text:
A-23.1 - Act respecting the National Assembly
Regulation 1
Alphanumerics
A-23.1, r. 1 Regulation respecting the survivor pension plan for Members of the National Assembly
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Updated to 29 November 2022
This document has official status.
chapter A-23.1
Act respecting the National Assembly
NATIONAL ASSEMBLY12December 18 198212December 18 1982
WHEREAS the people of Québec have a deep attachment to democratic principles of government;
Whereas the National Assembly is, through the elected representatives who compose it, the supreme and legitimate organ by which those principles are expressed and applied;
Whereas it behooves this Assembly, as the guardian of the historical and inalienable rights and powers of the people of Québec, to defend it against any attempt to despoil it of its rights and powers or to derogate from them;
Whereas it is befitting, therefore, that the perdurance, the sovereignty and the independence of the National Assembly be affirmed, and that its proceedings be protected against all interference,
HER MAJESTY, with the advice and consent of the National Assembly of Québec, enacts as follows:
ORGANIZATION AND OPERATION
COMPOSITION, TERM AND POWERS
1. The National Assembly is composed of the Members elected for each of the electoral divisions established in accordance with the Election Act (chapter E-3.3) and whose names have been transmitted to the Secretary General by the chief electoral officer in accordance with section 380 of the said Act.
1982, c. 62, s. 1; 1984, c. 51, s. 526; 1989, c. 1, s. 582.
2. The National Assembly and the Lieutenant-Governor form the Parliament of Québec. The Parliament of Québec assumes all the powers conferred on the Legislature of Québec.
No provision of this Act restricts the scope or exercise of those powers.
1982, c. 62, s. 2.
3. The Parliament has the exercise of the legislative power.
4. The Assembly has the power of supervision over all the acts of the Government and of its departments and agencies.
5. The Assembly is convoked, prorogued and dissolved by the Lieutenant-Governor.
6. A Legislature starts upon the receipt by the Secretary General, after a general election, of the list of the candidates declared elected transmitted by the Chief Electoral Officer pursuant to section 380 of the Election Act (chapter E-3.3).
A Legislature ends on 29 August of the fourth calendar year following the year that includes the most recent general election polling day.
However, if the publication provided for in the first paragraph of section 129.1 of the Election Act has been made, the Legislature ends instead on 27 February, or 28 February in the case of a leap year, of the fifth calendar year following the year that includes the most recent general election polling day.
Only the Lieutenant-Governor may dissolve the National Assembly before the expiry of a Legislature.
1982, c. 62, s. 6; 1984, c. 51, s. 527; 2013, c. 13, s. 9.
7. The Assembly sits in the territory of Ville de Québec; it may also sit at any other place in Québec.
1982, c. 62, s. 7; 1996, c. 2, s. 75.
8. The quorum of the Assembly or of a committee of the whole House is one-sixth of the Members, including the President.
However, when a committee of the Assembly is sitting, the quorum of the Assembly or of the committee of the whole House is one-tenth of the Members, including the President.
9. The rules of procedure of the Assembly are established by the Assembly, and it alone has authority to see that they are observed.
10. The National Assembly may appoint committees, composed of Members of the Assembly, to examine any matter within the jurisdiction assigned to them by the Assembly, and to carry out any mandate given to them by the Assembly.
11. The Assembly must appoint a committee on the Assembly to examine any matter referred to it by the Assembly.
The committee shall also carry out any other function vested in it by this Act.
12. A committee may form subcommittees composed of Members of the Assembly.
13. A committee or a subcommittee may sit even when the Assembly is not in session.
14. A committee or a subcommittee may sit anywhere in Québec, in accordance with the Standing Orders of the Assembly.
15. No Member may sit in the Assembly before making the oath provided in Schedule I.
1982, c. 62, s. 15; 1999, c. 40, s. 25.
16. A Member may resign his seat verbally at a sitting of the Assembly.
He may also resign in a writing countersigned by two other Members and sent to the President or the Secretary General of the Assembly.
If a Member resigns in writing, the President must so inform the Assembly at its next sitting.
17. The seat of a Member of the Assembly becomes vacant if he
(1) dies;
(2) resigns;
(3) becomes a candidate at a federal election or a provincial election in another province;
(4) is appointed to the Senate;
(5) is convicted of treason;
(6) is convicted of corrupt electoral or referendum practices;
(7) (subparagraph repealed);
(8) is sentenced to imprisonment for an indictable offence punishable by imprisonment for over two years;
(9) is in a situation that makes him disqualified within the meaning of the Election Act (chapter E-3.3), except the situation contemplated in subparagraph 4 of the second paragraph of section 235 of that Act.
A Member's seat also becomes vacant in the cases provided for in section 134 and in the Code of ethics and conduct of the Members of the National Assembly (chapter C-23.1).
1982, c. 62, s. 17; 1984, c. 51, s. 528; 1989, c. 1, s. 583; 1990, c. 4, s. 66; 1997, c. 8, s. 21; 2010, c. 30, s. 115.
18. If the seat of a Member of the Assembly becomes vacant at a time when the election held in the electoral division of the Member is contested or could still be contested within the legal limit of time, any subsequent election held in the electoral division during the same Legislature becomes null if, as a result of the contestation, the court declares elected a person other than the person proclaimed elected at the contested election or at a subsequent election.
THE PRESIDENT OF THE ASSEMBLY
19. At the beginning of its first sitting after a general election, the National Assembly shall elect a President and, subsequently, a first, a second and a third Vice-President from among its Members.
The first Vice-President and the second Vice-President shall be elected from among the Members forming the Government and the third Vice-President from among the Members forming the Official Opposition.
1982, c. 62, s. 19; 1999, c. 1, s. 1.
20. If the President is absent or unable to act, or at his request, one of the Vice-Presidents shall replace him and exercise his parliamentary duties.
21. If the President and the Vice-Presidents are absent or unable to act, the Secretary General shall notify the Assembly, and it shall designate a Member as interim President for his parliamentary duties.
22. If the office of President becomes vacant, the Secretary General shall inform the Assembly, and no business may be transacted until a new President is elected.
23. In addition to his duties under this Act, the President shall carry out the duties conferred on him by the Assembly.
24. On the dissolution of the Assembly, the President and the Vice-Presidents remain in office until they are replaced or reappointed by the new Assembly.
In such a case, they continue to receive the indemnity provided for in paragraphs 1 and 2 of section 7 of the Act respecting the conditions of employment and the pension plan of the Members of the National Assembly (chapter C-52.1).
1982, c. 62, s. 24; 2006, c. 10, s. 1.
DIVISION V
PARLIAMENTARY ASSISTANTS
25. The Government may appoint one or several Members as parliamentary assistants to assist a minister in the discharge of his duties. A parliamentary assistant may reply to questions addressed to the minister or take note of them on his behalf.
In no case, however, may the number of parliamentary assistants exceed twenty.
DIVISION VI
26. The National Assembly, on a motion of the Prime Minister, shall appoint a Secretary General and one or more associate secretaries general.
27. If the Secretary General is absent or unable to act, or in case of vacancy of the office of Secretary General, the President shall designate one of the associate secretaries general to replace the Secretary General during his absence or inability to act or during the vacancy of the office.
1982, c. 62, s. 27; 1984, c. 47, s. 8; 1999, c. 40, s. 25.
28. In addition to his duties under this Act, the Secretary General shall carry out the duties conferred on him by the Assembly.
LEGISLATIVE ACTS
29. The National Assembly passes the legislative Acts and the Lieutenant-Governor gives assent to them.
30. Any Member may present a bill.
However, only a minister may present a bill having as its object the commitment of public funds, the creation of a charge on the taxpayers, the remission of a debt owing to the Province or the alienation of property owned by the Province.
31. Every legislative bill must have the following introductory formula:
"The Parliament of Québec enacts as follows: ".
32. Upon the giving of assent to an Act, the Secretary General shall enter the date of assent therein. The entry forms part of the Act.
33. The Secretary General has custody of the originals of the Acts.
In case of the loss or destruction of an original, the Secretary General may substitute for it a certified true copy; thereafter, that copy serves as the original.
34. The Secretary General shall affix his seal to every copy of an Act that he certifies true.
35. After an Act has been assented to, the Secretary General shall send the Québec Official Publisher a copy certified true to the original enacted by Parliament.
36. Every year, the Québec Official Publisher shall publish a compilation of the statutes assented to during the preceding year.
37. The Office of the Assembly shall by by-law establish the conditions and modalities of printing, publication and distribution of the legislative Acts, the copies of the annual compilation of the statutes, the bills, and the other parliamentary documents.
The Secretary General shall provide printed copies of the Acts, free of charge, to the Lieutenant-Governor, government departments and public bodies, according to the rules established by the Office.
For the purposes of this section, a public body is a body to which the National Assembly, the Government or a minister appoints the majority of the members, to which, by law, the personnel is appointed in accordance with the Public Service Act (chapter F-3.1.1), or whose capital forms part of the domain of the State.
1982, c. 62, s. 37; 2010, c. 30, s. 116.
38. The Secretary General shall deliver a copy of the annual compilation of the statutes to the Lieutenant-Governor, and to the Registrar of Québec.
39. The Secretary General, or the person designated by him for that purpose, shall supply certified true copies of any Act to any person who applies therefor, upon payment of the cost fixed by the by-law of the Office of the Assembly. The designation has effect from its publication in the Gazette officielle du Québec.
Sums received under this section are paid into the Consolidated Revenue Fund.
40. A copy of an Act certified true by the Secretary General, or the person designated for that purpose, or the text of an Act published by the Québec Official Publisher is authentic and is proof of its existence and contents.
41. Every person who makes a request to the National Assembly for the passage of a private bill must pay to the Assembly the fees prescribed by regulation.
INDEPENDENCE OF THE NATIONAL ASSEMBLY
RIGHTS, PRIVILEGES AND IMMUNITIES
42. The Assembly has the power to protect its proceedings against all interference.
43. Every Member is vested with full independence for the carrying out of his duties.
44. No Member may be prosecuted, arrested or imprisoned by reason of anything said or done or any document tabled by him in the carrying out of his parliamentary duties in the Assembly or in any committee or subcommittee.
45. No Member may be bound to appear to answer a charge of contempt of court nor arrested or held for contempt of court while the Assembly or a committee or subcommittee in whose work he is taking part is sitting, or during the two preceding or two following days.
46. A Member is exempt from appearing as a witness in court or before any body or person empowered to summon witnesses while the Assembly or a committee or subcommittee in whose work he is taking part is sitting, and during the two preceding and two following days.
47. The President of the Assembly may exempt a member of the personnel of the Assembly from appearing as a witness in court, or before any body or person empowered to summon witnesses where he considers his presence required for the proper functioning of the Assembly and its services.
48. No person may be prosecuted for publishing or distributing an unedited report or official summary of the debates of the Assembly or of a committee or subcommittee, or for distributing, unedited, such debates or any document that has been submitted to them.
49. No person may be found guilty for publishing or distributing an abstract of the debates of the Assembly or of a committee or subcommittee, of a report or of an official summary of the debates, or of a document that has been submitted to them, or for giving an account of them, unless it is proved that he acted with malicious intent.
50. A copy of a written or audio-visual document contemplated in section 48 or 49 certified true by the Secretary General of the Assembly is admissible as evidence.
51. The Assembly or a committee may summon and compel the appearance before it of any person, either to answer questions put to him or to produce such papers and things as it may deem necessary for its acts, inquiries or proceedings.
52. The President or any Member of the Assembly or the chairman or any member of a committee or subcommittee may require a person appearing before it to make the oath provided in Schedule II.
53. In no case may a person's testimony before the Assembly or a committee or subcommittee be held against him in a court of law, unless he is being prosecuted for perjury.
54. No action may be instituted by reason of an official act performed in good faith by a person in the exercise of duties assigned to him under this Act or in carrying out an order of the Assembly or a committee or subcommittee.
55. No person may breach the privileges of the Assembly. The following acts, in particular, constitute breaches of the privileges of the Assembly:
(1) refusing to comply with an order of the Assembly, a committee or a subcommittee;
(2) giving false or incomplete testimony before the Assembly, a committee or a subcommittee;
(3) presenting a false document to the Assembly, a committee or a subcommittee with intent to deceive;
(4) forging, falsifying or altering, with intent to deceive, any document of the Assembly, a committee or a subcommittee or any document tabled or presented before it;
(5) creating a disturbance liable to disrupt the course of parliamentary proceedings;
(6) using or threatening to use force or using undue pressure to have a sitting cancelled or suspended;
(7) assaulting, interfering with, bullying or threatening Members of the Assembly in the carrying out of their parliamentary duties or members of the personnel of the Assembly in the carrying out of their parliamentary duties;
(8) defaming a Member of the Assembly or using abusive language about him;
(9) bribing or attempting to bribe a Member of the Assembly or a member of the personnel of the Assembly;
(10) attempting to influence the vote, opinion, judgment or action of a Member by means of deceit, threats or undue pressure;
(11) suborning or attempting to suborn or threatening a person in regard to any evidence to be given by him before the Assembly, a committee or a subcommittee;
(12) instituting an action with malicious intent against a Member;
(13) performing an act contrary to the parliamentary immunity conferred on a Member.
56. The person responsible for carrying out a warrant of the Assembly, a committee or a subcommittee may demand the assistance of a peace officer or any other person.
Refusal to give assistance when demanded constitutes a breach of the privileges of the Assembly.
Repealed, 2010, c. 30, s. 117.
57. (Repealed).
1982, c. 62, s. 57; 1988, c. 84, s. 544; 2010, c. 30, s. 117.
1982, c. 62, s. 59; 1999, c. 40, s. 25; 2010, c. 30, s. 117.
1982, c. 62, s. 66; 1983, c. 55, s. 161; 1999, c. 40, s. 25; 2000, c. 8, s. 242; 2010, c. 30, s. 117.
1982, c. 62, s. 68; 1986, c. 61, s. 66; 1988, c. 21, s. 66; 1997, c. 43, s. 38; 2010, c. 30, s. 117.
1982, c. 62, s. 71; 2002, c. 6, s. 84; 2010, c. 30, s. 117.
1982, c. 62, s. 73; 1986, c. 3, s. 1; 2010, c. 30, s. 117.
Heading repealed, 2010, c. 30, s. 117.
85. The bringing of a complaint before the Assembly by a Member against another Member without a serious reason constitutes a breach of the privileges of the Assembly.
DEFENCE COSTS, JUDICIAL COSTS, EXPENSES FOR COUNSEL AND INDEMNIFICATION
85.1. A Member or a former Member is entitled, subject to sections 85.2 to 85.4, to the payment of the defence costs and judicial costs arising out of proceedings brought against the Member or former Member by a third person for any act or omission in the performance of the Member's or former Member's duties of office.
The Member or former Member is also entitled to the payment of expenses incurred for counsel where the Member or former Member is summoned to appear at an inquiry, a preliminary inquiry or judicial or quasi-judicial proceedings in connection with the Member's or former Member's duties of office.
