal scallopini piccata, pasta primavera and roasted duck. Tuck in while enjoying stunning views of the Kanawha River and the skyline of downtown Charleston.
800 Smith Street, Charleston, West Virginia, 25301
Located in the popular Capitol Market, Soho's is a casual Italian eatery with a friendly atmosphere for friends and family alike. The menu includes specialty in-house soups, homemade desserts and Italian, American and Mediterranean dishes with a twist. Eat outside in the garden area, or inside the renovated 1800's train depot, with exposed brick, wood ceilings and steel beams.
Best things to do in Charleston
West Virginia State Capitol Building
1900 Kanawha Boulevard E, Charleston, West Virginia, 25305
Perhaps the most prominent building and attraction in Charleston, the grand exterior of this structure matches the importance of its residency – the governor of West Virginia. Explore the landscaped grounds or book a tour to learn about the history of the state and the building.
West Virginia State Museum
Located opposite the Governor's Mansion, the West Virginia State Museum focuses on the state's history, culture, art and even paleontology. The exhibitions change regularly but resident displays tell the story of West Virginia from prehistoric to current times. Don't forget a trip to the gift shop.
1 Clay Square, Charleston, West Virginia, 25301
The Clay Centre is a 240,000-square-foot venue, home to an art museum, an interactive discovery museum, a planetarium and a performance hall. There's something for everyone including live music, 3D shows, children's activities and art classes.

redpajama_set_name: RedPajamaCommonCrawl


text:
Sabic expanding global capacity for PEI and PPE resin production
Material supplier Sabic has announced projects in Asia and the Netherlands designed to increase global capacity for its Ultem polyetherimide (PEI) and Noryl polyphenylene ether (PPE) engineering resins.
To expand the production capacity of Ultem resins for polyetherimide materials, Sabic plans to expand its production site based on the existing blending production base in Singapore. Pending final approval of the government, new production facilities scheduled to be built in Singapore are expected to be completed in the first half of 2021.
In addition, the company's polyphenylene oxide (PPE) resin production line at the Bergen op Zoom plant in the Netherlands will also be re-enabled to produce the base resin PPE of Noryl resin and oligomer series. The retrofitting is expected to be completed by the end of 2019.
"Increased customer demand, especially in Asia, prompted the further capacity expansion plans," said Ernesto Occhiello, Sabic's executive vice president, specialties. "While interim capacity gains for both Ultem and Noryl resins have been achieved, global demand for both product lines has increased significantly, and Sabic is planning to expand its capacity to support our customers' growth aspirations."
Ultem resins are currently produced in two locations, Mt. Vernon, Indiana, and Cartagena, Spain. The planned operations in Singapore will localize supply for customers in Asia, reducing lead times, especially for shorter qualification cycle applications. When fully operational, the Singapore facility is expected to increase capacity by 50 per cent over a 2018 baseline. "With the addition of Singapore, Sabic will be the only petrochemical company with the ability to produce the high heat resin in Asia, the Americas and Europe, a significant advantage for the company's customers," Occhiello said.
The decision to recommission the Bergen op Zoom Noryl resins facility will give Sabic's customers a second source of PPE resins globally. When operational, the Bergen op Zoom facility is expected to add more than 40 per cent global capacity over a 2017 baseline. In the meantime, Sabic expects incremental manufacturing process improvements at the Selkirk, N.Y. and Mt. Vernon plants to provide increases in Noryl PPE resin and Ultem PEI resin production by the end of 2018.
GE invests in additional production capacity for "xtreme" Lexan resin
Sabic trims up to 20% of global engineering resin output
Sabic opens new pilot facility for high performance polymers
SABIC, SK Global Chemical partner to manufacture high-performance PE products
AceTRONIC Industrial Controls Inc.
Barrie-based Canplas Industries launches multimillion-dollar expansion
Magna debuts active air dam on 2019 Ram pickup


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package filter;

import entity.User;

import javax.servlet.*;
import javax.servlet.http.HttpServletRequest;
import javax.servlet.http.HttpServletResponse;
import javax.servlet.http.HttpSession;
import java.io.IOException;

/**
 * Created by Bruce-Jiang on 2016/10/9.
 * 过滤器配置类
 */
public class LoginFilter implements Filter {
    private FilterConfig config = null;
    public void destroy() {
        config = null;
    }

    public void doFilter(ServletRequest req, ServletResponse resp, FilterChain chain) throws ServletException, IOException {
        //把ServletRequest和ServletResponse转换成真正的类型
        HttpServletRequest request = (HttpServletRequest)req;
        HttpServletResponse response  = (HttpServletResponse)resp;
        HttpSession session = request.getSession();

        String loginStrings = config.getInitParameter("loginStrings");
        String includeStrings = config.getInitParameter("includeStrings");
        String redirectPath = request.getContextPath() + config.getInitParameter("redirectPath");

        String[] loginList = loginStrings.split(";");
        String[] includeList = includeStrings.split(";");

        //对指定后缀进行过滤
        if(!this.isContains(request.getRequestURI(),includeList)){
            chain.doFilter(req,resp);
            return ;
        }
        if (this.isContains(request.getRequestURI(), loginList)) {// 对登录页面不进行过滤
            chain.doFilter(request, response);
            return;
        }

        //是否登陆
        if(null == session.getAttribute("user")){
            response.sendRedirect(redirectPath);
            return ;
        }else{
            chain.doFilter(req,resp);
            return;
        }

    }

    public void init(FilterConfig config) throws ServletException {
        this.config = config;
        System.out.println("init Filter");
    }

    private  static boolean isContains(String container, String[] regx) {
        boolean result = false;

        for (int i = 0; i < regx.length; i++) {
            if (container.indexOf(regx[i]) != -1) {
                return true;
            }
        }
        return result;
    }
}

redpajama_set_name: RedPajamaGithub


Film Screening at the Guild!
We're really excited to share that in addition to the already packed Filmmakers Guild schedule, Guild attendees will get to be a part of an advance screening of the Erwin Brothers' new film I Can Only Imagine!
This special screening will take place on the evening of Wednesday, March 14th and is for Filmmakers Guild attendees only. Seating is limited on a first-come, first-served basis until the deadline on February 15th or until the theater is filled. So, make sure you get your Filmmakers Guild tickets soon!
Guild tickets that have already been purchased are automatically included in this event and Guild tickets purchased before the deadline or until seating capacity is met will be included. In other words, no additional tickets are required.
We are looking forward to all that the Lord is going to do this year at the Guild and the Festival! We hope to see you there!
This is great, but how are you able to show the movie, I can only imagine, when it releases in theaters that week?
The movie is wonderful. Just get your friends together and tissues and just go.


text:
Дионисиос Закитинос (, , о-в Кефалиния, Греция — , ) — греческий -византист, профессор.
Член Афинской академии (1966, её президент в 1974 году).
Членкор Британской академии (1976).
Иностранный член французской Академии надписей и изящной словесности (1981).

Окончил Афинский университет по философии (1927). Докторскую степень по византийской истории получил в Сорбонне под руководством Шарля Диля.

В 1937—46 годах директор Госархива Греции.
В 1939—1970 годах профессор Афинского университета.
В 1951—1965 годах профессор университета «Пантеон» (Афины).

Последователь школы Анналов.

Краткое время в 1963—1964 годах был министром во временном правительстве Иоанниса Параскевопулоса, а после падения диктатуры «чёрных полковников» — членом греческого парламента от «Новой демократии» в 1974—1977.

Примечания

Ссылки
 Zakythinòs, Dionỳsios в Итальянской энциклопедии
 Некролог в The Independent

Историки Греции
Выпускники философского факультета Афинского университета
Византинисты
Члены Афинской академии
Члены-корреспонденты Британской академии
Члены французской Академии надписей и изящной словесности
Министры Греции
Депутаты парламента Греции
Члены партии «Новая демократия» (Греция)

redpajama_set_name: RedPajamaWikipedia


Ukulele Underground Forums > Ukulele Boards > Uke Talk > Any Uke related happenings in Tampa, 2-7 to 2-9 ?
View Full Version : Any Uke related happenings in Tampa, 2-7 to 2-9 ?
I'm accompanying my wife to Tampa this weekend. Any uke related happenings? how bout stores with some nice ukes?
thanks Brian. I'll try to make the jam.


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The Cabaret Girl is a musical comedy in three acts with music by Jerome Kern and book and lyrics by George Grossmith, Jr. and P. G. Wodehouse. It was produced by Grossmith and J. A. E. Malone at the Winter Garden Theatre in London's West End in September 1922 and featured Dorothy Dickson, Grossmith, Geoffrey Gwyther, and Norman Griffin (later replaced by Leslie Henson) in the leading roles.

The first performance was originally scheduled for Thursday, 14 September 1922, with Henson in a leading role, but he fell ill on the morning of the scheduled opening, which was delayed to allow Griffin to prepare for the part.  The show finally opened the following Tuesday, 19 September. According to the reviewer in The Times, "Last night the piece received the warmest of receptions and thoroughly deserved it." The production ran for 361 performances, closing on 11 August 1923. Henson took over from Griffin in January 1924 and the latter then took the show on tour.

The Cabaret Girl was first given an American production in 2004 when San Francisco's 42nd Street Moon company produced a staged concert of the show.  Its first full American production was in 2008, when the Ohio Light Opera gave seven performances, between 26 June and 8 August, as part of their 30th Anniversary season. The same company released a commercial recording of the work in 2009 on Albany Records. The recording is the earliest work composed by Kern to be restored and recorded in its original form. The first New York City production was given in March 2009, in a concert staging by the semi-professional troupe Musicals Tonight!

Background
Actor-manager George Grossmith, Jr. and his partner Edward Laurillard bought the New Middlesex Theatre in London's West End in 1919 and, after its refurbishment, re-opened it as the Winter Garden Theatre.  The first production was Kissing Time, written by P. G. Wodehouse and Guy Bolton and starring Grossmith and Leslie Henson.  After Grossmith's partnership with Laurillard broke up two years later, Grossmith retained control of the Winter Garden where, between 1921 and 1926, in partnership with Pat Malone, he produced a series of shows, many of which were adaptations of imported shows and featured Henson. The first production by the Grossmith-Malone partnership was Sally, with music by Jerome Kern, a book by Bolton and some of the lyrics by Wodehouse, which was the London transfer of a Broadway hit. The second, with an original book, was The Cabaret Girl.

Kern and Wodehouse had both worked with Grossmith early in their careers and had, together with Bolton, created an innovative series of musicals for the Princess Theatre on Broadway. In his 1933 autobiography, Grossmith described how he, Wodehouse and Kern managed their collaboration. While the two writers travelled to New York, drafting the lyrics of the ensemble numbers and finales on the boat, Kern was already at work at his home in Bronxville, New York, composing the melodies. When the trio gathered at Bronxville, Kern began setting the completed lyrics to music, while Grossmith and Wodehouse prepared "dummy" lyrics for Kern's melodies, the actual lyrics being completed on the return voyage. The trio worked from piano or "fiddle" copies of the music, leaving Kern to follow them to London with the completed orchestration a few weeks later.

Synopsis
The hero, James ("Jim") Paradene is the nephew of the Marchioness of Harrogate. He has been left a small fortune by his father, on condition that he must marry a lady who meets with the approval of the Marchioness and her son, the Marquis of Harrogate. Unfortunately, Jim wishes to marry Marilynn Morgan, but his trustees disapprove of her because she is a chorus girl.

Act 1:  The Showroom of Messrs Gripps & Gravvins, Music Publishers, Bond Street

Jim comes to the offices of Gripps and Gravins looking for a song to sing at his local village concert. When Marilynn also arrives, to audition for a cabaret that Gripps and Gravvins are producing, Jim tries to persuade her to give up her career and settle with him in the country, but she refuses and suggests that they should part. Jim, however, has an idea: if he and Marilynn pretend to be married, his trustees will no longer be able to withhold their approval. Gravvins has "a little place in the country", "The Pergola" at Woollam Chersey, Hertfordshire, and invites the young couple to visit it.

