Proof Of Proposition Research Articles

On Newton’s shell theorem

In the present letter, Newton’s theorem for the gravitational field outside a uniform spherical shell is considered. In particular, a purely geometric proof of proposition LXXI/theorem XXXI of Newton’s Principia, which is suitable for undergraduates and even skilled high-school students, is proposed. Minimal knowledge of elementary calculus and three-dimensional Euclidean geometry are required.

Erratum: Linear Projections and Successive Minima

The proof of Proposition 1 and Theorem 2 in [3] is incorrect. Indeed, Sections 2.5 and 2.7 in [3] contain a vicious circle: the definition of the filtrationVi, 1 ≤i≤n, in Section 2.5 of that article depends on the choice of the integersni, when the definition of the integersniin Section 2.7 depends on the choice of the filtration(Vi). Thus, only Theorem 1 and Corollary 1 in [3] are proved. In the following we will prove another result instead of [3, Proposition 1].

Erratum: Linear Projections and Successive Minima

The proof of Proposition 1 and Theorem 2 in [3] is incorrect. Indeed, Sections 2.5 and 2.7 in [3] contain a vicious circle: the definition of the filtration Vi, 1 ≤ i ≤ n, in Section 2.5 of that article depends on the choice of the integers ni, when the definition of the integers ni in Section 2.7 depends on the choice of the filtration (Vi). Thus, only Theorem 1 and Corollary 1 in [3] are proved. In the following we will prove another result instead of [3, Proposition 1].

Erratum to: Rough Burgers-like equations with multiplicative noise

Unfortunately, the proof of Proposition 4.8 in [HW13] is not correct. In fact, equation (4.60) whereRθ is rewritten as a stochastic integral is not correct, because the integrand is not adapted to the filtration. Hence in (4.61) we are not allowed to apply the Burkholder-Davies-Gundy inequality. The argument was corrected in [HMW12, Lemma 3.6 and Corollary 3.7] in a more complicated situation. In order to transfer these statement to the situation in [HW13] some smaller changes are needed. First of all the regularity of the linearised process X should be measured in a Holder norm with slightly bigger index α∗ than the solution u. This can be done without further problems. Then the definition of the stopping time in (4.53) becomes. For K > 0 and for an α? ∈ (α, 1/2) we introduce the stopping time

Erratum to: An optimal gap theorem

Here we shall use the same notations of the original paper. The references to equations, lemmas, propositions, theorems are with respect to the paper unless stated otherwise. The argument provided for the proof of Proposition 3.1 is not complete. It overlooks the fact that the solution 2-form η(x, t) of the problem (3.1) may contain non-trivial (2,0) and (0,2) parts. Hence Proposition 3.1 needs a correction. We thank Gilles Carron for pointing out this oversight in the original proof of Proposition 3.1 and his comments. We also thank Luen-Fai Tam for helpful related discussions. We refer interested reader to [5] for an alternate proof of Theorem 1.2. Since the preservation of the positivity for solution 2-forms solving the problem (3.1) in the paper, without knowing that the 2-form being (1,1)-type, seems elusive (even for the (1,1) part projection of the 2-form solutions) we replace (3.1) with the following initial boundary value problem on (1,1)form η. Let Ω be a smooth bounded domain in M . For a given point p ∈ ∂Ω , let ν be the unit outward normal in a small neighborhood near p. We

Notes on “Bipolar fuzzy graphs”

In this note, we show by examples that Definitions 3.1, 5.6, and 5.7, Example 5.8, Propositions 5.9 and 5.10, and partial proof of Propositions 5.11 and 5.12 in a previous paper by Akram (Muhammad Akram, Bipolar fuzzy graphs, Inform. Sci. 181 (2011) 5548–5564) contain some flaws and then provide the correct versions. Finally, we introduce a generalized bipolar fuzzy graph.

Erratum to: An Equivalence of Fusion Categories

1. The author is grateful to Yi-Zhi Huang who has found the following gap in the proof of Proposition 3.6 and hence in the proof of the main Theorems 3.4, 4.3: In the second and third lines before the end of proof of Proposition 3.6 (page 265), it is stated (and proved) that one vector is proportional to another one. However, the coefficient of proportionality may vanish, hence in the notations of Proposition 3.6 it is only proved that L = H ′′ ⊂ H ′ = H. In order to prove that L = H ′′ = H ′ = H as stated in Proposition 3.6, we argue as follows: Theorem 4.4 is proved e.g. in [Fal94,Tel95]. It means that the K-rings of monoidal categories O±κ are based isomorphic. It means that dimL = dimH, and hence the above inclusion L ⊂ H must be an isomorphism. The proof of Proposition 3.6 (and hence Theorems 3.4, 4.3) is now complete. Note however that Theorem 4.4 is not deduced from the main Theorem 4.3 but instead is used in its proof. 2. The following minor corrections are also due to Yi-Zhi Huang: (a) Page 249, line 6 of the first paragraph of Introduction: replace “It is known” by “It is conjectured”, so that the sentence reads: “It is conjectured (see e.g. [MoSe] or [BFM]) that Ok has the structure of a rigid braided tensor category.” (b) Page 261, line 2: replace HomO±κ(V,W )×HomO±κ(W,V ) → C by HomO−κ(V,W )×HomO−κ(W,V ) → C. This pairing arises from rigidity of O−κ, and the corresponding pairing at the positive level would arise from rigidity of Oκ which is not yet established in Section 3, and is only proved in Section 4 as a result of tensor equivalence with

