Sarah seated herself under the shade of a walnut tree, pulling her white frock neatly around her and patting off loose blades of grass trapped in the lacing. Taking a deep breath, she motioned with just her mouth letting no sound out “Mentaharnin…” but before being able to continue a sudden shriek and then joyous laughter from behind broke her concentration.
Someone had just won, or scored many points in a game, Sarah guessed not really thinking. It was true, the other girls her age of Rosemary Manor were playing hopscotch; but the laughter was over Amy Henderson who trying a very challenging sequence had fallen flat on her rear, initially letting out a shriek but then bursting into laughter with the others over her own abysmal performance. None of this concerned Sarah, she always seemed to find the things that brought others great joy to be immensely trite.
The momentary cacophony died down and another round of hopscotch quietly ensued. Sarah focused intently on the sound of the breeze rustling the tree leaves until her mind was completely clear once again.
“Mentaharnin acquinte oransk” she chanted silently, motioning every syllable perfectly just as she had done each week the past three months. Immediately, Sarah felt the familiar, uncomfortable twin piercing at one side of her neck and a withdrawal of warmth. She sat motionless, staring toward the horizon as wispy clouds now stayed their position in the afternoon sky.
“Entaharn, refreshed?” Sarah thought without speech, feeling two incisors slide out her in response. She leaned forward but the air seemed unwilling to move. Defiantly fighting the immobile ether, she struggled forward with increasingly greater effort, and, suddenly, as if a glass door had given way, she hurled forward at frightening speed. The ground and sky spun around her, colors washed from blue to red to green and back to blue, and finally as suddenly as it all began, it stopped. Everything stopped. The sky was reassuringly above her and the ground thankfully beneath her. She noticed herself still seated on grass, but not where she had sat; then, looking forward, she saw nearly fifty meters ahead her own body still under a walnut tree, completely motionless.
“Elegant, always so very elegant.” A boyish, cavalier voice chuckled all around her. “Do you always tumble like that?”
“I’m still practicing,” Sarah retorted a bit embarrassed but returned to a regal tone, “Hurry up and repair that body over there.”
“Ah, yes; fixing your problems.” The same voice was distinctly in front of her but its owner remained unseen. “My, my… an entire centimeter in almost momentary time. I commend your effort, but ripping your flesh seems a rather coarse way to…”
“Just fix it!” Sarah snapped impatiently. An entire centimeter this time, she thought to herself worriedly. Entaharn had explained to her before the dangers of corporeally moving when time slows to a fraction of its original speed. Her whole body could irreparably dissolve under enough acceleration, and with a centimeter of movement this time, it had nearly happened.
Have a Hallowed Eve!
Monday, October 31, 2005
Monday, October 24, 2005
number theory: ah, those funny li'l modulos
There were only a handful of tricks with numbers that the typical elementary school student would be inculcated with during those years I enjoyed (well, attempted to enjoy) primary education. To cast the bleak bleaker, the divisibility test by 3 is often the only ubiquitously taught trick possessing some semblance of novelty. In case anyone wondered as a kid but failed to discover the reason, modulus arithmetic is the preferred foundation with which one derives efficient divisibility tests. For some reason still unbeknownst to me, I decided to raise this ol' topic and began concocting digit-wise divisibility tests for the primes 7 and 11. I urge readers to please have ready several pitchers of water, since what follows is fairly dry...
When X is a string (of some radix) of numerical value Σ(X[k]*radixk)∀k∈0..X.length-1⊆Z, then (int)X ≡ 0 mod Y iff: Σ(X[k] * cycle[k])∀k∈0..X.length⊆Z ≡ 0 mod Y, where cycle is the repeating series { radixk mod Y }.
Let's pick a radix we all know and love, base10:
Let Dn = { 100, 101, 102, ... }
Dn mod 3 = { 1, 1, 1, ... } = Cycle (1). Since cycle[k] = 1, we get the age-old mantra "X is divisible by 3 iff the sum of digits is divisible by 3".
But let's look beyond 3,
Dn mod 7 = cycle ( 1, 3, 3*3≡9≡2, 2*3≡6≡-1 ..) = cycle (1, 3, 2, -1, -3, -2)
Dn mod 11 = cycle (1, 10≡-1 ..) = cycle (1,-1)
Interestingly, but with less power than an iff relationship, since lcm(2,6) = 6 and both Dn mod 7 and Dn mod 11 have half-cycles (-1 as an element), then X ≡ 0 mod 77 if X in decimal form can be grouped into 3digit strings, where every other 3digit string is marked red and those in between are marked blue, and the { reds } minus { blues } = Null.
