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Wojciech Szymanski

Publications and source records attributed to Wojciech Szymanski.

At least 19 recordsLinked to original sources

Mind the Gap: Mesh-Guided Repair of Broken Vessels

Vessel segmentation is commonly optimized as voxel-wise classification, but small local errors can strongly disrupt vascular connectivity while having little effect on overlap scores. This is particularly problematic for downstream analyses that rely on centerlines, branches, connected components, or graph structure. We propose a mesh-guided post-processing framework for repairing broken vessel segmentations produced by nnU-Net. For each predicted binary mask, a deformable template mesh is fitted to the mask surface in physical space and used as a case-specific geometric scaffold. The fitted mesh is not voxelized as the final segmentation; instead, it guides conservative reconnection of disconnected components by proposing or validating thin bridge candidates under foreground-growth constraints. We evaluated this approach in three vascular anatomies using AortaSeg24 and SEGA for the aorta, TopCoW for the Circle of Willis, and PARSE for the pulmonary arteries. Performance is measured using Dice, connected-component Dice (ccDice), and the Betti-0 number. Across these datasets, repair substantially improved connectivity while preserving overlap: Dice remained nearly unchanged, whereas ccDice increased from 0.596 to 0.992 for aorta, from 0.722 to 0.835 for TopCoW, and from 0.028 to 0.862 for PARSE. The FOMAML meta-initialization further accelerated the fitting per-case, supporting practical mesh-based repair of the vascular topology. These results suggest that explicit mesh representations can provide a useful geometric prior for correcting topological failures in otherwise accurate voxel segmentations.

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The action of the Thompson group F on infinite trees

We construct an action of the Thompson group F on a compact space built from pairs of infinite, binary rooted trees. The action arises as an F-equivariant compactification of the action of F by translations on one of its homogeneous spaces, F/H_2, corresponding to a certain subgroup H_2 of F. The representation of F on the Hilbert space l^2(F/H_2) is faithful on the complex group algebra C[F].

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Endomorphisms of the Cuntz Algebras and the Thompson Groups

We investigate the relationship between endomorphisms of the Cuntz algebra ${\mathcal O}_2$ and endomorphisms of the Thompson groups $F$, $T$ and $V$ represented inside the unitary group of ${\mathcal O}_2$. For an endomorphism $λ_u$ of ${\mathcal O}_2$, we show that $λ_u(V)\subseteq V$ if and only if $u\in V$. If $λ_u$ is an automorphism of ${\mathcal O}_2$ then $u\in V$ is equivalent to $λ_u(F)\subseteq V$. Our investigations are facilitated by introduction of the concept of modestly scaling endomorphism of ${\mathcal O}_n$, whose properties and examples are investigated.

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On Conjugacy of MASAs in Graph $C^*$-Algebras

For a large class of finite graphs $E$, we show that whenever $α$ is a vertex-fixing quasi-free automorphism of the corresponding graph $C^*$-algebra $C^*(E)$ such that $α({\mathcal D}_E) \neq{\mathcal D}_E$, where ${\mathcal D}_E$ is the canonical MASA in $C^*(E)$, then $α({\mathcal D}_E)\neq w{\mathcal D}_E w^*$ for all unitaries $w\in C^*(E)$. That is, the two MASAs ${\mathcal D}_E$ and $α({\mathcal D}_E)$ of $C^*(E)$ are outer but not inner conjugate. Passing to an isomorphic $C^*$-algebra by changing the underlying graph makes this result applicable to certain non quasi-free automorphisms as well.

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On Endomorphisms of the Cuntz Algebra which Preserve the Canonical UHF-Subalgebra, II

It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms $λ_u$ of the Cuntz algebra $\mathcal{O}_n$ such that $λ_u (\mathcal{F}_n)\subseteq\mathcal{F}_n$ but $u\not\in\mathcal{F}_n$, and a question was raised if for such a $u$ there must always exist a unitary $v\in\mathcal{F}_n$ with $λ_u|_{\mathcal{F}_n} = λ_v|_{\mathcal{F}_n}$. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms $λ_u$ for which the relative commutant $λ_u(\mathcal{F}_n)'\cap\mathcal{F}_n$ is finite dimensional.

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Visualizing automorphisms of graph algebras

Graph C*-algebras have been celebrated as C*-algebras that can be seen, because many important properties may be determined by looking at the underlying graph. This paper introduces the permutation graph for a permutative endomorphism of a graph C*-algebra as a labeled directed multigraph that gives a visual representation of the endomorphism and facilitates computations. Combinatorial criteria have previously been developed for deciding when such an endomorphism is an automorphism, but here the question is reformulated in terms of the permutation graph and new proofs are given. Furthermore, it is shown how to use per- mutation graphs to efficiently generate exhaustive collections of permutative automorphisms. Permutation graphs provide a natural link to the textile systems representing induced endo- morphisms on the edge shift of the given graph, and this allows the powerful tools of the theory of textile systems developed by Nasu to be applied to the study of permutative endomorphisms.

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On Cohomology for Product Systems

A cohomology for product systems of Hilbert bimodules is defined via the Ext functor. For the class of product systems corresponding to irreversible algebraic dynamics, relevant resolutions are found explicitly and it is shown how the underlying product system can be twisted by the 2-cocycles. In particular, this process gives rise to cohomological deformations of the C*-algebras associated with the product system. Concrete examples of deformations of the Cuntz's algebra Q_N arising this way are investigated and we show they are simple and purely infinite.

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On Conjugacy of MASAs and the Outer Automorphism Group of the Cuntz Algebra

We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of MASAa in O_n. In particular, we exhibit an uncountable family of MASAs, conjugate to the standard MASA D_n via Bogolubov automorphisms, that are not inner conjugate to D_n.

