Tensor product semigroups Research Papers (original) (raw)

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT].

Any solution of the incompressible Navier-Stokes equations in three-dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix-Raviart (nonconforming P 1 ) elements and the pressure by piecewise constants. This method is stable for general meshes since the inf-sup condition is satisfied without minimal or maximal angle condition. The existence of solutions to the discrete problems follows. Consistency error estimates for the divergence equation are obtained for anisotropic tensor product meshes. As applications, the consistency error estimate for the Navier-Stokes solution and some discrete Sobolev inequalities are derived on such meshes. These last results provide optimal error estimates in the uniqueness case by the use of appropriately refined anisotropic tensor product meshes, namely, if N e is the number of elements of the mesh, we prove that the optimal order of convergence h ∼ N −1/3 e .

- by Serge Nicaise
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- Applied Mathematics, Tensor product semigroups, Navier Stokes, Three Dimensional
- by Ruud Pellikaan
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- Tensor product semigroups, Pure Mathematics, IT Value, Algebraic Geometry code

Explicit expressions for the generators of the quantum superalgebra U q [gl(n/m)} acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a GePfand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set of ^-number identities.

- by Nguyễn Anh Kỳ
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- Mathematical Physics, Algebra, Lie Algebra, Quantum Physics

We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra U q ( sl 2 ). Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin 1/2, and that the n-particle space has an RSOS-type structure rather than a simple tensor product of the 1-particle space. This agrees with the picture proposed earlier by Reshetikhin.

In this paper we present a very efficient Hermite subdivision scheme, based on rational functions, and outline its potential applications, with special emphasis on the construction of cubic-like B-splines -well suited for the design of... more

- by Carla Manni
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- Applied Mathematics, Numerical Analysis, Tensor product semigroups, Surface

Given a closed triangular mesh, we construct a smooth free-form surface which is described as a collection of rational tensor-product and triangular surface patches. The surface is obtained by a special manifold surface construction, which proceeds by blending together geometry functions for each vertex. The transition functions between the charts, which are associated with the vertices of the mesh, are obtained via subchart parameterization.

- by Myung-soo Kim
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- Engineering, Computer Aided Design, Topology, Tensor product semigroups

The concept, known as the Canard Rotor/Wing (CRW) unmanned aerial vehicle (UAV) combines the hover flight characteristic of a helicopter with high subsonic cruise of a fixed-wing aircraft. The longitudinal flight dynamic model of CRW was developed, and the trim result was derived in the full flight envelope. The linear model of CRW was derived by the Jacobian linear method, and it was nonlinearly dependent on the time-varying flight speed and altitude. The tensor-product (TP) model transformation was adopted to transform the model to a convex polytopic model form. Hence, a linear parameter-varying (LPV) synthesis method was used to design the flight control system of CRW in the rotary mode. The simulation results show that the desired performance objectives are achieved in the rotary flight stage.

- by Dawei Li
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- Atmospheric Modeling, Management Science, Vehicle Dynamics, Tensor product semigroups

Jordan ring is one example of the non-associative rings. We can construct a Jordan ring from an associative ring by defining the Jordan product. In this paper, we discuss the properties of non-associative rings by studying the properties of the Jordan rings. All of the ideals of a non-associative ring R are non-associative, except the ideal generated by the associator in R. Hence, a quotient ring R/ X can be constructed , where X is the ideal generated by associators in R. The fundamental theorem of the homomorphism ring can be applied to the non-associative rings. By a little modification, we can find that R/ X is isomorphic to S/ Y. Furthermore, we define a module over a non-associative ring and investigate its properties. We also give some examples of such modules. We show if M is a module over a non-associative ring R, then M is also a module over R/ X if X is contained in the annihilator of R. Moreover, we define the tensor product of modules over a non-associative ring. The tensor product of the modules over a non-associative ring is commutative and associative up to isomorphism but not element by element.

These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S = 1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.

