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Justia › US Law › Case Law › Arizona Case Law › Arizona Supreme Court Decisions › 1974 › Valley National Bank v. Brown
Valley National Bank v. Brown
Annotate this Case
110 Ariz. 260 (1974)
517 P.2d 1256
The VALLEY NATIONAL BANK of Arizona, a national banking association, Appellant, v. Edward BROWN and Lillian R. Brown, husband and wife, Appellees.
No. 11249-PR.
Supreme Court of Arizona, In Banc.
Rehearing Denied February 19, 1974.
*261 Rawlins Ellis Burrus & Kiewit, by Robert L. Gootsfield, Phoenix, for appellant.
Pain & Julian, by Fred J. Pain, Jr., Phoenix, for appellees.
HOLOHAN, Justice.
This cause is before us on a petition for review of a decision of the Court of Appeals, Valley National Bank v. Brown, 19 Ariz. App. 493, 508 P.2d 752 (1973), which reversed a judgment by the Superior Court of Maricopa County giving judgment for $5,000 to the plaintiffs Edward and Lillian R. Brown against the defendant Valley National Bank. The decision of the Court of Appeals is vacated.
On October 16, 1964 Home Savings and Loan Association obtained a judgment against Claude V. Brown and Lillian R. Brown, husband and wife, foreclosing a realty mortgage. The foreclosure sale did not produce sufficient funds to satisfy the debt and a deficiency judgment was obtained which was assigned by the loan association to Union Guaranty Co. for collection.
*262 An investigator employed by the attorneys for Union Guaranty reported that a Lillian R. Brown resided at an address on East Thomas and worked at Marrigay Foods Corp., 426 East Jackson, Phoenix, Arizona, and the investigator, in his oral report, maintained that the Lillian R. Brown he had discovered was the same Lillian R. Brown involved in the litigation.
Using the information from the investigator, the attorneys caused a writ of garnishment to be issued on May 21, 1968 against the 7th Avenue and Thomas Road Branch of the Valley National Bank for the accounts of the Browns. Concurrently a writ of execution was secured to be served at the address given in the investigatory report. Unfortunately, the Lillian R. Brown named in the judgment and the Lillian R. Brown residing at the address given for service of the writ of execution, who worked at Marrigay Foods, and who had an account at the 7th Avenue and Thomas Road Branch of the Bank were not the same individuals.
In preparing the information for the answer to the writ of garnishment, an experienced Bank Officer, Pauline Peters, found two accounts in the branch in the name of Lillian R. Brown. She noticed, however, one of the accounts was in the name of Edward and Lillian R. Brown rather than Claude V. and Lillian R. Brown, the judgment debtors. The officer questioned whether or not the correct account was being impounded. She relayed her fears to the creditor's attorney and attempted to clarify the possible confusion. This conversation resulted in the Bank being told to hold the funds. Being still concerned, Mrs. Peters notified the Bank's attorney of the conflict and the attorney requested that the creditor's counsel quash the writ. The request was refused, and the Bank was requested to impound the fund which it did. A total of $2,484.16 was held.
Pursuant to the writ of execution, a Maricopa County Deputy Sheriff attempted to seize an automobile belonging to the wrong Mrs. Brown. The officer felt that this person was not the right debtor, and he relayed this information to the creditor's attorney. He was told not to execute the writ, and the attorney called Mrs. Brown and apologized.
The Bank was not informed of the later events, and the day following the incident with the sheriff, the Bank advised Mrs. Brown that her funds had been impounded. Although she told the employees of the Bank that it was a mistake and asked them to call the judgment creditor's attorney, the Bank did nothing. The creditor's attorney finally notified the Bank and asked that the funds be released. The funds had been impounded for a fifteen-day period.
Plaintiffs brought suit against Valley National Bank, Union Guaranty Co., the firm of attorneys representing Union Guaranty, and the individual attorney in charge of the garnishment. The jury returned a verdict in favor of the plaintiffs and against the Bank and the individual attorney. The Bank was the only defendant to appeal from the judgment.
The recovery of plaintiffs against the Bank was based on liability in tort for the defendant's negligence in wrongfully impounding plaintiffs' account.
The Bank urges that it complied with the requirements of the garnishment statute (A.R.S. § 12-1595), and it cannot be liable for its actions. The statute in question provides:
"A. A bank deposit made in the names of two or more persons shall be subject to garnishment. "B. The answering garnishee shall, upon service of the writ, impound all funds then present in the bank account, and shall promptly notify each person who appears from the business records of the garnishee to have an interest in such bank account in addition to the defendant that such account has been impounded, the name of the plaintiff and defendant and the court in which the action is pending as stated on the writ. The notice may be made personally or by *263 registered mail, postage prepaid, addressed to each such person at his last known address as reflected by the business records of the garnishee. "C. The answer of the garnishee, in such case, shall state under oath the names of all persons who appear from the business records of the garnishee to have an interest in the bank account in addition to the defendant. Upon the filing of the answer the court shall join all persons who appear to have an interest in the bank account in addition to the defendant, as reflected by the answer of garnishee, and shall proceed to a determination of the interest of the defendant therein. "D. Upon entry of order based upon such proceeding all impounded funds, except those of defendant, shall be released by the garnishee. The matter shall otherwise proceed as in any other garnishment action. "E. All persons whose funds are impounded under the provisions of this section shall, together with the defendant, be considered to be joint and several obligees of the bond provided for by § 12-1572, but the liability of the sureties on such bond shall be limited to the amount of the penal sum named in the bond."
It is the contention of the Bank that it has no duty to determine the true ownership of the funds of an account, nor does it have any duty to resolve conflicting claims; the ownership and right to the deposits are matters for judicial determination. Valley Bank & Trust Co. v. Parthum, 47 Ariz. 496, 56 P.2d 1342, reh. den. 48 Ariz. 87, 59 P.2d 335 (1936).
As general propositions the contentions of the Bank are correct. The statute was designed for the protection of banks faced with garnishments affecting interests in joint accounts. The important point of distinction in this case is that the Bank did not follow the requirements of the statute; therefore it is not afforded the protection of the statute. The Bank did not hold the funds of the defendant, namely, Lillian R. Brown, the wife of Claude V. Brown. The Bank clearly knew from the writ that the judgment debtors were "Claude V. Brown and Lillian R. Brown, his wife," while the principal account garnished had a signature card naming Edward and/or Lillian R. Brown as joint tenants and defining their relationship as husband and wife. The Bank was on notice by its own records that, although the names appeared to be the same, the Browns named in the writ and the Browns named on the account records were not the same.
The Bank argues that it was placed in an untenable position in this case, i.e., liable if it held the funds; liable if it did not. The contention is not correct. The true answer to the writ was that the Bank was not indebted to Lillian R. Brown, the wife of Claude V. Brown. The Lillian R. Brown to whom it was indebted is the wife of Edward Brown, not Claude V. Brown. Such an answer would be correct, and the Bank would not be liable to the creditor.
The question remains that if the funds of the Browns should not have been held, were the actions of the Bank negligent? The case was tried on a negligence theory, and the Bank contends that it was entitled to a directed verdict. It was a question of fact whether the acts of the Bank under the evidence presented constituted negligence. With the information possessed by the Bank in its own records plus the subsequent information furnished by Mrs. Brown that the creditor's attorney acknowledged the mistake in identity, would an ordinary prudent person have acted in the same fashion? Sufficient evidence was presented at trial to raise a question as to whether the Bank met the standard.
"It is ... fundamental that a verdict will not be directed where the evidence on material facts is conflicting, or where on undisputed facts reasonable and fairminded men may differ as to the inferences and conclusions to be drawn, or where different conclusions might reasonably be reached by different *264 minds, and thus the question of negligence and proximate cause is one of fact to be submitted to the jury and not a question of law for the court; if, upon all the facts and circumstances, there is a reasonable chance or likelihood of the conclusions of reasonable men differing, then the question is one for the jury. Cope v. Southern Pacific Co., 66 Ariz. 197, 185 P.2d 772; Durham v. Firestone Tire & Rubber Co. of California, 47 Ariz. 280, 55 P.2d 648...." Figueroa v. Majors, 85 Ariz. 345 at 346-347, 338 P.2d 803 at 804 (1959).
The trial court was correct in submitting the matter to the jury.
The Bank next contends that the only damage recoverable by plaintiffs is the interest on the funds for the time they were impounded. This contention finds support in the older cases, but such cases generally dealt with actions brought on a contract theory. As a general rule damages for breach of contract are limited to those which arise naturally from the breach itself or those which may reasonably be supposed to have been within the contemplation of the parties at the time of making the contract. Jacob v. Miner, 67 Ariz. 109, 191 P.2d 734 (1948). The case at issue, however, was not based on a contract theory but one of tort, and the elements of damages can be different. 22 Am.Jur.2d Damages ¶ 80; Hall v. Motorists Insurance Corporation, 109 Ariz. 334, 509 P.2d 604 (1973).
Recovery in a tort action is limited to those damages which are the direct and proximate consequence of the defendant's wrongful acts. Under the theory upon which plaintiffs sought recovery they were entitled to be compensated for all detriment suffered as a proximate result of defendant's negligence.
Plaintiffs urge that it is not necessary for them to show any special elements of damages because damages are presumed in this type of case. Valley National Bank v. Witter, 58 Ariz. 491, 121 P.2d 414 (1942) is cited for the foregoing proposition. Witter is a case involving wrongful dishonor by a bank of a check drawn against it by one of its depositors. In the cited case this Court held that a depositor's credit is impeached when a bank wrongfully refuses payment of a check presented by the creditor of the depositor, and damages are presumed from the wrongful action of the bank. The dishonor of a check is distinguishable from the wrongful withholding of funds by a bank. In the former instance there is an actual "slur" on the depositor's credit and business reputation with at least one of the depositor's creditors, that is, the bank has caused the depositor to be held out to at least one creditor as not having sufficient funds to meet his financial obligations. In the instance of withholding funds there is not necessarily any imputation on the credit of the depositor. Thus, in the former instance there is a presumption of injury to the depositor's credit, but no such presumption in the latter instance. In the case at issue it was necessary for the plaintiffs to prove the elements of their damages without the aid of any presumption.
On the issue of damages, the Bank concedes that evidence was presented which might support at most $1,010.00 in damages, i.e., attorney's fees and costs to secure a court order releasing the impounded funds, plus cost of a return trip from New York to Phoenix by Mr. Brown to tend to the matter, and interest on the impounded funds, but the Bank objects to the remaining $3,990.00 in damages as unproven. The rule in this State is that an award of damages is within the province of the jury and will not be disturbed on appeal except where the verdict is so exorbitant as to show passion, prejudice, mistake or complete disregard of the evidence. Acheson v. Shafter, 107 Ariz. 576, 490 P.2d 832 (1971). The plaintiffs maintained at trial that their credit was damaged and a loss occurred in their business transactions and that there was an aggravation of an existing emotional disturbance.
*265 In examining this evidence, we find no proof of loss in any business transactions except for the bald testimony of Mr. Brown which is barren of any specifics. Mr. Baumgartner of the First National Bank, a witness called by plaintiffs, testified that Mr. Brown's credit was in good standing with his bank during this whole period of time. There was no evidence of loss of business reputation. The plaintiffs failed to prove this item of damages.
With but a few minor exceptions, the rule is that where no malice or intent is shown, no damages may be awarded for mental anguish or distress of mind. The exception to the rule occurs when it is shown that there is a physical invasion of a person or the person's security. Logan v. St. Luke's General Hospital, 65 Wash. 2d 914, 400 P.2d 296 (1965). No such physical invasion occurred here. In Witter, supra, we noted "no damages can be allowed for mental pain or suffering" in actions of this nature. There could be no compensation to the plaintiffs from the Bank for mental suffering or aggravation of existing emotional disturbance.
There was no competent evidence of any damages except for the $1,010.00 noted above. The $5,000 awarded by the jury was excessive and should be reduced to the proper figure supported by the evidence.
Finally, the Bank objects to certain instructions given by the trial court on pain and suffering as an element of damages. Although not overly clear, a reading of all the instructions shows that the trial court limited such elements of damages for consideration against the creditor's attorney if the jury found his conduct malicious, and the trial court instructed the jury in substance that there could be no recovery for mental pain or suffering by the plaintiffs from the Bank. There does not appear to have been any prejudice to the Bank in the instructions given.
The judgment of the trial court is modified by reducing the amount of damages assessed against the Bank to $1,010.00 plus costs in the trial court and interest from date of judgment. Each party will bear their own cost for this appeal.
Judgment as modified is affirmed.
HAYS, C.J., and STRUCKMEYER and LOCKWOOD, JJ., concur.
CAMERON, Vice Chief Justice (dissenting).
For the reasons stated in the opinion of the Court of Appeals, see Valley National Bank v. Brown, 19 Ariz. App. 493, 508 P.2d 752 (1973), I dissent.
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text:
Teofimo Lopez Positive For COVID-19; Entire Lopez-Kambosos Card Postponed Until August 14
By Keith Idec
Published Tue Jun 15, 2021, 01:57 PM EDT
George Kambosos Jr. will have to wait two more months to get his shot at Teofimo Lopez.
The 23-year-old Lopez has contracted COVID-19 and has been forced to postpone his fight against Australia's Kambosos. Their 12-round, 135-pound title fight was scheduled to headline a Triller Fight Club Pay-Per-View card Saturday night at loanDepot Park in Miami, the home stadium of Major League Baseball's Miami Marlins.
BoxingScene.com has confirmed that the entire card, which was set to include 12 fights, has been rescheduled for August 14 at a site to be determined. The Marlins have a home game scheduled against the Chicago Cubs the night of August 14, thus the Lopez-Kambosos card will have to take place at another venue.
The Athletic's Mike Coppinger, who is part of Triller Fight Club's broadcast team, was the first to report the news of Lopez's withdrawal from the fight Tuesday afternoon on Twitter. Ryan Kavanaugh, Triller's founder, told Coppinger that Lopez tested positive for coronavirus Tuesday morning and is symptomatic.
Lopez (16-0, 12 KOs) has been training for months in Las Vegas for a mandated defense of his IBF lightweight title. Sydney's Kambosos (19-0, 10 KOs) spent 10 weeks training in Miami for the biggest fight of his career.
Neither Lopez nor Kambosos have received COVID-19 vaccination shots while training for their fight.
BoxingScene.com has learned that Lopez began feeling ill late last week and consequently didn't attend a Top Rank show that ESPN televised Saturday night from The Theater at Virgin Hotels Las Vegas.
Tuesday's setback marked the second time this card has been postponed. It originally was scheduled for June 5 at loanDepot Park, but it was rescheduled last month for Saturday night.
Lopez, a Brooklyn native, is supposed to earn a career-high purse of $3,911,700 to battle Kambosos. The 28-year-old Kambosos, the IBF's number one-ranked lightweight contender, is contracted to earn a career-high payday of $2,106,300 for facing Lopez.
Triller Fight Club won the right to promote Lopez-Kambosos when it offered a whopping $6,018,000 at an IBF purse bid February 25.
Kavanaugh's company out-bid Bob Arum's Top Rank Inc., Lopez's promoter, which bid $2,315,000. Eddie Hearn's Matchroom Boxing bid $3,506,000 for the right to promote Lopez-Kambosos.
Keith Idec is a senior writer/columnist for BoxingScene.com. He can be reached on Twitter @Idecboxing.
Teofimo Lopez
George Kambosos
Lopez vs. Kambosos
Photos: Michael Conlan, Leigh Wood - Face To Face at Nottingham Presser

