In literature:
What's the odds if they do have to go back to their little Eighth avenue flat next week?
"Shorty McCabe on the Job" by Sewell Ford
It presented a very odd appearance.
"The Bush Boys" by Captain Mayne Reid
A rather odd-looking little person sat in a low armchair immediately inside it.
"Legacy" by James H Schmitz
It is an odd thing, how happily two people, if there are two, can live in a place where they have no acquaintance.
"The Works of Robert Louis Stevenson - Swanston Edition" by Robert Louis Stevenson
The kitchen of the Trellis House was oddly situated just opposite Mrs. Otway's sitting-room and at right angles to the dining-room.
"Good Old Anna" by Marie Belloc Lowndes
It was the odd way he did it which endeared him to us, as if apologizing for the kindness.
"A Labrador Doctor" by Wilfred Thomason Grenfell
It offended her oddly and jarred upon her good taste, as something to which she was not at all accustomed in her surroundings.
"Adam Johnstone's Son" by F. Marion Crawford
Oddly assorted people appear on this train.
"Pipefuls" by Christopher Morley
I wouldn't ask odds of anybody if I was the way I was before that train went off the track.
"Fair Harbor" by Joseph Crosby Lincoln
I was in a large room, oddly but elegantly furnished.
"The Scalp Hunters" by Mayne Reid
The trick elephant did several odd things rather clumsily.
"A Little Girl of Long Ago" by Amanda Millie Douglas
She had an odd little accent, too, but it gave her an added daintiness.
"A Little Girl in Old Boston" by Amanda Millie Douglas
I always thought him an odd old man, from the earliest time I can remember.
"Deerbrook" by Harriet Martineau
There are stalls set out on the pavements, heaped up with embroidery and odds and ends, including soap, which is manufactured here very largely.
"Round the Wonderful World" by G. E. Mitton
He was transported at meeting this odd, middle-aged, dry-skinned man, personally.
"The Rainbow" by D. H. (David Herbert) Lawrence
Wasn't it an odd thing for her to say?
"Is He Popenjoy?" by Anthony Trollope
But it was in a tone oddly detached, musing, that he went on.
"Where the Pavement Ends" by John Russell
The odd trick this game and it was her rubber.
"The Duchess of Wrexe" by Hugh Walpole
It is odd how soon a man acquires the air of a proprietor.
"Helena Brett's Career" by Desmond Coke
Then Hjoervarth advanced, and he and Odd had a stiff encounter; but Odd's silken shirt was so strong that no weapon could pierce it.
"Stories and Ballads of the Far Past" by Nora Kershaw
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In poetry:
An awful truth does here present,
Whoever think it odd;
In him thou art omnipotent,
Thy Husband is a God.
"The Believer's Jointure : Chapter I." by Ralph Erskine
What reverence is rightly paid
To a Divinity so odd
He lets the Adam whom He made
Perform the Acts of God?
"Friday's Child" by W H Auden
An' then agean,--he goes on th' rant;
Nah, that aw niver do;--
Aw allus mark misen content,
Wi' an odd pint or two.
"Latter Wit" by John Hartley
My lips are haunted by a kiss
There in my mirror -- it is odd!
We belong because of this.
No one knows why but God.
"Reflections" by Isobel Stone
It's the way o' the Warl' aye tae soun' weel the fame—
Nae odds hoo he gat it—o' the chiel wi' a name;
But the nameless, though giftit, are caul' i' the yird,
Ere a sang or a word i' their praise she wull mird!
"The Way of The Warl'" by Janet Hamilton
For never shall oblivion slight
The hearts that fight the People’s fight.
Much less, when, thro’ a life of stress,
One voice ’gainst countless odds has stood,
And won, in pain and bitterness,
The People’s good.
"William Henry Groom Vale`" by George Essex Evans
In news:
Odd pair offers one of the most surreal concerts of the year.
Odd, I thought, since I had only visited the city twice before.
Bushman 's story teaches a lesson that you can do what you hope to do, regardless of the odds.
Odds and ends from around Ashland County.
A Congolese singer performing Cuban dance music in Spanish may strike some people as odd, but to Ricardo Lemvo it makes perfect sense.
This month always has had odd weather but usually not involving this heat.
WASHINGTON – The St Louis Cardinals won a World Series championship last season by repeatedly defying the odds.
