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Home Parliamentary Business Committees Senate Committees Senate Standing Committees on Foreign Affairs Defence and Trade Completed inquiries and reports Completed Inquiries 1999-02 Chapter 3 - Humanitarian assistance and security matters
Chapter 3 - Humanitarian assistance and security matters
Humanitarian and development assistance
Initial aid activities
3.1 Mr Dawson, AusAID, told the Committee on 6 December 1999 that the violence associated with the ballot had necessitated the withdrawal of all project staff from East Timor and the suspension of all humanitarian development assistance activities.
3.2 In the aftermath of the post-ballot violence and before a presence could be re-established on the ground in East Timor, AusAID had worked closely with the World Food Program to deliver over 180,000 daily rations to isolated areas within East Timor using Australian Defence Force aircraft. In addition, warehouses were set up in Darwin to receive relief goods from international agencies and Australian NGOs; commercial air and sea transport were secured to move supplies quickly from Darwin to East Timor; and trucks and other vehicles were obtained for transporting supplies within East Timor. Mr Dawson said that 'these contributions of transport and logistical support had been widely acknowledged by international relief agencies as having greatly enhanced the effectiveness and the responsiveness of the international relief effort'.[1]
3.3 Once Interfet arrived in East Timor, aid agencies initially focussed on providing critical humanitarian needs for food, water, shelter and medical assistance.[2] As the major supply routes were secured and displaced East Timorese began to return from West Timor, international agencies and Australian NGOs began providing assistance to family groups to re-establish themselves in East Timor and ensure adequate shelter and care for people during the 1999-2000 wet season.[3] After an intense period of immediate humanitarian assistance, the focus shifted to planning for longer-term reconstruction and recovery.
Australian financial aid
3.4 In September and October 1999, the Australian Government announced $14 million in emergency and humanitarian assistance for East Timor through international relief agencies, especially the World Food Program, the United Nations High Commissioner for Refugees, the International Committee of the Red Cross and Australian NGOs. On 22 November 1999, Foreign Minister Downer announced that aid program funding for East Timor in 1999-2000 would be increased by a further $60 million, of which $23 million would be given to the consolidated international humanitarian appeal for East and West Timor.[4]
3.5 This appeal, which then totalled over $300 million, was used to meet identified urgent humanitarian needs over the following nine months and it provided a framework for donors to co-ordinate their assistance. Australia's contributions to the appeal focussed strongly on 'assisting return and resettlement of displaced East Timorese from West Timor, the restoration of basic health and education services, urgent work on water supply and sanitation, and support for peace-building and initiatives involving local East Timorese groups'.[5]
3.6 Australia's humanitarian assistance to the East Timor crisis for 1999-2000 totalled approximately $37 million, which was the largest Australian contribution ever made to an international humanitarian relief effort. AusAID also provided assistance with reconstruction for longer-term development needs.[6]
3.7 Foreign Minister Downer said on 18 December 1999 that Australia would contribute $A25 million to the Reconstruction and Development Trust Fund and the United Nations Trust Fund for East Timor during 1999-2000. He said Australia's total program of assistance to East Timor for 1999-2000 would be at least $A75 million.[7] As it turned out, Australian Government assistance for East Timor in 1999-2000 totalled $81 million.[8]
3.8 In the 2000-01 budget, the Government committed $150 million over the following four years towards the reconstruction of East Timor, with $40 million allocated for the current year.[9]
3.9 All aid provided by Australia was under the humanitarian program in close consultation with the United Nations co-ordinating authorities, which identified priorities and tried to match donor resources against those priorities. Anything done by Australia for longer-term development was done in accordance with the priorities that UNTAET set down through its consultative mechanisms with the East Timorese representatives.[10] Australia agreed to participate in World Bank Multilateral Trust Fund programming by providing community development, agriculture, infrastructure and social sector experts.[11]
3.10 Australia provided advice to UNTAET on a number of recovery and reconstruction matters, including assessment of immediate needs in water supply and sanitation, the provision of an expert adviser in that sector, and advice on telecommunications needs and other areas. Australia also assisted with building the capacity of local NGOs, redeploying some longer-term project staff and picking up work which was commenced before the ballot.[12]
3.11 Bishop Manning said that the Australian Catholic Church was giving support through Caritas Australia. The Church would help the East Timorese in the area of human rights if there were human rights abuses, and by helping them on a humanitarian basis to help themselves by, for example, helping them to grow their crops or develop water supplies.[13]
3.12 Mr Dawson told the Committee that:
the World Bank took the lead in organising a joint assessment mission comprising representatives of bilateral donors, UN organisations, the Asian Development Bank and East Timorese technical experts. The IMF also conducted a parallel assessment mission. This mission was a very important exercise in longer term planning.
The mission identified immediate economic priorities, including restarting the flow of goods and services, establishing payments and banking systems and organising a budget. The mission also identified a range of key development challenges, including restoring agriculture, re-establishing basic health and education services, rebuilding essential infrastructure, creating a new civil service and judicial system and training East Timorese in rebuilding communities.
The total cost of these activities over three years is estimated to be in the range of $US260 million to $US300 million.[14]
3.13 The results of the mission were discussed at a meeting of international donors from over 50 countries and international agencies, jointly chaired by the United Nations and the World Bank and hosted by the Government of Japan, in Tokyo on 17 December 1999. The Tokyo meeting exceeded expectations and gathered over $US522 million in pledges of grant funds for the following three years, of which $US373 million was allocated for the reconstruction, development and civil administration of East Timor, including $US31.5 million for UNTAET expenses. An amount of $US215 million was allocated to two trust funds, one of $US140 million to be administered by the World Bank and the Asian Development Bank (ADB), the other to be administered by UNTAET in consultation with East Timorese representatives. Another $US149 million was for humanitarian assistance. The $US158 million remaining from the fund for development and reconstruction was pledged for other bilateral and multilateral reconstruction assistance.[15]
3.14 The ADB drew upon the $US140 million trust fund, which it jointly administered with the World Bank for meeting infrastructure needs in East Timor, while the World Bank drew upon the fund for agriculture, health and educational development.[16] During a visit to Dili on 21 February 2000, Mr James Wolfensohn, President of the World Bank, signed the Bank's first project commitment for East Timor, $US21.5 million over two and a half years for a Community Empowerment and Local Governance Project (CEP).[17]
3.15 Japan pledged $US100 million over the following three years, of which $US28 million was for humanitarian assistance. Japan also promised to provide financial support to East Timorese students studying at Indonesian universities.[18] The Japan International Cooperation Agency (JICA) was expected to spend $US30 million by September 2000 on infrastructure repair and development, agriculture, health and community assistance. JICA-funded projects began in February 2000 with rehabilitation of the road network, water supply improvement in fifteen towns, assistance to rice producers in Lautem and Manatuto, and fishery training and development.[19]
3.16 The European Commission pledged 'at least' 60 million euros.[20] The United States Congress had voted $US25 million for aid for East Timor in the 1999 budget.[21] At the United Nations and the World Bank meeting on aid to East Timor in Tokyo on 17 December 1999, the United States pledged $US72 million for humanitarian and development aid.[22] The United States Congress approved a further $25 million aid for the fiscal year beginning 1 October 2000.[23]
3.17 On 1 March 2000, the World Bank country director for East Timor, Mr Klaus Rohland, said that, since September 1999, per capita income in the territory had fallen from $US380 to around $US190 a year. Although crops, such as wheat, rice and coffee, had been little affected by the September violence, many animals had been killed, including cattle and most chickens, the mainstay of the local subsistence economy. Therefore, a plan to increase the number of cattle and chickens was a priority project for rebuilding East Timor. Mr Rohland said that the most important task was to replace international aid with productive economic activities.[24]
3.18 In response to a plea from CNRT President, Mr Xanana Gusmão and Vice-President, Dr José Ramos-Horta, which was made on 5 February 2000, the Asian Development Bank (ADB), on 23 March, approved a $US1 million grant over two years to develop a capacity building plan for parliamentary business, justice administration and public sector management.[25] On 29 March, the ADB and UNTAET signed a $US 29.7 million grant agreement for roads, ports and electricity repair and maintenance.[26]
3.19 On 26 May 2000, Hedi Annabi, United Nations Assistant Secretary-General for Peacekeeping, said $US14.1 million in a United Nations trust fund of $US28.6 million had been disbursed but the World Bank, which had collected $US38.4 million of $US147 million pledged by donors in Tokyo in December 1999, had only spent $US2 million to date. However, he said the bank had now finished planning the health, education and agricultural projects it would fund and was expected to spend up to $US40 million in coming months.[27]
Co-ordination
3.20 One of the key functions of UNTAET was to establish mechanisms for setting aid priorities. The Office of the Secretary-General's Special Representative had a position dedicated to perform that function, in close consultation with East Timorese representatives through the established consultative processes.[28] Prioritisation of programs financed by the World Bank Multilateral Trust Fund of $US147 million is performed every six months by an East Timorese group in close consultation with UNTAET and the World Bank. Trust Fund work plans are discussed with donors to the Fund to avoid overlap with bilateral projects.[29]
3.21 Under the Office for the Co-ordination of Humanitarian Affairs (OCHA) working groups were set up for particular sectors. A representative of AusAID regularly attended the daily OCHA co-ordination meetings.[30]
The Oecusse enclave
3.22 The district of Oecusse, also known as Ambeno, forms an enclave on the north-west coast separated by some 80 kilometres of Indonesian territory from the rest of East Timor.[31] It is where the Portuguese first established themselves on Timor.[32] The present border was the product of the centuries of struggle between the Dutch and Portuguese. Mr Michael Grant explained:
To put the border in a brief historical context, the borders of East and West Timor reflect centuries of political compromise between Holland and Portugal, the two ex-colonial powers. The borders do not reflect distinct cultural or linguistic boundaries. Perhaps these days they reflect a psychological boundary, nor, apart from the enclave of Oekussi, do they reflect indigenous Timorese ethnic boundaries. The boundary of the enclave Oekussi was a natural indigenous, autonomous boundary of a kingdom that traditionally has not wanted to be dominated by either Dili or Kupang in West Timor.[33]
3.23 Mr Grant referred to the land swaps that had taken place during the period of colonial rule. Maucatar on the southern salient of the East/West border was part of Dutch West Timor up until 1916 when it was swapped for other parts of Timor, and there was an earlier land swap in 1859. Maubara, where much of the political violence has occurred, was once a Dutch enclave.[34] At one stage during the border negotiations from 1859 to 1916 between the Dutch and Portuguese, the Portuguese had wanted to exchange Oecusse for territory contiguous to the rest of their Timorese territory. The local ruler had refused to agree.[35] Professor James Fox referred to the historical importance of the kingdom of Ambeno which, as a nominal vassal of Portugal, had been able to preserve a large measure of Timorese independence from both the Dutch and Portuguese until the early part of the twentieth century.[36]
3.24 The predominant language in Oecusse is Dawan rather than Tetum, the official local language of East Timor.[37] Dawan is also the predominant language in the central part of West Timor.
3.25 During the post-referendum violence, some 50,000 of its 58,400 inhabitants were forced to flee or were driven away from their homes. The local TNI-sponsored Saukunar militia, led by Laurentinho 'Moko' Soares, was accused of conducting a rampage of arson, kidnapping and murder.[38] Interfet entered Ambeno on 22 October 1999 and, by 25 January 2000, more than 230 bodies had been found across the district by United Nations authorities, including the site at Passabe of possibly the largest single massacre in all of East Timor following the referendum.[39]
3.26 The district's isolated location invited the question of its continued viability as part of an independent East Timor.
3.27 Dr Peter Bartu saw long-term difficulties in defending the enclave: 'It was such a historical and geographic anomaly that it would be a challenge for any Dili based government to bring the enclave fully into East Timor.'[40] Mr Hamish McDonald said:
In terms of its viability, I think that would largely depend on the nature of the border that comes into being between East Timor and Indonesia. If it was what you might call a hard border, patrolled and with restricted crossing, then I think it would be very difficult for Oecussi to be supported from East Timor ... If it returns to the porous border that it was in Portuguese times, I think it could quite easily exist as an enclave.[41]
3.28 Mr McDonald explained that viability would rest on the assumption that there was fairly free movement of citizens on a local level across the border and free movement along the roads across West Timor territory back into East Timor. He referred to the two land routes: a central route, going south out of Oecusse to Kefamananu and then joining the central highway in West Timor, and a more recently constructed coast road, linking Oecusse with the East Timor border near Atapupu, and said, if those roads were open and there was a co-operative spirit on both sides of the border, there was no reason why Oecusse should not continue to be an East Timorese territory. He concluded:
Politically, I think it would be foolish for the East Timorese leadership to even talk about offering up Oecussi as a territorial concession at this stage, given that there are voices on the Indonesian side and on the pro-Indonesian side in East Timor still talking about a partition of the main part of East Timor and keeping the western districts as some kind of Indonesian buffer zone or even a separate territory.[42]
3.29 Dr Gerry van Klinken was of the view that, if relations remained hostile, it would be difficult for the enclave to remain part of an independent East Timor. He went on to point out that treatment of the enclave's population by the militia was identical to, if not worse than, the way the East Timorese had been treated in the rest of East Timor. This indicated that the pro-independence feeling in the enclave had been just as strongly in favour of independence as it had been elsewhere. The international community therefore owed it to the people of Oecusse to treat them in exactly the same way as they treated the rest of East Timor.[43]
3.30 Mr Abel Guterres told the Committee the enclave was not negotiable:
Oecussi was still part of the territory of East Timor ... Before the Indonesian invasion they were under Portuguese rule. There was access. Things will change. We hope to have a good, cooperative relationship with West Timor, with Kupang, with the Nusa Tenggara Timor government. We hope to have that good working relationship in the area with the whole of eastern Indonesia. That will happen. There was endeavour in terms of communicating with local West Timor leadership. It will be in their interest and in our interest to maintain that cooperative role.[44]
3.31 Professor Charles Sampford said that the residents of Oecusse should be asked if they wanted to stay part of East Timor, and that their wishes should be respected. He said it was better to work on creating an international order in which border differences did not mean as much:
changing those boundaries would be a real problem. 'Those who try to correct the accidents of history create some of its greatest tragedies.' I think it was best in this particular case to say to the Indonesians or anybody else that that was part of East Timor until they want to leave and just work with it.[45]
3.32 The Committee notes that the basic international instrument, the Declaration on the Granting of Independence to Colonial Countries and Peoples of December 1960, expressly asserts, 'Any attempt aimed at the partial or total disruption of the national unity and the territorial integrity of a country is incompatible with the purposes and principles of the Charter of the United Nations.'[46] This would appear to give strong support to those in Timor who desired to preserve the territorial boundaries as finally established in 1916.[47]
3.33 On 29 February 2000, during President Abdurrahman Wahid's visit to Dili, Indonesia and UNTAET reached in principle agreement for free access by land from Ambeno to the rest of East Timor by way of a 'special corridor'.[48] Further talks on this were held on 8 June when the Governor of Nusa Tenggara Timur (West Timor and adjacent islands), Piet Tallo, visited Dili for round table talks with UNTAET officials and East Timorese leaders, and on 5 and 6 July during UNTAET-Indonesia talks on the border in Surabaya.[49] It had not been established by October 2000 because of the fragile security situation in West Timor.
