text:
describe('Core canvas behavior', function() {
    let container = document.createElement('div');
    document.body.appendChild(container);
    const CANVAS_WIDTH = 600;
    const CANVAS_HEIGHT = 600;

    beforeEach(function() {
        container.innerHTML = `
            <canvas width=600 height=600 id=test></canvas>
            <canvas width=600 height=600 id=report></canvas>`;
    });

    afterEach(function() {
        container.innerHTML = '';
    });

    it('can draw an SkPicture', function(done) {
        LoadCanvasKit.then(catchException(done, () => {
            // This is taken from example.html
            const surface = CanvasKit.MakeCanvasSurface('test');
            expect(surface).toBeTruthy('Could not make surface');
            if (!surface) {
                done();
                return;
            }
            const spr = new CanvasKit.SkPictureRecorder();
            const rcanvas = spr.beginRecording(
                            CanvasKit.LTRBRect(0, 0, surface.width(), surface.height()));
            const paint = new CanvasKit.SkPaint();
            paint.setStrokeWidth(2.0);
            paint.setAntiAlias(true);
            paint.setColor(CanvasKit.Color(0, 0, 0, 1.0));
            paint.setStyle(CanvasKit.PaintStyle.Stroke);

            rcanvas.drawRoundRect(CanvasKit.LTRBRect(5, 35, 45, 80), 15, 10, paint);

            const font = new CanvasKit.SkFont(null, 20);
            rcanvas.drawText('this picture has a round rect', 5, 100, paint, font);
            const pic = spr.finishRecordingAsPicture();
            spr.delete();

            const canvas = surface.getCanvas();
            canvas.drawPicture(pic);

            reportSurface(surface, 'picture_test', done);
        }));
    });

    it('can compute tonal colors', function(done) {
        LoadCanvasKit.then(catchException(done, () => {
            const input = {
                ambient: CanvasKit.BLUE,
                spot: CanvasKit.RED,
            };
            const out = CanvasKit.computeTonalColors(input);

            expect(out.ambient).toEqual(CanvasKit.Color(0,0,0,1));

            const [r,g,b,a] = CanvasKit.getColorComponents(out.spot);
            expect(r).toEqual(44);
            expect(g).toEqual(0);
            expect(b).toEqual(0);
            expect(a).toBeCloseTo(0.969, 2);
            done();
        }));
    });

    // This helper is used for all the MakeImageFromEncoded tests.
    function decodeAndDrawSingleFrameImage(imgName, goldName, done) {
        const imgPromise = fetch(imgName)
            .then((response) => response.arrayBuffer());
        Promise.all([imgPromise, LoadCanvasKit]).then((values) => {
            const imgData = values[0];
            expect(imgData).toBeTruthy();
            catchException(done, () => {
                let img = CanvasKit.MakeImageFromEncoded(imgData);
                expect(img).toBeTruthy();
                const surface = CanvasKit.MakeCanvasSurface('test');
                expect(surface).toBeTruthy('Could not make surface');
                if (!surface) {
                    done();
                    return;
                }
                const canvas = surface.getCanvas();
                let paint = new CanvasKit.SkPaint();
                canvas.drawImage(img, 0, 0, paint);

                paint.delete();
                img.delete();

                reportSurface(surface, goldName, done);
            })();
        });
    }

    it('can decode and draw a png', function(done) {
        decodeAndDrawSingleFrameImage('/assets/mandrill_512.png', 'drawImage_png', done);
    });

    it('can decode and draw a jpg', function(done) {
        decodeAndDrawSingleFrameImage('/assets/mandrill_h1v1.jpg', 'drawImage_jpg', done);
    });

    it('can decode and draw a (still) gif', function(done) {
        decodeAndDrawSingleFrameImage('/assets/flightAnim.gif', 'drawImage_gif', done);
    });

    it('can decode and draw a still webp', function(done) {
        decodeAndDrawSingleFrameImage('/assets/color_wheel.webp', 'drawImage_webp', done);
    });

    it('can decode and draw an animated gif', function(done) {
        const imgPromise = fetch('/assets/flightAnim.gif')
            .then((response) => response.arrayBuffer());
        Promise.all([imgPromise, LoadCanvasKit]).then((values) => {
            const gifData = values[0];
            expect(gifData).toBeTruthy();
            catchException(done, () => {
                let aImg = CanvasKit.MakeAnimatedImageFromEncoded(gifData);
                expect(aImg).toBeTruthy();
                expect(aImg.getRepetitionCount()).toEqual(-1); // infinite loop
                expect(aImg.width()).toEqual(320);
                expect(aImg.height()).toEqual(240);
                expect(aImg.getFrameCount()).toEqual(60);

                const surface = CanvasKit.MakeCanvasSurface('test');
                expect(surface).toBeTruthy('Could not make surface');
                if (!surface) {
                    done();
                    return;
                }
                const canvas = surface.getCanvas();
                canvas.drawAnimatedImage(aImg, 0, 0);

                let c = aImg.decodeNextFrame();
                expect(c).not.toEqual(-1);
                canvas.drawAnimatedImage(aImg, 300, 0);
                for(let i = 0; i < 10; i++) {
                    c = aImg.decodeNextFrame();
                    expect(c).not.toEqual(-1);
                }
                canvas.drawAnimatedImage(aImg, 0, 300);
                for(let i = 0; i < 10; i++) {
                    c = aImg.decodeNextFrame();
                    expect(c).not.toEqual(-1);
                }
                canvas.drawAnimatedImage(aImg, 300, 300);

                aImg.delete();

                reportSurface(surface, 'drawDrawable_animated_gif', done);
            })();
        });
    });

    it('can create an image "from scratch" by specifying pixels/colorInfo manually', function(done) {
        LoadCanvasKit.then(catchException(done, () => {
            const surface = CanvasKit.MakeCanvasSurface('test');
            expect(surface).toBeTruthy('Could not make surface');
            if (!surface) {
                done();
                return;
            }
            const canvas = surface.getCanvas();
            canvas.clear(CanvasKit.WHITE);
            const paint = new CanvasKit.SkPaint();

            // This creates and draws an SkImage that is 1 pixel wide, 4 pixels tall with
            // the colors listed below.
            const pixels = Uint8Array.from([
                255,   0,   0, 255, // opaque red
                  0, 255,   0, 255, // opaque green
                  0,   0, 255, 255, // opaque blue
                255,   0, 255, 100, // transparent purple
            ]);
            const img = CanvasKit.MakeImage(pixels, 1, 4, CanvasKit.AlphaType.Unpremul, CanvasKit.ColorType.RGBA_8888);
            canvas.drawImage(img, 1, 1, paint);

            reportSurface(surface, '1x4_from_scratch', done);
        }));
    });

    it('can blur using ImageFilter or MaskFilter', function(done) {
        LoadCanvasKit.then(catchException(done, () => {
            const surface = CanvasKit.MakeCanvasSurface('test');
            expect(surface).toBeTruthy('Could not make surface');
            if (!surface) {
                done();
                return;
            }
            const canvas = surface.getCanvas();
            const pathUL = starPath(CanvasKit, 100, 100, 80);
            const pathBR = starPath(CanvasKit, 400, 300, 80);
            const paint = new CanvasKit.SkPaint();
            const textFont = new CanvasKit.SkFont(null, 24);

            canvas.drawText('Above: MaskFilter', 20, 220, paint, textFont);
            canvas.drawText('Right: ImageFilter', 20, 260, paint, textFont);

            paint.setColor(CanvasKit.BLUE);

            const blurMask = CanvasKit.SkMaskFilter.MakeBlur(CanvasKit.BlurStyle.Normal, 5, true);
            paint.setMaskFilter(blurMask);
            canvas.drawPath(pathUL, paint);

            const blurIF = CanvasKit.SkImageFilter.MakeBlur(8, 1, CanvasKit.TileMode.Decal, null);
            paint.setImageFilter(blurIF);
            canvas.drawPath(pathBR, paint);

            surface.flush();

            pathUL.delete();
            pathBR.delete();
            paint.delete();
            blurMask.delete();
            blurIF.delete();
            textFont.delete();

            reportSurface(surface, 'blur_filters', done);
        }));
    });

    it('can use various image filters', function(done) {
        const imgPromise = fetch('/assets/mandrill_512.png')
            .then((response) => response.arrayBuffer());
        Promise.all([imgPromise, LoadCanvasKit]).then((values) => {
            const pngData = values[0];
            expect(pngData).toBeTruthy();
            catchException(done, () => {
                let img = CanvasKit.MakeImageFromEncoded(pngData);
                expect(img).toBeTruthy();
                const surface = CanvasKit.MakeCanvasSurface('test');
                expect(surface).toBeTruthy('Could not make surface');
                if (!surface) {
                    done();
                    return;
                }
                const canvas = surface.getCanvas();
                canvas.clear(CanvasKit.WHITE);
                const paint = new CanvasKit.SkPaint();
                paint.setAntiAlias(true);
                paint.setColor(CanvasKit.Color(0, 255, 0, 1.0));
                const redCF =  CanvasKit.SkColorFilter.MakeBlend(
                        CanvasKit.Color(255, 0, 0, 0.1), CanvasKit.BlendMode.SrcOver);
                const redIF = CanvasKit.SkImageFilter.MakeColorFilter(redCF, null);
                const blurIF = CanvasKit.SkImageFilter.MakeBlur(8, 0.2, CanvasKit.TileMode.Decal, null);
                const combined = CanvasKit.SkImageFilter.MakeCompose(redIF, blurIF);

                // rotate 10 degrees centered on 200, 200
                const m = CanvasKit.SkMatrix.rotated(Math.PI/18, 200, 200);
                const rotated = CanvasKit.SkImageFilter.MakeMatrixTransform(m, CanvasKit.FilterQuality.Medium, combined);
                paint.setImageFilter(rotated);

                //canvas.rotate(10, 200, 200);
                canvas.drawImage(img, 0, 0, paint);
                canvas.drawRect(CanvasKit.LTRBRect(5, 35, 45, 80), paint);

                surface.flush();

                paint.delete();
                redIF.delete();
                redCF.delete();
                blurIF.delete();
                combined.delete();
                rotated.delete();
                img.delete();

                reportSurface(surface, 'combined_filters', done);
            })();
        });
    });

    it('can use various image filters on animated images', function(done) {
        const imgPromise = fetch('/assets/flightAnim.gif')
            .then((response) => response.arrayBuffer());
        Promise.all([imgPromise, LoadCanvasKit]).then((values) => {
            const gifData = values[0];
            expect(gifData).toBeTruthy();
            catchException(done, () => {
                let img = CanvasKit.MakeAnimatedImageFromEncoded(gifData);
                expect(img).toBeTruthy();
                const surface = CanvasKit.MakeCanvasSurface('test');
                expect(surface).toBeTruthy('Could not make surface');
                if (!surface) {
                    done();
                    return;
                }
                img.decodeNextFrame();
                img.decodeNextFrame();
                const canvas = surface.getCanvas();
                canvas.clear(CanvasKit.WHITE);
                const paint = new CanvasKit.SkPaint();
                paint.setAntiAlias(true);
                paint.setColor(CanvasKit.Color(0, 255, 0, 1.0));
                const redCF =  CanvasKit.SkColorFilter.MakeBlend(
                        CanvasKit.Color(255, 0, 0, 0.1), CanvasKit.BlendMode.SrcOver);
                const redIF = CanvasKit.SkImageFilter.MakeColorFilter(redCF, null);
                const blurIF = CanvasKit.SkImageFilter.MakeBlur(8, 0.2, CanvasKit.TileMode.Decal, null);
                const combined = CanvasKit.SkImageFilter.MakeCompose(redIF, blurIF);
                paint.setImageFilter(combined);

                const frame = img.getCurrentFrame();
                canvas.drawImage(frame, 100, 50, paint);

                surface.flush();

                paint.delete();
                redIF.delete();
                redCF.delete();
                blurIF.delete();
                combined.delete();
                frame.delete();
                img.delete();

                reportSurface(surface, 'animated_filters', done);
            })();
        });
    });

    it('can decode and draw an skp', function(done) {
        const skpPromise = fetch('/assets/red_line.skp')
            .then((response) => response.arrayBuffer());
        Promise.all([skpPromise, LoadCanvasKit]).then((values) => {
            const skpData = values[0];
            expect(skpData).toBeTruthy();
            catchException(done, () => {
                let pic = CanvasKit.MakeSkPicture(skpData);
                expect(pic).toBeTruthy();
                const surface = CanvasKit.MakeCanvasSurface('test');
                expect(surface).toBeTruthy('Could not make surface');
                if (!surface) {
                    done();
                    return;
                }
                const canvas = surface.getCanvas();
                canvas.clear(CanvasKit.TRANSPARENT);
                canvas.drawPicture(pic);

                reportSurface(surface, 'drawImage_skp', done);
            })();
        });
    });

    it('can use DecodeCache APIs', function(done) {
        LoadCanvasKit.then(catchException(done, () => {
            const initialLimit = CanvasKit.getDecodeCacheLimitBytes();
            expect(initialLimit).toBeGreaterThan(1024 * 1024);

            const newLimit = 42 * 1024 * 1024;
            CanvasKit.setDecodeCacheLimitBytes(newLimit);
            expect(CanvasKit.getDecodeCacheLimitBytes()).toEqual(newLimit);

