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Home » All People » 18th /19th Century People » Catherine the Great 1729 – 1796
Catherine the Great 1729 – 1796
Born – 2nd May 1729
Died – 16th November 1796
Father – Christian August, Prince of Anhalt-Zerbst (1690 – 1747)
Mother – Princess Johanna Elisabeth of Holstein-Gottorp (1712 – 1760)
Spouse – m. 1745 – Tsar Peter III of Russia (1728 – 1762)
Children – by Peter III – Paul I of Russia (1754 – 1801); illegitimate – by Peter of Holstein-Gottorp – Anna Petrovna (1757 – 1759); by Grigory Orlov – Aleksey Bobrinsky (1762 – 1813)
Empress of Russia – 1762 – 1796
Preceded by – Peter III – 5th Jan – 9th July 1762
Succeeded by Paul I – 1796 – 1801
1729 (2nd May)
Catherine the Great was born Princess Sophie Friederike Auguste of Anhalt-Zerbst in Stettin, Pomerania, Prussia (Szczecin, Poland) to Christian August, Prince of Anhalt-Zerbst and Princess Johanna Elisabeth of Holstein-Gottorp.
1735 (around)
Sophie was educated by a French governess and private tutors.
Sophie met Karl Peter Ulrich, son of Charles, Duke of Holstein-Gottorp and Anna Petrovna of Russia. Karl Peter, known as Peter, was heir to the Russian Empire. Catherine disliked him instantly.
Elizabeth, Empress of Russia and Frederick II of Prussia wanted to negotiate an alliance to be sealed with the marriage of Sophie to Peter of Holstein-Gottorp. However, Sophie's mother was not in favour of the match but was keen for her daughter to become Empress of Russia.
Sophie was sent to Russia and made a good impression on Empress Elizabeth. She also began learning Russian.
Sophie was taken ill with pneumonia.
Sophie converted to Russian Orthodoxy. Her name was changed to Catherine (Ekaterina).
Catherine was formally betrothed to heir to the Russian Empire, Peter of Holstein-Gottorp.
Catherine married Peter of Holstein-Gottorp in St Petersburg, Russia. They made their home in the palace of Oranienbaum.
1745 (after 21st August)
The marriage of Catherine and Peter was not happy and both took lovers.
A son, Paul (Pavel) Petrovich was born to Catherine and Peter of Holstein-Gottorp at the Winter Palace in St Petersburg.
A daughter, Anna Petrovna was born to Catherine and Peter of Holstein-Gottorp at the Winter Palace in St Petersburg. She may have been fathered by Stanislaw Poniiatowski.
Catherine's daughter, Anna Petrovna, died.
Catherine began a relationship with Grigory Orlov.
A son, Aleksey Bobrinsky was born to Catherine the Great and Grigory Orlov. The child was raised away from court.
The Empress of Russia, Elizabeth, died and Catherine's husband, Peter, became Emperor Peter III. The couple moved from the Palace of Oranienbaum to the Winter Palace.
The rule of Emperor Peter III was not popular and many of his decisions were unpopular with the military and the nobility. In spite of her husband, Catherine maintained good relations with these groups. She began plotting to depose Peter, aided by Grigory Orlov.
Emperor Peter III was visiting the Palace of Oranienbaum with a number of his favourite courtiers. He had left Catherine behind in the Winter Palace.
Catherine decided the time was right to depose her husband and take over the rule of Russia. she quickly gained the support of the Russian military and was ordained as Empress in her own right the same day. Peter III was arrested and forced to abdicate in his wife's favour.
Peter III of Russia died. Despite the cause of death being stated as a stroke, it was rumoured that he was murdered by Alexei Orlov, brother of Grigory Orlov who had helped Catherine overthrow the Emperor but this has never been proved.
1762 (22nd September)
Catherine the Great of Russia was crowned at the Assumption Cathedral, Moscow. Her crown, designed by Jeremie Pauzie, would become the coronation crown for all future Emperors.
The French government announced that Diderot was to stop publication of his Encyclopedie because it went against religion. On hearing of this, Catherine wrote to Diderot and offered her protection if he wished to complete his work in Russia.
Nikita Panin was appointed foreign minister.
Catherine began corresponding with the French enlightenment writer, Voltaire.
Catherine the Great was an 'enlightened monarch' and was interested in new teachings, literature, philosophies and art works. She founded The Hermitage Museum in St Petersburg to house her personal collection.
Catherine believed that a good education system would help to modernise Russia. She set up a commission comprised of educational pioneers to advise her. The commission reported that all Russians should be given schooling from the age of 5 to the age of 18 years with the exception of serfs.
The Smolny Institute was established for the education of the daughters of the nobility.
Northern Accord
This was a mutual defence alliance between Russia and Prussia.
Ivan VI of Russia, who had been deposed by Empress Elizabeth in 1741, tried to take the throne from Catherine, claiming he was the rightful ruler. He was murdered by Vasily Mirovich, a lieutenant in the army acting under instructions from Catherine.
Catherine made Stanislaw Augustus Poniatowski, one of her close favourites, ruler of Poland.
Free Economic Society
Catherine established this society for the sharing of ideas and innovations.
Catherine the Great agreed a treaty of commerce between Russia and Great Britain.
In a move towards following the principles of the Enlightenment, Catherine summoned representatives of all classes to a Grand Commission to discuss the wider needs of the Russian Empire.
The Cadet Corps began taking children from a younger age and educating them before they became soldiers.
Legislative Commission
Catherine began work on a statement of laws based on the principles of the Enlightenment. Called the Nakaz, Catherine wanted it to replace the established laws of Russia. She presented the work to a Legislative Commission.
Catherine the Great took control of Poland/Lithuania by establishing herself as protectof of the Polish-Lithuanian Commonwealth.
There were protests in Poland and Lithuania against Russia's takeover of their lands.
Russo-Turkish War
War broke out with Turkey over Catherine's increasing influence in Poland.
Pugachev's Rebellion
This was a rebellion against Catherine's rule. It was organised by Yemelyan Pugachev, an army lieutenant who wanted to see an end to serfdom.
Catherine's son, Paul married Whlhelmina Louisa of Hesse-Darmastadt.
Pugachev's Rebellion – Battle of Kazan
This was a battle between the forces of Pugachev and government forces. The rebels had some initial success but were eventually defeated. Pugachev was captured and executed.
The war ended with Russian victory. Russia had gained new territory in the south including Odessa.
1774 (early Summer)
Catherine began a relationship with Grigory Potemkin, who had been a commander in the Russo-Turkish War.
Treaty of Kucuk Kaynarca
This treaty between Russia and the Ottoman Empire gave Russia access to the Black Sea and made the Crimea an independent state under the protectorate of Russia.
The Statute for the Administration of the Provinces of the Russian Empire
This statute set out a blueprint for the efficient government of Russia.
A young woman claiming to be Princess Tarakanova, a daughter of Empress Elizabeth, appeared in Western Europe seeking support for her claim. Alexei Orlov feigned romantic interest in her and took her to Russia where she was imprisoned. She died from tuberculosis while in prison.
Catherine the Great's relationship with Grigory Potemkin ended.
Catherine began a relationship with Pyotr Zavadovsky.
Wilhelmina Louisa, Catherine's daughter-in-law, died from complications following a stillbirth.
Catherine's son, Paul, married Sophie Dorothea of Wurttemberg, who had taken the name Maria Feodorovna on conversion to Russian Orthodoxy.
Catherine's relationship with Pyotr Zavadovsky ended.
Catherine began a relationship with Ivan Rimsky-Korsakov ended.
Catherine's relationship with Ivan Rimsky-Korsakov ended after he was unfaithful to her.
Catherine the Great annexed the Crimea.
Treaty of Georgievsk
This was a treaty between Russia and Georgia where Russia agreed to protect Georgia in the event of an attack by Persia (Iran).
Charter to the Nobility
This charter established the nobility as a separate privileged class in Russia.
Catherine began a relationship with Alexander Dmitriev-Mamonov, a Count.
The Russian Statute of National Education established a two-tier system of schools. Primary and Secondary schools would be co-educational and pupils would follow a designated curriculum. Manuals were produced to instruct teachers on the best teaching methods.
War broke out again between Russia and the Ottoman Empire. The Turks wanted to reclaim the Crimea.
Russo-Swedish War
War broke out again when Sweden attacked Russia with the intention of deposing Catherine the Great. However, the Swedes were unable to defeat Russia.
Catherine's relationship with Alexander Dmitriev-Mamonov, ended.
Russia attempted to negotiate a trade deal with Japan but failed to get Japanese approval.
The war ended with victory for Russia.
Polish-Russian War
War broke out between Poland and Russia in a Polish bid to break free from Russian involvement. The Poles were not successful and Russia gained a greater control of Poland.
King agha Mohammad Khan of Persia invaded and took control of Georgia.
As agreed under the treaty of Georgievsk, Russia invaded Georgia to drive the Persians out.
Catherine the Great died following a stroke. She was succeeded by her son, Paul.
<–Peter III
Paul I–>
Published Oct 04, 2019 @ 9:30 pm – Updated – Nov 12, 2021 @ 9:23 pm
Heather Y Wheeler. (2019 – 2020). Catherine the Great 1729 – 1796 Available: https://www.totallytimelines.com/catherine-the-great-1729-1796 Last accessed January 15th, 2022
Categories 18th /19th Century People, All People Tags Leader, Russia, Russian Monarchy, Tsars of Russia


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CARTES News: Gemalto, IBM partner on banking security
By: Andrew Hudson
category: Digital ID, Financial
Gemalto, IBM to offer enhanced security platform for e-banking
Gemalto has announced a partnership with IBM that will bolster Gemalto's Ezio eBanking security platform.
Following the new partnership with IBM, Gemalto's Ezio eBanking Server platform has been further updated to support a wider range of end-user solutions. The platform has been tested and confirmed to work with more than 30 authentication devices from 20 different vendors, ranging from physical tokens to mobile applications.
Available worldwide directly from Gemalto or through IBM Security Services, the Ezio eBanking Server is the first of its kind to be compliant with both the leading proprietary technologies and the industry's most popular open standards. Ezio eBanking supports all types of two-factor authentication tokens, including OTP, PKI, OATH and software solutions.
For banks, the platform's interoperability across authentication infrastructures offers new opportunities in customer segmentation, multi-channel strategy, as well as provides wider choice of form-factors and use-cases. The Ezio eBanking Server comes in two versions: an off-the-shelf appliance and a software toolkit, which together will offer increased flexibility to a bank's existing back-end authentication architecture.
IBM Security Services offers end-to-end risk management solutions that protect against threats while reducing the costs and complexity of security.
Gemalto spotlights banking authenti... Gemalto provides multi-platform eBa... Gemalto authentication the solution...
Tags: Internet Security