3.34 Upon handing over to his Timorese successor at the end of July 2000, the first UNTAET district administrator of Ambeno, Mr Graham Day, said that it was essential for the future of the enclave that East Timor establish a transport link between Oecusse and Dili by sea that was entirely under its own control. 'Be it humble, be it infrequent, it must fly the Timorese flag, be controlled by Timorese authorities, and be subject to Timorese laws', he said. 'No resident of Oecusse is going to have confidence in a corridor open to the influence of the militia groups that continue to be active in the region.'[50]
3.35 UNTAET said that 'The takeover of parts of West Timor by militia groups makes it unlikely that a land corridor can be established anytime soon. Therefore, the only economic possibility for transport between Oecussi and East Timor proper is by sea', and announced that a passenger ferry service was being established to supplement the limited air service until a land corridor could be established. The East Timor Transitional Cabinet decided to provide a subsidy of $US5,000 per month to enable the Australia-based East Timor Shipping and Supply (ETSS) to operate the service. Early in 2001, there was speculation that another company might take over the ETSS passenger ferry service, which UNTAET called 'a temporary solution'.[51]
3.36 The Committee believes that the long-term future of Oecusse should be based on the wishes of the residents of the enclave. In the meantime, everything possible should be done to establish a secure land link between Oecusse and the rest of East Timor, free from the ravages and harassment of the militia or others yet to come to terms with an independent East Timor. Once security is restored in West Timor, and with co-operative Indonesian authorities, there is no reason to believe why Oecusse cannot operate successfully as part of East Timor.
AUSCIVPOL
3.37 The 2000-01 Budget provided for the maintenance of Australia's civilian police presence in UNTAET, funded from the $104.3 million over four years allocated to aid to East Timor. The Budget initiative provided for a continued civilian policing capacity of up to 80 deployed Australian civilian police (AUSCIVPOL) to meet UNTAET's requirement. A reserve pool of some 240 trained members was established, drawn from recently retired AFP and serving State and Territory police.[52]
3.38 The role of civilian police with UNTAET is to assist in every aspect of its peacekeeping and capacity-building role, from maintaining law and order at a community level to investigations to ensuring the security of prisons, airports, courts and harbours. They are also involved in the development and training of an East Timorese police force.[53]
3.39 On 15 February 2000, 28 State and Territory police were sworn in as Special Members of the Australian Federal Police to allow them to serve in East Timor. This was the first time that State or Territory police had served with a United Nations mission since 1976, when they served with the United Nations Force in Cyprus. Since then, although several police from Victoria and Queensland served as part of a United States-led multinational force in Haiti in 1995, the responsibility for contributing to United Nations missions had fallen solely to the AFP.[54] More than 250 Australian police served in East Timor between July 1999 and August 2000.[55]
3.40 Mr Alan Mills was selected by the United Nations to head the civilian police detachment (CIVPOL) which helped to conduct the 30 August 1999 ballot. Mr Mills headed a detachment of some 270 police, drawn from 15 countries, that liaised with the Indonesian police, who were responsible for security for the ballot.[56]
3.41 At the hearing of 15 September 1999, former Australian Federal Police agent, Mr Gary Wood, who was an accredited UN observer of the East Timor ballot, was critical of the conditions under which it was allowed to take place:
I think that, compared with formulas in the past, the formula of how they went was not quite right. If you take the example of Mozambique, there was a formula. People agree to go to different areas, the combatants go to different areas, the military come in, they then feed them and clothe them. The weapons are handed in, the military takes control and it is reasonably safe, then the military starts to withdraw, the civilian police start to come in and eventually there is some harmony on the ground. Then the electoral people come in with the humanitarian people ... and towards the end there is an election ... In 1994 I saw it work in Mozambique. East Timor was the opposite way around. It was like, 'Let's have an election and everything will be okay'... All the combatants were there. The anger, hostility and violence were still there. I thought that made the position of our people on the ground very difficult.[57]
3.42 Mr Wood referred to the very difficult situation unarmed CIVPOL officers consequently found themselves in when, following the ballot, they were required to protect people from the militias.
From the accounts of these good friends of mine, they said that when they had had enough - and you have to realise that these people are highly trained policeman who are used to being in a position of authority and having some power on the ground - and they had shouted at the militia, the police turned their firearms towards their stomachs and said, 'If you keep yelling at the militia, we will kill you.' That was on Tuesday morning, the day after the elections.[58]
3.43 Mr Wood said that people who had worked with the Australian CIVPOL as drivers and interpreters had been executed by the militias, in particular in the areas of Maliana and Ermera.[59]
3.44 Federal agent Sharon McCarthy, who received the International Policewoman's Association's Medal of Valour for her work in East Timor, took part in the evacuation from the UN's Liquiça compound when it came under fire from pro-Indonesian militia. She later described the experience:
There was militia on either side of the road. I remember seeing people in army uniforms with weapons. I remember weapons pointed at our vehicles, and I remember gunshots going off all around us. I remember a very large explosion in our vehicle—a strong smell of cordite ... When I later got out of the vehicle I realised that a round had actually come up through where my feet were in the vehicle and had gone through my backpack and up through the ceiling of the vehicle. We had shrapnel, sort of, all over us. We were picking it out of our skin for days. Not large chunks, but, you know, splinters of metal and glass.[60]
3.45 Public recognition of the vital role played by civilian police in securing the popular ballot in East Timor and in subsequent efforts to restore order to the territory was shown on 19 April 2000 when some 30 AFP members, who had served with UNAMET, Interfet and UNTAET, took part in the parade held in Sydney on that day for Australian Defence Force and AFP personnel who had served in East Timor.[61] Speaking in Sydney on 19 May 2000, Interfet Commander Major General Peter Cosgrove saluted the peacekeeping efforts of Australian Federal Police: 'Let there be no doubt the Australian Defence Force salutes the AFP for their great service in East Timor'.[62]
3.46 Presenting 16 members of the AFP with Police Overseas Service Medals and Clasp in Canberra on 8 August 2000, Justice and Customs Minister Senator Amanda Vanstone said:
AFP officers were amongst the first international contingents into East Timor, having served there since July 1999 when they played a significant role in the successful conduct of the self-determination ballot. Let's remember they were there first, they were there unarmed, they protected the Timorese while the ballot was being undertaken and they protected the ballot boxes to make sure that the will of the people was properly recorded.[63]
3.47 On 14 October 2000, Senator Vanstone announced that the United Nations would award its Service Medal for East Timor to the first detachment of Australian police officers who served in East Timor during the 1999 popular consultation. This involved the United Nations waiving the normal condition of award of a 90-day minimum period of service. The fourth detachment of 70 Australian police became the first to be presented with the medal in a ceremony in Dili on 14 October 2000. The first three detachments of about 170 police sent to East Timor since June 1999 had all been ineligible for the medal because of the 90-day condition. Senator Vanstone said she was pleased the UN had finally agreed to recognise the first and most deserving police sent to oversee the August 30 independence ballot. 'It was an accident of history that these people didn't serve the time normally required to achieve the UN medal,' she said. 'The men and women who went through the most dangerous time in Timor, who were there first, who were there unarmed before the army, are the ones who most deserve the medal.'[64]
3.48 The Committee believes that the police who served in East Timor as part of UNAMET had a more difficult and dangerous job than did the military as part of Interfet. They were unarmed and served there during the height of militia harassment and violence in the lead up to the 30 August poll and afterwards in the systemic destruction of the territory. Indonesia, which demanded and got responsibility for maintaining security in East Timor during the UNAMET period, abjectly failed in fulfilling that responsibility. There are obviously lessons to be learnt from this experience and the Australian Government should consider not acceding to such deployments in the future where the United Nations does not have responsibility for the security of its mission and where security arrangements are unlikely to be satisfactory.
United States role in East Timor
3.49 The United States had an important role to play in the processes that led to East Timor gaining its independence. Mr Alan Dupont emphasised this point in evidence to the Committee:
I think it is absolutely critical for the US to remain engaged politically and to be prepared to pressure the Indonesians, if necessary, because at the end of the day the US is the world's only superpower ... I think American political support is crucial to seeing ultimately a viable East Timorese state emerge from the ashes of the destruction of the last couple of months.[65]
3.50 Professor Hugh Smith agreed that the diplomatic role of the United States had been a key one, by providing diplomatic back-up and economic clout, through the IMF and other agencies, to create the right political and diplomatic atmosphere.[66]
3.51 Mr Tom Uren drew attention to the change in American policy toward East Timor. From 1975 until the Dili massacre, Indonesian actions in the territory had been accepted without question. Following that massacre, the Congress and Administration became concerned about human rights abuses and this changed to outright support for independence after the August 1999 ballot.[67] Mr Uren regretted that Australia had not worked to gain American support for East Timor's independence earlier in 1999, a view also put by Mr Robert Lowry, who said: 'I think that one of the great failures on Australia's part was not to mobilise support from the United States at that earlier period back in April or May'.[68]
3.52 Dr Harold Crouch referred to reports that senior American military and naval officers had spoken to Indonesian Armed Forces Chief, General Wiranto, during the crisis in early September 1999, to gain Indonesia's agreement to an international peacekeeping force in East Timor.[69] Following President Bill Clinton's condemnation of the Indonesian military's refusal to stop the violence in East Timor, the Commander-in-Chief of the United States Pacific Command, Admiral Dennis Blair, travelled to Jakarta, where he told General Wiranto the United States was cutting off the remainder of its military relations with Indonesia. He then left the country, cancelling a speech he was to deliver at the army staff college in Bandung. Meanwhile, Chairman of the Joint Chiefs of Staff, General Henry Shelton, spoke on three occasions with General Wiranto, finally on 10 September saying the United States would block the International Monetary Fund from delivering the funds that Indonesia needed to recover further from the Asian economic crisis unless either the violence in East Timor was halted or an international peacekeeping force was allowed into the territory. After visiting East Timor to inspect the situation there, General Wiranto telephoned General Shelton on 12 September to say he would recommend to President B.J. Habibie that he ask immediately for an international peacekeeping force.[70]
3.53 The United States provided vital support for Interfet, in particular in the form of state of the art intelligence equipment and expert personnel sent directly to Canberra in late September 1999 for installation at the Defence Signals Directorate headquarters at Russell Hill.[71]
3.54 Admiral Dennis Blair, Commander-in-Chief, Pacific (CINCPAC), said on 4 April 2000, during a visit to Jakarta, that Indonesia's armed forces needed to become more professional before military co-operation could resume. Admiral Blair said there were two main areas the Indonesian military needed to focus on. The first was that military personnel responsible for the rape, murder and destruction in East Timor in September 1999 should be brought to justice. The resumption of military links was also conditional on a peaceful solution being found to the refugee crisis in Indonesian West Timor, where United Nations officials estimated there were still about 120,000 East Timorese sheltering in camps, many of them being prevented from returning home by anti-independence militias. 'We need to see the disbanding and stopping of support to the militias', he said.[72]
3.55 Although Australia was instrumental in galvanising international support for the United Nations sponsored Interfet intervention in East Timor as a result of the post-ballot violence, it was the United States, using leverage that only it could bring into play, that finally persuaded Indonesia to support Interfet, thus allowing Interfet to operate in East Timor. The Committee believes that the United States has a continuing role to play in ensuring the future security of East Timor and the successful return of the remaining East Timorese held against their will by East Timorese militias in refugee camps in West Timor. With elements of the TNI still frustrating Indonesian Government efforts to resolve security and other problems in West Timor, United States involvement may be crucial in neutralising 'rogue elements'.
3.56 It was put to the Committee that the reduction of service to Indonesia by Radio Australia following the closure of the transmission station at Cox Peninsula near Darwin in June 1997 had had the unfortunate effect of reducing the flow of information into Indonesia and East Timor. Mr Bruce Haigh said: 'we withdrew the Radio Australia service, so how can they possibly know what is going on? We should have increased it, not decreased it. It is in our interests to have as much information flowing into the region as we can manage because, from their own sources, from their own newspapers, from their own radios, they will not get this information'.[73]
3.57 From 11 September 1999, Radio Australia was able to broadcast directly across Indonesia for the first time since 1997 because of a six-month contract the ABC negotiated for the lease of two transmitters in Taiwan and Singapore. This significantly increased broadcasting capacity, which had been limited to the transmission facilities at Shepparton, and enabled Radio Australia to be heard, on a not very good signal, in East Timor for about three hours or less per day.[74] Attempts by UNAMET to obtain transmission time on the Radio Australia transmitters were unsuccessful, and UNAMET finally turned to the Portuguese national broadcaster RTPI for access to their facilities.[75]
3.58 Former Radio Australia correspondent Ms Sue Downie told the Committee its importance for the East Timorese and Indonesian populations was to provide them with access to information about what was happening in their own territory, in Indonesia and the rest of the world.[76] Sister Susan Connelly said the re-installation of Radio Australia broadcasts in the region would allow its listeners an alternative view and could be welcomed by the new Indonesian Government.[77] Other witnesses also supported the case for an increase in Radio Australia services to East Timor and Indonesia.[78] Mr James Dunn said:
At this time, I think it is more important than ever that Radio Australia be opened. It is not only for the Timorese; it is also to give another side to the Indonesians who listen into radio quite a lot ... It should give our point of view and demonstrate over the airwaves, as Radio Australia was always good at, that Australians really care and are not just picking on Indonesia.[79]
3.59 Lieutenant General John Sanderson noted that explaining to the Indonesian people what Australia's actions, intentions and desires were would be equally important as explaining them to the East Timorese.[80]
3.60 Mr John Scott-Murphy agreed that it was important for Radio Australia to provide educational and general broadcasting services with an independent view to East Timor and for current services to be expanded. Caritas Australia had made a submission at the time of the proposed closing down of the service to that effect. He regarded Radio Australia as a crucial element of the provision of information to the local people, and for communication with people in general throughout South East Asia. He said: 'It really should be seen now as a terrible mistake to have reduced the service'.[81]
3.61 On 8 August 2000, the Australian Government announced that Radio Australia would receive up to $9 million over three years to extend its short-wave radio broadcasts. The Australian Government also would consider supporting commercial television broadcasts to the region. Minister for Communications, Information Technology and the Arts, Mr Richard Alston, and Minister for Foreign Affairs, Mr Alexander Downer, said the initiatives were a recognition of the importance of a credible, reliable and independent broadcaster in the region: 'Recent events have highlighted the value of Australia's international broadcasting activities in conveying accurate news and information to the region, as well as providing an Australian perspective'. ABC Chairman Donald McDonald said the ABC could now consider buying time at the Cox transmitter, which had been sold to the British charity broadcaster Christian Vision.[82]
3.62 In its Interim Report of September 1999, the Committee made three recommendations aimed at increasing Radio Australia transmissions to Indonesia. The Committee is pleased that the Australian Government has now enabled Radio Australia to improve its service to Indonesia, in accordance with the Committee's recommendations, even though it took the Government more than 10 months to come to that decision. In so doing, the Government has, at last, recognised the need to counter false and unfair criticisms of Australia, which have been broadcast by Indonesian broadcasters, and to put an Australian viewpoint direct to the Indonesian people.


