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Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal . In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces.

We introduce the two-parameter quantum affine algebra U r , s ( gl ˆ n ) via the RTT realization. The Drinfeld realization is given and the type A quantum affine algebra is proved to be a special subalgebra of our extended algebra.

Various statements and a variety of solutions to the problem of input-to-state stabilization of dynamic systems with disturbances are known. Methods based on the use of Lyapunov functions play an important role with regard to non-linear systems. When using these methods, the problem of finding an appropriate Lyapunov function arises. The Lyapunov functions redesign method provides a Lyapunov function for a certain subclass of affine systems with disturbances using transformation of the corresponding affine system without disturbances to the equivalent regular canonical form. The desired Lyapunov function is constructed as a quadratic form of the canonical variables. Further, the found Lyapunov function can be used to construct the input-to-state asymptotically stabilizing control. The limits of applicability of this approach remain unclear: in general, constructed on the basis of the transformation to the equivalent canonical form the Lyapunov function for the system without disturbances can both be and not be the Lyapunov function for the affine system with disturbance.In the paper, we study the possibility of using the described approach to second-order affine systems with scalar control and scalar disturbances for which the corresponding systems without disturbances are equivalent to regular systems of canonical form in the whole state space. We have obtained the easily verifiable conditions for construction of the Lyapunov function on the basis of the regular canonical form where the Lyapunov function for the system with control will be the function for the system with disturbances. Thus, the class of systems which can be stabilized by using the above method is defined. Examples of applications of the obtained conditions with regard to certain classes of second-order affine systems and the results of numerical simulation of the stabilization process of the zero equilibrium point in the presence of various disturbances for the particular two-dimensional system with disturbances are given.DOI: 10.7463/mathm.0315.0789645

We proposed a kind of naturally combined shape-color affine moment invariants (SCAMI), which consider both shape and color affine transformations simultaneously in one single system. In the real scene, color and shape deformations always exist in images simultaneously. Simple shape invariants or color invariants cannot be qualified for this situation. The conventional method is just to make a simple linear combination of the two factors. Meanwhile, the manual selection of weights is a complex issue. Our construction method is based on the multiple integration framework. The integral kernel is assigned as the continued product of the shape and color invariant cores. It is the first time to directly derive an invariant to dual affine transformations of shape and color. The manual selection of weights is no longer necessary, and both the shape and color transformations are extended to affine transformation group. With the various of invariant cores, a set of lower-order invariants are constructed and the completeness and independence are discussed detailed. A set of SCAMIs, which called SCAMI24, are recommended, and the effectiveness and robustness have been evaluated on both synthetic and real datasets.

Zonotopes are a useful set representation for bounded time reach set computation of affine hybrid systems because of their closure under Minkowski sum and matrix multiplication operations. For unbounded time reach set approximation of arbitrarily switched affine hybrid systems, template complex zonotopes and a corresponding invariant computation procedure were introduced, which utilized the possibly complex eigenstructure of the affine maps. But a major hurdle in extending the technique for computing invariants of more general affine hybrid systems, where switching is state dependent and controlled by linear constraints, is that the class of template complex zonotopes is not closed under intersection with linear constraints. In this paper, we use a more expressive set representation called augmented complex zonotopes, for which we propose an algebraic over-approximation of the intersection with linear constraints. This over-approximation is then used to derive a set of second order conic constraints for computing an augmented complex zonotopic positive invariant for discrete time affine hybrid systems with additive disturbance input and linear safety constraints. We demonstrate the efficiency of this approach by experimenting on some benchmark examples.

In this paper, we study affine transformations of normal bases and give an explicit formulation of the multiplication table of an affine transformation of a normal basis. We then discuss constructions of self-dual normal bases using affine transformations of traces of a type I optimal normal basis and of a Gauss period normal basis.

Refinable function-based affine frames and affine bi-frames have been extensively studied in the literature. All these works are based on some restrictions on refinable functions. This paper addresses what are expected from two general refinable functions. We introduce the notion of weak (quasi-) affine bi-frame; present a refinable function-based construction of weak (quasi-) affine bi-frames; and obtain a fast algorithm associated with weak affine bi-frames. An example is also given to show that our construction is optimal in some sense.

The paper offers generalizations of the Jensen-Mercer inequality for self-adjoint operators and generally convex functions. The obtained results are applied to define the quasi-arithmetic operator means without using operator convexity. The version of the harmonic-geometric-arithmetic operator mean inequality is derived as an example.

