The absence of residual property for strong exponents of oscillation of linear systems

In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of zeros, roots, hyperroots, strict and non-strict signs of non-zero solutions of linear homogeneous differential systems on the positive semi-axis. On the set of non-zero solutions of autonomous systems the relations between these exponents of oscillation are established. It is proved that all strong exponents of oscillations (unlike Sergeev's frequencies of sign changes, zeros and roots, as well as all the weak exponents of oscillations) considered as functions on the set of solutions to linear homogeneous $n$-dimensional differential systems with continuous coefficients on the semi-line are not residual (i.e. can be changed when changing solution on a finite interval). Besides, at any beforehand given natural $n\ge2$ we give the example of $n$-dimensional differential system, for some solution of which all strong oscillation exponents differ from corresponding weak exponents. In this case, all weak and all strong exponents on the chosen solution coincide with each other, respectively. When proving the results of this work, the case of parity and odd $n$ are considered separately.

Residually finite rationally p groups

In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so.

Part V Enforcement, 18 Enforcement of True Security Interests

This chapter explains how a number of general issues arise in connection with the enforcement of true security interests that will be taken before particular enforcement issues are dealt with under the relevant head. True security interests consist of the four nominate types of security recognized in English law, namely, the three consensual securities of pledge, mortgage, and charge, together with the non-consensual lien. The obligation of a secured party to account for any surplus obtained is implicit in the nature of security. In the case of a pledge, this is consistent with the pledgor’s residual property rights. This obligation is also consistent in the case of mortgages. Charges and mortgages may be taken together, given that they are assimilated in drafting practice and in judicial treatment as alike in recognizing the borrower’s equity of redemption.

Characterisation of Moisture Absorption Effects on the Strength of Composite Materials

In the present work, the variation of the tensile strengths of various polymers and polymer composites due to moisture ageing is investigated. The stacking sequences of the composites include 0o, 90o, ±45o and quasi-isotropic laminates. The variation in the strength of material is then fitted using a residual property model. Results show that the proposed model characterises well the variation in the strengths that exhibits a typical trend, where there is no obvious scattered data. In fibre dominated laminates (0o), the degradation parameter seems to be influenced by the type of material, whereas for matrix dominated laminates (90o), ageing temperature appears to be the dominant factor that affects the degradation trend. As for [±45]2S and adhesives, exponential decay trend is observed. This model could be further improved to include the material and temperature factors.

Corporate Ownership And Compact Costs

Using definitions borrowed from property law, this paper contend shareholders receive less than complete ownership because the contract written by the state is deficient. This suggests that an additional residual property benefit exists in the firm separate and apart from both equity-holders and bondholder claims. Thus, the value of the firm is V = E + D + r, where V, E, and D are the market values of the firm, equity, and debt, respectively, and r is a residual value. The authors refer to this compact costs since it stems from the states faulty title document, and it explains investors paying a premium seeking corporate control.