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Extra resources for Axial symmetry of solutions to semilinear elliptic equations in unbounded domains

Sample text

We also have the following. 2 A nontrivial vector space = over an infinite field - is not the union of a finite number of proper subspaces. Proof. Suppose that = ~ : r Ä r : , where we may assume that :  \ : r Ä r : Let \$  : ± ²: r Ä r : ³ and let # ¤ : . Consider the infinite set ( ~ ¸\$ b #    - ¹ which is the “line” through #, parallel to \$. We want to show that each : contains at most one vector from the infinite set (, which is contrary to the fact that = ~ : r Ä r : . This will prove the theorem.

Then    Ä , which implies that    for some . We can assume by reindexing if necessary that  ~   . Since  is irreducible  must be a unit. Replacing  by   and canceling  gives   Äc ~  Äc This process can be repeated until we run out of 's or 's. If we run out of 's first, then we have an equation of the form " Ä ~  where " is a unit, which is not possible since the  's are not units.  Fields For the record, let us give the definition of a field (a concept that we have been using).

Hence, 1) implies 2).  c Ä c !  44 Advanced Linear Algebra Then 2) implies that  ~  ~  and implies 3). " for all " ~ Á Ã Á  . Hence, 2) Finally, suppose that 3) holds. If  £ #  : q then # ~   : q £ s t  : and  where p ~  bÄb   : are nonzero. 5 Any matrix (  C can be written in the form (~   ²( b (! ³ b ²( c (! 1) where (! is the transpose of (. 1) is a decomposition of ( as the sum of a symmetric matrix and a skew-symmetric matrix. Since the sets Sym and SkewSym of all symmetric and skew-symmetric matrices in C are subspaces of C , we have C ~ Sym b SkewSym Furthermore, if : b ; ~ : Z b ; Z , where : and : Z are symmetric and ; and ; Z are skew-symmetric, then the matrix < ~ : c :Z ~ ; Z c ; is both symmetric and skew-symmetric.