In this work we provide three new characterizations of affine dual frames constructed from refinable functions. The first one is similar to Daubechies et al. (2003) [10, Proposition 5.2] but without any decay assumptions on the generators of a pair of affine systems. The second one reveals the geometric significance of the Mixed Fundamental function and the third one shows that the Mixed Oblique Extension Principle actually characterizes dual framelets. We also extend recent results on the characterization of affine Parseval frames obtained in Stavropoulos (2012) [27, Theorem 2.3] .

In this chapter we utilize extremum seeking (ES) to stabilize a large class of systems affine in control, including time-varying nonlinear systems. We work out the details of stability proofs and provide several numerical examples of applying ES for stabilization.

We suggest a method for solving the terminal control problem for multidimensional affine systems that are not linearizable by feedback. We prove a sufficient condition for the existence of a solution and present a numerical procedure for its construction.

This paper concerns affine infinitesimal generators of continuous semigroups of holomorphic functions on the polydisk. An easy-to-check necessary and sufficiently condition for an affine vector field to be an infinitesimal generator is given.

Information science focuses on understanding problems from the perspective of the stakeholders involved and then applying information and other technologies as needed. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces. This result is obtained through considering the basicquestion as to when the orthogonal complement of an afffine subspace in another one is still affine subspace. Motivated by the fundamental question as to whethor every affine subspace is singly-generated wavelet frame, we prove that every affine sub -space can be decomposed into the direct sum of a singly-generated afffine subspace.

This paper provides a detailed discussion of the second covariant derivative of a map and its role in the Lie group projection operator approach (a direct method for solving continuous time optimal control problems). We begin by briefly describing the iterative geometric optimal control algorithm and summarize the general expressions involved. Particular emphasis is placed on the expressions related to the search direction subproblem, writing them in a new compact form by using a new operator notation. Next, we show that the covariant derivative of a map between manifolds endowed with affine connections plays a key role in obtaining the required local quadratic approximations for the Lie group projection operator approach. We present a new result for computing an approximation of the parallel displacement associated with an affine connection which is an affine combination of two (or more) connections. As a corollary, an extremely useful approximation of the parallel displacement relative to the Cartan-Schouten (0) connection on Lie groups is obtained.

This paper describes a problem of interest in CSG modelling, namely the location of implicit polynomial surfaces in space. It is common for surfaces defined by implicits to be located using interval arithmetic. However, the method only gives conservative bounds for the values of the function inside a region of interest. This paper gives two possible ways of producing tighter bounds. One involves using a Bernstein–form representation of the implicit polynomials used as input to the method. The other fine–tunes the method itself by employing careful use of affine arithmetic—a more sophisticated version of interval arithmetic. As both methods contribute significant improvements, we speculate about combining the two into a fast and accurate method for surface location.

We study affine completeness of algebras. In the first part of the work, we use a generalized metric to prove an extension theorem. This extension theorem plays a key role in proving new results. In the second part, we show that in some situations, we can skip the extension theorem. This idea allows us to answer a question of Karrli and Pixley.

In this chapter, we present how to learn regression models on Lie groups and apply our formulation to visual object tracking tasks. Many transformations used in computer vision, for example orthogonal group and rotations, have matrix Lie group structure. Unlike conventional methods that proceed by directly linearizing these transformations, thus, making an implicit Euclidean space assumption, we formulate a regression model on the corresponding Lie algebra that minimizes a first order approximation to the geodesic error. We demonstrate our method on affine motions , however, it generalizes to any matrix Lie group transformations.

Equivalence of the metaplectic representation with a sum of affine representations is discussed for a class of subgroups of the symplectic group. The paper begins with a study of sums of wavelet transforms, and it is shown that the usual admissibility conditions for the wavelet transform apply to transforms by sums of affine representations as well. A construction of admissible vectors, bandlimited admissible vectors and frames for sums of affine representations by means of transversals is given. A class of subgroups of the symplectic group which are compact extensions affine groups are identified, and it is shown how subrepresentations of the metaplectic representation can be equivalent to affine representations. This equivalence is illuminated by several examples.

Motivated by long-time dynamic analysis of hybrid systems and safety verification problems, this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications. Necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long-time oscillatory dynamic analysis. A linear program based algorithm is proposed to verify these conditions, and its computational complexity is analyzed. The positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems. Further, the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets.

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering.. In this work, the notion of the trivariate generalized multiresolut- ion structure (TGMS) of subspace is proposed. The characteristics of trivariate affine pseudoframes for subspaces is investigated. Construction of a GMS of Paley-Wiener subspace of is studied. The pyramid decomposition scheme is obtained based on such a TGMS and a sufficient condition for its existence is provided. A constructive method for affine frames of based on a TGMS is established.

