Sugar cubes are placed in water. how will the mixture be affected by stirring the water, by breaking apart the sugar cubes, or by raising the temperature?

When the sugar cubes are placed in water, then, the mixture will be affected by stirring the water. 
By stirring the water, the sugar cubes dissolve in water. The molecules of sugar cubes interact with water molecules in the jar. But when more sugar is added to the water, there comes a saturation point when no more sugar will be dissolved and it will settle down. Then, by raising the temperature the settled sugar will be dissolved too.

The self-ionization of water is an ionization reaction in pure water or in an aqueous solution, in which a water molecule, H₂O, deprotonates to become a hydroxide ion, OH⁻. The hydrogen nucleus, H⁺, immediately protonates another water molecule to form hydronium, H₃O⁺.

Equation : In the attachment.

Hope it helps.