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The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
Interactions can be direct when physical contact is established or indirect, through intermediaries such as shared resources, territories, ecological services, metabolic waste, toxins or growth inhibitors. This type of relationship can be shown by net effect based on individual effects on both organisms arising out of relationship.
Direct reciprocity was proposed by Robert Trivers as a mechanism for the evolution of cooperation. [1] If there are repeated encounters between the same two players in an evolutionary game in which each of them can choose either to "cooperate" or "defect", then a strategy of mutual cooperation may be favoured even if it pays each player, in the short term, to defect when the other cooperates.
[3] However, some researchers, such as ecology professor Tim Clutton-Brock, assert that cooperation is a more complex process. They state that helpers may receive more direct, and less indirect, gains from assisting others than is commonly reported. These gains include protection from predation and increased reproductive fitness.
Direct realism, also known as naïve realism, argues we perceive the world directly. In the philosophy of perception and philosophy of mind, direct or naïve realism, as opposed to indirect or representational realism, are differing models that describe the nature of conscious experiences; [1] [2] out of the metaphysical question of whether the world we see around us is the real world itself ...
The direct sum and the direct product are not isomorphic for infinite indices for which the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory : the direct sum is the coproduct , and the direct product is the product.
An indecomposable module is a module that is not a direct sum of two nonzero submodules. Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings , then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory , is known as ...
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.