We present in this article inductive schemes that allow to classify all those affine hypersurfaces with affine normal parallel second fundamental (cubic) form, for every possible dimensional case greater or equal than two. The solution proposed is a follow up to previous works on the same topic by this author and it uses, firstly, the reduction of the problem, eminently geometric, to the classification of a certain class of solutions to an equation of Monge-Ampere type. Then it is applied the so-called “method of algorithmic sequence of coordinate changes”, in order to achieve the latter.

The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates (i.e., solves the word problem). This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted to those known from the assumptions and cases. This “subterm property” is proved by permuting suitably the order of application of the geometric rules. As an example of the decision method, it is shown that there cannot exist a derivation of Euclid’s fifth postulate if the rule that corresponds to the uniqueness of the parallel line construction is taken away from the system of plane affine geometry.

Frame theory has been the focus of active research for twenty years, both in theory and applications. In this work, the notion of the bivariate generalized multiresolution structure (BGMS) of subspace is proposed. The characteristics of bivariate affine pseudoframes for subspaces is investigated. The construction of a GMS of Paley-Wiener subspace of is studied. The pyramid decomposition scheme is obtained based on such a GMS and a sufficient condition for its existence is provided. A constructive method for affine frames of based on a BGMS is established.

In recent years, frames have been the focus of active research, both in theory and applications. In this paper, the notion of multiple affine pseudoframes for subspaces of space is introduced. The concept of a generalized multiresolution structure(GMRS) is proposed. The sufficient condition for the existence of a class of multiple pseudoframes with filter banks is obtained by virtue of a generalized multiresolution structure. An approach for constructing one GMRS of Paley-Wiener subspaces of is presented based on the pyramid decomposition scheme The characteristics of affine pseudoframes for subspaces of space is provided.

Affine moment invariants are important if one wants to recognize the surface of a plane in three dimensions when the orientation of the plane is not known beforehand and only two-dimensional information is available. The notion of generating function is introduced as a simple and straightforward way to derive various affine invariants. By this notion, we can get the explicit construction of much more affine moment invariants. Based on this conclusion, a large set of invariant polynomials can be generated automatically and immediately by the algorithm we have designed. These new affine moment invariants can be applied to recognize the image. Approaches in this paper will improve the practicability of affine invariants in object recognition applications.

In this paper, we investigate a class of affine nonlinear systems with a triangular-like structure and present its necessary and sufficient condition for global controllability, by using the techniques developed by Sun Yimin and Guo Lei recently. Furthermore, we will give two examples to illustrate its application.

This paper discusses two problems: one is some important theories and algorithms of affine bracket algebra; the other is about their applications in mechanical theorem proving. First we give some efficient algorithms including the boundary expanding algorithm which is a key feature in application. We analyze the characteristics of the boundary operator and this is the base for the implementation of the system. We also give some new theories or methods about the exact division, the representations and structure of affine geometry and so on. In practice, we implement the mechanical auto-proving system in Maple 10 based on the above algorithms and theories. Also we test about more than 100 examples and compare the results with the methods before.

We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and intersection commute between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces. Another sufficient condition is that the cosines of the angles between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces, be bounded away from one. Our results are then applied to a general infinite horizon, positive semi-definite, linear quadratic mathematical programming problem. Specifically, under suitable conditions, we show that optimal solutions exist and, modulo those feasible solutions with zero objective value, they are limits of optimal solutions to finite-dimensional truncations of the original problem.

We calculate the projection of the product of the Drinfeld currents on the intersection of the different Borel subalgebras in the current realization of the quantum affine algebra . This projection yields a universal weight function and has the structure of nested Bethe vectors.

This paper discusses the best affine approach (BAA) of multi-output m-valued logical functions. First, it gives the spectra of rate of accordance between multi-output m-valued logical functions and their affine functions, then analyzes the BAA of multi-output m-valued logical functions and finally gives the spectral characteristics of BAA of multi-output m-valued logical functions.

In this poster, we present methods for randomly generating hybrid automata with affine differential equations, invariants, guards, and assignments. Selecting an arbitrary affine function from the set of all affine functions results in a low likelihood of generating hybrid automata with diverse and interesting behaviors, as there are an uncountable number of elements in the set of all affine functions. Instead, we partition the set of all affine functions into potentially interesting classes and randomly select elements from these classes. For example, we partition the set of all affine differential equations by using restrictions on eigenvalues such as those that yield stable, unstable, etc. equilibrium points. We partition the components describing discrete behavior (guards, assignments, and invariants) to allow either time-dependent or state-dependent switching, and in particular provide the ability to generate subclasses of piecewise-affine hybrid automata. Our preliminary experimental results with a prototype tool called HyRG (Hybrid Random Generator) illustrate the feasibility of this generation method to automatically create standard hybrid automaton examples like the bouncing ball and thermostat.