In each case submitted to it, the Office of the National Assembly may, after obtaining the advice of the jurisconsult appointed under the Code of ethics and conduct of the Members of the National Assembly (chapter C-23.1), fix the maximum amount to be paid under the first and second paragraphs.
1998, c. 11, s. 1; 2010, c. 30, s. 118.
85.2. In the case of criminal proceedings, the defence costs and judicial costs shall be paid only if the case was withdrawn or dismissed or if the Member or former Member was acquitted by a judgment that has become res judicata, or was discharged.
85.3. Where a Member or former Member is found guilty of a penal offence in a judgment that has become res judicata, no costs or expenses may be paid and the Assembly shall recover any costs or expenses paid except where the Office, after obtaining the advice of the jurisconsult, is of the opinion that the Member or former Member had reasonable grounds for believing that the conduct in question was in conformity with the law. In the latter case, the Assembly shall assume the payment of any pecuniary penalty.
85.4. Where, in judgment in a civil suit that has become res judicata, a Member or former Member is held liable for damage by reason of an act or omission in the performance of the Member's or former Member's duties of office, no costs or expenses may be paid and the Assembly shall recover any costs or expenses paid if the Office, after obtaining the advice of the jurisconsult, is of the opinion that the Member or former Member acted in bad faith.
The Assembly shall, however, assume the payment of any pecuniary penalty arising out of a judgment in a civil suit, except where the Office, after obtaining the advice of the jurisconsult, is of the opinion that a gross fault was committed by the Member or former Member or that the judgment should be appealed by the Member or former Member.
ADMINISTRATION OF THE ASSEMBLY
THE OFFICE OF THE NATIONAL ASSEMBLY
86. An Office of the National Assembly is hereby established.
87. The President of the Assembly is the chairman of the Office. The Office is also composed of nine other Members.
For the duration of the 42nd Legislature, the first paragraph is amended by replacing "nine" by "eleven".
1982, c. 62, s. 87; 1999, c. 3, s. 1; 2018, c. 282018, c. 28, s. 11.
88. The members of the Office other than the chairman are designated by the Members of each party in the following manner:
(1) five from the Government party;
(2) four from the party of the Official Opposition or, where there are several opposition parties, three from the Official Opposition party and one from the party among the remaining opposition parties having obtained the greatest number of seats or, in case of equality of seats, from that having obtained the greatest number of valid votes.
For the duration of the 42nd Legislature, the first paragraph is amended
(1) by replacing "five" in paragraph 1 by "six";
(2) by replacing paragraph 2 by the following paragraph:
"(2) three from the Official Opposition party and one from each of the other opposition parties represented in the Assembly following the 1 October 2018 general election.".
89. Each of the designating parties shall also designate the same number of Members as substitute members of the Office; each of them may act in the place of a member who is absent or unable to act.
90. Within fifteen days from the beginning of a session, each party shall communicate to the President of the Assembly the names of the members and substitute members it has designated.
91. The President shall submit the list of the designated Members to the Assembly. The Assembly shall adopt or reject the list as a whole.
92. If a party fails to designate its representatives or if the composition of the Assembly does not allow the application of sections 88 and 89, the chairman shall himself designate the Members to complete the composition of the Office.
93. When the Assembly is prorogued, the members of the Office remain in office until they are replaced or designated again.
94. On the dissolution of the Assembly, the President and the Vice-Presidents of the Assembly shall perform the duties of the Office.
95. The Vice-Presidents of the Assembly may take part, without the right to vote, in the proceedings of the Office.
96. If the President is absent, or at the President's request, the Vice-President designated by the President shall act as chairman. Only the first Vice-President or the second Vice-President may be designated for that purpose.
If the President is unable to act or the office of President is vacant, the first Vice-President shall act as chairman during such inability or vacancy.
However, if the first Vice-President also is unable to act or the office of first Vice-President also is or also becomes vacant, the second Vice-President shall replace the first Vice-President in the circumstances described in the second paragraph.
1982, c. 62, s. 96; 1998, c. 54, s. 1; 1999, c. 3, s. 3; 1999, c. 40, s. 25.
97. Five members, including the chairman, constitute a quorum of the Office. In the event of a tie-vote, the chairman has a casting vote.
For the duration of the 42nd Legislature, the first paragraph is amended by replacing "Five" by "Six".
98. The Secretary General of the Assembly is the secretary of the Office. If the Secretary General is absent or unable to act, the Office shall designate an associate secretary general to replace him.
99. The Office shall establish its rules of procedure.
100. The Office shall have managerial and regulatory functions, in accordance with this Act.
It shall have such other functions as the Assembly may assign to it.
101. The Office shall give its opinion on any matter referred to it by the President.
102. The Office shall establish, by regulation, the terms and conditions, scales and modalities of reimbursement to the members, except members of the Conseil exécutif, members of the staff of the National Assembly and the persons contemplated in the first paragraph of section 124.2, of expenses incurred in carrying out official assignments requested by the President of the Assembly.
The Office may, according to the modalities, on the terms and conditions and for the time it determines, delegate to the person it designates the power to determine the amount of the expenses that may be reimbursed according to the fixed scale.
1982, c. 62, s. 102; 1984, c. 27, s. 33.
103. The Office shall, by regulation, establish the conditions, scales and modalities of payment of an attendance allowance to its members and to the members of and participants in a committee or subcommittee of the Assembly.
104. The Office shall, by regulation, establish the conditions, scales and modalities of payment to Members of, in particular,
(1) transportation allowances and travel expenses;
(2) the cost of renting premises in the electoral division of each Member to receive his electors as well as any other cost provided for by the Office in the regulations to ensure the proper administration of the Member's offices;
(3) allowances for the remuneration of their personnel and for payment of professional services;
(4) expenses for lodgings, in the territory of Ville de Québec or in the immediate vicinity, of any Member domiciled outside the territory formed by that of Ville de Québec or an electoral division adjacent to the territory of that city, or of the Member domiciled within the electoral division of Charlevoix–Côte-de-Beaupré and more than 50 kilometres from the Parliament Building by the shortest road;
(5) the cost of purchase or lease of property or services for communications.
The Office may, by regulation, in the cases, on the conditions and to the extent it determines, allow a Member to whom the first paragraph of section 124.1 applies to make transfers from the sums granted by the Office under subparagraphs 2 and 3 of the first paragraph to the sums granted under the first paragraph of section 104.2.
The Office may, by regulation, in the cases, on the conditions and to the extent it determines, pay the allowances or repay the expenses and other costs provided for in this section for a period, fixed in the regulation, between the day on which the seat of a Member becomes vacant or the Assembly is dissolved and the thirtieth day, or the sixtieth day as regards persons referred to in the first paragraph of section 124.1, after the day on which a poll is held to fill the vacancy or a poll is held following the dissolution of the Assembly.
1982, c. 62, s. 104; 1984, c. 27, s. 35; 1985, c. 19, s. 1; 1986, c. 3, s. 2; 1989, c. 22, s. 2; 1996, c. 2, s. 76; 1997, c. 13, s. 1; 1999, c. 40, s. 25; 2004, c. 19, s. 1; 2006, c. 10, s. 2; 2016, c. 5, s. 1.
104.1. The Office may, by regulation, provide for one or several categories of Members and establish the conditions, scales and modalities of payment to such Members of additional allowances for the same purposes as those paid under section 104.
104.2. The Office shall, by regulation, establish the conditions, scales and modalities of payment of the expenses connected with the administration of the offices of the persons contemplated in the first paragraph of section 124.1.
The Office may, by regulation, in the cases, on the conditions and to the extent it determines, allow those persons to make transfers from the sums granted by the Office under the first paragraph to the sums granted under subparagraphs 2 and 3 of the first paragraph of section 104.
1989, c. 22, s. 3; 2004, c. 19, s. 2.
104.3. The Office shall fix, by regulation, the conditions, rates and terms governing the payment of any amount pursuant to sections 85.1 to 85.4.
105. The Office shall fix the intervals for the payment of indemnities and expense allowances provided in the Act respecting the conditions of employment and the pension plan of the Members of the National Assembly (chapter C-52.1).
106. The Minister of Finance shall pay, for each Member participating in the plan, such portion of the premium of a group life insurance and disability insurance plan or of any other insurance plan as the Office may determine.
107. The Office shall, by regulation, determine the rules according to which the personnel and the financial resources are to be allocated to any committee or subcommittee of the Assembly.
108. The Office shall, by regulation, determine the moneys that may be received from the Assembly, for research and support purposes, by the political parties represented in the Assembly following the last general election and by independent Members, and the terms and conditions of payment thereof.
The Member who is the leader of the Government party and the Member who is the leader of the Official Opposition party may transfer the moneys required for the remuneration of the regular personnel hired to assist the party for research and support purposes to the budget granted, under subparagraph 3 of the first paragraph of section 104, to the offices referred to in section 124.1. The personnel hired to assist the party for such purposes forms part of the personnel of the offices so designated in the same manner as the other members of the personnel of those offices.
In the case of another party to which the first paragraph applies, the Member who is the leader of that party or the authorized Member may transfer the moneys required for the remuneration of the regular personnel hired to assist the party for research and support purposes to the budget granted to the Member under subparagraph 3 of the first paragraph of section 104. The personnel hired to assist the party for such purposes forms part of the personnel of that Member in the same manner as the other members of his personnel.
Independent Members may transfer the moneys required for the remuneration of regular personnel hired to assist them for research and support purposes to the budget granted to them under subparagraph 3 of the first paragraph of section 104. The personnel hired to assist independent Members for such purposes forms part of their personnel in the same manner as the other members of their personnel.
1982, c. 62, s. 108; 1985, c. 19, s. 2; 1986, c. 3, s. 3; 1989, c. 22, s. 4; 1994, c. 39, s. 1; 1999, c. 3, s. 5; 2012, c. 24, s. 1.
108.1. The Office shall, by regulation, determine the moneys that may be received from the Assembly, for the purposes of research and support, by a Member sitting as an independent Member on 15 June 1993 who is not a member of a political party represented in the Assembly, and the terms and conditions of payment thereof.
This section shall cease to have effect on 24 July 1994.
1992, c. 7, s. 1; 1993, c. 20, s. 1.
109. The President shall table in the Assembly the rules and regulations adopted by the Office within fifteen days of their adoption if the Assembly is in session or, if it is not sitting, within fifteen days of the opening of the next session or resumption.
MANAGEMENT OF THE ASSEMBLY
110. Subject to this Act, the Assembly shall continue to be managed within the scope of the Acts, regulations and rules applicable.
The Office may, however, by regulation, derogate from the applicable Acts, regulations and rules by specifically indicating the provisions derogated from and the provisions that are to apply in their place and stead.
110.1. Subject to this Act and for the purposes of this division, the Office may make any regulation it deems necessary for the management of the Assembly.
110.2. Subject to the second paragraph of section 110, Chapter III, Chapter IV with the exception of section 44, the second and fourth paragraphs of section 45, sections 46 and 53 and the third paragraph of section 57 and section 73 of the Public Administration Act (chapter A-6.01) apply to the National Assembly.
2000, c. 8, s. 103; 2011, c. 19, s. 28.
111. The Office may, by regulation, prescribe the rules governing the expenditures of the Assembly.
112. (Repealed).
113. The Office shall fix the maximum number of staff needed by the Assembly to administer its services and shall determine the apportionment of the staff.
The Office shall adopt the administrative organization plan of the Assembly.
114. The organization and use of the premises as well as the use of the equipment of the Assembly and its services must be approved by the Office.
SERVICES OF THE ASSEMBLY
115. The President of the Assembly shall direct and administer the services of the Assembly.
116. The President is responsible for the security of the buildings or premises occupied by the Members and the members of the personnel of the Assembly; he shall also provide protection for persons and property in the premises.
For that purpose, the President may establish an advisory committee to assist him with the examination and implementation of security and protective measures; the members of the committee are entitled, where such is the case, to the fees and other allowances determined by the Office.
117. If the President is absent, or at the President's request, the Vice-President designated by the President shall act as President. Only the first Vice-President or the second Vice-President may be designated for that purpose.
If the President is unable to act or the office of President is vacant, the first Vice-President shall act as President during such inability or vacancy.
1982, c. 62, s. 117; 1998, c. 54, s. 2; 1999, c. 3, s. 6; 1999, c. 40, s. 25.
118. The President may delegate some of the President's administrative responsibilities to the first or the second Vice-President who shall, within the limits of the delegation, have the same powers and duties as the President.
1982, c. 62, s. 118; 1999, c. 3, s. 6.
119. Under the responsibility of the President, the Secretary General of the Assembly has the supervision of the members of the personnel of the Assembly; he shall administer its day-to-day business and exercise the other functions assigned to him by the Office.
The orders of the Secretary General must be carried out in the same manner as those of the President.
120. Every member of the personnel of the Assembly, except a casual employee, is a member of the personnel of the civil service, whether appointed under the Public Service Act (chapter F-3.1.1) or by derogation by virtue of the second paragraph of section 110, unless, in the latter case, the Office excludes him therefrom.
The Secretary General has, in respect of the personnel of the Assembly, the powers vested in a deputy minister by the Public Service Act.
1982, c. 62, s. 120; 1983, c. 55, s. 161.
121. The Assembly may vest in the associate secretaries general, in their deeds of appointment, the rank and privileges of an assistant deputy minister.
The associate secretaries general are members of the personnel of the civil service.
122. The respective duties of the members of the personnel of the Assembly not expressly defined by law or by the Office are determined by the President.
123. No deed, document or writing binds the Assembly or may be attributed to the President unless it is signed by him, by the Secretary General or by another officer, and only, in this last case, to the extent determined by regulation of the Office.
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Q: Copy remote database views to local database as tables I currently have access to only the views on a remote database, and I'm trying to copy the data of those views locally so that I can work faster. I've tried to use mysqldump to create a .sql file and load that, but it doesn't seem to work.
To get the remote db:
$ mysqldump --single-transaction -u username -p -h somesite.com -P 32000 db_name > mysql.sql
To copy to local:
$ mysql -u root -p mydatabase < mysql.sql
But I get an error
ERROR 1146 (42S02) at line 668: Table 'db_name.some_view' doesn't exist

