Monotone Norm Research Articles

Location problem and inner product spaces

In this work we solve a problem that has been open for more than 110 years (see [21]). We prove that a real normed space (X,‖⋅‖) of dimension greater than or equal to three is an inner product space if and only if, for every three points a1,a2,a3∈X, the set of points at which the function x∈X→γ(‖x−a1‖,‖x−a2‖,‖x−a3‖) attains its minimum, intersects the convex hull of these three points, where γ is a symmetric monotone norm on R3.

Monotonic norms and orthogonal issues in multidimensional voting

We study issue-by-issue voting by majority and incentive compatibility in multidimensional frameworks where privately informed agents have preferences induced by general norms and where dimensions are endogenously chosen. We uncover the deep connections between dominant strategy incentive compatibility (DIC) on the one hand, and several geometric/functional analytic concepts on the other. Our main results are: 1) Marginal medians are DIC if and only if they are calculated with respect to coordinates defined by a basis such that the norm is orthant-monotonic in the associated coordinate system. 2) Equivalently, marginal medians are DIC if and only if they are computed with respect to a basis such that, for any vector in the basis, any linear combination of the other vectors is Birkhoff-James orthogonal to it. 3) We show how semi-inner products and normality provide an analytic method that can be used to find all DIC marginal medians. 4) As an application, we derive all DIC marginal medians for lp spaces of any finite dimension, and show that they do not depend on p (unless p=2).

A remark on a paper of P. B. Djakov and M. S. Ramanujan

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\{o}the matrices when every continuous linear operator between l-K\{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

Connections Between Spectral Properties of Asymptotic Mappings and Solutions to Wireless Network Problems

In this study we establish connections between asymptotic functions and properties of solutions to important problems in wireless networks. We start by introducing a class of self-mappings (called asymptotic mappings) constructed with asymptotic functions, and we show that spectral properties of these mappings explain the behavior of solutions to some maxmin utility optimization problems. For example, in a common family of max-min utility power control problems, we prove that the optimal utility as a function of the power available to transmitters is approximately linear in the low power regime. However, as we move away from this regime, there exists a transition point, easily computed from the spectral radius of an asymptotic mapping, from which gains in utility become increasingly marginal. From these results we derive analogous properties of the transmit energy efficiency. In this study we also generalize and unify existing approaches for feasibility analysis in wireless networks. Feasibility problems often reduce to determining the existence of the fixed point of a standard interference mapping, and we show that the spectral radius of an asymptotic mapping provides a necessary and sufficient condition for the existence of such a fixed point. We further present a result that determines whether the fixed point satisfies a constraint given in terms of a monotone norm.

Fixed-Point Theorems for Generalized Nonexpansive Mappings

Using the concept of asymptotic center we obtain the existence of fixed points having preassigned location for a wider class of asymptotic nonexpansive mappings in a uniformly convex Banach space. This generalization leads us to get a recent result of Alfuraidan and Khamsi for continuous monotone asymptotic nonexpansive mappings as well as the classical fixed-point result of Geobel and Kirk for asymptotic nonexpansive mappings in a uniformly convex Banach space. Also we prove a fixed-point theorem for order preserving continuous maps on a quasiordered closed convex subset of a uniformly convex Banach sapce having monotone norm.

GEOMETRIC AND FIXED POINT PROPERTIES IN PRODUCTS OF NORMED SPACES

Given two (real) normed (linear) spaces $X$ and $Y$, let $X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$, where $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$. It is known that $X\otimes _{1}Y$ is $2$-UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$-dimensional and $Y$ is $k$-UR, then $X\otimes _{1}Y$ is $(m+k)$-UR. In the other direction, we observe that if $X\otimes _{1}Y$ is $k$-UR, then both $X$ and $Y$ are $(k-1)$-UR. Given a monotone norm $\Vert \cdot \Vert _{E}$ on $\mathbb{R}^{2}$, we let $X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$ where $\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$. It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\Vert \cdot \Vert _{E}$ is strictly monotone, then $X\otimes _{E}Y$ has WFPP. Using the notion of $k$-uniform rotundity relative to every $k$-dimensional subspace we show that this result holds with a weaker condition on $X$.

