ABSTRACT We introduce a notion of labeled cycle for normal labeled spaces and prove a reduction theorem for Leavitt path algebras associated with labeled graphs. We show that every nonzero element can be reduced, by suitable left and right multiplication, either to a nonzero scalar multiple of a projection or to a polynomial supported on a labeled cycle without exits. This extends the classical reduction theorem for Leavitt path algebras of directed graphs and its analogues for ultragraph Leavitt path algebras and subshift algebras. As applications, we prove the graded uniqueness theorem and the Cuntz–Krieger uniqueness theorem for Leavitt path algebras associated with labeled graphs, and show that these algebras are semiprime and semiprimitive over fields.
ABSTRACT This paper proposes an alternated inertial subgradient extragradient algorithm with a new variable stepsize rule for solving variational inequality problems in real Hilbert spaces. Unlike existing methods whose stepsize rules are fixed, open‐loop, or based only on the current residual, the proposed rule combines the current residual and the next‐step residual through a parameter . This yields a flexible trade‐off between the two residuals and ensures a positive lower bound for the stepsize sequence. Moreover, we obtain the R‐linear convergence results of our algorithm under the condition that the relevant mapping is strongly pseudo‐monotone. Numerical experiments demonstrate the effectiveness of the proposed method.
ABSTRACT The paper is concerned with spectral problems for the Dirac‐type operator with arbitrary two‐point boundary conditions and complex‐valued potential summable over a finite interval. It is proved that under certain conditions the root function system of the considered operator with degenerate boundary conditions does not form a basis. We also establish a necessary condition of convergence of the corresponding spectral expansions.
ABSTRACT This paper considers the asymptotic problem for the Navier–Stokes–Poisson–Korteweg system with density‐dependent viscosity and cold pressure on the three‐dimensional torus , focusing on the incompressible limit and the vanishing capillarity limit. The convergence of the velocity field is examined via the action of the Leray operator, and the nonlinear terms are analyzed through the harmonic oscillator operator to show convergence in the distributional sense to gradient terms. Finally, we prove that the system converges to the incompressible Navier–Stokes equations.
ABSTRACT We study differential polynomial rings over ‐graded rings, where is an arbitrary group. We show that admits a ‐grading compatible with that of if and only if is a ‐derivation for some , and that this grading is unique once is fixed; if , then is itself uniquely determined by . We characterize the resulting graded ring by a universal property. We prove a characteristic‐free center criterion for gr‐simplicity whenever is a graded subring; in characteristic zero, gr‐simplicity is equivalent to ‐gr‐simplicity of and ‐outerness of , extending Jordan's simplicity criterion to the graded setting. We further show that is gr‐prime if and only if is ‐gr‐prime, and that gr‐Noetherianity of passes to , recovering a graded Hilbert basis theorem as a special case. When is abelian, gr‐simplicity and gr‐primality are shown to be invariants of homogeneous graded Morita equivalence, and every ring homogeneously graded equivalent to via a compatible idempotent is again a graded differential polynomial ring.
