Number Theory Sample Clauses

Number Theory. The prime counting function 2.4.1. For a positive integer s and a prime p, write sp for the p-part of s. Also, define lpp s = maxp prime sp to be the largest prime power divisor of s. Fix s ≥ 2, and let k = lpp s. By writing the prime factorization of s as s = kpr2 . . . prt , one immediately sees that s ≤ kδ(k), where δ(k) denotes the number of primes less than or equal to k. Hence, log s ≤ δ(k) log k. Also, it is proved in [46, Corollary 1] that
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Number Theory. Volume I: Tools and Diophantine equa- tions. Springer, 2007. [CP20] Xxxxxxxxx Xxxxxxxx and Xxxxxxxx Xxxxx. Entanglement in the family of division fields of elliptic curves with complex multiplication. arXiv preprint arXiv:2006.00883, 2020. [CR21] Xxxxxxx Xxxxxxx and Xxxxxx Xxxxx. Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves. Proceedings of the American Mathematical Society, 149(2):583–589, 2021. [Cre97] Xxxx X. Xxxxxxx. Algorithms for modular elliptic curves. Cambridge University Press, Cambridge, second edition, 1997. [CS19] Xxxxxxxxx Xxxxxxxx and Xxxxx Xxxxxxxxxxx. Cyclic reduction of ellip- tic curves. arXiv preprint arXiv:2001.00028, 2019. [Dav11] Agn`es Xxxxx. Borne uniforme pour les homoth´eties dans l’image de Xxxxxx associ´ee aux courbes elliptiques. Journal of Number Theory, 131(11):2175 – 2191, 2011. [Deu53] Xxx Xxxxxxx. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. Nachrichten von der Akademie der Wissenschaften G¨ottingen Mathematisch-Physikalische Klasse. IIa, Mathematisch- Physikalisch-Chemische Abteilung, 1953:85–94, 1953. [Deu58] Xxx Xxxxxxx. Die Klassenkorper der komplexen Multiplikation. In Enzyklop¨adie der mathematischen Wissenschaften, volume I2, Heft 10, Teil II. Teubner Verlag, Stuttgart, 1958. [DLM21] Xxxxxx X. Xxxxxxx, A´xxxxx Xxxxxx-Xxxxxxx, and Xxxxxxx X. Xxxxxx. Towards a classification of entanglements of Galois representations at- tached to elliptic curves. arXiv preprint arXiv:2105:02060, 2021. [DP16] Xxxxxxxxxx Xxxxx and Xxxxxxxxx Xxxxxxx. Reductions of algebraic in- tegers. Journal of Number Theory, 167:259–283, 2016. [EGH+11] Xxxxx X. Xxxxxxx, Xxxx Xxxxxxx, Xxxxxxxxx Xxxxxx, Xxxxxxx Xxx, Xxxx Xxxxxxxxxx, Xxxxxx Xxxxxxxx, and Xxxxx Xxxxxxxx. Introduction to representation theory. American Mathematical Society, 2011. [ES53] Xxxx Xxxxxxx and A Xxxxxx. U¨ ber injektive Moduln. Archiv der [Fle68] Xxxxxxx Xxxxxxxxx. A new construction of the injective hull. Canadian Mathematical Bulletin, 11(1):19–21, 1968. [GM20] Aur´xxxxx Xxxxxxxx and C´esar Mart´xxxx. Homoth´eties explicites des repr´esentations l-adiques. arXiv preprint arXiv:2006.07401, 2020. [Gou97] Xxxxxxxx X. Xxxxˆxx. p-adic Numbers. Springer, 1997. [Gre12] Xxxxx Xxxxxxxxx. The image of Galois representations attached to elliptic curves with an isogeny. American Journal of Mathematics, 134(5):1167–1196, 2012. [Gro91] Xxxxxxxx X. Xxxxx. Xxxxxxxxx’s work on modular elliptic curves. In L- functions ...
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Number Theory. Algebraic and complex geometry
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Number Theory. Academic Press, 1966. — 436 pp. [2] Cox X. Xxxxxx of the form x2 + ny2: Fermat, Class Field Theory and Complex Multiplication. — X.Xxxxx and Sons, 1989. — 363 pp.
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Related to Number Theory