Act 2: "The Pergola", Woollam Chersey

Jim and Marilynn arrive at "The Pergola" in the guise of a honeymoon couple. The plan is that Gravvins will invite the local aristocracy to a garden party, to meet the honeymoon couple, with the intention that the Marchioness will be impressed with Marilynn's social standing. But all the notables of the district are away on holiday, so the members of the Gripps & Gravvins cabaret troupe are enlisted to impersonate them. Gravvins himself takes the part of the local vicar, but the plot is unmasked when the real vicar appears. Marilynn, thoroughly embarrassed, admits her part in the deception and announces that she will have nothing more to do with Mr James Parradine, before fleeing the scene.

Act 3: "All Night Follies" at The Cabaret

Marilynn is performing in the Gripps & Gravvins production, "All Night Follies", at The Cabaret, where Jim comes looking for her. He has realised that he cannot expect Marilynn to give up the bright city lights and is prepared to go along with her wishes if she will agree to marry him. The curtain falls before the Marchioness and her son have given their approval, but as she has expressed admiration for Marilynn and he has fallen for the charms of Lily de Jigger, another member of the cast, a happy ending seems probable.

Cast
The original cast, in order of appearance, was:

 Wodehouse reused some of the names from The Cabaret Girl in his later works: "Flick" and Mr Paradene are characters in Bill the Conqueror (1924); in "Jeeves and the Impending Doom" (1926), Bertie Wooster's Aunt Agatha has a country estate at Woollam Chersey; so, too, does the hero of his play Good Morning, Bill (1928), Bill Paradene.

Musical numbers

Act 1
 Overture
 Opening chorus—Chopin ad lib - chorus and girls
 You want the best seats, we have 'em - Effie and girls
 Mr Gravvins—Mr Gripps - Gravvins and Gripps
 First rose of summer - Girls, Gripps and Jim
 Journey's end - Jim and Marilynn
 Whoop-de-oodle-do! - Gravvins and cabaret troupe
 Dancing time - Quibb, Miss Simmons, Gravvins and Marilynn
 My little place in the country - Gravvins and Gripps

Act 2
 The Pergola patrol - company
 Praise for our zeal - porters, Marilynn and Jim
 Shimmy with me - Marilynn
 Those days are gone forever - Gravvins
 Looking all over for you - Jim and Marilynn
 Nerves - Gravvins, Gripps and Ada
 Finale—Act 2 - Gravvins, Gripps, Jim, Marilynn, Vicar, chorus
Act 3
 Opening: Good evening - Jim and Ada
 London, dear old London - Jim and Burthen (George Grossmith)
 Kahlua - Marilynn
 Finale—Act 3: Oriental dreams - ensemble
 Dancing time - Marilynn, Jim and chorus (an alternative has this sung in Act 1, by Marilynn and Gravvins)

Many of the numbers in The Cabaret Girl derived from, or were later modified for, other works:
 "Chopin ad lib" and "First rose of summer" were adapted from Kern's 1919 Broadway show She's a Good Fellow, the book and lyrics of which were written by Anne Caldwell.
 The duet "Mr Gravvins – Mr Gripps" was a pastiche of a hit song, "Mister Gallagher and Mister Shean", from the 1922 Ziegfeld Follies.
 Kern revised the music of "Journey's end" when he included it in his 1925 Broadway show, The City Chap.
 "The Pergola patrol" reappeared, shortened and with some lyrics changed, as "Is this not a lovely spot?" in Sitting Pretty.
 "Shimmy with Me" introduced London's theatregoers to a new American dance craze, the "Shimmy". It, too, was reused in The City Chap, as "He Is the Type".
 Wodehouse had used a very similar version of "Nerves" in See You Later (1918).
 Kern borrowed "Oriental dreams" for Sweet Adeline.

Notes

References

1922 musicals
Musicals by Jerome Kern
Musicals by P. G. Wodehouse
Musicals set in the Roaring Twenties


text:
Ray White Whitsunday
CloseWhitsunday
Together with Ray White Marine, the leading prestige boat brokerage in Australia, we manage both boat and marina berth sales.
The east coast of Australia and the Whitsundays offers some of the world's best boating geography and attracts a vast range of Super Yachts, Cruisers and Sports Yachts, making for great viewing and buying opportunities.
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South east trade winds fan the Queensland coast from March – April and again in September – October, providing perfect conditions for sailing and cruising.
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Kameruka is know as the Whitsunday's best performing local boat (according to Whitsunday Times) and is known for winning the Cruising Spinnaker division in the Airlie Beach Race Week in 2014 and the Trailable Yachts division in the Hamilton Island Race Week in 2019.
You may have also seen Kameruka compete in the Whitsunday Sailing Club's Twilight Racing on Wednesday afternoons, where they race around the bay off Airlie Beach.
Kameruka includes large sail inventory, two trailers and is in excellent condition. Features also include, double bed, 2 x single beds, V berth, stove & toilet.
Priced at $43,000, contact Mark Beale on 0404498790 or mark.beale@raywhite.com for more details today.
BUSINESS FOR SALE – APOLLO ADVENTURE SAILING – WHITSUNDAY
67ft Maxi Yacht Apollo is one of Australia's racing legends. After competing in races all over the world and winning the famous Sydney-Hobart Race she was converted into the ultimate Whitsunday Sailing Adventure (Sail & Dive). Located in Airlie Beach, Apollo is the Ultimate Whitsunday Experience.
The business operates two 3 day / 2 night trips per week with a lay day on Thursday.
Apollo carries 27 Guests, 3 Crew and is a favourite must do adventure for 18-35 year old's visiting the Whitsundays.
The sale of this business includes, Apollo Maxi Yacht, Great Barrier Reef Marine Park Authority (GBRMPA) permit, National parks and Wildlife Service (NPWS) permit, Website (Domain), Email Addresses, a large range of spare parts including extra mainsail, head sail, engine parts, tender and stores and scuba equipment.
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Need a place to store your new vessel?
View our marina berth opportunities available in the Whitsundays.
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\section{Introduction}
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Throughout this note, let $\mathbb C^{m\times n}$ and  ${\mathbb C}_{{\rm H}}^{m}$ denote the
collections of all $m\times n$ complex matrices and all $m\times m$ complex Hermitian matrices,
respectively; the symbols $A^{*}$, $r(A)$ and ${\mathscr R}(A)$ stand for the conjugate
transpose, the rank and the range (column space) of a matrix $A \in
\mathbb C^{m \times n}$, respectively; $I_m$ denotes the identity
matrix of order $m$; $[\, A, \, B\,]$ denotes a row block matrix
consisting of $A$ and $B$.  The Moore--Penrose inverse of a matrix
$A \in {\mathbb C}^{m \times n}$,
denoted by $A^{\dag}$, is defined to be the unique matrix $X \in
{\mathbb C}^{n \times m}$ satisfying the matrix equations
$$
{\rm (i)} \  AXA = A,  \ \ {\rm (ii)} \ XA X = X, \ \ {\rm (iii)} \
(AX)^{*} = AX, \ \  {\rm (iv)} \ (XA)^{*} =XA.
$$
Further, let $E_A = I_m - AA^{\dag}$ and $F_A = I_n - A^{\dag}A$,
both of which are orthogonal projectors and their ranks are given
by $r(E_A) = m - r(A)$ and $r(F_A) = n - r(A)$. A well-known
property of the Moore--Penrose inverse is $(A^{\dag})^{*} =
(A^{*})^{\dag}$. Hence, if $A = A^{*}$, then both $A^{\dag} =
(A^{\dag})^{*}$ and $AA^{\dag} =A^{\dag}A$ hold.
The inertia of a matrix $A \in {\mathbb C}_{{\rm H}}^{m}$
 is defined to be the triplet ${\rm In}(A)= \{\, i_{+}(A), \, i_{-}(A), \, i_{0}(A)\,\},$
where $i_{+}(A)$, $i_{-}(A)$ and $i_0(A)$ are the numbers of the
positive, negative and zero eigenvalues of $A$ counted with
multiplicities, respectively. For a matrix $A\in
{\mathbb C}_{{\rm H}}^{m}$, both $r(A) = i_{+}(A) + i_{-}(A)$ and
$i_0(A) = m - r(A)$ hold.

The definitions of two well-known partial orderings on  matrices of the same size are given below.

\begin{definition} \label{TS11}
{\rm
\begin{enumerate}
\item[{\rm(a)}]  Two matrices $A, \, B \in \mathbb C^{m \times n}$ are said to satisfy the minus
partial ordering,  denoted by
 $B\leqslant^{-} A$, iff the rank subtractivity equality $r(\, A - B\,) = r(A) - r(B)$ holds,  or
  equivalently, both ${\mathscr R}(A - B) \cap {\mathscr R}(B) = \{0\}$ and
   ${\mathscr R}(A^{*} - B^{*}) \cap {\mathscr R}(B^{*})
   = \{0\}$ hold.

\item[{\rm(b)}] Two matrices $A,  \, B \in {\mathbb C}_{{\rm H}}^{m}$ are said to satisfy the
 L\"owner partial ordering,  denoted by $B\leqslant^{{\footnotesize{\rm L}}} A$,
  iff the difference $A - B$ is nonnegative definite$,$ or equivalently, $A - B = UU^{*}$ for some matrix $U.$
\end{enumerate}
}
\end{definition}

In this note, we consider the following two matrix inequalities
\begin{align}
& BXB^{*} \leqslant^{-} A,
\label{11}
\\
& BXB^{*} \leqslant^{{\footnotesize{\rm L}}} A
\label{12}
\end{align}
 induced from the minus and L\"owner partial orderings, and examine the relations of their solutions,
  where $A \in {\mathbb C}_{{\rm H}}^{m}$ and $B \in \mathbb C^{m \times n}$ are given, and
$X \in {\mathbb C}_{{\rm H}}^{n}$ is unknown. This consideration is motivated by some recent work
on  rank and inertia optimizations of $A -  BXB^{*}$ in \cite{LT,T-laa10,TL}. We shall derive general solutions
of (\ref{11}) and (\ref{12}) by using the given matrices and their generalized inverses,
and then discuss some algebraic properties of these solutions. In particular, we give solutions
of the following constrained rank and L\"owner partial ordering optimization problems
\begin{align}
& \max_{BXB^{*} \leqslant^{-} A} r(BXB^{*}), \ \ \ \ \ \ \ \ \
\min_{BXB^{*} \leqslant^{-} A}  r(\, A -  BXB^{*} \,),
\label{13}
\end{align}\\[-11mm]
\begin{align}
& \max_{\leqslant^{{\footnotesize{\rm L}}}} \{\, BXB^{*} \,| \, BXB^{*} \leqslant^{{\footnotesize{\rm L}}} A \,\},
\ \ \ \ \ \ \ \min_{\leqslant^{{\footnotesize{\rm L}}}} \{\, A - BXB^{*} \,| \, BXB^{*} \leqslant^{{\footnotesize{\rm L}}} A \,\}.
\label{14}
\end{align}
Eqs.\,(\ref{11}) and (\ref{12}) are equivalent to determining elements in the following matrix sets:
 \begin{align}
& {\cal S}_1 = \{\, Z  \in {\mathbb C}_{{\rm H}}^{m} \ | \  Z \leqslant^{-} A, \ \ \R(Z) \subseteq \R(B) \, \},
\label{15}
\\
&  {\cal S}_2 = \{\, Z  \in {\mathbb C}_{{\rm H}}^{m} \ | \ Z \leqslant^{{\footnotesize{\rm L}}} A,
\ \ \R(Z) \subseteq \R(B) \,\}.
\label{16}
\end{align}
The matrices $Z$ in (\ref{15}) and (\ref{16}) can be regarded as two constrained approximations of the matrix
$A$ in partial orderings. In particular, a matrix $Z \in {\cal S}_1$
that has the maximal possible rank is called a shorted matrix of $A$
relative to $\R(B)$ in the minus partial ordering (see \cite{Mi,MP}); while the maximal matrix
in ${\cal S}_2$ is called a shorted matrix of $A$ relative to $\R(B)$ in the L\"owner partial
ordering (see \cite{An,AT}). Our approaches to (\ref{11})--(\ref{14}) link some previous and
recent work in  \cite{An,AT,Go,Gr,HS,Mi,MBS,MP}  on shorted matrices of $A$ relative to given subspaces
in partial orderings, and some recent work on the rank and inertia of the matrix function $A - BXB^{*}$
in \cite{LT,T-laa10,TL}. It is obvious that there always exists a matrix $X$ that satisfies (\ref{11}),
say, $X =0$. Hence,  what we need to do is to derive a general expression
of $X$ that satisfies (\ref{11}). Eq.\,(\ref{12}) may have no solutions unless
the given matrices  $A$ and $B$ in (\ref{12}) satisfy certain conditions.