            // We cannot make any assumptions about this value,
            // so we just make sure it doesn't crash.
            CanvasKit.getDecodeCacheUsedBytes();

            done();
        }));
    });

    it('can combine shaders', function(done) {
        LoadCanvasKit.then(catchException(done, () => {
            const surface = CanvasKit.MakeCanvasSurface('test');
            expect(surface).toBeTruthy('Could not make surface');
            if (!surface) {
                done();
                return;
            }
            const canvas = surface.getCanvas();

            const rShader = CanvasKit.SkShader.Color(CanvasKit.Color(255, 0, 0, 1.0));
            const gShader = CanvasKit.SkShader.Color(CanvasKit.Color(0, 255, 0, 0.6));
            const bShader = CanvasKit.SkShader.Color(CanvasKit.Color(0, 0, 255, 1.0));

            const rgShader = CanvasKit.SkShader.Blend(CanvasKit.BlendMode.SrcOver, rShader, gShader);

            const p = new CanvasKit.SkPaint();
            p.setShader(rgShader);
            canvas.drawPaint(p);

            const gbShader = CanvasKit.SkShader.Lerp(0.5, gShader, bShader);

            p.setShader(gbShader);
            canvas.drawRect(CanvasKit.LTRBRect(5, 100, 300, 400), p);

            reportSurface(surface, 'combined_shaders', done);
        }));
    });

    it('exports consts correctly', function(done) {
        LoadCanvasKit.then(catchException(done, () => {
            expect(CanvasKit.TRANSPARENT).toEqual(0);
            expect(CanvasKit.RED).toEqual(4294901760);

            expect(CanvasKit.QUAD_VERB).toEqual(2);
            expect(CanvasKit.CONIC_VERB).toEqual(3);

            expect(CanvasKit.SaveLayerInitWithPrevious).toEqual(4);
            expect(CanvasKit.SaveLayerF16ColorType).toEqual(16);

            done();
        }));
    });

});


text:
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About 'Unstated'
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Unstated.name > Deliberate Fragments > (50 Pauses) So She Left (≅1,483)
(50 Pauses) So She Left (≅1,483)
She had a cat called Sid and a husband named James; she would introduce them as "Sid, James". In the main only older people got the joke though. Neither Sid nor James made her laugh much anyway.
James cycled to work every morning, spent the day carving – mainly funeral stuff – at a small stonemason's yard down the hill and two villages further along the valley, and cycled back home in the evening.
He walked the last bit up the hill, unless it was raining.
Grave stones and little statues and what-have-you were OK. It was regular enough. People will always need burying, same as they'll always need hairdressers. He didn't find the work macabre or depressing or maudlin.
Most evenings Sid would be waiting somewhere near the top of the hill. He'd trot up, apparently jauntily, and try to rub against James' legs while he pushed his bike through the gate and towards the shed.
Every time, he'd nearly get kicked. Every time, he'd jump out of the way just in time and up on to the garden wall.
From the wall he'd have to be rescued by James. Sid had grown old and nearly blind – with one eye lost in a fight and the other cloudy – and he struggled to get down from things himself.
If there wasn't anyone around he'd whine for help. No depth perception, so James reasoned.
She'd often see all this from the kitchen window. She didn't mind the routine. She didn't mind the predictability. It was boring. That was all. It could be worse.
At least James didn't tell her about "Sid's antics" anymore, in the same way as he didn't tell her about his day. He used to.
She owned the cottage they lived in; it had been her parents' place, before that her grandmother's and before that her great-grandparent's. Her family had lived in the village for donkey's years.
With no mortgage, James' money kept them comfortable enough. They didn't want for anything.
She was still kind to James but only out of habit now. She didn't put any effort into it.
He'd never been unkind to her.
She thought of it as a quiet, well-rooted English country life and would say to herself that there was nothing wrong with that.
Callum knew them both well but was her special friend. He lived in the next-door cottage.
Everyone who'd been in the village long enough knew that when he'd been young and she'd been even younger, he'd been very keen on her, but that was all way back when.
Opportunities are missed and times change. There was never going to be an affair, not even now, now Callum's wife had died. They both knew that.
She had wondered whether he'd move on now he was alone, whether living there would have too many memories because his wife had died quite young and unexpectedly, but Callum had said he was happy and thought all the reminders were positive.
One day Callum had mentioned that she could easily sell-up if she wanted to, now that commuters bought in to villages like theirs. He'd said the money would last a good few years.
He'd said it out of the blue, unbidden. They'd just been chatting about getting someone in to do the gutters; whether Callum knew anyone to do the job because James didn't and he didn't fancy doing it himself.
Later, after she'd thought about it, she'd been slightly surprised at Callum's suggestion and wondered why he'd been thinking about her leaving.
Once she'd thought about it a bit more, she'd realised it might not have been a suggestion. It might have just been a comment, an observation.
At the time she'd taken it merely as a passing remark he'd made; she hadn't even replied.
The gutters were fixed by one of the local farmer's sons, just starting out with his own roofing and guttering business. His father hadn't wanted him to go into farming, not now it had been ruined by the supermarkets and the government.
He'd done a good job for a good price. She'd used him again soon afterwards to re-do some slates on the west gable after they lifted and moved in a freakish gust during an October storm.
She felt sure the weather was changing, causing problems that it never used to. She'd said about it to Callum one evening as they'd stood outside in some warm late autumn sun.
They'd talked about the changes they'd seen and he'd said again about selling up and moving away. This time he'd said he'd buy her cottage if she'd like him to.
She knew he had money tucked away, that life assurances had paid out and that he'd inherited a bit from an Aunt too.
He'd said he could afford it; that James could stay there; that he'd only want a little bit in rent to cover keeping the place up.
This time, she'd been surprised but she didn't just dismiss the idea. She'd said she'd have to think it over. Callum didn't make much of it; he just said the offer was there if she wanted to take it up.
They'd both simply assumed she'd leave James behind.
Just two weeks later, when she saw Sid and James going through their evening performance yet again, she'd made up her mind: the boredom had to stop.
So she left. Callum kept his word and bought the cottage. There was no big bust-up with James. He didn't say much at all really. No tears and nothing silly. He did say he hoped she'd be happy.
At first she travelled England, spending most of her time on the coast, in the main staying in not-too-expensive hotels and decent bed-and-breakfasts. Almost always, she'd stay somewhere for at least a fortnight. She wasn't afraid of haggling for a good price.
After a while she started taking in Wales too and, later still, she included Scotland. As time went by she started visiting inland areas as well, particularly when the coast was busy. She never did like crowds.
She drove herself, unhurriedly and never during peak times. She travelled lightly and didn't once miss not having a home.
When she arrived somewhere new she'd take the trouble to try to buy a picture postcard if she could. She'd send it to 'Callum and James' but always addressed it to Callum's cottage.
She'd only put a few words on the cards, more-or-less bland pleasantries, but she'd also email them – often three or four times a week and always at least once.
Some emails would just be about practicalities: reassurances that she was well, details about where she was staying and for how long. In others, often sent late at night, she'd try to write about the things she'd seen that she thought would interest them.
When she started out she wasn't that good with words and struggled with descriptions but she took to it, enjoyed it and improved quickly: both Callum and James noticed. She bought a pocket thesaurus-and-dictionary to help
She did a lot of walking and she'd mention any well-known sites but she'd notice the humdrum and paid attention to the everyday as well – how people looked, local buildings, monuments and anything else; whatever caught her fancy or made an impression.
Callum and James rubbed along without any problems. They shared a grumbling mutual warmth and utter trust, and they both liked to hear from her.
They'd send her emails on high days and holidays but neither of them took easily to small talk and not a lot happened for them to write about, so the rest of the time they didn't try. She didn't mind.
More than three years after she'd left, she was staying near St Ives when James phoned her on New Year's Eve. He'd never phoned before so she knew it was going be bad news. He was calling to say Callum had died: he'd gone to bed and hadn't woken up.
He spoke hurriedly, the way she knew he always did when he was upset and uncertain. He hated uncertainty.
She walked on beaches and cliff tops in Cornwall for two days before deciding to go back, but she did. She phoned James and took control.
Callum had left everything to her, including both cottages. After a few weeks staying with her aunt she moved in to Callum's old place. She'd cried at his funeral but, like Callum, she was happy with the memories the rooms held.
James stayed where he was and carried on paying rent. He was happier with her back in the village.
She found that Sid had died too, quite a while ago, but that wasn't really a surprise. There was a life-sized stone cat marking his grave in the garden, just by the shed, beautifully carved.
James had found a replacement, a tabby, from a rescue centre and called it Babs.
Filed Under: Deliberate Fragments Tagged With: Affection, Love
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The following awards and trophies are listed here in the order found in the October 6, 2017 Annual Award Pamphlet. Each section will be listed by the latest name and description which applies. As you read through the below you will find that some contain more than one trophy due to the retiring and replacement of the trophy.
Obedience - All Awards and Names listed are in effect at the time the dog was awarded for all STCA awards.
Agility - All Awards and Names listed are in effect at the time the dog was awarded for all STCA awards.
Versatility - All Awards and Names listed are in effect at the time the dog was awarded for all STCA awards.
and Tracking. Performance, Outstanding Performance and Superior Performance Scottish Terrier award levels are tallied by the number of AKC venues and points accumulated by titles earned.
- All Awards and Names listed are in effect at the time the dog was awarded for all STCA awards.


text:
Zig Zag Wanderer
Stories From Here, Stories From There
Bell, Madison Smartt
Stretching from New York to Haiti to Paris to London, this deeply affecting story collection by National Book Award finalist Madison Smartt Bell takes readers on a crooked tour of many different psychological landscapes The 18 stories in this masterful collection are informed by Madison Smartt Bell's eye for revealing detail and empathy for his characters -- most of whom are outsiders in one way or another, sometimes obviously, sometimes not. All of the tales are influenced by music and Bell's abiding interest in the musicality of prose, and almost all take inspiration -- and their titles -- from particular songs. The title story plays on a theme from the Captain Beefheart classic "Zig-Zag Wanderer," and both here and elsewhere Bell uses the musician's apparently anarchic methods of composition to coax a vision of order out of apparent chaos. Drawn from a wide spectrum of race, class, and nationality, Bell's characters are nomads seeking elusive comfort in the socially fractured situations through which they must navigate. With the guidance of their author, they do their best to find new harmonies in the noise that surrounds us all.
Publisher: [New York] : Open Road Media, [2015]
Branch Call Number: FICTION Bel
Description: 1 online resource (136 pages)
Additional Contributors: Freading (Firm)
Read more reviews of Zig Zag Wanderer at iDreamBooks.com
Marginality, Social — Fiction
LibraryThing Series Information
No similar edition of this title was found at CRRL.
Try searching for Zig Zag Wanderer to see if CRRL owns related versions of the work.


