Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm

We define an infinite sequence of new invariants, δ_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the δ_n give obstructions to a 3-manifold fibering over S¹ and to a 3-manifold being Seifert fibered. Moreover, we show that the δ_n give computable algebraic obstructions to a 4-manifold of the form X×S¹ admitting a symplectic structure even when the obstructions given by the Seiberg-Witten invariants fail. There are also applications to the minimal ropelength and genera of knots and links in S³.     

关于共轭Goldberg问题的一点注记

对于(p,q)算子范数,我们考虑由Goldberg定义的常数c(称为Goldberg常数)的确定问题.该问题(即求常数c的精确值或它的一个可以达到的最小上界问题)由Moshe Goldberg于1986年提出,且至今仍未能彻底解决.本文给出了常数c的一个上界,改进了Goldberg在1986年所得出的结果.     

Attractors for strongly damped wave equations with critical exponent

We prove the existence of the global attractor for the semigroup generated by strongly damped wave equations when the nonlinearity has a critical growth exponent.     

The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms

The orthogonal Procrustes problem involves finding an orthogonal matrix which transforms one given matrix into another in the least-squares sense, and thus it requires the minimization of the Frobenius matrix norm. We consider, the solution of this problem for a family of orthogonally invariant norms which includes the Frobenius norm as a special case.     

Remarks for the axisymmetric Navier-Stokes equations

We are concerned with 3-D incompressible Navier-Stokes equations when the initial data and the domain are cylindrically symmetric. We show that there exists a solution in a weighted space and certain weighted norms of vorticity of the solution remain finite if they are finite initially. Consequently, we can estimate the growth rate of the solution both spatially and temporally.     

Geodesic restrictions for the Casimir operator

We consider the hyperbolic Casimir operator C defined on the tangent sphere bundle SY of a compact hyperbolic Riemann surface Y. We prove a non-trivial bound on the L²-norm of the restriction of eigenfunctions of C to certain natural hypersurfaces in SY. The result that we obtain goes beyond known (sharp) local bounds of L. Hörmander.     

Kernel sections for non-autonomous strongly damped wave equations

We prove the existence of compact kernel sections for the process generated by a non-autonomous strongly damped wave equation with homogeneous Dirichlet boundary condition. We show that the upper bound of the Hausdorff dimension of sections decreases as the damping grows for large strong damping.     

Twisted Alexander norms give lower bounds on the Thurston norm

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number greater than one, generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which are not determined by norms of McMullen and Turaev.

     

Mixed Norm Inequalities for Directional Operators Associated to Potentials

We study mixed norm inequalities for directional operators which appear applying the method of rotations to homogeneous operators with variable kernel and with the homogeneity of Riesz potentials. The results are sharp for a range of values of the parameter and for all its values when the inequalities are restricted to radial functions.     

Corrigendum to “On norms of composition operators acting on Bergman spaces” [J. Math. Anal. Appl. 291 (2004) 189-202]

A theorem and a corollary in the paper cited in the title were stated incorrectly, as was pointed out by Christopher Hammond. We now state correctly and prove both of them. These results still generalize and explain the geometric meaning of the Cowen-Hurst norm formula. We also include additional references and provide an example relevant for further study.     

Criteria for Validity of the Maximum Norm Principle for Parabolic Systems

We consider systems of partial differential equations, which contain only second derivatives in the x variables and which are uniformly parabolic in the sense of Petrovskii. For such systems we obtain necessary and, separately, sufficient conditions for the maximum norm principle to hold in the layer Rn × ( 0,T ] and in the cylinder × ( 0,T], where is a bounded subdomain of Rn. In this paper the norm is understood in a generalized sense, i.e. as the Minkowski functional of a compact convex body in Rm containing the origin. The necessary and sufficient conditions coincide if the coefficients of the system do not depend on t. The criteria for validity of the maximum norm principle are formulated as a number of equivalent algebraic conditions describing the relation between the geometry of the unit sphere of the given norm and coefficients of the system under consideration. Simpler formulated criteria are given for certain classes of norms: for differentiable norms, p-norms ( 1 p ) in Rm, as well as for norms whose unit balls are m-pyramids, m-bipyramids, cylindrical bodies, m-parallelepipeds. The case m = 2 is studied separately.     

Optimization of reservoirs in series on a river with a nonlinear storage curve for long-term regulation

We present in this paper a new approach to finding the monthly optimal operation of a multireservoir power system connected in series on a river. The hydroelectric power generation is a highly nonlinear function of the storage, and the conversion factor assigned to each power plant is also a nonlinear function of the storage. We use for both a quadratic function of the storage; the resulting problem has a highly nonlinear objective function and linear constraints. We propose a transformation such that the system equations are reduced to linear-quadratic form. Lagrange and Kuhn-Tucker multipliers are used to adjoin the equality and inequality constraints to the objective function. Numerical results are presented for a real system in operation consisting of two reservoirs in series on a river for widely different water conditions.This work was supported by the National Research Council of Canada, Grant No. A4146. The authors would like to acknowledge data obtained from B.C. Hydro.     

A New and Self-Contained Proof of Borwein’s Norm Duality Theorem

Borwein’s norm duality theorem establishes the equality between the outer (inner) norm of a sublinear mapping and the inner (outer) norm of its adjoint mappings. In this note we provide an extended version of this theorem with a new and self-contained proof relying only on the Hahn-Banach theorem. We also give examples showing that the assumptions of the theorem cannot be relaxed. This author is supported by Grant BES-2003-0188 from FPI Program of MEC (Spain).     

Bounds for the smallest norm in an ideal class

We develop a method for obtaining upper bounds for the smallest norm among all norms of integral ideals in an ideal class. Applying this to number fields of small degree, we are able to substantially improve on the best previously known bounds.

     

A new proof of the two weight norm inequality for the one-sided fractional maximal operator

We give a new proof of the two weight norm inequality for the one-sided, fractional maximal operator, , simplifying the original proof of Martín-Reyes and de la Torre.

     

Axiomatics for fuzzy rough sets

A fuzzy T-rough set consists of a set X and a T-similarity relation R on X, where T is a lower semi-continuous triangular norm. We generalize the Farinas-Prade definition for the upper approximation operator of a fuzzy T-rough set (X, R); given originally for the special case T = Min, to the case of arbitrary T. We propose a new definition for the lower approximation operator of (X,R). Our definition satisfies the two important identities and , as well as a number of other interesting properties. We provide axiomatics to fully characterize those upper and lower approximations.     

Improved Young and Heinz inequalities for matrices

We give refinements of the classical Young inequality for positive real numbers and we use these refinements to establish improved Young and Heinz inequalities for matrices.     

Composite norms and perfect conditioning

We study condition numbers for composite norms and analyze the situations where perfect conditioning occurs.