He plunged up the staircase, and stepping across the body of Michael as it lay on the landing, raced up the second flight of stairs.
"High Noon" by Anonymous
When she had gone once up and down "her" avenue, she could not move another step and asked to sit down on "her" seat.
"The works of Guy de Maupassant, Vol. 5 (of 8)" by Guy de Maupassant 1850-1893
When Clare stood up, he was stepping softly up behind Johnstone.
"Adam Johnstone's Son" by F. Marion Crawford
Step up, girls, and take your books.
"Shining Ferry" by Sir Arthur Thomas Quiller-Couch
Old Quimber caught Octavius by the arm to detain him a moment before he himself retraced his steps up street.
"Flamsted quarries" by Mary E. Waller
At the pier he turned the boat, and backed her up to the landing steps.
"Work and Win" by Oliver Optic
It was that tall, thin friar who is going up the steps.
"Romola" by George Eliot
Nooning is over, and there are hurrying steps up the wide alleyway.
"Floyd Grandon's Honor" by Amanda Minnie Douglas
You're the prettiest creatur' that ever stepped, but I wouldn't give up Sue for the Queen of England.
"Country Neighbors" by Alice Brown
He climbed quickly up the ladder and stepped out on the deck, gazing about him eagerly.
"A Prisoner of Morro" by Upton Sinclair
The young circus performer stepped up to Mr. Blair.
"Joe Strong, the Boy Fish" by Vance Barnum
A faint rustling came to his ears and then steps, and looking up, he saw his aunt Loraine coming down the stairs.
"Stubble" by George Looms
Then, half defiantly and half eagerly, she stepped up to the window.
"The Night Riders" by Ridgwell Cullum
Duvall stepped up to him quickly, and spoke in a pleasant voice.
"The Film of Fear" by Arnold Fredericks
Indeed, she did not seem to hear, but dragged herself up from step to step as though a weight was on her feet.
"The Destroyer" by Burton Egbert Stevenson
Mr. Phillips brought her up alongside a broad flight of white steps.
"The Island Mystery" by George A. Birmingham
She went quietly up the steps, after her short pause, and rang the bell.
"Shadows of Flames" by Amelie Rives
Jack stepped up to the window.
"A Bed of Roses" by W. L. George
So I groped and shuffled my way onward, rounding the banister and advancing step by step up the second stairway.
"The Man Who Couldn't Sleep" by Arthur Stringer
Irene had just stepped up to the place where they were standing.
"In Paradise" by Paul Heyse
***
When Rose the Red and White Lilly
Saw their twa loves were gane,
Then stopped ha they their loud, loud sang,
And tane up the still moarnin;
And their step-mother stood listnin by,
To hear the ladies' mean.
"Rose The Red And White Lily" by Andrew Lang
He weareth a dim and a shadowy form,
Like the varying clouds of a summer storm
With beauty like theirs;
And he speedeth o'er city and sea and land,
With a still soft step and a gentle hand,
Drying up tears.
"Academe" by Edwin Arnold
The sun was low in the eastern skies
When spoke the confident youth;
Sweet Psyche, all day, his hands and eyes
Wiled from the clay uncouth,
Nor ceased when the shadows came up like spies
That dog the steps of Truth.
"The Unseen Model" by George MacDonald
The grass, the very wilderness
On either side, breathed rapture of
Her passage: 'twas her hand or dress
That touched some tree--a slight caress--
That made the wood-birds sing above;
Her step that made the flowers up-press.
"Meeting In The Woods" by Madison Julius Cawein
One mounts up to beauty, serenely singing,
Forgetful of the steps that cry behind him;
One drifts slowly down from a waking dream.
One, foreseeing, lingers forever unmoving . . .
Upward and downward, past him there, we stream.
"The House Of Dust: Part 03: 08:" by Conrad Potter Aiken
Yet often I could fancy, where the pathway takes a turn,
I shall see you in a moment, coming round beside the burn,
Coming round beside the burn, with your swinging step and free,
And your face lit up with pleasure at the sudden sight of me.
"Come Back to St Andrews" by Robert Fuller Murray
Guests stepped up to the exhibits hoping that they would witness the animals doing something.
Iran Stepping Up Production of High-Grade Uranium.
PETMAN Steps Up For Humanoid Robot Challenge.
Backups Hendricks, Garrett step up in backfield for come-from-behind victory.
Reserves are stepping up and making an impact already for TCU.