At St Peter's School, we want every pupil to feel they have a voice which is heard and valued. Our 'School Council' is made up of children from across the school in Years 1 to 6. There are 12 members in total. Mr Cronin and Mrs Brady assist the children to run the meetings effectively.
At the beginning of the new academic year, children discuss in their classes the responsibilities and qualities that they think would be important in our school council. Children are voted in to the council by their peers and two representatives from each class are chosen. These children attend fortnightly meetings to discuss relevant matters of interest concerning school organisation and development. The School Council is actively involved in decision making within the school and regularly conducts surveys to gather views and opinions from their peers.
Each class also has a Class Council committee which is elected from their class. Class Council meetings allow children to discuss issues about which they are concerned. After discussion in class, suggestions may be taken to the School Council for discussion. A response will be made by the School Council either personally or during one of our school assemblies.
School Council also discuss events such as fundraising, assemblies, behaviour around the school, the playground, lessons, money matters, reward systems, the library, sun safety, the environment and any other matters that arise that we feel are important to discuss.


text:
Thomas Dekker
Thomas Dekker books and biography
Thomas Dekker (c. 1572 – August 25, 1632) was an Elizabethan dramatist and pamphleteer, a versatile and prolific writer whose career spanned several decades and brought him into contact with many of the period's most famous dramatists.
2.1 Drama
2.2 Prose
Little is known of Dekker's early life or origins. From references in his pamphlets, Dekker is believed to have been born in London around 1572, but nothing is known for certain about his youth. His last name suggests Dutch ancestry, and his work, some of which is translated from Latin, suggests that he attended grammar school.
Dekker embarked on a career as a theatre writer in the middle 1590s. His handwriting is found in the manuscript of Sir Thomas More, though the date of his involvement is undetermined. More certain is his work as a playwright for the Admiral's Men of Philip Henslowe, in whose account book he is first mentioned in early 1598. While there are plays connected with his name performed as early as 1594, it is not clear that he was the original author; his work often involved revision and updating. Between 1598 and 1602, he was involved in about forty plays for Henslowe, usually in collaboration. To these years belong the collaborations with Ben Jonson and John Marston which presumably contributed to the War of the Theatres in 1600 and 1601. Francis Meres includes Dekker in his list of notable playwrights in 1598.
For Jonson, however, Dekker was bumbling hack, a "dresser of plays about town"; Jonson lampooned Dekker as Demetrius Fannius in Poetaster and as Anaides in Cynthia's Revels. Dekker's riposte, Satiromastix, performed both by the Lord Chamberlain's Men and the child actors of Paul's, casts Jonson as an affected, hypocritical Horace.
Satiromastix marks the end of the "poetomachia"; in 1603, Jonson and Dekker collaborated again, on a pageant for coronation of James I. After this commission, however, the early Jacobean period was notably mixed for the author. In late 1602, he appears to have broken his association with Henslowe, for unknown reasons. He wrote for Worcester's Men for a time, then returned to the Admiral's Men (now patronized by Prince Henry) to produce The Honest Whore, an apparent success. But the failures of The Whore of Babylon (1607) and If This Be Not a Good Play, the Devil is in It (1611) left him crestfallen; the latter play was rejected by Prince Henry's Men before failing for Queen Anne's Men at the Red Bull Theatre.
In 1612, Dekker's lifelong problem with debt (he had earlier, 1599, been imprisoned in Poultry Compter) reached a crisis when he was imprisoned in the King's Bench Prison on a debt of forty pounds to the father of John Webster. He remained there for seven years, and despite the support of associates such as Edward Alleyn and Endymion Porter, these years were difficult; Dekker reports that the experience turned his hair white. He continued as pamphleteer throughout his years in prison.
On release, he resumed writing plays, now with collaborators both from his generation (John Day and John Webster) and slightly younger writers (John Ford and Philip Massinger). Among these plays is one, Keep the Widow Waking (1624, with Ford, Webster, and William Rowley), dramatized two recent murders in Whitechapel. In the latter half of the decade, Dekker turned once more to pamphlet-writing, revamping old work and writing a new preface to his most popular tract, The Bellman of London.
Dekker published no more work after 1632, and he is usually associated with the "Thomas Dekker, householder" who was buried at St. James's in Clerkenwell that year.
Throughout his career, Dekker wrote quickly, carelessly, and under pressure from creditors. His work has all the marks of such conditions: it is often repetitive, derivative, and sloppy. The view held by Jonson has stood, for many critics, ever since: Dekker (alongside Heywood) is seen as the consummate Renaissance hack, whose talents were perverted or stunted by the relentless need to produce. Dekker has also been labelled as a sentimentalist. More recent critics, while recognizing the mediocrity of much of his work, see Dekker as a spirited writer whose work explored the social and political tensions of his age.
When Dekker began writing plays, Thomas Nashe and Thomas Lodge were still alive; when he died, John Dryden had already been born. Like most dramatists of the period he adapted as well as he could to changing tastes; however, even his work in the fashionable Jacobean genres of satire and tragicomedy bears the marks of his Elizabethan training: its humor is genial, its action romantic. The majority of his surviving plays are comedies or tragicomedies.
Most of Dekker's work is lost. His apparently disordered life, and his lack of a firm connection (such as Shakespeare or Fletcher had) with a single company may have militated against the preservation or publication of manuscripts. Close to twenty of his plays were published during his lifetime; of these, more than half are comedies, with three significant tragedies, Lust's Dominion (presumably identical to The Spanish Moor's Tragedy, written with Day, Marston, and William Haughton, 1600) The Witch of Edmondton (with Ford and Rowley, 1621), and The Virgin Martyr (with Massinger, 1620).
The first phase of Dekker's career is documented in Henslowe's diary. His name appears for the first time in connection with "fayeton" (presumably, Phaeton) in 1598. There follow, before 1599, payments for work on The Triplicity of Cuckolds, The Mad Man's Morris, and Hannibal and Hermes. He worked on these plays with Robert Wilson, Henry Chettle, and Michael Drayton. With Drayton, he also worked on history plays on the French civil wars, Earl Godwin, and others. In 1599, he wrote plays on Troilus and Cressida, Agamemnon (with Chettle), and The Page of Plymouth. In that year, also, he collaborated with Chettle, Jonson, and Marston on a play about Robert II. 1599 also saw the production of three plays that have survived. It was during this year that he produced his most famous work, The Shoemaker's Holiday, or the Gentle Craft, categorised by modern critics as citizen comedy. This play reflects his concerns with the daily lives of ordinary Londoners. This play exemplifies his intermingling of everyday subjects with the fantastical, embodied in this case by the rise of a craftsman to Mayor and the involvement of an unnamed but idealised king in the concluding banquet. Old Fortunatus and Patient Grissel, the latter on the folkloric theme treated by Chaucer in The Clerk's Tale. In 1600, he worked on The Seven Wise Masters, Fortune's Tennis, Cupid and Psyche, and Fair Constance of Rome. The next year, in addition to Satiromastix, he worked on a play possibly about Sebastian of Portugal and Blurt, Master Constable, on which he may have worked with Thomas Middleton. In 1602 he revised two older plays, Pontius Pilate (1597) and the second part of Sir John Oldcastle. He also collaborated on Caesar's Fall, Jephthah, A Medicine for a Curst Wife, Sir Thomas Wyatt (on Wyatt's rebellion), and Christmas Comes But Once a Year.
Except for Blurt, which was performed by the Blackfriars Children, the earlier of these works were performed at the Admiral's Fortune Theatre. After 1602, Dekker split his attention between pamphlets and plays; thus, his dramatic output decreased considerably. He and Middleton wrote The Honest Whore for the Fortune in 1604, and Dekker wrote a sequel himself the following year. The Middleton/Dekker collaboration The Family of Love also dates from this general era. Dekker and Webster wrote Westward Ho and Northward Ho for Paul's Boys. The failures of the anti-Catholic Whore of Babylon and tragicomic If This Be Not... have already been noted. The Roaring Girl, a fanciful biography of Mary Frith, was a collaboration with Middleton in 1611.
During his imprisonment, Dekker did not write plays. After his release, he collaborated with Day on Guy of Warwick (1620), The Wonder of a Kingdom (1623), and The Bellman of Paris (1623). One other plays of this period, The Noble Spanish Soldier, may be a revision of earlier work with Day. With Ford, he wrote The Sun's Darling (1624), The Fairy Knight (1624), and The Bristow Merchant (1624). The Welsh Ambassador may be a reworking of The Noble Spanish Soldier. The Late Murder of the Son upon the Mother, or Keep the Widow Waking, a dramatization of two recent murders in Whitechapel, occasioned a suit for slander heard in the Star Chamber.
Dekker's plays of the 1620s were staged at the large amphitheaters on the north side of London, most commonly at the Red Bull; only two of his later plays were seen at the more exclusive, indoor Cockpit Theatre, and these two were presumably produced by Christopher Beeston, who operated both the Red Bull and the Cockpit. By the 1620s, the Shoreditch amphitheaters had become deeply identified with the louder and less reputable categories of play-goers, such as apprentices. Dekker's type of play appears to have suited them perfectly. Full of bold action, careless about generic differences, and always (in the end) complimentary to the values and beliefs of such audiences, his drama carried some of the vigorous optimism of Elizabethan dramaturgy into the Caroline era.
He exhibited a similar vigour in his pamphlets, which span almost his whole writing career, and which treat a great variety of subjects and styles.
Dekker's first spate of pamphleteering began in 1603, perhaps during a period when plague had closed the theaters. His first was The Wonderful Year, a journalistic account of the death of Elizabeth, accession of James I, and the 1603 plague. It succeeded well enough to prompt two more plague pamphlets, News From Gravesend and The Meeting of Gallants at an Ordinary. The Double PP (1606) is an anti-Catholic tract written in response to the Gunpowder Plot. News From Hell (1606) is an homage to and continuation of Nash's Pierce Penniless. The Seven Deadly Sins of London (1607) is another plague pamphlet.
After 1608, Dekker produced his most popular pamphlets: a series of "cony-catching" pamphlets that described the various tricks and deceits of confidence-men and thieves, including thieves' cant. These pamphlets, which Dekker often updated and reissued, include The Belman of London (1608), Lanthorne and Candle-light, Villainies Discovered by Candlelight, and English Villainies. They owe their form and many of their incidents to similar pamphlets by Robert Greene.
Other pamphlets are journalistic in form and offer vivid pictures of Jacobean London. The Dead Term (1608) describes Westminster during summer vacation. The Guls Horne-Booke (1609) describes the life of city gallants, including a valuable account of behaviour in the London theatres. Work for Armourers (1609) and The Artillery Garden (1616) (the latter in verse) describe aspects of England's military industries. London Look Back (1630) treats 1625, the year of James's death, while Wars, Wars, Wars (1628) describes European turmoil.
As might be expected, Dekker turned his experience in prison to profitable account. Dekker His Dreame (1620) is a long poem describing his despairing confinement; he contributed six prison-based sketches to the sixth edition (1616) of Sir Thomas Overbury's Characters; and he revised Lanthorne and Candlelight to reflect what he had learned in prison.
Dekker's pamphlets, even more than his plays, reveal signs of hasty and careless composition. Yet the best of them can still entertain, and almost all of them offer valuably precise depictions of everyday life in the Jacobean period.
Dekker came back into scene in the 20th century (although almost unnoticeably) when the Beatles included part of his ballad Golden Slumbers in their 1969 song of the same title.
Gregg, Kate. Thomas Dekker: A Study in Economic and Social Backgrounds. Seattle: University of Washington Press, 1924.
Hunt, Mary. Thomas Dekker: A Study. New York: Columbia University Press, 1911.
McLuskie, Kathleen. Dekker and Heywood: Professional Dramatists. New York: St. Martin's Press, 1993.
Wilson, F. P, editor. The Plague Pamphlets of Thomas Dekker. Oxford: Clarendon Press, 1925.
The Guls Horn Booke
By Thomas Dekker
The Wonderful Yeare


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text:
\section{Introduction}