Generalized vector valued paranormed sequence space using Orlicz function

In this paper, we introduced a vector valued paranormed space $$X(E,\Delta ^m,M,p,s)$$ using Orlicz function M. Class of vector valued sequences $$X(E,\Delta ^m,M,p,s)$$ is defined as $$\begin{aligned} X(E,\Delta ^m,M,p,s)&= \Big \{x\in W(E):\left( |\nu _k|^{-s} \left[ M\left( \frac{q(\Delta ^m x_k)}{\rho }\right) \right] ^{p_k}\right) \in X\\&\quad \, \, \mathrm{ for\, some}\, \rho >0,\, s\geqslant 0\Big \}, \end{aligned}$$where (E, q) is a seminormed space, $$p=(p_k)$$ is a bounded sequence of positive real numbers such that $$\displaystyle \inf _k p_k > 0$$, $$\begin{aligned} \Delta ^m x_k=\sum _{i=0}^m (-1)^i\left( \begin{array}{c} m \\ i \\ \end{array} \right) x_{k+i},\, m\in \mathbb{N}_0=\{0,1,2,3,\ldots \}, \end{aligned}$$$$W(E)=\{x=(x_k): x_k \in E\}$$ is a linear space under operations of vector addition and scalar multiplication, and $$\nu = (\nu_k)$$ is a bounded sequence of real or complex numbers such that $$\displaystyle \inf _k|\nu _k|>0$$. Further, let X be a normal sequence algebra with absolutely monotone norm $$\Vert \cdot \Vert _X$$ and having a Schauder basis $$(e_k)$$, where $$e_k=(0,0,\ldots ,0,1,0,\ldots )$$ with 1 in kth place. The topology on $$X(E,\Delta ^m,M,p,s)$$ is introduced with the help of paranorm g, which is given by $$\begin{aligned} g(x)=\sum _{i=1}^m q(x_i)+\inf \left\{ \rho ^{(p_n/H)}:\left\| \left( |\nu _k|^{-s} \left[ M\left( \frac{q\left( \Delta ^m x_k\right) }{\rho }\right) \right] ^{p_k}\right) \right\| _X ^{1/H} \leqslant 1,\, n \in \mathbb{N}\right\} , \end{aligned}$$where $$x\in X(E,\Delta ^m,M,p,s)$$ and $$\displaystyle H=\max (1,\sup _k p_k)$$. Completeness, normality, inclusion relations etc. for this space is obtained. The results of this paper includes, as particular case, some of the known scalar and vector valued sequence spaces.

COMPACT FACTORIZATION OF OPERATORS WITH λ-COMPACT ADJOINTS

AbstractLet λ be a symmetric, normal sequence space equipped with a k-symmetric, monotone norm ‖.‖λ. Also, assume that (λ, ‖.‖λ) is AK-BK. Corresponding to this sequence space λ, we study compactness of the operator ideal Kλ. We proved compactness, completeness and injectivity of the dual operator ideal Kλd. We also investigate the factorization of these operators.

Monotone norms and Finsler structures in noncommutative spaces

We study Orlicz functionals defined on noncommutative algebras and associated to a weight. We show that large class of them has a joint monotonicity property with respect to completely positive maps.

Making use of a partial order in solving inverse problems: II.

Mathematical formulations of applied inverse problems often involve operator equations in normed functional spaces. In many cases, these spaces can, in addition, be endowed with a partial order relation, which turns them into Banach lattices. The fact that two tools, such as a partial order relation and a monotone (with respect to this partial order) norm, are available to the researchers, gives them a clear advantage of having more freedom in the problem formulations. For instance, errors in the approximate data are sometimes easier to describe in terms of pointwise bounds. Inverse problems in partially ordered normed spaces (Banach lattices) have been studied before in the case when a compact set containing the unknown exact solution is available a priori. It turned out that under this assumption it is possible, even in the ill-posed case, to compute ‘pointwise’ bounds for the unknown exact solution (or rather, bounds by means of the appropriate partial order), thus providing an error estimate by means of the partial order. Also, a useful property of this approach was that one was able to quantify the uncertainty in the operator by means of linear inequalities that were included in the corresponding optimization problems as (linear) constraints and made the computations easier. However, the compactness assumption might sometimes be too demanding in practice. This paper aims at revealing the possibilities and advantages of using partial order in solving inverse problems in the case when no compact set of prior restrictions is available, concentrating on linear inverse (possibly ill-posed) problems.

Nonlinear extensions of the Perron–Frobenius theorem and the Krein–Rutman theorem

A unification version of the Perron–Frobenius theorem and the Krein–Rutman theorem for increasing, positively 1-homogeneous, compact mappings is given on ordered Banach spaces without monotonic norm. A Collatz-type minimax characterization of the positive eigenvalue with positive eigenvector is obtained. The power method in computing the largest eigenpair is also extended.

On nonlinear simultaneous approximation problems

This paper deals with best nonlinear simultaneous approximation problems to two continuous functions under a general monotonic norm on . Characterizations results of Kolmogorov type and of alternation type and uniqueness results for the best simultaneous approximation are established. Applications are given to the set of rational functions.

THE FIXED POINT PROPERTY IN DIRECT SUMS AND MODULUS

AbstractWe show that the direct sum $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M({X}_{i} )\gt 1$ for each $i= 1, \ldots , r$. In particular, $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ enjoys the fixed point property if Banach spaces ${X}_{i} $ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for $r= 2$: a direct sum ${X}_{1} {\mathop{\oplus }\nolimits}_{\psi } {X}_{2} $ of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended to asymptotically nonexpansive mappings in the intermediate sense.

