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Public International Law/International Criminal Law/International Criminal Courts and Tribunals

Author: Raghavi Viswanath "Required knowledge: Sources of international law; Jurisdiction; Law of Armed Conflict" "Learning objectives: to understand the foundation and purpose of international criminal justice."

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A. Introduction
This chapter introduces readers to a range of contemporary international criminal courts and tribunals, the political contexts in which they were set up, and the workings of such tribunals. It is in these sites that the substantive principles of international criminal law are applied on a case by case basis. The objective is to closely study the infrastructures and outputs of existing international criminal tribunals.

First, the chapter examines the International Criminal Court (ICC), a permanent and universal international criminal tribunal based in The Hague, arguably the most prominent international tribunal for criminal responsibility. It studies the procedural and substantive features of the ICC, as well as its defining contributions to international criminal law. The chapter then spotlights selected so-called hybrid criminal tribunals, the circumstances and motivations underscoring their establishment, the landmark jurisprudence of such tribunals, and their relationship to the ICC.

B. The International Criminal Court
The International Criminal Court is distinct for being the first permanent tribunal that applies international criminal law with jurisdiction in over 123 States. The idea of a permanent international criminal tribunal was mooted much before even the Nuremberg Tribunal was set up. In 1872, Gustav Moynier from the International Committee of the Red Cross articulated the concern that national judges would find it difficult to be impartial when prosecuting humanitarian law violations orchestrated by their own State. This apprehension developed into a request for a standing court. Following a study by the ILC, the UN General Assembly even prepared a draft code for such a court. This effort lost steam during the negotiations of the Genocide Convention. The demand was later picked up in 1989. The Prime Minister of Trinidad and Tobago approached the International Law Commission (ILC) to set up a court that would be able to prosecute drug crimes. The ILC, paying heed to the request, drafted a statute by 1994 and a separate conference was eventually held in Rome to deliberate the draft. The deliberations saw multiple States participating directly and contributions from the sidelines from NGOs, deliberations that were hailed as a gold standard of multilateralism. However, the jurisdiction of the Court (particularly war crimes) generated great controversy. Yet, the Court received the approval of 120 out of the 148 participating States. The Statute was adopted in 1998 and came into force on 1 July 2002 after ratifications by 60 States.

1.1. Composition and Organisation
The Court is composed of 4 organs – the Presidency, the Chambers, Office of the Prosecutor, the Registry (under article 34 of the Statute).

a. Presidency

The Presidency of the ICC oversees the constitution of the judicial chambers of the ICC. It is also the organ that liaises with States by concluding cooperation agreements and organizing outreach activities.

b. Chambers

The three Chambers – Pre-Trial, Trial, and Appeals - are responsible for various stages of the proceedings. Preliminarily, the inclusion of the Pre-Trial Chamber was emblematic of the confluence (arguably, also compromise) of common law and civil law systems, whose varying features the Court has drawn from. The Pre-Trial Chamber is tasked with determining whether the Prosecutor’s request for the opening of an investigation under Article 15 should be granted, and also for reviewing the Prosecutor’s decision not to open an investigation. The Pre-Trial Chamber is also in charge of confirming the charges pinned by the Prosecutor. The Trial Chamber’s jurisdiction is triggered after this stage is crossed. The Trial Chamber conducts the trial and, where required, awards the sentence. Appeals against the decisions of both the Pre-Trial Chamber and the Trial Chamber are heard and decided by the Appeals Chamber.

At any point of time, the Chambers are constituted by a total of 18 judges, who are elected for nine-year terms by signatories of the Rome Statute. Article 36(3) of the Rome Statute prescribes the qualification criteria for the nomination and appointment of judges. Notably, it requires nominated judges to either have relevant experience in criminal law (as a judge or a prosecutor), or in international law generally.

These conditions have come under fire on many occasions and for multiple reasons. Preliminarily, the Statute appears to equate diplomatic experience to judicial experience. This came under scrutiny following the requested removal of former Judge Saiga during the Katanga trial. Despite having served as an ambassador, Judge Saiga did not have the qualifications to be appointed as a criminal division judge in her home state, Japan. In fact, 15 out of the 48 judges who have served on the ICC bench have previously held some form of domestic diplomatic position. Similarly, amongst the judges currently serving the Court, there are judges who have many years of experience in legal academia, but not in litigation or the judiciary.

The issue of geographical representation has also been contentious. Article 36(8)(a) calls for equitable geographical representation determined through regional groupings (being the African States, Asia-Pacific States, Eastern European States, Latin American and Caribbean states, and Western Europe and Others Group) with only one judge of the same nationality eligible to sit at one time. However, issues of dual nationality distort this requirement. The Raising the Bar report identifies that minimum voting requirements in practice reflects “an alarming concentration of the ICC’s judiciary in only a small handful of states, as well as a decline in the engagement of States Parties in the judicial selection process over time.”

c. Office of the Prosecutor

The Office of the Prosecutor has been envisaged as an independent and impartial investigating authority, drawing on the Yugoslavia and Rwanda models. Under Article 15, the Prosecutor is empowered to initiate investigations in situations, based on information received from states, organs of the UN, intergovernmental and nongovernmental organisations or other reliable sources. Before doing this, the Prosecutor must obtain approval from the Pre-Trial Chamber. Under Article 15, when the Prosecutor decides not to open such an investigation, the Pre-Trial Chamber may order the Prosecutor to reconsider their decision.

d. Registry

The Registry helps the Court to conduct fair, impartial and public trials. The core function of the Registry is to provide administrative and operational support to the Chambers and the Office of the Prosecutor.

e. Trust Fund for Victims

The Fund, albeit independent from the Court, works in tandem with the Court’s outreach team with a specific focus on securing the physical, psychological, and emotional dignity of victims and their families (this is known as the assistance mandate of the Fund). The Fund also implements and follows up on the implementation of the Court’s material and symbolic reparations orders. This is typically done through the development of implementation plans. The Fund is sponsored in part by the ICC’s budget and in part through voluntary contributions. The monetary investment in the Fund generates increased stakes in the tangible success and effectiveness of the Fund’s assistance mandate.

f. Assembly of States Parties

The Assembly of States parties is an external and independent institution vested with administrative and legislative powers. The ASP, for instance, is responsible for deciding and revising the budget of the Court. The ASP is also the body that receives nominations for the Prosecutor positions and nominates said Prosecutor.

1.2. Seat
The International Criminal Court is located in Scheveningen in The Hague. Until 2015, the Court was housed in temporary premises – those of the bank L’Arche. Following an architectural competition, the Court moved to its permanent premises in Scheveningen. The move was intended to symbolically reflect the Court’s permanence and its universalist missions.

The selection of The Hague has faced great censure, given that it places significant distance between the Court and those it admittedly serves. Recently, the counsels for the defence in Bangladesh/Myanmar requested the ICC to move its seat within reasonable proximity of the affected populations. The Court rejected the request, citing reasons of prematurity and immobility during the pandemic. In this context, it is important to acknowledge that the statute of the Court does allow for the seat to be moved wherever deemed necessary. The new design of the Court has also been called out by critical scholars as not being encouraging for victims with its opaque setting, monochromatic colour scheme, less visible witness boxes – all of which impede the interests of reflexivity.

1.3. Applicable law
Article 21 of the Rome Statute prescribes the sources of law that the ICC can apply. Not only is it the first time that the applicable law of an international criminal tribunal has been codified, but it is also the first codified effort to specify the hierarchy between the sources. Earlier tribunals predominantly relied on custom and general principles as gap filling tools. This invited severe criticism about it impugning the principle of legality and vesting unreasonable law-making authority on the Court. The most important sources are the Statute, the Court’s Rules of Procedure and Evidence, and the Elements of Crimes. If this fails to yield an effective solution, then the Court may consult general principles of international law and failing that, rules derived from national legislations and human rights. Article 21 was inserted with the motive of restricting the Court’s discretion and ensuring that the principle of legality (nullum crimen sine lege) is respected. Naturally, it has been read and applied strictly. But elevating the legal status of the Statute and delimiting the Court’s sources could only be effected at the cost of epistemic exclusions. For one, the demand for Article 21 was backed and followed through mainly by the American delegation, for reasons of avoiding vagueness. The construction of Article 21 that the Statute finally retained does not create any room for oral sources, customs, or indigenous legal orders. It also imposes Western epistemologies governing the formation of “treaties” and “international legal principles” on Global South peoples who forge relationships with the Court. Substantively, Article 21 – as the Court’s own jurisprudence has demonstrated – has made it difficult for the Court to recognize the evolving nature of international criminal law and the victimhood triggered by crimes that the original Statute did not codify.

Some have rightly noted that the Court seldom applies external law. This can be explained by the density of its internal legal instruments. But the Court does not shy in relying on external aids to interpret internal law. Another aid that the Court typically resorts to is its own past jurisprudence. However, the Court’s jurisprudence, as is protocol in civil law systems, is not binding. It is merely of persuasive value.

1.4. Jurisdiction
There are four bases for the Court’s jurisdiction: personal, territorial/nationality, subject-matter, and temporal.

In terms of ratione materiae, the Court is authorized to exercise jurisdiction over ‘the most serious crimes of international concern’: genocide, crimes against humanity, war crimes and aggression (Article 5(1)). These crimes largely correspond to the elements discussed above. In particular, Article 7 of the Rome Statute recognizes murder, extermination, enslavement, deportation, rape and sexual violence, apartheid, torture, enforced disappearance, imprisonment, persecution, and other inhumane acts as crimes against humanity. In 2022, the Court convicted Dominic Ongwen of crimes including forced marriage as an "other inhumane act" under Article 7 of the Statute. The crime of aggression has also been recently activated within the Court's jurisdiction. The list of crimes over which the Court has jurisdiction continues to be under contention. Notably, negotiations are underway about the potential inclusion of ecocide as a fifth crime in the Rome Statute.

On ratione personae and tertii, the first condition is one of age. The Court can only try natural persons above the age of 18. The second is that of territoriality. Article 12 of the Rome Statute confers territorial jurisdiction on the Court in cases where “conduct in question” was committed on the territory of a state party to the Statute or by a national of a state party. This link to territory was created – against the wishes of States like Germany and Korea  – to distinguish the ICC from a domestic court with universal jurisdiction and to wed the Court to its treaty character. The Court’s reading of territoriality has been at the centre of recent litigation, particularly concerning the Bangladesh/Myanmar situation. In the former, the Prosecutor placed a proprio motu request for investigation over the crimes committed by the Tatmadaw forces against Rohingyas in Myanmar. The Pre-Trial Chamber of the Court authorized investigation. But a major roadblock was that Myanmar was not a State party to the Rome Statute. The Court read territoriality ‘objectively’, noting that the conduct was partly committed in Myanmar and partly in Bangladesh (a State that was party to the Statute). This offered a route for the Court to exercise jurisdiction over the crimes.