Let $n \geq 2$ and $\Omega \subset \R^n$ be open.
The Banach spaces $BV(\Omega)$ and $SBV(\Omega)$ consist, respectively, of functions of bounded variation \cite[3.1]{AMBROSIO.FUSCO.PALLARA} and special functions of bounded variation \cite[\S 4.1]{AMBROSIO.FUSCO.PALLARA}.
A long-standing open problem \cite[7.4]{ARCATA} is to provide a useful description of the dual of $BV(\Omega)$.
This seems to be still open, despite important contributions by N.G. Meyers and W.P. Ziemer \cite{MEY.ZIE.77}, N.C. Phuc and M. Torres \cite{PHU.TOR.17},  and N. Fusco and D. Spector \cite{FUS.SPE.18}.
Regarding the dual of $SBV(\Omega)$, the second author contributed the article \cite{DEP.98}.
In the present paper, we describe the dual of $SBV(\Omega)$ in a way
that is ``optimal'' in some specific universal sense.  \par For each
$u \in SBV(\Omega)$, the distributional gradient $Du$ of $u$
decomposes as $Du = \mathscr{L}^n \hel \nabla u + \mathscr{H}^{n-1} \hel j_u$, where
$\nabla u$ is the pointwise a.e. approximate gradient of $u$, and
$j_u$ is a vector field carried on the approximate discontinuity set
$S_u$ of $u$, on which we have $j_u = (u^+-u^-)\nu_u$, $\nu_u$ being a
unitary field normal to $S_u$, and $u^+, u^-$ the approximate limits
of $u$ on either sides of $S_u$.
We have
\begin{equation*}
  \|u\|_{SBV(\Omega)} = \int_\Omega |u| \, d \mathscr{L}^n + \int_{S_u} \|j_u\| \,
  d \mathscr{H}^{n-1} + \int_\Omega \|\nabla u\| \, d \mathscr{L}^n.
\end{equation*}
Thus, $u \mapsto (u, j_u, \nabla u)$ is an isometric embedding
\begin{equation*}
  SBV(\Omega) \to L^1(\Omega, \mathscr{B}(\Omega), \mathscr{L}^n) \times
  L^1(\Omega, \mathscr{B}(\Omega), \mathscr{H}^{n-1}; \R^n) \times L^1(\Omega,
  \mathscr{B}(\Omega), \mathscr{L}^n ; \R^n).
\end{equation*}
Since $(\Omega,\mathscr{B}(\Omega),\mathscr{L}^n)$ is $\sigma$-finite, the dual of the corresponding $L^1$ space is, of course, the corresponding $L^\infty$ space.
This remark does not apply to the middle measure space $(\Omega,\mathscr{B}(\Omega),\mathscr{H}^{n-1})$.
An attempt at understanding the dual of the corresponding $L^1$ space leads to a study of the canonical map
\begin{equation*}
\Upsilon \colon L^\infty(\Omega,\mathscr{B}(\Omega),\mathscr{H}^{n-1}) \to L^1(\Omega,\mathscr{B}(\Omega),\mathscr{H}^{n-1})^*\,.
\end{equation*}
This relates to the Radon-Nikod\'ym Theorem.
In this respect, we note that:
\begin{itemize}
\item $\Upsilon$ is not surjective. If $\mathscr{B}(\Omega)$ is replaced
  with $\mathscr{M}_{\mathscr{H}^{n-1}}(\Omega)$, whether $\Upsilon$ is surjective
  or not is undecidable in ZFC.
\item $\Upsilon$ is injective. If $\mathscr{B}(\Omega)$ is replaced with
  $\mathscr{M}_{\mathscr{H}^{n-1}}(\Omega)$, $\Upsilon$ is not injective.
\end{itemize}
Here, $\mathscr{M}_{\mathscr{H}^{n-1}}(\Omega)$ is the $\sigma$-algebra of
$\mathscr{H}^{n-1}$-measurable subsets of $\Omega$.
For a detailed treatment of these, see \cite{DEP.21}.
Our goal now is to stick with the axioms of ZFC.
\par
The problem is with measurability.
Let $\varphi \in L^1(\Omega,\mathscr{B}(\Omega),\mathscr{H}^{n-1})$ and let $E \subset \Omega$ be Borel such that $\mathscr{H}^{n-1}(E) < \infty$.
If $\iota_E \colon L^1(E,\mathscr{B}(E),\mathscr{H}^{n-1}) \to L^1(\Omega,\mathscr{B}(\Omega),\mathscr{H}^{n-1})$ is the obvious injection, then $\varphi \circ \iota_E$ belongs to the dual of $L^1(E,\mathscr{B}(E),\mathscr{H}^{n-1})$ and, by the classical Riesz Theorem, is represented by a Borel measurable function $g_E \colon E \to \R$, i.e. $(\varphi \circ \iota_E)(f) = \int_E fg_E d\mathscr{H}^{n-1}$ whenever $f \in L^1(E,\mathscr{B}(E),\mathscr{H}^{n-1})$.
The family $\langle g_E\rangle_E$ is compatible in the sense that $\mathscr{H}^{n-1}(E \cap E' \cap \{ g_E \neq g_{E'}\})=0$ for all $E, E'$.
There does not exist, in general, a {\it gluing} of this family, i.e. a Borel measurable function $g \colon \Omega \to \R$ such that $\mathscr{H}^{n-1}(E \cap \{ g \neq g_E \}) = 0$ for all $E$, and whether an $\mathscr{H}^{n-1}$-measurable gluing of $\langle g_E \rangle_E$ exists is undecidable.
\par
In fact, a ``proper'' gluing of $\langle g_E \rangle_E$ at $x \in \Omega$ depends both on $x$ and $E$.
However, it will depend on $(x,E)$ only according to the behavior of $E$ near $x$.
This prompts us to introduce a notion of approximate germ of $E$ at $x$ -- finer than the first order tangential behavior in case $E$ is, for instance, a smooth hypersurface.
Carrying out these ideas to describe the dual of $L^1(\Omega,\mathscr{B}(\Omega),\mathscr{H}^{n-1})$ presents technical difficulties that we avoid by considering a slightly different measure space.
The jump set $S_u$ of a function of bounded variation being countably
rectifiable, one has $\mathscr{H}^{n-1} \hel S_u = \mathscr{I}^{n-1}_\infty \hel
S_u$, where $\mathscr{I}^{n-1}_\infty$ is an integral geometric measure.
The application to studying $SBV(\Omega)^*$ is not affected either when we replace $\mathscr{I}^{n-1}_\infty$ by its semi-finite version $\mathscr{I}^{n-1}$.
Both measures are described in \ref{sf}.
What we have gained is that each set $E \in \mathscr{B}(\Omega)$ with $\mathscr{I}^{n-1}(E) < \infty$ has the property that $\Theta^{n-1}(E,x)=1$ for $\mathscr{I}^{n-1}$-almost every $x \in E$, according to the Structure Theorem.
One recognizes a density property.
This is an essential ingredient of our proof below and would not hold with $\mathscr{H}^{n-1}$ in place of $\mathscr{I}^{n-1}$.

\par
Notice that a family $\langle g_E \rangle_E$ is, in fact, the same thing as a function $g(x,E)$ that depends on both $x$ and $E$, as in \cite{DEP.98}, but the compatibility of the family $\langle g_E \rangle_E$ considered above means that there is then some redundancy in the corresponding space of variables $(x,E)$.
The gist of the present paper is to consider a special quotient of the space of pairs $(x,E)$ and to equip it with a structure of a measure space $(\hat{\Omega},\hat{\mathscr{A}},\hat{\mathscr{I}}^{n-1})$, see \ref{sf} below.
One can of course wonder if the quotient presented here is ``optimal'', i.e. the smallest one for the purpose of representing the dual of $L^1$, or if we are still left with some redundancy in the variables.
The optimality of the measure space $(\hat{\Omega},\hat{\mathscr{A}},\hat{\mathscr{I}}^{n-1})$ is stated in \cite{BOU.DEP.21} and established there, in an appropriate categorical setting.
Thus, the combination of \cite{BOU.DEP.21} and the present paper can
be considered a solution to describing the dual of $SBV(\Omega)$, see
Theorem~\ref{main}, as explicitly as one possibly can in ZFC.

\begin{Empty}[Notations]
  Our choice of notation is mostly compatible with that of \cite{GMT}
  and \cite{AMBROSIO.FUSCO.PALLARA}. As an exception, we let $\ominus$
  be the set theoretic symmetric difference.  In $\R^n$, the
  $(n-1)$-dimensional Hausdorff measure is $\mathscr{H}^{n-1}$ and the
  Lebesgue measure is $\mathscr{L}^n$. The $\sigma$-algebras consisting of
  Borel subsets and $\mathscr{H}^{n-1}$-measurable subsets of $E \subset
  \R^n$ are respectively denoted by $\mathscr{B}(E)$ and
  $\mathscr{M}_{\mathscr{H}^{n-1}}(E)$. The restriction symbol for measures is
  $\hel$ \cite[2.1.2]{GMT}. The lower and upper $(n-1)$-dimensional
  densities of $E \subset \R^n$ at $x \in \R^n$, denoted
  $\Theta_*^{n-1}(E,x)$ and $\Theta^{*\,n-1}(E,x)$, respectively, are
  defined in \cite[2.10.19]{GMT} (with $\mu = \mathscr{L}^n \hel E$). Both
  $x \mapsto \Theta_*^{n-1}(E,x)$ and $x \mapsto \Theta^{*\,n-1}(E,x)$
  are Borel measurable. When they coincide at $x$, the common value is
  denoted $\Theta^{n-1}(E,x)$.
\end{Empty}

\section{Results}

\begin{Empty}
  \label{para1}
  A set $E \subset \R^n$ is called \textbf{countably
    $(n-1)$-rectifiable} whenever there are countably many Lipschitz
  maps $f_k \colon \R^{n-1} \to \R^n$ such that
  \[
  \mathscr{H}^{n-1}\left( E \setminus \bigcup_{k = 0}^\infty f_k(\R^{n-1})
  \right) = 0.
  \]
  Also, $E$ is called \textbf{$(n-1)$-rectifiable} if it is
  $\mathscr{H}^{n-1}$-measurable, of finite $\mathscr{H}^{n-1}$-measure, and
  countably $(n-1)$-rectifiable.

  Throughout the paper, we fix an open subset $\Omega \subset
  \R^n$. For an $(n-1)$-rectifiable set $E \subset \Omega$, the
  following hold:
  \begin{enumerate}
  \item[(A)] $\Theta^{n-1}(E, x) = 1$ for $\mathscr{H}^{n-1}$-almost all $x
    \in E$, \cite[3.2.19]{GMT};
  \item[(B)] $\Theta^{n-1}(E, x) = 0$ for $\mathscr{H}^{n-1}$-almost all $x
    \in \Omega \setminus E$, \cite[2.10.19(4)]{GMT}.
  \end{enumerate}
  The set $\tilde{E} = \{x \in \Omega : \Theta^{n-1}(E, x) = 1\}$
  coincides $\mathscr{H}^{n-1}$-almost everywhere with $E$, i.e. $\mathscr{H}^{n-1}(\tilde{E} \ominus E) = 0$. It satisfies:
  \begin{enumerate}
  \item[(C)] $\tilde{E}$ is a Borel measurable $(n-1)$-rectifiable
    subset of $\Omega$;
  \item[(D)] A point $x \in \Omega$ is in $\tilde{E}$ iff
    $\Theta^{n-1}(\tilde{E}, x) = 1$.
  \end{enumerate}
\end{Empty}

\begin{Empty}
\label{sf}
For $S \subset \Omega$, we let
  \[
  \mathscr{I}^{n-1}(S) = \sup \left\{ \mathscr{H}^{n-1}(S \cap E) : E \subset
  \Omega \text{ is an } (n-1)\text{-rectifiable set} \right\}.
  \]
  It is easily checked that $\mathscr{I}^{n-1}$ is a Borel regular outer
  measure on $\Omega$. In fact, the measure space
  $(\Omega,\mathscr{B}(\Omega),\mathscr{I}^{n-1})$ is the semi-finite version, in
  the sense of \cite[213X(c)]{FREMLIN.II}, of
  $(\Omega,\mathscr{B}(\Omega),\mathscr{I}^{n-1}_\infty)$, where the integral
  geometric measure $\mathscr{I}^{n-1}_\infty$ is defined in
  \cite[2.10.5(1)]{GMT}. The latter is a consequence of the Structure
  Theorem \cite[3.3.14]{GMT} and \cite[3.2.26]{GMT}. The measure
  $\mathscr{I}^{n-1}$ is much more manageable than the Hausdorff measure
  $\mathscr{H}^{n-1}$, as it ignores purely unrectifiable sets. We notice
  that $\mathscr{H}^{n-1} \hel E = \mathscr{I}^{n-1} \hel E$ for any countably
  $(n-1)$-rectifiable Borel set $E \subset \Omega$. To find an
  integral representation for the elements of $SBV(\Omega)^*$, it is
  crucial to describe the dual of $L^1(\Omega, \mathscr{B}(\Omega),
  \mathscr{I}^{n-1})$. We will do this in Theorem~\ref{dualL1}. Prior to
  that, we must construct a particular measure space $(\hat{\Omega},
  \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$.
\end{Empty}

\begin{Empty}[The measure space $(\hat{\Omega}, \hat{\mathscr{A}},
    \hat{\mathscr{I}}^{n-1})$]
\label{fs}
    Let $\mathscr{E}$ be the collection of sets that
  satisfy conditions~(C) and~(D) of \ref{para1}.  For each
  $x \in \Omega$, we let $\mathscr{E}_x \mathrel{\mathop:}= \{E \in \mathscr{E} : x \in
  E\}$. We define an equivalence relation $\sim_x$ on $\mathscr{E}_x$:
  \[
  E \sim_x E' \text{ iff } \Theta^{n-1}(E \ominus E', x) = 0 \text{
    iff } \Theta^{n-1}(E \cap E', x) = 1.
  \]
  The equivalence class $[E]_x$ under $\sim_x$ of some $E \in \mathscr{E}_x$
  is called the {\em approximate germ of $E$ at $x$}. We will use
  repeatedly the following consequence of \ref{para1}(A):
  if $E, E' \in \mathscr{E}$, then $[E]_x = [E']_x$ for $\mathscr{H}^{n-1}$-almost
  all $x \in E \cap E'$.

