arXiv subjects

Adam Paszkiewicz

Publications and source records attributed to Adam Paszkiewicz.

10 recordsLinked to original sources

arXiv2024

The Convex Peano Curve Does Exist

We refer here to the surprising construction made by Giuseppe Peano in 1890. He gave an example of a continuous function (called now the Peano curve) from the unit interval to the whole unit square. We show here the existence of a more general space-filling curve with additional convexity properties. More precisely: by T ? R2 we denote a convex closed and bounded set and we show that there exists a continuous surjection f : [0; 1] 7! T for which the image of any interval is a convex set.

math.MG↗

arXiv2015

Strange products of projections

Let $H$ be an infinite dimensional Hilbert space. We show that there exist three orthogonal projections $X_1, X_2, X_3$ onto closed subspaces of $H$ such that for every $0\ne z_0\in H$ there exist $k_1, k_2,\dots \in \{1,2,3\}$ so that the sequence of iterates defined by $z_n= X_{k_n} z_{n-1}$ does not converge in norm.

math.FA↗

arXiv2014

Sums of compositions of pairs of projections

We give some necessary and sufficient conditions for the possibility to represent a Hermitian operator on an infinite-dimensional Hilbert space (real or complex) in the form $\sum_{i=1}^nQ_iP_i$, where $P_1,\dots,P_n$, $Q_1,\dots,Q_n$ are orthogonal projections. We show that the smallest number $n=n(c)$ admitting the representation $x=\sum_{i=1}^{n(c)}Q_iP_i$ for every $x=x^*$ with $\|x\|\leq c$ satisfies $8c+\frac83\leq n(c)\leq 8c+10$. This is a partial answer to the question asked by L. W. Marcoux in 2010.

math.FA↗

arXiv2014

On the stability of the existence of fixed points for the projection-iterative methods with relaxation

We consider an $\alpha$-relaxed projection $P_A^\alpha:H\to H$ given by $P_A^\alpha(x)=\alpha P_A(x)+(1-\alpha)x$ where $\alpha\in[0,1]$ and $P_A$ is the projection onto a non-empty, convex and closed subset $A$ of the real Hilbert space $H$. We characterise all the sets $F\subset[0,1]$ such that for some non-empty, convex and closed subsets $A_1,A_2,\dots,A_k\subset H$ the composition $P_{A_k}^\alpha P_{A_{k-1}}^\alpha\dots P_{A_1}^\alpha$ has a fixed point iff $\alpha\in F$. It proves, that if $\dim H\geq 3$ and $k\geq3$ then the class of the derscribed above sets $F$ of coefficients $\alpha$ is exactly the class of $F_\sigma$ subsets of $[0,1]$ containing $0$.

math.FA↗

arXiv2012

Large free linear algebras of real and complex functions

Let $X$ be a set of cardinality $\kappa$ such that $\kappa^\omega=\kappa$. We prove that the linear algebra $\mathbb{R}^X$ (or $\mathbb{C}^X$) contains a free linear algebra with $2^\kappa$ generators. Using this, we prove several algebrability results for spaces $\mathbb{C}^\mathbb{C}$ and $\mathbb{R}^\mathbb{R}$. In particular, we show that the set of all perfectly everywhere surjective functions $f:\mathbb{C}\to\mathbb{C}$ is strongly $2^\mathfrak{c}$-algebrable. We also show that the set of all functions $f:\mathbb{R}\to\mathbb{R}$ whose sets of continuity points equals some fixed $G_\delta$ set $G$ is strongly $2^\mathfrak{c}$-algebrable if and only if $\mathbb{R}\setminus G$ is $\mathfrak{c}$-dense in itself.

math.RA↗

arXiv2012

In the Amemiya-Ando problem 3 is enough

In any infinite dimensional Hilbert space $\mathcal H$ there exist orthogonal projections $Q_1$, $Q_2$ and $Q_3$, such that a sequence $(P_n... P_1(x))$ diverges in norm for some $P_1,P_2,...\in\{Q_1,Q_2,Q_3\}$ and $x\in\mathcal H$.

math.FA↗

arXiv2012

On quantum information

We investigate the following generalisation of the entropy of quantum measurement. Let H be an infinite-dimensional separable Hilbert space with a 'density' operator {\rho}, tr {\rho}=1. Let I(P)\in R be defined for any partition P = (P_1,...,P_m), P_1+ ... +P_m=1_H, P_i \in proj H$ and let I(P_i Qj, i \leq m, j \leq n) = I(P) + I(Q) for Q =(Q_1,..., Q_n), \sum Q_j = 1_H and P_iQ_j = Q_j P_i, tr {\rho} P_iQ_j = tr {\rho} P_i tr {\rho} Q_j (P, Q are physically independent). Assuming some continuity properties we give a general form of generalised information I.

cs.IT↗

arXiv2012

The Amemiya-Ando conjecture falls

In any infinite dimensional Hilbert space H, a sequence P_n...P_1 x diverges in norm for some x \in H and orthogonal projections P_n \in {Q_1,..., Q_5}.

math.FA↗

arXiv2012

Additive Entropies of Partitions

We provide, under minimal continuity assumptions, a description of \textsl{additive partition entropies}. They are real functions $I$ on the set of finite partitions that are additive on stochastically independent partitions in a given probability space.

cs.IT↗

arXiv2005

On complete characterization of coefficients of a.e. converging orthogonal series

We characterize sequences of numbers $(a_n)$ such that $\sum_{n\geq 1} a_nΦ_n$ converges a.e. for any orthonormal system $(Φ_n)$ in any $L_2$-space. In our criterion, we use the set $B =\{\sum_{m\geq n} |a_m|^2; n\geq 1\}$ and its information function $$h_B(t) = -\log_3(β-α)$$ for $t\in (α, β]$, $[α, β]\cap B =\{α, β\}.$

math.AP↗