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The Weyl group of the Cuntz algebra

The Weyl group of the Cuntz algebra O_n, with n finite, is investigated. This is (isomorphic to) the group of polynomial automorphisms of O_n, namely those induced by unitaries that can be written as finite sums of words in the canonical generating isometries and their adjoints. A necessary and sufficient algorithmic combinatorial condition is found for deciding when a polynomial endomorphism restricts to an automorphism of the canonical diagonal MASA. Some steps towards a general criterion for invertibility of such endomorphisms on the whole of O_n are also taken. A condition for verifying invertibility of a certain subclass of polynomial endomorphisms is given. First examples of polynomial automorphisms of O_n not inner related to permutative ones are exhibited, for every n. In particular, the image of the Weyl group in the outer automorphism group of O_n is strictly larger than the image of the reduced Weyl group analyzed in previous papers. Results about the action of the Weyl group on the spectrum of the diagonal are also included.

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Endomorphisms of graph algebras

We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D_E of C*(E). Our results pertain both automorphisms and proper endomorphisms. Firstly, the Weyl group and the restricted Weyl group of a graph C*-algebra are introduced and investigated. In particular, criteria of outerness for automorphisms in the restricted Weyl group are found. We also show that the restriction to the diagonal MASA of an automorphism which globally preserves both the diagonal and the core AF-subalgebra eventually commutes with the corresponding one-sided shift. Secondly, we exhibit several properties of proper endomorphisms, investigate invertibility of localized endomorphisms both on C*(E) and in restriction to D_E, and develop a combinatorial approach to analysis of permutative endomorphisms.

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The Cuntz Algebra Q_N and C*-Algebras of Product Systems

We consider a product system over the multiplicative semigroup N^x of Hilbert bimodules which is implicit in work of S. Yamashita and of the second named author. We prove directly, using universal properties, that the associated Nica-Toeplitz algebra is an extension of the C^*-algebra Q_N introduced recently by Cuntz.

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Automorphisms of the Cuntz algebras

We survey recent results on endomorphisms and especially on automorphisms of the Cuntz algebras O_n, with a special emphasis on the structure of the Weyl group. We discuss endomorphisms globally preserving the diagonal MASA and their corresponding actions. In particular, we investigate those endomorphisms of O_n which restrict to automorphisms of the diagonal. We review a combinatorial approach to the study of permutative endomorphisms. All the presented material is put in context with current research topics.

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Endomorphisms of the Cuntz Algebras

This mainly expository article is devoted to recent advances in the study of dynamical aspects of the Cuntz algebras O_n, with n finite, via their automorphisms and, more generally, endomorphisms. A combinatorial description of permutative automorphisms of O_n in terms of labeled, rooted trees is presented. This in turn gives rise to an algebraic characterization of the restricted Weyl group of O_n. It is shown how this group is related to certain classical dynamical systems on the Cantor set. An identification of the image in Out(O_n) of the restricted Weyl group with the group of automorphisms of the full two-sided n-shift is given, for prime n, providing an answer to a question raised by Cuntz in 1980. Furthermore, we discuss proper endomorphisms of O_n which preserve either the canonical UHF-subalgebra or the diagonal MASA, and present methods for constructing exotic examples of such endomorphisms.

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Labeled Trees and Localized Automorphisms of the Cuntz Algebras

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras $Ø_n$ which preserve both the diagonal and the core $UHF$-subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of ${\rm Aut}(Ø_n)$ and leads to numerous new examples. In particular, we completely classify all such automorphisms of ${\mathcal O}_2$ for the permutation unitaries in $\otimes^4 M_2$. We show that the subgroup of ${\rm Out}(Ø_2)$ generated by these automorphisms contains a copy of the infinite dihedral group ${\mathbb Z} \rtimes {\mathbb Z}_2$.

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The Restricted Weyl Group of the Cuntz Algebra and Shift Endomorphisms

It is shown that, modulo the automorphisms which fix the canonical diagonal MASA point-wise, the group of those automorphisms of the Cuntz algebra O_n which globally preserve both the diagonal and the core UHF-subalgebra is isomorphic, via restriction, with the group of those homeomorphisms of the full one-sided n-shift space which eventually commute along with their inverses with the shift transformation. The image of this group in the outer automorphism group of O_n can be embedded into the quotient of the automorphism group of the full two-sided n-shift by its center, generated by the shift. If n is prime then this embedding is an isomorphism.

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On Invariant MASAs for Endomorphisms of the Cuntz Algebras

The problem of existence of standard (i.e. product-type) invariant MASAs for endomorphisms of the Cuntz algebra O_n is studied. In particular endomorphisms which preserve the canonical diagonal MASA D_n are investigated. Conditions on a unitary in O_n equivalent to the fact that the corresponding endomorphism preserves D_n are found, and it is shown that they may be satisfied by unitaries which do not normalize D_n. Unitaries giving rise to endomorphisms which leave all standard MASAs invariant and have identical actions on them are characterized. Finally some properties of examples of finite-index endomorphisms of O_n given by Izumi and related to sector theory are discussed and it is shown that they lead to an endomorphism of O_2 associated to a matrix unitary which does not preserve any standard MASA.

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Endomorphisms of O_n which preserve the canonical UHF-subalgebra

Unital endomorphisms of the Cuntz algebra O_n which preserve the canonical UHF-subalgebra F_n of O_n are investigated. We give examples of such endomorphisms for which the associated unitary element in O_n does not belong to F_n. One such example, in the case where n=2, arises from a construction of a unital endomorphism on O_2 which preserves the canonical UHF-subalgebra and where the relative commutant of the image in O_2 contains a copy of O_2.

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