redpajama_set_name: RedPajamaCommonCrawl


In my interview with Dr. Norman Geisler, I asked him to briefly summarize the archaeological evidence supporting the New Testament.
"The noted Roman historian Colin J. Hemer, in The Book of Acts in the Setting of Hellenistic History, shows how archaeology has confirmed not dozens, but hundreds and hundreds of details from the biblical account of the early church," Geisler said. "Even small details have been corroborated, like which way the wind blows, how deep the water is a certain distance from the shore, what kind of disease a particular island had, the names of local officials, and so forth.
"Now, Acts was authored by the historian Luke. Hemer gives more than a dozen reasons why Acts had to have been written before AD 62, or about thirty years after Jesus' crucifixion. Even earlier, Luke wrote the gospel of Luke, which is substantially the same as the other biblical accounts of Jesus' life.
"Is Hemer a lone voice on that?" I asked.
Next Post With all the differences between the New Testament manuscripts, how can we trust the Bible?


text:
SENATE VOTES TODAY ON CHILD-CARE BILL
The Baltimore SunSUN-SENTINEL
WASHINGTON -- After a day of fierce debate over rival child-care bills intended to help low-income families, Senate Democrats and Republicans agreed to settle the matter today.
Wednesday's agreement says that the Senate will vote on a Republican alternative to the Democratic child-care bill. Then it will vote on the Democratic bill.
Senator Robert J. Dole, R-Kan., formally introduced the Republican alternative on Wednesday, then led the Republicans, backed by the White House, into a showdown with the Democrats over the two pieces of legislation.
The Democrats, he charged, wanted to create "another money-eating, bureaucratic sinkhole" with their bill. The Democratic proposal, Dole argued, would take the choice of child care away from parents and subject church and synagogue child-care centers to constitutional challenge.
In an angry rebuttal, Sen. Christopher J. Dodd, D-Conn., derided the GOP plan as a "bill crafted to get votes, not to provide child care."
Dole's position was reinforced by the White House, which released a policy statement reiterating its displeasure with the Democratic plan, known as the ABC bill, saying that the Dole proposal was "far more consistent with the president's principles." If the ABC bill were presented to the president for his signature, the statement read, "his senior advisers would recommend that he veto it."
Later, however, White House spokesman Marlin Fitzwater shied from confronting Democrats with an ultimatum, stating instead that the administration would be "willing to talk about alternatives and how this could be approved."
Christopher Dodd


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text:
Wikipedia disrespects experts says co-founder
Larry Sanger, who founded Wikipedia in 2001 with Jimmy Wales only to leave shortly afterwards, said that even as far back as 2001 the Wikipedia community "had no respect for experts".
Sanger said he told Jimmy Wales that unless he did something about the fact that no special role was provided for experts to reign in the various abusive elements like vandalism of articles, he was going to distance himself from the project permanently.
Years later Sanger went on to launch his own collaborative, community-based online encyclopedia called Citizendium.
As far back as 2001 Sanger noted that complaints were beginning to pour into the online community encyclopedia about the authenticity of contributions and edits.
The difference, he says, is that Citizendium has expert editors that nominate articles for approval and constables that essentially police the site, and make sure it runs well.
He hopes that Citizendium will become a credible competitor to Wikipedia but with the same expert reliability as a traditional print encyclopedia.
"Wikipedia, to put it plainly, lacks the sort of credibility and reliability that traditionally edited resources have.
"Experts are needed in the approval process, not necessarily in the process of creating the content. They are needed to make an efficient judgment that the article captures what is known about a subject, " he said.
Although Wikipedia is not for profit and is now very anti-advertising, Sanger pointed out that before the dotcom crash, Wikipedia did have advertising income, but following the drying up of money from that source it then became what it is today.
Sanger believes that such portals can be monetised while maintaining a professional presence as a resource tool. He is currently in the process of looking at ways to incorporate advertising in a non-intrusive way into Citizendium.
Apart from advertising, Sanger and co-founder Wales also disagree about who actually founded Wikipedia in the first place. Wales has previously claimed to be the sole founder of the online collaborative encyclopedia.
Maybe he should check his facts on Wikipedia which says that both men were identified as co-founders in 2001.
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In the dynamical theory of granular matter the so-called table problem consists instudying the evolution of a heap of matter poured continuously onto a bounded domain Ω⊂\R2. The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge-Kantorovich problem, is the object of this paper. Our main result is an integral representation formula for the solution, in terms of the boundary curvature and of the normal distance to the cut locus of Ω.


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text:
\section{Introduction and preliminaries}
\label{S1}

\subsection{Introduction}
\label{S1.1}

Non-Markovian processes bring in the novel future of phase transitions: Several measures can
share the same transition probabilities if these have a sufficiently strong dependence on
faraway future~\cite{brakal93,berhofsid05,hul06}.   Unlike the Markovian case, this coexistence
is not due to the partition of the space into non-communicating components ---the transition
probabilities are all strictly positive in the published examples--- but rather to the
persistence of the influence of past history into the infinite future.  Such transitions
parallel statistical mechanical first-order phase transitions, where different boundary
conditions lead, in the thermodynamic limit, to different consistent measures.  This suggests,
as advocated in~\cite{fermai05}, to take Gibbsian theory as a model for the study of multiple-chain
phase diagrams.

More directly, one may wonder whether chains can simply be treated as one-dimensional Gibbs measures.
If so, the usual theory of discrete-time processes ---geared towards the description of phenomena
characteristic of Markov processes--- could be supplemented by an appropriately transcribed Gibbs
theory ---tailored to the description of complicated phase diagrams. Our attempt in~\cite{fermai04}
was unable to reach the multiple-phase case.

From a complementary point of view, it is natural to search for conditions granting that phase
transitions do not occur, that is, granting that a given family of transition probabilities admit
only one consistent measure.  There exist, at present, a number of such uniqueness
criteria~\cite{har55,ber87,lal86,lal00,comferfer02,sten03,fermai05,hul06,johobe03,johobepol05}
involving different non-nullness and continuity hypotheses and yielding information on different
properties of the invariant measure (mixing properties, Markovian approximation schemes, regeneration
and perfect simulation procedures).

