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u(t)=\int_{a}^{b} G(t,s)f(s) \,\mathrm{d}s,

It will also be convenient to recall the following lemmas.

( cf. , e.g. , Theorem0.3.4 in [ ])

the set \{x_{k}(t) \mid k \in\mathbb{N}\} is bounded \forall t \in J ,

\bigl\vert x'_{k}(t)\bigr\vert \leq\alpha(t), \quad \textit{for almost all }t \in J, \forall k \in\mathbb{N}.

\{x_{k}\} ,

\{x'_{k}\} L^{1}(J,\mathbb{R}^{n}) x' .

( cf. [ ], p.88)

N_{F}(x) := \bigl\{ f \in L^{1}\bigl(J, \mathbb{R}^{n}\bigr) \mid f(t) \in F\bigl(t, x(t)\bigr), \textit{for almost all } t \in J \bigr\} ,

Let us consider at first the scalar Dirichlet problem for differential equations involving a dry friction ( ), where a , b , c , x_{0} , x_{T} and T >0 are real constants and p\colon J \rightarrow \mathbb{R} , J=[0,T] , is a Lebesgue integrable function.

Since the function \operatorname{sgn}(\cdot) is discontinuous in the spatial variable, problem ( ) need not have a Carathéodory solution , i.e. a function x \colon J \rightarrow\mathbb {R} with an absolutely continuous derivative, satisfying ( ), for almost all t \in J . Therefore, we need another notion of an appropriate solution, namely the one in the sense of Filippov. For this goal, we use the concept of the Filippov-like regularization (see [ ]) of spatially discontinuous maps. More precisely, applying Definition to the right-hand side involving spatial discontinuities, we can speak about a solution in the sense of Filippov of the original problem, provided it is a Carathéodory solution of a multivalued problem with a Filippov-like regularized right-hand side.

In our situation, the discontinuous function to be regularized is the function signum . On the basis of the Filippov-like regularization of \operatorname{sgn}(\cdot) , we obtain the multivalued mapping Signum defined in ( ), i.e. \operatorname{Sgn}(\cdot) .

One can readily check that the Signum mapping is u.s.c. with compact and convex values. Hence, after the described Filippov-like regularization, problem ( ) with a discontinuous function \operatorname{sgn}(\cdot) becomes multivalued, i.e. ( ).

If our musical expectations change according to context, then a number of important questions arise: How many different musical schemas can a listener maintain? How fast are listeners able to identify the context and invoke the appropriate schema? When the context changes, how fast are listeners able to switch from one schema to another? What cues signal the listener to switch schemas? How do listeners learn to distinguish different contexts? How are the expectations for one schema protected from novel information that pertains to a different schema? How does a listener assemble a totally new schema? What happens when the events of the world straddle two different schemas?

Once the music has begun, how fast are listeners able to recognize the musical context? In the case of music, dramatic changes in listeners' expectations arise depending on the style or genre of the music. Perrott and Gjerdingen (1999) have observed that listeners are very quick to identify different styles. When scanning the radio dial, listeners make split-second decisions regarding the style of music being played on each station. Perrott and Gjerdingen tested this observation by selecting random musical segments from samples of 10 different styles of music, including jazz, rock, blues, country western, classical, etc. They showed that listeners are adept at classifying the type of music in just 250 milliseconds. With just one second of exposure, ordinary listeners' abilities to recognize broad stylistic categories is nearly at ceiling; that is, further exposure to the musical work does not lead to a significant improvement in style identification. If we assume that identifying a schema is tantamount to activating the schema, then these observations suggest that experienced listeners can activate a schema appropriate to the genre of music they are hearing in a very short period of time.

What about the phenomenon of schema switching ? How rapidly can a listener switch from one schema to another? Although little research has been carried out pertaining to this question, suggestive evidence has come from the work of Krumhansl and Kessler (1982). Krumhansl and Kessler traced the speed with which a new key was established in modulating chord sequences. Modulations to related keys were "firmly established" within three chords lasting a few seconds (Krumhansl, 1990; p.221). However, some sense of the initial key was maintained throughout the modulating passage. Since modulation is common in Western music, this ability to switch rapidly between schemas might pertain only to key-related schemas. One might imagine that switching, say, from a Western string quartet to Beijing opera would take longer -- although perhaps not very long in absolute duration. Bi-lingual speakers differ in their abilities to switch rapidly between different languages. But this skill appears to be related to how often speakers must change language in their daily life.