- by Ferdinando Mancini
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- Quantum Mechanics, Quantum Monte Carlo, Tensor product semigroups, Ising Model
- by Enrique Castillo
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- Information Systems, Tensor product semigroups, Geometric model, Surfaces

This article presents a brief introduction to the classical geometry of ruled surfaces with emphasis on the Klein image and studies aspects which arise in connection with a computational treatment of these surfaces. As ruled surfaces are one parameter families of lines, one can apply curve theory and algorithms to the Klein image, when handling these surfaces. We study representations of rational ruled surfaces and get efficient algorithms for computation of planar intersections and contour outlines. Further, low degree boundary curves, useful for tensor product representations, are studied and illustrated at hand of several examples. Finally, we show how to compute efficiently low degree rational G 1 ruled surfaces. ᭧

This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of "positive space" and its rational powers. Positive spaces are 1-dimensional "semi-vector spaces" without the zero vector. A direct approach to this subject might be sufficient. On the other hand, a broader mathematical understanding requires the notions of sesqui and semi-tensor products between semi-vector spaces and vector spaces.

- by Raffaele Vitolo and +1
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- Tensor product semigroups, Dimensional, Vector Space

Several researchers, including Leonid Levin, Gerard 't Hooft, and Stephen Wolfram, have argued that quantum mechanics will break down before the factoring of large numbers becomes possible. If this is true, then there should be a natural set of quantum states that can account for all quantum computing experiments performed to date, but not for Shor's factoring algorithm. We investigate as a candidate the set of states expressible by a polynomial number of additions and tensor products. Using a recent lower bound on multilinear formula size due to Raz, we then show that states arising in quantum error-correction require n Ω(log n) additions and tensor products even to approximate, which incidentally yields the first superpolynomial gap between general and multilinear formula size of functions. More broadly, we introduce a complexity classification of pure quantum states, and prove many basic facts about this classification. Our goal is to refine vague ideas about a breakdown of quantum mechanics into specific hypotheses that might be experimentally testable in the near future. Because of the 'even if' clauses, the objections seem to us logically independent, so that there are 16 possible positions regarding them (or 15 if one is against quantum computing). We ignore the possibility that no speedup exists, in other words that BPP = BQP. By 'large quantum computer' we mean any computer much faster than its best classical simulation, as a result of asymptotic complexity rather than the speed of elementary operations. Such a computer need not be universal; it might be specialized for (say) factoring.

No infinite dimensional Banach space X is known which has the property that for m 2 the Banach space of all continuous m-homogeneous polynomials on X has an unconditional basis. Following a program originally initiated by Gordon and Lewis we study unconditionality in spaces of m-homogeneous polynomials and symmetric tensor products of order m in Banach spaces. We show that for each Banach space X which has a dual with an unconditional basis (x i *), the approximable (nuclear) m-homogeneous polynomials on X have an unconditional basis if and only if the monomial basis with respect to (x i *) is unconditional. Moreover, we determine an asymptotically correct estimate for the unconditional basis constant of all m-homogeneous polynomials on l n p and use this result to narrow down considerably the list of natural candidates X with the above property.

- by Domingo Garcia and +1
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- Functional Analysis, Tensor product semigroups, Pure Mathematics

An addition rule of impure density operators, which provides a pure state density operator, is formulated. Quantum interference including visibility property is discussed in the context of the density operator formalism. A measure of entanglement is then introduced as the norm of the matrix equal to the difference between a bipartite density matrix and the tensor product of partial traces. Entanglement for arbitrary quantum observables for multipartite systems is discussed. Star-product kernels are used to map the formulation of the addition rule of density operators onto the addition rule of symbols of the operators. Entanglement and nonlocalization of the pure state projector and allied operators are discussed. Tomographic and Weyl symbols (tomograms and Wigner functions) are considered as examples. The squeezed-states and some spin-states (two qubits) are studied to illustrate the formalism.

We m a t c h u p t ypes and processes by putting values in correspondence with events, coproduct with (noninteracting) parallel composition, and tensor product with orthocurrence. We then bring types and processes into closer correspondence by broadening and unifying the semantics of both using Chu spaces and their transformational logic. Beyond this point the connection appears to break down we pose the question of whether the failures of the corrrespondence are intrinsic or cultural. c 1997 Published by Elsevier Science B. V. Pratt events while Scott domains with their information ordering consist of states. The duality that NPW found between them is a true categorical duality in the sense that it reverses the morphisms of the respective categories.