The third condition is nationality. This is also prescribed by Article 12(2). Interestingly, the term nationality has not been defined in the Statute. The Court has implicitly imported the domestic understanding of nationality as the legal bond between the natural person and the sovereign State. Importantly, the Court’s jurisdiction cannot be activated through passive nationality (when only victims bear a nationality link to State parties). Nationality under Article 12(2)(b) is limited to active nationality.

The temporal starting point of the Court’s jurisdiction has been spelled out in Article 11. The provision notes that the Court’s jurisdiction is prospective and can be invoked only for crimes committed following the Statute’s coming into force on 1 July 2002.

Exceptionally, Article 12(3) allows non-State parties to file declarations accepting the Court’s jurisdiction on an ad hoc basis for crimes committed within their territories or by their nationals. This option – some argue – also offers the facility of circumventing the temporal limits of the Court’s jurisdiction. Palestine, for instance, has lodged an Article 12(3) declaration accepting the Court’s jurisdiction over crimes committed against its nationals prior to Palestine’s own accession of the Statute in 2015.

1.5. The triggering mechanisms
There are three ways to trigger the jurisdiction of the Court. a referral by a State Party, a referral by the Security Council acting under Chapter VII of the Charter of the United Nations, and the institution of an investigation by the Prosecutor acting on her own initiative (Article 13).

The first mode is a proprio motu investigation by the Prosecution. To do this, the Prosecutor must obtain the approval of the Pre-Trial Chamber by showing how and why the selected situation meets the admissibility and jurisdiction requirements prescribed by the Statute. The Prosecutor must also obtain the consent of the States implicated.

The second mode is self-referral. The bulk of the cases that the Court has heard have been self-referrals by the States in which the crimes were committed. A recurring concern with self-referrals has been that States have fashioned it into a tool to pursue retributive prosecutions of rebel non-State actors to bolster the “legitimacy of its own military operations”. Nouwen, and Werner have all shown how the States of Mali and Uganda self-selected certain crimes/criminals in their referrals and employed the rhetoric of self-referrals to exempt their own government forces from sanction.

Article 13(b) of the Statute also offers a route for the Security Council to intervene and refer cases to the ICC. This allows the Court – a treaty-based creature modeled to exercise jurisdiction purely basis nationality and territoriality – to extend jurisdiction over crimes and accused persons even in non-State parties. This route has been invoked twice. In March 2005, the Security Council, exercising its powers under Chapter VII of the UN Charter and acting on the recommendations of the UN fact finding commission, referred the situation in Darfur to the ICC. The Council found that war crimes and crimes against humanity were being committed by the Government forces and the Janjaweed militia in Sudan. The Darfur situation generated a string of decisions – famously, the ICC’s jurisprudence on the immunity of Omar Al-Bashir. Later, in 2011, through Resolution 1970, the Council referred the situation in Libya to the Court.

The Council referral route raises important questions about the legitimacy of the Court. The ICC originally postured itself as a mechanism to rectify the failures of past international criminal tribunals. The deliberations in Rome reveal that the drafters were clear about avoiding accusations of Eurocentric exercise of judicial discretion. However, the Security Council referral in the Statute suffers from the same vices. The Security Council referral departs from the nationality and territoriality based jurisdictional framework that the ICC otherwise rests on. As Talita Dias argues, the Security Council referral is not conceived as an exceptional tool that merely triggers the Court’s jurisdiction, it goes so far as to prescribe the contents of the Court’s substantive jurisdiction, elevating the referral to the same binding status as the Statute itself. This route of referral has faced much censure, primarily on account of its vulnerability to political misuse. Scholars argue that it offers a free pass to the permanent members to exercise “unilateral negative control” and exempt their own nationals from criminal responsibility for the same acts that they refer other individuals to the ICC for. The recent political clashes triggered by the Palestine and Afghanistan situations have shown that the Court still “reifies White supremacy” and “works to mask the core-periphery relations” that sustain economic and power inequalities.

1.6. Admissibility
Article 17 of the Statute lays down the substantive conditions for the admissibility of a case before the ICC. Admissibility at the Court hinges on two aspects. The first is complementarity. The deliberations at the Rome conference of 1998 show that complementarity was the most decisive in soliciting the support of the now States parties to the Statute. Complementarity requires an assessment of a) whether the referring State/host State is unwilling or unable to prosecute the case. The ICC, as a court of last resort, can be approached only once it is settled that the national judicial system cannot help. The Defence – in order to challenge admissibility – must demonstrate that the national jurisdiction is investigating and taking genuine steps to interrogate witnesses, collecting evidence, etc. The second part of the admissibility test relates to the analysis of the “gravity threshold”, in order to determine whether the case is of sufficient gravity to justify further action by the Court. The gravity threshold has been repeatedly debated by litigants. In the Al Hassan case, the Appeals Chamber listed some criteria that are relevant to assessments of gravity (while adding the caveat that the assessment necessarily varies from case to case). These criteria included the number of victims, qualitative factors such as the nature, scale, manner of commission of the crimes including human rights violated as a result, the impact on victims, and sentencing factors.

1.7. Procedure
The ICC – akin to its predecessors – has drawn great inspiration from domestic criminal procedures. Right from the deliberations in Rome, the Court leaned more towards inquisitorial rather than adversarial procedures. Some surmise that this was a reaction to the heavy influence of adversarial systems on the ad hoc tribunals. These procedures can be found both in the Rules of Procedure and Evidence, the Statute, and the Regulations of the Court.

Criminal investigations at the ICC begin with a referral – either by the Prosecutor, the State, or the Security Council. These referrals must meet the conditions of admissibility, gravity, and interests of justice. Whether or not these conditions are met, that is whether there is a reasonable basis to proceed with the investigation, is ascertained by the Pre-Trial Chamber.

The Chamber’s approval allows the criminal investigation to hit the ground. Criminal investigations in the ICC are typically spearheaded by the Prosecutor. They entail interviews with witnesses, forensic examinations, and collection of oral and documentary evidence. Importantly, the investigation relies heavily on cooperation of States given that neither the Office of the Prosecutor nor the ICC have their own police.

Based on the indictment prepared by the Prosecutor, the Court issues summons to the accused. This is followed by a confirmation of charges hearing. This feature is unique to the ICC and was introduced to add a judicial economy filter before cases go to trial. The confirmation of charges stage serves to give the accused a chance to challenge the charges pressed against them, the Prosecutor a chance to improve/modify/withdraw the changes, and the Chamber a chance to assess if the evidence is sufficient to proceed to trial.

The trial follows the confirmation of charges. The conduct of the trial is bound to respect for the accused’s right to fair trial is the fulcrum of the Rome Statute. Fair trial, which is a commitment binding on the States parties, the Court, and the OTP, works to recognize the accused person’s subjecthood. Fair trial has birthed multiple rights including the accused’s right to legal counsel, right to a hearing, right to cross-examination, and such. The Statute recognizes the presumption of innocence of the accused until proven guilty. The onus lies on the Prosecutor to prove the accused’s guilt “beyond reasonable doubt”.

The trial consists of opening statements, presentation of evidence, closing arguments, deliberations, and judgment. While it is generally the Prosecution which opens the trial (and the examination of evidence), the judges hold the discretion to modify the sequence if the case warrants it. Trials at the ICC usually last many months. But the Statute also contains several tools to promote judicial expeditiousness, such as plea bargaining/allowing the accused to admit to their guilt for reduced charges. Trials can yield three outcomes: no case to answer, conviction, or acquittal. The judgment delivered by the Chamber must be full and well-reasoned, it can contain majority and minority opinions.

1.8. Enforcement of judgments and State cooperation with the ICC
The primary challenge plaguing the ICC is its enforcement powers. Although decisions of the Court are binding on parties, the ICC does not possess its own enforcement infrastructures. Illustratively, the ICC does not have its own police that could accost those who are charged by the Court and bring them to the Court's premises in the Hague. The only recourse left for the Court is to rely on cooperation of the State parties to the Rome Statute. State parties to the Statute have an obligation to cooperate with the Court in all stages of the investigation and trial: from surrendering suspects/accused and seizing assets to detaining convicts. To prevent conflicts with other obligations, the Statute has also devised Article 98 agreements to seek prior agreement on crimes/individuals that are out of the Court’s reach. Despite the statutory obligation, the Court’s fiscal dependence on State parties has made their cooperation almost completely contingent on States’ political will and their own complicity in the cases brought before the ICC. The recent chequered history of the ICC is testament to this. Of the 36 arrest warrants issued by the Court, only 20 have been enforced. The Court’s warrants against Bosco Ntaganda, Simone Gbagbo, and Omar Al-Bashir were flouted for many months. The Court’s chiding of African States’ failure in Bashir’s case triggered a string of withdrawals (from Burundi, South Africa, and the Gambia). In all three cases, the withdrawals were intended to protect and immunize state officials, including sitting heads of State, from the ICC's reach. The Philippines also notified the ICC of its withdrawal, pushing back on the Prosecutor's efforts to investigate the drug war and former President Duterte's complicity in its violence. Withdrawals have becoming increasingly popular tools for States to express their discontents with the Court, and to curb the Court's prosecutorial reach. This pushback is seemingly quite alive to the Court's treaty-based character and the powers that such a design vests in treaty parties.

These challenges are compounded by the opposition to the ICC’s jurisdiction by powerful States. To impede the Court’s reach, the United States’ Congress has passed the American Service Members’ Protection Act in 2002, empowering the government to stop financial aid to the ICC’s state parties who surrender American nationals to the ICC. When the Prosecutor expressed her desire to prosecute CIA officials in relation to the opening of an investigation in Afghanistan, the US government also went so far as to issue sanctions against ICC officials. Similar non-cooperation quagmires have plagued the opening of investigations in Palestine against Israeli nationals and in Iraq against British nationals.

