Weak index pairs and the Conley index for discrete multivalued dynamical systems. Part II: properties of the index

Motivation to revisit the Conley index theory for discrete multivalued dynamical systems stems from the needs of broader real applications, in particular in sampled dynamics or in combinatorial dynamics. The new construction of the index in [B. Batko and M. Mrozek, {\em SIAM J. Applied Dynamical Systems}, 15(2016), pp. 1143-1162] based on weak index pairs, under the circumstances of the absence of index pairs caused by relaxing the isolation property, seems to be a promising step towards this direction. The present paper is a direct continuation of [B. Batko and M. Mrozek, {\em SIAM J. Applied Dynamical Systems}, 15(2016), pp. 1143-1162] and concerns properties of the index defined therin, namely Wa\.zewski property, the additivity property, the homotopy (continuation) property and the commutativity property. We also present the construction of weak index pairs in an isolating block

Weak index pairs and the Conley index for discrete multivalued dynamical systems

Motivated by the problem of reconstructing dynamics from samples we revisit the Conley index theory for discrete multivalued dynamical systems. We introduce a new, less restrictive definition of the isolating neighbourhood. It turns out that then the main tool for the construction of the index, i.e. the index pair, is no longer useful. In order to overcome this obstacle we use the concept of weak index pairs

On Approximate Solutions of Functional Equations in Vector Lattices

We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation F(x+y)+F(x)+F(y)≠0⇒F(x+y)=F(x)+F(y) in Riesz spaces, the Cauchy equation with squares F(x+y)2=(F(x)+F(y))2 in f-algebras, and the quadratic functional equation F(x+y)+F(x-y)=2F(x)+2F(y) in Riesz spaces

Stability of the exponential functional equation in Riesz algebras

We deal with the stability of the exponential Cauchy functional equation ( + ) = ( ) ( ) in the class of functions : → mapping a group ( , +) into a Riesz algebra . The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem

Linking combinatorial and classical dynamics : Conley index and Morse decompositions

We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field give rise to isomorphic objects in the multivalued map case

Venopunkcja żyły odłokciowej jako alternatywne podejście dla oceny stężenia CGRP w osoczu krwi u pacjentów z dysfunkcją układu ruchowego narządu żucia

Introduction: Calcitonin gene-related peptide is an important vasodilator. It plays an important role in the metabolism of chewing muscles. The aim of the study was to evaluate the plasma level of CGRP in patients with myofascial pain (RDC/TMD Ia) and myofascial pain with limited opening (RDC/TMD Ib) before and after occlusal splint therapy (Michigan splint). Material and methods: A randomised trial was performed including 39 patients (males = 3, females = 36). Blood samples were taken from the external jugular vein (JUG) and cubital vein (CUB) before and after 30 days of occlusal splint therapy. Plasma levels of CGRP were measured with ELISA KIT for Human Calcitonin Gene Related Peptide (CGRP) 96T (USCNK Business Co. Ltd.). Results: The results of the study show that the plasma CGRP level was higher in the external jugular vein (JUG1 = 5.07pg/mL [SD = 1.99]) than in cubital vein (CUB1 = 4.3 pg/mL [SD = 1.6]). After 30 days of the occlusal splint therapy the levels in both veins increased: JUG2 = 6.07 pg/mL (SD = 2.19), and CUB2 = 4.9 pg/mL (SD = 1.4). The CGRP plasma level increase was statistically significant only in the external jugular vein (JUG) (p < 0.05). Statistically significant pain intensity reduction was observed: VAS1 = 5.4 (SD = 2.08) decreased to VAS2 = 1.7 (SD = 2.07) after splint therapy (p < 0.05). Conclusions: Venepuncture of an external jugular vein is more precise, than venepuncture of a cubital vein in evaluating CGRP plasma level changes in patients with TMD.Wstęp: Peptyd pochodny genu kalcytoniny jest ważną substancja naczynio-rozkurczową. Odgrywa ważną role w metabolizmie mięśni żucia. Celem pracy była ocena stężenia osoczowego CGRP u pacjentów z bólem mięśniowo-powięziowym (RDC/TMD Ia) oraz z bólem mięśniowo-powięziowym z ograniczonym odwodzeniem (RDC/TMD Ib), przed i po terapii szyną okluzyjną (Szyna Michigan). Materiał i metody: Przeprowadzono badanie randomizowane, do którego włączono 39 pacjentów (mężczyźni = 3, kobiety = 36). Próbki krwi pobrano z żyły szyjnej zewnętrznej (JUG) oraz z żyły odłokciowej (CUB), przed leczeniem i po 30 dniach stosowania szyny okluzyjnej. Stężenie neuropeptydu CGRP oceniano za pomocą zestawu ELISA KIT for Human Calcitonin Gene Related Peptide (CGRP) 96T (USCNK Business Co. Ltd.). Wyniki: Stwierdzono, że stężenie neuropeptydu CGRP było wyższe w materiale pobranym z żyły szyjnej zewnętrznej JUG1 = 5,07pg/ ml (SD = 1,99), niż w materiale pobranym z żyły odłokciowej CUB1 = 4,3 pg/ml (SD = 1,6). Po 30 dniach terapii szyną okluzyjną średnie stężenie CGRP w obu grupach CUB i JUG wzrosły: JUG2 = 6,07 pg/ml (SD = 2,19) i CUB2 = 4,9 pg/ml (SD = 1,4). Wzrost stężenia CGRP w osoczu krwi był istotny statystycznie jedynie w materiale pobranym z żyły szyjnej zewnętrznej (JUG) (p < 0,05). Zaobserwowano także istotną statystycznie redukcję natężenia dolegliwości bólowych w skali VAS: VAS1 = 5,4 (SD = 2,08) redukcja do VAS2 = 1,7 (SD = 2.07), po przeprowadzonej szynoterapii (p < 0,05). Wnioski: Wenopunkcja żyły szyjnej zewnętrznej dostarcza bardziej precyzyjnych pomiarów stężenia neuropeptydu CGRP w osoczu niż wenopunkcja żyły odłokciowej, u pacjentów z bólową postacią dysfunkcji narządu żucia

