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This paper treats the strict semi-stability of the symmetric powers [Formula: see text] of a stable vector bundle [Formula: see text] of rank [Formula: see text] with even degree on a smooth projective curve [Formula: see text] of genus [Formula: see text]. The strict semi-stability of [Formula: see text] is equivalent to the orthogonality of [Formula: see text] or the existence of a bisection on the ruled surface [Formula: see text] whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of [Formula: see text] with strictly semi-stable [Formula: see text]. Moreover, it is shown that when [Formula: see text] is stable, every symmetric power [Formula: see text] is stable for all but a finite number of [Formula: see text] in the moduli of stable vector bundles of rank [Formula: see text] with fixed determinant of even degree on [Formula: see text].

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.

Let [Formula: see text] be a minimal parabolic subgroup of a real reductive Lie group [Formula: see text] and [Formula: see text] a closed subgroup of [Formula: see text]. Then it is proved by Kobayashi and Oshima that the regular representation [Formula: see text] contains each irreducible representation of [Formula: see text] at most finitely many times if the number of [Formula: see text]-orbits on [Formula: see text] is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite, where [Formula: see text] are complexifications of [Formula: see text], respectively, and [Formula: see text] is a Borel subgroup of [Formula: see text]. In this paper, we prove that the multiplicities of the representations of [Formula: see text] induced from a parabolic subgroup [Formula: see text] in the regular representation on [Formula: see text] are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite. For the proof of this claim, we also show the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic [Formula: see text]-modules.

The Young type inequality asserts that if [Formula: see text] are two positive real numbers, then for each [Formula: see text], we have [Formula: see text] In this paper, we obtain some new inequalities using two different weights. For example, if [Formula: see text], then [Formula: see text] [Formula: see text] where [Formula: see text] In addition, we refine some matrix inequalities for Unitarily invariant norms by applying the deduced numerical inequalities.

In this paper, we define a module [Formula: see text] to be [Formula: see text] if and only if intersection of each pair of [Formula: see text]-closed direct summands is also a direct summand of [Formula: see text]. We investigate structural properties of [Formula: see text]-modules and locate the implications between the other module properties which are essentially based on direct summands. We deal with decomposition theory as well as direct summands of [Formula: see text]-modules. We apply our results to matrix rings. To this end, it is obtained that the [Formula: see text] property is not Morita invariant.

In this paper, we introduce Fibonacci backward difference operator [Formula: see text] of fractional order l by the composition of Fibonacci band matrix [Formula: see text] and difference operator [Formula: see text] of fractional order l, defined by [Formula: see text] and introduce sequence spaces [Formula: see text] and [Formula: see text] We present some topological properties, obtain Schauder basis and determine [Formula: see text]-, [Formula: see text]- and [Formula: see text]-duals of the spaces [Formula: see text] and [Formula: see text] We characterize certain classes of matrix mappings from the spaces [Formula: see text] and [Formula: see text] to any of the space [Formula: see text] [Formula: see text] [Formula: see text] or [Formula: see text] Finally we compute necessary and sufficient conditions for a matrix operator to be compact on the spaces [Formula: see text] and [Formula: see text]