It's an odd combo – the crown and the camouflage – but as the current Miss Kentucky State University and a maintenance clerk in the Kentucky National Guard, Hardy's not too concerned with the status quo.
Odd legal move by Jerry Sandusky.
All odds were against her.
These reality stars have beaten the odds and showed America doesn't always get it right.
TCU's Boykin dealt with odd circumstance .
Some of the 80-odd acres of trees cut down on tribal land in Columbia Gorge Scenic area have already been replanted.
MONTPELIER – A federal scientist whose statements on global warming have put him at odds with the Bush administration is backing the state of Vermont and environmental groups in a lawsuit over the regulation of car emissions.
While it may seem odd to talk to a cloud, this is actually what is happening more and more in our industry.
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In science:
We deﬁne the odd Coxeter graph Γodd to be the subgraph of Γ obtained by removing all edges labelled by an even number or ∞.
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Suppose that ξ 6= ζ ∈ E I (r), ζ† = ξ†, m(ξ ◦, ξ†) = 2 = m(ζ ◦, ζ†) and m = m(ξ ◦, ζ ◦) is an odd integer (hence ξ ◦ and ζ ◦ are adjacent in Γodd ).
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Odd(r)⊥x := {y ∈ Odd(r) | m(x, y ) = 2} for x ∈ S .
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
For each x ∈ ∂EOdd(r), the graph Γodd Odd(r)⊥x is connected and contains K (or equivalently, Odd(r)⊥x is a pre-core of (Γodd )∼r ).
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Then W ⊥r is ﬁnitely generated if and only if either (Γodd )∼r does not contain a nontrivial cycle, or (Γodd )∼r contains a nontrivial cycle and Conditions 1, 3 and 4 in Theorem 4.1 are satisﬁed.
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Let Iodd (respectively, Ieven ) be the subsets of Odd(r) r {r} consisting of the elements x for which m(r, x) is an odd (respectively, even) integer.
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Now m(x, z) cannot be an odd integer; otherwise, a cycle (x, z, y, x) in (Γodd )∼r is not contained in the core K of (Γodd )∼r, a contradiction.
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Then W ⊥r is ﬁnitely generated if and only if m(x, y ) is an odd integer or ∞ for every x, y ∈ Odd(r), and m(x, y ) = ∞ for every x ∈ Odd(r) and y ∈ S r Odd(r) (i.e., W is the free product of WOdd(r) and WSrOdd(r) ).
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Then W ⊥r is ﬁnitely generated if and only if m(x, y ) is an odd integer or ∞ for every x, y ∈ Odd(r), and m(x, y ) = ∞ for every x ∈ Odd(r) and y ∈ S r Odd(r) (i.e., W is the free product of WOdd(r) and WSrOdd(r) ).
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Then Property 4 in Theorem 4.1 implies that Odd(r)⊥x is a pre-core of (Γodd )∼r ; in particular, there exist two distinct elements y, z ∈ Odd(r)⊥x for which m(y, z) is an odd integer.
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
For an element sr (c; ξ ) of Rr, we say that it is an inner generator (respectively, outer generator ) if ξ† ∈ Odd(r) (respectively, ξ† 6∈ Odd(r)).
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Now by Condition 4 in Theorem 4.1, the set K ′ is a pre-core of (Γodd )∼r, therefore we have c−1d ∈ π1 (Γodd ; ξ ◦, ζ ◦) = π1(Γodd K ′ ; ξ ◦, ζ ◦) by Lemma 2.5.
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
On the other hand, assume contrary that Odd(r)⊥x is not a pre-core of (Γodd )∼r for some x ∈ ∂EOdd(r).
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
Then there exist an element y ∈ Odd(r)⊥x and a nontrivial cycle P in (Γodd )∼r satisfying that V (P ) is not contained in (Γodd Odd(r)⊥x )∼y (note that Odd(r)⊥x 6=
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
On the other hand, by the deﬁnition of S-moves, for the case that m(ξ ◦, ξ†) = m(ζ ◦, ζ†) = 2, we have sr (c; ξ ) = sr (d; ζ ) if and only if ξ ◦ and ζ ◦ belong to the same connected component of Γodd Odd(r)⊥ξ† .
A characterization of finitely generated reflection subgroups of Coxeter groups orthogonal to a reflection
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