  We let $\hat{\Omega}$ be the set consisting of the pairs $(x,
  [E]_x)$, where $x \in \Omega$ and $[E]_x$ is some approximate germ
  at $x$. This set should be thought of as a fiber space over $\Omega$,
  with respect to the obvious projection map $p \colon \hat{\Omega} \to \Omega$
  that sends each $(x, [E]_x)$ to $x$. Below, we equip $\hat{\Omega}$
  with a $\sigma$-algebra and a measure.

  For each $E \in \mathscr{E}$, we let $\gamma_E \colon E \to \hat{\Omega}$
  be the map defined by $ \gamma_E(x) \mathrel{\mathop:}= (x, [E]_x)$. We define
  $\hat{\mathscr{A}}$ to be the finest $\sigma$-algebra on $\hat{\Omega}$ so
  that all maps $\gamma_E$ are $(\mathscr{M}_{\mathscr{H}^{n-1}}(E),
  \hat{\mathscr{A}})$-measurable, that is
  \[
  \hat{\mathscr{A}} \mathrel{\mathop:}= \left\{ A \subset \hat{\Omega} : \gamma_E^{-1}(A)
  \in \mathscr{M}_{\mathscr{H}^{n-1}}(E) \text{ for all } E \in \mathscr{E}\right\}.
  \]
  For all $E \in \mathscr{E}$, we denote by $\mu_E$ the measure on
  $(\hat{\Omega}, \hat{\mathscr{A}})$ defined by $\mu_E(A) \mathrel{\mathop:}=
  \mathscr{H}^{n-1}(\gamma_E^{-1}(A))$.  We let $\hat{\mathscr{I}}^{n-1}$ be the
  least upper bound of the measures $\mu_E$, with $E$ running over
  $\mathscr{E}$ (see e.g. \cite[1.68]{AMBROSIO.FUSCO.PALLARA}). The measure
  space $(\hat{\Omega}, \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$ is complete:
  a set $N \subset \hat{\Omega}$ is negligible iff $\gamma_E^{-1}(N)$
  is $\mathscr{H}^{n-1}$-negligible for all $E \in \mathscr{E}$.

  From the definition of the $\sigma$-algebra $\hat{\mathscr{A}}$, it is
  easy to show that a map $g \colon \hat{\Omega} \to Z$ with values in
  a measurable space $(Z, \mathscr{C})$ is $(\hat{\mathscr{A}}, \mathscr{C})$-measurable
  iff all maps $g \circ \gamma_E$ are $(\mathscr{M}_{\mathscr{H}^{n-1}}(E),
  \mathscr{C})$-measurable. As a consequence, the projection map $p$ is
  $(\hat{\mathscr{A}}, \mathscr{B}(\Omega))$-measurable, for the maps $p \circ
  \gamma_E$ are the inclusion maps $E \to \Omega$ and, therefore, are
  $\mathscr{H}^{n-1}$-measurable.
\end{Empty}

\begin{Lemma}
  \label{propMeasI}
  The following hold:
  \begin{enumerate}
  \item For all $A \in \hat{\mathscr{A}}$, one has $\hat{\mathscr{I}}^{n-1}(A) =
    \sup \left\{ \mu_E(A) : E \in \mathscr{E}
    \right\}$.

  \item $p_\# \hat{\mathscr{I}}^{n-1} = \mathscr{I}^{n-1}$.
  \item For $E \in \mathscr{E}$, we have $\hat{\mathscr{I}}^{n-1}(p^{-1}(E)
    \setminus \gamma_E(E)) = 0$
  \end{enumerate}
\end{Lemma}

\begin{proof}
  (1) Letting $\nu(A)$ denote the right hand side of the claimed equality, it suffices to check that $A \mapsto \nu(A)$ defines a
  measure. Clearly, $\nu$ is
  non-decreasing and $\sigma$-subadditive.

  It remains to check that $\nu(A_1 \cup A_2) \geq
  \nu(A_1) + \nu(A_2)$ for any pair of
  disjoint $A_1, A_2 \in \hat{\mathscr{A}}$. Let $E_1, E_2 \in
  \mathscr{E}$. Recalling the discussion in \ref{para1}, there is
  a set $E \in \mathscr{E}$ that coincides $\mathscr{H}^{n-1}$-almost everywhere
  with $E_1 \cup E_2$. For $i \in \{1, 2\}$, we have $[E]_x = [E_i]_x$
  for $\mathscr{H}^{n-1}$-almost all $x \in E \cap E_i$. From this, it
  follows that
  \[
  \gamma_{E_i}^{-1}(A_i) \setminus \gamma_E^{-1}(A_i) \subset (E_i
  \setminus E) \cup \{x \in E \cap E_i : [E]_x \neq [E_i]_x\}
  \]
  is $\mathscr{H}^{n-1}$-negligible. Hence, $\mu_E(A_i) \geq \mu_{E_i}(A_i)$.
  As $A_1$ and $A_2$ are disjoint, one has $\mu_E(A_1 \cup A_2) =
  \mu_E(A_1) + \mu_E(A_2) \geq \mu_{E_1}(A_1) + \mu_{E_2}(A_2)$.  The
  proof is then completed by taking the supremum over $E_1, E_2 \in
  \mathscr{E}$.

  (2) For all $B \in \mathscr{B}(\Omega)$, one has by~(1)
  \begin{align*}
  \hat{\mathscr{I}}^{n-1}(p^{-1}(B)) & = \sup \left\{
  \mathscr{H}^{n-1}(\gamma_E^{-1}p^{-1}(B)) : E \in \mathscr{E} \right\} \\ & =
  \sup \left\{ \mathscr{H}^{n-1}(B \cap E) : E \in \mathscr{E} \right\},
  \end{align*}
  which is $\mathscr{I}^{n-1}(B)$.

  (3) Let $E' \in \mathscr{E}$. The set
  \[
  \gamma_{E'}^{-1}(p^{-1}(E) \setminus \gamma_E(E)) = \{x \in E \cap
  E' : [E]_x \neq [E']_x \}
  \]
  has $\mathscr{H}^{n-1}$ measure zero. By the arbitrariness of $E'$, it
  follows that $p^{-1}(E) \setminus \gamma_E(E) \in \hat{\mathscr{A}}$ and
  $\hat{\mathscr{I}}^{n-1}(p^{-1}(E) \setminus \gamma_E(E)) = 0$.
\end{proof}

\begin{Empty}[Density points]
  We recall the following well-known facts regarding the existence of
  Lebesgue density points for functions defined on rectifiable
  sets. Let $E$ be an $(n-1)$-rectifiable set and $h \colon E \to \R$
  be a function in $L^1(E, \mathscr{M}_{\mathscr{H}^{n-1}}(E), \mathscr{H}^{n-1} \hel
  E)$, we have
  \[
  \frac{1}{\boldsymbol{\alpha}(n-1)r^{n-1}} \int_{E \cap B(x,r)} h \, d
  \mathscr{H}^{n-1} \xrightarrow[r \to 0]{} h(x)
  \]
  at $\mathscr{H}^{n-1}$-almost every $x \in E$. When the limit exists, $x$
  is called a density point of $h$. The proof of the next lemma is
  elementary.
\end{Empty}

\begin{Lemma}
  \label{sameDens}
  Let $E, E'$ be two Borel measurable $(n-1)$-rectifiable susbsets of
  $\Omega$, $h \in L^\infty(E, \mathscr{M}_{\mathscr{H}^{n-1}}(E), \mathscr{H}^{n-1}
  \hel E)$, $h' \in L^\infty(E', \mathscr{M}_{\mathscr{H}^{n-1}}(E'), \mathscr{H}^{n-1}
  \hel E')$ be two functions, and $x \in E \cap E'$. Assume that
  \begin{enumerate}
  \item[(A)] $h = h'$ $\mathscr{H}^{n-1}$-almost everywhere on $E \cap E'$;
  \item[(B)] $\Theta^{n-1}(E,x) = \Theta^{n-1}(E', x) = \Theta^{n-1}(E \cap E', x) = 1$;
  \item[(C)] $x$ is a density point of $h$.
  \end{enumerate}
  Then $x$ is a density point of $h'$ and
  \[
  \lim_{r \to 0} \frac{1}{\boldsymbol{\alpha}(n-1)r^{n-1}} \int_{E \cap B(x,r)} h
  \, d \mathscr{H}^{n-1} = \lim_{r \to 0} \frac{1}{\boldsymbol{\alpha}(n-1)r^{n-1}}
  \int_{E' \cap B(x,r)} h' \, d \mathscr{H}^{n-1}.
  \]
\end{Lemma}

We now come to our main result.

\begin{Theorem}
  \label{dualL1}
  The map $\Upsilon \colon L^\infty(\hat{\Omega}, \hat{\mathscr{A}},
  \hat{\mathscr{I}}^{n-1}) \to L^1(\Omega, \mathscr{B}(\Omega), \mathscr{I}^{n-1})^*$
  defined by
  \[
  \Upsilon(g)(f) = \int_{\hat{\Omega}} g(f \circ p) \, d\hat{\mathscr{I}}^{n-1}
  \]
  is an isometric isomorphism.
\end{Theorem}

\begin{proof}
  Let $g \in L^\infty(\hat{\Omega}, \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$
  and $f \in L^1(\Omega, \mathscr{B}(\Omega), \mathscr{I}^{n-1})$. Then
  \begin{equation*}
    |\Upsilon(g)(f)|  \leq \|g\|_\infty \int_{\hat{\Omega}} |f \circ p| \, d \hat{\mathscr{I}}^{n-1}
     = \|g\|_\infty \int_{\hat{\Omega}} |f| \, dp_{\#}\hat{\mathscr{I}}^{n-1} = \|g\|_\infty\|f\|_1,
  \end{equation*}
  by Lemma~\ref{propMeasI}(2). This shows that $\Upsilon$ is
  well-defined and $\|\Upsilon\| \leq 1$.

  Let $\varepsilon > 0$. The set $A \mathrel{\mathop:}= \{|g| > \|g\|_\infty -
  \varepsilon\}$ has positive $\hat{\mathscr{I}}^{n-1}$ measure, hence we
  infer the existence of an $(n-1)$-rectifiable set $E \in \mathscr{E}$ such
  that $\mathscr{H}^{n-1}(\gamma_E^{-1}(A)) > 0$. The function $g \circ
  \gamma_E$ is $\mathscr{H}^{n-1}$-measurable. As the measure space $(E,
  \mathscr{M}_{\mathscr{H}^{n-1}}(E), \mathscr{H}^{n-1} \hel E)$ is the completion of
  $(E, \mathscr{B}(E), \mathscr{H}^{n-1} \hel E)$, we can find a Borel measurable
  function $g_E \colon E \to \R$ that coincides almost everywhere with
  $g \circ \gamma_E$. We let $f \in L^1(\Omega, \mathscr{B}(\Omega),
  \mathscr{I}^{n-1})$ be the map
  \[
  f \colon x \mapsto \begin{cases} 1 & \text{if } x \in E \text{ and }
    g_E(x) > \|g\|_\infty - \varepsilon \\ -1 & \text{if } x \in E
    \text{ and } g_E(x) < -\left(\|g\|_\infty - \varepsilon \right)
    \\ 0 & \text{otherwise}
  \end{cases}
  \]
  of norm $\|f\|_1 = \mathscr{H}^{n-1}(\gamma_E^{-1}(A))$. As $p \circ
  \gamma_E$ is the inclusion map $E \to \Omega$, we have
  $\gamma_E(p(z)) = z$ for all $z \in \gamma_E(E)$. By
  Lemma~\ref{propMeasI}(3), it then follows that $\gamma_E(p(z)) = z$
  for $\hat{\mathscr{I}}^{n-1}$-almost all $z \in p^{-1}(E)$. Hence, $g$ and
  $g \circ \gamma_E \circ p$ coincide $\hat{\mathscr{I}}^{n-1}$-everywhere
  on $p^{-1}(E)$. Moreover, we have
  \[
  p^{-1}(E) \cap \{ g \circ \gamma_E \circ p \neq g_E \circ p\} \subset p^{-1}(\{g
  \circ \gamma_E \neq g_E\})
  \]
  By the Borel regularity of $\hat{\mathscr{I}}^{n-1}$ and
  Lemma~\ref{propMeasI}(2), we also infer that $g \circ \gamma_E \circ
  p =g_E \circ p$ almost everywhere on $p^{-1}(E)$. Whence
  \begin{align*}
  \Upsilon(g)(f) = \int_{p^{-1}(E)} g(f \circ p) \, d
  \hat{\mathscr{I}}^{n-1} & = \int_{p^{-1}(E)} (g_E \circ p) (f \circ p) \,
  d \hat{\mathscr{I}}^{n-1} \\ & = \int_E g_E f \, d
  \mathscr{I}^{n-1},
  \end{align*}
  Thus, by construction of $f$, we have $\Upsilon(g)(f) \geq
  (\|g\|_\infty - \varepsilon) \|f\|_1$. This shows, as $\varepsilon >
  0$ can be taken arbitrarily small, that $\|\Upsilon(g)\| = 1$ and
  $\Upsilon$ is an isometry.