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myDate           Location    Price
11/1/2013        South       10
11/1/2013        West        20
11/1/2013        East        10
11/2/2013        South       10
11/2/2013        West        20
11/2/2013        East        10
11/4/2013        South       10   <---- 11/3/2013 Data Missing
11/4/2013        West        30
11/4/2013        East        10

The way I tried to solve it was to find missing date in Access Database, and filled it with Null value using calender table.
myDate           Location    Price
11/1/2013        South       10
11/1/2013        West        20
11/1/2013        East        10
11/2/2013        South       10
11/2/2013        West        20
11/2/2013        East        10
11/3/2013                         <---- Null values
11/4/2013        South       10
11/4/2013        West        30
11/4/2013        East        10

But when I use crosstab query to get get price for each location, 11/3/2013 is skipped because it doesn't have any location related to it.
myDate    South   West   East
11/1/2013 10      20     10
11/2/2013 10      20     10    <---- 11/3/2013 Data skipped
11/4/2013 10      30     10

What I want my solution to be is following:
myDate    South   West   East
11/1/2013 10      20     10
11/2/2013 10      20     10
11/3/2013
11/4/2013 10      30     10

I thought I would be able to use NZ function when I'm calling this data from Access, and whenever I find null value, I can just put #N/A error in Excel spreadsheet so I would be able to catch what data is missing and fill it in later.
How would I accomplish this? Am I doing the right way or is there better way?
Any help or hint would be greatly appreciated.

A: Here's the way I would do it.
Don't include empty dates in the data table. Go back to the way you had it before, where the table only includes valid data.
Create a crosstab query on the table. This will not include the dates that have no data. (From your example:)
myDate    South   West   East
11/1/2013 10      20     10
11/2/2013 10      20     10    <---- 11/3/2013 Data skipped
11/4/2013 10      30     10

Save the crosstab query in your Access database. I'll call it qryXtab. Create a new query left joining your calendar table to qryXtab.
SELECT tblCalendar.cdate, South, West, East
FROM tblCalendar
LEFT JOIN qryXtab
ON tblCalendar.cdate = qryXtab.myDate
ORDER BY tblCalendar.cdate

Use this new query to populate your Excel file. It will include blank rows for dates that have no data.

redpajama_set_name: RedPajamaStackExchange


text:
Q: Calendar overlays sharepoint 2010 I have created calendar overlays with the colors I want by overriding CSS.
example following
How can I change the color dropdown list item names.
is it possible ?
 Screenshot attached.

A: Display the calendar that you wish you change the 'dropdown' name.
Click MODIFY VIEW
In the settings   VIEW NAME   change it to what you would like for it to show in the dropdown.
lq

A: I made a calendar in SharePoint 2013, where the user can select a category and e.G. category1 I changed the color to red. So I only needed one Calender without the overlay stuff.
If you need more information about it you can check my blogpost.
It is in German, but the scripts and the formular should work. You can give it a try, if this solution will work for you and test it in the SP2010 Environment.


The first step towards packing correctly during long distance moving is to harness the right packing materials. If you will be using the services of long distance moving companies in Montreal, then you should request the right packaging materials that will guarantee the safety of your items throughout the move. If you are not sure how to go about it, you can always task the moving company with the responsibility of packing as well, and they will do it professionally, but for an extra charge.
Marking the packing boxes will not only make the process easier, but also will ensure that the process of unpacking goes on very smoothly and all items are unpacked in the right location. In order to facilitate handling, use marker pens to clearly indicate what is packed, on the various boxes. For instance, clearly mark the boxes for utensils, electronics, clothes, etc.
Personal documents such as passports, birth certificates, social security cards and driver's licenses amongst others should be kept where you can easily reach them. During long distance moving, such documents face higher chances of getting lost or misplaced; hence you need to be very keen on how and where you keep them during the move. Furthermore, if you need to cross the US border, chances are, one or more of these documents will be necessary at a certain point during your trajectory.
Read on for more moving tips that will help you keep more money in your pocket?


text:
Published November 26
Rams' Aaron Donald believes referees are missing 'blatant' holding calls this year
Donald has nine sacks this season
By Ryan Gaydos | Fox News
Fox News Flash top headlines for November 26
Fox News Flash top headlines are here. Check out what's clicking on Foxnews.com.
Los Angeles Rams defensive tackle Aaron Donald is one of the best defenders in the NFL and, because of that, his opponents have to work extra hard to try and defend against him.
Donald said Wednesday he's had difficulty in the trenches and a lot of holding calls have been missed.
LIONS' MATTHEW STAFFORD JOINS EXCLUSIVE THANKSGIVING DAY CLUB WITH START AGAINST TEXANS
"Well, in my opinion they hold every play, but there's some blatant holds that we didn't get the call for," he said when asked about whether he thinks he's being blocked cleanly.
"You just have to keep playing. Hopefully, I'll get a couple. We just have to keep playing, that's all I can say. Hopefully, the refs will see the holding calls. If they don't, they don't and if they do, they do."
Donald was held without a tackle against the Tampa Bay Buccaneers for the second straight game. He is still among the leaders in sacks this season.
NFL DOCTOR ON STEELERS-RAVENS POSTPONEMENT: DECISIONS SHOULD BE 'RULED' BY SCIENCE, NOT SCHEDULING
Rams defensive coordinator Brandon Staley was asked about the lack of penalties against teams for holding Donald.
"Interesting about Aaron Donald that he has not drawn a holding call in I think three or four weeks, which we're going to have to take a look at because that's just saying that people are blocking him straight up, which we know is not the case," Staley said.
"So, holding calls just in general are down, but that's something that is a concern, because it happened in the Seattle game as well. Players like that, it's like Shaquille O'Neal back when he was playing – this guy, I think people take for granted how good he is and we need to make sure that people are seeing the game the way they need to."
Donald has 26 tackles and nine sacks this season.
Ryan Gaydos is the sports editor for FoxNews.com. Follow him on social media @Gaydos_

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d>0$ and $Q$ an $\mathfrak m$-primary ideal of $R$. Let $\lambda_R(M)$ denote the length of an $R$-module $M.$
The Hilbert-Samuel function of $Q$ is defined as $H(Q,n)=\lambda_R(R/Q^n)$ for $n\in\mathbb{Z}$. It is well known that $H(Q,n)$  coincides with a polynomial
$P(Q,n)$ of degree $d$ for all $n\gg 0.$ The polynomial $P(Q,x)$
is called the Hilbert-Samuel polynomial of $Q$. We may write
\begin{equation*}
P(Q,x)=e_0(Q)\binom{x+d-1}{d}-e_1(Q)\binom{x+d-2}{d-1}+\ldots+(-1)^de_d(Q)
\end{equation*}
for unique integers $e_i(Q)$ known as the Hilbert coefficients of $Q$.

 In the classical case of a Cohen-Macaulay
local ring, relations among various Hilbert coefficients and bounds for them have been explored by several authors. Northcott's inequality \cite{northcott}
$e_1(Q)\geq e_0(Q)-\lambda_R(R/Q)$ is one of the first results in this direction. It was improved by M. E. Rossi in \cite{rossi-reduction}.
Several bounds on $e_1(Q)$ in terms of $e_0(Q)$ exist in literature, see \cite{elias}, \cite{elias2}, \cite{hanumanthu-huneke}, \cite{RV} and \cite{RV2}.
For example, Rossi and Valla in \cite[Proposition 2.10]{RV} proved that $e_1(Q)\leq \binom{e_0(Q)-k+1}{2}$ where
 $Q\subseteq \mathfrak m^k.$
Such bounds are useful for examining the finiteness of Hilbert functions of ideals with fixed multiplicity, see \cite{srinivas-trivedi}.
With $R$ Cohen-Macaulay, it is known that $e_1(Q)$ and $e_2(Q)$ are non-negative, due to Northcott \cite{northcott} and Narita \cite{narita} respectively.
However the higher coefficients are not necessarily non-negative. Marley \cite[Example 2.3]{Marley} gave an example of a Cohen-Macaulay local ring and an
$\mathfrak m$-primary ideal $Q$ with $e_3(Q)<0$.
The higher coefficients are not yet explored much except when the associated graded ring have high depth. By depth of a standard graded ring with a
unique graded maximal ideal, we mean the grade of the unique maximal ideal. Let
$G(Q)=\mathop{\oplus}\limits_{n\geq 0} Q^n/Q^{n+1}$ denote the associated graded ring of $Q$. Marley \cite[Corollary 2.9]{Marley} showed
that in a Cohen-Macaulay local ring, $e_i(Q)\geq 0$ for $0\leq i\leq d$ provided depth of $G(Q)$ is at least $d-1$.

The case when $R$ is not Cohen-Macaulay is quite different. Let $Q$ be a parameter ideal of $R$ hereafter.
In this paper, we find uniform lower and upper bounds for the coefficients $e_i(Q)$ for $2\leq i\leq d$. In \cite[Theorem 3.5]{msv},
  Mandal, Singh and Verma  proved that $e_1(Q)\leq 0$.
 Mccune in \cite[Theorem 3.5]{Lori} showed that if $\depth R\geq d-1$ then $e_2(Q)\leq 0$. In the same paper, she proved that $e_i(Q)\leq 0$  for $2\leq i\leq d$
 provided $\depth G(Q)\geq d-1$.
We improve upon Mccune's result by relaxing the hypothesis to $\depth G(Q)\geq d-2$.
If $\depth G(Q)\geq d-1$, we provide a uniform lower bound for $e_i(Q)$ which is independent of $Q$.
 Let
$H_\mathfrak m^i(*)$ denote the $i$-th local cohomology functor with support in the maximal ideal $\mathfrak m$.
Goto and Ozeki \cite[Theorem 3.2]{goto-ozeki}
showed that in two dimensional local ring with $\depth R\geq 1$, $-\lambda_R(\h_{\mathfrak m}^1(R))\leq e_2(Q)\leq 0$. We extend their result to rings of  dimension
$d$ and $\depth$ at least $d-1$.  Our proofs are essentially inspired from the methods developed in \cite{goto-ozeki}.