For exampe, let red = { 123, 444, 812, 912, 083, 948, 020, 436 }
Thus, blue = { 123, 444, 812, 912, 083, 948, 020, 436 }
Let shuffle(blue) = { 444, 948, 912, 436, 083, 812, 123, 020 }
Then interlace(red, shuffle(blue)) = 123444 444948 812912 912436 083083 948812 020123 436020. This now yields a base10 string which is divisible by 77 (and obviously also by 2, 5, 7, 11 and all the other factors of their multiple 770): 123444444948812912912436083083948812020123436020
Too big a number to verify with a pocket-calculator? Here's an easy multiple of 77 to generate: 001001 = 1,001. A pocket calculator can verify it is 13*77.
But then, what's the ratio of densities between these simple multiples and the full set of multiples? In the case of 77, cardinality of the full set of multiples for base10 string X is (1/77) * (10X.length). Cardinality for the parlor-trick partial set is:
Let redi = ∪(X[6*i+0..6*i+2])
Let bluei = ∪(X[6*i+3..6*i+5])
Let g = X.length/6
(Π (φ(∃unique j s.t. bluei = redj))∀i∈0..g-1⊆Z) * (10X.length)
= (Π(1 - ((103-1)/103)k)∀k∈1..g⊆Z) * (10X.length)
= (Π(1 - 0.999k)∀k∈1..g⊆Z) * (10X.length)
limX.length→∞( Π(1 - 0.999k)∀k∈1..g⊆Z ) ≈ 7.4210E-713
Thus, the ratio of the two is (1/77) : 7.4210E-713 ⇒ 1 : 5.7142E-711. In other words, the parlor trick while being a very good way to generate multiples is a rather improbable way to test for divisibility by 77. The only rigorous divisibility test is the one where iff is ensured instead of the comparatively impotent if.
When X is a string (of some radix) of numerical value Σ(X[k]*radixk)∀k∈0..X.length-1⊆Z, then (int)X ≡ 0 mod Y iff: Σ(X[k] * cycle[k])∀k∈0..X.length⊆Z ≡ 0 mod Y, where cycle is the repeating series { radixk mod Y }.
Let's pick a radix we all know and love, base10:
Let Dn = { 100, 101, 102, ... }
Dn mod 3 = { 1, 1, 1, ... } = Cycle (1). Since cycle[k] = 1, we get the age-old mantra "X is divisible by 3 iff the sum of digits is divisible by 3".
But let's look beyond 3,
Dn mod 7 = cycle ( 1, 3, 3*3≡9≡2, 2*3≡6≡-1 ..) = cycle (1, 3, 2, -1, -3, -2)
Dn mod 11 = cycle (1, 10≡-1 ..) = cycle (1,-1)
Interestingly, but with less power than an iff relationship, since lcm(2,6) = 6 and both Dn mod 7 and Dn mod 11 have half-cycles (-1 as an element), then X ≡ 0 mod 77 if X in decimal form can be grouped into 3digit strings, where every other 3digit string is marked red and those in between are marked blue, and the { reds } minus { blues } = Null.
For exampe, let red = { 123, 444, 812, 912, 083, 948, 020, 436 }
Thus, blue = { 123, 444, 812, 912, 083, 948, 020, 436 }
Let shuffle(blue) = { 444, 948, 912, 436, 083, 812, 123, 020 }
Then interlace(red, shuffle(blue)) = 123444 444948 812912 912436 083083 948812 020123 436020. This now yields a base10 string which is divisible by 77 (and obviously also by 2, 5, 7, 11 and all the other factors of their multiple 770): 123444444948812912912436083083948812020123436020
Too big a number to verify with a pocket-calculator? Here's an easy multiple of 77 to generate: 001001 = 1,001. A pocket calculator can verify it is 13*77.