1.9. Victims’ rights
Criminal law – both domestically and in international criminal law – has historically been limiting for victims. The ICC was therefore conceived as a more expansive forum for victims’ participation. In large part, this was a function of the advocacy of non-governmental organizations which ensured that the Rules of Procedure and Evidence at the Court “contained strong provisions on victims’ rights”. Victims as a stakeholder are also defined generatively. As Rule 85 of the Court’s RPE spells out, the term victim under the Statute refers to any “natural persons who have suffered harm as a result of the commission of any crime within the jurisdiction of the Court”. The term is also said to include institutions and organizations whose properties sustain harm. Whether or not the applicants meet the conditions of victimhood is a determination made by the Court.

Victims’ rights at the Court fall within three main categories: right to reparation, right to protection, and right to participation. The right to representation is typically available at all stages of proceedings. Importantly, each stage of proceedings can see the participation of a different group of victims – whether victims of the situation or the convicted person or of the facts. On some occasions, participation translates into the right to legal representation. When victim groups have mutual interests, they can also be represented by common counsels. The right to protection, under Article 68(1) of the Statute, is effected through general preventative measures (available to all victims to secure their physical, psychological, and mental well-being upon testifying) and more ad hoc measures which are designed and customized to the real risk posed to each victim. The final bucket is the right to reparation. The exercise of this right is typically facilitated by the Registry and once invoked places the onus on the Court to direct the convicted person and/or the Trust Fund to offer reparations to the victims.

B. Hybrid (Mixed) Tribunals
Hybrid tribunals are those tribunals that are governed by and have the authority to apply both international and domestic laws. This is unlike the design of the ICC which is only authorized to apply the rules prescribed by an international statute. Predecessor tribunals of the ICC such as the International Criminal Tribunal for Rwanda and the International Criminal Tribunal for the former Yugoslavia also can be classified as international tribunals.

2.1. Special Court of Sierra Leone

The Special Court for Sierra Leone (SCSL) was established by treaty between Sierra Leone and the UN, to prosecute crimes committed during the 1991 civil war between militia and the governments in Sierra Leone and Liberia. The Court is independent of both the United Nations and the domestic legal system. The Court is composed of judges – the majority of whom are elected by the United Nations and the remaining by the government of Sierra Leone. The jurisdiction of the Court is circumscribed to crimes against humanity and war crimes committed in non-international armed conflicts. Like the ICC, the Court’s prosecutorial strategy is to prosecute those persons who are most responsible for serious violations of international humanitarian law and Sierra Leonese law. The Court commenced its work in 2002 and wrapped up in 2013, entrusting its pending cases to the Residual Court for Sierra Leone. The SCSL grapples with two key challenges: first, the award of blanket amnesties to the Lord Resistance Army combatants and the subsequent withdrawal of the amnesties. Second, the individual criminal responsibility of child soldiers in the Lord Resistance Army.

2.2. Kosovo Specialist Chambers

The Chambers – and the Specialist Prosecutor’s Office was established in 2011 following a report by the Parliamentary Assembly of the Council of Europe which shed light on the detention, torture, and enforced disappearances of Serbs and Kosovo Albanians during the 1999 conflict in Kosovo. The Specialist Chambers comprises two organs, the Chambers and the Registry. The Specialist Prosecutor’s Office is an independent office for the investigation and prosecution of the crimes within the jurisdiction of the Specialist Chambers. The Specialist Chambers and the Specialist Prosecutor’s Office are staffed with international judges, prosecutors and officers and have a seat in The Hague, the Netherlands.

The Chambers have been designed as a hybrid court which will apply the Constitution of the Republic of Kosovo and customary international law/international human rights instruments. Local Kosovar populations have been discomfited by the composition/applicable law of the Chambers, finding that is sidelines the local societal fractures and aspirations.

2.3. Extraordinary Chambers in the Courts of Cambodia

In 1997, the Cambodian government approached the United Nations to set up a tribunal to prosecute the crimes committed by the Khmer Rouge against political dissidents from 1975 to 1979. The Court was established through a 2003 agreement between the UN and Cambodia. The ECCC has been absorbed into the Cambodian domestic legal system, albeit supported by the United Nations. The jurisdiction of the ECCC extends to genocide, crimes against humanity, and war crimes (solely in international armed conflicts). The Cambodian government insisted that the Court be predominantly staffed by local judges and prosecutors. This demand was honored. Although the dominance of local staff has not inspired much confidence in the impartiality of the bench, all the judges and prosecutors are appointed by the Cambodian Supreme Council of Magistracy with the UN Secretary-General nominating international personnel.

C. The proposed African Criminal Court
Right from the mid-2000s when the ICC’s docket was almost completely populated by cases seeking prosecution of African rebel groups or heads of State, the African Union has voiced its strong objection to being disproportionately targeted by the ICC. Fair to say that the ICC found it difficult to retain the trust of the 34 African States who signed onto its Statute, with States like Burundi choosing to exit the Statute altogether.

The distrust in the ICC prompted the African Union to call for an African criminal court and dissuading African states from cooperating with the ICC. In 2014, the statute of this court – which came to be called the African Court of Justice and Human Rights – was passed. The jurisdiction of the African court and the ICC greatly overlap. Article 46Ebis of the African Criminal Court's Statute is different only in so far as it allows the Court to exercise jurisdiction when the victim is a national of a State party or when a State party’s vital interests have been threatened. The Court has jurisdiction over 14 unique offences, including the core crime but crimes outside the Rome Statute such as collective punishment.

Scholars have flagged that the glaring overlaps between the ICC and the African Criminal Court can lead to problematic consequences. It has been proposed that the overlap be avoided by either refashioning the complementarity principle or allocating transnational/less serious crimes to the African Criminal Court.

Summary
This section focusses on the workings and achievements of the ICC, the most prominent universal and permanent international criminal tribunal devoted to the adjudication of criminal responsibility. It also identifies the political pressures against the ICC. The final part of the chapter looks at contemporary hybrid tribunals.

Further Readings

 * Cryer, Robert, Hakan Friman, Darryl Robinson, and Elizabeth Wilmhurst. An Introduction to International Criminal Law and Procedure. 2d ed. Cambridge, UK: Cambridge University Press, 2010.
 * Werle, Gerhard. Principles of International Criminal Law. 2d ed. The Hague: TMC Asser, 2009.


text:
Public International Law/International Criminal Law/International Crimes

Author: Taxiarchis Fiskatoris "Required knowledge: International Criminal Law, Law of Armed Conflict" Learning objectives:

 * to understand the foundations and purpose of international criminal justice
 * to identify the most prominent international crimes
 * to recognize the content of international crimes and its dynamic evolution in time;
 * to apply the elements of international crimes to practical situations

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A. Introduction
A conceptual definition of international crimes does not exist in conventional or customary international law. The constitutive instruments of international or internationalised courts and tribunals enumerate their subject-matter jurisdiction without explicitly labelling the punishable offences as international crimes. The jurisdictional remit of such institutions cannot be considered a substitute for a comprehensive international criminal code, which does not exist. The Preamble to the Rome Statute implies that the ICC’s jurisdiction does not cover all “international crimes”. Scholars usually distinguish between “international crimes lato sensu (in the broad sense)”, and “international crimes stricto sensu (in the narrow sense)” or “core crimes”. According to the mainstream literature, the latter coincide to a great extent with the Rome Statute crimes.

B. Rome Statute crimes
The ICC Statute qualifies the offences within its jurisdiction as “the most serious crimes of concern to the international community as a whole”. Article 5 vests the ICC with jurisdiction over four categories of offences, which are analysed in more details in the ICC Elements of Crimes document. All Rome Statute crimes´ definitions have a similar structure, which consists of an exhaustive or indicative catalogue of individual underlying offences, and an introductory sentence about their contextual elements, also known with the French word "chapeau" (meaning "hat"). The underlying offences may overlap, but the contextual elements distinguish the crimes from one another, as well as from other international and ordinary crimes. Thus, a murder is an ordinary crime, which can take the form of a war crime, a crime against humanity or genocide, depending on what contextual elements are fulfilled. Additionally, according to Article 30, “unless otherwise provided”, the mental element of “intent and knowledge” applies to all offences within the ICC’s ambit. A legal analysis of the crimes that follows their historical evolution is deemed here the most appropriate, inasmuch as it highlights that their concept and content is not static but dynamic.

i. The nature of war crimes
War crimes is the oldest category among the four Rome Statute crimes. Individual accountability for war crimes has its origins in the process of progressive criminalisation of customary and conventional rules of the law of armed conflict.. War crimes generally pertain to the use of prohibited weapons and methods of warfare, and to attacks on protected persons or property.

ii. The underlying offences
All World War I peace treaties contained a clause regarding the prosecution of “persons accused of having committed violations of the laws and customs of war”. The Charter of the Nuremberg Military Tribunal provided a non-exhaustive catalogue. The exact list was reiterated in Article II(1)(b) of the Control Council Act No.10, which served as the legal basis for the post-war military tribunals of the occupying powers in Germany as well as in Principle VI of the so-called Nuremberg Principles, as formulated by the International Law Commission (ILC) in 1950.

In its 1951 “Draft Code of Offences against the Peace and Security of Mankind”, the ILC commented that war crimes were relevant not only in cases of declared war, but also in “any other armed conflict which may arise between two or more States, even if the existence of a state of war is recognized by none of them”. In 1968, a UN Convention provided that war crimes are not subjected to statutory limitations, and made a special mention to their relation with the “grave breaches” regime of the 1949 Geneva Conventions. The content of war crimes has continued being elaborated by the ILC in its review of the Draft Code, and in the Statutes of the ad hoc International Criminal Tribunals for the former Yugoslavia (ICTY) and Rwanda (ICTR). All of them confirmed that “grave breaches” of the Geneva Conventions give rise to individual criminal accountability. However, not every violation of the Geneva Conventions amounts to a war crime.

The ICTY had explicit jurisdiction over a restrictive list of “grave breaches of the Geneva Conventions of 1949” (art 2), and an illustrative catalogue of “violations of the laws or customs of war”, which followed key provisions of the 1899/1907 Hague Conventions (art.3). The ICTR’s jurisdiction over war crimes was different, given the predominantly internal nature of the Rwandan conflict. Its Statute referred to violations of common article 3 of the Geneva Conventions, and to the 1977 Additional Protocol II (AP II).

War crimes have also been incorporated into Article 8 (2) of the ICC Statute.