Let $\mathcal{R}=R(I)=\mathop\oplus\limits_{n=0}^\infty I^nt^n\subseteq R[t]$ and $\mathcal{R}^*=R^*(I)=\mathop\oplus\limits_{n\in\mathbb{Z}}I^nt^n\subseteq R[t,t^{-1}]$
denote the Rees algebra and the extended Rees algebra respectively of an ideal $I$.
 We put $\mathcal{M}=\mathfrak m \mathcal{R}+\mathcal{R_{+}}$ where $\mathcal{R_{+}}=\mathop\oplus\limits_{n>0}\mathcal{R}_n$
is the irrelevant ideal of the Rees algebra $\mathcal{R}$. Let $[T]_n$ denote the $n$-th graded piece of a graded $\mathcal{R}$-module $T$.
This paper is organized as follows.

In Section \ref{section-formula}, we discuss some lemmas concerning the local cohomology modules $\h_{\mathcal{M}}^i(G(I))$ of $G(I)$ with support in
$\mathcal{M}$ for an ideal $I$.
These results mainly develop the setting for the proofs of our main results in
subsequent sections. The content of this section is very basic but we include them for clarity.

In Section \ref{bounding-the-hilbert-coefficients}, we prove that $e_d(Q)\leq 0$ for a parameter ideal $Q$
if $[\h_{\mathcal{M}}^i(G(Q))]_n=0$ for all $n\leq -1$ and $0\leq i\leq d-3$ (Theorem \ref{theorem-main-(-1)}). Note that
this condition holds if $\depth G(Q)\geq d-2.$ As a consequence, we prove the following theorem.
\begin{theorem}
   Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 3$ and $\depth R\geq d-1$. Let $Q$ be a parameter ideal. Then $e_3(Q)\leq 0.$
\end{theorem}

For the higher coefficients $e_i(Q)$, we are able to prove the following result.
\begin{theorem}
  Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 2$ and $\depth R\geq d-1$. Let $Q$ be a parameter ideal of $R$.
  \begin{enumerate}
   \item Suppose $\depth(G(Q))\geq d-2$. Then $ e_i(Q)\leq 0$ for $2\leq i\leq d$.
   \item Suppose $\depth(G(Q))\geq d-1$. Then $ e_i(Q)\geq -\lambda_R(\h_{\mathfrak m}^{d-1}(R))$ for $2\leq i\leq d$.
  \end{enumerate}
\end{theorem}

In Section \ref{vanishing-of-coefficients}, we discuss the vanishing of the last coefficient $e_d(Q)$. In \cite[Theorem 3.2]{goto-ozeki}, Goto and Ozeki found a necessary and sufficient condition
for the vanishing of $e_2(Q)$ in rings of dimension two and depth at least one. We generalize their result in two directions. In Theorem \ref{theorem-vanishing-of-ed}, we find a necessary condition for the
equality $e_d(Q)=0$ provided $\depth R\geq d-1$. Let $Q=(x_1,\ldots,x_d)$ with
$x_1^*,\ldots,x_{d-1}^*$ a superficial sequence. Suppose $\depth G(Q)\geq d-2$ and $\h_{\mathcal{M}}^{d-2}(G(Q))$ is finitely graded, then we prove that
$e_d(Q)=0$ implies $x_1^l,\ldots,x_{d-1}^{l},x_d^{(d-1)l}$ is a $d$-sequence for all integers $l\geq 1$. Further in Theorem \ref{theorem-e2}, we extend \cite[Theorem 3.2]{goto-ozeki}
to dimension $d\geq 2$ by providing similar equivalent conditions for the vanishing of $e_2(Q)$. Our result on $e_2(Q)$ also includes \cite[Theorem 3.2]{Lori}.

We refer to \cite{bruns-herzog} for undefined terms.
\section{Preliminary results}\label{section-formula}
In this section, we discuss few lemmas which are the key steps for the results of subsequent sections.
For this section, let $I\subseteq \mathfrak m$ be an arbitrary ideal of $R$.  We have
$$H_{\mathcal{M}}^i(G(I))=H_{\mathcal{R}_{+}}^i(G(I))=H_{G_{+}}^i(G(I)).$$
It is well known that there exists $m_I\in\mathbb{Z}$  such that \begin{equation}\label{eqn-6-09}
                  [\h_{\mathcal{M}}^i(G(I))]_{n+1}=0 \text{ for all } n\geq m_I \text{ and for all } i\in\mathbb{Z}.
                 \end{equation}
For an element $0\neq x\in R$,
let $x^*$ denote the  initial form of $x$ in $G(I)$, i.e.
the image of $x$ in $G(I)_i$  where $i$ is the unique integer such that $x\in I^i\setminus I^{i+1}$.

\begin{lemma}\label{lemma-3}
Let $R$ be a Noetherian local ring and $I$ an ideal of $R$.
Let $x_1,\ldots,x_r\in I\setminus I^2$ such that $x_1^*,\ldots,x_r^*$ is a
regular sequence in $G(I)$. Then, for $1\leq j\leq r$,
\begin{equation*}
 [\h_{\mathcal{M}}^j(G(I))]_{m_I}\cong  [\h_{\mathcal{M}}^{j-1}(G(IR_1))]_{{m_I}+1}\cong\ldots\cong [\h_{\mathcal{M}}^0(G(IR_j))]_{m_I+j}
\end{equation*}
where $R_j=R/(x_1,\ldots,x_j)$. Moreover, for all $n\geq m_I+j$ and for all $i\in\mathbb{Z}$,
\begin{equation*}
[\h_{\mathcal{M}}^i(G(IR_j))]_{n+1}=0.
\end{equation*}
\end{lemma}
\begin{proof}
 We apply induction on $r$.  Consider the following exact sequence and the induced
 long exact sequence of local cohomology modules.
 \begin{align}
  0&\longrightarrow G(I)(-1)\xrightarrow{x_1^*} G(I)\longrightarrow G(IR_1)\longrightarrow 0\nonumber\\
  \ldots\longrightarrow \h_{\mathcal{M}}^i(G(I))&\longrightarrow \h_{\mathcal{M}}^i(G(IR_1))\longrightarrow  \h_{\mathcal{M}}^{i+1}(G(I))(-1)\longrightarrow \h_{\mathcal{M}}^{i+1}(G(I))
  \longrightarrow\ldots\label{long-exact-seq-1}
  \end{align}

 Since $[\h_{\mathcal{M}}^i(G(I))]_{n+1}=0$ for all $n\geq m_I$ and $i\in\mathbb{Z}$, we get from \eqref{long-exact-seq-1} that  for all $i\in\mathbb{Z}$,
 \begin{align}
[\h_{\mathcal{M}}^i(G(I))]_{m_I}\cong [\h_{\mathcal{M}}^{i-1}(G(IR_1))]_{m_I+1}\text{ and }\label{induction-eqn-1}\\
   [\h_{\mathcal{M}}^i(G(IR_1))]_{n+1}=0 \text{ for } n\geq m_I + 1.\nonumbe
 \end{align}
 By putting $i=1$ in \eqref{induction-eqn-1}, we get the result for $r=1$.

  Suppose $r>1$ and the assertion holds for $r-1$.
 Since $x_1^*,\ldots,x_r^*$ is a regular sequence in $G(I)$, we have $G(IR_1)\cong G(I)/x_1^*G(I)$ and $x_2^*,\ldots,x_r^*$ is
a regular sequence in  $G(IR_1)$. By induction hypothesis, for $1\leq k\leq r-1$,
 \begin{gather}
[\h_{\mathcal{M}}^k(G(IR_1))]_{m_I+1}\cong  [\h_{\mathcal{M}}^{k-1}(G(IR_2))]_{m_I+2}\cong \ldots\cong [\h_{\mathcal{M}}^0(G(IR_{k+1}))]_{m_I+1+k}\text{ and }\label{induction-eqn-3}\\
\text{[}\h_{\mathcal{M}}^i(G(IR_{k+1}))]_{n+1}=0 \text{ for all } n\geq m_I+1+k \text{ and for all } i\in\mathbb{Z}. \label{induction-eqn-4}
 \end{gather}
Now  let $1\leq j\leq r$.
We may assume that $j>1$. Then  \eqref{induction-eqn-1}, \eqref{induction-eqn-3}
and \eqref{induction-eqn-4} with $k=j-1$ give
\begin{gather*}
  [\h_{\mathcal{M}}^j(G(I))]_{m_I}\cong  [\h_{\mathcal{M}}^{j-1}(G(IR_1))]_{m_I+1}\cong\ldots\cong [\h_{\mathcal{M}}^0(G(IR_j))]_{m_I+j}\text{ and }\\
 \text{[}\h_{\mathcal{M}}^i(G(IR_j))]_{n+1}=0  \text{ for all } n\geq m_I+j \text{ and for all } i\in\mathbb{Z}.
  \end{gather*}
\end{proof}
The next lemma relates the local cohomology of Rees algebra and the associated graded ring.
\begin{lemma}\label{lemma-1}
Let $R$ be a Noetherian local ring and $I$ an ideal of $R$. Then
$$[\h_{\mathcal{R}_{+}}^i(\mathcal{R})]_n\cong [\h_{\mathcal{R}_{+}}^i(G(I))]_n$$ for all   $n>\max\{ m_I-1,-1\}$ and for all $i\in\mathbb{Z}$.
\end{lemma}
\begin{proof}
 Consider the following exact sequences with the canonical maps
 $$0\longrightarrow \mathcal{R}_{+}\longrightarrow \mathcal{R}\longrightarrow R\longrightarrow 0\text{ and } 0\longrightarrow \mathcal{R}_{+}(1)\longrightarrow \mathcal{R}\longrightarrow G(I) \longrightarrow 0$$
 and apply the functor $H^i_{\mathcal{R}_{+}}(*)$ to get
 \begin{gather}
  \hspace{-3cm}\ldots\longrightarrow H^{i-1}_{\mathcal{R}_{+}}(R)\longrightarrow H^i_{\mathcal{R}_{+}}(\mathcal{R}_{+})\longrightarrow H^i_{\mathcal{R}_{+}}(\mathcal{R})\longrightarrow H^{i}_{\mathcal{R}_{+}}(R)\longrightarrow \ldots\label{exact-seq1}\\
 \longrightarrow H^{i-1}_{\mathcal{R}_{+}}(G(I))\longrightarrow H^{i}_{\mathcal{R}_{+}}(\mathcal{R}_{+})(1)\longrightarrow H^{i}_{\mathcal{R}_{+}}(\mathcal{R})\longrightarrow
  H^{i}_{\mathcal{R}_{+}}(G(I))\longrightarrow H^{i+1}_{\mathcal{R}_{+}}(\mathcal{R}_{+})(1)\longrightarrow\label{exact-seq2}
 \end{gather}
Since $[H^{i}_{\mathcal{R}_{+}}(G(I))]_n=0$ for all $i\in\mathbb{Z}$ and for all $n>m_I$, we get from exact sequence \eqref{exact-seq2} that
\begin{equation}
[H^{i}_{\mathcal{R}_{+}}(\mathcal{R}_{+})]_{n+1}\cong[H^{i}_{\mathcal{R}_{+}}(\mathcal{R})]_n\label{eqn-chap-last-e1}
\end{equation}
for all $n> m_I$ and for all $i\in\mathbb{Z}$. Further by exact sequence \eqref{exact-seq1}, we have
\begin{equation}[H^{i}_{\mathcal{R}_{+}}(\mathcal{R}_{+})]_{n+1}\cong [H^{i}_{\mathcal{R}_{+}}(\mathcal{R})]_{n+1}\label{eqn-chap-last-e1-17/10-1}\end{equation}
for all  $n\geq 0$ and $i\in\mathbb{Z}$. By \eqref{eqn-chap-last-e1} and \eqref{eqn-chap-last-e1-17/10-1},
$[H^{i}_{\mathcal{R}_{+}}(\mathcal{R})]_n\cong [H^{i}_{\mathcal{R}_{+}}(\mathcal{R})]_{n+1}$ for all
 $n>\max\{m_I,0\}$.
Since $[H^{i}_{\mathcal{R}_{+}}(\mathcal{R})]_n=0$ for all $n\gg 0$,
$[H^{i}_{\mathcal{R}_{+}}(\mathcal{R})]_n=0$ for all  $n>\max\{m_I,0\}$. Therefore
by \eqref{eqn-chap-last-e1-17/10-1}, $[H^{i}_{\mathcal{R}_{+}}(\mathcal{R}_{+})(1)]_{n}=0$
for all   $n> \max\{m_I-1,-1\}$ and $i\in\mathbb{Z}$. Now by exact sequence \eqref{exact-seq2}, we get that
$$[\h_{\mathcal{R}_{+}}^i(\mathcal{R})]_n\cong [\h_{\mathcal{R}_{+}}^i(G(I))]_n$$
for all  $n> \max \{m_I-1,-1\}$ and for all $i\in\mathbb{Z}$.
\end{proof}
We set
\begin{align*}
 a_i\left(G(I)\right)&:= \sup\{n\in\mathbb{Z}: [\h_{\mathcal{R_+}}^i(G(I))]_n\neq0\}  \mbox{ and }\\
 b_i\left(G(I)\right)&:= \inf\{n\in\mathbb{Z}: [\h_{\mathcal{R_+}}^i(G(I))]_n\neq0\}.
\end{align*}
By convention, if $\h_{\mathcal{R_+}}^i(G(I))=0$ then we set $a_i(G(I))=-\infty$ and $b_i(G(I))=\infty$. Note that
$b_0(G(I))\geq 0$.
The next lemma plays crucial role in most of our proofs.
 We show that given a parameter ideal $Q$ with $b_i(G(Q))\geq 0$ for $0\leq i\leq d-3,$
 $G(Q^l)$ has high depth for all $l\gg 0$.