We build a differential calculus for subalgebras of the Moyal algebra on R 4 starting from a redundant differential calculus on the Moyal algebra, which is suitable for reduction. In some cases we find a frame of 1-forms which allows to realize the complex of forms as a tensor product of the noncommutative subalgebras with the external algebra Λ * .

- by G. Marmo
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- Mathematics, Physics, Tensor product semigroups, Mathematical Sciences

We construct a mapping from complex recursive linguistic data structures to spherical wave functions using Smolensky’s filler/role bindings and tensor product representations. Syntactic language processing is then described by the transient evolution of these spherical patterns whose amplitudes are governed by nonlinear order parameter equations. Implications of the model in terms of brain wave dynamics are indicated.

- by Dimitris Pinotsis
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- Physiology, Cognitive Science, Tensor product semigroups, Language Processing

In a series of papers TSIRELSON constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by ARVESON [4] for classifying E 0 -semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So we connect each continuous tensor product systems of Hilbert spaces with measure types of distributions of random (closed) sets in [0, 1] or R + . These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the TSIRELSON examples, that the classification scheme for product systems into types I n , II n and III is not complete. Moreover, based on a detailed study of this kind of measure types, we construct for each stationary factorizing measure type a continuous tensor product systems of Hilbert spaces such that this measure type arises as the before mentioned invariant.

We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some finiteness conditions). The

A constructive geometric approach to developable rational B6zier and B-spline surfaces is presented. It is based on the dual representation in the sense of projective geometry. By the principle of duality, projective algorithms for NURBS curves can be transferred to constructions for developable NURBS surfaces in dual rational B-spline form. We discuss the conversion to the usual tensor product representation of the obtained surfaces and develop algorithms for basic design problems arising in this context.

- by Gerald Farin and +1
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- Engineering, Tensor product semigroups, NURBS, Mathematical Sciences

This paper proposes an alternative physical interpretation and calculation of instantaneous power, using the concept of tensor product as a mathematical tool. "Instantaneous power tensor" is the single expression defined using this theory of power, which involves the two components of instantaneous power (active and reactive), in order to geometrically interpret the behavior of electrical phenomena, analogous to studies of deformation in the mechanics of solids. Additionally, a comparison is made between the definition of instantaneous power in the literature (vector analysis) and the proposed definition (tensor analysis).

- by HERNAN TACCA and +2
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- Tensor product semigroups, Electric Power Systems, Reactive Power, Electrical And Electronic Engineering

A solution strategy for plasticity and viscoplasticity models with isotropic yield surfaces depending upon all the principal invariants of the stress tensor is presented. Basically, it requires the inversion of a fourth-order positive de®nite tensor G both for the solution of the constitutive problem and for the evaluation of the consistent tangent operator. It is proved that the assumption of isotropic elastic behaviour and the isotropy of the yield criterion entail an explicit representation formula for G À1 as linear combination of dyadic and square tensor products. Further, an analogous representation formula for the consistent tangent operator is provided. By exploiting basic composition rules between dyadic and square tensor products along with Rivlin's identities for tensor polynomials, all tensor operations required to compute the coecients of the adopted representation formula for G À1 are carried out in intrinsic form. It is thus shown that the relevant computational burden essentially amounts to solving a linear system of order three. The performances of the proposed approach are illustrated by means of some numerical examples referred to the Argyris failure criterion. Ó

- by Luciano Rosati
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- Engineering, Tensor product semigroups, Finite Element, Mathematical Sciences

We survey some recent developments in the theory of Frechet spaces and of their duals. Among other things, Section 4 contains new, direct proofs of properties of, and results on, Fr ´ echet spaces with the density condition, and Section 5... more

- by Jose Bonet
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- Tensor product semigroups

Let 2tf and j f be Hilbert spaces, and let M and N be von Neumann algebras of operators on Jf and Jf respectively . Let Jf ® Jf denote the Hilbert space tensor product of 3V with X, and let M (g) N denote the von Neumann algebra on #? (g) Jf generated by the operators m ® « for meM, neN [2], We will denote the cornmutant of a von Neumann algebra M by M'. Although there were a number of earlier proofs of special cases, it was not until 1967 that Tomita [8, 6] gave a proof in full generality for: THEOREM 1. Let M and N be von Neumann algebras on 2tf and J f respectively. Then Received 11 May, 1974. t The research for this paper was conducted while we were both visiting at the University of Pennsylvania. We would like to thank the members of the Mathematics Department there for their warm hospitality during our visits.

