Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact.

Resection and postoperative radiation therapy for desmoid fibromatosis of the chest wall in a young woman

Background Surgery is an effective treatment for desmoid fibromatosis, but it may be difficult, depending on the location or local spread of the tumor, and the decision to perform surgery must be made carefully. We herein report a case of desmoid fibromatosis of the chest wall in a young woman suspected of having invasion to the 1st, 2nd and 3rd ribs. Case presentation A 35-year-old woman had been aware of dry cough and right chest pain, so she was referred to our hospital. Chest computed tomography showed a localized pleural tumor mainly at the first rib. Magnetic resonance imaging revealed a 75 × 65 × 27-mm tumor with a smooth surface, with partial contact from the first rib to third rib and partial extension to the 1st intercostal space. The tumor showed growth in the two months after the first visit, so resection was performed. The tumor was completely resected, and adjuvant radiation therapy (50 Gy) was performed for the small margin. The pathological diagnosis was desmoid fibromatosis. The postoperative course has been uneventful, without recurrence at 14 months after surgery. Conclusions In chest wall tumors located ventral of the pulmonary apex, we suggest that a combination of the Grunenwald method and Masaoka anterior approach may be a useful option. In cases where margin is not enough, adjuvant radiation therapy should be considered.

Cerebral Hydatid Cyst with Intraventricular Extension: A Case Report

Intracranial hydatid cyst is a rare entity, comprising about 2–3% of all hydatid cysts. Similarly, intracranial hydatid cysts account for 1–2% of all intracranial lesions. Clinical symptoms are generally nonspecific and patients usually present with symptoms of increased intracranial pressure. Cerebral hydatid cysts can be either primary or secondary to systemic hydatid disease. Primary cerebral hydatid cysts are usually solitary, unilocular with an intraparenchymal location. Intraventricular extension of hydatid cysts account for a limited percentage of all cerebral hydatid cysts with limited number of cases reported. Herein, we present the imaging and surgical findings of a primary cerebral hydatid cyst that is located in frontal lobe parenchyma with partial extension into the ventricular system.

Case of peroneus digiti quinti muscle

Lateral compartment muscles of the leg have shown variations like peroneus accessorius, peroneus quartus or peroneus digiti quinti with varied incidences and varied insertions. Here we report a case of peroneus digiti quinti which arose from peroneus brevis tendon and sheath covering it. The small belly moved forwards, diverting from peroneus brevis, midway between peroneus brevis and peroneus tertius, finally inserting on the dorsal digital expansion of fifth toe. Length of muscle belly and tendon were 2.01 cm and 6.5 cm respectively. It was also observed that when traction was applied to peroneus brevis, partial extension of the fifth toe was seen, suggesting functional need to evert and extend the little toe simultaneously.

Reverse Lexicographic Gröbner Bases and Strongly Koszul Toric Rings

Restuccia and Rinaldo proved that a standard graded $K$-algebra $K[x_1,\dots,x_n]/I$ is strongly Koszul if the reduced Gröbner basis of $I$ with respect to any reverse lexicographic order is quadratic. In this paper, we give a sufficient condition for a toric ring $K[A]$ to be strongly Koszul in terms of the reverse lexicographic Gröbner bases of its toric ideal $I_A$. This is a partial extension of a result given by Restuccia and Rinaldo. In addition, we show that any strongly Koszul toric ring generated by squarefree monomials is compressed. Using this fact, we show that our sufficient condition for $K[A]$ to be strongly Koszul is both necessary and sufficient when $K[A]$ is generated by squarefree monomials.

On the dimensions of a family of overlapping self-affine carpets

We consider the dimensions of a family of self-affine sets related to the Bedford–McMullen carpets. In particular, we fix a Bedford–McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford–McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman’s recent work on the dimensions of self-similar sets and measures.

EIGENVALUES AND LINEAR QUASIRANDOM HYPERGRAPHS

Let$p(k)$denote the partition function of$k$. For each$k\geqslant 2$, we describe a list of$p(k)-1$quasirandom properties that a$k$-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. For each of the quasirandom properties that is described, we define the largest and the second largest eigenvalues. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon, Hàn, Person, and Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung, Graham, and Wilson for graphs.

Selection in the monadic theory of a countable ordinal

A monadic formula ψ(Y) is a selector for a formula φ(Y) in a structure if there exists a unique subset P of which satisfies ψ and this P also satisfies φ. We show that for every ordinal α ≥ ωω there are formulas having no selector in the structure (α, <). For α ≤ ω1, we decide which formulas have a selector in (α, <) , and construct selectors for them. We deduce the impossibility of a full generalization of the Büchi-Landweber solvability theorem from (ω, <) to (ωω, <). We state a partial extension of that theorem to all countable ordinals. To each formula we assign a selection degree which measures “how difficult it is to select”. We show that in a countable ordinal all non-selectable formulas share the same degree.