In all these studies, there is an ignored aspect that deserves, in our opinion,
more careful consideration:  the role of the shift-invariance \emph{of the transition probabilities}.
All the phase transition examples involve shift-invariant transitions (and measures) and this is also
an ingrained feature in most of the proofs of existing uniqueness criteria.  The only exceptions are
the ``regeneration'' criterion of~\cite{comferfer02} and the ``one-side bounded uniformity'' proven
in~\cite{fermai05}. (For notational simplicity, a translation invariant setting was adopted
in~\cite{comferfer02}, but it is clear that the proof ---showing that every finite window can be
reconstructed from a finite past--- does not require shift invariance.)  It is legitimate to inquire
whether shift-invariance is an unavoidable requirement for the remaining criteria, or only an artifact
of the proof.

In this paper we illustrate some of the differences and similarities that arise when shift invariance is
lost.  We consider chains defined on the half line or, equivalent, on a ``time'' axis that is a countable
set with a total order and a maximal element.  We show, first, that in this setting
we can successfully complete the program initiated in~\cite{fermai04} and establish
a full correspondence between chains and Gibbs measures (Theorem \ref{th1} below).
We exploit the interplay
between Gibbsian and chain points of view to reveal a number of interesting facts:

\begin{itemize}
\item[(i)]
We borrow results by Dyson~\cite{dys69a} to show that the chains defined by
long range Ising models with couplings
decaying as power laws $|i-j|^{-p}$ ($p>1$ to ensure summability of the interactions) exhibit phase
transitions for $p<2$ for large values of the coupling parameters (``low temperatures'', see Theorem \ref{ising-res}).
The transcription of Dyson's approach amounts to a novel way to prove phase transitions in the context of chains,
namely by constructing a measure that is not mixing.  This implies that there should be at least two different
extremal consistent measures~\cite{fermai05}.
\item[(ii)]
The chains with $3/2<p<2$ do satisfy the square summability condition of Johansson and \"Oberg's uniqueness
criterion~\cite{johobe03} (see also~\cite{johobepol05}) and yet exhibit phase transitions (Remark~\ref{johobe-rk}
below).  This shows that such criterion ---which has been proven to be optimal in an appropriate
sense~\cite{berhofsid05}--- is false in general non-shift-invariant settings.
\item[(iii)]
Using the regeneration (chain) criterium of~\cite{comferfer02} we prove that, at least in the half line, a
conjecture by Kac and Thompson (mentioned in~\cite{dys69a}) is false (see Remark~\ref{ruel-rem} below).
\item[(iv)]
On the other hand, Gibbsian uniqueness criteria can be used to show that these models have a unique invariant
state at high temperatures.  The only chain criterion that is temperature-sensitive is one-sided
Dobrushin~\cite{fermai05,hul05} which, however, is not directly applicable to the Ising chains considered here.
\item[(v)] Present work implies a one-sided version of the Kozlov theorem~\cite{koz74} (Corollary \ref{koz-chains}
below): transition probabilities that are continuous and non-null are always defined by one-sided interactions,
albeit in an indirect manner that passes through an auxiliary specification.
\end{itemize}

\subsection{Notation and preliminary definitions}
\label{S1.2}

We consider a measurable space $({\mathcal A},{\mathcal E})$ where ${\mathcal A}$ is a finite alphabet and ${\mathcal E}$ is the
discrete $\sigma$-algebra. We denote $(\Omega,{\mathcal F})$ the associated product measurable space
with $\Omega={\mathcal A}^{{\mathbb L}}$, where ${\mathbb L}$ is a countable set with total order. In this paper we study
the case in which ${\mathbb L}$ has a maximal element, that is, ${\mathbb L}={\mathbb Z}^-$, but we shall also refer to
the usual unbounded case, where ${\mathbb L}={\mathbb Z}$.  For each $\Lambda\subset{\mathbb L}$ we denote
$\Omega_\Lambda = {\mathcal A}^{\Lambda}$ and $\sigma_\Lambda$ for the restriction of a configuration
$\sigma\in\Omega$ to $\Omega_\Lambda$, namely the family
$(\sigma_i)_{i\in\Lambda}\in {\mathcal A}^{\Lambda}$.  Also, ${\mathcal F}_\Lambda$ will denote the
sub-$\sigma$-algebra of $\mathcal{F}$ generated by cylinders based on $\Lambda$
(${\mathcal F}_\Lambda$-measurable functions are insensitive to configuration values outside $\Lambda$).
When $\Lambda$ is an interval, $\Lambda=[k,n]$ with $k,n\in{\mathbb L}$ such that $k\le n$, we use the
notation: $l_\Lambda=k$, $m_\Lambda=n$, $\Lambda_-=\{i\in{\mathbb L}:i<k\}$,
$\omega_{k}^{n}=\omega_{[k,n]}=\omega_{k},\ldots ,\omega_{n}$, $\Omega_k^n=\Omega_{[k,n]}$ and
${\mathcal F}_k^n={\mathcal F}_{[k,n]}$. For semi-intervals we denote also ${\mathcal F}_{\le n}:={\mathcal F}_{(-\infty,n]}$, etc.
The concatenation notation $\omega_\Lambda\,\sigma_\Delta$, where $\Lambda\cap\Delta=\emptyset$,
indicates the configuration on $\Lambda\cup\Delta$ coinciding with $\omega_i$ for $i\in\Lambda$
and with $\sigma_i$ for $i\in\Delta$. We denote ${\mathcal S}$ the set of finite subsets of ${\mathbb L}$ and ${\mathcal S}_b$
the set of finite intervals of ${\mathbb L}$. To lighten up formulas involving probability kernels, we will
freely use $\nu(h)$ instead of $E_\nu(h)$ for $\nu$ a measure on $\Omega$ and $h$ a
$\mathcal{F}$-measurable function. Also $\nu(\sigma_\Lambda)$ will mean
$\nu(\{\omega\in\Omega:\omega_\Lambda=\sigma_\Lambda\})$ for
$\Lambda\subset{\mathbb L}$ and $\sigma_\Lambda\in\Omega_\Lambda$.

For any sub-$\sigma$-algebra ${\mathcal H}$ of ${\mathcal F}$, we recall that a \emph{measure kernel} on ${\mathcal H}\times\Omega$
is a map $\pi(\,\cdot\mid\cdot\,)\colon$ ${\mathcal H}\times\Omega\to\mathbb{R}$ such that $\pi(\,\cdot\mid \omega)$
is a measure on $(\Omega,{\mathcal H})$ for each $\omega\in\Omega$ while $\pi(A\mid\cdot\,)$ is ${\mathcal F}$-measurable
for each event $A\in{\mathcal H}$.  If each $\pi(\,\cdot\mid \omega)$ is a probability measure the kernel is called
a \emph{probability kernel}.
For kernels
$\pi$ and $\widetilde\pi$, non-negative measurable functions $h$, measures $\nu$ on $(\Omega,{\mathcal H})$
and cylinders $C_{\omega_\Lambda}=\{\sigma\in\Omega: \sigma_\Lambda=\omega_\Lambda\}$, we shall denote:

\begin{itemize}
\item $\pi(\omega_\Lambda\mid\,\cdot\,)$ for $\pi(C_{\omega_\Lambda}\mid\,\cdot\,)$
\item $\pi(h)$ for the measurable function $\int_\Omega
  h(\eta)\,\pi(d\eta\mid\,\cdot\,)$.
\item $\pi\widetilde\pi$ for the composed kernel defined by
  $(\pi\widetilde\pi)(h) = \pi\big(\widetilde\pi(h)\big)$.
\item $\nu\pi$ for measure defined by $(\nu\pi)(h) =\nu\big(\pi(h)\big)$.
\end{itemize}

In the unbounded case, ${\mathbb L}={\mathbb Z}$, the (right) \emph{shift operator}
---$\tau:\Omega\to\Omega$, $(\tau \omega)_i=\omega_{i-1}$--- is an isomorphism that naturally induces
shift operations for measurable functions and kernels: $(\tau f)(\omega) = f(\tau^{-1}\omega)$,
$(\tau\pi)(h\mid\omega) = \pi(\tau^{-1}f\mid\tau^{-1}\omega)$.

\subsection{Chains and Gibbs measures}
\label{S1.2.2}\label{S1.2.1}

We start by briefly reviewing in parallel the well known notions of chains and Gibbs measures
in the spirit of~\cite{fermai05}.  The main difference is that, in chains, kernels apply only
to functions measurable with respect to the present and the past.

\bd{lis1}
A \embf{left singleton-specification (LSS)} (or system of transition probabilities) $f$
on $(\Omega,{\mathcal F})$ is a family of probability kernels $\left\{f_i\right\}_{i\in{\mathbb Z}}$ with
$f_i\colon{\mathcal F}_{\leq i}\times\Omega\to [0,1]$ such that for all $i\in{\mathbb Z}$,
\begin{itemize}
\item[\rm{(a)}] for each $A\in{\mathcal F}_{\leq i}$, $f_i(A\mid\cdot\,)$ is
${\mathcal F}_{\leq i-1}$-measurable;
\item[\rm{(b)}] for each $B\in{\mathcal F}_{\leq i-1}$ and $\omega\in\Omega$,
$f_i(B \mid \omega)=\rlap{\mbox{\small\rm 1}}\kern.15em 1_B(\omega)$.
\end{itemize}
The LSS $f$ is:
\begin{itemize}
\item[\rm{(i)}]
\embf{Continuous} if the functions
$f_i\left(\omega_i \mid \cdot\,\right)$ are continuous
for each $i \in {\mathbb L}$ and $\omega_i\in\Omega_i$;
\item[\rm{(ii)}]
\embf{Non-null} if the functions
$f_i\left(\omega_i \mid \cdot\,\right)$ are (strictly) positives for each
$i \in {\mathbb Z}$ and $\omega_i\in\Omega_i$;
\item[\rm{(iii)}]
\embf{Regular} if it is continuous and non-null;
\item[\rm{(iv)}]
\embf{Shift-invariant} if ${\mathbb L}={\mathbb Z}$ and $\tau f_i=f_{i+1}$, $i\in{\mathbb L}$.
\end{itemize}
\end{definition}
\bd{chain}
A probability measure $\mu$ on $(\Omega,\; \mathcal{F} )$ is said to be \embf{consistent}
with a LSS $f$ if for each $i \in{\mathbb Z}$,
\be{}
\mu f_i = \mu \quad\text{over }
{\mathcal F}_{\leq i}\;.
\end{equation}
The family of these measures will be denoted by $\mathcal{G}(f)$ and each $\mu\in\mathcal{G}(f)$
is called a {($f$-) \embf{chain}}. A measure $\mu$ is a \embf{regular chain} if there exists a
regular LSS $f$ such that $\mu\in{\mathcal G}(f)$.
\end{definition}

The singletons $f_i$ of a LSS define, through compositions, interval-kernels
\be{lss-k}
f_{[m,n]} \;=\; f_m\,f_{m+1}\,\cdots\, f_m
\end{equation}
for $m\le n\in{\mathbb L}$.  We observe that
\be{lis-mu}
\mu\in\mathcal{G}(f) \;\Longleftrightarrow\; \mu f_{[m,n]} = \mu
\quad\text{over }{\mathcal F}_{\leq n}\quad \forall m\le n\in{\mathbb L}\;.
\end{equation}
The family $\{f_{[m,n]} : m\le n\in{\mathbb L}\}$ ---called a LIS (\emph{Left Interval-Specification})
in~\cite{fermai04,fermai05}--- is the chain analogue of the notion of specification.