This note is organized as follows. In Section 2, we present some known results on ranks and inertias
of matrices and matrix equations, and then solve two homogeneous matrix equations with
symmetric patterns. In Section 3, we use the results obtained in Section 2 to derive the
general solution of (\ref{11}), and give an analytical expression for the shorted matrix of
$A$ relative to $\R(B)$ in the minus  partial ordering. In Section 4,
we derive necessary and sufficient conditions for (\ref{12}) to have
a solution,  and then give the general solution of (\ref{12}). We show in Section 5 an interesting
fact that the shorted matrices of $A$ relative to $\R(B)$  in  the minus and L\"owner partial
orderings are the same.

\section{Preliminary results}
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In order to characterize matrix equalities that involve the Moore--Penrose inverses, we need the
following rank and inertia expansion formulas.

\begin{lemma}[\cite{MS}] \label{T21}
\, Let $A \in \mathbb C^{m \times n}, \, B \in \mathbb C^{m \times
k}$ and $C \in \mathbb C^{l \times n}$ be given$.$  Then$,$ the following
rank expansion formulas hold
\begin{align}
r[\, A, \, B \,] & = r(A) + r(E_AB)  = r(B) + r(E_BA),
\label{21}
\\
r\!\left[\!\begin{array}{c} A \\ C \end{array} \!\right] & = r(A) +
r(CF_A) = r(C) + r(AF_C),
\label{22}
\\
 r\!\left[\!\begin{array}{cccc} A & B
\\
C & CA^{\dag}B
\end{array}\!\right] & =  r\!\left[\!\begin{array}{c} A \\ C \end{array} \!\right] +
r[\, A, \, B \,] - r(A).
\label{qq23}
\end{align}
\end{lemma}

\begin{lemma} [\cite{T-laa10}] \label{T22}
Let $A \in {\mathbb C}_{{\rm H}}^{m},$ $B \in \mathbb C^{m\times n},$
 and $D \in  {\mathbb C}_{{\rm H}}^{n}.$ Then$,$ the following
inertia expansion formulas hold
\begin{align}
&  i_{\pm}\!\left[\!\!\begin{array}{cc}  A  & B  \\ B^{*}  & 0 \end{array}
\!\!\right] = r(B) + i_{\pm}(E_BAE_B),
\label{qq24}
\\
& i_{\pm}\left[\!\!\begin{array}{cc}  A  & B  \\ B^{*}  & D \end{array}
\!\!\right] = i_{\pm}(A) + i_{\pm}(\,  D - B^{*}A^{\dag}B \,)\ \ for \   {\mathscr R}(B) \subseteq {\mathscr R}(A).
\label{qq25}
\end{align}
\end{lemma}

In order to solve  (\ref{11}) and (\ref{12}), we also need the following results on
solvability conditions and general solutions of two simple linear matrix equations.

\begin{lemma}  \label{T23}
Let $A\in \mathbb C^{m\times n}$  and $B\in {\mathbb C}^{m\times p}$ be given$.$ Then$,$ the following hold$.$
\begin{enumerate}
\item[{\rm(a)}] {\rm \cite{Pen}} The matrix equation $AX = B$ is consistent
if and only if ${\mathscr R}(B) \subseteq {\mathscr R}(A).$  In this case$,$ the general
solution can be written as $X = A^{\dag}B + F_AU,$ where $U \in {\mathbb C}^{n\times p}$ is
arbitrary$.$

\item[{\rm(b)}] {\rm \cite{KM}}  Under $B\in \mathbb C^{m\times n},$ the matrix equation $AX = B$
has a solution  $0\leqslant^{{\footnotesize{\rm L}}} X \in {\mathbb C}_{{\rm H}}^{n}$ if and
only if ${\mathscr R}(B) \subseteq {\mathscr R}(A),$ $AB^{*} \geqslant^{{\footnotesize{\rm L}}} 0$
and $r(AB^{*}) =r(B).$ In this case$,$ the
general nonnegative definite solution can be written as
\begin{equation}
X = B^{*}(AB^{*})^{\dag}B   + F_AUF_A,
 \label{26}
\end{equation}
where $0 \leqslant^{{\footnotesize{\rm L}}}  U\in {\mathbb C}_{{\rm H}}^{n}$ is arbitrary$.$
\end{enumerate}
\end{lemma}

\begin{lemma} \label{T24}
Let $A \in \mathbb C^{m \times n}$ and $B \in \mathbb C_{{\rm
H}}^{m}$ be given$.$ Then$,$  the following hold$.$
\begin{enumerate}
\item[{\rm(a)}] {\rm \cite{Gr}} The matrix equation
\begin{equation}
AXA^{*} = B
\label{27}
\end{equation}
has a solution $X \in \mathbb C_{{\rm H}}^{n}$ if and only if
${\mathscr R}(B) \subseteq {\mathscr R}(A),$ or equivalently$,$
$AA^{\dag}B = B.$

\item[{\rm(b)}] {\rm \cite{T-laa10}} Under $X \in \mathbb C_{{\rm H}}^{n},$ the general Hermitian solution of
 {\rm (\ref{27})} can be written in the following two forms
\begin{align}
X & = A^{\dag}B(A^{\dag})^{*} +  U- A^{\dag}A UA^{\dag}A,
\label{28}
\\
X & = A^{\dag}B(A^{\dag})^{*} +  F_AV + V^{*}F_A,
\label{29}
\end{align}
respectively$,$ where $U \in \mathbb C_{{\rm H}}^{n}$ and $V \in {\mathbb C}^{n \times n}$ are arbitrary$.$
\end{enumerate}
\end{lemma}

\begin{lemma}\label{T25}
Let $P \in \mathbb C^{m \times n}$ and $Q \in \mathbb C^{m \times k}$ be given$.$
Then$,$ the general solutions  $X \in {\mathbb C}_{{\rm H}}^{n}$ and
$Y \in {\mathbb C}_{{\rm H}}^{k}$ of the  matrix equation
\begin{align}
PXP^{*} = QYQ^{*}
 \label{210}
\end{align}
can be written as
\begin{align}
X = X_1WX^{*}_1 + X_2,  \ \ \ \ Y = Y_1WY^{*}_1 + Y_2,
 \label{211}
\end{align}
where $W  \in {\mathbb C}_{{\rm H}}^{m}$ is arbitrary$,$ and
$X_1 \in \mathbb C^{n \times m},$ $X_2 \in {\mathbb C}_{{\rm H}}^{n},$ $Y_1  \in \mathbb C^{k \times m}$
and $Y_2 \in {\mathbb C}_{{\rm H}}^{k}$ are the general solutions of the following
 matrix equations
\begin{align}
PX_1 = QY_1, \ \ \ PX_2P^{*} = 0, \ \ \  QY_2Q^{*} = 0,
\label{212}
\end{align}
or alternatively$,$ the general solution of {\rm (\ref{210})} can be written in the
following pair of parametric form
\begin{align}
X & = \widehat{I}_nF_HUF_{H}\widehat{I}^{*}_n
 + U_1 - P^{\dag}PU_1P^{\dag}P,
\label{213}
\\
Y & = \widetilde{I}_k F_HUF_{H}\widetilde{I}^{*}_k  + U_2 - Q^{\dag}QU_2Q^{\dag}Q,
\label{214}
\end{align}
where $H = [\, P, \, -Q \,],$  $\widehat{I}_n = [\, I_n, \, 0\,],$  $\widetilde{I}_k = [\, 0, \, I_k\,],$
and $U \in {\mathbb C}_{{\rm H}}^{n+k},$ $U_1 \in {\mathbb C}_{{\rm H}}^{n}$ and $U_2 \in {\mathbb C}_{{\rm H}}^{k}$
are arbitrary$.$
\end{lemma}

\noindent  {\bf Proof} \, It is easy to verify that the pair of matrices $X$ and $Y$ in (\ref{211})
 are both Hermitian. Substituting the pair of matrices into (\ref{210}) gives
$$
PXP^{*} = PX_1WX^{*}_1P^{*} = QY_1WY^{*}_1Q^{*} = QYQ^{*},
$$
which shows that (\ref{211}) satisfies (\ref{210}). Also, assume that
$X_0$ and $Y_0$ are any pair of solutions of (\ref{210}), and set
$$
W = (PX_0P^{*})^{\dag} = (QY_0Q^{*})^{\dag}, \ \ X_1 = P^{\dag}PX_0P^{*}, \ \
Y_1 = Q^{\dag}QY_0Q^{*},
$$
$$ X_2 = X_0 -  P^{\dag}PX_0P^{\dag}P, \ \ \ \  Y_2 = Y_0 -  Q^{\dag}QY_0Q^{\dag}Q.
$$
Then$,$ (\ref{211}) reduces to
\begin{align*}
X & = P^{\dag}PX_0P^{*}(PX_0P^{*})^{\dag}(P^{\dag}PX_0P^{*})^{*} + X_0 - P^{\dag}PX_0P^{\dag}P
\\
& =  P^{\dag}(PX_0P^{*})(PX_0P^{*})^{\dag}(PX_0P^{*})(P^{\dag})^{*} + X_0 - P^{\dag}PX_0P^{\dag}P
\\
& =  P^{\dag}PX_0P^{\dag}P + X_0 - P^{\dag}PX_0P^{\dag}P = X_0,
\\
Y&  = Q^{\dag}QY_0Q^{*}(QY_0Q^{*})^{\dag}(Q^{\dag}QY_0Q^{*})^{*} + Y_0 - Q^{\dag}QY_0Q^{\dag}Q
\\
& =  Q^{\dag}(QY_0Q^{*})(QY_0Q^{*})^{\dag}(QY_0Q^{*})(Q^{\dag})^{*}+ Y_0 - Q^{\dag}QY_0Q^{\dag}Q
\\
& =  Q^{\dag}QY_0Q^{\dag}Q + Y_0 - Q^{\dag}QY_0Q^{\dag}Q = Y_0,
\end{align*}
that is, any pair of solutions of (\ref{210}) can be represented by
(\ref{211}). Thus, (\ref{211}) is the general solution of (\ref{210}).