 Recall that a {\it reduction} of an ideal $I$ is an ideal $J\subseteq I$ such that $I^{n+1}=JI^n$ for some $n\geq 0.$ A {\it minimal reduction} of $I$ is a reduction of $I$ which is minimal with respect to inclusion.
For a minimal reduction $J$ of $I$, {\it reduction number} of $I$ with respect to $J$, denoted by $r_J(I)$, is the least non-negative integer $n$ such that $I^{n+1}=JI^n$.
For an ideal $I$, let $\mu(I)$ denote the minimal number of generators of $I$.
\begin{lemma}\label{lemma-4}
  Let $R$ be a Noetherian local ring of dimension $d\geq 2$ and
$\depth R\geq d-1$. Let $Q$ be parameter ideal such that $b_i(G(Q))\geq 0$ for $0\leq i\leq d-3$. Then the following assertions hold.
 \begin{enumerate}[label=(\alph*)]
 \item \label{lemma-4-item-0} For all $l\gg 0$,
 \begin{align}\label{31-aug-6-e1}
  a_i(G(Q^l)))\leq \begin{cases}
      0 &\text{ for all } i\in\mathbb{Z} ,\\
      -1 & \text{ for } i=d, d-2
 \end{cases}
\end{align}
 Furthermore, $\h_{\mathcal{M}}^i(G(Q^l))=0$  for  $0\leq i\leq d-3.$ In particular, $\depth G(Q^l)\geq d-2$ for $l\gg 0.$

\item \label{lemma-4-item-3} Suppose $\h_{\mathcal{M}}^{d-2}(G(Q))$ is finitely graded and  $[\h_{\mathcal{M}}^{d-1}(G(Q^{l}))]_0=0$ whenever $l$ is sufficiently large. Then $\depth G(Q^{l_0})\geq d-1$ for any sufficiently large integer $l_{0}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We may assume that the residue field $R/\mathfrak m$ is infinite.
 Let $Q=(x_1,\ldots,x_d)$ such that $x_1^*,\ldots,x_d^*$ is a superficial sequence. For $l\geq 1$, we put $I=Q^l$.
  For a rational number $a$, let $\lfloor a \rfloor=\max\{n\in\mathbb{Z}: n\leq a\}$.
Given that  $b_i(G(Q))\geq 0$ for $0\leq i\leq d-3$. Thus $b_i(G(I))\geq \lfloor b_i(G(Q))/l \rfloor \geq 0$ for $0\leq i\leq d-3$ by
\cite[Lemma 2.4]{Hoa}. Choose  $l>\max\{|a_i(G(Q))|:a_i(G(Q))\neq -\infty\}$
and $y_i=x_i^l$ for $1\leq i\leq d$. Then $y_1^*,\ldots,y_d^*$ is a superficial sequence with respect to
$I$ and
\begin{equation}\label{eqn-reduction-no-of-I}
 I^{d}=(y_1,\ldots,y_d)I^{d-1}.
\end{equation}
 To see the above equality, note that $I^d=Q^{ld}$ is generated by monomials in $x_1,\ldots,x_d$ of degree $ld$.
Let $m=x_1^{t_1}\ldots x_d^{t_d}$ with $t_1+\ldots+t_d=ld$ be a generator of $I^d$. Then at least one $t_i\geq l$, so
$m=x_i^{l}.m'$ where $m'=x_1^{t_1}\ldots x_i^{t_i-l}\ldots x_d^{t_d}\in Q^{ld-l}$. Hence
$m\in (x_1^l,\ldots,x_i^l,\ldots,x_d^l)Q^{l(d-1)}=(y_1,\ldots,y_d)I^{d-1}$ which gives
$I^d \subseteq(y_1,\ldots,y_d)I^{d-1}$.
 Since $J=(y_1,\ldots,y_d)$ is a reduction of $I$, $\mu(J)= d$. Hence $J$ is a minimal reduction of $I$ with $r_J(Q^l)\leq d-1$. So by
 \cite[Proposition 3.2]{trung-reduction}, $a_d(G(I))\leq r_J(I)-d.$

$\ref{lemma-4-item-0}$ It follows that $a_d(G(I))<0$ and  $a_i(G(I))\leq \lfloor a_i(G(Q))/l\rfloor \leq 0$ for $i\leq d-1$ by choice of $l$ and \cite[Lemma 2.4]{Hoa}.
In other words,
 \begin{align}\label{equation:bi(G(I))}
  [\h_{\mathcal{M}}^i(G(I))]_n=0\begin{cases}
                                  \text{ for all } n\geq 1 \text{~~and } i\in \mathbb{Z},\\
                                  \text{ for all } n\neq 0 \text{~~and } 0\leq i\leq d-3,\\
                                  \text{ for all } n\geq 0 \text{~~and } i=d.
                              \end{cases}
 \end{align}
 \textbf{Claim:} $[\h_{\mathcal{M}}^{i}(G(I))]_0=0 \text{ for } 0\leq i\leq d-2$.

 {\it {Proof of Claim.}}
   We apply induction on $i$. For $i=0,$
 $[\h_{\mathcal{M}}^0(G(I))]_0\cong[\h_{\mathcal{R}_+}^0(R(I))]_0=0$ by Lemma \ref{lemma-1} as $\depth R\geq 1$.
   Let $[\h_{\mathcal{M}}^{i}(G(I))]_0=0$ for $0\leq i\leq s$ for some $s\leq d-3$. Using \eqref{equation:bi(G(I))},
we get that $\h_{\mathcal{M}}^{i}(G(I))=0$ for $0\leq i\leq s$. So $\depth (G(I))\geq s+1$ and $y_1^*,\ldots,y_{s+1}^*$ is a regular sequence in $G(I)$. By
Lemma \ref{lemma-3},
\begin{align}
[\h_{\mathcal{M}}^{s+1}(G(I))]_0 &\cong [\h_{\mathcal{M}}^0(G(IR_{s+1}))]_{s+1} \text{ and }\label{eqn-chap-last-e2}\\
{[}\h_{\mathcal{M}}^i(G(IR_{s+1}))]_{n} &= 0 \text{ for all } n\geq s+2\text{ and }i\in\mathbb{Z}\nonumber
\end{align}
where $R_{s+1}=R/(y_1,\ldots,y_{s+1})$.
Thus using Lemma \ref{lemma-1} with $m_{IR_{s+1}}=s+1$, we get $[\h_{\mathcal{M}}^0(G(IR_{s+1}))]_{s+1}\cong [\h_{\mathcal{R}_+}^0(R(IR_{s+1}))]_{s+1}=0$
since $\depth R_{s+1}\geq 1$.
Thus $[\h_{\mathcal{M}}^{s+1}(G(I))]_0=0$ by \eqref{eqn-chap-last-e2}. This completes the proof of the claim. $\square$

By above claim, $a_{d-2}(G(Q^l))\leq -1$ and $\h_{\mathcal{M}}^i(G(I))=0$  for  $0\leq i\leq d-3$ which implies that $\depth G(Q^l)\geq d-2$ for all $l\gg 0$.

\ref{lemma-4-item-3}   For $d=2$ case, $a_{d-2}(G(Q^l))\leq -1$ implies $[\h_{\mathcal{M}}^{0}(G(I))]_n=0$ for
all $n\geq 0$.
Thus $\h_{\mathcal{M}}^{0}(G(I))=0$ and $\depth G(I)\geq 1.$ Now let $d\geq 3.$
We can choose $l_{0}> \max \{|b_{d-2}(G(Q))|, |a_i(G(Q))|:a_i(G(Q))\neq -\infty\}$ such that
$[\h_{\mathcal{M}}^{d-1}(G(Q^{l_0}))]_0=0$. By part $(a)$,  $\h_{\mathcal{M}}^{i}(G(Q^{l_0}))=0$ for $0\leq i \leq d-3$. If
we show that $\h_{\mathcal{M}}^{d-2}(G(Q^{l_0}))=0$, it will follow that $\depth G(Q^{l_0})\geq d-1.$ Let $I_{0}=Q^{l_0}$.
  Since $\h_{\mathcal{M}}^{d-2}(G(Q))$ is finitely graded,
 $b_{d-2}(G(I_0))\geq \lfloor b_{d-2}(G(Q))/l_0 \rfloor \geq -1$ by \cite[Lemma 2.4]{Hoa}. Thus, by part \ref{lemma-4-item-0},
\begin{equation*}
[\h_{\mathcal{M}}^{d-2}(G(I_0))]_n=0 \mbox{ for } n\neq -1.
\end{equation*}
Now it is enough to show that $[\h_{\mathcal{M}}^{d-2}(G(I_0))]_{-1}=0$.
Given that $[\h_{\mathcal{M}}^{d-1}(G(I_0))]_0=0$. By part \ref{lemma-4-item-0} and \eqref{equation:bi(G(I))}, we get that $[\h_{\mathcal{M}}^{i}(G(I_0))]_n=0$ for all $n\geq 0$
and $i\in\mathbb{Z}$. Also $\depth G(I_0)\geq d-2$ implies that $y_1^*,\ldots,y_{d-2}^*$ is a regular sequence in $G(I_0).$ Therefore by Lemma \ref{lemma-3},
\begin{align*}
[\h_{\mathcal{M}}^{d-2}(G(I_0))]_{-1}&\cong[\h_{\mathcal{M}}^{0}(G(I_0R_{d-2}))]_{d-3} \mbox{ and}\\
{[}\h_{\mathcal{M}}^{i}(G(I_0R_{d-2}))]_{n+1}&=0 \mbox{ for } n\geq d-3 \mbox{ and } i\in\mathbb{Z}.
\end{align*}
 So by Lemma \ref{lemma-1}, $[\h_{\mathcal{M}}^{0}(G(I_0R_{d-2}))]_{d-3}\cong [\h_{\mathcal{M}}^{0}(R(I_0R_{d-2}))]_{d-3}=0$
as $\depth R_{d-2}\geq 1$.  Hence $[\h_{\mathcal{M}}^{d-2}(G(I_0))]_{-1}=0$.
 \end{proof}

\section{Bounding the Hilbert coefficients}\label{bounding-the-hilbert-coefficients}
In this section, we obtain bounds on the coefficients $e_i(Q)$ for a parameter ideal $Q$ in a ring of depth at least $d-1$ with certain conditions on the local
cohomology modules of $G(Q)$. We show that the last coefficient $e_d(Q)\leq 0$ if $b_i(G(Q))\geq 0$ for $0\leq i\leq d-3$ (Theorem \ref{theorem-main-(-1)}) and that
$e_d(Q)\geq-\lambda_R(\h_{\mathfrak m}^{d-1}(R))$ if $b_i(G(Q))\geq 0$ for $0\leq i\leq d-2$ (Theorem \ref{theorem-main-2}). Note that the above conditions on $b_i(G(Q))$
holds if $\depth G(Q)$ is at least $d-2$ and $d-1$ respectively. Consequently for $2\leq i\leq d $, we obtain that (1) $e_i(Q)\leq 0$ provided $\depth G(Q)\geq d-2$
(Corollary \ref{corr-ei-lesseq-0}) and (2) $e_i(Q)\geq -\lambda_R(\h_{\mathfrak m}^{d-1}(R))$ provided $\depth G(Q)\geq d-1$ (Corollary \ref{corr-ei-grteq}).