  • Company’s Failure to Timely Deliver Securities If the Company shall fail, for any reason or for no reason, on or prior to the Share Delivery Date, if the Transfer Agent is not participating in FAST or such shares of Common Stock may not be issued without legends under the Securities Act, to issue and deliver to the Holder (or its designee) a certificate for the number of Warrant Shares to which the Holder is entitled and register such Warrant Shares on the Company’s share register or, if the Transfer Agent is participating in FAST and such shares of Common Stock may not be issued without legends under the Securities Act, to credit the balance account of the Holder or the Holder’s designee with DTC for such number of Warrant Shares to which the Holder is entitled upon the Holder’s exercise of this Warrant (as the case may be) (a “Delivery Failure”), then, in addition to all other remedies available to the Holder, (X) the Company shall pay in cash to the Holder on each day after the Share Delivery Date and during such Delivery Failure an amount in cash, as liquidated damages and not as a penalty, equal to 2.0% of the product of (A) the sum of the number of shares of Common Stock not issued to the Holder on or prior to the Share Delivery Date and to which the Holder is entitled, and (B) any trading price of the Common Stock selected by the Holder in writing as in effect at any time during the period beginning on the applicable exercise date and ending on the applicable Share Delivery Date, and (Y) the Holder, upon written notice to the Company, may void its Exercise Notice with respect to, and retain or have returned, as the case may be, any portion of this Warrant that has not been exercised pursuant to such Exercise Notice; provided that the voiding of an Exercise Notice shall not affect the Company’s obligations to make any payments which have accrued prior to the date of such notice pursuant to this Section 1(c) or otherwise. In addition to the foregoing, if on or after such Share Delivery Date the Holder purchases (in an open market transaction or otherwise) shares of Common Stock to deliver in satisfaction of a sale by the Holder of all or any portion of the number of shares of Common Stock issuable upon such exercise that the Holder is entitled to receive from the Company (a “Buy-In”), then, in addition to all other remedies available to the Holder, the Company shall, within three (3) Business Days after the Holder’s request and in the Holder’s discretion, either (i) pay cash to the Holder in an amount equal to the Holder’s total purchase price (including brokerage commissions and other out-of-pocket expenses, if any) for the shares of Common Stock so purchased (including, without limitation, by any other Person in respect, or on behalf, of the Holder) (the “Buy-In Price”), at which point the Company’s obligation to so issue and deliver such certificate (and to issue such shares of Common Stock) or credit the balance account of such Holder or such Holder’s designee, as applicable, with DTC for the number of Warrant Shares to which the Holder is entitled upon the Holder’s exercise hereunder (as the case may be) (and to issue such Warrant Shares) shall terminate, or (ii) promptly honor its obligation to so issue and deliver to the Holder a certificate or certificates representing such Warrant Shares or credit the balance account of such Holder or such Holder’s designee, as applicable, with DTC for the number of Warrant Shares to which the Holder is entitled upon the Holder’s exercise hereunder (as the case may be) and pay cash to the Holder in an amount equal to the excess (if any) of the Buy-In Price over the product of (A) such number of Warrant Shares multiplied by (B) the Closing Sale Price of the Common Stock on the Share Delivery Date with respect to the related Exercise Notice (the “Buy-In Payment Amount”). The Holder shall provide the Company written notice indicating the amounts payable to the Holder in respect of the Buy-In and, upon request of the Company, evidence of the amount of such loss. Nothing shall limit the Holder’s right to pursue any other remedies available to it hereunder, at law or in equity, including, without limitation, a decree of specific performance and/or injunctive relief with respect to the Company’s failure to timely deliver certificates representing shares of Common Stock (or to electronically deliver such shares of Common Stock) upon the exercise of this Warrant as required pursuant to the terms hereof.