The enumeration of war crimes in the context of non-international armed conflicts is modest in comparison to that of war crimes in international armed conflicts. By way of illustration, the war crime of “intentionally launching an attack in the knowledge that such attack will cause… widespread, long-term and severe damage to the natural environment…” can only be prosecuted by the ICC if linked to an international, and not an internal conflict. Through the amendment procedure of the Rome Statute, the number of punishable war crimes committed in non-international armed conflicts incrementally converges with that of war crimes perpetrated in international conflicts.

iii. The contextual elements
The jurisprudence of the ad hoc Tribunals further shed light on the contours of war crimes. In its first case, the ICTY clarified that the prerequisite for war crimes, the existence of an “armed conflict”, was fulfilled whenever “there is a resort to armed force between States or protracted armed violence between governmental authorities and organized armed groups or between such groups within a State”. That meant, essentially, that war crimes can be committed in both international and internal armed conflicts. A mere resort to force, such as in occasions of riots, does not meet the required level of intensity of “protracted armed violence”, and thus criminal conduct in such contexts does not constitute war crimes.

It should be highlighted that, even in the event of an armed conflict, not every offence is necessarily a war crime. The perpetrator’s ability or decision to commit the offence, the purpose for which it was committed, or the manner in which it was committed must be substantially linked to the conflict. This does not mean that the perpetrators of war crimes cannot be civilians.

Article 8(1) of the ICC Statute stipulates that “the Court shall have jurisdiction in respect of war crimes, in particular when committed as part of a plan or policy or as part of a large-scale commission of such crimes”. This intends to assist the ICC to assess the gravity of war crimes occurrences, but does not establish an additional contextual element. An isolated act may still theoretically amount to war crime. Each underlying war crime may have a higher mental element threshold. For instance, the accidental destruction of historic monuments may not qualify as a war crime, but “intentionally directing attacks” against them, provided they are not used for military purposes, most probably will.

i. The nature of crimes against humanity
The essential characteristic of crimes against humanity (CAH) is that humanity rather than the individual is their ultimate victim. Some CAH overlap with genocide and war crimes. They differ, though, from genocide because they lack the mental element of special intent to destroy a group, and from war crimes because they apply equally in wartime and peacetime.

ii. The underlying offences
The first codification, and successful prosecution of CAH occurred in Nuremberg. Article 6(c) of the IMT Charter provided an enumerative definition. Its primary rationale was the punishment of German officials responsible for atrocities against, and persecutions and deportations of German citizens of Jewish, Roma or Sinti descent. In that respect, the term engulfed what would soon become the crime of genocide.

The ICTY and ICTR Statutes expunged persecution on cultural grounds, but it could be argued that they rendered the CAH concept open-ended by appending “other inhumane acts”, which arguably is a broad term. Furthermore, both Statutes incorporated from the Draft Code the underlying offence of “torture”, and added on “rape”. Article 7 of the Rome Statute has an even longer list. It considerably expanded the crime of persecution to “any identifiable group or collectivity on political, racial, national, ethnic, cultural, religious, gender… or other grounds…”. Moreover, it associated with rape “sexual slavery, enforced prostitution, forced pregnancy, enforced sterilization, or any other form of sexual violence of comparable gravity”.

Enforced disappearance of persons, and the crime of apartheid were also included. However, “other inhumane acts” were restricted to “acts of similar character intentionally causing great suffering, or serious injury to body or to mental or physical health”. This did not stray far, though, from the jurisprudence of the ad hoc Tribunals.

iii. The dynamic evolution of the contextual element
In Nuremberg, the notion had to be linked with war crimes or crimes against peace. The stated reason was alignment with the principle of legality, given that, unlike war crimes, CAH did not exist as a criminal offence at the time they were committed. An additional pragmatic reason was that the victors of the Second World War wanted to dodge any allusions to their own colonial or deportation practices.

However, Control Council Law No 10 released the subsequent military tribunals in Germany from the obligation to establish a connection with other crimes under their jurisdiction, and broadened the scope of the concept. The Nuremberg Principles reinstated the mandatory link with war crimes and crimes against peace, which was dropped again by the 1954 ILC Draft Code of Offences against the Peace and Security of Mankind. The latter also detached the concept of genocide, which had already been recognised as an independent crime, thus being able to add persecution on cultural grounds to the CAH concept. The new condition was that CAH had to have been committed, instigated, or tolerated by State authorities. In 1968, a UN Convention established that statutory limitations do not apply to CAH “whether committed in time of war or in time of peace”.

In 1991, the ILC, reviewing the Draft Code, deemed it proper to rename this category of international crimes “systematic or mass violations of human rights”. It thus laid emphasis on a different distinctive criterion; the systematic and mass-scale nature of the crimes, which expressed their seriousness. The conflict nexus returned in the ICTY Statute due to the special nature of the Tribunal. However, the Tribunal itself found that such a nexus had become extinct in customary international law. Besides, the ICTR preferred the contextual element of “a widespread or systematic attack against any civilian population on national, political, ethnic, racial or religious grounds”. The ICTY held, though, that discriminatory intent is only necessary for the sub-category of persecutions. Accordingly, the Rome Statute did not adopt either conflict nexus as a contextual element, nor discriminatory intent as the general mental element for all CAH.

The contextual element of CAH in the Rome Statute is different from previous Statutes. They must be “committed as part of a widespread or systematic attack directed against any civilian population, with knowledge of the attack”. Article 7(2) further specifies that the “attack” must be “pursuant to or in furtherance of a state or organizational policy to commit such attack”. According to the Elements of Crimes, “attack” is not necessarily military, but understood as “involving the multiple commission” of an underlying offence. The “policy… requires that the state or organization actively promote or encourage such an attack”.

The proper interpretation of the contextual element has long been troubling judges and scholars. The phrase “any civilian population” excludes isolated acts, but does not mean that the target of the attack must be the entire population of a State or territory. The national, ethnical, racial etc. identity of the population is immaterial, and the perpetrator may share the same nationality with the victims. The victims must be civilians, but the presence of some combatants does not deprive the population of its civilian character.

The wording “widespread or systematic” is disjunctive, and the contextual element is met by the existence of either of the two. “Widespread” is usually interpreted as large scale conduct directed against many victims. “Systematic” points to an organised repetitive pattern, and excludes accidental or random occurrence. The ICC has attempted to set a series of criteria for identifying the policy requirement. The limits of the “organizational” requirement is a matter of contention.

To this date, apart from the Rome Statute, there is not any international convention on crimes against humanity. The ILC has concluded a Draft Treaty, but the General Assembly has not yet convened an international conference. The ICTY declared that CAH are part of customary international law, but there is definitely a number of major objector States.

i. Origin and definition
Genocide was explicitly recognised in the 1948 Genocide Convention as a “crime under international law” whether committed in time of war or peace. It is a compound word formed by the Greek “genos” (set of people with common ancestry or characteristics) and the Latin “cidium” (killing). The term was coined during the inter-war period by the Polish-Jewish jurist Raphael Lemkin to describe “a coordinated plan of different actions aiming at the destruction of essential foundations of the life of national groups, with the aim of annihilating the groups themselves”. The modern legal definition of genocide, formulated for the first time in the Genocide Convention, and verbatim replicated in the Statutes of most international and hybrid courts and tribunals, is both broader and narrower than Lemkin’s proposal. It is broader because it covers more than national groups, and narrower in that it does not address but physical and biological genocide, excluding acts that are often described as “cultural genocide”. “Hence, an enterprise attacking only the cultural or sociological characteristics of a human group in order to annihilate these elements which give to that group its own identity distinct from the rest of the community would not fall under the definition of genocide”. In strict legal terms, “[g]enocide means any of the following acts committed with intent to destroy, in whole or in part, a national, ethnical, racial or religious group, as such:

(a) killing members of the group;

(b) causing serious bodily or mental harm to members of the group;

(c) deliberately inflicting on the group conditions of life calculated to bring about its physical destruction in whole or in part;

(d) imposing measures intended to prevent births within the group;

(e) forcibly transferring children of the group to another group”.

ii. The material element (actus reus)
In other words, the material element of genocide may take the form of any one of five alternative prohibited acts, directed against any one of four alternative protected groups. The ICTR has attempted to define the four protected groups based on scientific criteria. However, jurisprudence has progressively accepted that whether one belongs to a protected group does not exclusively depend on objective facts, but also on the subjective perceptions of the victims or the perpetrators. In any case, prohibited acts committed against other groups, such as political, social or gender groups do not fall within the definition.

Causing the death of even one member of a protected group may amount to genocidal conduct, provided, of course, that the mens rea of the offence are also met. Normally, “[t]he conduct [takes] place in the context of a manifest pattern of similar conduct directed against that group”. But genocide is not confined to killing. Echoing the judgment of the first international genocide trial in history, the ICC Elements of Crimes accept that, among others, “torture, rape, sexual violence or inhuman or degrading treatment” may constitute underlying genocidal offences as causing serious bodily or mental harm. The inflicted harm does not need to be “permanent and irremediable”, but must nonetheless create “a grave and long-term disadvantage to a person’s ability to lead a normal and constructive life”.

The phrase “deliberately inflicting on the group conditions of life calculated to bring about its physical destruction in whole or in part” has been interpreted by the ICTR as referring to “the methods of destruction by which the perpetrator does not immediately kill the members of the group, but which, ultimately, seek their physical destruction”. Subjecting the group members to excessive work or to the bare minimum means of survival, such as food and water supplies, hygienic conditions, or medical services are characteristic cases in point. The ICC Elements of Crimes provide the additional example of “systematic expulsion from homes”. It is doubted, though, that forcible displacement of populations, also known with the expression “ethnic cleansing”, automatically falls under this category, although it may help establish genocidal intent.

The last two prohibited acts concern the destruction of a group through extinction of its future generations. According to the ICTR, the measures intended to prevent births within the group may be physical, such as sexual mutilation, sterilisation, forced birth control, separation of the sexes, prohibition of marriages, deliberate impregnation by a person belonging to another group so that the born child belongs to that other group. But they may also be mental. Traumatising acts, such as rape, can be a measure intended to prevent births when the person raped refuses subsequently to procreate. The wordings “intended to prevent” and “within a group” suggest that the conduct could still be labelled genocide even if the intended malicious outcome is not achieved or if the birth control does not apply to the whole group.