Accelerated predictive norm-optimal iterative learning control

This paper proposes a novel technique for accelerating the convergence of the previously published predictive norm-optimal iterative learning control (NOILC) methodology. The basis of the results is a formal proof that the predictive NOILC algorithm is equivalent to a successive projection algorithm between linear varieties in a suitable product Hilbert space. This leads to two proposed accelerated algorithms together with well-defined convergence properties. The results show that the proposed accelerated algorithms are capable of ensuring monotonic error norm reductions and can outperform predictive NOILC by more rapid reductions in error norm from iteration to iteration. In particular, examples indicate that the approach can improve the performance of predictive NOILC for the problematic case of non-minimum phase systems. Realization of the algorithms is discussed and numerical simulations are provided for comparative purposes and to demonstrate the numerical performance and effectiveness of the proposed methods.

On the fixed points of nonexpansive mappings in direct sums of Banach spaces

We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum of X and Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional.

Accelerated norm-optimal iterative learning control

This paper proposes a novel technique for accelerating the convergence of the previously published norm-optimal iterative learning control (NOILC) methodology. The basis of the results is a formal proof of an observation made by the first author, namely that the NOILC algorithm is equivalent to a successive projection algorithm between linear varieties in a suitable product Hilbert space. This leads to two proposed accelerated algorithms together with well-defined convergence properties. The results show that the proposed accelerated algorithms are capable of ensuring monotonic error norm reductions and can outperform NOILC by more rapid reductions in error norm from iteration to iteration. In particular, examples indicate that the approach can improve the performance of NOILC for the problematic case of non-minimum phase systems. Realisation of the algorithms is discussed and numerical simulations are provided for comparative purposes and to demonstrate the numerical performance and effectiveness of the proposed methods.

Fixed point theorems for nonexpansive mappings and Suzuki-generalized nonexpansive mappings on a Banach lattice

We prove the existence of fixed points for multivalued nonexpansive nonself-mappings on a weakly orthogonal reflexive Banach lattice with uniformly monotone norm. Moreover, for single-valued mappings, we extend Betiuk-Pilarska and Prus’s result [A. Betiuk-Pilarska, S. Prus, Banach lattices which are order uniformly noncreasy, J. Math. Anal. Appl. 342 (2008) 1271–1279] on the weak fixed point property to continuous mappings satisfying condition (C) on a w -weakly orthogonal OUNC Banach lattice.

Continuity of the generalized spectral radius in max algebra

Let ‖ · ‖ be an induced matrix norm associated with a monotone norm on R n and β be the collection of all nonempty closed and bounded subsets of n × n nonnegative matrices under this matrix norm. For Ψ , Φ ∈ β , the Hausdorff metric H between Ψ and Φ is given by H ( Ψ , Φ ) = max { sup A ∈ Ψ inf B ∈ Φ ‖ A - B ‖ , sup B ∈ Φ inf A ∈ Ψ ‖ A - B ‖ } . The max algebra system consists of the set of nonnegative numbers with sum a ⊗ b = max { a , b } and the standard product ab for a , b ⩾ 0 . For n × n nonnegative matrices A , B their product is denoted by A ⊗ B , where [ A ⊗ B ] ij = max 1 ⩽ k ⩽ n a ik b kj . For each Ψ ∈ β , the max algebra version of the generalized spectral radius of Ψ is μ ( Ψ ) = limsup m → ∞ [ sup A ∈ Ψ ⊗ m μ ( A ) ] 1 m , where Ψ ⊗ m = { A 1 ⊗ A 2 ⊗ ⋯ ⊗ A m : A i ∈ Ψ } . Here μ ( A ) is the maximum circuit geometric mean. In this paper, we prove that the max algebra version of the generalized spectral radius is continuous on the Hausdorff metric space ( β , H ) . The notion of the max algebra version of simultaneous nilpotence of matrices is also proposed. Necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well.

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.

On a type of stability of a multicriteria integer linear programming problem in the case of a monotone norm

A multicriteria integer linear programming problem with a finite number of admissible solutions is considered. The problem consists in finding the Pareto set. Lower and upper attainable estimates of the radius of strong stability of the problem are obtained in the case when the norm in the space of solutions is arbitrary, and the norm in the criteria space is monotone. Using the Minkowski-Mahler inequality, a formula for calculating this radius is derived in the case when the Pareto set consists of a single solution. Estimates of the radius are also found in the case of the Holder norm in the specified spaces. A class of problems is distinguished for which the radius of strong stability is infinite. As corollaries, certain results known earlier are derived. Illustrative numerical examples are also presented.