The most interesting result of this section is Theorem \ref{theorem-e3} which states that $e_3(Q)\leq 0$ for a parameter ideal $Q$ with $\depth R\geq d-1$.
In order to prove this, we first need the  non-positivity of $e_d(Q)$.
 \begin{theorem}\label{theorem-main-(-1)}
 Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 2$ and $\depth R\geq d-1$. Let $Q$ be a parameter ideal such that
 $b_i(G(Q))\geq 0$ for $0\leq i\leq d-3.$ Then
 $$e_d(Q)\leq 0.$$
\end{theorem}
\begin{proof}
We may assume that the residue field is infinite. Let $Q=(x_1,\ldots,x_d)$ such that $x_1^*,\ldots,x_d^*$ is a superficial sequence.
For an integer $l\gg 0$, we put $I=Q^l$. By Lemma \ref{lemma-4}$\ref{lemma-4-item-0}$,  \begin{equation}\label{equation:vanishing-of-n-pieces}
                                                                                [\h_{\mathcal{M}}^{i}(G(I))]_0=0 \text{ for } 0\leq i\leq d-2 \mbox{ and } i=d.
                                                                                \end{equation}
and by \cite[Theorem 3.8]{blancafort}, $[\h_{\mathcal{R}_+}^i(R(I)^*)]_0\cong [\h_{\mathcal{R}_+}^i(R(I))]_0$ for all $i\geq 0$. Thus
\cite[Theorem 4.1]{blancafort} yields that
 \begin{align}
(-1)^d e_d(I)&=P(I,0)- H(I,0)\nonumber\\
&=\sum_{i=0}^d(-1)^i\lambda_R([\h_{\mathcal{R}_+}^i(R(I)^*)]_0)\nonumber\\
&=\sum_{i=0}^d(-1)^i\lambda_R([\h_{\mathcal{R}_+}^i(R(I))]_0).\label{formula-for-ed}
\end{align}
By Lemma \ref{lemma-4}$\ref{lemma-4-item-0}$, $[\h_{\mathcal{M}}^i(G(I))]_n=0$ for all $n\geq 1$ and $i\in \mathbb{Z}$.
 Therefore by Lemma \ref{lemma-1} and \eqref{formula-for-ed}, we get
 \begin{align*}
 (-1)^de_d(I)=\sum_{i=0}^d(-1)^i\lambda_R([\h_{\mathcal{M}}^i(G(I))]_0)=(-1)^{d-1}\lambda_R([\h_{\mathcal{M}}^{d-1}(G(I))]_0)
\end{align*}
where the last equality holds due to \eqref{equation:vanishing-of-n-pieces}.
This implies
\begin{equation}\label{equation:ed-negative}
 e_d(Q)=e_d(I)=-\lambda_R([\h_{\mathcal{M}}^{d-1}(G(I))]_0)\leq 0.
\end{equation}
\end{proof}
In \cite[Corollary 4.5]{Lori}, Mccune proved that the coefficients
$e_i(Q)$, for $2\leq i\leq d$, are all non-positive when $\depth G(Q)\geq d-1$. In the next corollary, we improve upon her result by weakening the hypothesis
to $\depth G(Q)\geq d-2$.
\begin{corollary}\label{corr-ei-lesseq-0}
  Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 2$ and $\depth R\geq d-1$. Let $Q$ be a parameter ideal of $R$
  such that $\depth(G(Q))\geq d-2$. Then, for $2\leq i\leq d$,
 \begin{equation*}
 e_i(Q)\leq 0.
  \end{equation*}
\end{corollary}
\begin{proof}
We may assume that the residue field $R/\mathfrak m$ is infinite. Let $Q=(x_1,\ldots,x_d)$ such that  $x_1^*,\ldots,x_d^*$ is a superficial sequence.
Set $R_0=R$ and $R_i=R/(x_1,\ldots,x_i)$ for $1\leq i\leq d-2$.
Since $x_1^*,\ldots,x_{d-2}^*$ is a regular sequence in $G(Q)$, $G(QR_i)\cong G(Q)/(x_1^*,\ldots,x_i^*)G(Q)$ and $\depth G(QR_i)\geq d-i-2$ for $i\leq d-2$.
Hence $e_i(Q)=e_i(QR_{d-i})\leq 0$ for $2\leq i\leq d$ by Theorem \ref{theorem-main-(-1)}.
\end{proof}
 The following example emphasizes that the depth condition on the ring is necessary in the above corollary. This example is motivated by the idea presented in \cite{goto-ozeki}.
  \begin{example}\label{example-gcm-22-1}
   Let $(R,\mathfrak n)$ be a regular local ring of dimension $d\geq 2$ and $X_1,\ldots,X_d$ a regular system of parameters of $R$.
   We put $\mathfrak p_t=(X_1,\ldots,X_{d-t})$ for some $1\leq  t\leq d-1$ and $D=R/\mathfrak{p_t}.$ Let $A=R\ltimes D$ be the idealization
   of $D$ over $R.$ Then $A$ is a Noetherian local ring dimension $d$  with the maximal ideal $\mathfrak m=\mathfrak{n}\times D$ and $\depth A=t$.
   Consider the exact sequence of $A$-modules
   \begin{equation}\label{exact-seq-22-1}
    0\longrightarrow D\xrightarrow{j} A \xrightarrow{p} R\longrightarrow 0
   \end{equation}
where $j(x)=(0,x)$ and $p(a,x)=a$. Note that $D$ is an $A$-module via $p$.
Let $Q$ be a parameter ideal in $A$ and $q=p(Q)\subseteq R.$ Then
we have,
\begin{align}
 \lambda_A(A/Q^{n+1})&=\lambda_R(R/q^{n+1})+\lambda_R(D/q^{n+1}D)\nonumber\\
 &=e_0(q,R)\binom{n+d}{d}+\sum_{i=0}^t(-1)^ie_i(q,D)\binom{n+t-i}{t-i}\nonumber
\end{align}
for all $n\gg 0.$ This implies
 \begin{align}\label{eqn-22-1}
e_i(Q,A)=\begin{cases}
        e_0(q,R) & \text{ if }\ i=0\\
          0 & \text{ if }\ 1\leq i\leq d-t-1\\
          (-1)^{d-t}e_{i-d+t}(q,D) & \text{ if}\ d-t\leq i\leq d.
         \end{cases}
 \end{align}
In particular, let $d=4$ and $t=2$ so that
$D=R/(X_1,X_2)$. Let $q=(X_1,\ldots,X_d)$ and $Q=qA$. Then $\dim A=4$, $\depth A=2.$
Since $G(Q)=G(q)\ltimes G(qD)(-1)$, see \cite[Remark 2]{tony}, we have $\depth G(Q)=2$ but $e_2(Q,A)>0$ by \eqref{eqn-22-1}.
\end{example}
A noteworthy consequence of Theorem \ref{theorem-main-(-1)} is the following result.
\begin{theorem}\label{theorem-e3}
   Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 3$ and $\depth R\geq d-1$. Let $Q$ be a parameter ideal. Then $e_3(Q)\leq 0.$
\end{theorem}
\begin{proof}
 We may assume that $R/\mathfrak m$ is infinite. Then using reduction modulo superficial elements, it is enough to assume that $d=3$.
 The result now follows from Theorem \ref{theorem-main-(-1)}.
\end{proof}
The following example shows that the assumption on the depth of the ring can not be relaxed from Theorem \ref{theorem-e3}.
\begin{example}\label{example-22-1-chap-6}{\rm\cite[Example 4.7]{goto-ozeki-2}}
 Let $(S,\mathfrak{n})$ be a regular local ring of dimension $d=4$ with infinite residue field $S/\mathfrak{n}$. Let $X,Y,Z,W$ be a regular system
 of parameters of $S$ and $R=S/((X)\cap (Y^3,Z,W))$. Let $x,y,z,w$ be the images of $X,Y,Z,W$ in $R$ respectively and
 $\mathfrak m=(x,y,z,w)R$ be the maximal ideal of $R$. Then $\dim R=3,$ $\depth R=1$. Let $U=(x)$, $Q=(x-y,x-z,x-w)R$ and $T=R/(x)$.
 Then $T$ is a regular local ring with $\dim T=3$ and $QT=\mathfrak m T$. The following exact sequence
 $$0\longrightarrow (x)\longrightarrow R\longrightarrow R/(x)\longrightarrow 0$$ gives that
 \begin{align}
  \lambda_R(R/Q^{n+1}R)&=\lambda_R(T/\mathfrak m^{n+1}T)+\lambda_R(U/Q^{n+1}U)\nonumber\\
  &=\binom{n+3}{3}+e_0(Q,U)\binom{n+1}{1}-e_1(Q,U)\label{eqn-23-1-example}
 \end{align}
for all $n\gg 0.$ We have
 $(x)\cong R/I$ where $I=(y^3,z,w)R$ and $Q(R/I)=\mathfrak m (R/I)$, so $e_0(Q,U)=e_0(\mathfrak m,R/I)$ and $e_1(Q,U)=e_1(\mathfrak m,R/I)$. The
 Hilbert series of the associated graded ring $G(\mathfrak m (R/I))$ is
$$\frac{1+t+t^2}{1-t}$$
Hence  $e_0(Q,U)=e_0(\mathfrak m,R/I)=3$ and $e_1(Q,U)=e_1(\mathfrak m,R/I)=3.$ By \eqref{eqn-23-1-example}, we get
 $e_3(Q,R)=3>0.$
\end{example}
The following lemma is crucial for obtaining lower bound on $e_i(Q)$. We also obtain a necessary condition for the vanishing of $e_d(Q)$ in
Theorem \ref{theorem-vanishing-of-ed} as an application of this lemma.
\begin{lemma}\label{lemma-main-1} Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 2$ and $\depth R\geq d-1$. Let $I$ be an $\mathfrak m$-primary ideal such
that
$\depth G(I)\geq d-1$ and
 \begin{align}\label{eqn-lemma-main-1}
  a_i(G(I))\leq \begin{cases}
                 0 & \text{ for } i\in\mathbb{Z}, \\
		  -1 & \text{ for } i=d.
                \end{cases}
 \end{align}
Let $J=(y_1,\ldots,y_d)$ be a reduction of $I$ with $I^{d}=JI^{d-1}$ and $y_1^*,\ldots,y_{d-1}^*$
 is a superficial sequence. Then
 \begin{equation}\label{eqn-main-formula-for-ed}
  e_d(I)=-\lambda_R\left(\frac{((y_1,\ldots,y_{d-1}):y_d)\cap (I^{d-1}+(y_1,\ldots,y_{d-1}))}{(y_1,\ldots,y_{d-1})}\right).
 \end{equation}
\end{lemma}
\begin{proof}
Since $\depth G(I)\geq d-1$, $y_1^*,\ldots,y_{d-1}^*$ is a regular sequence in $G(I)$ and $\h_{\mathcal{M}}^i(G(I))=0$ for $0\leq i\leq d-2$. By Lemma \ref{lemma-1},
$[\h_{\mathcal{M}}^i(G(I))]_0\cong [\h_{\mathcal{R}_+}^i(R(I))]_0$ for all $i$. Therefore using \eqref{formula-for-ed},
\begin{align*}
 (-1)^de_d(I)&=\sum_{i=0}^d(-1)^i\lambda_R([\h_{\mathcal{R}_+}^i(R(I))]_0)\\
 &=\sum_{i=0}^d(-1)^i\lambda_R([\h_{\mathcal{M}}^i(G(I))]_0)
\end{align*}

Using \eqref{eqn-lemma-main-1} and Lemma \ref{lemma-3} respectively, we get that
\begin{equation}
e_d(I)=-\lambda_R([\h_{\mathcal{M}}^{d-1}(G(I))]_0)=-\lambda_R([\h_{\mathcal{M}}^0(G(IR_{d-1}))]_{d-1}) \label{eqn-18/10}
\end{equation}
where $R_{d-1}=R/(y_1,\ldots,y_{d-1})$.
Now consider the map $$\rho : \frac{((y_1,\ldots,y_{d-1}):y_d)\cap (I^{d-1}+(y_1,\ldots,y_{d-1}))}{(y_1,\ldots,y_{d-1})} \longrightarrow [\h_{\mathcal{M}}^0(G(IR_{d-1}))]_{d-1}$$
defined as $\rho(\bar{x})=\overline{\bar{x}t^{d-1}}$ where $\bar{x}$ and $\overline{\bar{x}t^{d-1}}$ are the images of $x\in R$ in $R/(y_1,\ldots,y_{d-1})$ and
$\bar{x}t^{d-1}\in R(IR_{d-1})$ in $G(IR_{d-1})$ respectively. It is now enough to show that $\rho$ is an isomorphism. To show surjectivity, let $\alpha=\overline{\bar{x}t^{d-1}}\in [\h_{\mathcal{M}}^0(G(IR_{d-1}))]_{d-1}$  with
$x\in I^{d-1}$. Then
\begin{equation}\label{eqn-pho-surjective-1}
 \bar{y}_dt~\cdot~\overline{\bar{x}t^{d-1}}=\overline{\overline{y_dx}t^d}\in [\h_{\mathcal{M}}^0(G(IR_{d-1}))]_{d}.
\end{equation}
Since $[\h_{\mathcal{M}}^0(G(IR_{d-1}))]_{d}=0$  by Lemma \ref{lemma-3} and
$I^d=JI^{d-1}$,
\eqref{eqn-pho-surjective-1} yields that
\begin{equation*}
y_dx\in(I^{d+1}+(y_1,\ldots,y_{d-1}))\cap I^d\subseteq (y_1,\ldots,y_{d-1})+I^{d+1}=(y_1,\ldots,y_{d-1})+y_dI^d.
\end{equation*}
 Let $y_dx= \sum_{i=1}^{d-1}r_iy_i+sy_d$ where $s\in I^{d}$.
This implies $y_d(x-s)\in (y_1,\ldots, y_{d-1})$. So $x-s\in ((y_1,\ldots,y_{d-1}):y_d)\cap I^{d-1}$ and $\rho(\overline{x-s})=\alpha$. Hence $\rho$ is surjective.