redpajama_set_name: RedPajamaCommonCrawl


Bag of 50 x Sempertex Balloons in colour: Mocha (code: 172) – All balloons come in 50 count bags.
SKU: code 172 (77). Category: 160 Fashion Solid. Tags: mocha 160's, modelling balloons, sempertex 160's, sempertex balloons, wholesale.
Bag of 50 x Sempertex Balloons in colour: Mocha (code: 172) – balloons come in 50 count bags.
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The Fall Of The Labour Party
The Labour party used to have principles. It used to stand for something. Nowadays it is indistinguishable from the Fianna Fail government it had vehemently opposed and its governing partner Fine Gael which it is supposed to counter balance.
The Irish Labour Party was founded a hundred years ago by Jim Larkin and James Connolly. Its' purpose was to represent the working class and trade unions. At the time poverty was rife in Ireland with Ireland's slums being the worst in Europe. Wages were low and unemployment was high. Larkin challenged the powerful in the epic struggle that was the Great Lockout.
Without a doubt he would be turning in his grave if he could see Labour today. Instead of fighting for the poor it makes statements Margaret Thatcher would be proud of. Labour's Minister for Social Protection, Joan Burton, claimed that the unemployed were making a "lifestyle choice". Donegal Senator Jimmy Harte caused controversy when he unjustly condemned an unemployed Polish woman and offered to pay her flight pack to Poland (it turns out the allegations against her were untrue and he had to apologise).
Harte had earlier been in the newspaper for claiming some families were receiving 90,000 a year in Social Welfare benefits. The example he gave was such an extreme case it's hard to believe anyone was in that situation. Allegedly, the father was on disability allowance, the mother was receiving carers allowance for looking after her daughter who was receiving special needs assistance, they received rent allowance (side note, according to the NESC only 11% of the unemployed receive rent allowance), and allowance for a child they had adopted. The family also happened to be Bosnian. This ridiculously extreme example is something you would expect Conservatives to drag up, not the supposedly main left wing party in Ireland.
I joined the Labour party because I thought it was the party that helped the poor and gave them an equal chance in life. I thought it cared about inequality, poverty and was the party of fresh new ideas. I was a proud member and canvassed for it in the 2009 local elections. But after the Budget I resigned in disgust. They were implementing the very same policies they had spent the last four years denouncing. As soon as they got into government they morphed into Fianna Fail, the very party they claimed to oppose.
In 2010 they voted against renewing the bank guarantee. Yet one year later in government, without a hint of shame they changed their position and supported the very thing they so strongly condemned in opposition. Tommy Broughan was the only TD who kept in principles and voted the same in government as in opposition. For this unforgivable act of principle he was expelled from the party.
They broke almost every promise they made from not raising student fees, to keeping Roscomon Hospital open, to protecting Social Welfare from cuts to ending the handouts to the banks. It makes you wonder why Labour opposed every Fianna Fail budget when its one is identical?
If you were to ask me what the Labour party stands for today, I couldn't tell you. They long ago dropped all principles and ideology in their quest for power. They have no new ideas or plans to change Ireland. They are nothing more than a power hungry few, willing to say anything to get votes. They are reduced to spouting empty words and meaningless clichés that not even they believe. Gilmore's recent speech at its conference was indistinguishable from anything said by Enda Kenny, Michael Martin or even Brian Cowen. If you close your eyes while listening to the Dail it is impossible to tell who is Fine Gael, Labour or Fianna Fail.
Those who sell their principles for power deserve neither. Labour will undoubtedly suffer the same fate as the Lib Dems in the UK, destroyed as disillusioned voters dissert the party. Recent polls show this happening already with Labour's support slowly eroding away to Sinn Fein. Just as the Spring Tide gains of 1992 were wiped out in the next election, so too will Labour's current gains be wiped out and it will return to the margins of politics.
Author Robert NielsenPosted on April 22, 2012 December 16, 2012 Categories PoliticsTags Fianna Fail, Fine Gael, Government, Ireland, Irish Politics, Jim Larkin, Labour, Politics
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Pirate Cove (Depoe Bay) during a king tide.\Photo by Bob Loewen.
The third and final round of the 2018-2019 King Tide Project, through which volunteer photographers document the reach of the winter's highest tides, is now taking place, February 18-20. (The first high-tide series the project covered took place Dec. 21-23, 2018; the second round occurred January 19-21.) The project reveals areas currently prone to flooding, but more important, gives us a preview of the coast in coming decades as sea level rises due to climate change.
The results from previous years and earlier rounds this winter are available for view on a special Flickr site.


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\section{Background and Motivation}

Let \(\Bbbk\) be a field, \(R=\Bbbk[x_1,x_2,...,x_n]\) a polynomial ring over \(\Bbbk\), \(Q\) be an ideal of \(R\), and \(S=R/Q\) its quotient ring.  By the Auslander-Buchsbaum-Serre theorem, we have that the free resolution of a module \(A\) over \(S\) is finite if and only if \(S\) is a regular ring.  When \(S\) fails to be regular, however, the complexity of the infinite free resolution of \(A\) depends on the structure of \(S\).

For example, if \(S\) is a Golod ring, monomial ring or a generic toric ring, the Poincar\'e-Betti series is known to be rational.  See \cite{MR2641236}, \cite{MR2158760}, and \cite{MR686351} for results in this direction.  Examples of modules with Poincar\'e-Betti series which are \emph{not} rational are also known, even in the setting of toric rings (see \cite{MR644015}, \cite{MR733933}, and \cite{MR1646972}.)

In the setting of monomial rings \(S=R/M\), where \(M\) is a monomial ideal, the Poincar\'e-Betti series and (frequently non-minimal) resolutions of \(\Bbbk\) over \(S\) are both known. Charalambous produced an explicit free resolution of a residue field \(\Bbbk\) over monomial rings \(S=R/M\), where \(M\) is an arbitrary monomial ideal (Theorem 1, \cite{MR1382031}.)  These resolutions of $\Bbbk$ over ring $S=R/M$ are not necessarily minimal (even for $R=\Bbbk[x,y]$), but do contain the minimal resolutions produced in this paper as a direct summand.  From work of Backelin~\cite{MR686351} and Avramov~\cite{MR2641236}, in the case of monomial rings it is known that the multigraded Poincar\'e-Betti series is a rational function and depends exclusively on the field \(\Bbbk\) and the lcm-lattice of the monomial ideal \(M\).  The denominator of this rational function has been interpreted combinatorially in \cite{MR2188858} and \cite{MR2290910}.

In this paper, we focus primarily on the case where \(R=\Bbbk[x,y]\) and our ring \(S=R/M\) is the quotient of \(R\) by an monomial ideal \(M\).  We partition monomial ideals in two variables into six types (one main case and five degenerate cases,) and explicitly construct a minimal free resolution of \(\Bbbk\) in each type. The denominator of the Poincar\'{e}-Betti series is given as an immediate corollary of the iterative construction of the syzygy modules.

In each of these cases, we use the recursively constructed free resolutions to produce the total Betti sequence \(\beta_i^S(k)\) for the residue fields of \(S=R/M\).  In particular, we completely classify which series can occur as the Poincar\'e-Betti sequence of \(\Bbbk\) over a monomial ring in two variables.  Specifically, when \(M\) is a monomial ideal with \(r\geq 2\) generators (not both pure powers) the Betti sequence of the resolution of the residue class field is given by the sequence \(\beta_0=1\), \(\beta_1=2\), and \(\beta_n=\beta_{n-1}+(r-1)\beta_{n-2}\) for \(n\geq 2\).

Our main case includes all monomial ideals \(M\) of the form
\[M=(x^{a_1}y^{b_1},...,x^{a_r}y^{b_r}),\]
where the \(x^{a_i}y^{b_i}\) form a minimal generating set and \(r>2\), or \(r=2\) and not of the form \((x^a,y^b)\).  The five degenerate cases on two or fewer generators, each with distinct resolution types, are described in the Appendix in Section~\ref{sec:degenerate}.

\begin{restatable*}[Resolutions of \(\Bbbk\) over Quotient Rings \(S\)]{theorem}{mainthm}\label{thm:main}  Let \(M\) be a monomial ideal with minimal generating set \(\{m_1,m_2,...,m_r\}\), where \(m_i=x^{a_i}y^{b_i}\) with \(a_1> a_2>\cdots>a_r\geq 0\) and \(0\leq b_1 <b_2<\cdots < b_r\) with \(r>2\), or \(r=2\) and \(M\) not generated by pure powers of \(x\) and \(y\).  A free resolution \({\mathcal F}\) of \(\Bbbk\) over \(S\) is given by
\[{\mathcal F}:  S\xleftarrow{\;\;\partial_1\;\;} F_1\xleftarrow{\;\;\partial_2\;\;} F_2\xleftarrow{\;\;\partial_3\;\;} F_3\xleftarrow{\;\;\partial_4\;\;} F_4\longleftarrow \cdots\]
where \(F_1\cong S^2\), \(F_2\cong S^{r+1}\), and \(F_3\cong S^{3r-1}\).

For \(i\geq 3\), the \((i+1)^{\text{st}}\) stage of the resolution for \(i\geq 3\) is constructed recursively in the following way:

Let \(F_i=F_1^{u_i}\oplus F_2^{v_i}\oplus F_3^{w_i}\) and \(\partial_i=\partial_1^{u_i}\oplus\partial_2^{v_i}\oplus\partial_3^{w_i}\) (with fixed \(u_i,v_i,w_i\) constructed via this method at a previous stage). Then the \((i+1)^{\text{st}}\)-stage of the minimal resolution is given by
\[F_i\cong F_1^{u_i}\oplus F_2^{v_i}\oplus F_3^{w_i}\xleftarrow{\partial_{i+1}} F_1^{(r-1)w_i}\oplus F_2^{u_i+rw_i}\oplus F_3^{v_i}\cong F_{i+1}\]
with \(\partial_{i+1}=\partial_3^{v_i}\oplus \partial_2^{w_i+ru_i}\oplus \partial_1^{(r-1)u_i}\).  Explicit formulas for the maps \(\partial_1\), \(\partial_2\), and \(\partial_3\) are given in Propositions~\ref{prop:syz1and2},~\ref{prop:syz3}, and~\ref{prop:syz4}
\end{restatable*}

With an explicit minimal free resolution in hand, a description of the Poincar\'{e}-Betti series is immediate.

\begin{restatable*}[Poincar\'{e}-Betti Series of \(\Bbbk\) over \(S\)]{theorem}{maincor}\label{cor:main}  Let \(M=(x^{a_1}y^{b_1},\cdots,x^{a_r}y^{b_r})\), with \(a_1> a_2>\cdots>a_r\geq 0\) and \(0\leq b_1 <b_2<\cdots < b_r\) with \(r>2\), or \(r=2\) and \(M\) not generated by pure powers of \(x\) and \(y\).  The total Betti numbers of the resolution of \(\Bbbk\) over \(S=R/M\) are given by
\begin{equation*}
\beta_i^S(\Bbbk)=\begin{cases}
1&\text{ if \(i=0\),}\\
2&\text{ if \(i=1\),}\\
\beta_{i-1}^S(\Bbbk)+(r-1)\beta_{i-2}^S(\Bbbk)&\text{ if \(i\geq 2\).}\\
\end{cases}
\end{equation*}
The Poincar\'e-Betti series then is
\[P_S(z)=\frac{1+z}{1-z+(1-r)z^2}.\]
\end{restatable*}