But then, what's the ratio of densities between these simple multiples and the full set of multiples? In the case of 77, cardinality of the full set of multiples for base10 string X is (1/77) * (10X.length). Cardinality for the parlor-trick partial set is:
Let redi = ∪(X[6*i+0..6*i+2])
Let bluei = ∪(X[6*i+3..6*i+5])
Let g = X.length/6
(
= (Π(1 - ((103-1)/103)k)∀k∈1..g⊆Z) * (10X.length)
= (Π(1 - 0.999k)∀k∈1..g⊆Z) * (10X.length)
limX.length→∞( Π(1 - 0.999k)∀k∈1..g⊆Z ) ≈ 7.4210E-713
Thus, the ratio of the two is (1/77) : 7.4210E-713 ⇒ 1 : 5.7142E-711. In other words, the parlor trick while being a very good way to generate multiples is a rather improbable way to test for divisibility by 77. The only rigorous divisibility test is the one where iff is ensured instead of the comparatively impotent if.
Wednesday, October 19, 2005
Manure
“Hello Professor Dumbledore,” I began friendlily but was cut off by the steely eyed headmaster obviously refusing to cow before a well-situated and grown up former student of his.
“Severus, yes”
Yes, first name basis will do, I thought grievously. My mind stalled for a moment to recollect his first name. It had been so long since seeing him, since even stepping foot back into Hogwarts. Then I was considered the short, pudgy son of a mediocre family; others knew not of my true legacy or the future legacy I would create. Today I towered in my custom tailored suit and polished leather shoes, and not because I sought to impress; oh, no, seeing in front of me in tawdry attire with a frail figure a man who once caused me anguish was a mere bonus, a treasure of fractional value to why I was here.
“Albus,” I finally remembered his first name as I gazed at the frays of his cuffs, “the ministry and I had a little chat earlier and since I have been elected to head…” but I was interrupted again,
“Bought to head” Albus finished. His boldly begun words waned into a whisper.
Perhaps it was only because of money I am here again; but money reigns supreme and I possessed a terrifying sum to now head the board of trustees for this school. Even Albus through his thick skull can see this, and through all his defiant pretensions, streaks of fear quiver through his voice and actions. Perhaps Albus need not be disposed after all, I mused. Fear for his job is sensible.
“I have heard a lot of discontent,” I continued, ignoring Albus’s throat clearings, “about the direction of the school and I cannot ignore the extent to which the dissatisfactions have grown. I haven’t invested three hundred million in this place to watch it meander its way into an abyss; I…”, noticing Albus’s open mouth my voice raised, body lurched forward and eyes widened to prevent an interruption, “I want to see this school succeed!”
The words were followed by a short silence, letting me lean back before I continued. “And I assume you do as well, so let us work together." I left a purposeful pause, letting my words sink in. Encouraged by the resounding silence and lack of any protest, I continued with a smirk, "I want to keep you around, but you cannot continue being insubordinate.” His eyes looked more resigned now, and, for once, the idiot had nothing to finish my sentences with.
“Severus, yes”
Yes, first name basis will do, I thought grievously. My mind stalled for a moment to recollect his first name. It had been so long since seeing him, since even stepping foot back into Hogwarts. Then I was considered the short, pudgy son of a mediocre family; others knew not of my true legacy or the future legacy I would create. Today I towered in my custom tailored suit and polished leather shoes, and not because I sought to impress; oh, no, seeing in front of me in tawdry attire with a frail figure a man who once caused me anguish was a mere bonus, a treasure of fractional value to why I was here.
“Albus,” I finally remembered his first name as I gazed at the frays of his cuffs, “the ministry and I had a little chat earlier and since I have been elected to head…” but I was interrupted again,
“Bought to head” Albus finished. His boldly begun words waned into a whisper.
Perhaps it was only because of money I am here again; but money reigns supreme and I possessed a terrifying sum to now head the board of trustees for this school. Even Albus through his thick skull can see this, and through all his defiant pretensions, streaks of fear quiver through his voice and actions. Perhaps Albus need not be disposed after all, I mused. Fear for his job is sensible.
“I have heard a lot of discontent,” I continued, ignoring Albus’s throat clearings, “about the direction of the school and I cannot ignore the extent to which the dissatisfactions have grown. I haven’t invested three hundred million in this place to watch it meander its way into an abyss; I…”, noticing Albus’s open mouth my voice raised, body lurched forward and eyes widened to prevent an interruption, “I want to see this school succeed!”
