Proofs of main theorems Sample Clauses

Proofs of main theorems. We start by proving Theorem 1 for m =0 and q = 1/2 using the approach in Sec. 7. It is inessential in our argument that the potential has form (5). It suffices to assume that v(x) is a smooth periodic function with a unique point of global maximum. We again stress that the symmetry x → —x is not required. We study the case of arbitrary q later (see more details in Appendix D). We now proceed to prove the theorems in Sec. 4. For simplicity, we assume that β = 1 without loss of generality. ^ Proof of Theorem 3. We first establish the estimates and asymptotic approximations for the band widths (i.e., dwell on the case q = 1/2). For this, we consider the operator H on the cylinder Z(2). According to Xxxxxxx 6, the operator H^ |Vν is determined by the matrix P = SI + W + O(e—2σ/h) ≡ S w12 + O(e—2σ/h).
Sample 1

Related to Proofs of main theorems

  • Proofs of Claim In case of the pendency of any receivership, insolvency, liquidation, bankruptcy, reorganization, arrangement, adjustment, composition or other judicial proceeding relating to the Company or the property of the Company or of such other obligor or its creditors, the Secured Party (irrespective of whether the Obligations shall then be due and payable as therein expressed or by declaration or otherwise and irrespective of whether the Secured Party shall have made any demand on the Company for the payment of the Obligations), subject to the rights of Previous Security Holders, shall be entitled and empowered, by intervention in such proceeding or otherwise: (i) to file and prove a claim for the whole amount of the Obligations and to file such other papers or documents as may be necessary or advisable in order to have the claims of the Secured Party (including any claim for the reasonable legal fees and expenses and other expenses paid or incurred by the Secured Party permitted hereunder and of the Secured Party allowed in such judicial proceeding), and (ii) to collect and receive any monies or other property payable or deliverable on any such claims and to distribute the same; and any custodian, receiver, assignee, trustee, liquidator, sequestrator or other similar official in any such judicial proceeding is hereby authorized by the Secured Party to make such payments to the Secured Party and, in the event that the Secured Party shall consent to the making of such payments directed to the Secured Party, to pay to the Secured Party any amounts for expenses due it hereunder.

  • Appointment to Fill a Vacancy in the Office of Trustee The Issuers, whenever necessary to avoid or fill a vacancy in the office of Trustee, will appoint, in the manner provided in Section 7.08, a Trustee, so that there shall at all times be a Trustee hereunder with respect to each series of Debt Securities.

  • Appointment to Fill Vacancy in Office of Trustee The Company, whenever necessary to avoid or fill a vacancy in the office of Trustee, will appoint, in the manner provided in Section 7.10, a Trustee, so that there shall at all times be a Trustee hereunder.

  • Appointment to Fill a Vacancy in Office of Trustee The Issuer, whenever necessary to avoid or fill a vacancy in the office of Trustee, will appoint, in the manner provided in Section 6.10, a Trustee, so that there shall at all times be a Trustee with respect to each series of Securities hereunder.

  • Control of Proceedings The Party responsible for the Tax under this Agreement shall control audits and disputes related to such Taxes (including action taken to pay, compromise or settle such Taxes). The Seller and the Buyer shall jointly control, in good faith with each other, audits and disputes relating to Straddle Periods. Reasonable out-of-pocket expenses with respect to such contests shall be borne by the Seller and the Buyer in proportion to their responsibility for such Taxes as set forth in this Agreement. Except as otherwise provided by this Agreement, the noncontrolling Party shall be afforded a reasonable opportunity to participate in such proceedings at its own expense.

  • Appointments to Fill Vacancies in Trustee’s Office The Company, whenever necessary to avoid or fill a vacancy in the office of Trustee, will appoint, in the manner provided in Section 7.09, a Trustee, so that there shall at all times be a Trustee hereunder.

  • Dissolution or Liquidation To the extent not previously exercised or settled, Options, SARs and Stock Units shall terminate immediately prior to the dissolution or liquidation of the Company.

  • Insolvency; Voluntary Proceedings The Borrower or any Material Subsidiary (i) ceases or fails to be solvent, or generally fails to pay, or admits in writing its inability to pay, its debts as they become due, subject to applicable grace periods, if any, whether at stated maturity or otherwise; (ii) voluntarily ceases to conduct its business in the ordinary course; (iii) commences any Insolvency Proceeding with respect to itself; or (iv) takes any action to effectuate or authorize any of the foregoing; or

  • No Action for Dissolution The Members acknowledge that irreparable damage would be done to the goodwill and reputation of the Company if any Member should bring an action in court to dissolve the Company under circumstances where dissolution is not required by Section 10.1. Accordingly, except where the Board has failed to cause the liquidation of the Company as required by Section 10.1 and except as specifically provided in Section 18-802, each Member hereby to the fullest extent permitted by law waives and renounces his right to initiate legal action to seek dissolution of the Company or to seek the appointment of a receiver or trustee to wind up the affairs of the Company, except in the cases of fraud, violation of law, bad faith, gross negligence, willful misconduct or willful violation of this Agreement.

  • Insolvency or Liquidation Proceedings Upon any payment or distribution of the assets of the Note Issuer or any other Grantor to creditors in connection with an Insolvency or Liquidation Proceeding: (a) the First Lien Claimholders shall be entitled to the prior indefeasible Discharge of First Lien Obligations in cash before the Second Lien Claimholders shall be entitled to receive any payment or distribution from any source (whether or not from the Notes Issuer or any other Grantor) of any kind (whether in cash, Equity Interests or otherwise) in respect of the Second Lien Obligations; (b) until the prior indefeasible Discharge of First Lien Obligations has occurred, any payment or distribution from any source (whether or not from the Notes Issuer or any other Grantor) of any kind (whether in cash, Equity Interests or otherwise) to which the Second Lien Claimholders would be entitled but for the subordination provisions of this Agreement shall be made to the First Lien Collateral Agent for the benefit of the First Lien Claimholders, except that Second Lien Claimholders may receive Permitted Junior Securities; provided, that, if such payments are in a form other than cash or cash equivalents (the “Non-Cash Consideration”), the First Lien Collateral Agent, for the benefit of the First Lien Claimholders, shall be authorized to monetize such Non-Cash Consideration (other than Permitted Junior Securities) in its sole discretion and any cash proceeds shall be applied to the First Lien Obligations as provided herein. The application of such cash proceeds shall reduce the First Lien Obligations only to the extent of the actual cash payment indefeasibly received by the First Lien Claimholders, net of fees, costs and commissions; and (c) if any payment or distribution from any source (whether or not from the Notes Issuer or any other Grantor) of any kind is made to the Second Lien Collateral Agent or any Second Lien Claimholders in respect of the Second Lien Obligations that, pursuant to this Agreement, should not have been made to them, such Second Lien Claimholders shall hold such payments or distributions in trust for the First Lien Claimholders and immediately pay and/or deliver such payments or distributions over to the First Lien Collateral Agent for the benefit of the First Lien Claimholders.