\br{g-function}
The particular case when $f$ and $\mu\in{\mathcal G}(f)$ are shift-invariant, reduces
to the study of $g$-functions and $g$-measures, respectively~\cite{kea72}.
Chains for general, non-shift-invariant singletons have also been called
$G$-measures~\cite{brodoo91,brodoo98}.
\end{remark}
\bd{specif}
A \embf{specification} $\gamma$ on $\left(\Omega,{\mathcal F}\right)$ is a family of probability
kernels $\left\{\gamma_\Lambda\right\}_{\Lambda\in{\mathcal S}}$ with
$\gamma_\Lambda:{\mathcal F}\times\Omega\rightarrow [0,1]$ such that for all $\Lambda$ in ${\mathcal S}$,
\begin{itemize}
\item[\rm{(a)}] for each $A\in{\mathcal F}$, $\gamma_\Lambda(A\mid\cdot)$ is ${\mathcal F}_{\Lambda^{c}}$-measurable;
\item[\rm{(b)}] for each $B\in{\mathcal F}_{\Lambda^{c}}$ and $\omega\in\Omega$,
$\gamma_\Lambda(B\mid\omega)=\rlap{\mbox{\small\rm 1}}\kern.15em 1_B(\omega)$;
\item[\rm{(c)}] for each $\Delta\in{\mathcal S}:\Delta\supset\Lambda$,
$\gamma_\Delta \gamma_\Lambda=\gamma_\Delta$.
\end{itemize}
A specification $\gamma$ is:
\begin{itemize}
\item[\rm{(i)}]
\embf{Continuous} if the functions
$\gamma_\Lambda\left(\omega_\Lambda \mid \cdot\,\right)$ are continuous
for each $\Lambda \in {\mathcal S}$ and $\omega_\Lambda\in\Omega_\Lambda$;
\item[\rm{(ii)}]
\embf{Non-null} if the functions
$\gamma_\Lambda\left(\omega_\Lambda \mid \cdot\,\right)$ are (strictly) positives
for each $\Lambda \in {\mathcal S}$ and $\omega_\Lambda\in\Omega_\Lambda$;
\item[\rm{(iii)}]
\embf{Gibbsian} if it is continuous and non-null;
\item[\rm{(iv)}]
\embf{Shift-invariant} if ${\mathbb L}={\mathbb Z}$ and $\tau \gamma_\Lambda=\gamma_{\Lambda+1}$, $\Lambda\in{\mathcal S}$.
\end{itemize}
\end{definition}
\bd{consist}
A probability measure $\mu$ on $(\Omega, \mathcal{F})$ is said to be \embf{consistent}
with a specification $\gamma$ if for each $\Lambda \in {\mathcal S}$,
\be{gibbs9}
\mu \,\gamma_{\Lambda}\;=\;\mu.
\end{equation}
The family of these measures will be denoted by ${\mathcal G}(\gamma)$.
A measure $\mu$ is a \embf{Gibbs measure} if there exists a Gibbsian specification $\gamma$
such that $\mu\in{\mathcal G}(\gamma)$.
\end{definition}

A celebrated theorem due to Kozlov~\cite{koz74} (see also~\cite{sul73} for an alternative
version in a different interaction space) shows that a specification $\gamma$ is Gibbsian if,
and only if, there exists a \embf{potential}, i.e.\ a family of functions
$\phi=(\phi_A)_{A\in{\mathcal S}}$ with each $\phi_A:\Omega\rightarrow{\mathbb R}$ being ${\mathcal F}_A$-measurable, that is
absolutely and uniformly summable in the sense
\be{abssum}
\sum_{A\ni i}\|\phi_A\|_\infty<\infty\,\,\forall\,i\in{\mathbb Z};
\end{equation}
such that $\gamma=\gamma^\phi$ where for all $\Lambda\in{\mathcal S}$ and $\omega,\sigma\in\Omega$
\be{gibbsspe}
\gamma_\Lambda^\phi(\sigma_\Lambda\mid\omega_{\Lambda^c})
= \frac{\exp\big[-H_{\Lambda,\omega}^\phi(\sigma)\big]}{Z_{\Lambda,\omega}^\phi}\;,
\end{equation}
 with
\be{gibbsspe*}
H_{\Lambda,\omega}^\phi(\sigma)
=\sum_{{A\in{\mathcal S}}\atop{A\cap\Lambda\neq\emptyset}}\phi_A(\sigma_\Lambda\,\omega_{\Lambda^{\rm c}})
\quad\text{and}\quad
Z_{\Lambda,\omega}^\phi
=\sum_{\sigma_\Lambda\in\Omega_\Lambda}\exp\Big[-H_{\Lambda,\omega}^\phi(\sigma)\Big].
\end{equation}
This is the original statistical mechanical prescription due to Boltzmann and Gibbs.

\section{Main Results}
\label{S2}

Let $\Omega={\mathcal A}^{{\mathbb L}}$ with ${\mathbb L}={\mathbb Z}^-:=\{\cdots,-2,-1,0\}$, be the configuration space on the half-line ending at $0$.

\subsection{Correspondence between regular LSS and Gibbsian specifications on half-spaces}
\label{S2.1}

Let
\be{1to1-sets}
\begin{aligned}
\Theta &= \Big\{\text{regular LSS on } \Omega\Big\}\\
\Pi &= \Big\{\text{Gibbsian specifications on } \Omega\Big\}\\
\end{aligned}
\end{equation}
and introduce the following maps
\be{1to1-maps}
b\colon\Theta\rightarrow\Pi,\,\, f\mapsto\gamma^f
\quad\text{and}\quad
c\colon\Pi\rightarrow\Theta,\,\, \gamma\mapsto f^\gamma
\end{equation}
defined by
\be{2.2.1}
\gamma^f_{[l,0]}\;=\; f_{[l,0]}\;;
\end{equation}
\be{1to1-maps*}
\gamma_\Lambda^f(\sigma_\Lambda\mid\omega_{\Lambda^{\rm c}})
=\frac{f_{[l_\Lambda,0]}\big(\sigma_\Lambda\,\omega_{[l_\Lambda,0]\cap\Lambda^{\rm c}}\mid \omega_{\Lambda_-}\big)}
{f_{[l_\Lambda,0]}\big(\omega_{[l_\Lambda,0]\cap\Lambda^{\rm c}}\mid \omega_{\Lambda_-}\big)}
\quad\text{with}
\qquad\omega\in\Omega\;,\,\Lambda\in{\mathcal S}\,\;,
\end{equation}
with $\Lambda$ strictly contained in $[l_\Lambda,0]$, and
\be{1to1-maps**}
f_i^\gamma\big(\omega_i\mid\omega_{-\infty}^{i-1}\big)
=\gamma_{[i,0]}\big(\omega_i\mid\omega_{-\infty}^{i-1}\big)
\qquad\forall\,i\in{\mathbb Z},\,\omega\in\Omega\;.
\end{equation}
We observe that, due to the consistency of $\gamma$,
\be{2.4.1}
f^\gamma_{[l,0]} \;=\; \gamma_{[l,0]}\;.
\end{equation}
\bt{th1}
The maps $b$ and $c$ establish a one-to-one correspondence between $\Theta$ and $\Pi$
that preserves consistency. More precisely,
\begin{itemize}
\item[{\rm 1)}]
\begin{itemize}
\item[{\rm (a)}] $f^\gamma\in\Theta$;
\item[{\rm (b)}] $\gamma^f\in\Pi$;
\end{itemize}
\item[{\rm 2)}]
\begin{itemize}
\item[{\rm (a)}] $b\circ c={\rm Id}_\Pi$;
\item[{\rm (b)}] $c\circ b={\rm Id}_\Theta$;
\item[{\rm (c)}] ${\mathcal G}\big(\gamma^f\big)={\mathcal G}(f)$, and
\item[{\rm (d)}] ${\mathcal G}\big(f^\gamma\big)={\mathcal G}(\gamma)$.
\end{itemize}
\end{itemize}
\end{theorem}

\bc{koz-chains}
For every regular LSS $f$ there exists an absolutely and uniformly summable potential
$\phi$ such that, for each $i\in{\mathbb L}$ and $\omega\in\Omega$:
\be{koz-f}
f_i(\omega_i\mid\omega_{-\infty}^{i-1}) \;=\;
\frac{\displaystyle \sum_{\sigma_{i+1}^0\in\Omega_{i+1}^0}
\exp\biggl[-\sum_{A\in{\mathcal S}: \atop A\cap[i,0]\neq\emptyset}
\phi_A\big(\sigma_{i+1}^{0}\,\omega_{-\infty}^{i}\big)\biggr]}
{\displaystyle \sum_{\sigma_{i}^0\in\Omega_{i}^0}
\exp\biggl[-\sum_{A\in{\mathcal S}: \atop A\cap[i,0]\neq\emptyset}
\phi_A\big(\sigma_{i}^{0}\,\omega_{-\infty}^{i-1}\big)\biggr]}
\end{equation}
($\sigma_1^0\equiv \emptyset$).
\end{corollary}

\subsection{Application: Long-range Ising ferromagnet chains}
\label{S2.2}

For the alphabet ${\mathcal A}=\{-1,1\}$, consider the \emph{long range Ising interaction potential} defined by
\be{pot}
\phi_A(\omega)=
\begin{cases}
-\beta\,J(i,j)\,\omega_i\,\omega_j &\text{if } A=\{i,j\},\, i\neq j,\\
0 &\text{otherwise,}
\end{cases}
\end{equation}
with $\sum_j\ |J(i,j)| < \infty$ for each
$i\in{\mathbb L}$ [c.f. \reff{abssum}].  The constants $J(i,j)$ are the \emph{couplings} and $\beta$ is an
overall factor interpreted as the \embf{inverse temperature} (high $\beta$ = low-temperature). The
potential is \embf{ferromagnetic} if $J(i,j)\ge 0$.  Such a potential defines a Gibbsian
\emph{Ising specification} $\gamma^\phi$ through the prescription \reff{gibbsspe}--\reff{gibbsspe*}
and a regular \emph{Ising LSS} $f^\phi$ through \reff{koz-f}.