Transferring children of one group to another can take the form of physical force, such as in the case of forced adoptions outside the original group. But it may equally “include threat of force or coercion, such as that caused by fear of violence, duress, detention, psychological oppression or abuse of power, against such person or persons or another person, or by taking advantage of a coercive environment”. This last form of genocidal act is the closest the legal definition gets to what is known as “cultural genocide”.

ii. The mental elements (mens rea)
The legal definition of genocide consists of two mental elements. First, the general intent to execute one of the underlying offences of the material element. However, genocide´s real distinctive feature is the second and more stringent mental element of a specific intent (dolus specialis) of the perpetrator to destroy “in whole or in part” a protected group “as such”. The actual destruction of the group is not required. The wording “in part” suggests that even the intention to destroy a small but “substantial part” of the group, not only in the sense of numeric size but also of emblematic prominence, counts as genocide. It is usually “difficult, even impossible” to unequivocally establish genocidal intent, especially when there are other reasonable explanations, although intent must be distinguished from the personal motive. Genocide is the only international crime which expressly covers instances of direct and public incitement, “even where such incitement failed to produce the result expected by the perpetrator”.

i. From crimes against peace to the crime of aggression
After the end of the First World War, the Allied Powers envisaged an international special tribunal to hold the German Emperor individually accountable, among others, for violations of the then applicable law on the use of force (jus ad bellum). The Kaiser evaded justice, but “crimes against peace” were officially prosecuted at the Nuremberg and Tokyo Trials, as well as by the post-war military tribunals established on the basis of the Control Council Act No.10.

The early version of the “crime of aggression” referred to "“planning, preparation, initiation or waging a war of aggression, or a war of violation of international treaties, agreements or assurances, or participation in a common plan or conspiracy for the accomplishment of any of the foregoing”."The Tokyo Charter added that the offence pertained to both “declared or undeclared” wars.

Crimes against peace formed part of the Nuremberg Principles of the International Law Commission. However, due to the exceptionally political nature of the matter, which interferes with the UN Security Council’s power to determine acts of State aggression or with diplomatic means of dispute resolution, the United Nations were unable to agree upon a definition. A definition of aggression was reached in 1974 with the UN General Assembly’s Resolution 3314(XXIX), but States would still find it unsuitable for the purposes of international criminal law. Hence, despite the inclusion of the crime of aggression in the ICC Statute in 1998, the Court remained unable to exercise jurisdiction over such offences until the Assembly of States Parties to the Statute agreed on a definition, which finally occurred at the 2010 Kampala Review Conference.

ii. Modern definition
Article 8bis of the Rome Statute provides that "“[f]or the purpose of this Statute, ´crime of aggression´ means the planning, preparation, initiation or execution, by a person in a position effectively to exercise control over or to direct the political or military action of a State, of an act of aggression which, by its character, gravity and scale, constitutes a manifest violation of the Charter of the United Nations”."It then follows the line of Resolution 3314(XXIX), defining an “act of aggression” as "“the use of armed force by a State against the sovereignty, territorial integrity or political independence of another State, or in any other manner inconsistent with the Charter of the United Nations […] regardless of a declaration of war”."The article concludes with an enumeration of specific aggressive acts.

The ICC’s jurisdiction was officially activated eight years after Kampala, with a more stringent jurisdictional regime than other ICC crimes. Yet, there hasn’t been any prosecution of the crime of aggression since the post-World War II Military Tribunals, which leaves the exact contours of the crime open to future judicial scrutiny. Nonetheless, a few preliminary remarks can already be made. First, the definition of the crime of aggression does not cover every breach of Article 2(4) of the UN Charter, but only the most “manifest violations”. Moreover, aggression is defined as a leadership crime, which renders the prosecution of  aggressive acts by non-State actors improbable, unless they have essentially substituted the State mechanism. By all means, aggressive acts must turn against other States and not the “enemy inside”.

I. General remarks
Depending on the definition of international crimes one adopts, the catalogue of international crimes can be much broader. For instance, Professor M. Cherif Bassiouni, one of the pioneers of modern international criminal law, having studied international conventions with penal characteristics, had compiled a list of no less than twenty-five international crimes in the broad sense.

Most of those offences, which share a transboundary element, are to be found in suppression conventions that establish for States parties a duty to domestically legislate against acts proscribed by the convention, as well as a right or duty to either prosecute or extradite the offenders and to cooperate in prosecution and punishment. A majority of modern scholars prefers to call such offences transnational crimes or treaty crimes.

Piracy, terrorism, mercenarism, corruption, money laundering, trafficking in persons, drugs and hazardous wastes, illicit exploitation of natural resources, and the crime of unconstitutional change of government appear on the Malabo Protocol, which establishes the subject-matter jurisdiction of a future African Criminal Court.

II. Piracy
Piracy was the first offence over which the principle of universal jurisdiction applied. It is now defined in the UN Convention on the Law of the Sea. It basically consists of "“any illegal acts of violence or detention, or any act of depredation, committed for private ends by the crew or passengers of a private ship or a private aircraft, and directed on the high seas against another ship or aircraft, or against persons or property on board such ship or aircraft”."

III. Human trafficking
Although “slave-trading counts among the first international crimes”, modern slavery in the form of trafficking in human beings, which is broader than the CAH of enslavement, is still considered a transnational crime. The 2000 UN “Protocol to Prevent, Suppress and Punish Trafficking in Persons, Especially Women and Children, supplementing the UN Convention against Transnational Organized Crime” obliges States Parties to domestically criminalise “trafficking in persons”, which it defines as "“the recruitment, transportation, transfer, harbouring or receipt of persons, by means of the threat or use of force or other forms of coercion, of abduction, of fraud, of deception, of the abuse of power or of a position of vulnerability or of the giving or receiving of payments or benefits to achieve the consent of a person having control over another person, for the purpose of exploitation. Exploitation shall include, at a minimum, the exploitation of the prostitution of others or other forms of sexual exploitation, forced labour or services, slavery or practices similar to slavery, servitude or the removal of organs”."

IV. Torture
Torture is an underlying offence of genocide, crimes against humanity, and war crimes. When it does not fulfil the contextual elements of the above offences, it may still be considered a discrete international crime, as proscribed under customary international law and the UN Convention against Torture.

V. Terrorism
The universal recognition of transnational terrorism as an independent international crime has long been on the way, with the lack of a solid definition usually being raised as an issue against. The champion of modern international criminal law, Antonio Cassese, as well as the Special Tribunal for Lebanon at the time of his presidency, have made the controversial argument that"“a customary rule of international law regarding the international crime of terrorism, at least in lime of peace, has indeed emerged. This customary rule requires the following three key elements: (i) the perpetration of a criminal act (such as murder, kidnapping, hostage-taking, arson, and so on), or threatening such an act; (ii) the intent to spread fear among the population (which would generally entail the creation of public danger) or directly or indirectly coerce a national or international authority to take some action, or to refrain from taking it; (iii) when the act involves a transnational element.”"

VI. Drug trafficking
Several multilateral conventions are dedicated to the suppression of trafficking in narcotic drugs. The Final Act of  the Rome Conference, where the establishment of the ICC was decided, recognizes that “the international trafficking of illicit drugs is a very serious crime, sometimes destabilizing the political and social and economic order in States”, and that a review conference should “consider the crimes of terrorism and drug crimes with a view to arriving at an acceptable definition and their inclusion in the list of crimes within the jurisdiction of the Court”.

VII. Ecocide
The relevance of international criminal law to the protection of the environment has been debated and occasionally put on the UN agenda at least since the 1970s. However, with the exception of the ICC Statute, where widespread, long-term and severe environmental damage is mentioned as an underlying war crime in international armed conflicts, international criminal law remains anthropocentric. In recent years, the recognition of environmental offences as international crimes worthy of prosecution at the international level has gained significant current. The connotative term “ecocide” is used in order to raise awareness. NGOs and eminent legal scholars have attempted to vest ecocide with a definition that could form the basis of consent for its becoming the fifth autonomous Rome Statute crime: "“For the purpose of this Statute, “ecocide” means unlawful or wanton acts committed with knowledge that there is a substantial likelihood of severe and either widespread or long-term damage to the environment being caused by those acts.”"

Summary

 * There is not any universally acceptable conceptual definition of international crimes. According to the mainstream literature, the core of the concept consists of the four crimes that can be prosecuted by the International Criminal Court, namely war crimes, crimes against humanity, genocide, and the crime of aggression. The content of these offences has been dynamically evolving mainly through their enumeration in the Statutes of the various international criminal courts and tribunals, as well as through their case law.
 * All Rome Statute crimes´ definitions have a similar structure, which consists of an exhaustive or indicative catalogue of individual underlying offences, and an introductory sentence about their contextual elements. The underlying offences may overlap, but the contextual elements distinguish the crimes from one another, as well as from other international and ordinary crimes.

 * War crimes generally pertain to the use of prohibited weapons and methods of warfare, and to attacks on protected persons or property taking place during an international or non-international armed conflict.

 * Crimes against humanity refer to a catalogue of human rights violations and other inhumane acts, occurring in times of armed conflict or peace, that presumably harm all of humanity when they fulfil the contextual element of having been committed as part of a widespread or systematic attack directed against any civilian population, with knowledge of the attack.

 * Genocide means the commission of five alternative prohibited acts, committed with intent to destroy, in whole or in part, a national, ethnical, racial or religious group, as such.

 * The crime of aggression has long been hard to determine, but finally defined in the ICC Statute as the planning, preparation, initiation or execution, by a person in a position effectively to exercise control over or to direct the political or military action of a State, of an act of aggression which, by its character, gravity and scale, constitutes a manifest violation of the Charter of the United Nations.

 * Some scholars have adopted broader definitions of the notion of international crimes, which extends to offences proscribed in international suppression conventions, such as piracy, torture, terrorism, and several forms of transnational trafficking. A novel crime of “ecocide” has recently gained significant traction in scholarship, activism and politics.