A correction to "Propagation of singularities for the wave equation on manifolds with corners"

We correct an error in the proof of Proposition 7.3 of the author’s paper on the propagation of singularities for the wave equation on manifolds with corners. The correction does not affect the statement of Proposition 7.3, and it does not affect any other part of the paper. There is a mistake in the proof of Proposition 7.3 of [1]; namely, a term was omitted in (7.9) so that the displayed equation after (7.15), as well as its analogues after (7.16), do not hold. The term omitted corresponds to the term |x|2 in (7.8) being differentiated by the term 2Aξ · ∂x in the Hamilton vector field appearing in (6.3). This mistake can be easily remedied as follows. First, after the displayed equation following (7.7) we specify one of the ρj slightly more carefully; namely, we require ρ1 = 1− τ|ζ|y. Note that dρ1 6= 0 at q0 for ζ 6= 0 there. Then |τ−1W ω0| ≤ C ′ 1ω 1/2 0 (ω 1/2 0 + |t− t0|) still holds. The argument of [1] proceeds with a motivational calculation, followed by the precise version of what is needed. We follow this approach here. So first the correct motivational calculation is presented. We still have p|x=0 = τ2 − |ξ|y − |ζ|y. Thus, the equation after (7.9) can be strengthened to τ|ξ|y ≤ C(τ−2|p|+ |x|+ ω 1/2 0 ), This work is partially supported by NSF grant DMS-0801226. c © 2013 Department of Mathematics, Princeton University. All equation and proposition numbers of the form (7.xx) or 7.xx refer to [1].

A comment on “welfare reducing licensing”

This comment points out the existence of flaws in the proof of the main proposition proposed by Faulí-Oller and Sandonís (2002) [Faulí-Oller, R., Sandonís, J., 2002. Welfare reducing licensing. Games Econ. Behav. 41, 192–205] and corrects them.

Erratum: “On a fourth order degenerate parabolic equation in higher space dimension” [J. Math. Phys. 50, 123524 (2009)

In this erratum we fix a flaw in the proof of Proposition 3.1 of Li [“On a fourth order degerate parabolic equation in higher space dimensions,” J. Math. Phys. 50, 123524 (2009)]10.1063/1.3272788.

Erratum to: Spectral Simplicity and Asymptotic Separation of Variables

The statement of Lemma A.3 in [HllJdg11] is false.1 This lemma is used in the proof of Lemma 8.2 which is also incorrect and is used in the proof of Proposition 9.1. Despite this gap, Proposition 9.1 is correct and, in order to rectify the situation, we provide here the correct estimates needed to derive Proposition 9.1. As a consequence, the statements of the main results of the article [HllJdg11] are correct. We would like to emphasize that the overall strategy of the proof of Proposition 9.1 remains unchanged. In particular, Proposition 9.1 is a statement about concentration properties of quasimodes for the quadratic form a t coming from separation of variables. More precisely, it relies on the fact that a quasimode of order t for a t at a non-critical energy cannot concentrate on the turning point (and thus must have some mass in the classically allowed region). In the exposition given in [HllJdg11], this non-concentration was hidden behind Lemmas 9.4 and 9.5. We will make it more transparent here by directly using the Langer-Cherry transform and the following estimate for solutions to the semiclassical Airy equation:2

Corrigendum: “Homogenized Spectral Problems for Exactly Solvable Operators: Asymptotics of Polynomial Eigenfunctions”

Here we provide a correct proof of Proposition 6 of [2]. No other results of the latter paper are affected.

Strange Attractors for Periodically Forced Parabolic Equations

Introduction Basic Definitions and Facts Statement of Theorems Invariant Manifolds Canonical Form of Equations Around the Limit Cycle Preliminary Estimates on Solutions of the Unforced Equation Time-$T$ Map of Forced Equation and Derived $2$-D System Strange Attractors with SRB Measures Application: The Brusselator Appendix A. Proofs of Propositions 3.1-3.3 Appendix B. Proof of Proposition 7.5 Appendix C. Proofs of Proposition 8.1 and Lemma 8.2 Bibliography

Erratum to: n-Fold implicative basic logic is Gödel logic

1. The proof of Proposition 5 is wrong. Equation (10) can be utilized only if the power of an element x is n, not if it is n 1 as is done in the proof. Moreover, an n ? 1 element Lukasiewicz chain is an explicit example that Proposition 5 does not hold: {1} is an n-fold implicative filter in that algebra, however, the Lukasiewicz product is not idempotent. 2. Corollary 6, Theorem 8 and Theorem 9 are based on Proposition 5, therefore also they do not hold. Since Godel logic trivially satisfies axiom (12), we can only say that Godel logic is an example of n-fold implicative basic logic, however, n-fold implicative basic logic is not, in general, Godel logic, contrary to the claim in Theorem 9. 3. The inaccuracy of Proposition 5 also implies that the problem of the relation of n-fold implicative basic logic and n-fold positive implicative basic logic remains open, contrary to what is alleged in the Conclusion.