Note that this implies that the Poincar\'e-Betti sequences of these resolutions depends only on the number of generators of \(M\), rather than on the degrees of or relations between the generators.

\begin{example}[2-generated ideals]\label{example:main} Consider the monomial ideals \((xy^2,y^4)\) and \((x^2y,xy^2)\).  The graded Betti diagram of the minimal resolutions of \(\Bbbk\) found over \(S_1=\Bbbk[x,y]/(xy^2,y^4)\) and \(S_2=\Bbbk[x,y]/(x^2y,xy^2)\) are:

\vspace{10pt}
\begin{center}
$\begin{matrix}
&0&1&2&3&4&5&6&\cdots\\
\text{total:}&1&2&3&5&8&13&21&\cdots\\
\text{0:}&1&2&1&\text{.}&\text{.}&\text{.}&\text{.}\\
\text{1:}&\text{.}&\text{.}&1&2&1&\text{.}&\text{.}\\
\text{2:}&\text{.}&\text{.}&1&3&4&3&1\\
\text{3:}&\text{.}&\text{.}&\text{.}&\text{.}&2&6&7\\
\text{4:}&\text{.}&\text{.}&\text{.}&\text{.}&1&4&9\\
\text{5:}&\text{.}&\text{.}&\text{.}&\text{.}&\text{.}&\text{.}&3\\
\text{6:}&\text{.}&\text{.}&\text{.}&\text{.}&\text{.}&\text{.}&1\\
      \end{matrix}\;\;\;\;\;\;\;\;\begin{matrix}
      &0&1&2&3&4&5&6&\cdots \\
      \text{total:}&1&2&3&5&8&13&21&\cdots \\
\text{0:}&1&2&1&\text{.}&\text{.}&\text{.}&\text{.}\\
\text{1:}&\text{.}&\text{.}&2&5&4&1&\text{.}\\
\text{2:}&\text{.}&\text{.}&\text{.}&\text{.}&4&12&13\\
\text{3:}&\text{.}&\text{.}&\text{.}&\text{.}&\text{.}&\text{.}&8\\
      \end{matrix}$
\end{center}
\vspace{10pt}

Note that while their graded Betti diagrams differ, the total Betti numbers are the same, given by \(\beta_i^{S_1}(\Bbbk)=\beta_i^{S_2}(\Bbbk)=f_{i+1}\), where \(f_{i+1}\) is the \((i+1)^{\text{st}}\) Fibonacci number.  The graded Betti numbers depend (in the main case) exclusively on the number and degrees of the generators \(m_i\).
\end{example}

\section{Staircase Diagram of Monomial Ideals in Two Variables}

Let \(M=(x^{a_1}y^{b_1},...,x^{a_r}y^{b_r})\) be a monomial ideal (given by its minimal generating set,) ordered such that \(a_1> a_2>\cdots>a_r\geq 0\) and \(0\leq b_1 <b_2<\cdots < b_r\).  A monomial ideal \(M\) in two variables can always be put into this form, which may be represented by a \emph{staircase diagram} (see \cite{MR2110098}).
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=0.5,auto=left,vertices/.style={circle, fill=black, inner sep=1.5pt}]

\foreach \xcoord/\ycoord in {1/2,0/4} {
\fill[-,fill=black!20] (\xcoord,\ycoord)--(\xcoord,6)--(6,6)--(6,\ycoord)--(\xcoord,\ycoord);
\fill[-,fill=black!10] (\xcoord,6)--(6,6)--(6,\ycoord)--(6.5,\ycoord)--(6.5,6.5)--(\xcoord,6.5)--(\xcoord,6);
};

\foreach \yval in {1,2,3,4,5,6} {
	\draw[-,black!40] (0,\yval)--(6.5,\yval);
};
\foreach \xval in {1,2,3,4,5,6} {
	\draw[-,black!40] (\xval,0)--(\xval,6.5);
};

\draw[->] (0,0)--(7,0);
\draw[->] (0,0)--(0,7);
\node[label=right:{x}] at (7,0) {};
\node[label=above:{y}] at (0,7) {};

\foreach \xcoord/\ycoord in {1/2,0/4} {
	\node[vertices] at (\xcoord,\ycoord){};
};

\node at (3.5,-1.5) {$M_1=(xy^2,y^4)$};
\end{tikzpicture}\;\;\;\;\;\;\begin{tikzpicture}[scale=0.5,auto=left,vertices/.style={circle, fill=black, inner sep=1.5pt}]

\foreach \xcoord/\ycoord in {2/1,1/2} {
\fill[-,fill=black!20] (\xcoord,\ycoord)--(\xcoord,6)--(6,6)--(6,\ycoord)--(\xcoord,\ycoord);
\fill[-,fill=black!10] (\xcoord,6)--(6,6)--(6,\ycoord)--(6.5,\ycoord)--(6.5,6.5)--(\xcoord,6.5)--(\xcoord,6);
};

\foreach \yval in {1,2,3,4,5,6} {
	\draw[-,black!40] (0,\yval)--(6.5,\yval);
};
\foreach \xval in {1,2,3,4,5,6} {
	\draw[-,black!40] (\xval,0)--(\xval,6.5);
};

\draw[->] (0,0)--(7,0);
\draw[->] (0,0)--(0,7);
\node[label=right:{x}] at (7,0) {};
\node[label=above:{y}] at (0,7) {};

\foreach \xcoord/\ycoord in {2/1,1/2} {
	\node[vertices] at (\xcoord,\ycoord){};
};

\node at (3.5,-1.5) {\(M_2=(x^2y,xy^2)\)};
\end{tikzpicture}
\end{center}

\caption{Staircase diagrams of \(M_1\) and \(M_2\) shown.\label{fig:staircase}}

\end{figure}

See Figure~\ref{fig:staircase} for the staircase diagrams of the monomial ideals in Example~\ref{example:main}. All monomials \(m\) corresponding to lattice points inside or on the boundary of the grey area are in the ideal \(M\), and all of the lattice points outside of the shaded area or its boundary are monomials in the ring \(S=\Bbbk[x,y]/M\).

\begin{definition}[Colon Ideals $M_x$ and $M_y$ in $S$]\label{defn:colon}Let \(M_x\) and \(M_y\) be the ideals \(0:(x)\) and \(0:(y)\) of $S$,
\begin{align*}
M_x &= \{ m : mx\in M, m\not\in M\}=0:(x),\text{ and}\\
M_y &= \{ m : my\in M, m\not\in M\}=0:(y).
\end{align*}
\end{definition}
The elements of \(M_x\) and \(M_y\) for the monomial ideals in Example~\ref{example:main} are shown in white in Figure~\ref{fig:halo}.

\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=0.4,auto=left,vertices/.style={circle, draw, fill=black, inner sep=1.5pt}, dotvertices/.style={circle, draw, fill=white, inner sep=1.5pt}]

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\node[label=right:{x}] at (7,0) {};
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\node at (3.5,-1.5) {\(M_2:(x)=(xy,y^2)\)};
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\node[label=right:{x}] at (7,0) {};
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\node at (3.5,-1.5) {\(M_1:(y)=(xy,y^3)\)};
\end{tikzpicture}\;\;\;\;\;\;\begin{tikzpicture}[scale=0.4,auto=left,vertices/.style={circle, draw, fill=black, inner sep=1.5pt},dotvertices/.style={circle, draw, fill=white, inner sep=1.5pt}]

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\draw[->] (0,0)--(7,0);
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\node[vertices] at (\xcoord,\ycoord){};
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\node[label=right:{x}] at (7,0) {};
\node[label=above:{y}] at (0,7) {};

\node at (3.5,-1.5) {\(M_2:(y)=(x^2,xy)\)};
\end{tikzpicture}
\end{center}
\caption{Elements of \(M_x\) and \(M_y\). \label{fig:halo}}

\end{figure}

We will use the ideals \(M_x\) and \(M_y\) frequently when constructing our syzygies of \(\Bbbk\) over \(S=\Bbbk[x,y]/M\).

\begin{proposition}[First and Second Syzygy Modules of \(\Bbbk\) over \(S\)]\label{prop:syz1and2}
Let \(M=(x^{a_1}y^{b_1},...,x^{a_r}y^{b_r})\) be a monomial ideal given by its minimal set of generators, ordered so that \(a_1> a_2>\cdots>a_r\geq 0\) and \(0\leq b_1 <b_2<\cdots < b_r\).  Let either \(r>2\) or \(r=2\) and \(M\neq(x^n,y^m)\).  The first three stages of a (graded) minimal resolution of \(\Bbbk\) over \(S=\Bbbk[x,y]/M\) are given by

\[{\mathcal F}: \Bbbk \longleftarrow F_0 \xleftarrow{\;\;\;\partial_1\;\;\;} F_1\xleftarrow{\;\;\;\partial_2\;\;\;} F_2.\]
These syzygy modules and their respective bases are:
\begin{center}
\begin{tabular}{l c l}
Syzygy Module &\;\;\; & Basis\\ \hline
\(F_0\cong S\)&\;\;\;& \(\{e_1\}\)\\
\(F_1=S(-1)^2\cong S^2\)&\;\;\;&\(\{e_x,e_y\}\)\\
\(F_2=\left(\bigoplus_{1\leq i\leq r} S(-a_i-b_i)\right)\oplus S(-2)\cong S^{r+1}\)&\;\;\;&\(\{e_{f_1},e_{f_2},...,e_{f_r},e_{f_{r+1}}\}\)
\end{tabular}
\end{center}

\noindent The map \(F_1\xrightarrow{\partial_1} F_2\) is given by
\begin{align*}
\partial_1(e_x)&=x\cdot e_1\\
\partial_1(e_y)&=y\cdot e_1.
\end{align*}
The map \(F_2\xrightarrow{\partial_2}F_1\) is given by:

\begin{enumerate}[{\bf Case }1:]
\item If all generators of \(M\) are divisible by \(x\) (i.e. \(a_r\geq 1\)), then the second syzygy map \(\partial_2\) is given by
\begin{equation*}
\partial_2(e_{f_i})=\begin{cases}
x^{a_{i}-1}y^{b_{i}}\cdot e_x & \text{if \(1\leq i\leq r\),}\\
-y\cdot e_x+x\cdot e_y&\text{if \(i=r+1\).}
\end{cases}
\end{equation*}
\item If the final generator of \(M\) is \(m_r=y^{b_r}\), then the second syzygy map \(\partial_2\) is given by
\begin{equation*}
\partial_2(e_{f_i})=\begin{cases}
x^{a_{i}-1}y^{b_{i}}\cdot e_x & \text{if \(1\leq i\leq r-1\),}\\
y^{b_r-1}\cdot e_y & \text{if \(i=r\),}\\
-y\cdot e_x+x\cdot e_y&\text{if \(i=r+1\).}
\end{cases}
\end{equation*}
\end{enumerate}

\end{proposition}

\begin{proof}
Let \(M\) be a monomial ideal satisfying the conditions of the theorem.  Note that \(r>2\) or \(r=2\) and \(M\neq(x^n,y^m)\) for \(n,m\geq 0\) implies that \(x,y\) are nonzero elements in \(S\).  Both \(x,y\in S\) will map to zero in the residue field \(\Bbbk\), so \(x\) and \(y\) must span our set of syzygies in \(F_0=S\). We choose \(e_1\) as a generator of \(F_0\), and \(e_x,e_y\) as generators for our first syzygy module \(F_1\cong S(-1)^2\), and we define \(\partial_1(e_x)=x\cdot e_1\) and \(\partial_1(e_y)=y\cdot e_1\).