Further Readings

 * Ambos K, ‘Judicial Creativity at the Special Tribunal for Lebanon: Is there a Crime of Terrorism under International Law?’ (2011) 24 LJIL 655
 * Bassiouni M C, Crimes against Humanity: Historical Evolution and Contemporary Application (CUP 2011)
 * Bassiouni M C, International Criminal Law Conventions and their Penal Provisions (Transnational Publishers 1997)
 * Boister N, An Introduction to Transnational Criminal Law (2nd ed, OUP 2018)
 * Cassese A, ‘The Multifaceted Criminal Notion of Terrorism in International Law’ (2006) 4 JICJ 933
 * Cassese A et al, Cassese´s International Criminal Law (3rd ed, OUP 2013)
 * Cryer R, Prosecuting International Crimes: Selectivity and the International Criminal Law Regime (CUP 2005)
 * Fabijanić Gagro S, ‘Mental and Material Elements of Genocide‘ (2021) 11(1) TLQ 1
 * Fiskatoris T, ‘The Global South and the Drafting of the Subject-Matter Jurisdiction of the ICC’ in Jeßberger F, Steinl L, Mehta K (eds), International Criminal Law: A Counter-Hegemonic Project? (TMC Asser Press 2023)
 * Gallagher A, The International Law of Human Trafficking (CUP 2010)
 * Heller K J, ‘Retreat from Nuremberg: The Leadership Requirement in the Crime of Aggression’ (2007) 18(3) EJIL 477
 * Jalloh C C; ‘A Classification of the Crimes in the Malabo Protocol’ in Jalloh C C, Clarke K M, Nmehielle V O (eds), The African Court of Justice and Human and Peoples’ Rights in Context (CUP 2019) 225-256
 * Klamberg M (ed), Commentary on the Law of the International Criminal Court (TOAEP 2017)
 * Kreβ C, ‘On the Outer Limits of Crimes against Humanity: The Concept of Organization within the Policy Requirement: Some Reflections on the March 2010 ICC Kenya Decision’ (2010) 23 LJIL 855
 * Kreß C, ‘The Crime of Genocide under International Law’ (2006) 6 ICLR 461
 * Kreß C and Barriga S (eds), The Crime of Aggression: A Commentary (CUP 2017)
 * Kreß C and von Holtzendorff L, ‘The Kampala Compromise on the Crime of Aggression’ (2010) 8 JICJ 117
 * Lingaas C, 'Defining the Protected Groups of Genocide through the Case Law of International Courts' (2015) ICD Brief 18, 12/2015
 * Novic E, The Concept of Cultural Genocide: An International Law Perspective (OUP 2016)
 * Schabas W, Genocide in International Law: The Crime of Crimes (2nd ed, CUP 2009)
 * Stahn C, A Critical Introduction to International Criminal Law (CUP 2018)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Public International Law/International Criminal Law/Domestic Prosecution of International Crimes

Author: Raghavi Viswanath "Required knowledge: International criminal courts and jurisdiction; International Law and Domestic Law" "Learning objectives: To understand the ways in which domestic courts apply international criminal law; to understand universal jurisdiction and its applications"

Introduction
Given the jurisdictional and budgetary constraints faced by the ICC, not all cases that trigger international criminal law can be tried by the ICC alone. This is why the Rome Statute rests on the principle of complementarity, whereby the ICC can only be invoked once domestic recourses have been exhausted. Domestic courts try cases using different jurisdictional routes: such as active nationality, passive nationality, territory, and even when the effects of some crimes are felt within their territory. These modes are discussed at length in the jurisdiction chapter. States now also recognize certain crimes over which any State court - regardless of nationality or territory - can exercise jurisdiction. This mode of jurisdiction is called universal jurisdiction. This chapter maps such routes for domestic activation of jurisdiction over international crimes.

A.     Complementarity
The ICC rests on the principle of complementarity. By application of this principle, the ICC's jurisdiction can only be triggered once it is reasonably established that domestic courts are unable or unwilling to try the case. According to this principle, States enjoy the first right (and the first obligation) to prosecute individuals charged with war crimes, genocide, crimes against humanity, and aggression.[1] Of course, this means that the requirements of national legal jurisdiction – whether nationality or territory or effects – would have to be met (as discussed in the Jurisdiction chapter).

B.      Universal jurisdiction: History and rationale
Compared to the route of transferring cases to the ICC, domestic courts can exercise universal jurisdiction over certain crimes. Universal jurisdiction allows the prosecution of certain crimes by any state, unconnected to the commission of the crime, the place it occurred, the accused or the victim because the conduct is of universal concern.[2] Universal jurisdiction does not replace domestic prosecutions of crimes recognized under domestic law but simultaneously having an equivalent in the Rome Statute. It elevates certain crimes because of their seriousness and ensures that impunity is eliminated for such crimes. It is also implicit in this rationale that powerful States actively shield their senior officials who commit core crimes, this would hold them back from prosecuting such actors[3].

Universal jurisdiction was conceived as a way out of such conflicts of interest. Universal jurisdiction was first recognized for the crime of piracy. Ever since, a longer list of crimes can now trigger universal jurisdiction. The 1948 Genocide Convention, for instance, enjoins all State parties to punish and prosecute perpetrators of genocide. The 1984 Convention against Torture codifies universal jurisdiction for the crime of torture. Crimes against humanity,[4] apartheid,[5] and enforced disappearance[6] have also been added to this list.

Many scholars laud universal jurisdiction for creating a globalized jurisprudence, involving transnational networks.[7] This does not mean that universal jurisdiction is not political. This is evident in the statistics compiled by TRIAL International annually. Although universal jurisdiction has acquired much traction in terms of geographical reach (almost 92 States initiated universal jurisdiction cases in 2021-22), these prosecutions are concentrated on crimes committed mostly in the Global South. The African Union has been vocal in its opposition to such exercise of jurisdiction. It has instead adopted a Model Law calling on African States to legislate on universal jurisdiction and prosecute “international crimes, trafficking, and terrorism crimes”.[8] This addition of terrorism and trafficking departs from the internationally recognized list of crimes warranting universal jurisdiction.

C. Sources of universal jurisdiction
When seen from a positivist lens, the validity of exercises of universal jurisdiction rest majorly on the source which confers such jurisdiction. At least three sources of universal jurisdiction have emerged in case law so far.

I. Domestic statute
In the trial of Anwar R in 2022 before the Koblenz Higher Regional court, the applicants invoked the 2002 International Criminal Code (VStGB), which vests the Office of the Federal Prosecutor with the authority to try individuals accused of committing international crimes anywhere in the world.[9] Similarly, in the trial of Colonel Lama in the UK in 2016 on two counts of torture, the English court also derived jurisdiction from Section 134 of the Criminal Justice Act.[10] Although both trials were met with varying degrees of interest and coverage, one challenge that both courts encountered is the validity of universal jurisdiction sanctioned by a domestic law. In other words, is it defensible for a domestic law to empower a domestic court to sanction a national of a different State for a crime committed in the territory of a different State? In this framing, universal jurisdiction seemingly pushes the boundaries of sovereignty.

II. International treaty law
When confronted with the limitations of domestic law, courts and applicants typically resort to international treaty obligations. In 2011, the Brussels Criminal Court arrested Ernest Gakwaya and Emmanuel Nkunduwimye and opened the trial against them for sexual crimes committed during the 1994 genocide in Rwanda.[11] The trial and the 2003 Belgian Criminal Code are both grounded in Belgium’s obligations under the Genocide Convention. The landmark trial of Gibril Massaquoi for war crimes and crimes against humanity committed in Sierra Leone and Liberia during the 1991-2002 civil war[12] also invokes the Finish Criminal Code and Finland’s obligations under the 1949 Geneva Conventions.[13]

However, citing international treaty obligations to support universal jurisdiction is not as common as one would expect. Case in point is the trial of former General Pinochet by the UK House of Lords for his alleged crimes in Chile and Argentina during his 1973-1990 dictatorship. Although the case was an instance of universal jurisdiction, the Court did not place any emphasis on whether Chile has ratified the Convention against Torture.[14]

III. Customary law
It is not surprising that international treaties suffer from gaps. The corpus of war crimes, for example, is quite large. However, the 1949 Geneva Conventions do not prescribe universal jurisdiction for the full list of war crimes. The deficiencies of treaty law are best illustrated by the Eichmann case. The 1961 trial of Adolf Eichmann, the Head of Section (Referant) for Jewish Affairs in Nazi Germany, on counts of war crimes, crimes against humanity, saw the Supreme Court of Israel invoke universal jurisdiction because of its customary law status. The Court also eschewed the exclusive reliance on treaty law for evidence of universal jurisdiction.[15] The ICJ’s Arrest Warrant decision also stressed the importance of customary bases for universal jurisdiction.[16] That said, customary law sources for universal jurisdiction are not all that common because it is often not in States’ interests to encourage universal jurisdiction. As a result, it is difficult to pin down opinio juris for universal jurisdiction for certain crimes in order to take recourse to the customary law route.

[1] Article 17, Rome Statute of the International Criminal Court (2002).

[2] Kenneth C Randall, ‘Universal Jurisdiction Under International Law’ (1988) 66 Texas Law Review 785, 788 as cited in Steven W Becker, ‘Universal Jurisdiction: How Universal Is It? A Study of Competing Theories’ (2002–3) 12 Palestine Yearbook of International Law 49, 50; Roger O’Keefe, ‘Universal Jurisdiction: Clarifying the Basic Concept’ (2004) 2(3) Journal of International Criminal Justice 735.

[3] Comments From Kenya, The Scope and Application of the Principle of Universal Jurisdiction: The Report of the Sixth Committee A/64/452-Res64/117 (2018).

[4] Charles Jalloh, Universal criminal jurisdiction, A/73/10 (2018), https://legal.un.org/ilc/reports/2018/english/annex_A.pdf.

[5] Convention against Apartheid (1973).

[6] Convention against Enforced Disappearance (2006).

[7] A. M. Slaughter, A New World Order (2004) 150.

[8] African Union (Draft) Model National Law on Universal Jurisdiction over International Crimes exp/min/Legal/vi, adopted at Twenty-First Ordinary Session of the Executive Council, 9–13 July 2012, Addis Ababa Ethiopia.

[9] https://www.justiceinitiative.org/litigation/federal-prosecutors-office-v-anwar-r

[10] R v. Kumar Lama, Case no. 2013/05698 (Central Criminal Court, London, 2016).

[11] https://trialinternational.org/wp-content/uploads/2022/03/TRIAL_International_UJAR-2022.pdf.

[12] https://trialinternational.org/latest-post/gibril-massaquoi/

[13] Chapter 1, Section 7 of the Criminal Code of Finland, 39/1889, amendments up to 766/2015 included, Translation from Finnish by Ministry of Justice, Finland (hereinafter CC); Asetus rikoslain 1 luvun 7 §:n  soveltamisesta (unofficial translation: Decree on the Application of Chapter 1, Section 7 of the Criminal Code), 16.8.1996/627, 1996.

[14] R v. Bow Street Stipendiary Magistrate and Others, Ex parte Pinochet (No. 3), [1999] 2 All ER 97.

[15] Supreme Court of Israel 336/31, Attorney General v. Eichmann, 36 ILR 28.

[16] Arrest Warrant of 11 April 2000 (Democratic Republic of the Congo v. Belgium), Judgment, 14 February 2002, ICJ Reports (2002) 3, Joint Separate Opinion of Judges Higgins, Kooijmans and Buergenthal, at 63.


