Given Q > 0 and writing Ac = Ω \ A, we let P0 (f , Q) = {∃∆ ∈ D(f , ω ), ∆ ⊆ [−Q/ǫ, Q/ǫ]}c , be the set of realizations of the random ﬁeld that neither give rise to a positive nor to a negative elongation in the interval [−Q/ǫ, Q/ǫ].
One-dimensional random field Kac's model: localization of the phases
The power transformation φ(D) = Dq with 0 < q < 1 is not rectiﬁable nor bounded.
Robust Estimation through Schoenberg transformations
Note also that P ∪ R[w , a] ∪ T ∪ S [b, x] and Q ∪ R[y , c] ∪ U ∪ S [d, z ] are odd circuits A and B , respectively, for neither G[Hi−1 ∪ P ] nor G[Hi−1 ∪ Q] is an even-circuit-connected graph.
Even circuits of prescribed clockwise parity
As a consequence, neither ¯Q{γ · D, B+ }Q nor ¯Q[γ · D, E− ]Q is redundant.
New lattice action for heavy quarks
Any choice of p, 0 < p < (q − 1)/q now gives a proof of the existence of µ(k , d) such that if n ≥ µ(d, k) and C is an [n, k , d] code, then neither C nor its dual C ⊥ are good for error detection.
Codes for error detection, good or not good
If one changes a(f ) to be a(f ) = A → A , then ⇉ is a sort⊲ is not irreﬂexive: invariant, terminating relation on T that is not T-monotonic but /0-monotonic; whereas neither ⊲ Q.e.d. (Lemma B.7) nor (⇉ ∪⊲ST )+ (in contrast to DST [ /0] ↿id ◦ (⇉ ∪⊲ST )+ ) are irreﬂexive.
Syntactic Confluence Criteria for Positive/Negative-Conditional Term Rewriting Systems
Note that neither the degree nor inﬂuences of Q(X ) depends on the actual basis selected in (9), hence we can write deg f = deg Q, Inf i f = Inf i Q(X ) and Inf ≤d f = Inf i Q≤d (X ).
Maximally Stable Gaussian Partitions with Discrete Applications
Let A, B , C, D be the sets of vertices joined to p and q , p but not q , q but not p, and neither p nor q , respectively.
Overgroups of the Automorphism Group of the Rado Graph
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