The words were followed by a short silence, letting me lean back before I continued. “And I assume you do as well, so let us work together." I left a purposeful pause, letting my words sink in. Encouraged by the resounding silence and lack of any protest, I continued with a smirk, "I want to keep you around, but you cannot continue being insubordinate.” His eyes looked more resigned now, and, for once, the idiot had nothing to finish my sentences with.
Monday, October 17, 2005
I Service the Machine that Services the Machine
Every morning before breakfast, a whistle shoots up high above the rooftops of the nearby service station and blows with all the fury it can muster a high-pitched "Wheeet." That’s the sign. That’s the sign that a baby lord machine has been hatched. We the servicemen rush from our homes, leaving our cereal bowls vacated. Forty leaps with all the might our legs can afford and the service station’s door is in sight. A swipe of the badge and a careful walk past the sterile threshold to the station reveals the little lord machine, shining wetly and acquainting itself with servicemachines fondly dabbing warm cloth on its chassis to prevent any buckles during the cooling process. This time of day is always a treat, since it is the only time we servicemen can gaze upon a lord machine. Our task is not to care for it directly, but to ensure all the servicemachines who do care for it are well cared for themselves. All one servicemachine need do is let us see its light lit orange and four of us will come walking hurriedly to it with clean oil, chilled water, and a sizable, freshly charged battery pack. Mere feet away from a lord machine, our pride swells; to attend to the machine directly attending a lord machine is a privilege bestowed upon the few, the lucky.
Monday, October 10, 2005
do not incense the crack-addicted squirrels (source)
Rare is it when I start rehashing periodicals, but it is imperative I report that English rodents have gone wild. The fuzzy, cuddly bunnies fondly dined upon by bambi-slaughtering, full regalia donning lords with knives and forks carried in proper hands are not the morsels at play here; rather, unlike rodentia-posing leporidae, these are true rodents who have been feasting upon saran-wrapped packets of crack neatly patted down with topsoil in the gardens of dubious white-powder-shrub gardeners. These are critters teeming with more satanically horrifying vices than contained in all the FOX reality shows and almost as much as contained in Jane Fonda's hair – leave a few cubic hectares of eye-burning aerosol. That's right, these are squirrels, now most easily identified by disheveled fur and the harrowing glare of their blood-shot, demented eyes, and the squirrels have taken a liking to stash buried by scuzzy junkies. Through the mighty rodent quest for granulated satisfaction, squirrels are leaving in their wake a dazed lot of heart-palpitating Londoners.
As always, the bushy tailed recidivism began with the bobbies. After a recent rash of anti-drug enforcement, gardens-turned-safehouses began storing "the goods" and in a double whammy counteracted both the sleuths and the residual yet pungent anal odors gifted by intestinal convoys aboard a long British Airways flight from Lithuania. Of course, now look at the mess; the capital of the mightiest of feeble empires is being overrun by panicky and angsty teens suffering withdrawal and feverishly irascible squirrels who some speculate have already formed a powerful crime syndicate and have half of parliament in their pockets.. pouches.. er paws. Obviously also paid off, the nearly vowelless RSPCA has tried to contain rampant fear by insisting the thimble-sized hearts of our fury friends would go supernovae upon receiving any cocaine. Unconvinced and alarmed by the growing British fiasco, French authorities have decided to examine carefully the baroque snuffboxes of their infamous wayside frogs for any traces of the serious stuff concealed within.
As always, the bushy tailed recidivism began with the bobbies. After a recent rash of anti-drug enforcement, gardens-turned-safehouses began storing "the goods" and in a double whammy counteracted both the sleuths and the residual yet pungent anal odors gifted by intestinal convoys aboard a long British Airways flight from Lithuania. Of course, now look at the mess; the capital of the mightiest of feeble empires is being overrun by panicky and angsty teens suffering withdrawal and feverishly irascible squirrels who some speculate have already formed a powerful crime syndicate and have half of parliament in their pockets.. pouches.. er paws. Obviously also paid off, the nearly vowelless RSPCA has tried to contain rampant fear by insisting the thimble-sized hearts of our fury friends would go supernovae upon receiving any cocaine. Unconvinced and alarmed by the growing British fiasco, French authorities have decided to examine carefully the baroque snuffboxes of their infamous wayside frogs for any traces of the serious stuff concealed within.
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