- by Alfons Van Daele
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- Tensor product semigroups, Pure Mathematics, von Neumann algebra

We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to di erent classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, some forms of Projection Pursuit Regression and several types of neural networks. We propose to use the term Generalized Regularization Networks for this broad class of approximation schemes that follow from an extension of regularization. In the probabilistic interpretation of regularization, the di erent classes of basis functions correspond to di erent classes of prior probabilities on the approximating function spaces, and therefore to di erent types of smoothness assumptions.

ruled surfaces of degree 2m, whose shape is guided by m ϩ 1 control lines and m frame lines. This is an advantage In this paper, geometric design problems for rational ruled surfaces are studied. We investigate a line geometric control over the method of Ravani and Wang, which results in structure and its connection to the standard tensor product Bsurfaces of degree 3m from the same data.

In this paper we first develop the notion of topological tensor products for topological semigroups and we will show that topological tensor product of topological groups S and T is an extension of S and T. Also, we study the universal... more

- by Hamidreza Rahimi
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- Tensor product semigroups

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graph on four vertices representing entangled states. It turns out that for some of these graphs the value of the concurrence is exactly fractional.

- by Samuel Braunstein
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- Quantum Physics, Tensor product semigroups, Pure Mathematics, Density Matrix

We introduce a new algebraic concept of a difierential graded al- gebra which is "almost" commutative (ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some flniteness conditions). The theory is su-ciently general

- by Max Karoubi
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- Tensor product semigroups, Hochschild Homology, Homotopy Type Theory

In this paper we establish a decomposition theorem for an ordinary representation of a finite group G in any category C which expresses a suitable irreducible representation of G as the tensor product of two projective ones. The celebrated theorem due to Clifford for a linear representation turns out to be a particular case of it. For that purpose, a definition of projective extension of an ordinary representation of a normal subgroup of G is introduced, as well as a tensor product between two of them.

Recent development brings new results on the interplay of states on operator algebras and axiomatics of quantum mechanics. Neither hidden space in the sense of Kochen and Specker nor approximate hidden variables exist on von Neumann algebras. Tracial properties of states are connected with dispersions. The axioms on composite systems simplify to state extension properties.

- by Jan Hamhalter
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- Theoretical Physics, Quantum Theory, Quantum Mechanics, Tensor product semigroups

Cover: An extensive green roof system installed atop the NYC Department of Parks and Recreation's (DPR) Five Borough Building on Randall's Island. This modular system is one of six variations installed on the roof and covers 800 square feet, consisting of two-foot by two-foot trays with six inches of mineral soil and over 1,500 sedum plugs. DPR has installed 25 green roof systems covering over 29,000 square feet on the Five Borough Building rooftop to feature different types and depths of growing medium and plant selection.

In this paper we describe the ring of invariants of the space of m-tuples of n×n matrices, under the action of SL(n)×SL(n) given by (A, B)·(X 1 , X 2 , · · · , X m ) → (AX 1 B t , AX 2 B t , · · · , AX m B t ). Determining the ring of invariants is the first step in the geometric approach to finding multiplicities of representations of the symmetric group in the tensor product of rectangular shaped representations. We show that these invariants are given by multi-determinants and can also be described in terms of certain magic squares. We compute the null cone for this action. We also study a birational subring of invariants and an analysis thereof results into a different proof of the Artin-Procesi theorem for the ring of invariants for several matrices under the simultaneous conjugation action of SL(n).

- by Bharat G Adsul
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- Tensor product semigroups, Symmetric group

In this book I treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a new picture. Matrices allow two products linked by transpose. Biring is algebra which defines on the set two correlated structures of the ring. As in the commutative case, solutions of a system of linear equations build up right or left vector space depending on type of system. We study vector spaces together with the system of linear equations because their properties have a close relationship. As in a commutative case, the group of automorphisms of a vector space has a single transitive representation on a frame manifold. This gives us an opportunity to introduce passive and active representations. Studying a vector space over a division ring uncovers new details in the relationship between passive and active transformations, makes this picture clearer. Considering of twin representations of division ring in Abelian group leads to the concept of D-vector space and their linear map. Based on polylinear map I considered definition of tensor product of rings and tensor product of D-vector spaces.