My local database is completely empty except me creating an empty one using the command create database mydatabase; in the mysql shell.
Am I getting this problem because I can't use this method in the first place since I only have access to the views in the remote database?

A: Did you taken the data from the remote system correctly.If not try this command once you connect through remotely to database server,the command is

*

*mysqldump -u username -h 192.168.X.X -P password --routines>xxxx.sql

It will ask the password of the database.Till now data is only in database server.From your local system use this command to take the data from database server,the command is

*

*rsync -vaH username@192.168.X.X:/datalocation /yourlocation

Now the data is on your local system.Then after create one database in your system.

*

*mysql

*create database;

*grant all privileges on databasename .*to databaseschema@'%' identified by 'password';

*flush privileges;

I think this may help you,if not sorry bro.

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}
G. Kaniadakis \cite{kaniadakis:2001PhA,kaniadakis:2001PhLA,kaniadakis:2002PhRE,kaniadakis:2005PhRE}, based on arguments from Statistical Physics and Special Relativity, has defined the \emph{$\kappa$-deformed exponential} for each $x \in \mathbb{R}$ and $-1 < \kappa < 1$ to be
\begin{equation}
  \kexpof x = \expof{\int_0^x \frac{dt}{\sqrt{1+\kappa^2 t^2}}},
\end{equation}
with special cases
\begin{align}
\kexpof x &= \begin{cases}
\left(\kappa x + \sqrt{1 + \kappa^2 x^2}\right)^{\frac1\kappa}, &\text{if $\kappa \ne 0$,}\\
\exp x, &\text{if $\kappa = 0$,}
\end{cases}
\end{align}
and derivation formul\ae
\begin{align}
  \Kexp'(x) &= (1+\kappa^2 x^2)^{-1/2} \kexpof x > 0 \\
  \Kexp''(x) &= \frac{\sqrt{1+\kappa^2x^2} - \kappa^2 x}{1+\kappa^2x^2} \Kexp'(x)>0
\end{align}
For each $\kappa\ne0$, $y = (\kexpof x)^\kappa$ and $x$ are related by the polynomial equation
\begin{equation} \label{eq:master}
  y^2 - 2\kappa xy - 1 = 0
\end{equation}
Therefore, the graph of $(\Kexp)^\kappa$ is the upper branch of a hyperbola:
\begin{equation}
  \label{eq:hyperbola}
  x = \frac1{2\kappa}\left(y - \frac1y\right), \quad y>0.
\end{equation}

For each given $\kappa$, the function $\Kexp$ maps $\mathbb{R}$ onto $\mathbb{R}_>$, it is strictly increasing and it is strictly convex. If $\kappa \ne 0$, its inverse function is
\begin{equation}\label{eq:klog}
  \klnof y = \frac{y^\kappa - y^{-\kappa}}{2\kappa}, \quad y > 0,
\end{equation}
with derivative
\begin{equation}
  \label{eq:Klnderiv}
  \Kln'(y) = \frac{y^\kappa+y^{-\kappa}}{2}\frac1y,
\end{equation}
and it is called \emph{$\kappa$-deformed logarithm}. The function $\Kln$ maps $\mathbb{R}_>$ unto $\mathbb{R}$, is strictly increasing and is strictly concave. Both deformed exponential and logarithm functions $\Kexp$ and $\Kln$ reduce to the ordinary $\exp$ and $\ln$ functions when $\kappa \to 0$. Moreover, the algebraic properties of the exponential and logarithmic function are partially preserved, because
\begin{equation} \label{eq:kexpofminus}
  \kexpof x \kexpof{-x} = 1, \quad \klnof y + \klnof {y^{-1}} = 0.
\end{equation}