To construct the second syzygy module, we note that we are looking for a generating set for all elements \(w\cdot e_x+z\cdot e_y\in F_1\) such that \(wx+zy=0\in S\).

By construction of \(M_x=\{ m : mx\in M, m\not\in M\}=0:(x)\) and \(M_y=\{ m : my\in M, m\not\in M\}=0:(y)\) in the previous section, we have that \(\{m\cdot e_x : m\in M_x\}\) and \(\{m'\cdot e_y : m'\in M_y\}\) are nonzero elements of \(F_1\) in the kernel of \(\partial_1\). We also have that \(-yx+xy=0\), so \(-y\cdot e_x+x\cdot e_y\) is another element of \(\ker(\partial_1)\). We now show that all elements in
\begin{align*}
\ker(\partial_1)&=\{w\cdot e_x+z\cdot e_y : wx+zy=0\}\\
&=\{w\cdot e_x+z\cdot e_y : wx=-zy\in S\}
\end{align*}
 may be written as a linear combination of syzygies of these three types:
 \[\underbrace{\left\{x^{a_i-1}y^{b_i}\cdot e_x\right\}_{i=1}^r}_{\text{Type I}}\bigcup\underbrace{\left\{x^{a_i}y^{b_i-1}\cdot e_y\right\}_{i=1}^r}_{\text{Type II}}\bigcup \underbrace{\left\{-y\cdot e_x+x\cdot e_y\right\}}_{\text{Type III}}\]

Given a syzygy \(w\cdot e_x+z\cdot e_y\), it follows that \(wx=-zy\in S\). This implies that either
\begin{enumerate}[(i)]
\item \(wx=zy=0\in S\) or
\item \(wx=-zy=s\in S\), where \(s\neq 0\in S\).
\end{enumerate}

If condition (i) holds, we have \(wx,zy\in M\), which implies that \(w\in M_x\) and \(z\in M_y\).  So \(w\cdot e_x+z\cdot e_y\) is a linear combinations of syzygies of the first two types.

If condition (ii) holds, we can factor out a \(y\) from \(w\) and an \(x\) from \(z\), giving
\begin{align*}
s & = wx=-zy\\
& = w'xy=-z'xy.
\end{align*}
So \(s\) is a monomial divisible by \(xy\), and our original syzygy can be obtained as a multiple of the syzygy \(-y \cdot e_x+x\cdot e_y\),
\begin{align*}
-\frac{s}{xy}(-y\cdot e_x+x\cdot e_y)&=\frac{s}{x}\cdot e_x-\frac{s}{y}\cdot e_y=w\cdot e_x+z\cdot e_y.
\end{align*}
So every syzygy of \(F_1\) can be written as a linear combination of syzygies of the three types above. We now reduce this set of syzygies to a minimal generating set.

Assume \(b_i\geq 1\), so the syzygy \(x^{a_i}y^{b_i-1}\cdot e_y\) exists. If \(b_i=0\), there are no syzygies to consider. If \(a_i\geq 1\), we may write
\begin{align*}
x^{a_i}y^{b_i-1}\cdot e_y & = \left(x^{a_i-1}y^{b_i}-x^{a_i-1}y^{b_i}\right)\cdot e_x+x^{a_i}y^{b_i-1}\cdot e_y\\
& = x^{a_i-1}y^{b_i}\cdot e_x+\left(-x^{a_i-1}y^{b_i}\cdot e_x+x^{a_i}y^{b_i-1}\cdot e_y\right)\\
& = x^{a_i-1}y^{b_i}\cdot e_x+x^{a_i-1}y^{b_i-1}\left(-y\cdot e_x+x\cdot e_y\right)
\end{align*}
Note that this syzygy is a linear combination of syzygies of the first and third type.
In Case 1, \(a_i\geq 1\) for all generators \(x^{a_i}y^{b_i}\) of \(M\). We may then remove all syzygies of Type II, \(m'\cdot e_y\), from our basis, rewriting them as a linear combination of syzygies of the first and third type.  This gives us a kernel minimally spanned by
\[\left\{x^{a_i-1}y^{b_i}\cdot e_x\right\}_{i=1}^r\bigcup \left\{-y\cdot e_x+x\cdot e_y\right\}.\]

In Case 2, we have \(a_r=0\), and our kernel is spanned by
\[\left\{x^{a_i-1}y^{b_i}\cdot e_x\right\}_{i=1}^{r-1}\bigcup\left\{y^{b_r-1}\cdot e_y\right\}\bigcup\left\{-y\cdot e_x+x\cdot e_y\right\}.\]
This is also minimal, as the \(x^{a_i-1}y^{b_i}\) are the minimal generating set of \(M_x\), and by construction, \(y^{b_r-1}\cdot e_y\) cannot be written as a linear combination of any of the other syzygies.

Our basis for \(F_2\) is then given by \(\{e_{f_1},e_{f_2},...,e_{f_r},e_{f_{r+1}}\}\), with \(\partial_2(e_{f_i})\) given as in the statement of the theorem, with the degree-shifts of \(F_2\) calculated appropriately.

\end{proof}

We verify that by construction, \(\partial_1\circ\partial_2(e_{f_i})=0\), so
\[F_0\xleftarrow{\;\;\;\partial_1\;\;\;} F_1\xleftarrow{\;\;\;\partial_2\;\;\;}F_2\] as defined form a complex:
\begin{equation*}
\partial_1\circ \partial_2(e_{f_i})=\begin{cases}
\partial_1(x^{a_i-1}y^{b_i}\cdot e_x)&= x^{a_i-1}y^{b_i}\cdot \partial_1(e_x)\\
&=x^{a_i}y^{b_i}\cdot e_1=0 \hspace{51pt} \text{ if \(a_i\geq 1\) and \(1\leq i\leq r\),}\\
\partial_1(y^{b_r-1}\cdot e_y)&= y^{b_r-1}\cdot \partial_1(e_y)\\
&=y^{b_r}\cdot e_1=0 \hspace{65pt}\text{ if \(a_r\geq 0\) and \(i=r\),}\\
\partial_1(-y\cdot e_x+x\cdot e_y)&=-y\cdot \partial_1(e_x)+x\cdot \partial_1(e_y)\\
&=(-yx+xy)\cdot e_1=0 \hspace{25pt} \text{if \(i=r+1\).}\\
\end{cases}
\end{equation*}

\section{Third Syzygy Modules in Main Case}

For an \(r\)-generated monomial ideal \(M\) in \(x,y\), we have so far that the resolution of \(\Bbbk\) over \(S=\Bbbk[x,y]/M\) has second syzygy module \[F_2=\left(\bigoplus_{1\leq i\leq r} S(-a_i-b_i)\right)\oplus S(-2)\cong S^{r+1}\]
with basis \(\{e_{f_1},e_{f_2},...,e_{f_r},e_{f_{r+1}}\}\), and a second syzygy map:

\begin{equation*}
\partial_2(e_{f_i})=\begin{cases}
x^{a_{i}-1}y^{b_{i}}\cdot e_x & \text{if \(a_i\geq 1\) and \(1\leq i\leq r\),}\\
y^{b_r-1}\cdot e_y & \text{if \(a_r=0\) and \(i=r\),}\\
-y\cdot e_x+x\cdot e_y&\text{if \(i=r+1\)}.
\end{cases}
\end{equation*}

\noindent Before constructing the third syzygy module, we begin with a useful lemma describing resolutions of ideal $M_x$ over $S$.

\begin{lemma}[Syzygy Module of $M_x$ over $S$]\label{lem:syzMx} Let $M=(x^{a_1}y^{b_1},...,x^{a_r}y^{b_r})\subset \Bbbk[x,y]=R$ where $0\leq b_1<\cdots<b_r$ and $a_1>\cdots > a_r\geq 0$, and let $S=R/M$. Let $M_x=0:(x)$ be the module over $S$ from Definition~\ref{defn:colon}.
Let $\{e_{f_i}\}$ be the minimal generating set for $M_x$ as a module over $S$, and set $M_x\xrightarrow{\partial_0} S$ where $\partial_0(e_{f_i})=x^{a_i-1}y^{b_i}\in S$.
Then the first syzygy module of $M_x$, $\text{Syz}_1(M_x)\cong\ker(\partial_0)$, is minimally generated by
\[\left\{e_{g_i}\right\}_{i=1}^{r}\cup\left\{e_{g_i'}\right\}_{i=1}^{r-1}\text{ if $a_r>0$,}\]
\[\left\{e_{g_i}\right\}_{i=1}^{r-1}\cup\left\{e_{g_i'}\right\}_{i=1}^{r-1}\text{ if $a_r=0$,}\]
where
\begin{align*}
\left\{\partial_1(e_{g_i})=x\cdot e_{f_i}\right\}_{i=1}^{r}&\cup\left\{\partial_1(e_{g_i'})=y^{b_{i+1}-b_{i}}\cdot e_{f_i}\right\}_{i=1}^{r-1}& \text{ for $a_r>0$}\;\\
\left\{\partial_1(e_{g_i})=x\cdot e_{f_i}\right\}_{i=1}^{r-1}&\cup\left\{\partial_1(e_{g_i'})=y^{b_{i+1}-b_{i}}\cdot e_{f_i}\right\}_{i=1}^{r-1}& \text{ for $a_r=0.$}
\end{align*}
\end{lemma}

\begin{proof}
Note that $M_x$ is minimally generated by the following monomials:
\begin{equation*}
M_x=\begin{cases}
(x^{a_1-1}y^{b_1},\ldots, x^{a_r-1}y^{b_r}) & \text{ if $a_r>0$,}\\
(x^{a_1-1}y^{b_1},\ldots, x^{a_{r-1}-1}y^{b_{r-1}}) & \text{ if $a_r=0$.}
\end{cases}
\end{equation*}
As $M_x$ is a monomial ideal over a monomial quotient ring, all sygyzies must take one of two forms:
\begin{align*}
m_i\cdot x^{a_i-1}y^{b_i}-m_j\cdot x^{a_j-1}y^{b_j}&=0\text{ for $m_i,m_j\neq 0\in S$}\;\;\;&\text{(Type I)}\;\;\\
m_i\cdot x^{a_i-1}y^{b_i}&=0\text{ for $m_i\neq 0\in S$}\;\;\;\;\;&\text{(Type II)}.
\end{align*}
As $x\cdot x^{a_i-1}y^{b_i}=0$ in $S$ for all generators $x^{a_i-1}y^{b_i}\in M_x$, we add the elements $\left\{e_{g_i}:\genfrac{}{}{0pt}{}{1\leq i\leq r \text{ if $a_r>0$}}{1\leq i\leq r-1 \text{ if $a_r=0$}}\right\}$ to a basis for $\text{Syz}_1(M_x)$ with syzygy map $\partial_1(e_{g_i})=x\cdot e_{f_i}$. All syzygies of Type II with $x|m_i$ will be in the $S$-span of the image of $\partial_1$.

