Complexity of Linear Problems with a Fixed Output Basis

We use an information-based complexity approach to study the complexity of approximation of linear operators. We assume that the values of a linear operator may be elements of an infinite dimensional space G. It seems reasonable to look for an approximation as a linear combination of some elements g_i from the space G and compute only the coefficients c_i of this linear combination. We study the case when the elements g_i are fixed and compare it with the case where the g_i can be chosen arbitrarily. We show examples of linear problems where a significant output data compression is possible by the use of a nonlinear algorithm, and this nonlinear algorithm is much better than all linear algorithms. We also provide an example of a linear problem for which one piece of information is enough whereas an optimal (minimal cost) algorithm must use information of much higher cardinality.     

Aglianò-Montagna type decomposition of linear pseudo hoops and its applications

We decompose every linear pseudo hoop as an Aglianò-Montagna type of ordinal sum of linear Wajsberg pseudo hoops which are either negative cones of linear ?-groups or intervals in linear unital ?-groups with strong unit. We apply the decomposition to present a new proof that every linear pseudo BL-algebra and consequently every representable pseudo BL-algebra is good. Moreover, we show that every maximal filter and every value of a linear pseudo hoop is normal, and every σ-complete linear pseudo hoop is commutative.     

On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261–267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, q ^h ) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, q ^h ).     

On approximation of approximately linear mappings by linear mappings

Assume A is a normed linear space, B is a Banach space, and f: A → B is a mapping “approximately linear.” We solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.”     

Probability of unique integer solution to a system of linear equations

We consider a system of m linear equations in n variables Ax = d and give necessary and sufficient conditions for the existence of a unique solution to the system that is integer: x ∈ {−1, 1}ⁿ. We achieve this by reformulating the problem as a linear program and deriving necessary and sufficient conditions for the integer solution to be the unique primal optimal solution. We show that as long as m is larger than n/2, then the linear programming reformulation succeeds for most instances, but if m is less than n/2, the reformulation fails on most instances. We also demonstrate that these predictions match the empirical performance of the linear programming formulation to very high accuracy.     

Linear independence of digamma function and a variant of a conjecture of Rohrlich

Let ψ(x) denote the digamma function. We study the linear independence of ψ(x) at rational arguments over algebraic number fields. We also formulate a variant of a conjecture of Rohrlich concerning linear independence of the log gamma function at rational arguments and report on some progress. We relate these conjectures to non-vanishing of certain L-series.     

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.     

Modèles linéaires généralisés fonctionnels avec dérivée

We consider the functional generalized linear model whose response function is a linear operator depending on an explanatory variable X belonging to a functional space. It has been studied, among others, by Cardot and Sarda [4]. In this paper, we consider the functional generalized linear model with derivative component, denoted MLGFD in the following, whose response function depends on a linear operator of X and on its derivative. We propose estimators for the unknown functional parameters and provide convergence rates.     

Boolean-rank-preserving operators and Boolean-rank-1 spaces

We study the extent to which certain theorems on linear operators on field-valued matrices carry over to linear operators on Boolean matrices. We obtain analogues and near analogues of several such theorems. One of these leads us to consider linear spaces of m × n Boolean matrices whose nonzero members all have Boolean rank 1. We obtain a structure theorem for such spaces that enables us to determine the maximum Boolean dimension of such spaces and their maximum cardinality.     

On linear maps preserving generalized invertibility and related properties

Let H be an infinite-dimensional complex Hilbert space, B(H) be the algebra of all bounded linear operators on H. We study surjective linear maps on B(H) preserving generalized invertibility. We also investigate surjective linear maps preserving Fredholm (respectively, semi-Fredholm) operators. Our results improve those of Mbekhta, Rodman and Šemrl.     

Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution

In this paper we extend one direction of Fröberg?s theorem on a combinatorial classification of quadratic monomial ideals with linear resolutions. We do this by generalizing the notion of a chordal graph to higher dimensions with the introduction of d-chorded and orientably-d-cycle-complete simplicial complexes. We show that a certain class of simplicial complexes, the d-dimensional trees, correspond to ideals having linear resolutions over fields of characteristic 2 and we also give a necessary combinatorial condition for a monomial ideal to be componentwise linear over all fields.     

Coefficient quantization for frames in Banach spaces

Let (ei) be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces.     

The geometry of linear separability in data sets

We study the geometry of datasets, using an extension of the Fisher linear discriminant to the case of singular covariance, and a new regularization procedure. A dataset is called linearly separable if its different clusters can be reliably separated by a linear hyperplane. We propose a measure of linear separability, easily computed as an angle that arises naturally in our analysis. This angle of separability assumes values between 0 and π/2, with high [resp. low] values corresponding to datasets that are linearly separable, resp. inseparable.     

On Linearly Kleiman Groups

Let V denote a finite-dimensional vector space over some field. We say that a linear group G ≤ GL(V) is a linearly Kleiman group if, for every pair of linear subspaces v, u ≤ V, there is an element g ∈ G such that the subspaces g(v), u are in general position. The main result of this paper is the classification of connected linear algebraic groups over a field of characteristic zero which are linearly Kleiman. We also consider some properties of linearly Kleiman groups.     

Linear systems on tropical curves

A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve Γ analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from Γ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to deg(D) when |D| is base point free. The tropical convex hull of the image realizes the linear system |D| as a polyhedral complex. We show that curves for which the canonical divisor is not very ample are hyperelliptic. We also show that the Picard group of a ${\mathbb{Q}}$ -tropical curve is a direct limit of critical groups of finite graphs converging to the curve.     

Asymptotic efficiency of ridge estimator in linear and semiparametric linear models

The linear model with a growing number of predictors arises in many contemporary scientific endeavor. In this article, we consider the commonly used ridge estimator in linear models. We propose analyzing the ridge estimator for a finite sample size n and a growing dimension p. The existence and asymptotic normality of the ridge estimator are established under some regularity conditions when p→∞. It also occurs that a strictly linear model is inadequate when some of the relations are believed to be of certain linear form while others are not easily parameterized, and thus a semiparametric partial linear model is considered. For these semiparametric partial linear models with p>n, we develop a procedure to estimate the linear coefficients as if the nonparametric part is not present. The asymptotic efficiency of the proposed estimator for the linear component is studied for p→∞. It is shown that the proposed estimator of the linear component asymptotically performs very well.     

About computation of the linear complexity of generalized cyclotomic sequences with period p n+1

We propose a computation method for linear complexity of series of generalized cyclotomic sequences with period p ⁿ⁺¹. This method is based on using the polynomial of the classic cyclotomic sequences of period p. We found the linear complexity of generalized cyclotomic sequences corresponding to the classes of biquadratic residues and Hall sequences.     

Robustness of general dichotomies

For a linear equation v^′=A(t)v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates ecρ(t) for some function ρ(t). This includes as a special case the usual exponential behavior when ρ(t)=t. We also consider the general case of nonuniform exponential dichotomies. We establish the robustness of the exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy.     

Linear mappings preserving ρ-orthogonality

We define a ρ-orthogonality in a real normed space and we consider the class of linear mappings preserving this relation. We show that a linear mapping preserving ρ-orthogonality has to be a similarity, i.e., a scalar multiple of an isometry. As a result, we give a characterization of smooth spaces in terms of this orthogonality.     

Center manifolds for impulsive equations under nonuniform hyperbolicity

We establish the existence of smooth center manifolds under sufficiently small perturbations of an impulsive linear equation. In particular, we obtain the C¹ smoothness of the manifolds outside the jumping times. We emphasize that we consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential trichotomy.