Solving the latter two equations in (\ref{212}) by Lemma  \ref{T24}(b) yields the following general solutions
\begin{align}
X_2 = U_1 - P^{\dag}PU_1P^{\dag}P,  \ \ \ \ \ Y_2 = U_2 -
Q^{\dag}QU_2Q^{\dag}Q,
\label{215}
\end{align}
where $U_1 \in {\mathbb C}_{{\rm H}}^{n}$ and $U_2\in {\mathbb C}_{{\rm H}}^{k}$ are arbitrary. To solve the first
equation in (\ref{212}),  we rewrite it as $[\,P, \, -Q\,]\!\left[\! \begin{array}{c} X_1  \\ Y_1
\end{array} \!\right]  =0$. Solving this equation by Lemma \ref{T23}(a) gives the general solution
$\left[\! \begin{array}{c} X_1  \\ Y_1 \end{array} \!\right] =
F_HV_1,$ where $V_1$ is an arbitrary matrix. Hence, the general
expressions of $X_1$ and $Y_1$ can be written as
\begin{align}
X_1 = \widehat{I}_n F_HV_1, \ \ Y_1 = \widetilde{I}_k F_HV_1.
\label{216}
\end{align}
Substituting (\ref{215}) and  (\ref{216}) into (\ref{211}) gives
(\ref{213}) and (\ref{214}). \qquad $\Box$

\begin{lemma}\label{T26}
 Let $B \in \mathbb C^{m \times n}$ and $A \in {\mathbb C}_{{\rm H}}^{m}$ be given$.$
  Then$,$ the general solution $X \in {\mathbb C}_{{\rm H}}^{n}$ of the quadratic matrix equation
\begin{align}
(BXB^{*})A(BXB^{*}) = BXB^{*}
\label{217}
\end{align}
can be expressed in the following parametric form
\begin{align}
X = U(U^{*}B^{*}ABU)^{\dag}U^{*} +  V - B^{\dag}BVB^{\dag}B,
\label{218}
\end{align}
where $U \in \mathbb C^{n \times n}$ and $V \in {\mathbb C}_{{\rm H}}^{n}$ are
arbitrary$.$
\end{lemma}

\noindent {\bf Proof} \, Substituting (\ref{218}) into $BXB^{*}$
gives $BXB^{*} = BU(U^{*}B^{*}ABU)^{\dag}U^{*}B^{*}$. It is easy to
verify by the definition of the Moore--Penrose inverse that
\begin{align*}
(BXB^{*})A(BXB^{*}) & = BU(U^{*}B^{*}ABU)^{\dag}U^{*}B^{*}ABU(U^{*}B^{*}ABU)^{\dag}U^{*}B^{*}
\\
& =BU(U^{*}B^{*}ABU)^{\dag}U^{*}B^{*}  = BXB^{*}.
\end{align*}
Hence, (\ref{218}) satisfies (\ref{217}). On the other hand, for
any Hermitian solution $X_0$ of (\ref{217}), set $U
=B^{\dag}BX_0B^{\dag}B$ and $V = X_0$ in (\ref{218}). Then$,$
(\ref{218}) reduces to
\begin{align*}
X & = B^{\dag}BX_0B^{\dag}B(B^{\dag}BX_0B^{*}ABX_0B^{\dag}B)^{\dag}B^{\dag}BX_0B^{\dag}B + X_0 - B^{\dag}BX_0B^{\dag}B
\\
& = B^{\dag}BX_0B^{\dag}B(B^{\dag}BX_0B^{\dag}B)^{\dag}B^{\dag}BX_0B^{\dag}B + X_0 - B^{\dag}BX_0B^{\dag}B
\\
& = B^{\dag}BX_0B^{\dag}B + X_0 - B^{\dag}BX_0B^{\dag}B = X_0.
\end{align*}
This result indicates that all solutions of (\ref{217}) can be
represented through (\ref{218}). Hence, (\ref{218}) is the general
solution of (\ref{217}). \qquad $\Box$

\section{General solution of $BXB^{*} \leqslant^{-} A$}
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A well-known necessary and sufficient condition for the rank subtractivity equality in Definition 1.1
to hold is
\begin{equation}
r(\, A - B\,) = r(A) - r(B) \Leftrightarrow  \R(B) \subseteq \R(A), \
 \R(B^{*}) \subseteq \R(A^{*})  \ \text{and}  \ BA^{\dag}B = B,
\label{31}
\end{equation}
see \cite{MS}. Applying (\ref{31}) to  (\ref{11}), we can convert (\ref{11}) to a system of matrix equations.

\begin{lemma}\label{T31}
Eq.~{\rm (\ref{11})} is equivalent to the following system of
matrix equations
\begin{align}
 BXB^{*} = AYA,  \ \  (BXB^{*})A^{\dag}(BXB^{*}) = BXB^{*},
 \label{32}
\end{align}
where $Y\in {\mathbb C}_{{\rm H}}^{m}$ is an unknown matrix$.$
\end{lemma}

\noindent {\bf Proof} \, From (\ref{31}), the minus partial order $BXB^{*} \leqslant^{-} A$ in (\ref{11}) is
equivalent to
\begin{align}
\R(BXB^{*})  \subseteq  \R(A) \ \  {\rm and} \ \   (BXB^{*})A^{\dag}(BXB^{*}) = BXB^{*}.
 \label{33}
\end{align}
By Lemma \ref{T24}(a), the first range inclusion in (\ref{33}) holds
if and only if the first matrix equation in  (\ref{32}) is solvable
for $Y$. Thus,  (\ref{32}) and  (\ref{33}) are
equivalent. \qquad $\Box$

\begin{theorem}\label{T32}
Let $A \in {\mathbb C}_{{\rm H}}^{m}$ and $B \in \mathbb C^{m\times n}$ be given$,$
and ${\cal S}_1$ be as given in {\rm (\ref{16})}$.$ Also define
$$
M = \left[\! \begin{array}{cc}  A  & B  \\ B^{*} & 0 \end{array}\!\right]\!, \ \
H = [\, B, \, -A \,], \ \
 \widehat{I}_n = [\, I_n, \, 0\,], \ \ \widehat{B} =[\, B, \, 0\,], \ \ A_1 = E_BA, \ \ B_1 = E_AB.
$$
Then$,$  the following hold$.$
\begin{enumerate}
\item[{\rm(a)}] The general Hermitian solution of the inequality
\begin{equation}
BXB^{*}  \leqslant^{-} A
\label{34}
\end{equation}
can be written as
\begin{align}
& X = \widehat{I}_nF_HU(U^{*}F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_HU)^{\dag}U^{*}F_{H}\widehat{I}^{*}_n
+ V - B^{\dag}BVB^{\dag}B,
\label{35}
\end{align}
where  $U\in {\mathbb C}^{(m +n)\times (m +n)}$ and $V \in {\mathbb C}_{{\rm H}}^{n}$  are arbitrary$.$

\item[{\rm(b)}]  The general expression of the matrices in {\rm (\ref{15})} can be written as
\begin{align}
 Z = \widehat{B}F_HU(U^{*}F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_HU)^{\dag}
U^{*}F_{H}\widehat{B}^{*},
\label{36}
\end{align}
where $U\in {\mathbb C}^{(m +n)\times (m +n)}$ is arbitrary$.$ The global maximal and minimal inertias and ranks
of $Z$ in {\rm (\ref{36})} and the corresponding $A -Z$  are given by
\begin{align}
& \max_{Z\in {\cal S}_1} i_{\pm}(Z)  =  i_{\mp}(M) + i_{\pm}(A) - r[\, A,\,  B \,],
\label{37}
\\
& \max_{Z\in {\cal S}_1} r(Z)  = r(M) + r(A) - 2r[\, A,\,  B \,],
\label{38}
\\
& \min_{Z\in {\cal S}_1}  i_{\pm}( \, A - Z \, ) = r[\, A,\,  B \,] -  i_{\mp}(M),
\label{39}
\\
& \min_{Z\in {\cal S}_1} r( \, A - Z \, ) = 2r[\, A,\,  B \,] - r(M).
\label{310}
\end{align}
The shorted matrix of $A$ relative to $\R(B),$ denoted by $\phi^{-}(\, A \,|\, B\,),$
which is a matrix $Z$ that satisfies {\rm (\ref{38})}$,$ is given by
\begin{align}
\phi^{-}(\, A \,|\, B\,) = \widehat{B}F_H(F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H)^{\dag}F_{H}\widehat{B}^{*}.
\label{311}
\end{align}
\end{enumerate}
\end{theorem}

\noindent {\bf Proof} \, Applying Lemma \ref{T25} to the first equation in (\ref{32}),
we obtain the general solutions of $X$ and $Y$ as follows
\begin{align}
X = \widehat{I}_nF_HTF_{H}\widehat{I}^{*}_n +  V - B^{\dag}BVB^{\dag}B, \ \
Y = \widehat{I}_mF_HTF_{H}\widehat{I}^{*}_m +  W - A^{\dag}AWA^{\dag}A,
\label{312}
\end{align}
where $T \in {\mathbb C}_{{\rm H}}^{m+n}$, $V  \in {\mathbb C}_{{\rm H}}^{n}$ and
$W \in {\mathbb C}_{{\rm H}}^{m}$ are arbitrary$.$ Substituting (\ref{312}) into the second equation
in (\ref{32}) leads to the following quadratic matrix equation
$$
(\widehat{B}F_HTF_{H}\widehat{B}^{*})A^{\dag}(\widehat{B}F_HTF_{H}\widehat{B}^{*})
= \widehat{B}F_HTF_{H}\widehat{B}^{*}.
$$
By Lemma \ref{T26}, the general solution of this quadratic matrix equation is given by
\begin{align*}
T& = U(U^{*}F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_HU)^{\dag}U^{*} +
 W_1 - (\widehat{B}F_H)^{\dag}(\widehat{B}F_H)W_1(\widehat{B}F_H)^{\dag}(\widehat{B}F_H),
\end{align*}
where $U\in \mathbb C^{(m+n) \times (m+n)}$ and $W_1 \in {\mathbb C}_{{\rm H}}^{m+n}$ are arbitrary.
Substituting this $T$ into  the matrix $X$ in (\ref{312}) gives
\begin{align}
X & =  \widehat{I}_nF_HU(U^{*}F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_HU)^{\dag}U^{*}F_{H}\widehat{I}^{*}_n
\nonumber
\\
&  \ \ + [\, \widehat{I}_nF_HW_1F_{H}\widehat{I}^{*}_n -
\widehat{I}_nF_H(\widehat{B}F_H)^{\dag}(\widehat{B}F_H)W_1(\widehat{B}F_H)^{\dag}(\widehat{B}F_H)F_{H}\widehat{I}^{*}_n \,]
  + V - B^{\dag}BVB^{\dag}B.
\label{313}
\end{align}
It is easy to verify from $B\widehat{I}_n = \widehat{B}$ that
\begin{align*}
& B[\, \widehat{I}_nF_HW_1F_{H}\widehat{I}^{*}_n -
\widehat{I}_nF_H(\widehat{B}F_H)^{\dag}(\, \widehat{B}F_H)W_1(\widehat{B}F_H \,)^{\dag}
(\widehat{B}F_H)F_{H}\widehat{I}^{*}_n\,]B^{*}
\\
& =\widehat{B}F_HW_1F_{H}\widehat{B}^{*} -
(\widehat{B}F_H)(\widehat{B}F_H)^{\dag}(\, \widehat{B}F_H)W_1(\widehat{B}F_H \,)^{\dag}
(\widehat{B}F_H)(\widehat{B}F_{H})^{*} =0.
\end{align*}
This fact shows that the second term on the right-hand side of (\ref{313}) is a solution to $BXB^{*} =0$. Also, note
 from Lemma \ref{T24}(b) that $V - B^{\dag}BVB^{\dag}B$ is the general solution to $BXB^{*} =0$.
 Hence, the second term on the right-hand side of (\ref{313}) can be represented by the third term of the same side,
 so that (\ref{313}) reduces to  (\ref{35}).

