Her own appearance harmonised admirably with her surroundings.
"Captain Desmond, V.C." by Maud Diver
They were both very good, but their characters did not harmonise.
"Fantômas" by Pierre Souvestre
Sympathy is a great harmoniser.
"The Higher Powers of Mind and Spirit" by Ralph Waldo Trine
There remained mud which harmonised tonelessly with our uniforms.
"A Padre in France" by George A. Birmingham
Incisive he was not ordinarily; caution of his type harmonises ill with incisiveness.
"Life of Charles Darwin" by G. T. (George Thomas) Bettany
Friends that would harmonise with his gloves and umbrella he had none as yet.
"The Belovéd Vagabond" by William J. Locke
In both instances, Nature and Circumstance have harmonised between the subject and the background.
"The Book of Khalid" by Ameen Rihani
How are we to make our imagination of facts of this kind harmonise with the reality?
"Introduction to the Study of History" by Charles V. Langlois
But the common rendering appears to me to harmonise best with the preceding portion of it.
"Notes and Queries, Number 218, December 31, 1853" by Various
Chapman's strain is higher than Tennyson's, but they harmonise.
"Flowers of Freethought" by George W. Foote
Characters such as Johnson's harmonise best with the enthusiastic and easily influenced.
"A Book of Sibyls" by Anne Thackeray (Mrs. Richmond Ritchie)
A variety of the same Tones harmonised for four voices, but so as to preserve unaltered the original melodies; 3.
"Australia, its history and present condition" by William Pridden
The two are built on entirely different lines, and they don't seem to harmonise.
"From Sea to Sea" by Rudyard Kipling
Black crapes too, and sable hues, to harmonise with a world of sorrow and darkness.
"Fragments of an Autobiography" by Felix Moscheles
Egypt harmonised all three of them.
"The Wave" by Algernon Blackwood
Herminie's sentiments certainly harmonise with her charming and noble face, do they not, mamma?
"Pride" by Eugène Sue
All this was not in the least bizarre in effect, but harmonised perfectly.
"The Song of Songs" by Hermann Sudermann
How would he harmonise with the humble middle-class English setting to which he was on the point of confiding himself?
"Mrs. Fitz" by J. C. Snaith
Many chimes in country churches played the psalm tunes that he had harmonised.
"Milton's England" by Lucia Ames Mead
The sweet, melodious tone of her voice harmonised wonderfully with the bright child-like smile that accompanied the words.
"The Breaking of the Storm, Vol. I." by Friedrich Spielhagen
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Keywords and phrases: Stable random ﬁelds, harmonisable ﬁelds, excursion sets, Euler characteristic, intrinsic volumes, geometry..
Excursion sets of stable random fields
For example, whereas in the Gaussian case many stationary processes have both a moving average representation (with respect to white noise) as well as a harmonisable representation, in the stable case moving average and harmonisable processes belong to quite distinct families.
Excursion sets of stable random fields
With this in hand, in Sections 4–6 we start with the new results, for sub-Gaussian, harmonisable, and concatenated-harmonisable random ﬁelds, deriving asymptotic formulas for the expected values of the Euler characteristics of their excursion sets.
Excursion sets of stable random fields
However, they do not represent a particularly rich class of stable ﬁelds. A much richer class of stable ﬁelds is given by the stationary, symmetric, α-stable (SαS ) harmonisable ones.
Excursion sets of stable random fields
This will enable us, as in the sub-Gaussian case, to use conditional Gaussian arguments to prove the following result, in which we establish the asymptotics of the expected Euler characteristic of the excursion sets of the real harmonisable stable ﬁelds.
Excursion sets of stable random fields
Let f be a harmonisable, SαS , random ﬁeld as in (18) or (19), deﬁned on the N -rectangle T of (2).
Excursion sets of stable random fields
Comparing, for example, (23) with its Gaussian counterpart (12) (take j = 0 there) we see that while the leading term in the Gaussian case comes from the volume term LN (M ), in the harmonisable stable case it comes from the two lowest Lipschitz-Killing curvatures, L0 (M ) and L1 (M ).
Excursion sets of stable random fields
Consider the representation (19) for harmonisable stable ﬁelds.
Excursion sets of stable random fields
The harmonisable case, however, is somewhat more complicated.
Excursion sets of stable random fields
Since these are, again, harmonisable ﬁelds with compactly supported control measures they are all continuous.
Excursion sets of stable random fields
We actually discovered them by looking for a class of examples which ‘interpolated’ between the sub-Gaussian ones, for which all the Lj appear in the asymptotic formula for the mean Euler characteristic of excursion sets, and the harmonisable ones, for which only L0 and L1 appear.
Excursion sets of stable random fields
This random ﬁeld is quite different from that of the simple random wave generated by the ﬁrst term of the harmonisable processes, and so the arguments there, based on examples as in Figure 1, are not going to carry over easily to the current situation.
Excursion sets of stable random fields
Let f be a concatenated-harmonisable, SαS random ﬁeld as in (39), deﬁned on the N -rectangle T of (2).
Excursion sets of stable random fields
The arguments that worked in the harmonisable case also work here to show that the conditionally Gaussian process satisﬁes all the conditions of Theorem 3.1.
Excursion sets of stable random fields
Let f be a harmonisable, or concatenated-harmonisable, SαS random ﬁeld satisfying the conditions of Theorem 5.1, or Theorem 6.1, respectively.
Excursion sets of stable random fields
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