  • Adjustment in Number of Securities Upon each adjustment of the Exercise Price pursuant to the provisions of this Section 8, the number of Warrant Securities issuable upon the exercise at the adjusted exercise price of each Warrant shall be adjusted to the nearest full amount by multiplying a number equal to the Exercise Price in effect immediately prior to such adjustment by the number of Warrant Securities issuable upon exercise of the Warrants immediately prior to such adjustment and dividing the product so obtained by the adjusted Exercise Price.

  • Adjustment in Number of Rights The Company may elect on or after the date of any adjustment of the Purchase Price to adjust the number of Rights, in substitution for any adjustment in the number of one one-hundredths of a Preferred Share issuable upon the exercise of a Right. Each of the Rights outstanding after such adjustment of the number of Rights shall be exercisable for the number of one one-hundredths of a Preferred Share for which a Right was exercisable immediately prior to such adjustment. Each Right held of record prior to such adjustment of the number of Rights shall become that number of Rights (calculated to the nearest one-hundredth) obtained by dividing the Purchase Price in effect immediately prior to adjustment of the Purchase Price by the Purchase Price in effect immediately after adjustment of the Purchase Price. The Company shall make a public announcement of its election to adjust the number of Rights, indicating the record date for the adjustment, and, if known at the time, the amount of the adjustment to be made. This record date may be the date on which the Purchase Price is adjusted or any day thereafter, but, if the Right Certificates have been issued, shall be at least ten (10) days later than the date of the public announcement. If Right Certificates have been issued, upon each adjustment of the number of Rights pursuant to this Section 11.9, the Company may, as promptly as practicable, cause to be distributed to holders of record of Right Certificates on such record date Right Certificates evidencing, subject to Section 14, the additional Rights to which such holders shall be entitled as a result of such adjustment, or, at the option of the Company, shall cause to be distributed to such holders of record in substitution and replacement for the Right Certificates held by such holders prior to the date of adjustment, and upon surrender thereof, if required by the Company, new Right Certificates evidencing all the Rights to which such holders shall be entitled after such adjustment. Right Certificates so to be distributed shall be issued, executed and countersigned in the manner provided for herein (and may bear, at the option of the Company, the adjusted Purchase Price) and shall be registered in the names of the holders of record of Right Certificates on the record date specified in the public announcement.

  • No Fractional Warrants Other Than as Part of Units The Company shall not issue fractional Warrants other than as part of the Units, each of which is comprised of one share of Common Stock and one-half of one Public Warrant. If, upon the detachment of Public Warrants from Units or otherwise, a holder of Warrants would be entitled to receive a fractional Warrant, the Company shall round down to the nearest whole number the number of Warrants to be issued to such holder.

  • o Check if Transfer is Pursuant to Other Exemption (i) The Transfer is being effected pursuant to and in compliance with an exemption from the registration requirements of the Securities Act other than Rule 144, Rule 903 or Rule 904 and in compliance with the transfer restrictions contained in the Indenture and any applicable blue sky securities laws of any State of the United States and (ii) the restrictions on transfer contained in the Indenture and the Private Placement Legend are not required in order to maintain compliance with the Securities Act. Upon consummation of the proposed Transfer in accordance with the terms of the Indenture, the transferred beneficial interest or Definitive Note will not be subject to the restrictions on transfer enumerated in the Private Placement Legend printed on the Restricted Global Notes or Restricted Definitive Notes and in the Indenture. This certificate and the statements contained herein are made for your benefit and the benefit of the Company. [Insert Name of Transferor] By: Name: Title: Dated: 1. The Transferor owns and proposes to transfer the following: (a) o a beneficial interest in the:

  • No Creation of a Partnership or Exclusive Purchase Right Nothing contained in this Agreement, and no action taken pursuant hereto shall be deemed to constitute the relationship created hereby between the Note Holders as a partnership, association, joint venture or other entity. No Note Holder shall have any obligation whatsoever to offer to any other Note Holder the opportunity to purchase a participation interest in any future loans originated by such Note Holder or its Affiliates and if any Note Holder chooses to offer to any other Note Holder the opportunity to purchase a participation interest in any future mortgage loans originated by such Note Holder or its Affiliates, such offer shall be at such purchase price and interest rate as such Note Holder chooses, in its sole and absolute discretion. No Note Holder shall have any obligation whatsoever to purchase from any other Note Holder a participation interest in any future loans originated by such Note Holder or its Affiliates.