Now let $x\in ((y_1,\ldots,y_{d-1}):y_d)\cap I^{d-1}$ such that $\rho(\bar{x})=\overline{\bar{x}t^{d-1}}=0$ in $[G(IR_{d-1})]_{d-1}$.
Then \begin{equation*}
x\in ((y_1,\ldots,y_{d-1}):y_d)\cap (I^d+(y_1,\ldots,y_{d-1}))=(y_1,\ldots,y_{d-1})+\left(((y_1,\ldots,y_{d-1}):y_d)\cap I^d\right).
     \end{equation*}
\textbf{Claim:} Let $n\geq d$ be an integer. Then \begin{equation*}
((y_1,\ldots,y_{d-1}):y_d)\cap I^n\subseteq (y_1,\ldots,y_{d-1})+((y_1,\ldots,y_{d-1}):y_d)\cap I^{n+1}).
                                    \end{equation*}
{\it{Proof of Claim.}} Let $x\in ((y_1,\ldots,y_{d-1}):y_d)\cap I^n$, then $y_dx\in (y_1,\ldots,y_{d-1})$. So
$$\bar{y}_dt~\cdot~\overline{\bar{x}t^n}=\overline{\overline{y_dx}t^{n+1}}=0 \text{~in~} [G(IR_{d-1})]_{n+1}$$
which implies
$\overline{\bar{x}t^n}\in [G(IR_{d-1})]_n$ is annihilated by some power of $\mathcal{M}$. Thus
$\overline{\bar{x}t^n}\in [\h_{\mathcal{M}}^0(G(IR_{d-1}))]_n=0$. Recall that $[\h_{\mathcal{M}}^0(G(IR_{d-1}))]_n=0$ for all $n\geq d$ by Lemma \ref{lemma-3}. This gives
that $x\in (y_1,\ldots,y_{d-1})+I^{n+1}$. So $x\in (y_1,\ldots,y_{d-1})+((y_1,\ldots,y_{d-1}):y_d)\cap I^{n+1})$.$\square$

By the above claim, $x\in (y_1,\ldots,y_{d-1})+((y_1,\ldots,y_{d-1}):y_d)\cap I^{n})\subseteq(y_1,\ldots,y_{d-1})+I^{n}$ for all $n\geq d$. This implies
$x\in (y_1,\ldots,y_{d-1})$ and $\rho$ is injective. Thus by \eqref{eqn-18/10}
  $$e_d(I)=-\lambda_R\left(\frac{((y_1,\ldots,y_{d-1}):y_d)\cap (I^{d-1}+(y_1,\ldots,y_{d-1}))}{(y_1,\ldots,y_{d-1})}\right).$$
This completes the proof.
\end{proof}
\begin{theorem}\label{theorem-main-2}
  Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 2$ and $\depth R\geq d-1$. Let $Q$ be a parameter ideal of $R$ such that
  $b_i(G(Q))\geq 0$ for $0\leq i\leq d-2$.
 Then
 \begin{equation*}
  -\lambda_R(\h_{\mathfrak m}^{d-1}(R))\leq e_d(Q).
 \end{equation*}
\end{theorem}
\begin{proof}
 We may assume that $R/\mathfrak m$ is infinite. Let $Q=(x_1,\ldots,x_d)$ such that $x_1^*,\ldots,x_d^*$ is a superficial sequence.
 For an integer $l\gg 0$, let $I=Q^l$ and $y_i=x_i^l$ for $1\leq i\leq d$. Then $y_1^*,\ldots,y_d^*$ is a superficial sequence with respect to $I$ and
\begin{equation}\label{equation:reduction-no-of-I}
 I^{d}=(y_1,\ldots,y_d)I^{d-1}.
\end{equation}
By Lemma \ref{lemma-4}, for $l\gg 0,$ $\depth G(I)\geq d-2.$
Since $b_{d-2}(G(Q))\geq 0$, we get $b_{d-2}(G(I))\geq \lfloor b_{d-2}(G(Q))/l \rfloor\geq 0$ by
\cite[Lemma 2.4]{Hoa}. Using \eqref{31-aug-6-e1}, we get that $\h_{\mathcal{M}}^{d-2}(G(I))=0$ for $l\gg 0$. Hence $\depth G(I)\geq d-1.$
By Lemma \ref{lemma-main-1},
\begin{align}
e_d(Q)=e_d(I)&=-\lambda_R\left(\frac{((y_1,\ldots,y_{d-1}):y_d)\cap (I^{d-1}+(y_1,\ldots,y_{d-1}))}{(y_1,\ldots,y_{d-1})}\right)\nonumber\\
 &\geq-\lambda_R\left(\frac{((y_1,\ldots,y_{d-1}):y_d)}{(y_1,\ldots,y_{d-1})}\right)\nonumber\\
 &\geq-\lambda_R(\h_{\mathfrak m}^0(R/(y_1,\ldots,y_{d-1})))\label{eqn-ed-lower-bound}
\end{align}
where the last inequality holds since

  \begin{align*}
 \frac{((y_1,\ldots,y_{d-1}):y_d)}{(y_1,\ldots,y_{d-1})}\subseteq \h_{\mathfrak m}^0(R/(y_1,\ldots,y_{d-1}))
 \end{align*}
Now let $R_i=R/(y_1,\ldots,y_{i})$ for $1\leq i\leq d-1$ and $R_0=R$. Note that $y_1,\ldots,y_{d-1}$ is a regular sequence in $R$ and $\depth R_i\geq d-i-1$.
For $0\leq i\leq d-2$, the exact sequence $$0\longrightarrow R_i\xrightarrow{y_{i+1}} R_i\longrightarrow R_{i+1}\longrightarrow 0$$
gives the long exact sequence of local cohomology modules
$$0\longrightarrow \h_{\mathfrak m}^{d-i-2}(R_{i+1})\longrightarrow \h_{\mathfrak m}^{d-i-1}(R_{i})\xrightarrow{~y_{i+1}} \h_{\mathfrak m}^{d-i-1}(R_{i})\longrightarrow\ldots.$$
Thus for $0\leq i\leq d-2$,
\begin{equation}\label{eqn-length-of-H-m-decreasing}
 \lambda_R(\h_{\mathfrak m}^{d-i-2}(R_{i+1}))\leq \lambda_R(\h_{\mathfrak m}^{d-i-1}(R_{i})).
 \end{equation}
Putting the values of $i$ successively, we get
$$\lambda_R(\h_{\mathfrak m}^0(R_{d-1}))\leq \lambda_R(\h_{\mathfrak m}^1(R_{d-2}))\leq\ldots\leq \lambda_R(\h_{\mathfrak m}^{d-1}(R)).$$
Hence $e_d(Q)\geq -\lambda_R(\h_{\mathfrak m}^{d-1}(R))$ by \eqref{eqn-ed-lower-bound}.
\end{proof}
\begin{corollary}\label{corr-ei-grteq}
  Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 2$. Let $Q$ be a parameter ideal of $R$
  such that $\depth G(Q)\geq d-1$.
 Then for $2\leq i\leq d$,
 \begin{equation}\label{eqn-23-3-corr}
   -\lambda_R(\h_{\mathfrak m}^{d-1}(R))\leq e_i(Q).
  \end{equation}
\end{corollary}
\begin{proof}
We may assume that the residue field $R/\mathfrak m$ is infinite. Let $Q=(y_1,\ldots,y_d)$ such that  $y_1^*,\ldots,y_d^*$ is a superficial sequence
in $G(Q)$.
Let $R_i=R/(y_1,\ldots,y_i)$ for $1\leq i\leq d-1$ and $R_0=R$.
Since $y_1^*,\ldots,y_{d-1}^*$ is a regular sequence in $G(Q)$, $G(QR_{d-i})\cong G(Q)/(y_1^*,\ldots,y_{d-i}^*)G(Q)$ and
$\depth G(QR_{d-i})\geq i-1.$
Hence
by Theorem \ref{theorem-main-2},
\begin{equation}\label{eqn-bound-ei}-\lambda_R(\h_{\mathfrak m}^{i-1}(R_{d-i}))\leq e_i(Q) \text{~~for ~~} 2\leq i\leq d.\end{equation}
Since $y_1,\ldots,y_{d-1}$ is a regular sequence in $R$, we have from \eqref{eqn-length-of-H-m-decreasing} that
$$\lambda_R(\h_{\mathfrak m}^{i-1}(R_{d-i}))\leq \lambda_R(\h_{\mathfrak m}^{i}(R_{d-i-1}))$$
for  $2\leq i\leq d-1$. This implies
$$\lambda_R(\h_{\mathfrak m}^{i-1}(R_{d-i}))\leq \lambda_R(\h_{\mathfrak m}^{i}(R_{d-i-1}))\leq \ldots \leq \lambda_R(\h_{\mathfrak m}^{d-1}(R)).$$
Therefore by \eqref{eqn-bound-ei}, we get
\begin{equation*}
    -\lambda_R(\h_{\mathfrak m}^{d-1}(R))\leq e_i(Q) \mbox{ for } 2\leq i\leq d.
\end{equation*}
\end{proof}
We include an example where \eqref{eqn-23-3-corr} does not hold.
\begin{example}
 We recall Example \ref{example-gcm-22-1} with $\dim R=d=5$ and $t=2$. Then $\dim A=5$ and $\depth A=2$.
 By \eqref{eqn-22-1}, $e_3(Q,A)<0$ for a parameter ideal $Q$ of $A$ where as $-\lambda_R(\h_{\mathfrak m}^{d-1}(A))=0.$
\end{example}
\section{Vanishing of coefficients}\label{vanishing-of-coefficients}
In this section, we generalize partially a result of Goto and Ozeki \cite[Theorem 3.2]{goto-ozeki} on the vanishing of $e_2(Q)$ in two dimensional local rings.
In Theorem \ref{theorem-vanishing-of-ed}, we obtain a necessary condition similar to that given in \cite{goto-ozeki} for the vanishing of $e_d(Q)$ for a
parameter ideal $Q$ with certain conditions.
Further, we characterize the vanishing of
$e_2(Q)$ in rings of dimension $d\geq 2$ and $\depth $ at least $d-1$ in Theorem \ref{theorem-e2} which extends \cite[Theorem 3.2]{goto-ozeki} to
dimension $d$.

Let $x_0=0$. A sequence $x_1,\ldots,x_r\in\mathfrak m$  is called a $d$-sequence if (i) $x_i$ is not in the ideal generated by the rest of the $x_j$ and (ii)
 $((x_0,\ldots,x_i): x_{i+1}x_j)=((x_0,\ldots,x_i): x_j)$ for $0\leq i\leq r-1$ and for all $j\geq i+1$. See \cite{huneke-d} and \cite{trung-abs-sup}
 for the theory of $d$-sequences. Note that the hypothesis of the following theorem is satisfied if $\depth G(Q)\geq d-1.$
\begin{theorem}\label{theorem-vanishing-of-ed}
 Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 2$ and $\depth R\geq d-1$. Let $Q=(x_1,\ldots,x_d)$ be a parameter ideal of $R$ such that
  $x_1^*,\ldots,x_{d-1}^*$ is a superficial sequence. Let $b_i(G(Q))\geq 0$ for $0\leq i\leq d-3$ and $[\h_\mathcal{M}^{d-2}(G(Q))]_n=0$ for all $n\ll 0$ i.e.
  $\h_\mathcal{M}^{d-2}G(Q)$ is finitely graded.
 Suppose $$e_d(Q)=0.$$ Then $x_1^l,\ldots,x_{d-1}^l,x_d^{(d-1)l}$ is a $d$-sequence in $R$ for all integers $l\geq 1$.
\end{theorem}
\begin{proof}
 For $l\gg 0,$ let $I=Q^l$ and $J=(x_1^l,\ldots,x_d^l).$ Then ${(x_1^l)}^*,\ldots,{(x_d^l)}^*$ is a superficial sequence with respect to $I$ and $I^d=JI^{d-1}.$
 Suppose $e_d(Q)=0$, then \eqref{equation:ed-negative} implies
$[\h_{\mathcal{M}}^{d-1}(G(I))]_0=0$. Hence by Lemma \ref{lemma-4},  $\depth G(I)\geq d-1$.