- by Aleks Kleyn
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- Tensor product semigroups, Linear Algebra, Vector Space, Linear Equations
- by Barbara Russo
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- Algebraic Geometry, Tensor product semigroups

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product [12] of Gray categories. Moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations, and we illustrate this by giving an expedited account of many aspects of Cheng's analysis [11] of Trimble's definition of weak n-category. By way of application we obtain an "Ekmann-Hilton" result for braided monoidal 2-categories, and give the construction of a tensor product of A-infinity algebras.

- by Michael Batanin
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- Category Theory, Tensor product semigroups
- by Michael Batanin and +1
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- Category Theory, Tensor product semigroups, Pure Mathematics, Numerical Analysis and Computational Mathematics

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction-diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a special mesh in space, condensing the mesh points in the boundary layer regions. In this paper three different meshes of Shishkin, Bahkvalov and Vulanovic type are used, proving the uniform convergence with respect to the diffusion parameter. The analysis of the uniform convergence is based on a new study of the asymptotic behavior of the solution of the semidiscrete problems, which are obtained after the time discretization by the Euler method. Some numerical results are showed corroborating in practice the theoretical results on the uniform convergence and the order of the method.

Letĝ be an untwisted affine Kac-Moody algebra. The quantum group U q (ĝ) is known to be a quasitriangular Hopf algebra (to be precise, a braided Hopf algebra). Here we prove that its unrestricted specializations at odd roots of 1 are braided too: in particular, specializing q at 1 we have that the function algebra F H of the Poisson proalgebraic group H dual of G (a Kac-Moody group with Lie algebraĝ) is braided. This in turn implies also that the action of the universal R-matrix on the tensor products of pairs of Verma modules can be specialized at odd roots of 1.

- by Fabio Gavarini
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- Mathematics, Lie Algebra, Tensor product semigroups, Pure Mathematics

The theory of Nevanlinna-Pick and Carathéodory-Fejér interpolation for matrix-and operator-valued Schur class functions on the unit disk is now well established. Recent work has produced extensions of the theory to a variety of multivariable settings, including the ball and the polydisk (both commutative and noncommutative versions), as well as a time-varying analogue. Largely independent of this is the recent Nevanlinna-Pick interpolation theorem by P.S. Muhly and B. Solel for an abstract Hardy algebra set in the context of a Fock space built from a W * -correspondence E over a W * -algebra A and a * -representation σ of A. In this review we provide an exposition of the Muhly-Solel interpolation theory accessible to operator theorists, and explain more fully the connections with the already existing interpolation literature. The abstract point evaluation first introduced by Muhly-Solel leads to a tensor-product type functional calculus in the main examples. A second kind of point-evaluation for the W * -correspondence Hardy algebra, also introduced by Muhly and Solel, is here further investigated, and a Nevanlinna-Pick theorem in this setting is proved. It turns out that, when specified for examples, this alternative point-evaluation leads to an operator-argument functional calculus and corresponding Nevanlinna-Pick interpolation. We also discuss briefly several Nevanlinna-Pick interpolation results for Schur classes that do not fit into the Muhly-Solel W * -correspondence formalism.

- by Joseph Ball
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- Functional Analysis, Tensor product semigroups, Fock Space, Functional calculus
- by Oliver Staadt
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- Tensor product semigroups, Level Of Detail (LOD), Wavelet Transform, Look up Table

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz 50 (2005)

- by Radek Kucera
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- Applied Mathematics, Tensor product semigroups, Fast Fourier Transform

The first part of the paper is an overview of the theory of approximation of wave equations by Galerkin methods. It treats convergence theory for linear second order evolution equations and includes studies of consistency and eigenvalue approximation. We emphasize differential operators, such as the curl, which have large kernels and use L 2 stable interpolators preserving them. The second part is devoted to a framework for the construction of finite element spaces of differential forms on cellular complexes. Material on homological and tensor algebra as well as differential and discrete geometry is included. Whitney forms, their duals, their high order versions, their tensor products and their hp-versions all fit.