Substituting  (\ref{35}) into $BXB^{*}$
gives
\begin{align}
Z = BXB^{*} = \widehat{B}F_HU(\,U^{*}F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_HU\,)^{\dag}
U^{*}F_{H}\widehat{B}^{*},
\label{314}
\end{align}
as required for (\ref{36}). Note further that this $Z$ satisfies
\begin{align}
(U^{*}F_{H}\widehat{B}^{*}A^{\dag})Z(A^{\dag}\widehat{B}F_HU)  = U^{*}F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_HU.
\label{315}
\end{align}
Both  (\ref{314}) and  (\ref{315}) imply
\begin{align}
i_{\pm}(Z)= i_{\pm}(U^{*}F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_HU) \leqslant
i_{\pm}(\,F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H\,)
\label{316}
\end{align}
and
\begin{align}
\max_{Z\in {\cal S}_1}i_{\pm}(Z) = i_{\pm}(F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H), \ \ \
\max_{Z\in {\cal S}_1}r(Z) = r(F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H).
\label{317}
\end{align}
Recall that the inertia of a Hermitian matrix does not change under Hermitian congruence operations.
 Applying (\ref{qq24}) to $F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H$ and simplifying by
  Hermitian congruence operations, we obtain
\begin{align}
i_{\pm}(F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H) & =
i_{\pm}\!\left[\! \begin{array}{cc} \widehat{B}^{*}A^{\dag}\widehat{B} & H^{*}
\\
H &  0 \end{array} \!\right] - r(H) \nonumber
\\
& = i_{\pm}\!\left[\! \begin{array}{cccc} B^{*}A^{\dag}B & 0  & B^{*}
\\
0 & 0 & -A
\\
B & -A & 0  \end{array} \!\right] - r[\, A, \, B \,]  \nonumber
\\
& = i_{\pm}\!\left[\! \begin{array}{cccc} 0 & \frac{1}{2}B^{*}A^{\dag}A  & B^{*}
\\
\frac{1}{2}A^{\dag}A B  & 0 & -A
\\
B & -A & 0  \end{array} \!\right] - r[\, A, \, B \,] \nb
\\
&  = i_{\pm}\!\left[\! \begin{array}{cccc} 0 & 0  & B^{*}
\\
0 & A & -A
\\
B & -A & 0  \end{array} \!\right] - r[\, A, \, B \,] \nonumber
\\
& = i_{\pm}\!\left[\! \begin{array}{cccc} 0 & 0  & B^{*}
\\
0 & A & 0
\\
B & 0 & -A  \end{array} \!\right] - r[\, A, \, B \,] \nonumber
\\
& = i_{\mp}\!\left[\! \begin{array}{cccc} A  & B
\\
B^{*} & 0  \end{array} \!\right] + i_{\pm}(A)  - r[\, A, \, B \,].
\label{318}
\end{align}
Substituting  (\ref{318}) into  (\ref{317}) leads to  (\ref{37}) and (\ref{38}).  Also, note that
$$
\min_{X \in {\cal S}_1} i_{\pm}(\, A - Z\,)
= i_{\pm}(A) - \max_{X \in {\cal S}_1} i_{\pm}(Z).
$$
Thus, (\ref{39}) and  (\ref{310}) follow from  (\ref{37}) and (\ref{38}).  \qquad $\Box$

\section{General solution of $BXB^{*} \leqslant^{{\footnotesize{\rm L}}} A$}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{section}{4}
\setcounter{equation}{0}

In this section, we derive an analytical expression for the general solution of (\ref{12}) by
using generalized inverses of matrices, and show some algebraic properties of the solution.

\begin{theorem} \label{T41}
Let $A \in {\mathbb C}_{{\rm H}}^{m}$ and $B \in {\mathbb C}^{m\times n}$ be given$,$
  and let ${\cal S}_2$ be as given in {\rm (\ref{16})}$.$
Then$,$  the following hold$.$
\begin{enumerate}
\item[{\rm(a)}]  There exists an $X \in {\mathbb C}^{n}_{{\rm H}}$ such that
\begin{equation}
BXB^{*}  \leqslant^{{\footnotesize{\rm L}}} A
\label{41}
\end{equation}
if and only if
\begin{equation}
E_BAE_B \geqslant^{{\footnotesize{\rm L}}} 0 \ \ \ and  \ \ \  r(E_BAE_B)= r(E_BA),
\label{42}
\end{equation}
or equivalently$,$
\begin{equation}
i_{+}\!\left[\!\! \begin{array}{cc} A & B \\ B^{*} & 0 \end{array}\!\!\right] = r[\, A, \, B\,] \ \ \
and  \ \ \ i_{-}\!\left[\!\! \begin{array}{cc} A & B \\ B^{*} & 0 \end{array}\!\!\right] = r(B).
\label{43}
\end{equation}
 In this case$,$ the general Hermitian solution of {\rm (\ref{41})} can be written in
 the following parametric form
\begin{align}
& X  =  B^{\dag}A(B^{\dag})^{*} - B^{\dag}AE_B(E_BAE_B)^{\dag}E_BA(B^{\dag})^{*} - UU^{*}
 + F_BV + V^{*}F_B,
\label{44}
\end{align}
where $U, \ V \in {\mathbb C}^{n \times n}$ are arbitrary$.$  Correspondingly$,$  the general expression of
the matrices in ${\cal S}_2$ can be written as
\begin{align}
Z  =  A - AE_B(E_BAE_B)^{\dag}E_BA  - BUU^{*}B^{*}.
\label{45}
\end{align}

\item[{\rm(b)}] Under {\rm (\ref{42})}$,$ the shorted matrix of $A$ relative to $\R(B),$ denoted
by $\phi^{{\rm L}}(\, A \,|\, B\,),$ which is the maximizer in ${\cal S}_2,$
can uniquely be written as
\begin{align}
\phi^{{\rm L}}(\, A \,|\, B\,) = A - AE_B(E_BAE_B)^{\dag}E_BA.
\label{46}
\end{align}
The rank and inertia of $\phi^{{\rm L}}(\, A \,|\, B\,)$ and $A - \phi^{{\rm L}}(\, A \,|\, B\,)$ satisfy
\begin{align}
& i_{+}[\,\phi^{{\rm L}}(\, A \,|\, B\,) \,] = i_{+}(A) + r(B) - r[\, A, \, B \,], \ \
\label{47}
\\
& i_{-}[\,\phi^{{\rm L}}(\, A \,|\, B\,) \,]   = i_{-}(A),
\label{8}
\\
& i_{+}[\, A - \phi^{{\rm L}}(\, A \,|\, B\,) \,]  =
 r[\, A - \phi^{{\rm L}}(\, A \,|\, B\,) \,]  = r[\, A, \, B\,] - r(B).
\label{49}
\end{align}
\end{enumerate}
\end{theorem}

\noindent {\bf Proof} \, It is obvious that (\ref{41}) is equivalent to
\begin{equation}
BXB^{*} = A - YY^{*}
\label{410}
\end{equation}
for some matrix $Y$.  In other words, (\ref{41}) can be relaxed to a matrix equation with two
unknown matrices. From Lemma \ref{T24}(a), (\ref{410}) is solvable for $X \in {\mathbb C}^{n}_{{\rm H}}$
 if and only if $E_B(\,A - YY^{*} \,) = 0,$ that is,
\begin{equation}
E_BYY^{*} = E_BA.
\label{411}
\end{equation}
From Lemma \ref{T23}(b), (\ref{411}) is solvable for $YY^{*}$ if and only if
$E_BAE_B \geqslant^{{\footnotesize{\rm L}}} 0$
and $r(E_BAE_B)= r(E_BA)$, establishing (\ref{42}), which is further equivalent to (\ref{43})
by (\ref{21}) and (\ref{qq24}). In this case, the general nonnegative definite solution of
(\ref{411}) can be written as
\begin{equation}
YY^{*} =  AE_B(E_BAE_B)^{\dag}E_BA +  BB^{\dag}WBB^{\dag},
\label{412}
\end{equation}
where $0 \leqslant^{{\footnotesize{\rm L}}} W \in {\mathbb C}_{{\rm H}}^{m}$ is
arbitrary. Substituting the $YY^{*}$ into (\ref{410}) gives
\begin{equation}
BXB^{*} = A -AE_B(E_BAE_B)^{\dag}E_BA -  BB^{\dag}WBB^{\dag}.
\label{413}
\end{equation}
By Lemma \ref{T24}(b), the general Hermitian solution of (\ref{413}) can be written as
\begin{equation}
X = B^{\dag}A(B^{\dag})^{*} - B^{\dag}AE_B(E_BAE_B)^{\dag}E_BA(B^{\dag})^{*} - B^{\dag}W(B^{\dag})^{*}
 + F_BV + V^{*}F_B,
\label{414}
\end{equation}
where $V \in \mathbb C^{n \times n}$ is arbitrary$.$ Replacing the matrix
$0 \leqslant^{{\footnotesize{\rm L}}} B^{\dag}W(B^{\dag})^{*} \in {\mathbb C}_{{\rm H}}^{n}$ in (\ref{414})
 with a general matrix $0 \leqslant^{{\footnotesize{\rm L}}} U \in {\mathbb C}_{{\rm H}}^{n}$ yields
 (\ref{44}), which is also the general Hermitian solution of (\ref{41}). Substituting (\ref{44}) into
 $BXB^{*}$ gives (\ref{45}).

 Eq.\,(\ref{46})  follows from  (\ref{45}) by noticing $BUU^*B^{*} \geqslant^{{\footnotesize{\rm L}}} 0$.

It follows from (\ref{42}) that $\R(E_BAE_B)  = \R(E_BA)$. In this case, applying (\ref{qq25}) to (\ref{46})
and simplifying  by Hermitian congruence transformations, we obtain
\begin{align*}
i_{\pm}[\,\phi^{{\rm L}}(\, A \,|\, B\,)\,] & =
i_{\pm}[\,  A - AE_B(E_BAE_B)^{\dag}E_BA \,]
\\
& = i_{\pm}\!\left[\!\begin{array}{cc} E_BAE_B &  E_BA
\\
AE_B & A \end{array}\!\right] - i_{\pm}(E_BAE_B)
\\
& = i_{\pm}\!\left[\!\begin{array}{cc} 0 & 0 \\  0 & A \end{array}\!\right] - i_{\pm}(E_BAE_B) \nonumber
\\
& = i_{\pm}(A) - i_{\pm}(E_BAE_B),
\\
i_{\pm}[\, A - \phi^{{\rm L}}(\, A \,|\, B\,)\,]  &= i_{\pm}[\,AE_B(E_BAE_B)^{\dag}E_BA\,]
\\
& = i_{\pm}\!\left[\!\begin{array}{cc} - E_BAE_B &  E_BA
\\
AE_B & 0 \end{array}\!\right] - i_{\mp}(E_BAE_B)
\\
&  = i_{\pm}\!\left[\!\begin{array}{cc} 0 &  E_BA
\\
AE_B & 0 \end{array}\!\right] - i_{\mp}(E_BAE_B)
\\
&  = r(E_BA) - i_{\mp}(E_BAE_B).
\end{align*}
Hence, we further find from that (\ref{21}) and (\ref{42}) that
\begin{align*}
 i_{+}[\phi^{{\rm L}}(\, A \,|\, B\,)] &  = i_{+}(A) -  i_{+}(E_BAE_B) = i_{+}(A) -  r(E_BA)  = i_{+}(A) + r(B) - r[\, A, \, B \,],
 \\
i_{-}[\,\phi^{{\rm L}}(\, A \,|\, B\,)\,] &  = i_{-}(A) -i_{-}(E_BAE_B)  = i_{-}(A),
\\
i_{+}[\,A - \phi^{{\rm L}}(\, A \,|\, B\,)\,] &  = r(E_BA) - i_{-}(E_BAE_B) =  r(E_BA)  =
 r[\, A, \, B \,] - r(B),
\\
i_{-}[\,A - \phi^{{\rm L}}(\, A \,|\, B\,)\,] &  = r(E_BA) - i_{+}(E_BAE_B) =0,
\end{align*}
establishing (\ref{47})--(\ref{49}).
\qquad $\Box$

\section{An equality for the shorted matrices of $A$ relative to $\R(B)$ in the minus and L\"owner partial orderings}

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{section}{5}
\setcounter{equation}{0}

Since ${\cal S}_1$ and ${\cal S}_2$ in  (\ref{15}) and  (\ref{16}) are defined
from different matrix inequalities, the two sets are not necessarily the same, as demonstrated
in Theorems \ref{T32}(b) and \ref{T41}(a).  However, they may have some common matrices.
In this section, we show an interesting fact that the shorted matrices
of $A$ relative to $\R(B)$ in the minus and L\"owner partial orderings are the same.