Consider any remaining syzygies of Type II not in this span, i.e. syzygies with $y^{d_i}\cdot x^{a_i-1}y^{b_i}=0$ where $x^{a_i-1}y^{b_i+(d_i-1)}\neq 0\in S$. Note that then $x^{a_i-1}y^{b_i+d_i}\in M$, and must in particular be divisible by $x^{a_i-1}y^{b_{i+1}}$. So we must have $y^{d_i}=y^{b_{i+1}-b_i}$, giving us new minimal syzygies of the forms $y^{b_{i+1}-b_i}\cdot e_{f_i}.$

\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=0.4,auto=left,vertices/.style={circle, draw, fill=black, inner sep=1pt}, dotvertices/.style={circle, draw, fill=white, inner sep=1.5pt}]

\foreach \xcoord/\ycoord in {5/1,1/4} {
\fill[-,fill=black!15] (\xcoord,\ycoord)--(\xcoord,6)--(8,6)--(8,\ycoord)--(\xcoord,\ycoord);
\fill[-,fill=black!5] (\xcoord,6)--(8,6)--(8,\ycoord)--(8.5,\ycoord)--(8.5,6.5)--(\xcoord,6.5)--(\xcoord,6);
};

\foreach \yval in {0,1,2,3,4,5,6} {
	\draw[-,black!40] (0,\yval)--(8.5,\yval);
};
\foreach \xval in {0,1,2,3,4,5,6,7,8} {
	\draw[-,black!40] (\xval,0)--(\xval,6.5);
};

\foreach \xcoord/\ycoord in {6.5/3}{
\draw[black!15, fill=black!15, thick,rounded corners] (\xcoord-1.3,\ycoord-1.5) rectangle (\xcoord+1.3,\ycoord-0.5);
}

\foreach \xcoord/\ycoord in {4/5.9}{
\draw[black!15, fill=black!15, thick,rounded corners] (\xcoord-2.5,\ycoord-1.5) rectangle (\xcoord+1.3,\ycoord-0.5);
}

\foreach \xcoord/\ycoord in {2.5/3.4}{
\draw[white, fill=white, thick,rounded corners] (\xcoord-1.4,\ycoord-1.3) rectangle (\xcoord+1.5,\ycoord-0.7);
}

\foreach \xcoord/\ycoord in {2/1.6}{
\draw[white, fill=white, thick,rounded corners] (\xcoord-1.6,\ycoord-1.5) rectangle (\xcoord+1.6,\ycoord-0.5);
}

\draw[->] (4,1)--(4.8,1);
\node at (4.5,0.5){\tiny $x$};
\draw[->] (4,1)--(4,3.9);
\node at (2.5,2.4){\tiny $y^{b_{i+1}-b_i}$};

\node[vertices,label=above right:{\footnotesize $x^{a_i}y^{b_i}$}] at (5,1){};
\node[vertices,label=above right:{\footnotesize $x^{a_{i+1}}y^{b_{i+1}}$}] at (1,4){};
\node[dotvertices] at (4,1){};
\node at (2,0.6){\footnotesize $x^{a_i-1}y^{b_i}$};
\end{tikzpicture}
\end{center}
\caption{Syzygies of Type I on $M_x$ over $S$}\label{fig:syzygiesMx}
\end{figure}
Considering syzygies of Type I, we note that $\deg(m_i),\deg(m_j)\geq 1$. If $x|m_i$ or $x|m_j$, then we may rewrite one or both terms as sums of syzygies of Type II. The remaining term (if either $m_i$ or $m_j$ was a pure power of $y$) must be a syzygy in the $S$-span of $y^{b_{i+1}-b_i}\cdot e_{f_i}$. Hence all syzygies of $M_x$ must be in the $S$-span of the forms described in the statement of the Lemma.
\end{proof}

We now construct our third syzygy module $F_3$ over \(S\).

\begin{proposition}[Third Syzygy Module of \(\Bbbk\) over \(S\)]\label{prop:syz3}  Let \(M=(x^{a_1}y^{b_1},...,x^{a_r}y^{b_r})\) be a monomial ideal given by its minimal set of generators, ordered so that \(a_1> a_2>\cdots>a_r\geq 0\) and \(0\leq b_1 <b_2<\cdots < b_r\).  Let either \(r>2\) or \(r=2\) and \(M\neq(x^n,y^m)\).  The third syzygy module of \(\Bbbk\) over \(S=\Bbbk[x,y]/M\) is of dimension \(3r-1\) with basis
\[\bigl\{e_{c_i^x}\bigr\}_{i=1}^r\cup\bigl\{e_{c_i^y}\bigr\}_{i=1}^r\cup\bigl\{e_{d_i}\bigr\}_{i=1}^{r-1}.\]
\begin{enumerate}[{\bf Case }1:]
\item If all generators of \(M\) are divisible by \(x\) (i.e. \(a_r\geq 1\)), then the third syzygy map \(\partial_3\) is given by
\begin{align*}
\partial_3(e_{c_i^x})&=x\cdot e_{f_i} \hspace{95pt} \text{ for \(1\leq i\leq r\)}\\
\partial_3(e_{c_i^y})&=y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}} \hspace{10pt}\text{ for \(1\leq i\leq r\)}\\
\partial_3(e_{d_i})&=x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}}\hspace{32pt}\text{ for \(1\leq i\leq r-1\)}.
\end{align*}
\item If the final generator of \(M\) is \(m_r=y^{b_r}\), then the third syzygy map \(\partial_3\) is given by
\begin{align*}
\partial_3(e_{c_i^x})&=\begin{cases} x\cdot e_{f_i} &\hspace{15pt}\text{for \(1\leq i\leq r-1\)}\\
x\cdot e_{f_r}-y^{b_r-1}\cdot e_{f_{r+1}} &\hspace{15pt}\text{for \(i=r\),}
\end{cases}\\
\partial_3(e_{c_i^y})&=\begin{cases}y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}} &\hspace{3pt} \text{for \(1\leq i\leq r-1\)}\\
y\cdot e_{f_{r}} &\hspace{3pt} \text{for \(i=r\),}
\end{cases}\\
\partial_3(e_{d_i})&=x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}} \hspace{43pt}\text{ for \(1\leq i\leq r-1\)}.
\end{align*}
\end{enumerate}
\end{proposition}

We wish to construct a generating set for all syzygies
$$g_1\cdot e_{f_1}+ g_2\cdot e_{f_2}+\cdots+g_re_{f_r}+g_{r+1}e_{f_{r+1}},$$
for $g_1,...,g_r,g_{r+1}\in S$, where any given syzygy must satisfy the equations
\begin{align}
\label{eqn:1syz3}\text{Case }1:\;\;\;\;& \left({\displaystyle \sum_{i=1}^{r}g_ix^{a_{i}-1}y^{b_i}}\right)-g_{r+1}y=0 \\
\label{eqn:2syz3}& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,g_{r+1}x=0\\
\label{eqn:3syz3}\text{Case }2:\;\;\;\;& \left({\displaystyle \sum_{i=1}^{r-1}g_ix^{a_{i}-1}y^{b_i}}\right)-g_{r+1}y=0 \\
\label{eqn:4syz3}& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,g_{r}y^{b_r-1}+g_{r+1}x=0.
\end{align}

\begin{proof}[Proof of Case 1]
In Case 1, Equation~\ref{eqn:2syz3} forces $g_{r+1}=0$ or $g_{r+1}\in M_x.$

\noindent If $g_{r+1}=0$, it suffices to find a basis for syzygies
$$g_1\cdot e_{f_1}+ g_2\cdot e_{f_2}+\cdots+g_re_{f_r},$$
where $${\displaystyle \sum_{i=1}^{r}g_ix^{a_{i}-1}y^{b_i}}=0.$$
By Lemma~\ref{lem:syzMx}, we know that a basis for syzygies of this form is given by
$$\left\{x\cdot e_{f_i}\right\}_{i=1}^r\cup\left\{y^{b_{i+1}-b_i}\cdot e_{f_i}\right\}_{i=1}^{r-1}.$$
We will rewrite the syzygies $y^{b_{i+1}-b_i}\cdot e_{f_i}$ as sums of other basis elements later in the proof.

If $g_{r+1}\neq 0$, then $g_{r+1}\in M_x$. As these syzygies are the kernel of a map with monomial entries over a monomial quotient $S=R/M$, we must have minimal syzygies of the form
$$g_ix^{a_i-1}y^{b_i}-g_{r+1}y=0$$
for some $g_i\in S$ and $g_{r+1}\in M_x$.

If $g_i=0$, this forces $g_{r+1}\in M_x\cap M_y$, as then $x\cdot g_{r+1}=y\cdot g_{r+1}=0$. So $g_{r+1}=x^{a^i-1}y^{b_{i+1}-1}$. We have one such syzygy for each $g_{r+1}\in M_x\cap M_y$.

\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=0.4,auto=left,vertices/.style={circle, draw, fill=black, inner sep=0.75pt}, dotvertices/.style={circle, draw, fill=white, inner sep=1pt},bigdotverticesy/.style={star, star points=4, draw, fill=black!30, inner sep=1.5pt},dotverticesy/.style={star, star points=4, draw, fill=black!30, inner sep=0.75pt}]

\foreach \xcoord/\ycoord in {1/12,2/11,5/10,6/8,10/6,13/1} {
\fill[-,fill=black!15] (\xcoord,\ycoord)--(\xcoord,14)--(16,14)--(16,\ycoord)--(\xcoord,\ycoord);
\fill[-,fill=black!5] (\xcoord,14)--(16,14)--(16,\ycoord)--(16.5,\ycoord)--(16.5,14.5)--(\xcoord,14.5)--(\xcoord,14);
};

\foreach \yval in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14} {
	\draw[-,black!40] (0,\yval)--(16.5,\yval);
};
\foreach \xval in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16} {
	\draw[-,black!40] (\xval,0)--(\xval,14.5);
};

\foreach \xcoord/\ycoord in {1/12,2/11,5/10,6/8,10/6,13/1} {
\node[vertices] at (\xcoord,\ycoord){};
}

\foreach \xcoord/\ycoord in {1/11,4/10,5/9,9/7,12/5}{
\node[dotvertices] at (\xcoord,\ycoord){};
\node[bigdotverticesy] at (\xcoord,\ycoord){};
}

\foreach \xcoord/\ycoord in {0/12,0/13,0/14,1/11,4/10,5/9,5/8,9/7,9/6,12/5,12/4,12/3,12/2,12/1}{
\node[dotvertices] at (\xcoord,\ycoord){};
}

\foreach \xcoord/\ycoord in {2/10,3/10,6/7,7/7,8/7,10/5,11/5,13/0,14/0,15/0,16/0}{
\node[dotverticesy] at (\xcoord,\ycoord){};
}

\node[dotvertices, label=right:{\footnotesize Elements of $M_x$}] at (18,8){};
\node[dotverticesy, label=right:{\footnotesize Elements of $M_y$}] at (18,6.5){};
\node[bigdotverticesy] at (18,5){};
\node[dotvertices, label=right:{\footnotesize Elements of $M_x\cap M_y$}] at (18,5){};
\node[vertices, label=right:{\footnotesize Generators of $M$}] at (18,9.5){};

\end{tikzpicture}
\end{center}
\caption{Diagram of $M_x, M_y$ for $M=(xy^{12},x^2y^{11},x^5y^{10},x^6y^8,x^{10}y^6,x^{13}y)$}\label{fig:pictureMx}
\end{figure}

Assuming $g_i\neq 0$ and $g_{r+1}\in M_x$ for some syzygy, we may without loss of generality consider only syzygies where $x\nmid g_i$ (as syzygies where $x|g_i$ are in the $S$-span of syzyzgies with $x\cdot e_{f_i}$ and $x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}}$.)
Considering first syzygies with $g_i=y$, we have
$$y\cdot x^{a_i-1}y^b_i-g_{r+1}\cdot y=x^{a_i-1}y^{b_i+1}-g_{r+1}\cdot y=0,$$
giving us that $g_{r+1}=x^{a_i-1}y^{b_i-1}$.

