An Inverse Result of Approximation by Sampling Kantorovich Series

In the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns out to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

An Investigation Into The Effect Of Isothermal Annealing On Magnetic Properties In The Alloy: Fe64Co10Y6B20

This work presents the results of investigations into the structural and magnetic properties of the bulk amorphous alloy: Fe64Co10Y6B20. The structure, thermal stability and magnetic properties of the alloy were studied using: X-ray diffractometry, differential scanning calorimetry (DSC), and a vibrating sample magnetometer (VSM), respectively. The investigations were performed on samples of the alloy in both the ‘as-cast’ state, and the state resulting from a process of isothermal annealing at a temperature of 750 K for 30 minutes.The aim of the conducted studies was to relax the structure and improve the soft magnetic properties of the investigated alloy. The results show that annealing the alloy at a temperature well below its crystallisation temperature leads to an increase in the value of the saturation magnetisation and a decrease in the value of the coercivity. Utilising the ‘approach to the ferromagnetic saturation’ theorem, the nature of structural defects within the investigated material has been established. For both ‘as-cast’ and isothermally-annealed samples, the magnetisation process has been found to be connected with the existence of linear structural defects.

Pointwise Saturation Theorem of Bi-Continuous Semigroups

By means of Riemann-Stieltjes stochastic process, moment-generating functions and operator-Valued mathematical expectation,the problem of probabilistic approximation for bi-continuous semigroups was studied and the saturation theorem of probabilistic representations of semigroups are obtained .

Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

The saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong.

Cluster complexes via semi-invariants

We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the (n−1)-sphere.