Kaniadakis \cite{kaniadakis:2002PhRE} defines, among others, two commutative group operations $(\mathbb{R}, \stackbin[]{\kappa}{\oplus})$ and $(\mathbb{R}_>,\stackbin[\kappa]{}{\otimes})$ in such a way that $\Kexp$ is a group isomorphism from $(\mathbb{R},+)$ onto $(\mathbb{R}_>,\stackbin[\kappa]{}{\otimes})$ and also from $(\mathbb{R},\stackbin[]{\kappa}{\oplus})$ onto $(\mathbb{R}_>,\times)$:
\begin{align}
  \kexpof{x_1 \stackbin[]{\kappa}{\oplus} x_2} &= \kexpof{x_1} \kexpof{x_2}, \\
  \kexpof{x_1 + x_2} &= \kexpof{x_1} \stackbin[\kappa]{}{\otimes} \kexpof{x_2},
\end{align}
and, equivalently,
\begin{align}
  \klnof{y_1 \stackbin[\kappa]{}{\otimes} y_2} &= \klnof{y_1} + \klnof{y_2}, \\
  \klnof{y_1 y_2} &= \klnof{y_1} \stackbin[]{\kappa}{\oplus} \klnof{y_2}.
\end{align}
Therefore, the binary operations $\stackbin[]{\kappa}{\oplus}$ and $\stackbin[\kappa]{}{\otimes}$ are defined by
\begin{align}
  x_1 \stackbin[]{\kappa}{\oplus} x_2 &= \klnof{\kexpof{x_1}\kexpof{x_2}} \label{eq:kplus-def}\\
  y_1 \stackbin[\kappa]{}{\otimes} y_2 &=\kexpof{\klnof{y_1} + \klnof{y_2}} \label{eq:kprod-def}
\end{align}
Because of \eqref{eq:kexpofminus}, the deformed operation have the same inverse that the usual operations:
\begin{equation}
  \label{eq:kinverse}
  x_1 \stackbin[]{\kappa}{\oplus} (-x_2) = 0, \quad y_1 \stackbin[\kappa]{}{\otimes} y_2^{-1} = 1.
\end{equation}
The operation $\stackbin[\kappa]{}{\otimes}$ is defined on positive real numbers. However, \eqref{eq:kprod-def} can be extended by continuity to non-negative real numbers:
\begin{equation}
  \label{eq:kprod-by-zero}
  0 \stackbin[\kappa]{}{\otimes} y_2= y_1 \stackbin[\kappa]{}{\otimes} 0 = 0 \stackbin[\kappa]{}{\otimes} 0 = 0
\end{equation}

We want to derive defining relations for the $\kappa$-deformed operations in polynomial form. This is obtained by repeated use of \eqref{eq:master} followed by algebraic elimination of the unwanted indeterminate. Symbolic computations have been done with \cite{CocoaSystem}. First, we want to find $x=x_1\stackbin[]{\kappa}{\oplus} x_2$, i.e. such that $\kexpof x = \kexpof{x_1}\kexpof{x_2}$.
Let $y_1 = (\kexpof{x_1})^\kappa$ and $y_2 = (\kexpof{x_2})^\kappa$. As
\begin{equation}
  (\kexpof x)^\kappa = (\kexpof{x_1}\kexpof{x_2})^\kappa = y_1y_2,
\end{equation}
Equation \eqref{eq:master} gives a system of three quadratic equations in the indeterminates $x_1,x_2,y_1,y_2,x,\kappa$. Algebraic elimination of $y_1,y_2$ gives the polynomial equation
\begin{equation}
x^{4} - 2\left(2 \kappa^{2}x_{1}^{2}x_{2}^{2} + x_{1}^{2} + x_{2}^{2}\right)x^{2} + \left(x_{1}^{2} - x_{2}^{2}\right)^2 = 0,
\end{equation}
whose solution gives
\begin{equation}
   x_1 \stackbin[]{\kappa}{\oplus} x_2 = x_1 \sqrt{1+\kappa^2 x_2^2} + x_2 \sqrt{1+\kappa^2 x_1^2}.
\end{equation}
We will use later on the derivation formula
\begin{equation}\label{eq:11}
  \frac{\partial (x_1 \stackbin[]{\kappa}{\oplus} x_2)}{\partial x_1} = \sqrt{1+\kappa^2 x_2^2} - \frac{\kappa^2 x_1x_2}{\sqrt{1+\kappa^2 x_1^2}}.
\end{equation}

Second, we want to find $z = \left(y_1\stackbin[\kappa]{}{\otimes} y_2\right)^\kappa$. As before, $y_1 = (\kexpof{x_1})^\kappa$, $y_2 = (\kexpof{x_2})^\kappa$, while
\begin{equation}
  z = (\kexpof{x_1 + x_2})^\kappa.
\end{equation}
Equation \eqref{eq:master} gives three quadratic equations in the indeterminates $x_1,x_2,y_1,y_2,z,\kappa$. Elimination of $x_1,x_2$ gives the polynomial equation
\begin{equation}
  y_{1}y_{2}z^{2} +(1 - y_{1}y_{2})(y_{1} + y_{2}) z - y_{1}y_{2} = 0
\end{equation}
It is remarkable that this equation does not depend on $\kappa$. An explicit solution is obtained by solving the quadratic equation. A possibly more suggestive solution is obtained as follows. First, we reduce to the monic equation
\begin{equation}
  \label{eq:kprod-2}
  z^{2} + \left(1 - \frac1{y_{1}y_{2}}\right)(y_{1} + y_{2}) z - 1 = 0
\end{equation}
and denote the two solutions as $z >0$ and $-1/z$. Therefore,
\begin{equation} \label{eq:15}
z - \frac1z = \left(y_1 - \frac1{y_1}\right) + \left(y_2 - \frac1{y_2}\right).
\end{equation}

The $\kappa$-logarithm defined in \eqref{eq:klog} is reminiscent of a family of transformation well known in Applied Statistics under the name of Box-Cox transformation \cite{box|cox:64} or power transform. For data vector $y_1,\dots, y_n$, with $y_i > 0$ for all $i$, the power transform is:
\begin{equation}
  y_i^{(\lambda)} \propto \frac{y_i^\lambda -1}{\lambda}
\end{equation}
The parameter $\lambda$ is to be estimated in order to get the best fit to the Normal distribution of the transformed data vector. When compared with the power transform, the $\kappa$-deformed logarithm $x = \klnof y$ has the extra feature of the symmetry induced by the term $-y^{-\kappa}$ and it would be interesting to study it from the point of view of transformations to normality. We are not further discussing this issue here.

J. Naudts \cite{naudts:2002PhysA,naudts:2004-332,naudts:2004JIPAM,naudts:2008} has presented a general discussion of a class of deformed exponentials that contains $\Kln$ and $\Kexp$, with applications in Information Theory and Statistical Physics, based on a notion of generalised entropy. We are not discussing generalised entropies in this paper. The purpose of the present paper is to extend to $\kappa$-deformations the non-parametric and geometric approach to statistical manifolds as it was developed by the Author and co-workers in \cite{pistone|sempi:95,gibilisco|pistone:98,pistone|rogantin:99,cena|pistone:2007,pistone:2008}. Such an approach is designed to present in a non-parametric and fully geometric way work by S-i. Amari on Information Geometry, see \cite{amari:82} and the joint monograph with H. Nagaoka \cite{amari|nagaoka:2000}. According to those references, the the power transform is applied to probability densities and re-named $\alpha$-embedding, $\alpha=1-2/\lambda$.

To explain the idea of $\alpha$-embedding, we consider a general setting. Let $\Omega$ be any set, $\mathcal F$ a $\sigma$-algebra of subsets, $\mu$ a reference probability measure, e.g. the uniform distribution. A density $p$ of the measure space $(\Omega,\mathcal F, \mu)$ is a non-negative random variable such that $\int p \,d\mu = 1$. We write the expected value of a random variable $U$ with respect to the probability measure $p \cdot \mu$ as $\expectat p U = \int Up\,d\mu$. For $a > 1$, the mapping $p \mapsto p^{1/a}$ maps $p$ into the unit ball of the Lebesgue space $L^a(\Omega,\mathcal F, \mu)$. Other parametrization of the same setting are: $\lambda = 1/a$ (Box-Cox), $\kappa = 1/a$ (Kaniadakis), $\alpha=(a-2)/a$ (Amari). The basic idea is to pull-back the structure of the unit ball of $L^a$ to construct a Banach manifold on the set of densities $\mathcal M_{\ge}$. The actual construction is not straightforward in the infinite dimensional case, because the image set of the $\alpha$-embedding has empty interior in the Lebesgue space because of the non-negativity constraint $p^{1/a} \ge 0$. However, the approach works perfectly well in the cases of either a parametric model or a general model over a finite state space.

Variants of the $\alpha$-embedding has been studied by \cite{gibilisco|pistone:98}, \cite{burdet|combe|nencka:2001}, \cite{copas|eguchi:2005}. There is also literature covering the non-commutative (quantum) case, which is not treated here, e.g. \cite{gibilisco|isola:1999}, \cite{streater:2004orlicz}, \cite{jencova:2006}.

The present paper is aimed to discuss what happens if we use $\Kln$ instead of the $\alpha$-embedding in the construction of the statistical manifold. The related issue of the algebro-statistical aspects of finite state space models is also discussed. See the books \cite{pistone|riccomagno|wynn:2001,pachter|sturmfels:2005,drton|sturmfels|sullivan:2009} on Algebraic Statistics and \cite{AGMS} on the relations between Algebraic Statistics and Information Geometry.

Section 2 is an introduction to both the algebraic and the geometric features of the $\kappa$-deformed Gibbs model. Section 3 introduces a system of charts based on the $\kappa$-exponential; the resulting differentiable manifold is modeled on the space of centered and $1/\kappa$-integrable random variables. Section 4 discusses some properties of the tangent bundle; only a few basic results are presented. Section 5 is a short discussion on how the present construction relates to our formalism of exponential statistical manifolds and to other Authors' work. A full account the totality of the existing literature on Information Geometry is outside the scope of the present paper and will be published elsewhere.
\section{$\kappa$-Deformed Gibbs model}
On a finite state space $\Omega$, equipped with the energy function $U \colon \Omega \rightarrow \mathbb{R}_\ge$, we want to discuss the $\kappa$-deformation of the standard Gibbs model. There are two options, as there are two different presentation of the normalizing constant.

The first option would be to consider the statistical model
\begin{align}
  p(x;\theta) &= \frac{\kexpof{\theta U(x)}}{Z(\theta)} \label{eq:var-kgibbs}
\\ &= \kexpof{\theta U(x) \stackbin[]{\kappa}{\oplus} \klnof{\frac1{Z(\theta)}}} \notag
\end{align}
or, with $\widetilde\psi_\kappa(\theta) = \Kln Z(\theta)$,
\begin{multline}
  \Kln p(x;\theta) = \\ \theta U(x) \sqrt{1+\kappa^2 (\widetilde\psi_\kappa(\theta))^2} - \widetilde\psi_\kappa(\theta) \sqrt{1 + \kappa^2 \theta^2 U(x)^2}
\end{multline}
To the best of our knowledge, this model is not considered in the literature, therefore we are not discussing it here.

The second option is to define the generalised model as
\begin{align}
  p(x;\theta) &= \kexpof{\theta U(x) - \psi_\kappa(\theta)} \label{eq:kgibbs}\\
              &= \kexpof{\theta U(x)} \stackbin[\kappa]{}{\otimes} \kexpof{- \psi_\kappa(\theta)} \notag,
  \end{align}
where $\psi_\kappa(\theta)$ is the unique solution of the equation
\begin{equation}\label{eq:gibbs-psi}
  \sum_{x\in \Omega} \kexpof{\theta U(x) - \psi_\kappa(\theta)} = 1.
\end{equation}
This is the model that has been considered in \cite{kaniadakis:2002PhRE}, on the basis of the classical maximum entropy argument. It is a particular case of more general models as discussed by \cite{grunvald|dawid:2004} and by \cite{naudts:2004JIPAM,naudts:2008} and . We focus here on the algebraic and geometrical features of the model, not on its derivation from general principles.