  • Compensation for Buy-In on Failure to Timely Deliver Conversion Shares Upon Conversion In addition to any other rights available to the Holder, if the Company fails for any reason to deliver to the Holder such Conversion Shares by the Share Delivery Date pursuant to Section 4(c)(ii), and if after such Share Delivery Date the Holder is required by its brokerage firm to purchase (in an open market transaction or otherwise), or the Holder’s brokerage firm otherwise purchases, shares of Common Stock to deliver in satisfaction of a sale by the Holder of the Conversion Shares which the Holder was entitled to receive upon the conversion relating to such Share Delivery Date (a “Buy-In”), then the Company shall (A) pay in cash to the Holder (in addition to any other remedies available to or elected by the Holder) the amount, if any, by which (x) the Holder’s total purchase price (including any brokerage commissions) for the Common Stock so purchased exceeds (y) the product of (1) the aggregate number of shares of Common Stock that the Holder was entitled to receive from the conversion at issue multiplied by (2) the actual sale price at which the sell order giving rise to such purchase obligation was executed (including any brokerage commissions) and (B) at the option of the Holder, either reissue (if surrendered) this Debenture in a principal amount equal to the principal amount of the attempted conversion (in which case such conversion shall be deemed rescinded) or deliver to the Holder the number of shares of Common Stock that would have been issued if the Company had timely complied with its delivery requirements under Section 4(c)(ii). For example, if the Holder purchases Common Stock having a total purchase price of $11,000 to cover a Buy-In with respect to an attempted conversion of this Debenture with respect to which the actual sale price of the Conversion Shares (including any brokerage commissions) giving rise to such purchase obligation was a total of $10,000 under clause (A) of the immediately preceding sentence, the Company shall be required to pay the Holder $1,000. The Holder shall provide the Company written notice indicating the amounts payable to the Holder in respect of the Buy-In and, upon request of the Company, evidence of the amount of such loss. Nothing herein shall limit a Holder’s right to pursue any other remedies available to it hereunder, at law or in equity including, without limitation, a decree of specific performance and/or injunctive relief with respect to the Company’s failure to timely deliver Conversion Shares upon conversion of this Debenture as required pursuant to the terms hereof.

  • Compensation for Buy-In on Failure to Timely Deliver Certificates Upon Conversion In addition to any other rights available to the Holder, if the Company fails for any reason to deliver to the Holder such certificate or certificates by the Share Delivery Date pursuant to Section 4(c)(ii), and if after such Share Delivery Date the Holder is required by its brokerage firm to purchase (in an open market transaction or otherwise), or the Holder’s brokerage firm otherwise purchases, shares of Common Stock to deliver in satisfaction of a sale by the Holder of the Conversion Shares which the Holder was entitled to receive upon the conversion relating to such Share Delivery Date (a “Buy-In”), then the Company shall (A) pay in cash to the Holder (in addition to any other remedies available to or elected by the Holder) the amount, if any, by which (x) the Holder’s total purchase price (including any brokerage commissions) for the Common Stock so purchased exceeds (y) the product of (1) the aggregate number of shares of Common Stock that the Holder was entitled to receive from the conversion at issue multiplied by (2) the actual sale price at which the sell order giving rise to such purchase obligation was executed (including any brokerage commissions) and (B) at the option of the Holder, either reissue (if surrendered) this Debenture in a principal amount equal to the principal amount of the attempted conversion (in which case such conversion shall be deemed rescinded) or deliver to the Holder the number of shares of Common Stock that would have been issued if the Company had timely complied with its delivery requirements under Section 4(c)(ii). For example, if the Holder purchases Common Stock having a total purchase price of $11,000 to cover a Buy-In with respect to an attempted conversion of this Debenture with respect to which the actual sale price of the Conversion Shares (including any brokerage commissions) giving rise to such purchase obligation was a total of $10,000 under clause (A) of the immediately preceding sentence, the Company shall be required to pay the Holder $1,000. The Holder shall provide the Company written notice indicating the amounts payable to the Holder in respect of the Buy-In and, upon request of the Company, evidence of the amount of such loss. Nothing herein shall limit a Holder’s right to pursue any other remedies available to it hereunder, at law or in equity including, without limitation, a decree of specific performance and/or injunctive relief with respect to the Company’s failure to timely deliver certificates representing shares of Common Stock upon conversion of this Debenture as required pursuant to the terms hereof.

