It is ordinarily accepted among nations that territorial jurisdiction is based on a crime that is committed within the territory of the sovereign state. However, sometimes states also exercise jurisdiction because the crime was committed abroad by one of their nationals or because the victim of the crime was a national of the state.

For instance, in the federal court case United States v. Fawaz Yunis, in which five men hijacked a Jordanian airliner. Among these passengers were two American citizens who were apparently not the target of the terrorist activity. Hijackers latter released the passengers and blew up the aircraft. The defendant was lured into international waters and brought to the United States, where he was convicted of conspiracy, hostage taking, and air piracy (Smit1118). On appeal the defendant affirmed that as a matter of domestic law, the federal hostage taking and air piracy statutes do not authorize assertion of federal jurisdiction over him. Yunis also suggested that a contrary structure of these statutes would conflict with established principles of international law, and so should be avoided by the court (1121).
Thus, the United States sustained jurisdiction under the Hostage Taking Statute of the providing jurisdiction on the nationality of the victim. Also, under sect. 32 of the Aircraft Piracy Statute adopted from traditional international legal bases of the universal and passive personality principle (1120).
However, subsection (b) of the Hostage Taking Act sets forth the terms for Congress to impose legal responsibility in which “the offender is found in the United States” (1120). In this case, Yunis was not found in the United States within the meaning of the statute, but rather apprehended and forcibly brought to the U.S. for trial. Thus, bringing someone to the United States by force to be charged with a crime over which the United States had jurisdiction only when he was inside of the country is unreasonable. In fact, the abduction of a suspected terrorist is contrary to United States domestic law of the Due Process clause of the Unites States Constitution (5th & 14th amendment). This case raises the novel issues of domestic and international law, thus, the United Sates went to the extreme extent of holding the criminal jurisdiction under statute.
In the Wildenhus’ case in which a Belgian native was accused of homicide against another Belgian on board a Belgian vessel anchored in the port of Jersey City (1498). The Belgian government relied on the 1880 treaty with the U.S for a writ of habeas corpus under Article IX of the treaty which would provide the Belgian government consuls with “exclusive charge of internal order of the merchant vessels of their nation” and that the State’s authorities would not exercise jurisdiction. However, the circuit court refused to deliver the prisoner to the consul (Smit1498). The Supreme Court of the United States sustained the jurisdiction of the lower courts ruled that the Belgian nationals would remain in American custody for trial. Jurisdiction was based on the "tranquility of the port" principle, which is found in same treaty. This general principle of international law is based on “comity”, though, not completely defined in regards to the meaning of disorders” that “disturb[s] the tranquility and public order on shore, or in port” (1498). Thus, in this case, there would be concurrent jurisdiction. Belgium could have exercised jurisdiction because the place is Belgian, since the vessel has Belgian nationality and its victim was a Belgium national.
In conclusion, both cases demonstrate a deviation from the principles of customary international law, while a sovereign cannot exercise no legislative jurisdiction outside its boundaries as a judgment rendered in accordance with a statute attempting to assert jurisdiction where it does not possess.

It is hard to comment on the content without knowing where the paper is heading. If this is an introduction, so full of details, it is hard to see this case developed into more detailed substance. But for an introduction, your second paragraph is surprisingly detailed.

The concept of jurisdiction in international law is based on the idea that a sovereign state has the authority to exercise legal control and enforce its laws within its territory. Typically, territorial jurisdiction is accepted as the basis for a state's jurisdiction over a crime committed within its borders. However, there are instances where states may exercise jurisdiction even if the crime was committed abroad or involved foreign nationals.

In the case of United States v. Fawaz Yunis, the United States exercised jurisdiction over the defendant because the crime involved the hijacking of a Jordanian airliner, which had two American citizens on board. The United States sustained jurisdiction under the Hostage Taking Statute, which allows for jurisdiction based on the nationality of the victim. Additionally, the Aircraft Piracy Statute is based on the universal and passive personality principles of international law.

However, there was a question raised regarding the defendant being brought to the United States by force for trial. The Hostage Taking Act states that legal responsibility is imposed when "the offender is found in the United States." In this case, Yunis was not found in the United States but was apprehended and brought to the country against his will. The abduction of a suspected terrorist was argued to be contrary to the Due Process clause of the United States Constitution, raising novel issues of domestic and international law.

In the Wildenhus case, a Belgian national was accused of a homicide that occurred on a Belgian vessel anchored in the port of Jersey City. The Belgian government argued that under a treaty with the United States, their consuls should have exclusive control over the internal order of their merchant vessels. However, the circuit court refused to deliver the prisoner to the consul. The Supreme Court of the United States upheld the jurisdiction of the lower courts, citing the "tranquility of the port" principle found in the treaty. This principle is based on the concept of comity in international law and refers to disturbances that disrupt public order and tranquility on shore or in a port.

Both of these cases deviate from the principles of customary international law. While a sovereign state cannot exercise legislative jurisdiction outside its boundaries, judgments attempted to assert jurisdiction where it does not legally possess are not in line with international law.