The one-parameter statistical models \eqref{eq:var-kgibbs} and \eqref{eq:kgibbs} are different unless $\kappa=0$. This fact marks an important difference between the theory of ordinary exponential models and $\kappa$-deformed exponential models. For example, the derivative with respect to $\theta$ of the left hand side of \eqref{eq:gibbs-psi} is
\begin{equation}
  \sum_{x\in \Omega} \frac{U(x) - \psi_\kappa'(\theta)}{\sqrt{1+\kappa^2 \left(\theta U(x) - \psi_\kappa(\theta)\right)^2}} \kexpof{\theta U(x) - \psi_\kappa(\theta)},
\end{equation}
therefore
\begin{equation}
  \expectat \theta {\frac{U - \psi'(\theta)}{\sqrt{1+\kappa^2 \left(\theta U - \psi_\kappa(\theta)\right)^2}}} = 0,
\end{equation}
where $\expectat \theta V = \sum_x V(x) p(x;\theta)$. If $\kappa=0$, we have the usual formula $\psi'(\theta) = \expectat \theta U$.

From the geometrical point of view, the second approach has the advantage that the model for the $\Kln$-probabilities is linear. Here, by geometry we mean differential geometry of statistical models, i.e. the construction of an atlas of charts representing the subset of probability densities unto open sets of a Banach space, see \cite{lang:1995}. Before moving into this approach, we discuss the duality of the Gibbs model and its algebraic features.

Let $V = \spanof{1,U}$ and $V^\perp$ the orthogonal space, i.e. $v \in V^\perp$ if, and only if,
$\sum_{x} v(x) = 0$ and $\sum_{x} v(x) U(x) = 0$. It follows from Equation \eqref{eq:kgibbs} that
\begin{equation}\label{eq:kgibbsorth}
    \sum_{x \in {\Omega}} v(x) \klnof{p(x;\theta)} = 0, \qquad v \in V^\perp
\end{equation}
Vice versa, if a strictly positive probability density function $p$ is such that $\Kln p$ is orthogonal to $V^\perp$, then $p$ belongs to the $\kappa$-Gibbs model for some $\theta$.

For each $v \in V^\perp$, we can take its positive part $v^+$ and its negative part $v^-$, so that $v = v^+ - v^-$ and $v^+ v^- = 0$. Equation \eqref{eq:kgibbsorth} can be rewritten as
\begin{equation}
\label{eq:kgibbsbin}
 \sum_{x\colon v(x) > 0} v^+(x) \klnof{p(x)} = \sum_{x\colon v(x) < 0} v^-(x) \klnof{p(x)}
\end{equation}
The interpretation of \eqref{eq:kgibbsbin} is the following. As $\sum_x v(x)=0$, we have
\begin{equation}
  \sum_{x \in {\Omega}} v^+(x) =   \sum_{x \in {\Omega}} v^-(x) = \lambda.
\end{equation}
It follows that $r_1 = v^+/\lambda$, and $r_2 = v^-/\lambda$ are probability densities (states) with disjoint support, so that \eqref{eq:kgibbsbin} can be restated by saying that a positive density $p$ belongs to the $\kappa$-Gibbs model if, and only if,
\begin{equation}
  \expectat {r_1}{\klnof{p}} = \expectat {r_2}{\klnof{p}}
\end{equation}
for each couple of densities $r_1$, $r_2$ such that $r_1r_2=0$ and $\expectat {r_1} U = \expectat {r_2} U$, where $\expectat r V$ denotes the mean value of $V$ with respect to $r$, i.e. $\sum_x V(x) r(x)$.

If $v \in V^\perp$ happens to be integer valued, using the $\kappa$-algebra and the notation
\begin{equation}
\stackrel{\text{$n$ times}}{\overbrace{x \stackbin[\kappa]{}{\otimes} \cdots \stackbin[\kappa]{}{\otimes} x}} = x^{\stackbin[\kappa]{}{\otimes} n},
\end{equation}
we can write \eqref{eq:kgibbsbin} as
\begin{equation} \label{eq:kgibbs-zero}
 \stackrel[\kappa]{}{\bigotimes}_{x\colon v(x)>0} p(x)^{\stackbin[\kappa]{}{\otimes} v^+(x)} = {\stackrel[\kappa]{}{\bigotimes}_{x\colon v(x)<0} p(x)^{\stackbin[\kappa]{}{\otimes} v^-(x)}},
\end{equation}

It should be noted that \eqref{eq:kgibbs-zero} is continuous function of $p(x)$, $x \in \Omega$ and it does not require the strict positivity of each $p(x)$, $x \in \Omega$. Therefore, the same equation is satisfied by all limits (if any) of the model \eqref{eq:kgibbs}. Moreover, \eqref{eq:kgibbs-zero} is a \emph{$\kappa$-polynomial invariant} for the $\kappa$-Gibbs model, cf. in \cite{pistone:2008} the discussion of the case $\kappa=0$. Vice versa, each positive probability satisfying \eqref{eq:kgibbs-zero} belongs to the $\kappa$-Gibbs model.

The set of all $\kappa$-polynomial equations \eqref{eq:kgibbs-zero} is not a finite set, because each equation depends on the choice of an integer valued vector $v$ in the orthogonal space $V^\perp$. Accurate discussion of this issue requires tools from Commutative Algebra. If the energy function $U$ takes its values on a lattice $0, \Delta, 2\Delta, \dots$, $\Delta > 0$, we can choose integer valued random variables $v_1,\dots,v_{N-2}$ to be a linear basis of the orthogonal space $V^\perp$. In such a case, we have a finite system of binomial equations
\begin{equation}\label{eq:kgibbs-basis}
   \stackrel{\kappa}{\bigotimes}_{x\colon v_j(x)>0} p(x)^{\stackbin[\kappa]{}{\otimes} v^+_j(x)} =   \stackrel{\kappa}{\bigotimes}_{x\colon v(x)<0} p(x)^{\stackbin[\kappa]{}{\otimes} v^-_j(x)},
\end{equation}
$j = 1,\dots,N-2$, which is equivalent to the original model \eqref{eq:kgibbs}. However, it is not generally true that the finite system \eqref{eq:kgibbs-basis} is equivalent to the infinite system \eqref{eq:kgibbs-zero}.

In the classical case $\kappa=0$, the polynomial invariants of the Gibbs model form a polynomial ideal $I$ in the polynomial ring $\Q[p(x)\colon x \in \Omega]$. The ideal I admits, because of the Hilbert Theorem, a finite generating set. The discussion of various canonical forms of such a generating set is one of the issues of Algebraic Statistics.

 \paragraph{Example} We specialize our discussion to the toy example which is discussed in \cite{pistone:2008} for $\kappa=0$. Let ${\Omega} = \set{1,2,3,4,5}$, $U(1) = U(2) = 0$, $U(3) = 1$, $U(4)= U(5) = 2$. The following display shows a set of integer valued $v_j$, $j=1,2,3$ of the orthogonal space $V^\perp$.
\begin{equation}
\bordermatrix[{[]}]{%
& 1 & U  & v_1 & v_2 & v_3 \cr
1& 1 & 0 & 1 & 0 & 1 & \cr
2& 1 & 0 & -1 & 0 & 1 \cr
3& 1 & 1 & 0 & 0 &  -4\cr
4& 1 & 2 & 0 & 1 & 1 \cr
5& 1 & 2 & 0 & -1 & 1 \cr
}%
\end{equation}
Equation \eqref{eq:kgibbs-basis} becomes:
\begin{equation} \label{eq:1}
  \left\{
    \begin{aligned}
      &p(1) = p(2) \\
      &p(4) = p(5) \\
      &p(1)\stackbin[\kappa]{}{\otimes} p(2)\stackbin[\kappa]{}{\otimes} p(4)\stackbin[\kappa]{}{\otimes} p(5) = p(3)^{\stackbin[\kappa]{}{\otimes} 4}
    \end{aligned}
\right.
\end{equation}
A strictly positive probability density belongs to the $\kappa$-Gibbs model \eqref{eq:kgibbs} if and only if it satisfies \eqref{eq:1}. The set of all polynomial invariants of the Gibbs model is a polynomial ideal and the question of finding a set of generators is intricate. A non strictly positive density that is a solution of \eqref{eq:1} is either $p(1)=p(2)=p(3)=0$, $p(4)=p(5)=1/2$, or $p(1)=p(2)=1/2$, $p(3)=p(4)=p(5)=0$. These two solutions are the uniform distributions on the sets of values that respectively maximize or minimize the energy function.

From \eqref{eq:15} we can derive another form of the last equation in the system \eqref{eq:1}:
\begin{multline}
 \left(p^\kappa(1) - \frac1{p^\kappa(1)}\right)+\left(p^\kappa(2) - \frac1{p^\kappa(2)}\right)+ \\ \left(p^\kappa(4) - \frac1{p^\kappa(4)}\right)+\left(p^\kappa(5) - \frac1{p^\kappa(5)}\right) = \notag \\ 4\left(p^\kappa(3) - \frac1{p^\kappa(3)}\right), \notag
\end{multline}
which is another algebraic form of the $\kappa$-Gibbs model.

Jet another algebraic presentation is available. In the model \eqref{eq:kgibbs}, we introduce the new parameters
\begin{align}
  \zeta_0 &= \kexpof{- \psi_\kappa(\theta)}, \\
  \zeta_1 &= \kexpof{\theta},
\end{align}
so that
\begin{align}
  p(x;\theta) &= \kexpof{\theta U(x)} \stackbin[\kappa]{}{\otimes} \kexpof{- \psi_\kappa(\theta)}, \\
              &= \zeta_0 \stackbin[\kappa]{}{\otimes} \zeta_1^{\stackbin[\kappa]{}{\otimes} U(x)}.
\end{align}
The probabilities are $\kappa$-monomials in the parameters $\zeta_0, \zeta_1$, e.g.:
\begin{equation}\label{eq:toric-gibbs}
  \left\{
      \begin{aligned}
        p(1) &= p(2) = \zeta_0 \\ p(3) &= \zeta_0 \stackbin[\kappa]{}{\otimes} \zeta_1 \\ p(4) &= p(5) = \zeta_0 \stackbin[\kappa]{}{\otimes} \zeta_1^{\stackbin[\kappa]{}{\otimes} 2}
      \end{aligned}
\right.
\end{equation}

In algebraic terms, such a model is called a \index{toric model}\emph{toric model}. It is interesting to note that in \eqref{eq:toric-gibbs} the parameter $\zeta_0$ is required to be strictly positive, while the parameter $\zeta_1$ could be zero, giving rise the uniform distribution on $\set{1,2} = \set{x \colon U(x)=0}$. The other limit solution is not obtained by \eqref{eq:toric-gibbs}. The algebraic elimination of the indeterminates $\zeta_0, \zeta_1$ in \eqref{eq:toric-gibbs} will produce back polynomial invariants. For example, from $(\zeta_o\stackbin[\kappa]{}{\otimes}\zeta_1)^{\stackbin[\kappa]{}{\otimes} 2} = (\zeta_0)\stackbin[\kappa]{}{\otimes}(\zeta_0\stackbin[\kappa]{}{\otimes}\zeta_1^{\stackbin[\kappa]{}{\otimes} 2})$, we get $p(3)^{\stackbin[\kappa]{}{\otimes} 2} = p(2)\stackbin[\kappa]{}{\otimes} p(5)$.

