{\displaystyle 1\to \operatorname {Homeo} _{0}(\mathbf {T} ^{n})\to \operatorname {Homeo} (\mathbf {T} ^{n})\to \operatorname {MCG} _{\operatorname {TOP} }(\mathbf {T} ^{n})\to 1.} The mapping class group of higher genus surfaces is much more complicated, and an area of active research. Coloring a torus
The toruss chromatic number is seven, meaning every graph that can be embedded on the torus has a chromatic number of at most seven.
Expressive language disorder is a communication disorder in which there are difficulties with verbal and written expression. It is a specific language impairment characterized by an ability to use expressive spoken language that is markedly below the appropriate level for the mental age, but with a language comprehension that is within normal limits. There can be problems with vocabulary, producing complex sentences, and remembering words, and there may or may not be abnormalities in articulation.Careful diagnosis is also important because "atypical language development can be a secondary characteristic of other physical and developmental problems that may first manifest as language problems". Models of language production
Willem Levelt outlined the currently accepted theory of speech production. Words are produced after the concept waiting to be produced is conceptualized, related words are selected, encoded and the sound waves of speech are produced. Association with language networks
There is also a lot of debate about whether specific language impairments, such as expressive language disorder, are caused by deficits in grammar or by a deficit in processing language information. However, an alternative hypothesis to the cause of SLIs has been posited, called the Procedural Deficit Hypothesis. The Procedural Deficit Hypothesis opines that we can explain language impairments due to abnormal development of brain structures that are involved in procedural memory, our memories that remember how to perform different cognitive and motor tasks.