\begin{theorem} \label{T51}
Let $A \in {\mathbb C}_{{\rm H}}^{m}$ and $B \in \mathbb C^{m\times n}$ be given$,$  and
${\cal S}_1$ and ${\cal S}_2$  be as given in {\rm (\ref{15})} and  {\rm (\ref{16})}$.$
If {\rm (\ref{41})} has a solution$,$ then the two shorted matrices in  ${\cal S}_1$ and ${\cal S}_2$
are the same, namely$,$
\begin{align}
\phi^{-}(\, A \,|\, B\,)  = \phi^{{\rm L}}(\, A \,|\, B\,).
\label{51}
\end{align}
\end{theorem}

\noindent {\bf Proof} \, Note from (\ref{311}) and (\ref{46}) that (\ref{51}) holds  if and only if
\begin{align}
 \widehat{B}F_H(F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H)^{\dag}F_{H}\widehat{B}^{*} =
  A - AE_B(E_BAE_B)^{\dag}E_BA.
\label{52}
\end{align}
It is easy to derive from (\ref{22}) that
\begin{align}
r(\,\widehat{B}F_H \,) = r\!\left[\!\begin{array}{c} B \\ H \end{array} \!\right] - r(H)
= r(A) + r(B) - r[\, A, \, B \,].
\label{53}
\end{align}
 Under (\ref{42}), (\ref{317}) reduces to
\begin{align}
r(\,F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H \,)  = r(M) + r(A) - 2r[\, A, \, B \,] =  r(A) + r(B) -
 r[\, A, \, B \,].
\label{54}
\end{align}
Both (\ref{53}) and (\ref{54}) imply that
$\R(\,F_{H}\widehat{B}^{*}\,) = \R(\,F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H \,).$  In this case,
applying (\ref{qq25}) to the difference of  both sides of (\ref{52}) and simplifying by elementary
matrix operations, we obtain
\begin{align*}
& r[\,   A - AE_B(E_BAE_B)^{\dag}E_BA  -
\widehat{B}F_H(F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H)^{\dag}F_{H}\widehat{B}^{*} \,]  \nb
\\
& = r\!\left[\!\begin{array}{cc} F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H
& F_{H}\widehat{B}^{*}  \\
\widehat{B}F_H &   A - AE_B(E_BAE_B)^{\dag}E_BA   \end{array}\!\right]
- r(F_{H}\widehat{B}^{*}A^{\dag}\widehat{B}F_H)
\\
& = r\!\left[\!\begin{array}{ccc} \widehat{B}^{*}A^{\dag}\widehat{B}
& \widehat{B}^{*}  & H^{*} \\
\widehat{B} &  A - AE_B(E_BAE_B)^{\dag}E_BA  & 0
\\
H & 0 & 0  \end{array}\!\right] - 2r(H) - r(A) - r(B) + r[\, A, \, B \,] \ \  \mbox{(by (\ref{qq24}))}
\\
& = r\!\left[\!\begin{array}{cccc} B^{*}A^{\dag}B & 0 & B^{*}  & B^{*}
\\
0  & 0  & 0 & -A
\\
B & 0& A - AE_B(E_BAE_B)^{\dag}E_BA  & 0
\\
B & -A & 0 & 0  \end{array}\!\right]  - r(A) - r(B) - r[\, A, \, B \,]
\\
& = r\!\left[\!\begin{array}{cccc} B^{*}A^{\dag}B & 0 & B^{*}  & 0
\\
0  & 0  & 0 & -A
\\
B & 0& A - AE_B(E_BAE_B)^{\dag}E_BA  & 0
\\
0 & -A & 0 & 0  \end{array}\!\right]  - r(A) - r(B) - r[\, A, \, B \,]
\\
& = r\!\left[\!\begin{array}{cccc} B^{*}A^{\dag}B & B^{*}
\\
B & A - AE_B(E_BAE_B)^{\dag}E_BA  \end{array}\!\right] + r(A) - r(B) - r[\, A, \, B \,]
\\
& = r\left(\!\left[\!\begin{array}{cccc} B^{*}A^{\dag}B & B^{*}
\\
B & A \end{array}\!\right] - \!\left[\!\begin{array}{cccc} 0 & 0
\\
0 &  AE_B(E_BAE_B)^{\dag}E_BA  \end{array}\!\right] \right) + r(A) - r(B) - r[\, A, \, B \,]
\\
& = r\!\left[\!\begin{array}{cccc} B^{*}A^{\dag}B & B^{*} & 0
\\
B & A  &  AE_B
\\
0 &  E_BA &  E_BAE_B
\end{array}\!\right]  - r(E_BAE_B)  + r(A) - r(B) - r[\, A, \, B \,] \ \  \mbox{(by (\ref{qq25}))}
\\
& = r\!\left[\!\begin{array}{cccc} B^{*}A^{\dag}B & B^{*} & 0
\\
B & A  & 0
\\
0 &  0 &  0
\end{array}\!\right] - r(E_BA)  +  r(A) - r(B) - r[\, A, \, B \,]
\\
& = r\!\left[\!\begin{array}{cccc} B^{*}A^{\dag}B & B^{*}
\\
B & A
\end{array}\!\right]  +  r(A) - 2r[\, A, \, B \,] = 0 \ \ \ \  \mbox{(by (\ref{qq23}))},
\end{align*}
which means that (\ref{52}) is an equality.  \qquad $\Box$

\medskip

The minus and L\"owner partial orderings in Definition 1.1 can accordingly be defined for linear operators
on a Hilbert space. Also, note that the results in this note are derived from some ordinary algebraic
operations of the given matrices and their Moore--Penrose inverses. Hence, it is no doubt that most of the
conclusions in this note can be extended to operator algebra, in which the Moore--Penrose
inverses of linear operators were defined.

redpajama_set_name: RedPajamaArXiv


text:
Castle Mound
A Scheduled Monument in Upper Slaughter, Gloucestershire
OS Grid: SP156232
Mapcode National: GBR 4Q6.87H
Mapcode Global: VHB1W.699V
Entry Name: Castle Mound
Scheduled Date: 16 June 1948
English Heritage Legacy ID: GC 179
County: Gloucestershire
Civil Parish: Upper Slaughter
Traditional County: Gloucestershire
Lieutenancy Area (Ceremonial County): Gloucestershire
Church of England Parish: Upper Slaughter St Peter
Church of England Diocese: Gloucester
Motte and bailey castle 160m ESE of Lower Farm.
This record was the subject of a minor enhancement on 10 July 2015. The record has been generated from an "old county number" (OCN) scheduling record. These are monuments that were not reviewed under the Monuments Protection Programme and are some of our oldest designation records.
The monument includes a motte and bailey castle situated on the summit of a low hill within a meander of the River Eye and in the settlement of Upper Slaughter. The motte survives as a circular flat topped mound of up to 22m in diameter and 2.7m high surrounded by a partially buried outer ditch which excavation showed to be up to 8m wide and 4m deep and to the north and west a 90m long scarp with a flat platform above the river indicating a bailey. Partial excavations in 1877 revealed a well and the footings for a building on the summit of the motte and 12th to 13th century pottery finds.
Motte and bailey castles are medieval fortifications introduced into Britain by the Normans. They comprised a large conical mound of earth or rubble, the motte, surmounted by a palisade and a stone or timber tower. In a majority of examples an embanked enclosure containing additional buildings, the bailey, adjoined the motte. Motte castles and motte-and-bailey castles acted as garrison forts during offensive military operations, as strongholds, and, in many cases, as aristocratic residences and as centres of local or royal administration. Built in towns, villages and open countryside, motte and bailey castles generally occupied strategic positions dominating their immediate locality and, as a result, are the most visually impressive monuments of the early post-Conquest period surviving in the modern landscape. Over 600 motte castles or motte-and-bailey castles are recorded nationally, with examples known from most regions. As one of a restricted range of recognised early post-Conquest monuments, they are particularly important for the study of Norman Britain and the development of the feudal system. Although many were occupied for only a short period of time, motte castles continued to be built and occupied from the 11th to the 13th centuries, after which they were superseded by other types of castle.
Despite partial early excavation the motte and bailey castle 160m ESE of Lower Farm survives well and will contain archaeological and environmental evidence relating to its construction, development, longevity, social and political significance, strategic importance, domestic arrangements, abandonment and overall landscape context.
PastScape 330390
Bowl barrow known as `Wagborough Bush round barrow' 100m south west of Manor Farm
Upper Slaughter
Chessels Roman site
Lower Slaughter
Bourton Bridge Roman settlement
New Court Ground Roman villa (S of new buildings)
Bowl barrow forming part of a round barrow cemetery and a long barrow, 500m north east of Eyford Hil
Lower Harford medieval settlement
Two bowl barrows 80m north of Arch Hollow
Four bowl barrows forming part of a round barrow cemetery, and a long barrow 550m NNE of Eyford Hill