\bt{ising-res}
Consider an Ising chain and let $J(r):=\sup \{ |J(i,j)|: |i-j|=r\}$.
\begin{itemize}
\item[\rm (a)]
If the chain is ferromagnetic, with decreasing couplings such that
\be{phtr-cond}
\sum_{r\ge 1} \frac{\log\log(r+4)}{r^3 J(r)} \; <\; \infty
\end{equation}
then there are multiple consistent measures at low temperatures and only one at high temperatures
\item[\rm (b)]
If
\be{uniq-cond}
\sum_{j\ge 1} \exp\Bigl[-C\sum_{r\ge 1} (j\wedge r)\, J(r)\Bigr] \;=\;\infty
\quad \forall C>0
\end{equation}
then there is a unique consistent chain at all temperatures.
\end{itemize}
\end{theorem}
\smallskip

\br{ruel-rem}
Kac and Thompson conjectured (in 1968) that a necessary and sufficient condition for absence of phase
transitions is
\be{kac-tho.for}
\sum_{r\ge 1} r\,J(r) \;<\;\infty
\end{equation}
Part (b) of the precedent theorem shows that the conjecture is false in the half line.  Consider for
instance $J(r)\propto (r^2\log r)^{-1}$.
\end{remark}

As an application, consider the  \embf{power-law Ising LSS}:
\be{pli}
J(i,j)\;=\; \frac{1}{|i-j|^p} \quad,\quad p>1\;,
\end{equation}
\bp{th3}
The power law Ising ferromagnet LSS $f$ is well defined if and only $p>1$.  Furthermore:
\begin{itemize}
\item[\rm (i)]
$|{\mathcal G}(f)|=1$ at high temperature, or at all temperatures if $p> 2$.
\item[\rm (ii)]
If $1<p<2$, then $|{\mathcal G}(f)|>1$ at low temperature.
\end{itemize}
\end{proposition}
The marginal case $p=2$ leads to a very special phase transition in the full line~\cite{frospe82},
and lies outside the scope of our analysis for the half line.

A second application, included for historical reasons, is the \embf{hierarchical Ising chain}.
Its specification version was introduced by Dyson~\cite{dys69a} as a tool for the study
of one-dimensional ferromagnetic Ising models. The hierarchical character of the model
allows for a number of explicit computations that
make its study easier. Moreover, phase transitions for the hierarchical and
the power-law Ising models are related.  The hierarchical chain is defined by considering blocks of
sizes $2^p$, $p\ge 1$ placed consecutively to the left of the origin.  Spins within the same $2^p$-block
interact through a coupling $2^{-2p+1}\,b_p$, for a suitable sequence of positive numbers $b_p$.  Thus,
\be{hier-int}
J(i,j)\;=\; \sum_{n\ge p(i,j)} \frac{b_q}{2^{2q-1}}\;,
\end{equation}
where $p(i,j)$ is the smallest $p$ such that $i$ and $j$ belongs to the same $2^p$ block.

Dyson's results can be transcribed in the following form.
Denote
\be{bpcond}
\Sigma(b)=\sum_{p\geq 1} 2^{-2p+1}\, b_p(2^p-1)
\quad\text{and}\quad
\Sigma^\star(b)=\sum_{p\geq 1}\big(\log(1+p)\big)b_p^{-1}\;.
\end{equation}
\bp{th2}
Assume that $\Sigma(b)<\infty$. Then, the LSS $f$ defined by \reff{koz-f}--\reff{pot} satisfy
\begin{itemize}
\item[{\rm (i)}] $|{\mathcal G}(f)|=1$ at all temperatures satisfying $\beta\Sigma(b)<1$ or
at all temperatures when $b_p$ are bounded.
\item[{\rm (ii)}] $|{\mathcal G}(f)|>1$ if $\beta>8 \Sigma^\star(b)$.
\item[{\rm (iii)}] For $b_p=2^{(2-\alpha)p}$ the LSS admits admits several
hierarchical chains at low temperature if and only if $1<\alpha<2$.
\end{itemize}
\end{proposition}

\subsection{The uniqueness issue}
\label{S2.3}
Published uniqueness criteria for chains refer to the following four different ways of measuring
the sensitivity of a LSS to changes in the past:
\begin{eqnarray}
{\rm var}_k(f_i)
&=&\sup_{\omega\,,\sigma}\Bigl | f_i\bigl(\omega_i\mid\omega_{\le i-1}\bigr)-
f_i\big(\omega_i\mid\omega_{k+1}^{i-1}\,\sigma_{\le k}\big)\Bigr |
\;;\label{var}\\
{\rm osc}_k(f_i)
&=&\sup_{\omega\,,\sigma}\Bigl | f_i\bigl(\omega_i\mid\omega_{\le i-1}\bigr)-
f_i\bigl(\omega_i\mid\omega_{k+1}^{i-1}\,\sigma_k\,
\omega_{\le k-1}\bigr)\Bigr |\;;\label{osc}\\
a_k(f_i)&=&\inf_{\sigma} \sum_{\xi_i\in\Omega_i}\inf_{\omega}
f_i\bigl(\xi_i\mid\sigma_k^{i-1}\,\omega_{\le k-1}\,\bigr)\;;\label{const-a}\\
b_k(f_i)&=&\inf_{\omega,\sigma} \sum_{\omega_i\in{\mathcal A}}
f_i\bigl(\omega_i\mid\omega_{\le i-1}\big)\wedge f_i\big(\omega_i\mid
\omega_{k}^{i-1}\,\sigma_{\le k-1}\bigr) \;.
\label{const-b}
\end{eqnarray}
The first and second quantity are called, respectively, the \embf{$k$-variation} and the
\embf{$k$-oscillation} of the kernel $f_i$.  They are related by the obvious inequalities
\be{varoscine}
{\rm osc}_k(f_i) \;\leq\; {\rm var}_k(f_i)
\;\leq\; \sum_{j<k}{\rm osc}_j(f_i)\;.
\end{equation}
Let us summarize further relations valid for the Ising  case.
\bp{rel-IC}
\begin{itemize}
\item[\rm (a)] If $|{\mathcal A}|=2$, then for each $k<i\in{\mathbb L}$
\be{a=b}
a_k(f_i)\;=\;b_k(f_i)\;=\;1-{\rm var}_k(f_i)\;.
\end{equation}
\item[\rm (b)] If $\{f_i\}$ is an Ising LSS, then for each $k<i\in{\mathbb L}$:
\be{prop-3-0}
{\rm osc}_k(f_{i})\leq \beta \sum_{j=i}^0 |J(j,k)|
\quad,\quad
{\rm var}_k(f_{i})\leq \beta \sum_{j=i}^0 \sum_{\ell\le k+1}|J(j,\ell)|\;.
\end{equation}
In particular, if $|J(i,j)|\geq |J(k,l)|$ as soon as $|i-j|\leq |k-l|$,
\be{prop-3}
{\rm osc}_k(f_{i})\leq \beta \,(i+1)\, |J(i,k)|
\quad,\quad
{\rm var}_k(f_{i})\leq \beta \,(i+1) \sum_{j\leq k+1}|J(i,j)|\;.
\end{equation}
\end{itemize}
\end{proposition}

The following are the only chain uniqueness criteria proven without a shift-invariance hypothesis.
\bp{unicLSS}
A continuous LSS $f$ admits exactly one consistent chain if it satisfies one of the following assertions:
\begin{itemize}
\item[\rm{(a)}] \emph{CFF \cite{comferfer02}:} $f$ non-null
and
\be{comferfer}
\sum_{j< i}\prod_{k=j}^{i-1} a_k(f_i)=\infty\;,\quad \forall\,i\in{\mathbb L}\;;
\end{equation}
\item[\rm{(b)}] \emph{One-sided boundary-uniformity \cite{fermai05}:}
There exists $C>0$ satisfying:  For every $m\in{\mathbb Z}$ and every cylinder set $A\in{\mathcal F}_{\leq m}$
there exists $n<m$ such that $\displaystyle f_{[n,m]}(A\mid\xi)\geq C f_{[n,m]}(A\mid \eta)$
for all $\xi,\eta\in\Omega$.  In particular, this condition is satisfied if
\be{osc-nonshift}
\sum_{j< i} {\rm var}_j(f_i) \;<\; \infty\;,\quad \forall\,i\in{\mathbb L}\;.
\end{equation}
\end{itemize}
\end{proposition}

For comparison purposes, let us list the uniqueness criteria proven for shift-invariant LSS.
\bp{unicgfunct}
A continuous $g$-function $f_0$ admits exactly one consistent chain if it satisfies one of the following criteria:
\begin{itemize}
\item[\rm{(a)}] \emph{Harris \cite{har55,sten03}:} $f_0$ non-null and
$\displaystyle\sum_{j<0}\prod_{k=j}^{-1}\bigg(1-\frac{|{\mathcal A}|}{2}{\rm var}_k(f_0)\bigg)=\infty$.
\item[\rm{(b)}] \emph{Stenflo \cite{sten03}:} $f_0$ non-null and
$\displaystyle\sum_{j< 0}\prod_{k=j}^{-1} b_k(f_0)=\infty$.
\item[\rm{(c)}] \emph{Johansson-\"Oberg \cite{johobe03}:}
$f_0$ non-null and
$\displaystyle\sum_{j< 0}{\rm var}_j^2(f_0)<\infty$;
\item[\rm{(d)}] \emph{One-sided Dobrushin \cite{fermai05}:}
$\displaystyle \sum_{j< 0}{\rm osc}_j(f_0)\;<\;1$.
\end{itemize}
\end{proposition}

It is natural to ask whether these criteria admit a non-shift-invariant version, in which
each $f_0$-condition is replaced by a similar $f_i$-condition valid for all $i\in{\mathbb L}$,
\emph{without asking uniformity with respect to $i$} (that is, without imposing conditions
on $\sup_i f_i$).  For our Ising  examples, (a) of Proposition~\ref{rel-IC} shows that the
non-shift invariant versions of the first two criteria in Proposition~\ref{unicgfunct} coincide
with the CFF criteria.   We are unable to test the corresponding version for one-sided
Dobrushin due to the factor $i+1$ in the leftmost bound in~\reff{prop-3} (though the ``$\sup f_i$" version
of the specification Dobrushin criterium can be applied and yields uniqueness at high
temperature).  On the other hand, the rightmost bound in~\reff{prop-3} allows us to conclude about
the remaining criterion.
\br{johobe-rk}
The power law Ising ferromagnetic LSS $f$ with $3/2<p<2$ show that
the non-shift-invariant version of the Johansson-\"Oberg criterion is false
in general.  Indeed, these LSS do exhibit phase transitions, by Theorem \ref{th3},
but
$\sum_{j\leq i} {\rm var}_k^2(f_i)<\infty$ for all $ i\in{\mathbb Z}$.
\end{remark}

\section{Proofs}
\label{S3}

\subsection{Proof of Theorem \ref{th1} and Corollary \ref{koz-chains}}
\label{S3.1}

Corollary \ref{koz-chains} is a direct consequence of Theorem \ref{th1} and Kozlov theorem \cite{koz74}.
The proof Theorem \ref{th1} runs as follows.
\smallskip