The monomial parametric representation of the type \eqref{eq:toric-gibbs}, together with their ability to parametrize the limit cases is discussed in \cite{pistone:2008}. The notion of toric model and the related discussion could be applied to the model \eqref{eq:var-kgibbs} which we are not discussing here.
\section{Charts}
\label{sec:chartintro}
We use the coordinate-free formalism of differential geometry of \cite{lang:1995}. Let us fix a $\kappa \in ]0,1[$. In order to construct an atlas of charts, i.e. mappings from the set of strictly positive probability densities $\mathcal M_{>}$ to some model vector space, we consider each chart as associated to a density $p \in \mathcal M_{>}$. Such a $p$ is used as a reference for every other density $q$ of a suitable subset of $\mathcal M_{>}$, via the statistical notion of likelihood $q/p$. The theory is not restricted to a finite sample space. However, we do not discuss here the technicalities involved in the non finite case, but we use a general notation.

We first define a $\kappa$-divergence. If $q, p \in \mathcal M_{>}$ satisfy the condition
\begin{equation} \label{eq:100}
  \left(\frac qp\right)^\kappa, \left(\frac pq\right)^\kappa \in L^1(p),
\end{equation}
then the $\kappa$-divergence is defined to be
\begin{multline} \label{eq:divergence}
  D_\kappa(p \Vert q) = \expectat p {\klnof{\frac pq}} \\ = \frac1{2\kappa} \expectat p {\left(\frac pq\right)^\kappa - \left( \frac qp \right)^\kappa}.
\end{multline}
The first of the two conditions in \eqref{eq:100} is always satisfied because
\begin{equation}
  \label{eq:2}
  \expectat p {\left(\frac qp\right)^\kappa} \leq \left(\expectat p {\frac qp}\right)^\kappa = 1.
\end{equation}
The second condition is non-trivial unless the state space is finite, because
we are assuming
\begin{equation}
  \label{eq:3}
\expectat p {\left(\frac pq\right)^\kappa} = \int \frac {p^{\kappa+1}}{q^\kappa} \,d\mu < +\infty.
\end{equation}
When such a condition is not satisfied the value of the expectation in \eqref{eq:divergence} is $+\infty$. We are not interested in this case here.

 The strict convexity of $-\Kln$ implies
\begin{multline}
  D_\kappa(p \Vert q) = \expectat p {-\klnof{\frac qp}} \geq \\ -\klnof{\expectat p {\frac qp}} = \klnof 1 = 0.
\end{multline}
with equality if, and only if $q=p$.

The manifold we want to define has to be modelled on the Lebesgue space of centered $(1/\kappa)$-$p$-integrable random variables $L^{1/\kappa}_0(p)$, i.e. $v \in L^{1/\kappa}_0(p)$ if, and only if, $\expectat p {|v|^{1/\kappa}} = \int |v|^{1/\kappa}p\,d\mu < +\infty$ and $\expectat p v = \int vp\,d\mu=0$. At each $p$ there is a different model space, so that an isomorphism between them has to be provided. The simplest isometric identification between $L^{1/\kappa}(p_1)$ and $L^{1/\kappa}(p_2)$ is
\begin{equation} \label{eq:isometry}
  L^{1/\kappa}(p_1) \ni u \mapsto \left(\frac{p_1}{p_2}\right)^\kappa u \in L^{1/\kappa}(p_2)
\end{equation}
In fact,
\begin{equation}
  \label{eq:21}
  \int \left|\left(\frac{p_1}{p_2}\right)^\kappa u\right|^{1/\kappa} p_2\,d\mu = \int |u|^{1/\kappa} p_1\,d\mu.
\end{equation}

We use a variation of the formalism used in the exponential case \cite{pistone|sempi:95}. The new manifold will be called $\kappa$-\emph{statistical manifold}. We define the subset $\mathcal E_p$ of $\mathcal M_{>}$ by
\begin{align}
\mathcal E_{p} &= \setof{q \in \mathcal M_{>}}{\left(\frac qp\right)^\kappa, \left( \frac pq\right)^\kappa \in L^{1/\kappa}(p)} \notag \\
              &= \setof{q \in \mathcal M_{>}}{\frac qp, \frac pq \in L^1(p)} \notag \\ &= \setof{q \in \mathcal M_{>}}{\frac pq \in L^1(p)} \label{eq:4}
\end{align}
The assumption in \eqref{eq:4} is stronger than that in \eqref{eq:3}, because here we want $\expectat p {p/q} = \int p^2/q \, d\mu < +\infty$, and $\expectat p {p^\kappa/q^\kappa} \le (\expectat p {p/q})^\kappa$. Indeed, we are assuming more than the mere existence of $D_\kappa(p \Vert q)$, unless the state space is finite. It should be noticed that the set $\mathcal E_p$ does not depend on $\kappa \in ]0,1[$. The condition $\int p^2/q \, d\mu < +\infty$ can be interpreted with the aid of the isometry \eqref{eq:isometry}. As
\begin{equation}
  \left(\frac pq\right)^\kappa \in L^{\frac1\kappa}(p)
\end{equation}
 by definition, we can map it to $L^{\frac1\kappa}(q)$ by multiplying it by itself, to get
 \begin{equation}
   \left(\frac pq\right)^{2\kappa} \in L^{\frac1\kappa}(q),\quad \text{or} \quad \left(\frac pq\right)^2 \in L^1(q).
 \end{equation}
Each of the sets $\mathcal E_p$, $p\in \mathcal M_{>}$, is going to be the domain of a chart, and this will define an atlas of charts because of the covering $\mathcal M_{>} = \cup_p \mathcal E_p$; a connected component of the manifold will be the union of overlapping $\mathcal E$'s.

If $q \in \mathcal E_{p}$, then $q$ is almost surely positive and we can write it in the form $q = \kexpof v p$, where
\begin{equation}
  v = \klnof{\frac qp} = \frac{\left(\frac qp\right)^{\kappa} - \left(\frac pq\right)^{\kappa}}{2\kappa} \in L^{1/\kappa}(p)
\end{equation}
The expected value at $p$ of $v = \klnof{\frac qp}$ is
\begin{equation}
\expectat p{\klnof{\frac qp}} = - D_{\kappa}(p \Vert q),
\end{equation}
 so that we can write every $q \in \mathcal E_{p}$ as
\begin{equation} \label{eq:7}
  q = \kexpof{u - D_\kappa(p \Vert q)} p,
\end{equation}
where $u$ is a uniquely defined element of the set of centered $(1/\kappa)$-$p$-integrable random variables $L^{1/\kappa}_0(p)$, namely
\begin{align}
  \label{eq:011}
  u &= \klnof{\frac qp} - \expectat p {\klnof{\frac qp}} \notag \\ &= \klnof{\frac qp} + D_\kappa(p \Vert q).
\end{align}

Conversely, given any $u \in L^{1/\kappa}_0(p)$, the real function $\psi \mapsto \expectat p {\kexpof{u - \psi}}$ is continuous and strictly decreasing from $+\infty$ to 0, therefore there exists a unique value of $\psi$, say $\psi_{\kappa,p}(u)$, such that
\begin{equation} \label{eq:8}
  \expectat p {\kexpof{u - \psi_{\kappa,p}(u)}} = 1,
\end{equation}
so that
\begin{equation} \label{eq:6}
  q = \kexpof{u - \psi_{\kappa,p}(u)} p \in \mathcal E_{p} \subset \mathcal M_{>}.
\end{equation}

The 1-to-1 mapping
\begin{equation}
  \label{eq:5}
  \mathcal E_p \ni q \leftrightarrow u \in L^{1/\kappa}_0(p)
\end{equation}
is our chart. By comparing \eqref{eq:6} with \eqref{eq:7} we obtain $\psi_{\kappa,p}(u)=D_\kappa(p \Vert q)$, where $u$ is the image of $q$ in the chart at $p$. I.e. the $p$-chart representation of the functional $q \mapsto D_\kappa(p \Vert q)$ is $u \mapsto \psi_{\kappa,p}(u)$. A second example is the representation of $q \mapsto D_\kappa(q\Vert p)$:
\begin{equation}
  \label{eq:other-divergence}
  D_\kappa(q\Vert p) = \expectat q {\klnof{\frac qp}} = \expectat q u - \psi_{\kappa,p}(u).
\end{equation}

The functional $\psi_{\kappa,p}$ is of key importance in the case $\kappa=0$, where it is the cumulant functional of the random variable $u$:
\begin{equation}
  \psi_p(u) = \ln \expectat p {\mathrm{e}^u}
\end{equation}

The proof of differentiability properties, in the non-finite case with $\kappa=0$ is not trivial, see e.g. \cite{cena|pistone:2007}, because the domain of  $\psi_p$ functional has a non-trivial description in that case. For $\kappa \ne 0$, the domain of $\psi_{\kappa,t}$ is the full space $L^{1/\kappa}_0(p)$, and we can compute the directional derivatives
\begin{equation}
  \label{eq:9}
  D \psi_{\kappa,p}(u) v = \left.\frac{d}{dt} \psi_{\kappa,p}(u + tv)\right|_{t=0}
\end{equation}
and
\begin{equation}
  \label{eq:800}
  D^2 \psi_{\kappa,p}(u) v w = \left.\frac{d^2}{ds dt} \psi_{\kappa,p}(u + sv + tw)\right|_{s=t=0}.
\end{equation}

For $\kappa=0$, the Fr\'echet derivatives of $\psi_p$ are
\begin{align}
  D \psi_p(u) v &= \expectat q v, \label{eq:Dpsi0}\\
  D^2 \psi_p(u) v w &= \covat q v w \label{eq:D2psi0},
\end{align}
where $v$ and $w$ are the directions of derivation and $q = \mathrm{e}^{u - \psi_p(u)}  p$.

For $\kappa \ne 0$ the computation of the directional derivative of \eqref{eq:8} gives
\begin{multline}
  \label{eq:90}
  \expectat p {\Kexp'\left(u - \psi_{\kappa,p}(u)\right)(v -D \psi_{\kappa,p}(u) v)} = \\ \expectat q {\frac{v -D \psi_{\kappa,p}(u) v}{\sqrt{1+\kappa^2(u - \psi_{\kappa,p}(u))^2}}} = 0.
\end{multline}
It follows from \eqref{eq:90} with $u=0$ that $D \psi_{\kappa,p}(0) = 0$.

Otherwise, let $q|p$ denote the density proportional to
\begin{multline}
  \label{eq:22}
  \Kexp'(u - \psi_{\kappa,p}(u)) p = \\ \frac{q}{\sqrt{1+\kappa^2(u - \psi_{\kappa,p}(u))^2}} \\ =   \frac{q}{\sqrt{1+\kappa^2\left(\klnof{\frac qp}\right)^2}},
\end{multline}
see \cite{naudts:2008}, where such a density is called escort probability. The explicit expression for the derivative is
\begin{equation}
  \label{eq:23}
  D \psi_{\kappa,p}(u) v = \expectat {q|p} v,
\end{equation}
which is the same as \eqref{eq:Dpsi0}, but the expectation is computed with respect of the escort density $q|p$. Later we will give a geometric interpretation of $q|p$.

