‎In this paper‎, ‎by using the sequence of multipliers‎, ‎we introduce frames with algebraic bounds in Hilbert pro-$ C^* $-modules‎. ‎We investigate the relations between frames and $ ast $-frames‎. ‎Some properties of $ ast $-frames in Hilbert pro-$ C^* $-modules are studied‎. ‎Also‎, ‎we show that there exist two differences between $ ast $-frames in Hilbert pro-$ C^* $-modules and Hilbert $ ...

In this work, we proposed an effective method based on cubic and pantic B-spline scaling functions to solve partial differential equations of fractional order. Our method is based on dual functions of B-spline scaling functions. We derived the operational matrix of fractional integration of cubic and pantic B-spline scaling functions and used them to transform the mentioned equations to a syste...

We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation. Based on this definition, we prove duality between controllability/reachability and determinability/observability for switched differential-algebraic equations. The switched system is viewed as a (linear) time-varying system, i.e. the switching signal is assumed to be fixed. In particula...

An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is char...

Macaulay dual spaces provide a local description of an affine scheme and give rise to computational machinery that is compatible with the methods of numerical algebraic geometry. We introduce eliminating dual spaces, use them for computing dual spaces of quotient ideals, and develop an algorithm for detection of embedded points on an algebraic curve.

An algebraic quantum group is a multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterize...

We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show that the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash [2]. This result implies that if there is a du...

In this paper, we introduce a method of generating the dual graph of the minimal normal resolution of the algebraic curve singularities. Using factorization of polynomials over algebraic extension field, we can calculate the dual graph from the coefficients of the given polynomial exactly. The computing process includes no approximation. We constructed the system to generate the dual graph on R...

Algebraic GIFS is a class of graph-directed iterated function systems (IFS) on R with algebraic parameters. A dual IFS of an algebraic GIFS can be constructed, and the duality between the two systems and has been investigated from various points of view. We review the results of dual IFS concerning the open set condition, purely periodic codings and the Rauzy-Thurston tilings, and their relatio...