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User:Hijuthyuu

hi there so um im a user i like minecraft and roblox

title: User:Hijuthyuu


text:
Stardew Valley/Characters

There are many characters is Stardew Valley. Some have special routines and they all have favorite gifts and disliked gifts. Some characters are single and you can eventually date and even marry them. (The game does restrict you to dating people of a specific gender based on the gender of your player character.) You're friendship level for single characters is 8 hearts; but can be increased if you begin dating.

Villagers
There are 28 villagers; all of them are listed in the Social tab, and you can become friendly with them.

 * /Abigail/ - Single, female.
 * Abigail is the daughter of Caroline and Pierre. She lives at Pierre's General Store and can sometime be seen working there.

 * /Alex/ - Single, male.
 * Alex is the grandson of Evelyn and George. He lives at 1 River Road and his hobby is bodybuilding.

 * /Caroline/ - Married, female.
 * Caroline is Abigail's mother and Pierre's wife. She lives at Pierre's General Store.

 * /Clint/ - Non-datable, male.
 * Clint lives and works at the Blacksmith's.

 * /Demetrius/ - Married, male.
 * Demetrius is Maru's father, Sebastian's step-father, Robin's husband. He lives at the Carpenter's Shop where he works as a scientist.

 * /Elliott/ - Single, male.
 * Elliott is a writer. He lives in Elliott's Cabin but can be seen walking around town gathering inspiration.

 * /Emily/ - Single, female.
 * Emily lives with her sister Haley at 2 Willow Lane. She works at the Saloon afternoons.

 * /Evelyn/ - Married, female.
 * Evelyn is Alex's grandmother and George's wife. She lives at 1 River Road and her hobby is baking.

 * /George/ - Married, male.
 * George is Alex's grandfather and Evelyn's husband. He lives at 1 River Road.

 * /Gus/ - Non-datable, male.
 * Gus lives in the Saloon, which he owns and operates.

 * /Haley/ - Single, female.
 * Haley lives with her sister Emily at 2 Willow Lane. Her hobby is photography.

 * /Harvey/ - Single, male.
 * Harvey is the local doctor and you can purchase medical supplies from his shop when it's open. He works and lives at Harvey's Clinic, but can often be seen walking outdoors.

 * /Jas/ - Child, female.
 * Jas is Marnie's Niece. She is often seen with Vincent, and receives tutoring from Penny.

 * /Jodi/ - Married, female.
 * Jodi is Sam and Victor's mother, and Kent's wife. She lives at 1 Willow Lane and spends most of the time caring for her children.

 * /Kent/ - Married, male.
 * Jodi is Sam and Victor's father, and Jodi's husband. He lives at 1 Willow Lane but can often be seen walking around town. (Kent does not appear in the game until the second year.)

 * /Leah/ - Single, female.
 * Leah is a sculptor and painter. She lives in Leah's Cottage, but often visits different locations in the valley to set up her easel.

 * /Lewis/ - Non-datable, male.
 * Lewis lives in the Mayor's Manor. He can often be seen in the town square.

 * /Linus/ - Non-datable, male.
 * Linus is practically homeless and lives in a Tent.

 * /Marnie/ - Non-datable, female.
 * Marnie is Shane and Vincent's aunt. She lives on Marnie's Ranch and you can purchase animals and related supplies when she's there.

 * /Maru/ - Single, female.
 * Maru is the daughter of Robin and Demetrius. She lives in the Carpenter's Shop, but works part time at Harvey's Clinic.

 * /Pam/ - Non-datable, female.
 * Pan is Robin's mother. She lives in the Trailer, but she spends afternoons at JojaMart and evenings in the Saloon.

 * /Penny/ - Single, female.
 * Penny is the daughter of Pam. She lives in the Trailer and tutors Jas and Vincent, the neighborhood children.

 * /Pierre/ - Married, male.
 * Pierre is Abigail's father and Caroline's husband. He lives at Pierre's General Store, which he owns and operates.

 * /Robin/ - Married, female.
 * Robin is Maru and Sebastian's mother, and Demetrius' wife. She lives at the Carpenter's Shop which she owns and operates.

 * /Sam/ - Single, male.
 * Sam is the son of Jodi and Kent, and the brother of Vincent. He works part-time at Jojamart.

 * /Sebastian/ - Single, male.
 * Sebastian is Robin's son and Demetrius' step-son. He works as a freelance computer programmer.

 * /Shane/ - Single, male.
 * Shane is Marnie's Nephew. He lives at Marnie's Ranch and works at Jojamart; he spends evenings in the Saloon.

 * /Vincent/ - Child, male.
 * Vincent is Jodi's and Kent's son, and Sam's brother. He is often seen with Jas, and receives tutoring from Penny.

 * /Willie/ - Non-datable, male.
 * Willie lives in the Fish Shop, which he owns and operates. He can often be seen fishing around the valley.

Friendable Non-Villagers
In addition to villager, there are other characters in the game you can become friends with.
 * /Dwarf/
 * The Dwarf lives in a secret location near town.

 * /Krobus/
 * The Krobus lives in a secret location near town.

 * /Sandy/
 * Sandy lives in the Calico Desert, which can't be reached without the bus.

 * /Wizard/
 * The Wizard lives in the Wizard's Tower where he practices his magical arts.

Other characters in Stardew Valley
There are a few other characters in Stardew Valley which you can interact with, but are not listed in the Social tab.
 * Gunther
 * Gunther runs the Stardew Library Museum & Library.

 * Marlon
 * Marlon runs the Adventurer's Guild.

 * Gil
 * Spends his time in the Adventurer's Guild.

 * Morris
 * Morris runs the JojaMart and is the local representative of Joja Corporation.

Visitors
Finally, there are some characters which are only seen on special occasions.
 * Grandpa
 * This is your grandfather who left you his farm.

 * Governor
 * This is governor of the region where Stardew Valley is location. He visits once a year.

title: Stardew Valley/Characters


text:
Calculus/Resources

Single variable calculus

 * Derivative calculator with detailed steps
 * Integral calculator with detailed steps

title: Calculus/Resources


text:
User talk:152.32.107.17

Notice
Please do not create any more test pages. -- Jules (Mrjulesd) 11:10, 13 April 2022 (UTC)

title: User talk:152.32.107.17


text: Template:Tlx/sandbox

title: Template:Tlx/sandbox


text:
User talk:151.188.105.159

--Xania   talk 14:20, 13 April 2022 (UTC)

title: User talk:151.188.105.159


text:
Calculus/Calculus on matrices

This section gives an overview on how calculus can be applied to matrices. Note that a general understanding of Linear Algebra is expected - you're expected to be familiar with the common ways of manipulating matrices.

The problem
Consider a n by n matrix $$\textbf{A}$$ and a n by 1 vector $$\textbf{x}$$. How can we, for instance, find $$\frac{d}{d\textbf{x}} \textbf{A} \textbf{x}$$? Now, if you were to naively apply single-variable calculus rules, a plausible answer would be

$$\frac{d}{d\textbf{x}} \textbf{A} \textbf{x} = \textbf{A}$$

After all, the corresponding scalar form of the problem $$\frac{d}{dx} (ax)$$ would indeed be a. And indeed the answer to the vector form is $$\textbf{A}$$. But now consider the following problem: $$\frac{d}{d\textbf{A}} \textbf{A} \textbf{x}$$. If you were to take the scalar form, you'd probably think that the answer would be $$\textbf{x}$$. But that isn't right - the answer is actually $$\textbf{x}^T$$, where T refers to the transpose of the vector x.

The purpose of this section is to scratch the surface of this beautiful field - because it's not something that the average Calculus 3 or linear algebra course at university will teach - yet it has its own quirks. And what is it used for? Matrix calculus is widely used in machine learning and also in other fields, such as computational finance. It can also help us avoid having to take (potentially nasty) Lagrangians and effectively reduce the problem to a single-variable scenario!

Derivative with respect to a vector
In this section, we consider problems that involve differentiating with a vector x. As with above, we assume that x is a column vector.

One way to think about this problem is to reduce this to a problem of scalars. Notice that we can consider x to be a collection of scalars $$\textbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ ... \\ x_n \end{pmatrix}$$. Now take the individual partial derivatives $$\frac{\partial}{\partial x_i}$$ for $$1 \leq i \leq n$$. Finally put them together. We're essentially finding $$\nabla f(\textbf{x})$$ after all - the steps are the same (only that before, the size of x was 2 or 3 that represented the i, j and k coordinate frame).

So let's try this from the above example. We want to find, for all $$1 \leq i \leq n$$,

$$\frac{\partial}{\partial x_i} (\textbf{A} x_i)$$

... which is A. And that's the same for every i.

Now what would get if you would combine all the partial derivatives? Just like how you'd find $${\displaystyle \nabla f({\textbf {x}})}= \nabla f\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ ... \\ x_n \end{pmatrix} = \nabla \begin{pmatrix} \textbf{A} x_1 \\ \textbf{A} x_2 \\ \textbf{A} x_3 \\ ... \\ \textbf{A} x_n \end{pmatrix}$$, you'll get $$\nabla f(\textbf{x}) = \begin{pmatrix} \textbf{A} \\ \textbf{A} \\ \textbf{A} \\ ... \\ \textbf{A} \end{pmatrix}$$. This is just A! Indeed, that's why $$\frac{d}{d\textbf{x}} \textbf{A} \textbf{x} = \textbf{A}$$.

A first step towards matrices
Now we return back to the other problem: $$\frac{d}{d\textbf{A}} \textbf{A} \textbf{x}$$.

Let's assume that A is a 2 by 2 matrix, and represent A as $$\textbf{A} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$. Using the same notation for x, perform matrix multiplication:

$$\textbf{A} \textbf{x} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} a_{11} \times x_1 + a_{12} \times x_2 \\ a_{21} \times x_1 + a_{22} \times x_2 \end{pmatrix}$$

Take the partial derivatives with respect to each of the elements of A (this is equivalent to finding the Jacobian). For $$1 \leq i, j \leq 2$$, find $$\frac{\partial}{\partial A_{ij}}$$:

$$\frac{\partial}{\partial A_{11}} = \begin{pmatrix} x_1 \\ 0 \end{pmatrix} $$, $$\frac{\partial}{\partial A_{12}} = \begin{pmatrix} x_2  \\ 0 \end{pmatrix} $$, $$\frac{\partial}{\partial A_{21}} = \begin{pmatrix} 0  \\ x_1 \end{pmatrix} $$ and $$\frac{\partial}{\partial A_{22}} = \begin{pmatrix} 0  \\ x_2 \end{pmatrix} $$

But then what do you do with that? How do you "combine" the result? Clearly, we are missing something.