  Now we turn to establishing the surjectivity of $\Upsilon$. For any $E \in
  \mathscr{E}$, we define a map $\alpha_E \in L^1(E, \mathscr{B}(E), \mathscr{H}^{n-1}
  \hel E)^*$ that is the localized version of $\alpha$: for an
  integrable function $f \in L^1(E, \mathscr{B}(E), \mathscr{H}^{n-1} \hel E)$, we
  set $\alpha_E(f) \mathrel{\mathop:}= \alpha(\overline{f})$, where $\overline{f}$
  is the extension by zero of $f$ to $\Omega$. Since $(E, \mathscr{B}(E),
  \mathscr{H}^{n-1} \hel E)$ is a finite measure space, the standard duality
  between $L^1$ and $L^\infty$ spaces holds, so there is a function
  $g_E \in L^\infty(E, \mathscr{B}(E), \mathscr{H}^{n-1} \hel E)$ such that
  \[
  \alpha_E(f) = \int_E g_E f \, d\mathscr{H}^{n-1} \text{ for all } f \in
  L^1(E, \mathscr{B}(E), \mathscr{H}^{n-1} \hel E).
  \]
  Obviously, for any $E, E' \in \mathscr{E}$, we have
  \[
  \int_{E\cap E'} g_E f \, d \mathscr{H}^{n-1} = \int_{E \cap E'} g_{E'} f
  \, d \mathscr{H}^{n-1}
  \]
  for any integrable function $f$ on $E \cap E'$, which implies that
  $g_E = g_{E'}$ almost everywhere on $E \cap E'$.

  Let $g \colon \hat{\Omega} \to \R$ be the function partially defined
  by
  \[
  g(x, [E]_x) \mathrel{\mathop:}= \lim_{r \to 0} \frac{1}{\boldsymbol{\alpha}(n-1) r^{n-1}}
  \int_{E \cap B(x,r)} g_E \, d \mathscr{H}^{n-1}
  \]
  whenever $x$ is a density point of $g_E$. The compatibility between
  the local Radon-Nikod\'{y}m derivatives $g_E$, $E \in \mathscr{E}$,
  together with Lemma~\ref{sameDens}, guarantees that $g$ is
  well-defined. Denoting by $N \subset \hat{\Omega}$ the set of points
  at which $g$ is not defined, we readily see that, for any $E \in
  \mathscr{E}$, the set $\gamma_E^{-1}(N)$ consists of the $x \in E$ that
  are not density points of $g_E$. We readily have
  $\mathscr{H}^{n-1}(\gamma_E^{-1}(N)) = 0$, and this shows that
  $\hat{\mathscr{I}}^{n-1}(N) = 0$. We can extend $g$ to $\hat{\Omega}$ by
  sending the elements of $N$ to an arbitrary value. Then $g$ is
  $\hat{\mathscr{A}}$-measurable. Indeed, for any $E \in \mathscr{E}$ the function
  $g \circ \gamma_E$ coincides $\mathscr{H}^{n-1}$-almost every with $g_E$,
  and therefore is $\mathscr{H}^{n-1}$-measurable.

  Finally, we claim that $\alpha = \Upsilon(g)$. Let $f \in L^1(\Omega,
  \mathscr{B}(\Omega), \mathscr{I}^{n-1})$. Choose $\varepsilon > 0$ and set $E
  \mathrel{\mathop:}= \{|f| > \varepsilon\}$. As $\mathscr{I}^{n-1}(E) < \infty$, we can
  suppose, up to a modification of $f$ on a negligible set, that $E
  \in \mathscr{E}$. Let $f_E$ be the restriction of $f$ to $E$. Then,
  recalling that $\mathscr{H}^{n-1} \hel E = \mathscr{I}^{n-1} \hel E$, we have
  \[
  \alpha(\ind_{\{|f| > \varepsilon\}} f) = \alpha_E(f_E) = \int_E g_E f_E \,
  d\mathscr{I}^{n-1}.
  \]
  By Lemma~\ref{propMeasI}(2), we deduce that
  \[
  \alpha(\ind_{\{|f| > \varepsilon\}} f) = \int_{p^{-1}(E)} (g_E \circ
  p)(f_E \circ p) \, d \hat{\mathscr{I}}^{n-1}.
  \]
  Clearly, $f_E \circ p = f \circ p$ on $p^{-1}(E)$ and $g_E\circ p =
  g$ almost everywhere on $\gamma_E(E)$. By Lemma~\ref{propMeasI}(3),
  this ensures that $g_E \circ p = g \circ p$ almost everywhere on
  $p^{-1}(E)$. Thus,
  \[
  \alpha(\ind_{\{|f| > \varepsilon\}} f) = \int_{p^{-1}(E)} g (f \circ p)
  \, d \hat{\mathscr{I}}^{n-1} = \int \ind_{\{|f \circ p| > \varepsilon\}} g (f
  \circ p) \, d \hat{\mathscr{I}}^{n-1}.
  \]
  Letting $\varepsilon \to 0$ yields $\alpha(f) = \Upsilon(g)(f)$.
\end{proof}

\begin{Remark}
  The above proof exhibits a construction of the density $g$, by
  gluing the local Radon-Nikod\'ym derivatives $g_E$ in the measure
  space $(\hat{\Omega}, \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$. The gluing
  is not possible within the measure space $(\Omega, \mathscr{B}(\Omega),
  \mathscr{I}^{n-1})$, as it is not localizable (if $\Omega$ is non
  empty). It was proven in \cite[Section 11]{BOU.DEP.21} that
  $(\hat{\Omega}, \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$ is the ``minimal''
  measure space in which a compatible family of functions defined on
  subsets of $\Omega$ can be glued. Using the concept introduced in
  \cite{BOU.DEP.21}, $(\hat{\Omega}, \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$
  is the strictly localizable version of the completion of $(\Omega,
  \mathscr{B}(\Omega), \mathscr{I}^{n-1})$.

  Assuming that $(\hat{\Omega}, \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$ is
  localizable, we can give an alternate proof of Theorem~\ref{dualL1}:
  the map $\iota \colon L^1(\Omega, \mathscr{B}(\Omega), \mathscr{I}^{n-1}) \to
  L^1(\hat{\Omega}, \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$ that sends $f$ to
  $f \circ p$ is an isometry, as $\|f \circ p\|_1 = \|f\|_1$ by
  Lemma~\ref{propMeasI}(3). By the localizability of $(\hat{\Omega},
  \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$, the dual of $L^1(\hat{\Omega},
  \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$ is $L^\infty(\hat{\Omega},
  \hat{\mathscr{A}}, \hat{\mathscr{I}}^{n-1})$, in such a way that $\Upsilon$ is
  just the adjoint map of $\iota$. Reasoning as before, $\Upsilon$ is
  injective, which entails that $\iota$ and then $\Upsilon$ are
  isometric isomorphisms.
\end{Remark}

\begin{Empty}[Approximate tangent cone of an approximate germ]
  \label{para2.9}
  Let $E \in \mathscr{E}$ and $x \in E$. The approximate tangent cone
  $\rmTan^{n-1}(E, x)$ (see \cite[3.2.16]{GMT}) is unchanged if we
  substitute $E$ with any $E' \in \mathscr{E}_x$ such that $E \sim_x
  E'$. Because of this, we can define the approximate tangent cone
  $\rmTan^{n-1}([E]_x) \mathrel{\mathop:}= \rmTan^{n-1}(E, x)$ of an approximate
  germ. We recall that $\rmTan^{n-1}(E, x)$ is an $(n-1)$-plane at
  $\mathscr{H}^{n-1}$-almost every point $x \in E$, and that the function $x
  \mapsto \rmTan^{n-1}(E, x) \in \mathbf{G}(n-1, n)$, defined almost
  everywhere on $E$, is $\mathscr{H}^{n-1}$-measurable. Here $\mathbf{G}(n-1, n)$
  denotes the Grassmann manifold of hyperplanes in $\R^n$.
\end{Empty}

\begin{Theorem}
\label{main}
  Let $\varphi \colon SBV(\Omega) \to \R$ be a continuous linear
  functional. There are fields $f \in L^\infty(\Omega, \mathscr{B}(\Omega),
  \mathscr{L}^n)$, $g \in L^\infty(\hat{\Omega}, \hat{\mathscr{A}},
  \hat{\mathscr{I}}^{n-1} ; \R^n)$ and $h \in L^\infty(\Omega,
  \mathscr{B}(\Omega), \mathscr{L}^n ; \R^n)$ such that
  \begin{equation}
    \label{dualSBV}
    \varphi(u) = \int_\Omega fu \, d \mathscr{L}^n + \int_{\hat{\Omega}} g \cdot (j_u \circ
    p) d \hat{\mathscr{I}}^{n-1} + \int_\Omega h \cdot \nabla u \, d \mathscr{L}^n
  \end{equation}
  where the distributional gradient $Du$ of $u$ is decomposed into its
  Lebesgue part $\mathscr{L}^n \hel \nabla u$ and its jump part $D^j u = j_u
  \, \mathscr{H}^{n-1} \hel S_u$. Moreover, one can select $g$ in such a way
  that $g(x, [E]_x)$ is normal to $\rmTan^{n-1}([E]_x)$ for
  $\hat{\mathscr{I}}^{n-1}$-almost all $(x, [E]_x) \in \hat{\Omega}$.
\end{Theorem}

\begin{proof}
  For each $u \in SBV(\Omega)$, we have
  \[
  \|u\|_{SBV(\Omega)} = \int_\Omega |u| \, d \mathscr{L}^n + \int_{S_u} \|j_u\| \,
  d \mathscr{H}^{n-1} + \int_\Omega \|\nabla u\| \, d \mathscr{L}^n.
  \]
  Here, the approximate discontinuity set $S_u$ (see
  \cite[3.63]{AMBROSIO.FUSCO.PALLARA}) is Borel measurable and
  countably $(n-1)$-rectifiable \cite[3.64(a) and
    3.78]{AMBROSIO.FUSCO.PALLARA}. We recall that $j_u = 0$ outside
  $S_u$, whereas $j_u = (u^+ - u^-) \nu_u$ almost everywhere on
  $S_u$. In particular, $\mathscr{H}^{n-1} \hel j_u =  \mathscr{I}^{n-1} \hel j_u$. Therefore, $u \mapsto (u, j_u, \nabla u)$ is an
  isometric embedding
  \[
  SBV(\Omega) \to L^1(\Omega, \mathscr{B}(\Omega), \mathscr{L}^n) \times
  L^1(\Omega, \mathscr{B}(\Omega), \mathscr{I}^{n-1}; \R^n) \times L^1(\Omega,
  \mathscr{B}(\Omega), \mathscr{L}^n ; \R^n).
  \]
  Hence, $\varphi$ can be split into three parts $\varphi \colon u
  \mapsto \varphi_1(u) + \varphi_2(j_u) + \varphi_3(\nabla u)$, where
  $\varphi_1 \in L^1(\Omega, \mathscr{B}(\Omega), \mathscr{L}^n)^*$, $\varphi_2
  \in L^1(\Omega, \mathscr{B}(\Omega), \mathscr{I}^{n-1}; \R^n)^*$, and $\varphi_3
  \in L^1(\Omega,\mathscr{B}(\Omega), \mathscr{L}^{n-1}; \R^n)^*$. The
  duality theorem~\ref{dualL1} provides an integral representation for the term
  $\varphi_2$ that acts on the jump part of the derivative and yields
  the formula~\eqref{dualSBV}.