Zapalenie palców jako objaw chorobowy występujący nie tylko w spondyloartropatii zapalnej

  W praktyce reumatologicznej najczęściej spotykamy się z zapaleniem palców w przebiegu spondyloartropatii zapalnej; nie możemy jednak zapomnieć, że nie jest to jedyna jednostka chorobowa, w której występuje ten objaw. To ważne, ponieważ mimo wspólnego mianownictwa leczenie jest różne w zależności od etiologii. Niniejsza praca ma na celu zapoznanie czytelników z innymi przyczynami zapalenia palców. Forum Reumatol. 2016, tom 2, nr 3: 106–110  

Identifying Nonlinear Dynamics with High Confidence from Sparse Data

We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics

Unfavorably altered fibrin clot properties in patients with eosinophilic granulomatosis with polyangiitis (Churg-Strauss Syndrome) : association with thrombin generation and eosinophilia

Given reports on the increased prevalence of thromboembolic incidents in patients with eosinophilic granulomatosis with polyangiitis (EGPA; Churg-Strauss syndrome), we investigated whether fibrin clot properties are unfavorably altered in EGPA.Ex vivo plasma fibrin clot characteristics, including clot permeability, turbidimetry and efficiency of fibrinolysis using two assays, were investigated in 34 consecutive patients with remission in EGPA according to the Birmingham Vasculitis Activity Score version 3 (23 female, 11 male), aged 48 (range, 21-80) years. The control group comprised 34 age- and sex- matched volunteers.Compared with controls, patients with EGPA were characterized by denser fiber clots (estimated pore size, Ks, 7.30±0.93 vs 10.14±1.07 10-9 cm2), faster fibrin polymerization (lag phase in a turbidimetric curve, 41.8±3.6 vs 47.4±2.9 s), thicker fibrin fibers (maximum absorbance, ΔAbs, 0.87±0.09 vs 0.72±0.07), higher maximum levels of D-dimer released from clots (DDmax 4.10±0.46 vs 3.54±0.35 mg/L), and prolonged clot lysis time (t50%; 9.50±1.45 vs 7.56±0.87 min); all p<0.0001. Scanning electron microscopy images confirmed denser plasma fibrin networks composed of thinner fibers formed in EGPA. Antineutrophil cytoplasmic antibody status and C-reactive protein did not affect clot variables. Multivariate analysis adjusted for fibrinogen showed that Ks was predicted by eosinophil count, peak thrombin generation, factor VIII, and soluble CD40 ligand, whereas eosinophil count, peak thrombin generation and antiplasmin predicted t50%.This study is the first to show that EGPA is associated with prothrombotic plasma fibrin clot phenotype, which may contribute to thromboembolic manifestations reported in this disease