Our minimal syzygies must be of the form
$$\partial_2(g_i\cdot e_{f_i}+x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}})=g_i\cdot x^{a_i-1}y^{b_i}-x^{a_i-1}y^{b_{i+1}}=0.$$
So $g_i=y$ and $g_{r+1}=x^{a_i-1}y^{b_i}$.

Finally, we remove all elements $y^{b_{i+1}-b_i}\cdot e_{f_i}$ from our generating set for syzygies $g_1e_{f_1}+\cdots+g_re_{f_r}+g_{r+1}e_{f_{r+1}}$, by noting that each of these are in the $S$-span of elements $y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}}$ and $x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}}$:
\begin{align*}
y^{b_{i+1}-b_i}\cdot e_{f_i}&=y^{b_{i+1}-b_i-1}\left(y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}}-x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}}\right)\\
&=y^{b_{i+1}-b_i-1}\left(y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}}\right)-y^{b_{i+1}-b_i-1}\left(x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}}\right)\\
&=y^{b_{i+1}-b_i-1}\left(y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}}\right)-x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}}
\end{align*}
Our third syzygy module is then generated by $e_{c_i^x}$, $e_{c_i^y}$, and $e_{d_i}$, with the described $\partial_3$.
\end{proof}
\begin{proof}[Proof of Case 2]
In Case 2, consider separately the cases where $g_r=0$ and $g_r\neq 0$. If $g_r=0$, then we have the equations
\begin{align}
\left({\displaystyle \sum_{i=1}^{r-1}g_ix^{a_{i}-1}y^{b_i}}\right)-g_{r+1}y&=0\\
g_{r+1}x&=0.
\end{align}
Using Lemma~\ref{lem:syzMx} and recreating the proof on Case 1,  we have that a generating set for all syzygies mapping to $S^{r+1}$ under $\partial_3$ with $g_r=0$ are of the forms:
\begin{align*}
\partial_3(e_{c_i^x})&=x\cdot e_{f_i} \hspace{95pt} \text{ for \(1\leq i\leq r-1\)}\\
\partial_3(e_{c_i^y})&=y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}} \hspace{10pt}\text{ for \(1\leq i\leq r-1\)}\\
\partial_3(e_{d_i})&=x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}}\hspace{32pt}\text{ for \(1\leq i\leq r-1\)}.
\end{align*}
If $g_r\neq 0$, then as we are resolving a monomial ring over a monomial quotient our syzygies must satisfy either
\begin{enumerate}[(i)]
\item $g_{r+1}=0$, $g_ix^{a_i-1}y^{b_i}=0$, and $g_ry^{b_r-1}=0$, or
\item $g_{r+1}\neq 0$, $g_{r+1}y=0$, and $g_ry^{b_r-1}-g_{r+1}x=0$.
\end{enumerate}
If condition (i) holds, we have the syzygy $g_i=0$ and $g_r=y$ generates all such new syzygies (all syzygies with $g_i\neq 0$ may be rewritten as sums of $y\cdot e_{r}$ and $x\cdot e_{f_i}$ for $1\leq i\leq r-1$.) In the second case we have $g_{r+1}\in M_y$, and for $g_r\neq 0$, we have new minimal syzygy $x\cdot e_{f_r}-y^{b_r-1}\cdot e_{f_{r+1}}.$ The proof that adding these two syzygies to our generating set gives us a complete $S$-spanning set for syzygies is similar to that given in Case 1 and omitted here.

As in Case 1, our third syzygy module has been shown to be generated by $e_{c_i^x}$, $e_{c_i^y}$, and $e_{d_i}$, with the described $\partial_3$.
\end{proof}

A careful examination of the degrees of the map $\partial_4$ produces the appropriate twists for $F_3$ in a graded resolution:

\[F_3=\left(\bigoplus_{1\leq i\leq r} S(-a_i-b_i-1)^2\right)\oplus\left(\bigoplus_{1\leq i\leq r-1} S(-a_i-b_{i+1})\right)\cong S^{3r-1}\]

\section{Fourth Syzygy Modules in Main Case}

Finally, we construct our fourth syzygy module $F_4$ and fourth syzygy map \(\partial_4\) in a minimal free resolution of \(\Bbbk\) over \(S\).

\begin{proposition}[Fourth Syzygy Module of \(\Bbbk\) over \(S\)]\label{prop:syz4} Let \(M=(x^{a_1}y^{b_1},...,x^{a_r}y^{b_r})\) be a monomial ideal given by its minimal set of generators, ordered so that \(a_1> a_2>\cdots>a_r\geq 0\) and \(0\leq b_1 <b_2<\cdots < b_r\).  Let either \(r>2\) or \(r=2\) and \(M\neq(x^n,y^m)\). The fourth syzygy module \(F_4\cong F_2^r\oplus F_1^{r-1}\) and \(\partial_4=\partial_2^r\oplus\partial_1^{r-1}\). We may choose as a generating set
\[\bigl\{e_{h_j^x},e_{h_j^y}\bigr\}_{j=1}^{r-1}\cup \biggl\{\bigl\{e_{k_{ij}}\bigr\}_{i=1}^{r+1}\biggr\}_{j=1}^r.\]
\begin{enumerate}[{\bf Case }1:]
\item If all generators of \(M\) are divisible by \(x\) (i.e. \(a_r\geq 1\)), then the fourth syzygy map \(\partial_4\) is given by
\begin{align*}
\partial_4(e_{h_j^x})&=x\cdot e_{d_j} \hspace{95pt} \text{ for \(1\leq j\leq r-1\)}\\
\partial_4(e_{h_j^y})&=y\cdot e_{d_j} \hspace{95pt} \text{ for \(1\leq j\leq r-1\)}\\
\partial_4(e_{k_{ij}})&=\begin{cases}
x^{a_i-1}y^{b_i}\cdot e_{c_{j}^x} &\hspace{20pt}\text{ for \(1\leq i,j\leq r\)}\\
-y\cdot e_{c_j^x}+x\cdot e_{c_j^y} &\hspace{20pt}\text{ if $i=r+1$ and \(1\leq j\leq r\)}.
\end{cases}
\end{align*}
\item If the final generator of \(M\) is \(m_r=y^{b_r}\), then the fourth syzygy map \(\partial_4\) is given by
\begin{align*}
\partial_4(e_{h_j^x})&=x\cdot e_{d_j} \hspace{95pt} \text{ for \(1\leq j\leq r-1\)}\\
\partial_4(e_{h_j^y})&=y\cdot e_{d_j} \hspace{95pt} \text{ for \(1\leq j\leq r-1\)}\\
\partial_4(e_{k_{ij}})&=\begin{cases}
x^{a_i-1}y^{b_i}\cdot e_{c_{j}^x} &\hspace{20pt}\text{ for \(1\leq i\leq r-1\) and \(1\leq j\leq r\)}\\
y^{b_r-1}\cdot e_{c_{j}^y} &\hspace{20pt}\text{ for \(i=r\) and \(1\leq j\leq r\)}\\
-y\cdot e_{c_j^x}+x\cdot e_{c_j^y} &\hspace{20pt}\text{ if $i=r+1$ and \(1\leq j\leq r\)}.
\end{cases}
\end{align*}
\end{enumerate}
\end{proposition}

\begin{proof}[Proof of Case 1]
From Proposition~\ref{prop:syz3}, our third syzygy module $F_3$ had generators
\[\bigl\{e_{c_i^x}\bigr\}_{i=1}^r\cup\bigl\{e_{c_i^y}\bigr\}_{i=1}^r\cup\bigl\{e_{d_i}\bigr\}_{i=1}^{r-1},\]
where the $\partial_3$ was given by
\begin{align*}
\partial_3(e_{c_i^x})&=x\cdot e_{f_i} \hspace{95pt} \text{ for \(1\leq i\leq r\)}\\
\partial_3(e_{c_i^y})&=y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}} \hspace{10pt}\text{ for \(1\leq i\leq r\)}\\
\partial_3(e_{d_i})&=x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}}\hspace{32pt}\text{ for \(1\leq i\leq r-1\)}.
\end{align*}
Beginning with syzygies on the $e_{d_j}$ terms, we note that $x\cdot x^{a_j-1}y^{b_{j+1}-1}=y\cdot x^{a_j-1}y^{b_{j+1}-1}=0\in S$, so we have minimal generators $\{x\cdot e_{d_j},y\cdot e_{d_j}\}_{j=1}^{r-1}$. We now find a generating set for remaining syzygies \emph{only} involving the $e_{c_j^x},e_{c_j^y}$.
Given a syzygy $g_1^x\cdot e_{c_1^x}+g_1^y\cdot e_{c_1^y}+\cdots+g_r^y\cdot e_{c_r^x}+g_r^y\cdot e_{c_r^y}$ for $g_i^x,g_i^y\in S$, we must have
\begin{align*}
g_1^x(x\cdot e_{f_1})&+g_1^y(y\cdot e_{f_1}+x^{a_1-1}y^{b_1}\cdot e_{f_{r+1}})+\cdots\\
&+g_r^x(x\cdot e_{f_1})+g_r^y(y\cdot e_{f_r}+x^{a_r-1}y^{b_r}\cdot e_{f_{r+1}})=0,
\end{align*}
So we have the relations
\begin{align*}
g_1^xx&+g_1^yy=0,\\
g_2^xx&+g_2^yy=0,\\
\vdots \\
g_r^xx&+g_r^yy=0,\text{ and}\\
\sum_{i=1}^rg_i^y &x^{a_i-1}y^{b_i}=0.
\end{align*}
Note that all syzygies of the field over a monomial quotient are generated by either $m\cdot e_{\ast}$ for some generator $e_{\ast}\in F_3$ or by pairs $m_1\cdot e_{\ast,1}+m_2\cdot e_{\ast,2}$ for $e_{\ast,1},e_{\ast,2}$ generators of $F_3$. As $g_j^xx+g_j^yy=0$ must hold for all $j$, it will suffice to consider syzygies on pairs $e_{c_j^x}$ and $e_{c_j^x}$.