Dimensions of ∇f
Let's take a step back and ask the question: given a vector $$\textbf{f} = \textbf{A} \textbf{x} $$, what should be the dimension of $$\nabla \textbf{f} $$?

Consider the example above. We have two variables: A with 4 elements (2x2) and x with 2 elements, and we want to find $$\frac{d \textbf{f}}{d\textbf{A}}$$. It is straightforward to show that the dimension of f is a column vector with $$\textbf{f} = \binom{f_1}{f_2}$$, where $$\binom{f_1}{f_2} = \begin{pmatrix} a_{11} \times x_1 + a_{12} \times x_2 \\ a_{21} \times x_1 + a_{22} \times x_2 \end{pmatrix}$$. So we also need to consider the derivatives with respect to $$f_1$$ and $$f_2$$. In other words, the dimension of $$\frac{df_1}{d\textbf{A}}$$ and $$\frac{df_2}{d\textbf{A}}$$ is a 2x2 vector, corresponding to the partial derivatives of each element of the matrix A.

How many elements would $$\frac{d \textbf{f}}{d\textbf{A}}$$ have in total? For each of the two scalars that comprise f, there are four partial derivatives. This results in (2 * (2 * 2)) = 8 elements in total - actually a tensor (which can be thought of as a higher-order matrix). This is where things start to get messy, but fortunately this is a simple enough example.

Getting a solution
So let's use this above observation to solve the problem.

First consider $$\frac{df_1}{d\textbf{A}}$$, where $$f_1 = a_{11} \times x_1 + a_{12} \times x_2 $$. Compute the individual partial derivatives: $$\frac{\partial f_1}{\partial A_{11}} = x_1, \frac{\partial f_1}{\partial A_{12}} = x_2, \frac{\partial f_1}{\partial A_{21}} = 0$$ and $$\frac{\partial f_1}{\partial A_{22}} = 0$$. Similarly, $$\frac{\partial f_2}{\partial A_{11}} = 0, \frac{\partial f_2}{\partial A_{12}} = 0, \frac{\partial f_2}{\partial A_{21}} = x_1$$ and $$\frac{\partial f_2}{\partial A_{22}} = x_2$$.

Now, how do we combine this? The issue is that $$\nabla \textbf{f} $$ is a tensor, but we can display only a 2D representation using matrices. So let's take the "face" that corresponds to $$\nabla f_1$$. We can represent the collection of partial derivatives (that is, the Jacobian) we found above in a matrix: $$\nabla f_1 = \begin{pmatrix} x_1 & x_2 \\ 0 & 0 \end{pmatrix}$$. Similarly, $$\nabla f_2 = \begin{pmatrix} 0 & 0 \\ x_1 & x_2 \end{pmatrix}$$. What do we observe? This is simply $$\textbf{x}^T$$ (notice the change from column to row form) ! And indeed, that's how we show that $$\frac{d}{d\textbf{A}} \textbf{A} \textbf{x} = \textbf{x}^T$$.

In practice
In practice, you won't have to do all this work every time you want to find the derivative of a matrix. Instead, there are many matrix cookbooks (sometimes also called as a cheatsheet), which give a table of common derivatives with respect to matrices, and that's what you're likely to require in practice. Here's one.

An example
Consider the Markowitz problem. Assume that we have n stocks, and we want to assign weights $$w_i$$. The inter-element covariance between each stock is $$\sigma_{ij}$$, for all $$1 \leq i, j \leq n$$. Suppose we want to solve this problem in a traditional way. Then the optimisation problem is to minimise $$\sum_{i = 1}^{n} {\sum_{j = 1}^n {w_i \sigma_{ij} w_j}}$$ subject to constraints, while important, we won't mention here.

Solving this problem is likely to be messy given the double summation. Let's try matrix calculus. Let w be a N times 1 vector and $$\Sigma$$ be a N times N matrix. The above problem can be reduced to minimising $$\textbf{w}^T \Sigma \textbf{w}$$, and all you need to do is to take derivatives with respect to w! As shown above using a matrix calculus cookbook, $$\frac{d}{d \textbf{w}} \textbf{w}^T \Sigma \textbf{w} = 2 \textbf{w}^T \Sigma$$, which is much more elegant than trying to compute the individual partial derivatives.

title: Calculus/Calculus on matrices


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User:Veronica Glass Educational Psychology

I currently trying to develop my projects steps by steps, what is going to be resources as writings in a magazine.

wikidata - > Veronica In Dream (Q109796032)


text:
User talk:2001:48F8:9:D29:2C58:5764:3823:4A8A

You cannot edit for 1 month due to disruptive editing. You are welcome to contribute constructively when editing privileges are restored. You may appeal this action with a reasonable argument by adding the text below.

--Xania   talk 01:22, 14 April 2022 (UTC)


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Circuit Theory/Quadratic Equation Revisited

The Quadratic Equation
The roots to the quadratic polynomial

$$ y = ax^2 + bx +c $$

are easily derived and many people memorized them in high school:

$$ \begin{align} x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \end{align} $$

Derivation of the Quadratic Equation
To derive this set $$y = 0 $$ and complete the square:

$$ \begin{align} 0 &= ax^2 + bx +c \\ 0 &= a \left( x^2 + \frac{b}{a} x + \frac{c}{a} \right)\\ 0 &= x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 + \frac{c}{a} - \left(\frac{b}{2a}\right)^2 \\ 0 &= \left( x + \frac{b}{2a} \right)^2 + \left(\frac{c}{a} - \left(\frac{b}{2a}\right)^2 \right). \end{align} $$

Solving for $$x$$ gives

$$ \begin{align} 0 &= \left( x + \frac{b}{2a} \right)^2 + \left(\frac{c}{a} - \left(\frac{b}{2a}\right)^2\right) \\ \left( x + \frac{b}{2a} \right)^2 &= -\left(\frac{c}{a} - \left(\frac{b}{2a}\right)^2\right) \\ &= \frac{b^2 - 4ac }{\left(2a\right)^2}. \end{align} $$

Taking the square root of both sides and putting everything over a common denominator gives

$$ \begin{align} x + \frac{b}{2a} &= \pm \sqrt{\frac{b^2 - 4ac }{\left(2a\right)^2}} \\ x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \end{align} $$

Numerical Instability of the Usual Formulation of the Quadratic Equation
Middlebrook has pointed out that this is a poor expression from a numerical point of view for certain values of $$a$$, $$b$$, and $$c$$.

[Give an example here]

Middlebrook showed how a better expression can be obtained as follows. First, factor $$-\frac{b}{2a}$$ out of the expression:

$$ \begin{align} x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ &= \left( -\frac{b}{2a} \right) \left( 1 \pm \sqrt{1 - \frac{4ac}{b^2}} \right). \end{align} $$

Now let

$$ Q^2 = \frac{ac}{b^2}. $$

Then

$$ \begin{align} x &= \left( -\frac{b}{2a} \right) \left( 1 \pm \sqrt{1 - 4Q^2} \right) \end{align} $$

A More Numerically Stable Formulation of the Negative Root
Considering just the negative square root we have

$$ \begin{align} x_1 &= \left( -\frac{b}{2a} \right) \left( 1 - \sqrt{1 - 4Q^2} \right). \end{align} $$

Multiplying the numerator and denominator by $$1 + \sqrt{1 - 4Q^2}$$ gives

$$ \begin{align} x_1 &= \left( -\frac{b}{2a} \right) \left( 1 - \sqrt{1 - 4Q^2} \right) \left( \frac{1 + \sqrt{1 - 4Q^2}}{1 + \sqrt{1 - 4Q^2}} \right) \\ &= \left( -\frac{b}{2a} \right) \left( \frac{1 - 1 + 4Q^2}{1 + \sqrt{1 - 4Q^2}} \right) \\ &= -\frac{2 b Q^2}{a} \left( \frac{1}{1 + \sqrt{1 - 4Q^2}} \right) \\ &= -\frac{2 b \left( \frac{ac}{b^2} \right)}{a} \left( \frac{1}{1 + \sqrt{1 - 4Q^2}} \right) \\ &= -\frac{2 c}{b} \left( \frac{1}{1 + \sqrt{1 - 4Q^2}} \right) \\ &= -\frac{c}{b} \left( \frac{1}{\frac{1}{2} + \frac{1}{2}\sqrt{1 - 4Q^2}} \right). \end{align} $$

By defining

$$ F = \frac{1}{2} + \frac{1}{2}\sqrt{1 - 4Q^2} $$

we can write

$$ \begin{align} x_1 &= -\frac{c}{bF}. \end{align} $$

Note that as $$Q \to 0$$, $$F \to 1$$.

Finding the Positive Root Using the Same Approach
Turning now to the positive square root we have

$$ \begin{align} x_2 &= \left( -\frac{b}{2a} \right) \left( 1 + \sqrt{1 - 4Q^2} \right) \\ &= \left( -\frac{b}{a} \right) \left( \frac{1}{2} + \frac{1}{2}\sqrt{1 - 4Q^2} \right) \\ &= -\frac{bF}{a}. \end{align} $$

Using the two roots $$x_1$$ and $$x_2$$, we can factor the quadratic equation

$$ \begin{align} y &= a x^2 + b x + c \\ &= a \left( x^2 + \frac{b}{a} + \frac{c}{a}\right) \\ &= a \left( x - x_1\right) \left( x - x_2\right) \\ &= a \left( x + \frac{c}{bF}\right) \left( x +\frac{bF}{a}\right). \end{align} $$

Accuracy for Low $$Q$$
For values of $$Q \le 0.3$$ the value of $$F$$ is within 10% of 1 and we may neglect it. As noted above, the approximation gets better as $$Q \to 0$$. With this approximation the quadratic equation has a very simple factorization:

$$ \begin{align} y &= a x^2 + b x + c \\ &\approx a \left( x + \frac{c}{b}\right) \left( x +\frac{b}{a}\right), \end{align} $$

an expression that involves no messy square roots and can be written by inspection. Of course, it is necessary to check the assumption about $$Q$$ being small before using the simplification. Without this simplification, $$F$$ needs to be calculated and the roots are slightly more complicated.

[Explore the consequences if $$Q > 0.5 $$.]