The second derivative of $u \mapsto \kexpof{u - \psi_{\kappa,p}(u)}$ in the directions $v$ and $w$ is the first derivative in the direction $w$ of $u \mapsto \Kexp'(u - \psi_{\kappa,p}(u))(v - D\psi_{\kappa,p}(u) v)$, therefore it is equal to
\begin{multline}
  \label{eq:30}
\Kexp''(u - \psi_{\kappa,p}(u))(v-D\psi_{\kappa,p}(u)v)(w-D\psi_{\kappa,p}(u)w) \\
-\Kexp'(u - \psi_{\kappa,p}(u)) D^2\psi_{\kappa,p}(u) vw.
\end{multline}
The random variable in \eqref{eq:30} has zero $p$-expectation, so that
\begin{multline}
  D^2\psi_{\kappa,p}(u) vw = \\ \frac{\expectat p {\Kexp''(u - \psi_{\kappa,p}(u))(v-D\psi_{\kappa,p}(u)v)(w-D\psi_{\kappa,p}(u)w)}}{\expectat p {\Kexp'(u - \psi_{\kappa,p}(u))}}.
\end{multline}
If $w=v\ne0$, then $D^2\psi_{\kappa,p}(u)vv > 0$, therefore the functional $\psi_{\kappa,p}$ is strictly convex. For $u=0$ we obtain
\begin{equation}
 D^2 \psi_{\kappa,p}(0) vw = \covat p u v.
\end{equation}
We do not have a similar interpretation for $u \ne 0$, but see the discussion of parallel transport below.
\section{The $\kappa$-statistical manifold and its tangent bundle}
\label{sec:kstat}
Assume now we want to change of chart, that is we want to change the reference density from $p$ to $\bar p$ to represent a $q$ that belongs both to $\mathcal E_{p}$ and to $\mathcal E_{\bar p}$. From now on, we skip the discussion of the non-finite case; such a discussion will be published elsewhere. In the finite state space case, the integrability conditions of \eqref{eq:4} are always satisfied, so that all the chart's domains $\mathcal E_p$ are equal to $\mathcal M_{>}$. The application of \eqref{eq:011} and \eqref{eq:6} to the change
\begin{equation}
  \label{eq:300}
  L^{1/\kappa}_0(p) \ni u \mapsto q \mapsto \bar u \in L^{1/\kappa}_0(\bar p)
\end{equation}
gives
\begin{align}
  \bar u &= \klnof{\frac {q}{\bar p}} - \expectat {\bar p}{\klnof{\frac {q}{\bar p}}} \notag \\
      &= \klnof{\kexpof{u - \psi_{\kappa,p}(u)}\frac {p}{\bar p}} \notag \\
      &\qquad - \expectat{\bar p}{\klnof{\kexpof{u - \psi_{\kappa,p}(u)}\frac {p}{\bar p}}} \notag \\
      &= \left(u - \psi_{\kappa,p}(u)\right) \stackbin[]{\kappa}{\oplus} \klnof{\frac {p}{\bar p}} \notag \\
      &\qquad - \expectat{\bar p}{\left(u - \psi_{\kappa,p}(u)\right) \stackbin[]{\kappa}{\oplus} \klnof{\frac {p}{\bar p}}}
    \end{align}
For $\kappa=0$ the change of chart is an affine function:
\begin{equation}
  \label{eq:change0}
\bar u =  u + \lnof{\frac {p}{\bar p}} - \expectat{\bar p}{u + \lnof{\frac {p}{\bar p}}}
\end{equation}
with linear part
\begin{equation}
  \label{eq:13}
  v \mapsto v - \expectat {\bar p} v.
\end{equation}

For $\kappa \neq 0$, the derivative in the direction $v$ of the change of chart at $u$ is obtained from the derivation formula \eqref{eq:11}. It has the form $A - \expectat {\bar p} A$, with
\begin{multline}
  \label{eq:derivchange}
 A = \\ \left(\sqrt{1+\kappa^2 \left(\klnof{\frac {p}{\bar p}}\right)^2} - \frac{\kappa^2 \klnof{\frac {p}{\bar p}} \left(u - \psi_{\kappa,p}(u)\right)}{\sqrt{1+\kappa^2 \left(u - \psi_{\kappa,p}(u)\right)^2}}\right) \\ \times\left(v - D \psi_{\kappa,p}(u) v\right).
\end{multline}

We are then led to the study the tangent spaces of the $\kappa$-statistical manifold. Let $p_\theta$, $\theta \in ]-1,1[$, be a curve in $\mathcal E_{p}$,
    \begin{equation} \label{eq:curve}
      p_\theta = \kexpof{u_\theta -\psi_{\kappa,p_0}(u_\theta)} p_0.
    \end{equation}
In the chart at $p$ the velocity vector is given by
    \begin{equation}
      \dot{u_\theta} \in L^{1/\kappa}_0(p).
    \end{equation}

We identify the tangent space at $p_0$ with the space of the random variables $\dot{u_0}$. In general, the tangent space at $p \in \mathcal M_{>}$ is defied to be $T_p = L^{1/\kappa}_0(p)$. Derivation with respect to $\theta$ of \eqref{eq:curve} gives
  \begin{equation}
\label{eq:velocity}
    \frac{\dot{p_\theta}}{p_\theta} = (1 + \kappa^2(u_\theta -\psi_{\kappa,p}(u_\theta)^2)^{-1/2} (\dot{u_\theta} -D\psi_{\kappa,p}(u_\theta)(\dot{u_\theta})).
  \end{equation}
In particular, $\dot{p_0}/p_0 = \dot{u_0}$. To extend this to the general $\theta$, let us observe first that \eqref{eq:90}, with $p=p_0$, $u = u_\theta$, $v=\dot{u_\theta}$, shows that $\expectat {p_0}{\dot{p_\theta}/p_\theta} = 0$. The second factor in the right end side of \eqref{eq:velocity} is, because of \eqref{eq:23},
\begin{equation}
  \dot{u_\theta} -D\psi_{\kappa,p}(u_\theta)(\dot{u_\theta}) = \dot{u_\theta} - \expectat {p_\theta|p_0}{\dot{u_\theta}}.
\end{equation}
The first factor is
\begin{multline}
  \left(1 + \kappa^2(u_\theta -\psi_{\kappa,p}(u_\theta)^2\right)^{-1/2} = \\ \left(1 + \kappa^2\left(\klnof{\frac{p_\theta}{p_0}}\right)^2\right)^{-1/2}
\end{multline}

Let us define a parallel transport $U^\kappa_{p,\bar p}$ mapping the tangent space at $p$, i.e. $T_{p} = L^{1/\kappa}_0(p)$, on the tangent space at $\bar p$, i.e. $T_{\bar p} = L^{1/\kappa}_0(\bar p)$ as follows. If $u \in L^{1/\kappa}_0(p)$, then the random variable
\begin{equation}
  \label{eq:connection}
  \bar u = \frac{u - \expectat {\bar p|p}{u}}{\sqrt{1 + \kappa^2\left(\klnof{\frac {\bar p}{p}}\right)^2}}
\end{equation}
belongs to $T_{\bar p} = L^{1/\kappa}_0(\bar p)$, and we define $\bar u = U^\kappa_{p,\bar p}(u)$. In fact, $\expectat {\bar p} {\bar u} = 0$, and, moreover,
\begin{equation}
  \left(1 + \kappa^2\left(\klnof{\frac {\bar p}{p}}\right)^2\right)^{-1/2} \le 2 \left(\frac {\bar p}{p}\right)^\kappa,
\end{equation}
so that the conclusion
\begin{equation}
  \expectat {\bar p}{\vert U^\kappa_{p,\bar p}(u)\vert^{1/\kappa}} \le 2^{1/\kappa} \expectat {p}{\vert u\vert^{1/\kappa}}
\end{equation}
follows from the isometry \eqref{eq:isometry}.

The quantity $\dot{p_\theta}/p_\theta$ in \eqref{eq:velocity} has the following remarkable interpretation in terms of the parallel transport:
\begin{equation} \label{eq:parallel}
  \frac{\dot{p_\theta}}{p_\theta} = U^\kappa_{p_0,p_\theta} (\dot{u_\theta}),
\end{equation}
i.e. it is the velocity vector at $\theta$, transported to the tangent space at $p_\theta$.

In this setting, we can define $\kappa$-exponential models in a non parametric way as
\begin{equation}\label{eq:kexpmodel}
 q = \kexpof{u - \psi_{\kappa,p}(u)} p, \quad u \in V,
\end{equation}
where $V$ is a linear sub-space of $L_0^{1/\kappa}(p)$. Each $v \in V$ is called a canonical variable of the $\kappa$-exponential model. The implicit representation of the exponential model \eqref{eq:kexpmodel} is
\begin{equation}
\expectat p {\klnof{\frac qp} v} = 0, \quad v \in V^\perp .
\end{equation}
where $V^\perp \subset L^{1-1/\kappa}_0(p)$ is the orthogonal of $V$. In Statistics, a $v \in V^\perp$ is called a constraint of the log-linear model. We could derive, as we did in the $\kappa$-Gibbs model example, for lattice-valued constraint variables $v$, the relevant polynomial-type equations based of the deformed product operation.

In particular, a one dimensional exponential model is characterised by a one-dimensional space $V = \spanof{u}$, $u \in L^{1/\kappa}_0(p)$. Let us show that such a model satisfies a differential equation on the manifold. Given $u \in L^{1/\kappa}_0(p_0)$, for each $q \in \mathcal E_p$, we can define the mapping
\begin{equation}
  \label{eq:field}
  \mathcal E_p \ni q \mapsto U^\kappa_{p,q}(u) \in T_q,
\end{equation}
which is a vector field of the $\kappa$-manifold. The velocity of a curve $p_\theta=\kexpof{u_\theta - \psi_{\kappa,p_0}(u_\theta)} p_0$ is represented in the chart at $p_\theta$ by $\dot{p_\theta}/{p_\theta}$ because of \eqref{eq:parallel}, therefore we can consider the differential equation
\begin{equation}
  \label{eq:differential}
  \frac{\dot{p_\theta}}{p_\theta} =  U^\kappa_{p_0,p_\theta}(u),
\end{equation}
whose solution is
\begin{equation}
  p_\theta = \kexpof{\theta u - \psi_{\kappa,p}(\theta u)} p_0,
\end{equation}
cf. \eqref{eq:velocity}.
Equation \eqref{eq:differential} implies that the one dimensional exponential model has tangent vectors which are transported into each other by the parallel transport $U^\kappa$, i.e. the model is $U^\kappa$-auto-parallel.