Fix a $j\in\{1,...,r\}$ and consider syzygies on $\{e_{c_j^x},e_{c_j^y}\}$. The equalities $g_j^x x+g_j^yy=0$ and $g_j^y x^{a_j-1}y^{b_j}=0$ hold. From the second equality, we must have $g_j^y=0$, $g_j^y=x$, or $g_j^y=y^{b_{j+1}-b_j}$ for a degree minimal relation. As $g_x^xx+y^{b_{j+1}-b_j+1}\neq 0$ for any $g_j^x$ (by $y^d\not\in M$ for any $d\in{\mathbb N}$ in Case 1,) we must have either $g_j^y=0$ or $g_j^y=x$.

If $g_j^y=0$, then $g_j^x=0\in S$, or $g_j^x\in M_x$. This gives syzygies of the form $g_j^x=x^{a_i-1}y^{b_i}$ and $g_j^x=0$ for all $1\leq i\leq r$. If $g_j^y=x$, then $g_j^xx+xy=0$. The syzygy with $g_j^x=-y$ and $g_j^y=x$ is a minimal generator for all such syzygies.

So $\partial_4$ is of the form given in the statement of the theorem. Note that for each $1\leq j\leq r-1$, we have
\begin{align*}
F_4|_{\{e_{k_j^x},e_{h_j^y}\}}&\cong F_1\\
\partial_4|_{\{e_{h_j^x},e_{h_j^y}\}}&\cong \partial_1,
\end{align*}
and for each $1\leq j\leq r$ we have
\begin{align*}
F_4|_{\{e_{k_{ij}}\}_{i=1}^{r+1}}&\cong F_2\\
\partial_4|_{\{e_{k_{ij}}\}_{i=1}^{r+1}}&\cong \partial_2.
\end{align*}
So we have $F_4=F_1^{r-1}\oplus F_2^r$ and $\partial_4=\partial_1^{r-1}\oplus\partial_2^r$, completing our proof.
\end{proof}

\begin{proof}[Proof of Case 2]
In Case 2, the map $\partial_3$ is given by
\begin{align*}
\partial_3(e_{c_i^x})&=\begin{cases} x\cdot e_{f_i} &\hspace{15pt}\text{for \(1\leq i\leq r-1\)}\\
x\cdot e_{f_r}-y^{b_r-1}\cdot e_{f_{r+1}} &\hspace{15pt}\text{for \(i=r\),}
\end{cases}\\
\partial_3(e_{c_i^y})&=\begin{cases}y\cdot e_{f_i}+x^{a_i-1}y^{b_i}\cdot e_{f_{r+1}} &\hspace{3pt} \text{for \(1\leq i\leq r-1\)}\\
y\cdot e_{f_{r}} &\hspace{3pt} \text{for \(i=r\),}
\end{cases}\\
\partial_3(e_{d_i})&=x^{a_i-1}y^{b_{i+1}-1}\cdot e_{f_{r+1}} \hspace{43pt}\text{ for \(1\leq i\leq r-1\)}.
\end{align*}
The proof in Case 2 is identical to that in Case 1 for $\partial_4(e_{h_j^x})$ and $\partial_4(e_{h_j^y})$. When constructing $\partial_4(e_{k_{ij}})$ in Case 2, for any syzygies we have the equalities
\begin{align*}
g_1^xx+g_1^yy&=0,\\
g_2^xx+g_2^yy&=0,\\
\vdots \\
g_{r}^xx+g_{r}^yy&=0,\text{ and}\\
\left(\sum_{i=1}^{r-1}g_i^y x^{a_i-1}y^{b_i}\right)+g_r^xy^{b_r-1}&=0.
\end{align*}
After fixing a $j\in\{1,...,r\}$ and considering syzygies on pairs $\{e_{c_j^x},e_{c_j^y}\}$, we have minimal syzygies of the forms
\[\left\{x^{a_i-1}y^{b_i}\cdot e_{c_j^x}\right\}_{i=1}^{r-1}\bigcup\left\{y^{b_r-1}\cdot e_{c_j^y}\right\}\bigcup\left\{-y\cdot e_{c_j^x}+x\cdot e_{c_j^y}\right\},\]
via arguments similar to those in Case 1. As in Case 1, we now have that $\partial_4$ is of the form given in the statement of the theorem, with $F_4\cong F_1^{r-1}\oplus F_2^r$ and $\partial_4=\partial_1^{r-1}\oplus\partial_2^r$.
\end{proof}

\section{Proof of Main Theorem and Corollary~\ref{cor:main}}

We now return to the main proof of our theorem.

\mainthm

\begin{proof}
From Propositions~\ref{prop:syz1and2},~\ref{prop:syz3}, and~\ref{prop:syz4}, we  have that the first four stages of the resolution of $\Bbbk$ over $S$ may be written as
\begin{equation*}
{\mathcal F}:\;\;\;\Bbbk\longleftarrow S\longleftarrow F_1\longleftarrow F_2\longleftarrow F_3\longleftarrow\mathop{\oplus}^{\displaystyle F_2^r}_{\displaystyle F_1^{r-1}}.
\end{equation*}
We prove inductively that we may decompose the $F_{i}$ syzygy module in our resolution to a direct sum of the form $F_{1}^{u_i}\oplus F_{2}^{v_i}\oplus F_3^{w_i}$.

For $F_4$, we have by Proposition~\ref{prop:syz4} that $F_4=F_1^{r-1}\oplus F_2^{r}\oplus F_3^0$. Let $F_i=F_1^{u_i}\oplus F_2^{v_i}\oplus F_3^{w_i}$ be the $i^{\text{th}}$-syzygy module of $\Bbbk$ over $S$ for $i\geq 4$. Then a resolution of each component of the direct sum is given by the modules and maps
\begin{align*}
F_1^{u_i}&\xleftarrow{\;\;(\partial_2)^{u_i}\;\;} F_2^{u_i}\\
F_2^{v_i}&\xleftarrow{\;\;(\partial_3)^{v_i}\;\;} F_3^{v_i}\\
F_3^{w_i}&\xleftarrow{\;\;(\partial_1)^{(r-1)w_i}\oplus(\partial_2)^{rw_i}\;\;} F_1^{(r-1)w_i}\oplus F_2^{rw_i},
\end{align*}
so the $(i+1)^{\text{st}}$ syzygy module is $F_{i+1}=F_1^{(r-1)w_i}\oplus F_2^{u_i+rw_i}\oplus F_3^{v_i}$.
\end{proof}

\maincor

\begin{proof}
From Propositions~\ref{prop:syz1and2} and \ref{prop:syz3}, we have that
\begin{align*}
\beta_2^S(\Bbbk)&=r+1=2+(r-1)\text{ and}\\
\beta_3^S(\Bbbk)&=3r-1= (r+1)+2\cdot(r-1).
\end{align*}
To calculate $\beta_i^S(\Bbbk)$ for $i\geq 4$, we note that $F_4=F_2^r\oplus F_1^{r-1}$. So the $i^{\text{th}}$ stage of the resolution for $i\geq 4$ may be rewritten as $F_i=F_{i-2}^{r}\oplus F_{i-3}^{r-1}$, giving us a third-order linear recursion formula,
\[\beta_i^S(\Bbbk)=r\beta_{i-2}^S(\Bbbk)+(r-1)\beta_{i-3}^S(\Bbbk),\]
or equivalently, the second-order linear recursion,
\[\beta_i^S(\Bbbk)=\beta_{i-1}^S(\Bbbk)+(r-1)\beta_{i-2}^S(\Bbbk).\]

Our Poincar\'{e}-Betti series then is
\[P_S(z)=\frac{(1+z)^2}{1-rz^2+(1-r)z^3}=\frac{1+z}{1-z+(1-r)z^2}.\]
\end{proof}

\section{Conclusions and Future Work}
Between Theorem~\ref{thm:main} and Appendix~\ref{sec:degenerate}, we have a complete classification of all resolutions of $\Bbbk$ over monomial quotient rings $S=\Bbbk[x,y]/M$. Note that interestingly, the recursion formula appearing in the actual construction of syzygy modules \(F_i\) is
\begin{equation*}
F_i=(F_{i-2})^r\oplus(F_{i-3})^{r-1}\text{ if \(i\geq 4\).}
\end{equation*}
This third-order linear recursion formula matches the Poincar\'{e}-Betti series denominator calculated in~\cite{MR2188858},
\[P_S(z)=\frac{(1+z)^2}{1-rz^2+(1-r)z^3}=\frac{(1+z)^2}{b_S(z)},\]
where $b_S(z)$ has been interpreted combinatorially in terms of dimensions of the homologies of intervals in the lcm-lattice of $M$.

Whether all resolutions of $\Bbbk$ over $S=\Bbbk[x_1,...,x_n]/M$ for monomial ideals $M$ generated in degree 2 or higher, not all pure powers, can be decomposed via a similar recursive formula for $F_i$ is unknown. All monomial ideals $M$ in $\Bbbk[x,y,z]$ and $\Bbbk[x,y,z,w]$ that the author has examined so far, however, have had decompositions of their syzygy modules $F_i$ matching the formula $b_S(z)$ for the denominator of the Poincar\'{e}-Betti series produced in~\cite{MR2188858} and~\cite{MR2290910}.

\begin{openquestion}[Resolutions of $\Bbbk$ over Monomial Quotient Rings] Let $M=(m_1,...,m_n)\subset R=\Bbbk[x_1,...,x_k]$ with $\deg(m_i)\geq 2$ for all $i$ and \emph{not} all $m_i$ pure powers of the variables. Let $b_S(z)$ be the Poincar\'{e}-Betti denominator of $\Bbbk$ over $S=$ produced in~\cite{MR2188858} and~\cite{MR2290910}, setting $b_S(z)=1-c_2z^2-\cdots-c_dz^d$. Does there exist an $n\geq d+1$ and a decomposition of $F_i$ such that $F_i=\left(F_{i-2}\right)^{c_2}\oplus\cdots\oplus \left(F_{i-d}\right)^{c_d}$ for all $i\geq n$?
\end{openquestion}

Thanks to Brandon Stone and Courtney Gibbons for noting that the examples $M=(x^{\alpha},y^{\beta},z^{\gamma})$ for $\alpha,\beta,\gamma\geq 1$ are degenerate cases similar to those in Appendix~\ref{sec:degenerate}. In particular, this conjecture excludes complete intersections $M=(x_{i_1}^{\alpha_1},x_{i_2}^{\alpha_2},...,x_{i_r}^{\alpha_r})\subseteq \Bbbk[x_1,...,x_k]=R$ and $S=R/M$, as the Poincar\'{e}-Betti series of $\Bbbk$ over $S$ in these cases are \emph{not} of the form $b_S(z)=1-c_2z^2-\cdots-c_dz^d.$

