In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdiierential mapping of a real-valued locally Lipschitz function is a minimal weak cusco. We then use this characterisation to deduce some new results concerning Lips-chitz functions with minimal subdiierential mappings. In the papers 7] and 1], the authors investigate a class of locally Lipschitz functions ...

We study hierachical segmentation in the framework of edgeweighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical edgesegmentations.

The Caenorhabditis elegans gene gld-1 is essential for oocyte development; in gld-1 (null) hermaphrodites, a tumor forms where oogenesis would normally occur. We use genetic epistasis analysis to demonstrate that tumor formation is dependent on the sexual fate of the germline. When the germline sex determination pathway is set in the female mode (terminal fem/fog genes inactive), gld-1 (null) g...

We propose a method of designing equiripple linear-phase FIR filters with linear constraint by carrying out the Remèz exchange algorithm. A novel technique is derived to convert a linearly constrained problem into an equivalent unconstrained one. We proposed a technique to modify the original desired frequency response so that the original linear constraint can be reduced to a simpler one (the ...

Let μ and λ be two positive bounded measures on the same meaurable space (Ω,F). We call μ and λ equivalent, and write μ ≡ λ, if they have the same null sets— so, if they were probability measures, the notion of “a.s.” would be the same for both. More generally, we call λ absolutely continuous (AC) w.r.t. μ, and write λ μ, if μ(A) = 0 implies λ(A) = 0, i.e., if every μ-null set is also λ-null. W...

Abstract We study four-dimensional Einstein–Maxwell fields for which any higher-order corrections to the field equations effectively reduces just a rescaling of gravitational and cosmological constant. These configurations are thus simultaneous solutions (virtually) modified theory gravity coupled (possibly non-minimally) electrodynamics. In case non-null electromagnetic we provide full charact...

We prove a linear operator T acting between lp-type spaces attains its norm if, and only if, there exists a not weakly null maximizing sequence for T . For 1 < p 6= q we show that any not weakly null maximizing sequence for a norm attaining operator T : lp → lq has a norm-convergent subsequence. We also prove that for any fixed x0 in lp, the set of operators T : lp → lq that attain their norm a...

In this paper we propose a test for a set of linear restrictions in a Vector Autoregressive Moving Average (VARMA) model. This test is based on the autoregressive metric, a notion of distance between two univariate ARMA models, M0 and M1, introduced by Piccolo in 1990. In particular, we show that this set of linear restrictions is equivalent to a null distance d(M0,M1) between two given ARMA mo...

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-...

Because X is a modification of Y , the right-hand side is a union of finitely many P -null sets, hence is itself a P -null set. A and B each belong to F , so P (A4B) = 0. Because P (A4B) = 0, P (A) = P (B), i.e. P (Xt1 ∈ A1, . . . , Xtn ∈ An) = P (Yt1 ∈ A1, . . . , Ytn ∈ An). 1We have not assumed that (Ω,F , P ) is a complete measure space, so we must verify that a set is measurable before spea...