text:
The Philippines »
Philippine Business News »
24th BSP-DepEd Annual Oratorical Contest
« DIGITAL PAYMENTS LEADERS SUMMIT | Rules on the issuance of long-term negotiable certificates of time deposit »
Author Topic: 24th BSP-DepEd Annual Oratorical Contest (Read 228 times)
mikeortegaligalig
Augustinian Abbey School, Division of Las Piñas, Wins First Place in 24th BSP-DepEd Annual Oratorical Contest
Press Release:: The Bangko Sentral ng Pilipinas (BSP), in collaboration with the Department of Education (DepEd), held the 24th BSP-DepEd Oratorical Contest, the culminating activity of the 30th National Statistics Month, last 29 October 2019 at the BSP Assembly Hall. Sixteen 10th Grade students from the National Capital Region (NCR) divisions showcased their oratorical skills and insights on the importance of the innovative use of statistics in advancing the country's development agenda, in line with the theme, "Data Innovation: Key to a Better Nation."
Of the sixteen finalists, Mr. Shawn Kelly B. Lizardo from the Augustinian Abbey School in Las Piñas won this year's first prize. He was awarded a cash prize of ₱25,000 and a trophy by BSP Monetary Board Member Felipe M. Medalla and DepEd Director John Arnold S. Siena. The oratorical coach of Mr. Lizardo was Ms. Izah Rizza T. San Francisco.
Mr. Paul Resty V. Arcega from Rizal High School in Pasig came in second, while Ms. Sofia Marie G. Marquito from Navotas National High School came in third in the competition.
The thirteen finalists in this year's oratorical contest were:
(1) Mr. Judgeon Haendel P. Boniel from Caloocan National Science and Technology High School, Caloocan;
(2) Ms. Merry Joy G. Tablanda from Benigno "Ninoy" S. Aquino High School, Makati;
(3) Ms. Dinelin Love T. Demdam from Malabon National High School, Malabon;
(4) Ms. lannah Chelsea L. Belonia from Mataas na Paaralang Neptali A. Gonzales, Mandaluyong;
(5) Ms. Chenelle Ashley G. Palileo from Lakan Dula High School, Manila;
(6) Mr. Julian Miguel G. Tecson from Marikina Heights High School, Marikina;
(7) Ms. Rona Joy P. Tomondong from Muntinlupa National High School, Muntinlupa;
( Ms. Sharlla Mae S. Cerilles from Parañaque National High School, Parañaque;
(9) Ms. Ezra Andrei A. Ezpañola from Pasay City West High School, Pasay;
(10) Ms. Althea Pableo from San Bartolome High School, Quezon City;
(11) Mr. Shirbert T. Villanueva from San Juan National High School, San Juan;
(12) Mr. Jumar D. Pata from Taguig Science High School, Taguig Pateros; and
(13) Ms. Ma. Loraine D. Labrador from Malanday National High School, Valenzuela.
The orators passed through different stages of elimination conducted by the DepEd in the 16 NCR divisions. Winners of the division level eliminations represented their respective divisions in the oratorical finals.
For more than two decades, the oratorical contest has been a platform for instilling greater awareness and appreciation of the crucial role of statistics in our country's development, especially among the youth.
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In honor of the Dog Days issue, the dogs of the Source weigh in on keeping up with their humans at the park, on the bike path and at the concert grounds.
Still, there are times when their love for us—and for checking out all the good stuff Central Oregon has to offer—gets in the way of us being the wild and free beasts our ancestors meant us to be. In honor of the Dog Days of Summer issue, and in the spirit of our humans' 20/40/60 monthly feature, we submit these accounts of what it's like to be dogs whose humans make us do human things, a LOT.
I have a flowing, glowing mane that handles -30 F temps, beefy muscles to pull sleds and can handle virtually anything thrown my way — even a Satanic cat that these hoomans make me live with. I mean, I'm named after "Cool Runnings," star, Sanka, who apparently was a badass Olympic Jamaican bobsledder... though I didn't quite understand the film.
But I'm just barely a teenager. I recently just learned that pooping on carpet is not the same thing as pooping on grass. Or that eating chocolate from the counter or these tiny snack things called "Advil" from the floor isn't allowed (I tried, anyway).
And c'mon, I've been bred historically by Siberians, aka, Russians, so I can handle some strenuous, kooky, off-the-wall stuff. Except vodka. I accidently just tasted it when the girl with the round face who sometimes I call "Mom" left out her "seltzer" water. It was NOT just seltzer. Trust.
But "Ya ne mogu poverit', chto eti lyudi nastol'ko glupy!" Oh, sorry, the Russian came out. It happens when I'm baffled.
Why am I so perplexed? These humans put my still-growing self onto a paddleboard at Elk Lake the other week, because they said it would be "fun." Do they KNOW how heavy this thick and luxurious fur is when wet? I could drown! Plus, my skin is PINK and susceptible to burning at these high altitudes. I shouldn't even be up there (OK, the brief stop at Devil's Lake to play in that white snuff....err, stuff, was pretty cool).
So I pulled a sly Soviet move. I jumped ship and nearly toppled "Mom" over as I swam over to "Uncle Chadd." I'm pretty sure I won't be stepping foot on those oversized surf-boards again. I mean, I thought dealing with Russians was cray-cray but...try a Polish-Canadian. She tops the cake.
Yoda content while cruising...though you couldn't tell.
As an 11-pound mini poodle/Jack Russell rescue who could only be described as a "grumpy old man in training," I admit: riding in a basket on the back of my human's bike is not exactly my ideal way to spend an evening—but with places like Crux practically inviting my kind to post up and sniff butts every evening, it happens a lot. The wind in my ample head hair is nice, and since riding in that basket, with a bungee cord holding me in, means I get to stay with my human, I can live with it. Sure, we may roll up to an outdoor concert where loud noises force me to hide my nose in my human's armpit, but it's better than being left alone at home.
What really chaps my dog behind, however, is being anthropomorphized into a human skater dude. I am a grumpy old man-dog in training, not a young punk who wants to sport a rainbow mohawk around town, practically inviting every dog lover around to pet me. Did you see me at Pride? Jeesh, people. Paws off! And get off my lawn!
Since this is my soapbox, I conclude with this: If you're going to fawn over my mohawk at the next outdoor event, at least have the decency to bring me a treat.
I don't know why these crazy ladies keep shoving me into t-shirts. I'm an old lady PomChi and I just want to be free to lounge around in front of the heater/air conditioner. The first time the black-haired one shoved my little sausage body into a shirt it was because I had a hot spot from scratching. Sometimes you just have an itch to scratch and she wouldn't let me! Back then it was winter time and I didn't mind quite so much, though going to the loo was a bit inconvenient.
Again, this week the black-haired one outfitted me with a t-shirt that belongs to the tiniest human. It says "Sleep Under the Stars." Why would I want to do that!? I prefer sleeping under the covers, as close to my mom's face as caninely possible. Sure, the black-haired one feeds me and scratches me behind the ears endlessly (just the way I prefer, I often force my little face into her hand while she's watching the big thing on the wall with moving pictures). But that doesn't mean I want to wear a toddler t-shirt.
I'm not going to say I don't love partaking in human activities. I love riding in the car in what mom and the black-haired one call my "fortress of solitude" (the space between the backseat and the window). I love eating anything the tiny one drops on the floor. But when the black-haired one busts out a t-shirt for me to sport, preventing me from living my gosh darn life the way I want to, I'm not into it. Sheesh, I wish my mom's roommate would dress up the cat. That young fella deserves some of the torture, too.
Look for the human version of the 20/40/60 feature, featuring three generations of Source staffers trying out a new activity, the third week of each month.
A Millennial, a Xennial and an old hippie endeavor to alt-commute to work. Who will log the most miles?


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Results for «pacific»:
Total: 2000 results - 0.05 seconds
Allergy Diagnostic 100%
Allergy Diagnostic Asia-Pacific Allergy Diagnostic Market 2017 Industry Trend and Forecast 2022 www.orbisresearch.com;
https://www.pdf-archive.com/2017/11/15/allergy-diagnostic/
Asia-Pacific CAD CAM Dental Systems 98%
Asia Pacific CAD CAM Dental Systems Asia-Pacific CAD/ CAM Dental Systems Market 2017 Industry Trend and Forecast 2022 www.orbisresearch.com;
https://www.pdf-archive.com/2017/11/15/asia-pacific-cad-cam-dental-systems/
Rickey Henderson Checklist 1000-1999 98%
Rickey Henderson Checklist 1000 1999 Rickey Henderson Checklist 1000-1999 YEAR SET CARD# 1998 Donruss Silver Press Proofs #118 Rickey Henderson 1998 Donruss Silver Press Proofs #236 Rickey Henderson 1998 Fleer Tradition #430 Rickey Henderson 1998 Fleer Tradition Vintage '63 #110 Rickey Henderson 1998 Fleer Tradition Vintage '63 Classic #110 Rickey Henderson 1998 Leaf #34 Rickey Henderson 1998 Leaf Fractal Diamond Axis #34 Rickey Henderson 1998 Leaf Fractal Foundations #34 Rickey Henderson 1998 Leaf Fractal Materials #34 Rickey Henderson NX 1998 Leaf Fractal Materials Die Cuts #34 Rickey Henderson NX 1998 Leaf Fractal Materials Z2 Axis #34 Rickey Henderson N 1998 Leaf Fractal Matrix #34 Rickey Henderson BX 1998 Leaf Fractal Matrix Die Cuts #34 Rickey Henderson BX 1998 Leaf Rookies and Stars #24 Rickey Henderson 1998 Leaf Rookies and Stars Longevity #24 Rickey Henderson 1998 Leaf Rookies and Stars Longevity Holographic #24 Rickey Henderson 1998 Leaf Rookies and Stars True Blue #24 Rickey Henderson 1998 Pacific #9 Rickey Henderson 1998 Pacific Invincible Team Checklists #1 R.Henderson/Erstad/Salmon 1998 Pacific Omega #177 Rickey Henderson 1998 Pacific Omega Red #177 Rickey Henderson 1998 Pacific Online #533 Rickey Henderson 1998 Pacific Online Red #533 Rickey Henderson 1998 Pacific Online Web Cards #533 Rickey Henderson 1998 Pacific Online Winners #533 Rickey Henderson 1998 Pacific Platinum Blue #9 Rickey Henderson 1998 Pacific Red Threatt #9 Rickey Henderson 1998 Pacific Silver #9 Rickey Henderson 1998 Paramount #80 Rickey Henderson 1998 Paramount Copper #80 Rickey Henderson 1998 Paramount Gold #80 Rickey Henderson 1998 Paramount Holographic Silver #80 Rickey Henderson 1998 Paramount Platinum Blue #80 Rickey Henderson 1998 Paramount Red #80 Rickey Henderson 1998 Pinnacle Inside Stand-Up Guys #21AB R.Hend/Molit/Ryno/Murray 1998 Pinnacle Inside Stand-Up Guys #21CD Molit/Ryno/Murr/Hend 1998 Pinnacle Inside Stand-Up Guys Samples #21AB R.Hend/Molitor/Ryno/Murray 1998 Pinnacle Inside Stand-Up Guys Samples #21CD Molitor/Ryno/Murray/R.Hend
https://www.pdf-archive.com/2017/04/01/rickey-henderson-checklist-1000-1999/
pacific palms holidays near elizabeth1304 96%
pacific palms holidays near elizabeth1304 pacific palms holidays near elizabeth Three hours north of Sydney and if you are heading down from far north New South Wales, the scenic township of Pacific Palms is situated just twenty minutes from Forster Tuncurry.
https://www.pdf-archive.com/2014/07/04/pacific-palms-holidays-near-elizabeth1304/
et16pge1971 flight dishwasher report final 95%
Jeff Beresini Pacific Gas and Electric Company Prepared By:
https://www.pdf-archive.com/2017/06/24/et16pge1971-flight-dishwasher-report-final/
Asia Pacific (APAC) Bioplastics Market.PDF 95%
Asia Pacific (APAC) Bioplastics Market Asia Pacific (APAC) Bioplastics Market – Forecasts from 2017 to 2022 DESCRIPTION Asia pacific (APAC) bioplastics market was valued at US$X.XX billion in 2016 and is projected to expand at a CAGR of XX.XX% over the forecast period to reach US$X.XX billion by 2022.
https://www.pdf-archive.com/2017/08/23/asia-pacific-apac-bioplastics-market/
Asia Pacific (APAC) Proximity Sensor Market.PDF 95%
Asia Pacific (APAC) Proximity Sensor Market Asia Pacific (APAC) Proximity Sensor Market – Forecasts from 2017 to 2022 DESCRIPTION Asia Pacific (APAC) proximity sensor market was valued at US$X.XX billion in 2016 and is projected to expand at a CAGR of XX.XX% over the forecast period to reach US$X.XX billion by 2022.
https://www.pdf-archive.com/2017/09/07/asia-pacific-apac-proximity-sensor-market/
Asia Pacific (APAC) Intelligent Transportation System Market.PDF 95%
Asia Pacific (APAC) Intelligent Transportation System Market Asia Pacific (APAC) Intelligent Transportation System Market – Forecasts from 2017 to 2022 DESCRIPTION Asia Pacific intelligent transportation system market was valued at US$X.XX billion in 2016 and is projected to expand at a CAGR of XX.XX% over the forecast period to reach US$X.XX billion by 2022.
https://www.pdf-archive.com/2017/09/11/asia-pacific-apac-intelligent-transportation-system-market/
Asia Pacific Coil Coatings Market.PDF 95%
Asia Pacific Coil Coatings Market Asia Pacific Coil Coatings Market – Forecasts from 2017 to 2022 DESCRIPTION Asia Pacific Coil coatings market was valued at US$X.XX billion in 2016 and is projected to expand at a CAGR of XX.XX% over the forecast period to reach US$X.XX billion by 2022.
https://www.pdf-archive.com/2017/09/12/asia-pacific-coil-coatings-market/
Asia Pacific Minimally Invasive Surgical Instruments Market.PDF 95%
Asia Pacific Minimally Invasive Surgical Instruments Market Asia Pacific Minimally Invasive Surgical Instruments Market - Forecasts from 2017 to 2022 DESCRIPTION The Asia Pacific Minimally Invasive Surgical Instruments Market was valued at US$X.XX billion in 2016 and is projected to expand at a CAGR of XX.XX% over the forecast period to reach US$X.XX billion by 2022.
https://www.pdf-archive.com/2017/12/21/asia-pacific-minimally-invasive-surgical-instruments-market/
Asia Pacific (APAC) Car Rental Market.PDF 95%
Asia Pacific (APAC) Car Rental Market Asia Pacific (APAC) Car Rental Market – Forecasts from 2017 to 2022 DESCRIPTION Asia Pacific (APAC) car rental market was valued at US$X.XX billion in 2016 and is projected to expand at a CAGR of XX.XX% over the forecast period to reach US$X.XX billion by 2022.
https://www.pdf-archive.com/2017/08/18/asia-pacific-apac-car-rental-market/
pacific palms accommodation for vacations1469 95%
pacific palms accommodation for vacations1469 pacific palms accommodation for vacations Been to Pacific Palms region in north coast New South Wales?
https://www.pdf-archive.com/2014/10/05/pacific-palms-accommodation-for-vacations1469/
Asia Pacific Advanced Ceramics Market.PDF 94%
Asia Pacific Advanced Ceramics Market Asia Pacific Advanced Ceramics Market – Forecasts from 2017 to 2022 DESCRIPTION Asia Pacific advanced ceramics market is projected to witness the growth at a CAGR of XX.XX% during the forecast period to reach a total market size of US$XX.XX billion by 2022, increasing from US$XX.XX billion in 2017.
https://www.pdf-archive.com/2018/01/09/asia-pacific-advanced-ceramics-market/
APAHE 2017 handout 94%
APAHE 2017 handout USE OF DATA VISUALIZATION TOOLS TO UNDERSTAND AND ADVOCATE FOR UNDERREPRESENTED ASIAN PACIFIC AMERICAN GROUPS Asian Pacific Americans in Higher Education (APAHE) Conference • Oakland, CA • April 7, 2017 TONY LOSONGCO, Academic Support Specialist, Learning Center YOSHIKO TAKAHASHI, Ph.D., Associate Professor, Department of Criminology BACKGROUND  Over 1 million Asian Pacific Islander (API) students in college  Overall, API young adults have high rates of college enrollment and graduation  In the past 20 years, national-level data have revealed differences within API o Differences between Asians and Pacific Islanders in college enrollment and graduation rates o Enrollment reporting for only Asian subgroups TIMELINE ❶ 1997 ❷ ❸ 2002 2007 ❹ ❺ 2012 2017 ❶ U.S.
https://www.pdf-archive.com/2017/04/01/apahe-2017-handout/
20151027 HK airport Cargo Departures (eng) 20151027 94%
20151027 HK airport Cargo Departures (eng) 20151027 Real Ti m e Fl i ght I nf or m at i on 25 OCTOBER 2015 16:40 CX 017 Chennai Cathay Pacific Dep 10:46 (27/10/2015) Anchorage Dallas FT.Worth Cathay Pacific Dep 01:41 (27/10/2015) 26 OCTOBER 2015 09:05 CX 3217 Delhi Cathay Pacific Dep 09:16 (27/10/2015) 23:40 CX 041 Delhi Cathay Pacific Est 16:50 (27/10/2015) 23:45 CX 050 Shanghai/PVG Cathay Pacific
https://www.pdf-archive.com/2015/10/27/20151027-hk-airport-cargo-departures-eng-20151027/
20151027 HK airport Cargo Arrivals (eng) 20151027 94%
20151027 HK airport Cargo Arrivals (eng) 20151027 Real Ti m e Fl i ght I nf or m at i on 26 OCTOBER 2015 06:45 CX 068 Frankfurt Milan/MXP New York/JFK Anchorage Dallas FT.Worth Atlanta Anchorage Miami Houston Anchorage Los Angeles Mexico City Don Miguel Anchorage Delhi Cathay Pacific At gate 04:02 (27/10/2015) Cathay Pacific At gate 01:46 (27/10/2015) Cathay Pacific Est at 12:50 (27/10/2015) Cathay Pacific At gate 11:36 (27/10/2015) Cathay Pacific At gate 12:22 (27/10/2015)
https://www.pdf-archive.com/2015/10/27/20151027-hk-airport-cargo-arrivals-eng-20151027/
Asia Pacific Smart Glass Market.PDF 94%
Asia Pacific Smart Glass Market Asia Pacific Smart Glass Market – Forecasts from 2017 to 2022 DESCRIPTION Asia-Pacific Smart Glass market is estimated to attain a CAGR of 16.56%, reaching US$1.239 Billion in 2022, increasing from US$0.494 billion 2016.
https://www.pdf-archive.com/2017/11/29/asia-pacific-smart-glass-market/
Asia-Pacific Fleece Jackets Market Report 94%
Asia Pacific Fleece Jackets Market Report Asia-Pacific Fleece Jackets Market In this report, the Asia-Pacific Fleece Jackets market is valued at USD XX million in 2016 and is expected to reach USD XX million by the end of 2022, growing at a CAGR of XX% between 2016 and 2022.
https://www.pdf-archive.com/2017/05/22/asia-pacific-fleece-jackets-market-report/
Developments in Outer Space 94%
Asia Pacific and Singapore Prepared by Elliott Tan Intern, Defence Policy Office Ministry of Defence (Singapore) 1 UNCLASSIFIED UNCLASSIFIED Table of Contents Executive Summary..............................................................................................................................................
https://www.pdf-archive.com/2016/12/12/developments-in-outer-space/
Asia Pacific Aerosol Cans Market.PDF 94%
Asia Pacific Aerosol Cans Market Asia Pacific Aerosol Cans Market – Forecasts from 2017 to 2022 DESCRIPTION Asia Pacific aerosol cans market is projected to witness the growth at a CAGR of XX.XX% during the forecast period to reach a total market size of US$XX.XX billion by 2022, increasing from US$XX.XX billion in 2017.
https://www.pdf-archive.com/2018/01/02/asia-pacific-aerosol-cans-market/
Pacific Grove Museum of Natural History 94%
Pacific Grove Museum of Natural History Pacific Grove Museum of Natural History Hosts 57th Annual Wildflower Show April 20-22 One of the largest wildflower shows in the Northern and Western Hemispheres returns April 20 through April 22 to The Pacific Grove Museum of Natural History.
https://www.pdf-archive.com/2018/03/24/pacific-grove-museum-of-natural-history/
Asia-Pacific Cocktail Table Market Report 94%
Asia Pacific Cocktail Table Market Report Asia-Pacific Cocktail Table Market In this report, the Asia-Pacific Cocktail Table market is valued at USD XX million in 2016 and is expected to reach USD XX million by the end of 2022, growing at a CAGR of XX% between 2016 and 2022.
https://www.pdf-archive.com/2017/05/22/asia-pacific-cocktail-table-market-report/
Asia-Pacific Digital Piano Market Report 94%
Asia Pacific Digital Piano Market Report Asia-Pacific Digital Piano Market Report In this report, the Asia-Pacific Digitsal Piano market is valued at USD XX million in 2016 and is expected to reach USD XX million by the end of 2022, growing at a CAGR of XX% between 2016 and 2022.
https://www.pdf-archive.com/2017/05/22/asia-pacific-digital-piano-market-report/
25 Colloquial Jakartan Indonesian 93%
25 Colloquial Jakartan Indonesian Colloquial Jakartan Indonesian Pacific Linguistics 581 Pacific Linguistics is a publisher specialising in grammars and linguistic descriptions, dictionaries and other materials on languages of the Pacific, Taiwan, the Philippines, Indonesia, East Timor, southeast and south Asia, and Australia.
https://www.pdf-archive.com/2016/12/18/25-colloquial-jakartan-indonesian/
Delightful Day Fancy Fold Card PDF 93%
Pacific Point:
https://www.pdf-archive.com/2019/06/28/delightful-day-fancy-fold-card-pdf/