\section*{Acknowledgements}
The author would like to thank James Parson for many interesting conversations on infinite free resolutions and Poincar\'{e} series, and to thank Neil Epstein for initially suggesting this problem. The author was partially supported by a Hood College Board of Associates (BOA) grant for Summer 2013.

redpajama_set_name: RedPajamaArXiv


The ongoing construction work and the preparations of the subsequent planning authorisation and operational documentation are entering the final phase. After intensive discussions with the key contract partners, performed on the basis of the necessary planning or equally on the basis of the work still outstanding, Flughafen Berlin Brandenburg GmbH has now fixed a schedule for the binding completion of construction, as agreed within the basic principles.
The schedule includes both the residual work and the subsequent expert assessments, which form the prerequisites for the formal start of the control and principle inspection at BER as required by the planning law. The schedule envisages a completion of these activities before the end of August 2018. This work will be followed by the intensive testing of the final official control and principal inspection, group tests for operational processes, and ORAT (Operational Readiness and Airport Transfer) commissioning.
Prof. Dr.-Ing. Engelbert Lütke Daldrup, CEO, emphasized that this decisive phase, from now until final completion of construction and technical commissioning of the new airport, is of great public interest and therefore, the steps up to and including its final completion must be explained clearly. "This year, FBB will state the opening date of BER. The prerequisite for this is that the input parameters should be valid and that all service providers involved in the project respect the schedule as agreed. This schedule, up to the start of the formal control and principle inspection, required under planning law, therefore forms a binding basis of work for all those participating in the project including construction companies."
In March 2017, FBB introduced quality control and quality assurance of its BER scheduling in order to independently assess and control the progress of construction, which is used to monitor and control both the plausibility of individual plans as well as their interaction.


text:
The Feinstein Center's oral history project consists of 48 interviews which were conducted by staff and volunteers between 1996 and 1999. They sought to capture and record the experiences of Jews who played a significant role in business, political, cultural, educational, philanthropic, and other aspects of the community.
Debra Kodish discusses her experience of anti-Semitism in Cleveland suburbs, tension with her parents over political and cultural views, and her discovery of Jewish communities. She elaborates on the start of the Philadelphia Folklore Project, her plans for life after the Project and her sense of Jewish identity and tradition.
Debra Kodish Oral History, April 20, 1998
Kodish, Debra
Philadelphia Folklore Project (Philadelphia, Pa.), University of Texas, American Folklore Society


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text:
Q: Unix: Delete File code giving some process I have coded java code to delete the file from the particular path. We are able to see the process running using the below command and we find only one entry.
root@ad4d # ps -ef | grep Del

root 27896 27895   0   Jun 24 pts/5       0:49 java -jar /oracle/php/online/jars/dist/DelFile.jar

But when the Administrator checks the process which is running from the command below we can see many child threads and parent thread process below
 root@almt4d # ps -aAdeflcjLPyZ | grep Del

 S   global    root 27896 27895 26891 26891     1   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891     2   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891     3   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891     4   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891     5   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891     6   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891     7   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891     8   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891     9   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    10   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    11   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    12   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    13   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    14   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    15   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    16   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    17   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    18   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    19   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    20   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    21   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    22   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    23   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    24   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    25   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    26   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    27   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    28   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    29   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    30   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    31   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    32   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    33   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    34   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    35   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    36   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    37   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    38   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    39   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    40   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    41   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    42   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    43   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    44   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    45   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    46   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    47   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    48   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    49   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    50   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    51   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    52   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    53   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    54   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    55   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    56   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    57   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    58   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    59   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    60   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    61   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    62   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    63   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    64   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    65   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    66   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    67   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    68   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    69   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    70   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    71   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    72   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    73   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    74   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    75   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    76   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    77   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    78   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    79   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    80   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    81   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    82   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    83   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    84   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    85   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    86   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    87   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    88   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:04 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    89   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    90   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    91   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    92   -    96   TS  58 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    93   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    94   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    95   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:45 java -jar /oracle/php/online/jars/d
 S   global    root 27896 27895 26891 26891    96   -    96   TS  59 37144 116392        ?   Jun 24 pts/5       0:00 java -jar /oracle/php/online/jars/d

Can anyone please explain from where these kind of child and parent thread process started executing.
I have executed only one DelFile.java which has only one thread
 Is there anyway to control the child threads which are visible to the Administrator using below command
root@almt4d # ps -aAdeflcjLPyZ | grep Del

DelFile.java
package DelFile;
import java.io.File;
import java.io.IOException;
import java.util.Date;
import org.apache.log4j.FileAppender;
import org.apache.log4j.Logger;
import org.apache.log4j.PatternLayout;
import org.apache.log4j.SimpleLayout;
public class DelFile extends Thread{
     static Logger log =Logger.getLogger(DelFile.class);
     FileAppender app = null;
     public DelFile()throws IOException{
         app = new FileAppender(new SimpleLayout(), "/oracle/php/online/jars/dist/del_file.log");

     }
     public void logSet(){
        app.setLayout(new PatternLayout(("%d{yyyy-MM-dd HH:mm:ss}  %-5.5p  [%t] %m%n")));
        log.addAppender(app);

    }
     boolean flage=true;
public void run(){

    while(flage){
         final String str="xls";
         String add="/oracle/php/online/jars/dist/";

         String[] paths;
         File f = null;

         try{
         f = new File(add);
         paths=f.list();
         for(String path:paths)
         {

            String temp="";
            String temp1="";
            int count=0;

                for(int i=path.length()-1;i>=0;i--){
                     if(path.charAt(i)!='.'){
                        count++;
                         temp+=path.charAt(i);//pmbs
                    }else{
                    for(int j=temp.length()-1;j>=0;j--){
                        temp1+=temp.charAt(j);
                        }
                        if(str.length()==count){
                        break;
                        }
                     }
                }
                if(str.equals(temp1)){
                        long time= new File(add+path).lastModified();
                        long utime=time+600000l;
                        while(utime>System.currentTimeMillis()){
                           Thread.sleep(601000);
                        }
                           logSet();
                           log.info(path +"  :  File is deleted");
                           new File(add+path).delete();
                }
         }
        }catch(NullPointerException np){

            try{
                Thread.sleep(600000);
            }catch(InterruptedException ip) {}

        }catch(InterruptedException ie){
        }
    }
}
public static void main(String[] args)throws IOException{
           DelFile r=new DelFile();
           r.start();
            }
}

A: boolean flage=true;
public void run(){
    while(flage){
         // flage not referenced below, no break, no return
         ...
    }

This is called an endless loop. It causes the thread to run for ever, which explains the many entries seen by ps.
Later
The LWPs (or threads) you see are JVM internal threads - not threads created by your program. I thought you were worried why they'd remain in the machine, apparently for ever, which is indeed due to the endless loop: the program doesn't terminate even after all files are gone.

redpajama_set_name: RedPajamaStackExchange


Reeta Chakrabarti is a BBC News presenter and correspondent. She presents the One, Six and Ten O'Clock News on BBC1 and also reports at home and abroad. She has previously held many prestigious reporting roles within the BBC and years reporting politics from Westminster.
Reeta has chaired and hosted many corporate events, including several Education Television Award Dinners at the Royal Television Society, conferences on climate change, the criminal justice system, the NHS, and the travel industry.


text:
Atlantic Sapphire chairman and CEO, Johan Andreassen
Photo: Trine Forsland
"The pandemic will lead to enormous changes in the market and consumer patterns"
Atlantic Sapphire CEO and founder Johan Andreassen believes the coronavirus can be an advantage for the land-based salmon.
"The pandemic will lead to enormous changes in the market and consumer patterns. Here in the United States, we see a big push to focus more on proper nutrition and good health. People note that there is a link between overall health and how sick one becomes from the virus. Seafood consumption in general in the US is significantly up after the coronavirus," Andreassen told the newspaper Dagens Næringsliv.
Read more: This is Atlantic Sapphire boss' new summer residence
Andreassen added that he believes that increasingly more expensive air freight during the pandemic reinforced his argument that being close to the market will produce good earnings.
After eight years of establishing, the land-based project of Atlantic Sapphire in Florida rounded off a milestone in late September when the first salmon were harvested. Now the company is harvesting weekly, with gradually increased volumes, according to Andreassen. But he did not want to comment on the price they were receiving for the salmon.
Atlantic Sapphire has ambitions to harvest 220,000 tonnes of salmon by 2031, which will make it one of the world's largest Atlantic salmon farmers.


Dr. Xiaoguang Qi, Associate Professor from International Business School Suzhou (IBSS) at Xi'an Jiaotong-Liverpool University (XJTLU), has been appointed as a Supervisor (Advisor) of Chinese Students & Scholars Association in Cambridge (CSSA Cambridge) since 2015.
He hosted a talent programme, 'Jiangsu Explore', at the University of Cambridge, accompanied by Professor Zhang (the programme head) from Jiangsu after the appointment.
'Jiangsu Explore' is organised by the China Zhi Gong Party of Jiangsu Province and funded by the provincial government. 'Jiangsu Explore' is approved by the Central Committee of the Communist Party China (CPC), CPC Jiangsu Provincial Committee and Government leaders such as Mr. Xueyong Li, Governor of Jiangsu Province.
Professor Zhang introduced the programme and policies on behalf of 'Jiangsu Explore'. Several rounds of discussions were carried out on the implementation of the programme and policies and it was well received at Cambridge with many Chinese scholars and students expressing great interests to join this programme in order to develop their career and serve Jiangsu Province.
'Jiangsu Explore' has achieved many great successes during the past five years. Supported by many enterprises within Jiangsu Province, 8,443 employment opportunities have been provided to overseas Chinese scholars and students throughout many continents, such as. America, Europe, Asia and Oceania. By the end of 2014, there were more than 70 overseas Chinese returning to China to develop their career in Jiangsu via this programme.
During recent years, the 'Jiangsu Explore' has won awards for outstanding achievements in community service. Its previous activities have been widely reported by the People's Daily (overseas edition), the Central Committee of the CPC's "Thousand Person Plan" Website, the People's Political Consultative Conference newspaper, the United Daily, Xinhua daily, Jiangsu TV and other Central and mainstream Medias in Jiangsu province.
"The CSSA-CAM is a not-for-profit and non-political association officially registered at the University of Cambridge and supported by Cambridge University and Chinese Embassy in the UK. Its main mission is to serve the needs of the Chinese students, scholars as well as thousands of local Chinese residents in and around Cambridge", said Dr. Qi.
"CSSA-CAM provides welfare and community support to them by facilitating the exchange of information, ideas, opportunities, friendship and cultural activities among Cambridge's Chinese, British and many other ethnic communities. CSSA-CAM is keen on collaboration with 'Jiangsu Explore' and contributing talent pool for better serving Jiangsu Province in particular and China in general", added Dr. Qi.
XJTLU, as an independent Sino-Foreign cooperative university, it captures the essence of both prestigious parent universities and is the first and only one of its kind approved by the Ministry of Education in China, is dedicated to regional and international community service. The faculties in IBSS and XJTLU bring the most dynamic and international insights and contributions to various communities by their global citizenship.