The one dimensional exponential model is one of the simplest example of differential equation on the $\kappa$-statistical manifold. The treatment of evolution equations for densities is one of the main motivation for introducing a manifold structure on the set of densities, cf. \cite{otto:2001}, \cite{ay|erb:2005}, \cite{ohara|wada:2008}.
\section{Statistical manifolds}
In this final section, we go back to the general setting and briefly discuss how the $\kappa$-statistical manifolds we have defined relate with the previous construction of the exponential statistical manifold.
$(\Omega,\mathcal F, \mu)$ is a generic probability space, $\mathcal M^1$ is the set of real random variables $f$ such that $\int f\,d\mu =1$, $\mathcal M_{\ge}$ the convex set of probability densities, $\mathcal M_{>}$ the convex set of strictly positive probability densities:
\begin{equation}
  \mathcal M_{>} \subset \mathcal M_{\ge} \subset \mathcal M^1
\end{equation}

In the classical case, i.e. $\kappa=0$, differentiable manifolds are defined on both $\mathcal M_{>}$ and $\mathcal M^1$. Such manifolds are both modeled on suitable Orlicz spaces, see \cite{rao|ren:2002}. Orlicz spaces are a generalization of Lebesgue spaces, where the norm is defined through a symmetric, null at zero, non-negative, convex, with more than linear growth, function called Young function. Let $\Phi$ be any Young function with growth equivalent to $\exp$, e.g. $\Phi(x) = \cosh(x)-1$, with convex conjugate $\Psi$, e.g. $\Psi(y) = (1+\absoluteval y) \logof{1+\absoluteval y} - \absoluteval y$. The relevant Orlicz spaces are denoted by $L^{\Phi}$ and $L^{\Psi}$, respectively. Both these Banach spaces appear naturally in Statistics. A random variable $u$ belongs to the space $L^\Phi$ if, an only if, its Laplace transform exists in a neighborhood of 0, that is, the one dimensional exponential model $p(\theta) \propto \mathrm{e}^{\theta u}$ is defined for values of the parameter in a neighborhood of 0. A density function $f$ has finite entropy if, and only if, it belongs to the space $L^\Psi$. We denote by $L_0^{\Phi}$, $L_0^{\Psi}$ the sub-spaces of centered random variables. If the sample space is not finite, then the exponential Orlicz space is not separable and the closure $M^{\Phi}$ of the space of bounded functions is different from $L^{\Phi}$. There is a natural separating duality between $L^{\Phi}_0$ and $L^{\Psi}_0$, which is given by the bi-linear form
\begin{equation}
  (u,v) \mapsto \int uv\,d\mu.
\end{equation}
It can be proved that the cumulant generating functional $\psi_p(u) = \expectat p {\mathrm{e}^u}$, $u \in \L^\Psi_0(p)$, $p \in \mathcal M_{>}$ is positive, strictly convex, analytic. The interior of the proper domain of $\psi_p$,
\begin{equation}
  \mathcal S_p = \setof{u \in \L^\Psi_0(p)}{\psi_p(u) < +\infty}^\circ
\end{equation}
 defines the so called maximal exponential model
 \begin{equation}
   \mathcal E_{0,p} = \setof{\mathrm{e}^{u-\psi_p(u)} p}{u \in \mathcal S_p}
 \end{equation}

In our case, i.e. $0 < \kappa  < 1$, we could consider
\begin{multline}
  \kcoshof x - 1 = \frac12(\kexpof{x}+\kexpof{-x}) - 1 \\
             = \frac12\left(\mathrm{e}^{\int_0^x \frac{dt}{\sqrt{1+\kappa^2 t^2}}}+\mathrm{e}^{\int_0^{-x} \frac{dt}{\sqrt{1+\kappa^2 t^2}}}\right) - 1 \\
             = \cosh\left(\int_0^x \frac{dt}{\sqrt{1+\kappa^2 t^2}}\right) - 1.
\end{multline}
We obtain a Young function equivalent to $|x|^{1/\kappa}$, so that the related Orlicz space is just $L^{1/\kappa}$ and the $\kappa$-statistical manifolds appears as a generalisation of the exponential construction.

If $q \in \mathcal E_{0,p}$, then $q = \expof{u - \psi_p(u)} . p$. If $-u \in \mathcal S_p$, then $q \in \mathcal E_p$, so that $q = \kexpof{v - \psi_{k,p}(v)}$ for a suitable $v \in L^{1/\kappa}_0(p)$. It follows that
\begin{equation}
  \expof{u - \psi_p(u)} = \kexpof{v - \psi_{k,p}(v)},
\end{equation}
therefore the e-coordinate $u$ and $\kappa$-coordinate $v$ are related by the equation
\begin{equation}
  u - \psi_p(u) = \int_0^{v - \psi_{k,p}(v)} \frac{dt}{\sqrt{1 + \kappa^2 t^2}}.
\end{equation}
\section{Conclusion}
We have presented a non parametric construction of the statistical manifold based on the use of the centered $\Kln$-likelihood as a functional coordinate. The basics of the formalism are discussed in Section 3 and 4. A notable difference from the standard case $\kappa=0$ is the absence of simple formul{\ae} for the functional $\psi_{\kappa,p}$ and its derivatives. It has been shown that the derivation of such quantities is related with a suitably defined parallel transport of the tangent bundle; for such a parallel transport the exponential models are auto-parallel. This, in turn, should lead to a theory of evolution equations on the $\kappa$-manifold. The algebraic features of the $\kappa$-exponential models for lattice contrasts have been described on an example. The relation of this construction to $\alpha$-geometries of \cite{amari|nagaoka:2000} is not discussed here nor is the relation with other geometric construction based on different divergences, e.g. Tsallis entropy \cite{tsallis:1988}. When comparing our formalism with similar construction of the statistical manifold based on other transformations derived from the power transform, it should be noticed that the $\kappa$-logarithm has a distinctive advantage of having range $\mathbb{R}$ while retaining a simple algebraic character.
\begin{acknowledgement}
This piece of research was started in occasion of the SigmaPhi 2008 conference, Kolympari, GR. The Author thanks the organizers of the symposium Fisher Information and Geometry for the invitation to present a paper there and to R. Trasarti-Battistoni for the conversations during the conference. A preliminary version of this paper has been discussed with G. Kaniadakis and was presented at MSRI Berkeley CA. The work has been supported by DIMAT, Politecnico di Torino, by MSRI Berkeley CA, and by SAMSI, Research Triangle Park NC.
\end{acknowledgement}

redpajama_set_name: RedPajamaArXiv


text:
Alcohol Distributor Southern Glazer's Launches Digital Marketing Consulting Company
DRAM Agency will help alcohol brands and suppliers better leverage new tech to reach their customers.
Written byAbel Rodriguez
Published on Dec. 09, 2021
Abel Rodriguez
News Reporter at
Abel Rodriguez is a Built In news reporter covering the technology industry in Chicago, Seattle and other markets.
Photo: Southern Glazer's Wine & Spirits / Linked In
Southern Glazer's Wine & Spirits — a national, multi-generational alcohol distributor co-headquartered in Miami and Dallas — recently announced the formation DRAM Agency, a new affiliated company that will offer clients marketing consulting services.
DRAM, which stands for digital revolution of alcohol marketing, will consult brands partnered with Southern Glazer's Wine & Spirits and offer services such as data and e-commerce education. DRAM will also assist with online content and display advertising among other marketing needs, according to a prepared statement.
"Similar to the origins of the word 'dram' referring to a single pour of whisky, DRAM Agency is a single source of best practices in online marketing for the wine and spirits industry," according to the statement.
Also in DFWEvolve Biologics Announces New Dallas Facility, Plans To Hire 300
Although the formation of DRAM was recently announced, the company has been busy at work over the last year. It has already worked with approximately 200 brands to maximize media effectiveness. According to the company, DRAM will continue working with brands to meet their e-commerce needs and will be based remotely to support Southern Glazer's national footprint.
"With the explosive growth in e-commerce for beverage alcohol over the last two years, we saw an opportunity to accelerate our wine and spirits supplier partners' journey in the channel. While the largest suppliers had teams and resources in place, other partners were starting from scratch," Nathan Mansperge, Southern Glazer's VP of e-commerce, said in an email to Built In. "The mountain of requests for access to digital marketing capabilities highlighted the need for us to create DRAM Agency to provide these kind of incremental services."
With the new service offerings, Southern Glazer's Wine & Spirits will be able to help brands and suppliers leverage new technologies, market themselves and grow its sales, according to the company.
"Southern Glazer's has invested significantly in the digital marketing space by bringing in the best available talent and curating innovative e-commerce partnerships to create the strongest capability in the industry," David Chaplin, Southern Glazer's chief growth officer, said in the statement. "The new DRAM Agency brings a distinctive service offering to our suppliers to ensure they win at the digital point of purchase and grow sales and market share across e-retail channels."
Southern Glazer's Wine & Spirits

redpajama_set_name: RedPajamaCommonCrawl


text:
Barrick's Production Rose as Costs Fell in the Fourth Quarter
Steven Frank
Jan 14 2021, 5:52 PM Jan 15 2021, 5:03 AM January 14 2021, 5:52 PM January 15 2021, 5:03 AM
(Bloomberg) -- Barrick Gold Corp. produced more of the precious metal with lower costs in the fourth quarter than the previous period as it met its full-year gold guidance amid the global pandemic.
Production in the quarter matched analysts' expectations, while all-in sustaining costs are expected to be 3% to 5% lower than the third quarter, Toronto-based Barrick said Thursday in a statement.
Gold production in the quarter was 1.21 million ounces, based on preliminary data, the company said. Output rose from the third quarter largely because of strong performance from its Pueblo Viejo mine, a ramp-up of operations in Bulyanhulu and improvements at Turquoise Ridge.
The world's second-largest gold miner lowered its 2020 guidance in May to a range of 4.6 million to 5 million ounces because of a conflict with Papua New Guinea's government that halted production at Porgera mine in April. Barrick's full-year output was 4.8 million ounces, the midpoint of its annual forecast.
Gold prices have been strengthening in the past two years, helping lift miners' cash flow and stock prices. Spot gold prices were up 26% on average in the fourth quarter from a year earlier, and in August they breached the record set in September 2011.
Barrick shares climbed 23% last year in New York, lagging the 46% gain of a Bloomberg Intelligence index of senior gold miners.
News about Barrick.


BRIDGETON, N.J. — The third annual event saw votes for 879 area businesses.
BRIDGETON, N.J. — Dusty's Custom Golf Carts and Auto Detailing was recognized as Best Auto Detailing at the Bridgeport Chamber of Commerce's Best of West & South Cumberland event, according to NJ.com.
People voted for their favorite businesses in several categories, including Best Overall Business, Best Dentist, Best Graphic Design, Best Farm Supplies, Best Breakfast, Best Fitness Center, Best Pharmacy, Best Landscaping and more, noted the article.
The winners will be recognized at a celebration at a later date, stated the article. The contest was sponsored by the chamber of commerce, Cumberland Development Corp. (CDC) and South Jersey Times.
The event is held annually, reported the article. The chamber received 272 ballots consisting of 7,721 votes for 879 businesses, shared Tony Stazione, executive director of the chamber and CDC, in the article.
"On behalf of the Chamber of Commerce, CDC and South Jersey Times, we thank everyone who voted, and we congratulate the first place businesses and invite the winners and the public to attend the Best of West and South Cumberland Celebration [to] be held at Gia's Catering on Mulford Drive in Bridgeton," said Susan Jones, chairperson of the Chamber and the Best of West and South Cumberland Committee, in the article.


text:
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redpajama_set_name: RedPajamaGithub