  Regarding the second assertion, we claim that $j_u \circ p(x, [E]_x)
  = j_u(x)$ is normal to $\rmTan^{n-1}([E]_x)$ for almost all $(x,
  [E]_x) \in \hat{\Omega}$. For this, we need to show that, for all $E
  \in \mathscr{E}$,
  \begin{multline*}
    \{x \in E : j_u(x) \text{ not normal to } \rmTan^{n-1}(E, x)\}
    \\ = \{x \in E \cap S_u : j_u(x) \text{ not normal to }
    \rmTan^{n-1}(E, x)\}
  \end{multline*}
  is $\mathscr{H}^{n-1}$-negligible. This is clear, once we remark that
  $\rmTan^{n-1}(E, x) = \rmTan^{n-1}(S_u, x)$ for almost every $x \in
  E \cap S_u$ and that $j_u$ is normal to $S_u$ almost everywhere on
  $S_u$. Hence, we can replace the vector field $g$ by its normal part
  $\tilde{g}$ (that is, $\tilde{g}(x, [E]_x)$ is the orthogonal
  projection of $g(x, [E]_x)$ onto $\rmTan^{n-1}([E]_x)^\perp$). The
  measurability of $\tilde{g}$ follows from the discussion in
  Paragraph~\ref{para2.9}.
\end{proof}

\bibliographystyle{amsplain}

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text:
La historia de los judíos en el Reino Unido es tan antigua que incluso abarca mucho antes de la formación del Reino Unido como tal en 1707, por lo que está muy arraigada en la historia misma del país.

La primera comunidad judía registrada en Gran Bretaña fue llevada a Inglaterra en 1070 por el rey Guillermo I de Inglaterra, quien creía que sus habilidades comerciales harían que su país recién conquistado fuese más próspero. Doscientos años más tarde los judíos ya no eran bienvenidos. El rey Eduardo I de Inglaterra aprobó el Estatuto de la judería (Statutum de Judaísmo) en 1275, la cual restringía las actividades de la comunidad, más notablemente declarar ilegal la práctica de la usura (cobro de intereses). Cuando, 15 años después, Eduardo encontró que muchas de estas disposiciones fueron ignoradas, expulsó a los judíos de Inglaterra. Ellos emigraron a países como Polonia, donde eran protegidos por la ley. Una pequeña comunidad inglesa persistió en la clandestinidad a pesar de la expulsión. Los judíos no fueron prohibidos de Escocia, que hasta 1707 era un reino independiente.

El Reino Unido es el hogar de la segunda mayor población judía en Europa, después de Francia, y asimismo alberga la quinta comunidad judía más grande en todo el mundo. El número de personas que se identifican como judíos en Inglaterra y Gales ha aumentado ligeramente durante la última década, lo que indica una estabilidad que contrasta con la impresión habitual de descenso implacable de la diáspora. El incremento se debe en gran parte al rápido crecimiento de la comunidad jaredí.

Asimismo, durante la década de 2010 Gran Bretaña ha visto un aumento considerable de inmigrantes de origen judío, provenientes de Europa continental, principalmente de Francia, país que ha sido testigo de un aumento considerable de antisemitismo en los últimos años.

Referencias

Reino Unido
Judaísmo en el Reino Unido
judíos
Judios

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Mrs. Jacqueline Miriam Gift, 78, of Concord, passed away on Thursday, Sept. 2, 2010 at her residence. Mrs. Gift was born on Jan. 21, 1932, in Windsor, Pa. and was the daughter of the late John Emory Snyder and Ada Viola Houser Snyder. Mrs. Gift was a graduate of Red Lion Senior High School in Red Lion, Pa. She...
Jack Plummer Safley
Jack Plummer Safley, 81, died Monday, Oct. 25, 2010, at the North Carolina State Veterans Home. Born July 4, 1929, he was the son of the late Flora Melinda Plummer Safley and David Frank Safley of Salisbury and was a lifelong resident of the city. On March 10, 1957, Mr. Safley married Barbara Ann Carter of Lincolnton. Mr. Safley was...
Richard "Rick" Edwin Carver, Jr.
Richard Edwin Carver, Jr. 'Rick' Mr. Carver, 52, of Charlotte, passed away Sunday, December 19, 2010, at the Levine & Dickson Hospice House of Huntersville. Mr. Carver was born on August 13, 1958 in Mecklenburg County, a son of Edna Howell Carver and the late Richard Edwin Carver, Sr. Mr. Carver was a 1976 graduate of South Mecklenburg High School....


Yes, it's that time again, kiddies.
But first, let me regale my reader with some great Cat stuff.
And speaking of cats, who knew Pope Benedict was a life-long cat lover? He's no John Paul II, but this Protestant has finally given him some points for character! Now a new book about his life has been released and, oddly, it appears to have been written by one of his cats. Which makes me wonder. Just exactly where does the Pope keep the litter box?
from PewForum.org All News Feeds \ Ok, now. Who the hell thought this was a good idea?
But what about Joshua Bibles? Or Carlos Santana bibles? Could this be a moment when a ruling was, perhaps, too specific?
A Roman Catholic priest who floated off under hundreds of helium party balloons was missing Monday off the southern coast of Brazil.
"Show your library colleagues how much you will miss them by giving a demonstration and lecture on the Reader's Guide to Periodical Literature as your retirement party speech. Then, ride off into the sunset on your booktruck."
And high time, too. I can't tell you how many times a toddler has come up to me with a puzzled look and asked "Why are Mommy's boobies bigger now?"
HEY! You saying aging by itself ain't a thinga beauty?
"Yes, there is another library-themed anime series: Library War. The Anime News Network links to a commercial and a brief trailer for the series. I love that the series is focused on libraries battling a government that is trying to censor information...true greatness. The series debuted on Japanese television on April 10th and you can already find versions of it for (illegal) download all over the web. Not that I encourage that type of behavior. Oh no."
Well! Many Thank yous for referencing the Sock Snark!
The priest story was too funny. I mean, I'm sorry he's missing but someone really did that... wow.
Did they ever find the priest? I'm worried about him.


text:
Q: 403 forbidden for presigned_url upload I have 403 error when uploading presigned url
At first, I exec s3.generate_presigned_post in local
    temp = s3.generate_presigned_post(
        "my-resource-bucket-v","test"
    )

and get the url and fields
{'url': 'https://my-resource-bucket-v.s3.amazonaws.com/', 'fields': {'key': 'test', 'AWSAccessKeyId': 'AKIAZ3YPMLASV6736UA4', 'policy': 'eyJleHBpcmF0aW9uIjogIjIwMjItMDUtMDlUMDY6MDU6MTNaIiwgImNvbmRpdGlvbnMiOiBbeyJidWNrZXQiOiAic2kyLXMzLXNidS1qb2ItaHVudGluZy1keC10b2t5by1qeGMtc3RhdGljLXJlc291cmNlLXYifSwgeyJrZXkiOiAidGVzdCJ9XX0=', 'signature': 'F7E074YWIVwy4ZL2zXSv8YVTbyE='}}"

then I try this to upload
curl -v -X POST \
-F key="{'key': 'test', 'AWSAccessKeyId': 'AKIAZ3YPMLASV6736UA4', 'policy': 'eyJleHBpcmF0aW9uIjogIjIwMjItMDUtMDlUMDY6MTE6MjhaIiwgImNvbmRpdGlvbnMiOiBbeyJidWNrZXQiOiAic2kyLXMzLXNidS1qb2ItaHVudGluZy1keC10b2t5by1qeGMtc3RhdGljLXJlc291cmNlLXYifSwgeyJrZXkiOiAidGVzdCJ9XX0=', 'signature': 'GNUthYj0cec9uIQjeJsuap7OTfk='}" \
-F policy="{'key': 'test', 'AWSAccessKeyId': 'AKIAZ3YPMLASV6736UA4', 'policy': 'eyJleHBpcmF0aW9uIjogIjIwMjItMDUtMDlUMDY6MTE6MjhaIiwgImNvbmRpdGlvbnMiOiBbeyJidWNrZXQiOiAic2kyLXMzLXNidS1qb2ItaHVudGluZy1keC10b2t5by1qeGMtc3RhdGljLXJlc291cmNlLXYifSwgeyJrZXkiOiAidGVzdCJ9XX0=', 'signature': 'GNUthYj0cec9uIQjeJsuap7OTfk='}" \
-F file=myimage.png https://my-resource-bucket-v.s3.amazonaws.com/

It returns like this.
It looks like uploading is success but denied by auth?
I think bucket policy of S3 is not relevant with presigned_url, So,, where the permission error occurs??
Note: Unnecessary use of -X or --request, POST is already inferred.
*   Trying 52.219.196.109:443...
* Connected to my-resource-bucket-v.s3.amazonaws.com (52.219.196.109) port 443 (#0)
* ALPN, offering h2
* ALPN, offering http/1.1
* successfully set certificate verify locations:
*  CAfile: /etc/ssl/cert.pem
*  CApath: none
* TLSv1.2 (OUT), TLS handshake, Client hello (1):
* TLSv1.2 (IN), TLS handshake, Server hello (2):
* TLSv1.2 (IN), TLS handshake, Certificate (11):
* TLSv1.2 (IN), TLS handshake, Server key exchange (12):
* TLSv1.2 (IN), TLS handshake, Server finished (14):
* TLSv1.2 (OUT), TLS handshake, Client key exchange (16):
* TLSv1.2 (OUT), TLS change cipher, Change cipher spec (1):
* TLSv1.2 (OUT), TLS handshake, Finished (20):
* TLSv1.2 (IN), TLS change cipher, Change cipher spec (1):
* TLSv1.2 (IN), TLS handshake, Finished (20):
* SSL connection using TLSv1.2 / ECDHE-RSA-AES128-GCM-SHA256
* ALPN, server did not agree to a protocol
* Server certificate:
*  subject: CN=*.s3.amazonaws.com
*  start date: Dec 15 00:00:00 2021 GMT
*  expire date: Dec  3 23:59:59 2022 GMT
*  subjectAltName: host "my-resource-bucket-v.s3.amazonaws.com" matched cert's "*.s3.amazonaws.com"
*  issuer: C=US; O=Amazon; OU=Server CA 1B; CN=Amazon
*  SSL certificate verify ok.
> POST / HTTP/1.1
> Host: my-resource-bucket-v.s3.amazonaws.com
> User-Agent: curl/7.77.0
> Accept: */*
> Content-Length: 941
> Content-Type: multipart/form-data; boundary=------------------------5a06682e8a44d108
>
* We are completely uploaded and fine
* Mark bundle as not supporting multiuse
< HTTP/1.1 403 Forbidden
< x-amz-request-id: TBGPSXDQGM1JRAKV
< x-amz-id-2: NPU+clavFOAAUKCsc0wVb2Bi1Aeh86ce+k8J4vq6YSv+SCLC4bAzWSdbcgzhcYObU765goqg/Jw=
< Content-Type: application/xml
< Transfer-Encoding: chunked
< Date: Mon, 09 May 2022 05:17:55 GMT
< Server: AmazonS3
<
<?xml version="1.0" encoding="UTF-8"?>
* Connection #0 to host my-resource-bucket-v.s3.amazonaws.com left intact
<Error>AccessDenied<Message>Access Denied</Message><RequestId>TBGPSXDQGM1JRAKV</RequestId><HostId>NPU+clavFOAAUKCsc0wVb2Bi1Aeh86ce+k8J4vq6YSv+SCLC4bAzWSdbcgzhcYObU765goqg/Jw=</HostId></Error>%