  • No Suspension of Trading in or Delisting of Common Shares; Other Events The trading of the Common Shares (including without limitation the Shares) shall not have been suspended by the Commission, the Principal Market or FINRA and the Common Shares (including without limitation the Shares) shall have been approved for listing or quotation on and shall not have been delisted from the Nasdaq Stock Market, the New York Stock Exchange or any of their constituent markets. There shall not have occurred (and be continuing in the case of occurrences under clauses (i) and (ii) below) any of the following: (i) trading or quotation in any of the Company’s securities shall have been suspended or limited by the Commission or by the Principal Market or trading in securities generally on either the Principal Market shall have been suspended or limited, or minimum or maximum prices shall have been generally established on any of such stock exchanges by the Commission or the FINRA; (ii) a general banking moratorium shall have been declared by any of federal or New York, authorities; or (iii) there shall have occurred any outbreak or escalation of national or international hostilities or any crisis or calamity, or any change in the United States or international financial markets, or any substantial change or development involving a prospective substantial change in United States’ or international political, financial or economic conditions, as in the judgment of the Agent is material and adverse and makes it impracticable to market the Shares in the manner and on the terms described in the Prospectus or to enforce contracts for the sale of securities.

  • Compensation for Buy-In on Failure to Timely Deliver Warrant Shares Upon Exercise In addition to any other rights available to the Holder, if the Company fails to cause the Transfer Agent to transmit to the Holder the Warrant Shares in accordance with the provisions of Section 2(d)(i) above pursuant to an exercise on or before the Warrant Share Delivery Date, and if after such date the Holder is required by its broker to purchase (in an open market transaction or otherwise) or the Holder’s brokerage firm otherwise purchases, shares of Common Stock to deliver in satisfaction of a sale by the Holder of the Warrant Shares which the Holder anticipated receiving upon such exercise (a “Buy-In”), then the Company shall (A) pay in cash to the Holder the amount, if any, by which (x) the Holder’s total purchase price (including brokerage commissions, if any) for the shares of Common Stock so purchased exceeds (y) the amount obtained by multiplying (1) the number of Warrant Shares that the Company was required to deliver to the Holder in connection with the exercise at issue times (2) the price at which the sell order giving rise to such purchase obligation was executed, and (B) at the option of the Holder, either reinstate the portion of the Warrant and equivalent number of Warrant Shares for which such exercise was not honored (in which case such exercise shall be deemed rescinded) or deliver to the Holder the number of shares of Common Stock that would have been issued had the Company timely complied with its exercise and delivery obligations hereunder. For example, if the Holder purchases Common Stock having a total purchase price of $11,000 to cover a Buy-In with respect to an attempted exercise of shares of Common Stock with an aggregate sale price giving rise to such purchase obligation of $10,000, under clause (A) of the immediately preceding sentence the Company shall be required to pay the Holder $1,000. The Holder shall provide the Company written notice indicating the amounts payable to the Holder in respect of the Buy-In and, upon request of the Company, evidence of the amount of such loss. Nothing herein shall limit a Xxxxxx’s right to pursue any other remedies available to it hereunder, at law or in equity including, without limitation, a decree of specific performance and/or injunctive relief with respect to the Company’s failure to timely deliver shares of Common Stock upon exercise of the Warrant as required pursuant to the terms hereof.

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