Differentiation Of Norms Research Articles

Integration and differentiation of norms of the Russian civil legislation as the most essential trends in its reforming

Integration and differentiation of norms of the Russian civil legislation as the most essential trends in its reforming

The characterization of complex continuous norms of reaction

Recent focus on the array of phenotypes expressed under differing environmental conditions, or phenotypic plasticity, has led to increased understanding of its genetic basis as well as its adaptive significance. However, the quantification of plasticity has proven difficult, hampered by both the limited number of environments over which plasticity may typically be assessed and by the need to assume, a priori, the general form of reaction norms under study. Our understanding of the shapes of continuous norms of reaction and, consequently, the subtle differences that may exist in shapes among genotypes or populations is rudimentary. Here, we propose the use of the loess smoothing function to analyze complex norms of reaction and to quantify total plasticity over many environments. A thermogradient incubator offers an ideal means to provide many environments for a demonstration of the use of the loess method. We test seed germination in three populations of two monocarpic plant species for population differentiation in plasticity to temperature. First, we test for differentiation in norms of reaction to 30 temperature environments among three populations of the monocarpic perennial, Lobelia inflata. The second demonstration assesses plasticity to eight temperature environments of three populations of the arctic‐alpine annual, Koenigia islandica. Our demonstration shows that the loess technique can detect significant genetic differentiation among populations in complex norms of reaction for both species studied, and suggests that the use of this procedure should be considered where the form of norms of reaction might be complex. The general applicability of the approach is discussed.

The characterization of complex continuous norms of reaction

Recent focus on the array of phenotypes expressed under differing environmental conditions, or phenotypic plasticity, has led to increased understanding of its genetic basis as well as its adaptive significance. However, the quantification of plasticity has proven difficult, hampered by both the limited number of environments over which plasticity may typically be assessed and by the need to assume, a priori, the general form of reaction norms under study. Our understanding of the shapes of continuous norms of reaction and, consequently, the subtle differences that may exist in shapes among genotypes or populations is rudimentary. Here, we propose the use of the loess smoothing function to analyze complex norms of reaction and to quantify total plasticity over many environments. A thermogradient incubator offers an ideal means to provide many environments for a demonstration of the use of the loess method. We test seed germination in three populations of two monocarpic plant species for population differentiation in plasticity to temperature. First, we test for differentiation in norms of reaction to 30 temperature environments among three populations of the monocarpic perennial, Lobelia inflata. The second demonstration assesses plasticity to eight temperature environments of three populations of the arctic-alpine annual, Koenigia islandica. Our demonstration shows that the loess technique can detect significant genetic differentiation among populations in complex norms of reaction for both species studied, and suggests that the use of this procedure should be considered where the form of norms of reaction might be complex. The general applicability of the approach is discussed.

On \(\beta\)-differentiability of norms

In this note we give some characterizations for the differentiability with respect to a bornology of a continuous convex function. The special case of seminorms is treated. A characterization of this type of differentiability in terms of the subgradient of the function is also obtained.

Geometric conditions of differentiability for a regular locally Lipschitz function

The differentiability of a norm of a Banach space may be characterized by its unit sphere. This paper generalizes these geometric conditions of norm's differentiability to the case of a regular locally Lipschitz function.

A note on cotype of smooth spaces

For sufficiently smooth Banach spaces weak cotype 2 implies Hilbert space. It is well known that differentiability of the norm on a given Banach space X has a significant impact on the structure of X. Speaking about higher order differentiability of norms, it was shown in [2] that a Banach space X is isomorphic to a Hilbert space if on both X and X* there exist real valued functions ~ with bounded nonempty support and such that ~o' are locally Lipschitzian. It also is known that a Banach space X which admits a real valued twice continuously differentiable function ~o with bounded nonempty support is isomorphic to a FIilbert space provided any infinite dimensional subspace of X contains an infinite dimensional subspace which is isomorphic to a Hilbert space [9]. There are many spaces which do not admit real valued twice differentiable functions with bounded support and yet have norms with differential which is Lipschitzian on the sphere [12], I4], [9]. In fact the result in this note puts the spaze of Tsirelson's kind [19], constructed by Johnson and Figiel and described in [5], Ex. 5.3 to this family of spaces. Indeed, this space is of type 2 with unconditional basis and superreflexive (of. [1], [5], [8], [15]) and as such it admits an equivalent norm I1' II whose modulus of smoothness is of power type 2 [3]. Thus the differential I1" I1' of the norm If" II is Lipsehitzian on the sphere [2]. *Research supported in part by NSERC (Canada)

Essential supremum norm differentiability

The points of Gateaux and Fréchet differentiability in L∞(μ, X) are obtained, where (Ω, ∑, μ) is a finite measure space and X is a real Banach space. An application of these results is given to the space B(L1(μ, ℝ), X) of all bounded linear operators from L1(μ, ℝ) into X.

Supremum norm differentiability

The points of Gateaux and Fréchet differentiability of the norm in C(T, E) are obtained, where T is a locally compact Hausdorff space and E is a real Banach space. Applications of these results are given to the space ℓ∞(E) of all bounded sequences in E and to the space B(ℓ1, E) of all bounded linear operators from ℓ1 into E

The operator of minimal continuance of linear functionals

In terms of weak differentiability of norms and algebraic complementation of subspaces, the single-valuedness of the Hahn-Banach continuation is characterized, as well as its linearity as an operator. The connection is established between this continuation and the best approximation by the annihilator of the given subspace.

Principles of controlling traffic noise: Solving a multi-objective decision problem

The main objectives for a government to control traffic noise are such as economic efficiency, a reasonable welfare distribution and acceptable noise standards. The instruments available for arriving at these objectives are traffic control, emission regulation and environment protection. This paper analyzes how to make trade-offs between such objectives and how to choose a combination of noise control instruments. Special attention is then made to one welfare distribution objective, namely the ‘polluters-pay-principle’. That principle demands using the vehicle tax to finance investments to protect environment. And in order to stimulate the development of less noisy vehicles such a tax has to be progressively increased for noisy cars. The analysis is demonstrated on the situation of Sweden where a differentiation of norms as well as of vehicle taxes is proposed.

Principles of controlling traffic noise Solving a multi-objective decision problem

The main objectives for a government to control traffic noise are such as economic efficiency, a reasonable welfare distribution and acceptable noise standards. The instruments available for arriving at these objectives are traffic control, emission regulation and environment protection. This paper analyzes how to make trade-offs between such objectives and how to choose a combination of noise control instruments. Special attention is then made to one welfare distribution objective, namely the ‘polluters-pay-principle’. That principle demands using the vehicle tax to finance investments to protect environment. And in order to stimulate the development of less noisy vehicles such a tax has to be progressively increased for noisy cars. The analysis is demonstrated on the situation of Sweden where a differentiation of norms as well as of vehicle taxes is proposed.