Graffiti Graffiti Graffiti (both singular and plural; the singular "graffito" is very rare in English except in archeology) is writing or drawings made on a wall or other surface, usually without permission and within public view. Graffiti ranges from simple written words to elaborate wall paintings, and it has existed since ancient times, with examples dating back to ancient Egypt, ancient Greece, and the Roman Empire. In modern times, paint (particularly spray paint) and marker pens have become the most commonly used graffiti materials. In most countries, marking or painting property without the property owner's permission is considered defacement and vandalism,
Singular homology on singular "n"-chains. The extension, called the boundary operator, written as is a homomorphism of groups. The boundary operator, together with the formula_25, form a chain complex of abelian groups, called the singular complex. It is often denoted as formula_26 or more simply formula_27. The kernel of the boundary operator is formula_28, and is called the group of singular "n"-cycles. The image of the boundary operator is formula_29, and is called the group of singular "n"-boundaries. It can also be shown that formula_30. The formula_31-th homology group of formula_32 is then defined as the factor group The elements of formula_34