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Register a new userSolvability and uniqueness criteria for generalized Sylvester-type equations
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We provide necessary and sufficient conditions for the generalized $star$-Sylvester matrix equation, $AXB + CX^star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $star$-Sylvester equations.
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We define the Ladyzhenskaya-Lions exponent $alpha_{rm {tiny sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-Delta)^{alpha}$ in ${Bbb R}^n$, for all $ngeq 2$. We review the proof of strong global solvability when $alphageq alpha_{rm {tiny sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1over 2} | abla u |^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $alpha<alpha_{rm {tiny sc l}} (n)$. The energy is critical under scale transformations only for $alpha=alpha_{rm {tiny sc l}} (n)$.
The matrix equation $AX-XB=C$ has a solution if and only if the matrices [A&C0&B] and [A &00 & B] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) obtained an analogous criterion for the matrix equation $X-AXB=C$ over a field. We extend these criteria to the matrix equations $AX-widehat XB=C$ and $X-Awidehat XB=C$ over the skew field of quaternions with a fixed involutive automorphism $qmapsto hat q$.
We consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $star$-Sylvester equations with $ntimes n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized $star$-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition, and leads to a backward stable $O(n^3r)$ algorithm for computing the (unique) solution.
This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them.
An ultragraph gives rise to a labelled graph with some particular properties. In this paper we describe the algebras associated to such labelled graphs as groupoid algebras. More precisely, we show that the known groupoid algebra realization of ultragraph C*-algebras is only valid for ultragraphs for which the range of each edge is finite, and we extend this realization to any ultragraph (including ultragraphs with sinks). Using our machinery, we characterize the shift space associated to an ultragraph as the tight spectrum of the inverse semigroup associated to the ultragraph (viewed as a labelled graph). Furthermore, in the purely algebraic setting, we show that the algebraic partial action used to describe an ultragraph Leavitt path algebra as a partial skew group ring is equivalent to the dual of a topological partial action, and we use this to describe ultragraph Leavitt path algebras as Steinberg algebras. Finally, we prove generalized uniqueness theorems for both ultragraph C*-algebras and ultragraph Leavitt path algebras and characterize their abelian core subalgebras.
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