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Q: setText() not working on inflated layout I'm trying to inflate a LinearLayout to build a customized section on my screen.
String txt = "testing";
LinearLayout rowLink = (LinearLayout)getLayoutInflater().inflate(R.layout.additional_link, null);
TextView tvCsAddLink = (TextView)findViewById(R.id.tvCsAddLink);
tvCsAddLink.setText(txt);

// adding this inflated layout to an existing layout
myLinearLayout.addView(rowLink);

The content of my additional_link.xml is as follows:
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    android:id="@+id/llCsAddLink"
    android:orientation="horizontal"
    android:layout_width="fill_parent"
    android:layout_height="wrap_content"
    android:background="@drawable/white_row"
    android:clickable="true"
    >
    <ImageView
        android:layout_width="wrap_content"
        android:layout_height="fill_parent"
        android:src="@drawable/youtube_icon" />
    <TextView
        android:id="@+id/tvCsAddLink"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:paddingLeft="10px"
        android:paddingRight="20px"
        android:textSize="16dp"
        android:textAppearance="?android:attr/textAppearanceSmall"
        android:layout_gravity="center_vertical" />
</LinearLayout>

I'm getting a NullPointerException on the line:
tvCsAddLink.setText(txt);

A: Use this to get TextView from layout.
TextView tvCsAddLink = (TextView)rowLink.findViewById(R.id.tvCsAddLink);

A: you have inflated your XML on your linearLayout rowLink  , and you try to find the TextView from the layout of your Activity ( usualy main.xml ) , so you need to find your textView on your layout rowLink like this  :
TextView tvCsAddLink = (TextView)rowLink.findViewById(R.id.tvCsAddLink);

A: Call findViewById(R.id.tvCsAddLink) on your inflated LinearLayoutrowLink, now you are searching it in the current activity.

redpajama_set_name: RedPajamaStackExchange


Swap amount every day will be debited or credited to the broker's customer account as a cash entry.
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text:
Q: What's the Best Way to Grind XP in Dynasty Warriors 6 I'm trying to max out the level of all Heroes to get the Achievement: Completist.
I'm not sure if I have to only max out heroes I can play in story mode, or all heroes.  Does this affect the strategy? Is there one for story characters, and one for non-story characters?

A: There's an eHow article outlining one method (however you need a level 50 character to utilize it):

*

*Bring one character up to level 50. That's as high as you can go, so your character will have all their abilities. Ideally, you'll want to level a character that has a powerful Area of Effect attack. Yue Ying, Sun Shang Xiang, Diao Chan and Zhen Ji all work well. Of course, the best is Lu Bu, since he has the highest stats of any character.

*Train a pair of horses that have the Find Weapon ability. This isn't hard to get and if you've played the game for more than a couple hours, you've probably already got at least one. Make sure one of them is very fast, with a speed of 400+.

*Take your level 50 character and whoever you want to power level to the Battle of He Fei Castle, Wei side. Make sure your level 50 character is set as Player 2. Set the difficulty to Hard or something higher in order to get the best results.

*Have Player 1 ride around and attempt to prevent the enemy from taking any of the objectives. At a minimum, ride in circles so you don't die.

*Have Player 2 (your uber character) ride down and attack Sun Quan as quickly as possible. He has a swarm of officers around him, so beware. If you manage to kill him quickly, you will both get a boatload of XP and some of the best weapons to be had!


Women and men have equality for the law. The government signed the Convention on the Elimination of All Forms of Discrimination against Women (CEDAW) in 1985. Nearly 3 out of ten females reported having experienced sexual violence before the age of 18.  The prevalence of female genital mutilation has decreased. School girls are reinstated back to school after delivery. The Police Force administration strives to separate the Gender Desks from normal police operations to enhance confidentiality of the processing of women victims of abuse. Most of the abuses and violence against women and children occurs at the family level. The Constitution of Tanzania requires that women to constitute at least 30% of all elected members of National Assembly. The gender differences in education and training have implications later in life of these women and girls. Unemployment is higher for females than for males. The right of a female employee to maternity leave is guaranteed in labour law.
^ "Tanzania". Oxford Dictionaries Online. Retrieved October 28, 2018.
^ a b c "OEC - Tanzania (TZA) Exports, Imports, and Trade Partners". atlas.media.mit.edu. Retrieved 2019-04-09.
^ a b "Tanzania". Global Hunger Index - Official Website of the Peer-Reviewed Publication. Retrieved 2019-03-26.
^ a b "Tanzania: Nutrition Profile". www.usaid.gov. Retrieved 2018-10-18.
^ a b "UNICEF Tanzania—Nutrition—The situation". www.unicef.org. Retrieved 2018-10-28.
^ Makoye, Kizito. "Survey finds most Tanzanians go hungry, despite government denials". U.S. Retrieved 2018-11-20.
^ a b Makoye, Kizito. "Survey finds most Tanzanians go hungry, despite government denials". U.S. Retrieved 2018-10-18.
^ a b c "UN World Food Programme". Retrieved 2018-11-08.
^ a b "Nutrition | Tanzania | Save the Children". tanzania.savethechildren.net. Retrieved 2018-11-14.


There's something a bit off about being a sports fan.
As Jerry Seinfeld told it, there's something a bit off about being a sports fan. With players coming and going and so much turmoil and turnover, when you break it down, you're essentially shouting for an item of clothing.