By Lemma \ref{lemma-main-1}, we get that
$$e_d(Q)=-\lambda_R\left(\frac{((x_1^l,\ldots,x_{d-1}^l):x_d^l)\cap (I^{d-1}+(x_1^l,\ldots,x_{d-1}^l))}{(x_1^l,\ldots,x_{d-1}^l)}\right).$$
 $e_d(Q)=0$ implies that
$((x^l_1,\ldots,x^l_{d-1}):x^l_d)\cap (I^{d-1}+(x^l_1,\ldots,x^l_{d-1}))=(x^l_1,\ldots,x^l_{d-1})$ for all $l\gg 0$.
Let $N\geq 1$ be an integer such that for all  $l\geq N$,
\begin{align}
((x^l_1,\ldots,x^l_{d-1}):x^l_d)\cap I^{d-1}\subseteq (x^l_1,\ldots,x^l_{d-1}).\label{eqn-claim-is-true-for-l>N}
\end{align}
{\bf{Claim:}} For all $l\geq 1$
\begin{equation}\label{eqn-containment-for-all-l}
((x^l_1,\ldots,x^l_{d-1}):x^l_d)\cap I^{d-1} \subseteq (x^l_1,\ldots,x^l_{d-1}).
\end{equation}
{\it {Proof of Claim.}}
Let $1\leq l< N$ and $y\in ((x^l_1,\ldots,x^l_{d-1}):x^l_d)\cap I^{d-1}$. Then
\begin{align*}
x_d^N\cdot x_1^{N-l}\cdot x_2^{N-l}\cdots x_{d-1}^{N-l}\cdot y&=x_d^{N-l}\cdot x_1^{N-l}\cdot x_2^{N-l}\cdots x_{d-1}^{N-l}\cdot x_d^ly\\
&\in x_d^{N-l}\cdot x_1^{N-l}\cdot x_2^{N-l}\cdots x_{d-1}^{N-l}(x_1^l,\ldots,x_{d-1}^l)\\
&\subseteq(x_1^N,\ldots,x_{d-1}^N).
\end{align*}
This implies \begin{align}
              x_1^{N-l}\cdot x_2^{N-l}\cdots x_{d-1}^{N-l}\cdot y&\in((x_1^N,\ldots,x_{d-1}^N):x_d^N)\cap Q^{(N-l)(d-1)}I^{d-1}\nonumber\\
              &=((x_1^N,\ldots,x_{d-1}^N):x_d^N)\cap Q^{N(d-1)}\nonumber\\
              &\subseteq(x^N_1,\ldots,x^N_{d-1})\label{eqn-d-seq}
      \end{align}
 where the last containment is due to \eqref{eqn-claim-is-true-for-l>N}.
 Now we show by induction on $d$ that if $y$ is such that $\eqref{eqn-d-seq}$ holds then $y\in (x^l_1,\ldots,x^l_{d-1})$.
 Note that $x_1,\ldots,x_{d-1}$ is a regular sequence. For $d=2$, $~x_1^{N-l}y\in (x_1^N)\implies y\in (x_1^l)$.
 Let $d>2$ and $\eqref{eqn-d-seq}$ holds. Set $y^{\prime}= x_1^{N-l}y$ and $R_1=R/(x_1^N)$. Let $\overline{\alpha}$ denote the image of an element
 $\alpha\in R$ in $R_1$. Then
\begin{align*}
 &x_2^{N-l}\cdot x_3^{N-l}\cdots x_{d-1}^{N-l}\cdot y^{\prime}
 \in (x^N_1,\ldots,x^N_{d-1})\\
 \implies&\overline{x}_2^{N-l}\cdot \overline{x}_3^{N-l}\cdots\overline{x}_{d-1}^{N-l}\cdot\overline{y^{\prime}}\in (\overline{x}_2^N,\ldots,\overline{x}^N_{d-1})R_1\\
 \implies& \overline{y^{\prime}}\in (\overline{x}_2^l,\ldots,\overline{x}^l_{d-1})R_1 \hspace{3cm} \text{ [by induction hypothesis]}\\
 \implies&   x_1^{N-l}y=y^{\prime}\in (x_2^l,\ldots,x_{d-1}^l)+(x_1^N)\\
 \implies& y\in (x_1^l,\ldots,x_{d-1}^l)
\end{align*}
where the last statement holds since $x_1^{N-l}$ is regular in $R/(x_2^l,\ldots,x_{d-1}^l)$. $\square$

To see that $x_1^l,\ldots,x_{d-1}^l,x_d^{(d-1)l}$ is a $d$-sequence in $R$, we use \eqref{eqn-containment-for-all-l} repeatedly. For this purpose, let $l\geq 1$ and
\begin{align*}
  r &\in ((x_1^l,\ldots,x_{d-1}^l):x_d^{(d-1)l})\cap (x_d^{(d-1)l})\\
 \implies rx_d^{(d-2)l} &\in  ((x_1^l,\ldots,x_{d-1}^l):x^l_d)\cap I^{d-1} \subseteq (x^l_1,\ldots,x^l_{d-1}) \\
  \implies rx_d^{(d-3)l}&\in  ((x_1^l,\ldots,x_{d-1}^l):x^l_d)\cap I^{d-1} \subseteq (x^l_1,\ldots,x^l_{d-1})\\
 &\vdots&\\
 \implies r&\in ((x_1^l,\ldots,x_{d-1}^l):x^l_d)\cap I^{d-1} \subseteq (x^l_1,\ldots,x^l_{d-1}).
\end{align*}
This implies that for $l\geq 1,$ $$((x_1^l,\ldots,x_{d-1}^l):x_d^{(d-1)l})\cap (x_1^l,\ldots,x_{d-1}^l,x_d^{(d-1)l})=(x^l_1,\ldots,x^l_{d-1}).$$
Since $x_1^l,\ldots,x_{d-1}^l$ is a regular sequence in $R$, it follows that
$x_1^l,\ldots,x_{d-1}^l,x_d^{(d-1)l}$ is a $d$-sequence for $l\geq 1$.
\end{proof}
In the next theorem, we obtain equivalent conditions for the equality $e_2(Q)=0$ which recovers \cite[Theorem 3.2]{goto-ozeki}.
It also unifies the necessary and sufficient conditions given by Mccune \cite[Theorem 3.5]{Lori} for $e_2(Q)=0.$
Recall that the postulation number of $Q$, denoted
 by $\eta(Q)$, is defined as
\begin{equation*}
\eta(Q):=\text{~min}\{i~|~H(Q,n)=P(Q,n) \text{  for all } n> i\}.
\end{equation*}
\begin{theorem}\label{theorem-e2}
   Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 2$ and $\depth R \geq d-1$. Let $Q=(x_1,\ldots,x_d)$ be a parameter ideal of $R$ such that $x_1^*,\ldots,x_d^*$ is a superficial sequence.
   Then the following assertions hold.
   \begin{enumerate}
\item \label{theorem-e2-item1} $-\lambda_R(\h_{\mathfrak m}^{d-1}(R))\leq e_2(Q)\leq 0$.
   \item \label{corr-e2-part-2} The following statements are equivalent:
   \begin{enumerate}[label=(\alph*)]
    \item \label{corr-e2-a} $e_2(Q)=0$;
    \item \label{corr-e2-b0} $x_1,\ldots,x_{d-2},x_{d-1}^l,x_d^l$ is $d$-sequence in $R$ for all integers $l\geq 1$;
    \item \label{corr-e2-b} $x_1,\ldots,x_d$ is $d$-sequence in $R$;
    \item \label{corr-e2-c} $\depth G(Q)\geq d-1$ and $\eta(Q)< 2-d$.
   \end{enumerate}
\item \label{theorem-e2-item3} $e_2(Q)=0\implies e_i(Q)=0$ for $2\leq i\leq d$.
   \end{enumerate}
\end{theorem}
\begin{proof}
\eqref{theorem-e2-item1}
Set $R_i=R/(x_1,\ldots,x_{i})$ for $1\leq i\leq d-1$ and $R_0=R$. Then $e_2(Q)=e_2(QR_{d-2})$ and by Theorems
\ref{theorem-main-(-1)} and \ref{theorem-main-2}, $-\lambda_R(\h_{\mathfrak m}^{1}(R_{d-2}))\leq e_2(Q) \leq 0$. From \eqref{eqn-length-of-H-m-decreasing}, we have that
$$\lambda_R(\h_{\mathfrak m}^1(R_{d-2}))\leq \lambda_R(\h_{\mathfrak m}^2(R_{d-3}))\leq\ldots\leq \lambda_R(\h_{\mathfrak m}^{d-1}(R)).$$
\eqref{corr-e2-part-2} \ref{corr-e2-a}$\implies$ \ref{corr-e2-b0}
 $e_2(Q)=0\implies e_2(QR_{d-2})=0$.
Therefore for all $l\geq 1$, the images of $x_{d-1}^l,x_d^l$ in $R_{d-2}$ is a $d$-sequence by
Theorem \ref{theorem-vanishing-of-ed}. Since $x_1,\ldots,x_{d-1}$ is a
regular sequence in $R$, it follows that $x_1,\ldots,x_{d-2},x_{d-1}^l,x_{d}^l$
is a $d$-sequence in $R$.

\ref{corr-e2-b0}$\implies$\ref{corr-e2-b} It is obvious.

\ref{corr-e2-b}$\implies$\ref{corr-e2-c}
Since the images of $x_{d-1},x_d$
  in $R_{d-2}$ is a $d$-sequence, we have that $\h_{\mathcal{M}}^0(G(QR_{d-2}))\cong \h_{\mathfrak m}^0(R_{d-2})$, see \cite[Proposition 3.4(5)]{goto-ozeki}. Since
  $\depth R_{d-2}\geq 1$, we get $\h_{\mathcal{M}}^0(G(QR_{d-2}))=0$ which implies  $\depth G(QR_{d-2})\geq 1$.
  Thus by Sally-machine,
  $\depth G(Q)\geq d-1$.
  Since the image of $x_d$ in $R_{d-1}$ is a $d$-sequence, by
  \cite[Proposition 3.4(3)]{goto-ozeki}, $\eta(QR_{d-1})\leq 0$.
    Since $x_1^*,\ldots, x_{d-1}^*$ is a regular sequece in $G(Q)$ by \cite[Lemma 1.3]{RV} , we
  get $\eta(Q)=\eta(QR_{d-1})-(d-1)\leq 1-d$ by \cite[Lemma 2.8]{Marley}.

\ref{corr-e2-c}$\implies$\ref{corr-e2-a}
  $\eta(Q)\leq 1-d$, then $P(Q,n)=H(Q,n)=0$ for $n=0,-1,\ldots,2-d$. By putting the values of
  $n$ into $P(Q,n)$ successively, we easily get that $e_i(Q)=0$ for $2\leq i\leq d$.

\eqref{theorem-e2-item3} It follows from part \eqref{corr-e2-part-2}.
\end{proof}
The depth condition on the ring is necessary as evidenced by the following example.
\begin{example}
 In Example \ref{example-gcm-22-1}, let $\dim R=d\geq 4$ and $t=d-3$ so that $D=R/(X_1,X_2,X_3)$ and $A=R\ltimes D.$ Let $q=(X_1,\ldots,X_d)$
  and $Q=qA$. Then $e_2(Q,A)=0$ but  $\depth G(Q)=d-3$ and $e_3(Q,A)\neq 0$ by \eqref{eqn-22-1}. In this case, $\depth A=d-3.$
\end{example}
\nocite{*}

\noindent {\it Acknowledgement}: The second author would like to thank Indian Institute of Technology Guwahati for providing fellowship during the period when the research was carried out.

\addcontentsline{toc}{section}{References}
\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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text:
New Order's "Regret" Lyrics Meaning
by SMF · January 29, 2020
New Order's "Regret" is said to be about the challenges that come along with being a celebrity, particularly in terms of said person finding it difficult to function like a normal human being.
In the first verse, the singer seems to come off as if losing touch of 'everyone he's ever known' isn't that much of a big deal. Or the way the scenario is presented is that some people may be offended that he no longer communicates with them. However, he admits that this is 'nothing he regrets'. And considering that we know this track is premised as stated above, we can also conclude that the reason he is no longer in direct contact with these individuals is in some way, shape or form related to him blowing up.
Now whereas the first verse may come off as if he is addressing say someone he knew before making it big, the second verse is apparently directed at a certain segment of his fandom. But the sentiment is more or less the same. What the singer is telling this individual(s) is that he is not privy on getting too close to people he doesn't know. And he realizes that some may try to diss him for conducting himself so. However, he then alludes to the idea that they can't judge him. And why? Because the life he lives is completely different from their own.
Primary Sentiment of "Regret"
But that being said, the ultimate sentiment expressed is not one whereas the singer is touting his celebrity status or is criticizing other people per se. Rather he too desires a normal life. And by a normal life, he's talking about one in which he could "have a conversation on the telephone" and "wake up every day". These statements of course are not meant to be taken literally. Instead what they obviously allude to, all things considered, is once again the singer being unable to partake in standard routines, as his lifestyle does not afford him such. Moreover in terms of his fandom, he acknowledges that these are people he actually does not know, even if they do know him.
So conclusively, perhaps the narrator can be viewed by some as coming off as smug on this track. He understands that certain people would like to be close to him but then goes on to basically say that the feeling is not mutual. But focusing on his conceit misses the ultimate message of the tune. And such is that his alienation is a direct result of his celebrity. And being that the people he may offend by such actions are not celebrities themselves, they cannot empathize with the negative interpersonal and even internal effects of being a star which compels him to behave in such a manner.
Facts about "Regret"
"Regret" came out on 5 April 1993 as the lead single from New Order's sixth-studio album. The band titled that album "Republic". And the label that put the song out is London Records.
"Regret" was written by New Order members along with songwriter/producer Stephen Hague.
The band also co-produced the track as a singular entity, in conjunction with Hague.
New Order performed this song live for the long-standing British music show "Top of the Pops" in 1993. However, instead of doing so in a studio they actually filmed the performance in California on the set of "Baywatch", with the series' star, David Hasselhoff, even making a cameo appearance.
"Regret" topped Billboard's Alternative Songs and Dance Club Songs charts. It also broke the top 40 of the Hot 100 itself.
In New Order's native country the song peaked at number four on the UK Singles Chart. And outside of the US and UK, it also charted in eight other nations. Indeed "Regret" is considered to be the last true hit of the band's heyday.
Tags: New OrderRegretRepublic
New Order's "Blue Monday" Lyrics Meaning
by Jessica Shelton · Published September 4, 2019
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by SMF · Published November 29, 2022
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