THE SEPARATION OF POWERS, COURT‐CURBING AND JUDICIAL LEGITIMACY ERRATUM

In “The Separation of Powers, Court Curbing, and Judicial Legitimacy” (American Journal of Political Science 53 (4): 971-89 (2009)), there is a slight error in the proof of Proposition 3. Specifically, Congress’s (C ’s) payoff from deviating to a pure strategy, !(! = L ) = h should be given by (1 − m)bc − mbc − e. C ’s strategy from playing a pure strategy !(! = L ) = l is given by e − bc . Thus, the mixing probability that makes C indifferent between playing!(! = H) = l and!(! = H) = h is m = bc−e bc . This probability holds for all bc > e. Thus, the equilibrium characterized in Proposition 3 exists for all bc > e. Note that this correction does not affect the substantive interpretation of the model or the empirical implications derived. In the article, analysis was constrained to the case where bc 2e, it was asserted there exists a pooling equilibrium in which the Court never makes a constrained decision upon observing Court curbing. The comparative static derived was that as bc increases—as the Court and Congress become more ideologically divergent—the Court should be less likely to make a constrained decision upon observing Court curbing. The algebraic correction reveals that the semi separating equilibrium holds for all b > e, and the restriction that bc < 2e is not necessary. However, the probability with which the Court makes an unconstrained decision upon observing Court curbing (! = h), m = bc−e bc , is increasing in bc . Thus, the prediction that judicial restraint in response to Court curbing should decrease as Congress becomes more ideologically divergent from the Court continues to hold.

Fuzzy data envelopment analysis models with assurance regions: A note

In a recent paper in this journal by Liu and Chuang [Liu, S. -T., & Chuang, M. (2009). Fuzzy efficiency measures in fuzzy DEA/AR with application to university libraries. Expert Systems with Applications, 36(2), 1105–1113]. Proposition 1 is proved for finding lower and upper bounds of the α-cut of E ∼ r . However, we find that the above proof omit the verification of constraints of assurance regions and the models (11) and (12) omit the constraints of DMUr, which makes it incredible. In this paper, we correct the models and proof of Proposition 1 in Liu and Chuang (2009). We also propose a generalized fuzzy data envelopment model with assurance regions, whose lower and upper bounds at given levels could be obtained similarly.

Replenishment run time problem with machine breakdown and failure in rework

This paper studies the optimal replenishment run time for a production system with stochastic machine breakdown and failure in rework. In most real world manufacturing processes, generation of defective items and random breakdown of equipment are inevitable. This research addresses such reliability issues by examining a production system with a Poisson breakdown rate and an imperfect reworking process. When machine breaks down, no-resumption policy is adopted. Under such a policy, the present production run is aborted and will be restarted only when all on-hand inventories are depleted. Mathematical modeling is used. Derivations of production- inventory cost functions and proofs of theorems and proposition are presented, with the objective of determination of the optimal replenishment run time and lot size that minimize the expected costs per unit time. An illustrating example is provided to show its practical usage.

A Note on “Lot-sizing with fixed charges on stocks: The convex hull”

We update and complete the proof of Proposition 7 in Van Vyve and Ortega (2004) [1], which states that the projection of a facility location reformulation of an uncapacitated lot-sizing problem with fixed charges on stocks (ULSW) to the original space is equivalent to that of the tight shortest path reformulation of ULSW. Their proof is interesting and consists of two cases, only first of which is analyzed in detail. We show that the second case exhibits several challenges not present in the first one and necessitates an updated proof.

Comments on "Partial channel feedback schemes maximizing overall efficiency in wireless networks" [Apr 08 1306-1314

The proof of Proposition 2 in the above-mentioned paper was incorrect due to some flaws in the prerequisite analysis of the proposed hybrid feedback scheme. We fix the flaws and correct the proof in this comment. From the corrected proof, we arrive at a new observation that the hybrid feedback scheme has exactly the same feedback overhead as the pure opportunistic feedback scheme if the threshold is adaptively set for a target utilization. Simulation results demonstrate the correctness of our analysis.

Correction to “A cartesian presentation of weakn–categories”

In this note, I provide a correction to the proof of Proposition 6.6 of [Rez10]. It forms a key part of the proof of the main Theorem 1.4 of that paper. The error lies in the use of Proposition 2.19, which is simply false as stated. In fact, it is easy to construct counterexamples to 2.19 when the poset P does not have contractible nerve. The error the original paper was pointed out to me by Bill Dwyer, who also suggested the idea which forms the basis of the proof given here. I thank Bill Dwyer for his help. The author was supported under NSF grant DMS-0505056. I assume context and notation of [Rez10].