1)(a) This is a direct consequence of (\ref{1to1-maps**}).
\smallskip

1)(b) Observe that the Gibbsianness of $\gamma^f$ follows directly from the
definition (\ref{1to1-maps*}). Moreover, $\gamma_{\Lambda}^{f}(A\mid \cdot\,)$
is clearly $\mathcal{F}_{\Lambda^{c}}$-measurable for every $\Lambda\in\mathcal{S}$
and every $A \in \mathcal{F}$. Condition (b) of Definition \ref{lis1} together
with the presence of the indicator function $\rlap{\mbox{\small\rm 1}}\kern.15em 1\{\omega_{[l_\Lambda,0]\cap\Lambda^{\rm c}}\}$
in the numerator of \reff{1to1-maps*} ensure that $\gamma_{\Lambda}^{f}(B \mid \cdot \, )
= \rlap{\mbox{\small\rm 1}}\kern.15em 1_B(\, \cdot\,)$ for every $\Lambda \in \mathcal{S}$ and every $B \in
\mathcal{F}_{\Lambda^{c}}$. To conclude the proof that $\gamma^f$ is a Gibbsian specification,
it suffices to show that
\be{th1*-3}
\sum_{\omega_{\Delta \setminus
\Lambda}}\gamma_{\Lambda}^{f}\left( \omega_{\Lambda} \mid
\omega_{\Lambda^{c}} \right) \gamma_{\Delta}^{f}\left( \omega_{\Delta
\setminus \Lambda} \mid \omega_{\Delta^{c}} \right) \;=\;
\gamma_{\Delta}^{f}\left( \omega_{\Lambda} \mid
\omega_{\Delta^{c}}\right)
\end{equation}
for each $\Lambda, \Delta \in \mathcal{S}$ such that $\Lambda \subset
\Delta$ and each $\omega \in \Omega$. Define
$G_\Lambda\colon{\mathcal F}\rightarrow[0,1]$, $\Lambda\in{\mathcal S}$, by
\be{Gfunction}
G_\Lambda\big(\,\cdot\mid\omega_{\Lambda^{{\rm c}}}\big)
=f_{[l_\Lambda,0]}\Big(\,\cdot\,\rlap{\mbox{\small\rm 1}}\kern.15em 1_{\omega_{[l_\Lambda,0]\cap\Lambda^{{\rm c}}}}
\mid\omega_{\Lambda_{-}}\Big).
\end{equation}
By \reff{1to1-maps*}
\be{th1*-5}
\gamma_{\Lambda}^f\big(\omega_\Lambda \mid \omega_{\Lambda^{c}}\big)
\;=\;\frac{G_{\Lambda}\left( \rlap{\mbox{\small\rm 1}}\kern.15em 1_{\omega_{\Lambda}} \mid
\omega_{\Lambda^{c}}\right)}{G_{\Lambda}\left( \Omega_\Lambda \mid
\omega_{\Lambda^{c}}\right)}\;.
\end{equation}
Using that $f_{[l,n]}=f_{[l,m]}f_{[m+1,n]}$ for all $l\leq m<n\leq 0$,
we obtain
\be{th1*-7}
G_{\Delta}\big(\rlap{\mbox{\small\rm 1}}\kern.15em 1_{\omega_{\Delta \setminus \Lambda}}\mid\omega_{\Delta^{c}}\big)
\;f\;_{[l_\Delta,l_{\Lambda}-1]}\Big(\rlap{\mbox{\small\rm 1}}\kern.15em 1\big\{\omega_{l_\Delta}^{l_\Lambda-1}\big\}
\mid\omega_{\Delta_{-}}\Big)
\times G_{\Lambda}\big(\Omega_\Lambda\mid\omega_{\Lambda^{c}}\big)
\end{equation}
and
\be{th1*-9}
G_{\Delta}\big(\rlap{\mbox{\small\rm 1}}\kern.15em 1_{\omega_{\Delta}}\mid\omega_{\Delta^{c}}\big)\;=\;
f_{[l_\Delta,l_{\Lambda}-1]}\Big(\rlap{\mbox{\small\rm 1}}\kern.15em 1\big\{\omega_{l_\Delta}^{l_\Lambda-1}\big\}
\mid\omega_{\Delta_{-}}\Big) \times
G_{\Lambda}\big(\rlap{\mbox{\small\rm 1}}\kern.15em 1_{\omega_{\Lambda}}\mid\omega_{\Lambda^{c}}\big)
\;.
\end{equation}
Therefore
\be{th1*-11}
\frac{G_{\Lambda}\big(\rlap{\mbox{\small\rm 1}}\kern.15em 1_{\omega_{\Lambda}}\mid \omega_{\Lambda^{c}}\big)}
{G_{\Lambda}\big(\Omega_\Lambda \mid \omega_{\Lambda^{c}}\big)}
\times \frac{G_{\Delta}\big(\rlap{\mbox{\small\rm 1}}\kern.15em 1_{\omega_{\Delta \setminus
\Lambda}} \mid \omega_{\Delta^{c}}\big)}{G_{\Delta}\big(\Omega_\Delta \mid
\omega_{\Delta^{c}}\big)}
=\frac{G_{\Delta}\big(\rlap{\mbox{\small\rm 1}}\kern.15em 1_{\omega_{\Delta}}\mid\omega_{\Delta^{c}}\big)}
{G_{\Delta}\big(\Omega_\Delta\mid \omega_{\Delta^{c}}\big)}.
\end{equation}
Identity \reff{th1*-3} follows from \reff{th1*-5} and \reff{th1*-11}.
\smallskip

2)(a) For all $\Lambda\in{\mathcal S}$ and $\omega\in\Omega$
\be{th1-5}
\gamma_\Lambda^{f^\gamma}\big(\omega_\Lambda\mid\omega_{\Lambda^{\rm c}}\big)
\;=\;\frac{\displaystyle\gamma_{[l_\Lambda,0]}\big(\omega_{l_\Lambda}^0
\mid\omega_{\Lambda_{-}}\big)}
{\displaystyle\sum_{\sigma_\Lambda\in\Omega_\Lambda}
\gamma_{[l_\Lambda,0]}\big(\sigma_\Lambda\omega_{[l_\Lambda,0]\cap\Lambda^{\rm c}}
\mid\omega_{\Lambda_{-}}\big)}\;.
\end{equation}
By the consistency of $\gamma$ [Definition \ref{lis1} (c)]
\be{th1-7}
\gamma_{[l_\Lambda,0]}\big(\omega_{l_\Lambda}^0\mid\omega_{\Lambda_{-}}\big)
\;=\;\gamma_\Lambda\big(\omega_\Lambda\mid\omega_{\Lambda^{\rm c}}\big)
\sum_{\sigma_\Lambda\in\Omega_\Lambda}
\gamma_{[l_\Lambda,0]}\big(\sigma_\Lambda\omega_{[l_\Lambda,0]\cap\Lambda^{\rm c}}
\mid\omega_{\Lambda_{-}}\big)\;.
\end{equation}
From \reff{th1-5} and \reff{th1-7} we obtain that
$\gamma_\Lambda^{f^\gamma}\big(\omega_\Lambda\mid\omega_{\Lambda^{\rm c}}\big)
=\gamma_\Lambda\big(\omega_\Lambda\mid\omega_{\Lambda^{\rm c}}\big)$.
\smallskip

2)(b) For all $\Lambda\in{\mathcal S}_b$ and $\omega\in\Omega$
\be{th1-9}
f_\Lambda^{\gamma^f}\big(\omega_\Lambda\mid\omega_{\Lambda_{-}}\big)
\;=\;\gamma_{[l_\Lambda,0]}^f\big(\omega_\Lambda\mid\omega_{\Lambda_{-}}\big)
\;=\;\frac{f_{[l_\Lambda,0]}\big(\omega_\Lambda\mid\omega_{\Lambda_{-}}\big)}
{f_{[l_\Lambda,0]}\big(\Omega_{l_\Lambda}^0\mid\omega_{\Lambda_{-}}\big)}
\;=\;f_\Lambda\big(\omega_\Lambda\mid\omega_{\Lambda_{-}}\big).\;
\end{equation}
\smallskip

2)(c) We use \reff{2.4.1} and consistency to obtain the following two strings of inequalities.
If $\mu\in{\mathcal G}(\gamma)$ and $\Lambda\in{\mathcal S}_b$,
\be{th1-11}
\mu \,f_\Lambda^\gamma
\;=\;\mu \,\gamma_{[l_\Lambda,0]}
\;=\;\mu\;.
\end{equation}
If $\mu\in{\mathcal G}\big(f^\gamma\big)$ and $\Lambda\in{\mathcal S}$,
\be{th1-13}
\mu\,\gamma_\Lambda
=\bigl(\mu \,f_{[l_\Lambda,0]}^{\gamma}\bigr)\gamma_\Lambda
=\mu \bigl(\gamma_{[l_\Lambda,0]}\,\gamma_\Lambda\bigr)
=\mu \,\gamma_{[l_\Lambda,0]}
=\mu \,f_{[l_\Lambda,0]}^{\gamma}
=\mu\;.
\end{equation}
Together, \reff{th1-11} and \reff{th1-13} show that ${\mathcal G}\big(f^\gamma\big)={\mathcal G}(\gamma)$.
\smallskip

2)(d) The following two displays are a consequence of \reff{2.2.1} and consistency.
For any $\mu\in{\mathcal G}(f)$ and $\Lambda\in{\mathcal S}$,
\be{th1-15}
\mu\,\gamma_\Lambda^f
\;=\;\bigl(\mu\, f_{[l_\Lambda,0]}\bigr)\gamma_\Lambda^f
\;=\;\mu\bigl (\gamma_{[l_\Lambda,0]}^f\,\gamma_\Lambda^f\bigr)
\;=\; \mu\,\gamma_{[l_\Lambda,0]}^f
\;=\; \mu\,f_{[l_\Lambda,0]}
\;=\;\mu\;.
\end{equation}
For any $\mu\in{\mathcal G}\big(\gamma^f\big)$ and $\Lambda\in{\mathcal S}_b$,
\be{th1-17}
\mu\, f_\Lambda
\;=\;\Big(\mu\,\gamma_{[l_\Lambda,0]}^f\Big)f_\Lambda
\;=\;\mu\bigl(f_{[l_\Lambda,0]}\,f_\Lambda\bigr)
\;=\;\mu\, f_{[l_\Lambda,0]}
\;=\;\mu\,\gamma_{[l_\Lambda,0]}^f
\;=\;\mu\;.
\end{equation}
The combination of both lines shows that ${\mathcal G}\big(\gamma^f\big)={\mathcal G}(f)$.
{\hspace*{\fill}$\halmos$\medskip}