According to @BoarGules helps.
I set the bucket policy as like this below
    {
        "Effect": "Allow",
        "Principal": {
            "AWS": "*"
        },
        "Action": "s3:*",
        "Resource": "arn:aws:s3:::si2-s3-sbu-job-hunting-dx-tokyo-jxc-static-resource-v"
    },

redpajama_set_name: RedPajamaStackExchange


A lovely, spacious Finnish sauna with aromatic steam. It will give you back your energy that you have lost during the sports activities, and will relieve your skin as well as your mind. There is an opportunity to cool yourself down by running outside in frost and snow. Then you will gain new strength in a spacious, cosy rest room.
If our masseuse is present, you can indulge yourself in oil massage of your whole body, lava stones, and chocolate or detoxicating honey massage.
Enjoy us taking care of your hands, the paraffin hand wrap will treat, soften and smooth your dry skin. While the paraffin is taking effect, your muscles and joints will relieve, and that is why paraffin wraps are used as a part of rehabilitative cure, especially in case of joint illness.


text:
Ariana Grande Engaged To 'SNL' Star Pete Davidson
By J'na Jefferson
J'na Jefferson
Plus Icon jnajefferson
More Stories by J'na
Come As You Are: PJ Morton On New Album, Individuality And Journey Towards Nirvana
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As She Should: Megan Thee Stallion Reportedly Files To Trademark 'Hot Girl Summer'
Pop superstar Ariana Grande and Saturday Night Live repertory player Pete Davidson are reportedly engaged, according to People. The couple, both 24-years-old, is in the relatively early stages of their relationship, and have only been dating for a few weeks.
"It's a recent engagement," a source told the magazine. "They're just two people who found love quickly and make each other happy all the time. They both started talking about it this past weekend. It's nothing they've been hiding," While they haven't commented publicly on their engagement, Grande has been favoriting fan tweets congratulating her.
Previously, the "No Tears Left To Cry" songbird was in a relationship with rapper Mac Miller. Despite being together for nearly two years, the musical couple officially called it quits a few weeks ago. On Twitter, Grande clapped back at someone for commenting on her decision to move on from the Divine Feminine MC. She detailed that their relationship was "toxic," but she wishes him the best.
"I am not a babysitter or a mother and no woman should feel that they need to be," she wrote. "I have cared for [Miller] and tried to support his sobriety & prayed for his balance for years (and always will of course) but shaming/blaming women for a man's inability to keep his sh*t together is a very major problem."
A post shared by Ariana Grande (@arianagrande) on Jun 9, 2018 at 10:32am PDT


The next documentary project for Professor Greg Mellott and his students in the film and video program will begin shooting in the spring.
It will be based on the book "The Grand Energy Transition" by Robert A. Hefner III.
Hefner is a lifetime natural gas explorer, and the documentary will be about his life and his book.
Hefner is sometimes referred to as the "father of natural gas" and this story will depict how he fostered that title, Mellott said.
Mellott and Gray Frederickson, OCCC's artist-in-residence and Academy Award-winning producer of the "Godfather" trilogy, will film with Hefner.
The project also will include interviews, stock footage, and archival footage, Mellott said. He said it will probably be under an hour in length.
Mellott has been a professor at OCCC for four years.
He, along with Frederickson, won Emmys for their documentary "Dream No Little Dream," a biography of Oklahoman Robert S. Kerr.
Mellott said all but one person involved in the documentary was either a graduate of or a student in OCCC'S film program.
Hefner said he became interested in making the documentary after seeing "Dream No Little Dream," Mellott said.
Mellott and Frederickson saw it as a great idea for the students in the program and also a way to bring Hefner's story to the screen.
Recording, editing, and set productions are just a few of the components necessary in making a successful documentary, such as "Dream No Little Dream" and "The Grand Energy Transition," Mellott said.
OCCC film students are capable of doing all of these, he said.
OCCC's film program has grown enormously since its birth from the Oklahoma Film Institute, Mellott said.
He said he hopes it will continue to grow through films such as this upcoming documentary.
For more information about the OFI, contact Mellott by e-mail at gmellott@occc.edu or by phone at 405-682-1611, ext. 7793.


CEMEX de Puerto Rico, Inc.
-Administer, develop, recommend, and advise quality control and environmental programs to comply with state and federal laws and regulations, and in compliance with shareholder goals and expectations. -Work with assigned budget to manage quality assurance, quality control, and environmental personnel to guarantee that all CEMEX products are in compliance with high quality standards, and in compliance with the applicable environmental regulations. -Responsible for the search, permit process, delivery logistics and price negotiations of Alternative Fuels to be consumed at plant, as part of CEMEX's Sustainability Process. -Develop, implement and administer quality and environmental management systems (ISO 9001 and ISO 14001). -Developed and conducted training in order to instruct managers, supervisors and workers in environmental pollution control. -Administer over 2MM USD in Capital Investment for the acquisition and operation of quality and environmental monitoring equipment, air pollution control, BMP's, etc.


text:
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    <declare-styleable name="TinderRecyclerView">
        <attr name="spanCount" format="integer" />
        <attr name="aspectRatio" format="float" />
        <attr name="numberColoum" format="integer"/>
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</resources>

redpajama_set_name: RedPajamaGithub


I'll be honest: I simply don't understand the bull case for Coca-Cola (NYSE:KO). The KO stock price has held up reasonably well in recent years, admittedly. But Coca-Cola stock isn't cheap. There are obvious risks to demand going forward. Add on disappointing growth, and the numbers here don't seem to add up.
I concede that I've long been a skeptic toward KO stock. I called it "expensive" 20 months ago at roughly the same price, and it's at least held up. On this site, Luke Lango made a "buy the dip" case last month; a few days later, Josh Enomoto highlighted the company's opportunity in China.
With all due respect, I disagree. The issue from here is that many investors are valuing Coca-Cola for what it was: a wonderful business that produced steady growth and lockstep increases in its dividend. Coca-Cola stock famously has made billions of dollars for Warren Buffett and Berkshire Hathaway (NYSE:BRK.A, NYSE:BRK.B). It's also been a great long-term investment for the rest of us.
But times change. And those changes have notably shifted the investment case here.
At this point, it seems increasingly difficult to make the fundamental case for Coca-Cola stock. A multi-year transformation — including a "refranchising" of its bottling operations — has hit revenue in recent years, but it was supposed to create a leaner, more profitable company.
That's a key reason why the KO stock price fell over 8% after disappointing earnings last month — a huge move for a typically low-volatility stock. Guidance for 2019 was much weaker than expected. Coca-Cola expects earnings growth of -1% to 1% against 2018's $2.08.
Investors were hoping for much more after the refranchising. The problem goes beyond results for a single year, however, as 2019 guidance implies EPS of $2.06 to $2.10. In 2013, Coca-Cola's non-GAAP EPS was $2.08. Over six years, including a massive transformation, earnings per share will barely move, if at all.
Coca-Cola stock, like so many other U.S.-based investments, has benefited from a lower tax rate. The underlying (i.e., adjusted) tax rate in 2013 was 23%. It's estimated to be 19.5% in 2019. That lower rate provides a 4.5% benefit to net earnings. Billions of dollars in share buybacks boost EPS as well. Considering this, Coca-Cola has 4.5% fewer shares outstanding than it did in 2013. In other words, KO is making less pre-tax profit than it did six years ago (the figure has declined about 8.7% by my math).
And yet, Coca-Cola stock trades for 22x the midpoint of 2019 EPS guidance. That multiple seems incompatible with the performance over the past few years.
To be fair, a stronger dollar has been an issue, hitting revenues and profits overseas. Coca-Cola is projecting a significant currency headwind, with 6% to 7% impact on operating income in 2019 alone. This comes after a 4% hit in 2018. Essentially, much of the potential benefit of the refranchising has been swallowed by currency effects.
That said, it's not as if the dollar is guaranteed to get weaker going forward. KO stock still looks reasonably expensive against even currency-neutral growth. Meanwhile, risks are rising.
Soda consumption continues to decline in the U.S., dropping by 20%-plus over two decades according to one report. Coca-Cola has tried to diversify, acquiring Costa Coffee and sparkling water manufacturer Topo Chico last year. But a $5 billion coffee deal — let alone a $220 million bottled water purchase — doesn't move the needle much against a $200 billion market cap.
The trend here is consistently negative, particularly in terms of diet soda. Sparkling waters from Nestle (OTCMKTS:NSRGY), National Beverage (NASDAQ:FIZZ) (even with some recent trouble), and private companies like Polar and Spindrift are taking share from diet soda. Neither Coke with its Dasani brand, nor Pepsi (NASDAQ:PEP) with its Aquafina, have been able to win much in that market.
Meanwhile, HSBC Securities highlighted an interesting stumbling block for Coca-Cola's plans for expansion. Coca-Cola has looked to new extensions, including flavored Diet Coke and an orange-vanilla offering for its full-calorie brand. But, as HSBC pointed out, those efforts are likely to upset the same bottlers who have taken over Coke's operations.
Smaller products have high startup costs, as well as long payback periods. Coke's refranchising may slow it from following the brand expansion strategy that's currently popular among consumer companies. It's a bit analogous to the risk facing Coke customers McDonald's (NYSE:MCD) and Restaurant Brands International (NYSE:QSR). Those companies have shifted costs to their franchisees too. However, those franchisees may rebel as their parents look to compete solely on price or otherwise push the limits of their profitability.
From here, zero growth isn't worth the risks facing the industry. As such, KO stock looks far too expensive. I asked last year if Coca-Cola might be the next giant to stumble after consumer heavyweights Anheuser-Busch InBev (NYSE:BUD), Kraft Heinz (NASDAQ:KHC), and Altria (NYSE:MO) saw big declines.
It hasn't happened yet, even with the post-Q4 selloff. But I wouldn't be shocked if it did. In fact, I'd be less surprised if KO stock fell sharply than if Coca-Cola somehow figured out how to grow in an industry destined for long-term declines.
Article printed from InvestorPlace Media, https://investorplace.com/2019/03/coca-cola-stock-ko-stock-price-too-expensive/.


text:
Facebook's Libra Increasingly Under Scrutiny From US Government
Home » Facebook's Libra Increasingly Under Scrutiny From US Government
Facebook may have expected to make a big splash with its Libra cryptocurrency project, and it certainly has. But the attention that Libra has garnered has been largely negative, particularly from government officials in the US and around the world. There's a very real chance that the move to clamp down on Libra and Facebook could end up harming other cryptocurrencies too.
One of the prime threats to Libra is that it could end up being classified as a security under existing US law, not as a currency. That's the opinion both of cryptocurrency analysts and former government officials such as former CFTC chairman Gary Gensler. That's in part because Libra, even though it sees itself as a stablecoin, will be backed at least partially by government bonds. And because government bonds are securities, anything backed by them is therefore by definition a security too.
That could make it much more difficult for Libra to get off the ground, as it would require regulatory approval from the Securities and Exchange Commission (SEC). Given the amount of opposition Facebook is facing from regulators worldwide, that approval is definitely not certain. But if Libra ends up being categorized as a security rather than as a currency, it would take a lot of pressure off Bitcoin and other cryptocurrencies.
Facebook is facing increasing opposition from lawmakers and regulators in the US and abroad, including from Members of Congress who want the company to stop development of Libra altogether.
Then there are the recent statements from Treasury Secretary Steve Mnuchin, who continues to express the Trump administration's concerns with Bitcoin and cryptocurrencies, particularly their use by criminals for purposes of money laundering. The irony, of course, is that the paper cash printed by the Treasury is far more popular for money laundering and criminal activity around the world.
As tempting as it may be for many people to use a cryptocurrency that's associated with the Facebook account they probably already use on a regular basis, the regulatory pressures against Facebook may very well doom the project. Anyone looking to capitalize on the popularity and growth potential of cryptocurrencies would do well to stick to proven investments like Bitcoin rather than wait for hyped up projects that may never reach their full potential.
Who Can I Trust to Open a Crypto IRA?
« President Trump Admits He's "Not a Fan" of Bitcoin: How Will That Impact Investors?
Ransomware That Hit US Spreads to China »


PATTERSON, Iowa (AP) — Authorities say a man and woman who'd gone ice fishing on a rural Madison County pond have drowned.
A fire department dive team pulled the bodies from the pond on Monday night. The pond is about four miles north of Patterson.
Authorities identified the pair as 64-year-old Linda Jones, of rural Prole, and 80-year-old George Pierce, of Des Moines.
Jones' husband, Larry Jones, told authorities that he'd gone to the pond to check on his wife and Pierce but found a hole in the ice that was several feet wide.
Chief Deputy Jason Barnes of the Madison County Sheriff's department says the hole was about 10 feet from shore and that the water there was 7 to 8 feet deep.


The published collection of books by Andrea and containing her anthologies are a must read for everyone! Whether you have an extra hour or a day to kill, Andrea's books are worth the read!
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