Dragan agreed to step up his efforts to help, and work with the health department to find a solution.
Steps Up Pressure to Oust Assad.
HHS Steps Up Efforts to Curb Health Insurance Premium Hikes.
Avid Studio is a welcomed step up for advanced video editing enthusiasts chafing at the limitations of consumer tools.
If he's going to be on the floor in the NBA, he's got to be able to step up and get after it.
Renner wins Sony Step Up gold.
Ronnie Renner wins gold in Moto X Step Up.
A Concordia College chef steps up to save the Continental Region's honor.
Midtown Memphis' new Mexican spot Frida's must step up its game.
Ric Fogel For ESPN.comDemetrious Johnson steps up in a big way when he faces Miguel Torres at UFC 130.
***
For l ≤ j ≤ Ln we can step to the right till we reach (Ln , −(n + 1)) and then step up to (Ln , −n).
Degenerate random environments
The ﬁrst step up will be at a point (j ′ , −(n + 1)) with Ln ≤ j ′ ≤ Un , so by induction the point (j ′ , −n) we reach will lie in Bo .
Degenerate random environments
However, there is not a ﬁxed number of theory formation steps set up for this process, since this varies depending on the domain, i.e. some domains need more theory formation steps than others.
Discovery of Invariants through Automated Theory Formation
The second builds up the matrix in a step-by-step way.
A Short Course on One-Dimensional Random Schr\"odinger Operators
The recurrence (3) holds since a path ending at (m, n) can be obtained from either a path ending at (m − 1, n) followed by a unit right (→) step (1, 0) or a path ending at (m, n − 1) followed by a unit up (↑) step (0, 1).
Redundant generating functions in lattice path enumeration
Then this problem asks about the number Cp(m, n) of lattice paths from (1, 0) to (m, n), (where m > pn) with unit up (↑) steps (0, 1) and unit right (→) steps (1, 0), that never touch the line x = py .
Redundant generating functions in lattice path enumeration
Let Dp (m, n) be the number of lattice paths from (0, 0) to (m, n) (where n ≤ pm), with unit up (↑) steps (0, 1) and unit right (→) steps (1, 0), that never cross the line y = px.
Redundant generating functions in lattice path enumeration
From the path transformation, we can also derive the number of lattice paths starting at a point that is below the line y = px and ending on the line y = px, with unit up (↑) steps (0, 1) and unit right (→) steps (1, 0), that never cross the line y = px.
Redundant generating functions in lattice path enumeration
Since a path cannot have four consecutive up (ր) steps, such a generalized Dyck path ending at (m, n) can be obtained uniquely from a generalized Dyck path ending at (m − i − 1, n − i + 1) followed by i consecutive up (ր) steps for 0 ≤ i ≤ 3 and a down (ց) step.
Redundant generating functions in lattice path enumeration
The transformed path does not go below the x-axis because when the path R takes a unit right (→) step (1, 0) the y -coordinate difference between the path R and the line y = 2x increases by 2, whereas when the path R takes a up (↑) step (0, 1) the y -coordinate difference decreases by 1.
Redundant generating functions in lattice path enumeration
To do this let us consider a lattice path S from (0, 0) to (m, n) (where n ≤ pm), with unit up (↑) steps (0, 1) and unit right (→) steps (1, 0), that never crosses the line y = px.
Redundant generating functions in lattice path enumeration
The transformed path does not go below the x-axis because when the path S takes a unit right (→) step (1, 0) the y -coordinate difference between the path S and the line y = px increases by p, whereas when the path S takes a up (↑) step (0, 1) the y -coordinate difference decreases by 1.
Redundant generating functions in lattice path enumeration
Let Dp (m, n) be the number of lattice paths from (0, 0) to (m, n) (where n ≤ pm), with unit up (↑) steps (0, 1) and unit right (→) steps (1, 0), that never cross the line y = px.
Redundant generating functions in lattice path enumeration
For a ﬁxed h, p ∈ P, let Ep,h(m, n) be the number of lattice paths from (0, 0) to (m, n) (where n ≤ pm + h), with unit up (↑) steps (0, 1) and unit right (→) steps (1, 0), that never cross the line y = px + h.
Redundant generating functions in lattice path enumeration
I.e., transform (X, D) by the Step 3: Make (X, Dred ) semi-snc. blowings-up needed to make (X, Dred ) semi-snc.) The algorithm for Step 3 is given following Step 4 below.
Resolution of singularities of pairs preserving semi-simple normal crossings
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