\subsection{Proof of Proposition \ref{rel-IC}}
\label{S3.1.1}

We need the following well known bound, whose proof we present for completeness
(we follow the approach of~\cite[Lemma V.1.4]{sim93}).
\bl{simlem}
Let $\gamma^\varphi$ be a Gibbsian specification for some absolutely summable
potential $\varphi$. Then, for any $\Lambda\in{\mathcal S}$, $h\in{\mathcal F}_\Lambda$ such that
$\|h\|_\infty<\infty$ and $\omega,\sigma\in\Omega$,
\be{simlem-3}
\bigg|\int_{\Omega_\Lambda}h(\xi)\Big(
\gamma_\Lambda^\varphi\big(d\xi\mid\omega_{\Lambda^{\rm c}}\big)
-\gamma_\Lambda^\varphi\big(d\xi\mid\sigma_{\Lambda^{\rm c}}\big)\Big)\bigg|
\leq \|h\|_\infty \sup_{\xi\in\Omega_\Lambda}
\Big|H_{\Lambda,\omega}^\varphi(\xi)
-H_{\Lambda,\sigma}^\varphi(\xi)\Big|,
\end{equation}
where $H^\varphi$ is the Hamiltonian associated to $\varphi$.
\end{lemma}

\begin{proof}
For all $\Lambda\in{\mathcal S}$, $\omega,\sigma,\xi\in\Omega$ and $0<\theta<1$, define
$\Gamma_{\Lambda,\omega,\sigma}^{\varphi,\theta}\colon\Omega_\Lambda\rightarrow(0,1)$ by
\be{simlem-11}
\Gamma_{\Lambda,\omega,\sigma}^{\varphi,\theta}(\xi)
=\frac{\displaystyle\exp\Big[\theta H_{\Lambda,\omega}^{\varphi}(\xi)
+(1-\theta)H_{\Lambda,\sigma}^{\varphi}(\xi)\Big]}
{\displaystyle\sum_{\eta\in\Omega_\Lambda}
\exp\Big[\theta H_{\Lambda,\omega}^{\varphi}(\eta)
+(1-\theta)H_{\Lambda,\sigma}^{\varphi}(\eta)\Big]}.
\end{equation}
Then, to prove (\ref{simlem-3}), it suffices to see that
\be{simlem-13}
\begin{aligned}
&\bigg|\int_{\Omega_\Lambda}h(\xi)\Big(
\gamma_\Lambda^\varphi\big(d\xi\mid\omega_{\Lambda^{\rm c}}\big)
-\gamma_\Lambda^\varphi\big(d\xi\mid\sigma_{\Lambda^{\rm c}}\big)\Big)\bigg|\\
&\quad\leq \int_{0}^{1}\bigg|\frac{d}{d\theta}\bigg[\int_{\Omega_\Lambda} h\,
d\Gamma_{\Lambda,\omega,\sigma}^{\varphi,\theta}\bigg]\bigg|\, d\theta\\
&\quad=\int_{0}^{1}\bigg|\int_{\Omega_\Lambda} h\Big(H_{\Lambda,\omega}^{\varphi}
-H_{\Lambda,\sigma}^{\varphi}\Big) d\Gamma_{\Lambda,\omega,\sigma}^{\varphi,\theta}
-\int_{\Omega_\Lambda} h\, d\Gamma_{\Lambda,\omega,\sigma}^{\varphi,\theta}
\int_{\Omega_\Lambda} \Big(H_{\Lambda,\omega}^{\varphi}
-H_{\Lambda,\sigma}^{\varphi}\Big) d\Gamma_{\Lambda,\omega,\sigma}^{\varphi,\theta}
\bigg|\, d\theta\\
&\quad\leq \|h\|_\infty\int_{0}^{1}\int_{\Omega_\Lambda} \bigg|\Big(H_{\Lambda,\omega}^{\varphi}
-H_{\Lambda,\sigma}^{\varphi}\Big) -\int_{\Omega_\Lambda} \Big(H_{\Lambda,\omega}^{\varphi}
-H_{\Lambda,\sigma}^{\varphi}\Big)d\Gamma_{\Lambda,\omega,\sigma}^{\varphi,\theta} \bigg|\,
d\Gamma_{\Lambda,\omega,\sigma}^{\varphi,\theta}\, d\theta\\
&\quad\leq \|h\|_\infty\sup_{\xi_\Lambda\in\Omega_\Lambda}\big|H_{\Lambda,\omega}^{\varphi}
-H_{\Lambda,\sigma}^{\varphi}\big|.
\end{aligned}
\end{equation}
\end{proof}

\noindent
\emph{Proof of Proposition \ref{rel-IC}}.
\smallskip

(a) For any $k<i$,
\be{th3-60}
\begin{aligned}
a_k(f_i)
&=\inf_{\sigma,\omega,\xi}\Bigl[
f_i(1\mid\sigma_k^{i-1}\,\omega_{\le k-1})
+f_i(-1\mid\sigma_k^{i-1}\,\xi_{\le k-1})\Bigr]\\
&=1-\sup_{\sigma,\omega,\xi}\Bigl[
-f_i(1\mid\sigma_k^{i-1}\,\omega_{\le k-1})
+f_i(1\mid\sigma_k^{i-1}\,\xi_{\le k-1})\Bigr]\\
&=1-{\rm var}_k(f_i).
\end{aligned}
\end{equation}
Likewise,
\begin{eqnarray}
\label{th3-60.1}
b_k(f_i)&=&\inf_{\sigma,\omega,\xi}\Bigl[
f_i(1\mid\sigma_k^{i-1}\,\omega_{\le k-1})\wedge
f_i(1\mid\sigma_k^{i-1}\,\xi_{\le k-1})\nonumber\\
&&\qquad {}+\bigl[1-f_i(1\mid\sigma_k^{i-1}\,\omega_{\le k-1})\bigr] \wedge
\bigl[1 - f_i(1\mid\sigma_k^{i-1}\,\xi_{\le k-1})\bigr]
\Bigr]\nonumber\\
&=&1-\sup_{\sigma,\omega,\xi}\Bigl[
f_i(1\mid\sigma_k^{i-1}\,\omega_{\le k-1})\vee
f_i(1\mid\sigma_k^{i-1}\,\xi_{\le k-1})\\
&&\qquad\qquad {}- f_i(1\mid\sigma_k^{i-1}\,\omega_{\le k-1})\wedge
f_i(1\mid\sigma_k^{i-1}\,\xi_{\le k-1})\Bigr]\nonumber\\
&=&1-{\rm var}_k(f_i)\;.\nonumber
\end{eqnarray}
\smallskip

(b) Applying the previous lemma for $\Lambda=[i,0]$ and $h(\xi)=1$ if $\xi_i=\omega_i$
and 0 otherwise, we obtain that, for any $\omega_{\le i-1}, \sigma_{\le i-1}\in \Omega_{\le i-1}$,
\be{simlem-5}
\begin{aligned}
\Bigl|f_i\bigl(\omega_i\mid\omega_{\le i-1}\bigr)
-f_i\bigl(\omega_i\mid\sigma_{\le i-1}\bigr)\Bigr|
\;&=\; \Bigl|\gamma^\phi_{[i,0]}(h\mid\omega_{\le i-1}) -
\gamma^\phi_{[i,0]}(h\mid\sigma_{\le i-1})\Bigr|\\
\;&\le\;
\sup_{\xi}\Bigl|H_{[i,0]}^\phi\bigl(\xi_{i}^{0}\mid\omega_{\le i-1}\big)
-H_{[i,0]}^\phi\big(\xi_{i}^{0}\mid\sigma_{\le i-1}\bigr)\Bigr|\;.
\end{aligned}
\end{equation}
Both inequalities in \reff{prop-3-0} are an immediate consequence.  {\hspace*{\fill}$\halmos$\medskip}

\subsection{Proof of Theorem \ref{ising-res}}
\label{S3.2}

(a) This is a direct consequence of Theorem \ref{th1} and Theorem 1 in \cite{dys69a}.
\smallskip

(b) We show the validity of the CFF condition [Proposition \ref{unicLSS}(a)].
Pick an  $\alpha>1$.  If $x$ is small enough, $1-x\ge \exp(-\alpha\,x)$.  Hence,
there exists $j_0<i$ small enough and $K>0$ such that
\be{th3-61}
\sum_{j<i}\prod_{k=j}^{i-1}a_k(f_i)
\;\geq\; K\sum_{j\leq j_0}\exp\Bigg[-\alpha\sum_{k=j}^{j_0}{\rm var}_k(f_i)\Bigg].
\end{equation}
Then, by (\ref{uniq-cond}) and (\ref{prop-3}),
\begin{eqnarray}
\label{th3-63}
\sum_{j<i}\prod_{k=j}^{i-1}a_k(f_i)
&\geq&K\sum_{j\leq j_0}\exp\Bigg[-\alpha\beta K(i)\sum_{k=j}^{j_0}\, \sum_{r\geq |k|}J(r)\Bigg]
\nonumber\\
&\geq& K\sum_{j\geq|j_0|}\exp\Bigg[-\alpha\beta K(i)\sum_{r\geq |j_0|}(j\wedge r)J(r)\Bigg]\\
&=&\infty
\end{eqnarray}
by hypothesis. {\hspace*{\fill}$\halmos$\medskip}

redpajama_set_name: RedPajamaArXiv


text:
Michigan: NCAA looking into potential football…
Michigan: NCAA looking into potential football infractions
Michigan head coach Jim Harbaugh watches his players prior to the Fiesta Bowl NCAA college football semifinal playoff game against TCU, Saturday, Dec. 31, 2022, in Glendale, Ariz. (AP Photo/Rick Scuteri)
PUBLISHED: January 6, 2023 at 2:41 p.m. | UPDATED: January 6, 2023 at 2:42 p.m.
By LARRY LAGE
ANN ARBOR, Mich. (AP) — Michigan announced Friday that it has received notice from the NCAA that it is looking into potential rules infractions in the Wolverines football program.
Harbaugh: "I expect" to be back at Michigan next season
Panthers owner, Jim Harbaugh discuss coaching job
No. 3 TCU upsets No. 2 Michigan 51-45 in wild CFP semifinal
Athletic director Warde Manuel said the "draft allegations" were received Thursday.
"We have cooperated and will continue to cooperate with this investigation," Manuel said. "Out of respect to the NCAA's enforcement process, we will not offer further comments."
The investigation is believed to involve impermissible texts and calls — including some by coach Jim Harbaugh — to high school prospects during part of a pandemic-related dead period for contact with potential recruits, a person with knowledge of the situation told The Associated Press on Thursday.
The NCAA was also looking at whether a member of Michigan's off-field football staff violated rules by doing on-the-field coaching during practice, the person said, speaking on condition of anonymity because Michigan officials were not making details of the investigation public.
FILE -Michigan coach Jim Harbaugh watches during an NCAA college football game against Michigan State, Saturday, Oct. 30, 2021, in East Lansing, Mich. According to a person with knowledge of the process, New York Giants defensive coordinator Patrick Graham will interview at Vikings headquarters on Tuesday followed by Michigan head coach Jim Harbaugh on Wednesday, Feb. 2, 2022 . (AP Photo/Al Goldis, File)
The announcement came a day after Harbaugh said he expects to coach at Michigan next season, posting a statement on the team's social media accounts following reports of his interest in returning to the NFL.
"I am aware of the rumors and speculation over the past few days," Harbaugh said. "College and NFL teams have great interest in all our personnel, from players to coaches to staff, and I truly believe that is a testament to the strength of our University of Michigan football program."
Carolina Panthers owner David Tepper had a conversation with Harbaugh about the team's head coaching position, according to a second person familiar with the situation who spoke to the AP on condition of anonymity Tuesday because the Panthers do not typically release details of their coaching search.
Pickett, Penn State blister Michigan 83-61
Edey scores 38 as No. 1 Purdue overwhelms Spartans 77-61
Michigan State edges Iowa 63-61 with late buckets
College Notebook: MSU gymnast, OU freshman earn weekly honors
Harbaugh also has been mentioned as a possible candidate to coach in Denver and Indianapolis, one of four teams he played quarterback for in the league from 1987 to 2000.
"As I stated in December, while no one knows what the future holds, I expect that I will be enthusiastically coaching Michigan in 2023," Harbaugh said. "I have spoken with president Santa Ono and athletic director Warde Manuel and appreciate their support of me and our program."
Harbaugh coached the San Francisco 49ers from 2011-2014, winning two NFC West titles and reaching the NFC championship game three times in four seasons. He was 44-19-1 during that span and helped the 49ers get to the Super Bowl a decade ago, where they lost 34-31 to the Baltimore Ravens, led by his brother, John.
Jim Harbaugh left the 49ers after going 8-8 in 2014 — the only season the 49ers didn't make the postseason under his direction — to take the job at Michigan.
The 59-year-old Harbaugh has a 74-25 record in eight seasons with the Wolverines, who were 13-0 this season before getting beat by TCU in a College Football Playoffs semifinal. Harbaugh also spent four seasons as head coach of Stanford and three more at San Diego.
Harbaugh was the AP NFL Coach of the Year in 2011 and AP College Coach of the Year in 2021.
AP College Football Writer Ralph D. Russo in New York and AP Sports Writer Steve Reed in Charlotte, North Carolina, contributed.
AP college football: https://apnews.com/hub/college-football and https://twitter.com/ap_top25 Sign up for the AP Top 25 newsletter here: https://link.apnews.com/join/6nr/morning-wire-newsletter-footer-internal-ads
More in College Sports