Vectorial Boolean functions are used as substitution boxes in cryptosystems. Designing inequivalent functions resistant to known attacks is one of the challenges in cryptography. In doing this, finding a fast technique for determining whether two given functions are equivalent is a significant problem. A special class of the equivalence called restricted extended affine (REA) equivalence is studied in this paper. We update the verification procedures of the REA-equivalence types given in the recent work of Budaghyan and Kazymyrov (2012). In particular, we solve the system of linear equations simultaneously in the verification procedures to get better complexity. We also present the explicit number of operations of the verification procedures of these REA-equivalence types. Moreover, we construct two new REA-equivalence types and present the verification procedures of these types with their complexities.

This is a review paper on some new inequalities for convex functions of one and several variables. The most important result presented for convex functions of one variable is the extension of Jensen’s inequality to affine combinations. The most interesting results presented for convex functions of several variables refer to inequalities concerning simplexes and its cones. Mathematics subject classification (2010): 26A51, 26B25, 52A20, 52B11.

This paper extends the modified affine arithmetic in matrix form method for\ud bivariate polynomial evaluation and algebraic curve plotting in 2D to modified affine\ud arithmetic in tensor form for trivariate polynomial evaluation and algebraic surface\ud plotting in 3D. Experimental comparison shows that modified affine arithmetic in\ud tensor form is not only more accurate but also much faster than standard affine\ud arithmetic when evaluating trivariate polynomials.

We analyze foundations of the construction of the direct product of affine partial linear spaces, as defined by Johnson and Ostrom. Fundamental preservation theorems are proved, and the role of some of frequently used affine axioms in the context of this theory is discussed.

We define affine pseudo-planes as one class of Q-homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line A 1 . Moreover, we show that there exists an infinite-dimensional family of the universal coverings of affine psendo-planes with a cyclic group acting as the Galois group, which have the equivariant non-cancellation property. Our family contains the surfaces without the cancellation property, due to Danielewski-Fieseler and torn Dieck.

The minimization control problem of the quadratic functionals for the class of affine nonlinear systems in the hypothesis of nilpotent associated Lie algebra is analyzed. The optimal control correspondent to the first-, second- and third-order nilpotent operator is determined. H. Bourdache-Siguerdinljane has applied the method of Lie algebras to the study of optimal control regulation of satellites. S. Banks and M. Yew have studied the optimal control of energy consumption minimization for a class of bilinear systems and J.S. Liu et al., have generalized this result for the class of affine nonlinear systems. The optimal control determination of nonlinear system with a nilpotent structure minimizing the quadratic functionals generalizes the results of J.S. Liu, K. Yuan, W.S. Lin, and S.P. Banks, and M.K. Yew regarding the energy minimization of the affine nonlinear and bilinear systems.

In this paper, we will introduce the concept of affine frame in wavelet analysis to the field of p-adic number, hence provide new mathematic tools for application of p-adic analysis.

A lively example to use in a first course in linear algebra to clarify vector space notions is the space of square matrices of fixed order with its subspaces of affine, coaffine, doubly affine, and magic squares. In this note, the projection theorem is illustrated by explicitly constructing the orthogonal projections (in closed forms) of any matrix U onto these subspaces. The results follow directly from a canonical decomposition of U.

We introduce and study the equiaffine symmetric {\bf hyperspheres}. For the first step we consider the locally strongly convex ones. In fact, by the idea used by Naitoh, we provide in this paper a direct proof of the complete classification for those affine symmetric hyperspheres. Then, via an earlier result of the first author, we are able to provide an alternative proof for the classification theorem of the affine hypersurface with parallel Fubini-Pick forms, which has already been established by Z.J. Hu et al in a totally different way., 19 pages, already submitted in May, 2014. arXiv admin note: substantial text overlap with arXiv:1303.4278, arXiv:1401.0860

We introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine W-algebra via the complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible λ-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we describe explicit forms of free generators and compute λ-brackets between them. Provided some assumptions on a classical affine fractional W-algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.

This paper is meant to be a short review and summary of recent results on the structure of finite and affine classical W-algebras, and the application of the latter to the theory of generalized Drinfeld-Sokolov hierarchies.

Affine spheres are discussed in the context of loop groups. We show that every affine sphere can be obtained by solving two ordinary differential equations followed by an application of a generalized Birkhoff Decomposition Theorem (which we proof in the Appendix). A geometric interpretation of the coefficients of the ODE is given. Finally the method is applied to construct all ruled surfaces.