Mr. Anandraj Govindaraj is a lecturer at Sunway College KL. He graduated with a Bachelor of Engineering (Hons.) majoring in Telecommunication in 2007 and Master of Applied Psychology in Coaching in 2015. He began teaching Physics for Australian Matriculation Programme at Sunway College since 2012 and Engineering Studies since 2018.
Being a professional certified coach by ICF-ACTP and expert in education technology, he strives to engage and support digital transitions within the programme and college. He continues to explore new technological tools and provide professional development trainings to equip educators with technology literacy. He also empowers educator to enrich the teaching and learning experiences in classroom, though integration of technology.


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Real Living is proud to announce that an astonishing 18 Real Living network brokerages won a 2018 QE Award – more than any other real estate brand this year. The QE Award recognizes America's Top 35 real estate firms based on service excellence.
Sine & Monaghan Real Living, Mich.
Real Living Kee Realty, Rochester, Mich.
Real Living Realty Professionals, Mass.
BOWES Real Estate Real Living, Arlington, Mass.
Real Living Pittman Properties, Raleigh, N.C.
Real Living Property For You, Peoria, Ariz.
Real Living Capital City, Atlanta, Ga.
Real Living 1st Choice Realty, Coral Springs, Fla.
Real Living Real Estate Experts, Warren, Pa.
Real Living Georgia Life Realty, Villa Rica, Ga.
Real Living Casa Fina Realty, Tampa Fla.
Real Living Cornerstone, Farmville, Va.
Real Living Gateway Real Estate, Hasbrouck Heights, N.J.
Real Living Now Real Estate, St. Louis Metro Area, Mo.
Real Living Real Estate Professionals, Redding, Calif.
Real Living Northwest REALTORS®, Bothell, Wash.
Real Living Palm West Home Realty, Inc., Palm Coast, Fla.
According to QSC, the QE Award is the combined overall Customer Satisfaction Rating of all returned surveys of real customers, where every past customer has been surveyed without selectivity, editing, deletion, cleansing or manipulation.


text:
require 'optparse'
require_relative 'octokit_utils'

options = {}
options[:oauth] = ENV['GITHUB_COMMUNITY_TOKEN'] if ENV['GITHUB_COMMUNITY_TOKEN']
parser = OptionParser.new do |opts|
  opts.banner = 'Usage: release_planning.rb [options]'

  opts.on('-c', '--commit-threshold NUM', 'Number of commits since release') { |v| options[:commits] = v.to_i }
  opts.on('-g', '--tag-regex REGEX', 'Tag regex') { |v| options[:tag_regex] = v }
  opts.on('-m', '--time-threshold DAYS', 'Days since release') { |v| options[:time] = v.to_i }
  opts.on('-n', '--namespace NAME', 'GitHub namespace. Required.') { |v| options[:namespace] = v }
  opts.on('-r', '--repo-regex REGEX', 'Repository regex') { |v| options[:repo_regex] = v }
  opts.on('-t', '--oauth-token TOKEN', 'OAuth token. Required.') { |v| options[:oauth] = v }
  opts.on('-v', '--verbose', 'More output') { options[:verbose] = true }
end

parser.parse!

missing = []
missing << '-n' if options[:namespace].nil?
missing << '-t' if options[:oauth].nil?
missing << '-m or -c' if options[:time].nil? and options[:commits].nil?
if not missing.empty?
  puts "Missing options: #{missing.join(', ')}"
  puts parser
  exit
end

options[:repo_regex] = '.*' if options[:repo_regex].nil?
options[:tag_regex] = '.*' if options[:tag_regex].nil?

util = OctokitUtils.new(options[:oauth])
repos = util.list_repos(options[:namespace], options)

repo_data = []

repos.each do |repo|
  begin
    latest_tag = util.fetch_tags("#{options[:namespace]}/#{repo}", options).first
    tag_ref = util.ref_from_tag(latest_tag)
    date_of_tag = util.date_of_ref("#{options[:namespace]}/#{repo}", tag_ref)
    commits_since_tag = util.commits_since_date("#{options[:namespace]}/#{repo}", date_of_tag)

    repo_data << { 'repo' => "#{options[:namespace]}/#{repo}", 'date' => date_of_tag, 'commits' => commits_since_tag }
  rescue
    puts "Unable to fetch tags for #{options[:namespace]}/#{repo}" if options[:verbose]
  end
end

if options[:commits]
  due_by_commit = repo_data.select { |x| x['commits'] > options[:commits] }
end

if options[:time]
  threshold = Time.now - options[:time]
  due_by_time = repo_data.select { |x| x['date'] < threshold }
end

if due_by_commit and due_by_time
  due_for_release = due_by_commit & due_by_time
elsif due_by_commit
  due_for_release = due_by_commit
else
  due_for_release = due_by_time
end

due_for_release.each do |entry|
  puts "#{entry['repo']} is due for release. Last release was tagged on #{entry['date']} and there have been #{entry['commits']} commits since then."
end

redpajama_set_name: RedPajamaGithub


Max: The Curse of Brotherhood is rated 4.0 out of 5 by 32.
When Max wishes for his annoying little brother to be whisked away he gets more than he bargained for... Armed with only his trusty Magic Marker, Max must journey to a hostile and unforgiving world to rescue his kidnapped kid brother, Felix.
Draw your way through lantern-lit bogs, ancient temples and lush-green-forests, as you take on Mustacho's henchmen. Use the marker to overwhelm your enemies, define new pathways and protect you on your quest.


text:
Q: from 9 sets of 12 elements, arrange 12 groups of the 9 elements, selecting 1 element from each set I'm trying to take 9 sets of 12 elements in each set and selecting 1 element from each set to create 12 unique groups.  Now do this 9 times without pairing any 2 elements more than once. The first 12 groups could  be:

*

*S1E1   S2E1   S3E1   S4E1   S5E1   S6E1   S7E1   S8E1   S9E1

*S1E2   S2E2   S3E2   S4E2   S5E2   S6E2   S7E2   S8E2   S9E2

*S1E3   S2E3   S3E3   S4E3   S5E3   S6E3   S7E3   S8E3   S9E3

*S1E4   S2E4   S3E4   S4E4   S5E4   S6E4   S7E4   S8E4   S9E4
.....

*S1E12  S2E12  S3E12  S4E12  S5E12  S6E12  S7E12  S8E12  S9E12

Second group not too difficult:

*

*S1E1   S2E2   S3E3   S4E4   S5E5   S6E6   S7E7   S8E8   S9E9

*S1E2   S2E3   S3E4   S4E5   S5E6   S6E7   S7E8   S8E9  S9E10

*S1E3   S2E4   S3E5   S4E6   S5E7   S6E8   S7E9  S8E10  S9E11

*S1E4   S2E5   S3E6   S4E7   S5E8   S6E9  S7E10  S8E11  S9E12
.....

*S1E12   S2E1   S3E2   S4E3   S5E4   S6E5   S7E6   S8E7   S9E8

Now I'm stuck since an arithmetic progression leads to pairing of 2 elements more than once:

*

*S1E1   S2E3   S3E5   S4E7   S5E9  S6E11   S7E1(13 mod 12)

Has anyone solved this particular problem?  Arithmetic progression works great for 9 sets of 11 elements and 9 sets of 13 elements (because 11 and 13 are prime numbers?).  Any ideas?
Thanks,
Jim Brown
I tried offsetting the elements each time the progression exceeded 12 and then using modulus but I still end up with duplicate pairs.

A: I am going to take a bit of a leap here.
The more I look at this, the more I think it is actually mathematically impossible.
The solution for 9 sets of 12 elements, with elements grouped into 108 sets of 9 elements with no pairing of elements in any set repeated is that there is no solution.
I would conjecture that in general for s sets of e elements with elements grouped into s x e sets of s elements with no pairing of elements in any set repeated is only doable if e is prime and s <= e. Mutual primality between s and e is not sufficient.
But I could be wrong.
You may wish to take the problem to the math forum ...
[EDIT 20221109] Correction to conjecture: e must be prime.

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Haftar welcomes the formation of a new executive authority
Haftar's spokesperson expresses his party's hope that the new appointed executive authority will deliver on the promises it made and prepare the country for elections
BY Libyan Express Feb 07, 2021 - 04:41
Khalifa Haftar joins the many regional and international voices welcoming the formation of Libya's interim government. [Photo: Getty Images]
Field Marshal Khalifa Haftar welcomed the outcome of the Libyan Political Dialogue Forum's run-off vote in Geneva and the appointment of a new executive authority to lead the country in the period preceding to the national elections.
In a statement delivered by Haftar's spokesperson Ahmed Al-Mismari, the military leader congratulated the winners of the LPDF vote in Switzerland and called the people selected as 'national figures'
He also expressed their party's hope that the newly appointed leaders will work to provide services and prepare the country to hold the national elections as stipulated in the LPDF roadmap in Tunis.
The outcome of the vote by members of the LPDF was welcomed by international bodies of governance such as the Arab League, US, UK, France, Germany, Italy and others.
The Libyan House of Representatives is expected to approve the newly selected executive authority in 21 days in keeping with the established LPDF roadmap in Tunis.
If approval is not reached, the decision will go to the members of the Libyan Political Dialogue Forum.
Abdul Hamid DebaibaGenevaKhalifa HaftarLPDFMohamed MenfiUNSMIL
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For: Haftar welcomes the formation of a new executive authority


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