Free Loop Research Articles

On invariant generating sets for the cycle space

Consider a unimodular random graph, or just a finitely generated Cayley graph. When does its cycle space have an invariant random generating set of cycles such that every edge is contained in finitely many of the cycles? Generating the free Loop O ( 1 ) O(1) model as a factor of iid is closely connected to having such a generating set for FK-Ising cluster. We show that geodesic cycles do not always form such a generating set, by showing for a parameter regime of the FK-Ising model on the lamplighter group the origin is contained in infinitely many geodesic cycles. This answers a question by Angel, Ray and Spinka. Then we take a look at how the property of having an invariant locally finite generating set for the cycle space is preserved by Bernoulli percolation, and apply it to the problem of factor of iid presentations of the free Loop O ( 1 ) O(1) model.

Umbilical Artery Doppler and Adverse Outcomes in Severe Preeclampsia Without Fetal Growth Restriction: A Retrospective Cohort Study.

Background and objective Severe preeclampsia may be managed expectantly before 34 weeks gestation with close surveillance.Utilized in fetal growth restriction (FGR), evidence supports umbilical artery (UA) Dopplerpreventing neonatal morbidity from hypertensive disease and predicting adverse outcomes in preeclampsia. We evaluated the association of abnormal UA Doppler waveforms with early delivery (before 34 weeks gestation) and adverse maternal-fetal outcomes in patients with early severe preeclampsia without FGR. Methodology This is a retrospective cohort study of singleton pregnancies with International Classification of Diseases (ICD) Ninth or Tenth Revision, defined severe preeclampsia diagnosed before 34 weeks gestation without FGR from January 1, 2018, through January 27, 2023, at a large tertiary care center where S/D ratios were calculated from UA Doppler interrogation of a free loop of cord at least once weekly. This study was approved by the IRB (ID:00002216) and granted a full Health Insurance Portability and Accountability Act (HIPAA) waiver of consent. Exclusion criteria were major congenital anomalies, congenital infection, aneuploidy, leaving against medical advice >24 hours, and patient instability on admission defined as condition(s) precluding expectant management by the American College of Obstetrics and Gynecology. The primary outcome was delivery before 34 weeks gestation. Secondary outcomes were the mode of delivery and maternal/fetal complications. Patient characteristics and outcomes for normal versus abnormal UA Doppler groups were compared with chi-square, t-tests, and Fisher's exact test. Odds ratios and relative risks were calculated to compare outcomes. Results Of 194 patients with severe preeclampsia, 107 met inclusion criteria. Thirty-four patients had abnormal UA Doppler studies. There were no differences in demographic and clinical data between patients with normal and abnormal UA Doppler studies. Patients with abnormal UA Doppler studies were more likely to deliver before 34 weeks (OR=3.91; 95%CI 1.24-12.33) for worsening severe features (OR=3.85; 95% CI 1.42-10.41), and were less likely to deliver vaginally (OR=0.12; 95% CI 0.03-0.54). Abnormal UA Doppler studies were associatedwith an increased risk of neonatal complications (OR=6.46; 95% CI 1.42-29.42) and respiratory distress syndrome (RDS) (OR=4.75; 95%CI 1.32-17.16). Abnormal UA Doppler subgroups were divided into patients with elevated S/D >95% Acharya (N=22) and absent end-diastolic flow (EDF) (N=10). The elevated S/D group tended to deliver before 34 weeks gestation for worsening severe features (OR=3.71, 95% CI 1.144-12.050) and had a higher risk of neonatal complications (RR 1.404; 95% CI 1.213-1.624). The absent EDF subgroup was more likely to deliver before 34 weeks (RR=1.52; 95%CI 1.29-1.79) for abnormal fetal testing (OR=6.92; 95%CI 1.71-28.08) and undergo primary cesarean delivery (OR=7.23; 95%CI 1.43-36.61). Conclusion Pregnancies with severe preeclampsia without FGR displayed a high incidence of abnormal UA Doppler waveforms associated with loss of clinical stability and adverse fetal outcomes. The groups with more impedance to umbilical artery flow tended to deliver earlier, and as the Doppler shifted from elevated S/D to absent end-diastolic flow, the mode of delivery shifted to cesarean delivery with increased risk of abnormal fetal testing. These results support the utility of UA Doppler surveillance in severe preeclampsia.

A Pilot Study of Ultrasound Assessment of Umbilical Cord and Placental Vascular Flow for Cord Accident Stillbirth Prevention.

Our objective was to determine the feasibility and interobserver reliability of umbilical cord and placental arteriolar flow assessment in low-risk pregnancies near term. This was a prospective pilot study in low-risk pregnancies at 36 weeks with anterior placentas. We excluded any with an indication for antenatal testing or delivery before 39 weeks. Each participant underwent two ultrasounds by different examiners, which included arterial and venous velocimetry at three cord sites (fetal, free loop, and placental) in addition to maternal and fetal placental arterioles. The interobserver reliability was quantified using the Pearson correlation coefficient with that of standard clinical parameters serving as a benchmark for interpretation. Among 53 participants scanned at 356/7-371/7 weeks, the mean examination duration was 20.5 ± 4.2 minutes. Ascertainment success was high for measures at the free loop, placental cord insertion, and fetal placental arterioles (range 90.6%-99.1%) and was lower at the fetal cord insertion and maternal spiral arterioles (range 47.2%-87.7%). Interobserver reliability estimates for free-loop systolic/diastolic and pulsatility index ranged from 0.38 to 0.44. Interobserver reliability for experimental parameters varied by measurement site, and all were poor at the fetal insertion and in placental arterioles. Parameters had significant variation across cord sites (range 4.3%-21.7%). In our cohort, flow assessments of the free loop, placental insertion, and placental arterioles are feasible, but interrater reliability varies by measurement type and cord site. Future studies are needed to establish feasibility and reliability in nonanterior placentation and to assess clinical relevance.

Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

Chromatic Cardinalities via Redshift

Abstract Using higher descent for chromatically localized algebraic $K$-theory, we show that the higher semiadditive cardinality of a $\pi $-finite $p$-space $A$ at the Lubin–Tate spectrum $E_{n}$ is equal to the higher semiadditive cardinality of the free loop space $LA$ at $E_{n-1}$. By induction, it is thus equal to the homotopy cardinality of the $n$-fold free loop space $L^{n} A$. We explain how this allows one to bypass the Ravenel–Wilson computation in the proof of the $\infty $-semi-additivity of the ${T(n)} $-local categories. “Cardinalis sinuatus – the Pyrrhuloxia” by Dick Culbert licensed under CC BY 2.0.

On multiplicative spectral sequences for nerves and the free loop spaces

We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology algebra of the classifying space of a category internal to the category of topological spaces. By applying the machinery to a Borel construction, we explicitly determine the mod p cohomology algebra of the free loop space of the real projective space for each odd prime p. This example is emphasized as an important computational case. Moreover, we represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold M with differential forms on the universal cover of M via Chen's iterated integral map.

Порівняння функціональних результатів і якості життя після хірургічного лікування стресового нетримання сечі в жінок

Sling surgery is a modern and effective surgical treatment for stress urinary incontinence in women. However, there is insufficient data on the comparative analysis of long-term results and quality of life of patients after various options of such operations. Purpose - to compare the functional results and quality of life of women after two variants of sling operations for stress urinary incontinence. Materials and methods. 84 women with stress urinary incontinence were examined, who were divided into two groups. In the Group I, a plastic free synthetic loop (TVT) was performed, in the Group II - an operation using the transobturator location of the prolene loop (TVT-Obturator - TVT-О). Functional outcomes of treatment and quality of life were assessed 2 years after surgery using the ICIQ-SF questionnaire, the Patient General Impression of Improvement (PGI-I) scale, and the Patient Satisfaction Scale. An objective indicator was the absence of leakage of urine during a cough test. Results. Treatment effectiveness, according to objective criteria, was 87.9% and 67.9% 2 years after TVT and TVT-O surgery, respectively (p=0.045). When evaluating subjective indicators, no statistically significant differences between groups were found, however, according to the ICIQ-SF questionnaire, the frequency of improvement in the quality of life was reliably 1.3 times higher in the case of using the TVT technique (odds ratio 3.33; 95% confidence interval 1.14-9.78). The frequency of urinary incontinence recurrence did not depend on the method of sling operation (p>0.05). Data analysis was carried out using The Statistical Package for Social Sciences. Conclusions. When evaluating the results of treatment 2 years after the intervention, the effectiveness of TVT surgery, according to objective criteria, is statistically significantly higher than the effectiveness of TVT-O surgery, although the latter technique is associated with the absence of bladder etching. The research was carried out in accordance with the principles of the Helsinki Declaration. The study protocol was approved by the Local Ethics Committee of the participating institution. The informed consent of the patient was obtained for conducting the studies. No conflict of interests was declared by the authors.

Upper bounds for the critical values of homology classes of loops

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected n-dimensional manifold of positive Ricci curvature Ric≥n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Ric}\\ge n-1$$\\end{document} has length ≤nπ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le n \\pi .$$\\end{document} This improves the bound 8π(n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi (n-1)$$\\end{document} given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).

Families of relatively exact Lagrangians, free loop spaces and generalised homology

AbstractWe prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy $$\psi ^1$$ ψ 1 of a symplectic manifold $$(M, \omega )$$ ( M , ω ) fixing a relatively exact Lagrangian L setwise must act trivially on $$R_*(L)$$ R ∗ ( L ) , where $$R_*$$ R ∗ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over $${\mathbb {Z}}/2$$ Z / 2 and over $${\mathbb {Z}}$$ Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, $$\psi ^1|_L$$ ψ 1 | L is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that $$\psi ^1|_L$$ ψ 1 | L acts trivially on $$R_*({\mathcal {L}}L)$$ R ∗ ( L L ) , where $${\mathcal {L}}L$$ L L is the free loop space of L. From this we deduce that when L is a surface or a $$K(\pi , 1)$$ K ( π , 1 ) , $$\psi ^1|_L$$ ψ 1 | L is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over $${\mathbb {Z}}/2$$ Z / 2 .

The string topology coproduct on complex and quaternionic projective space

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.

A cocyclic construction of $S^1$-equivariant homology and application to string topology

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to S^1 -equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) S^1 -equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an S^1 -equivariant version of operadic Deligne’s conjecture.

A(∞)-Algebra Structure in the Cohomology and Cohomologies of a Free Loop Space

A(∞)-Algebra Structure in the Cohomology and Cohomologies of a Free Loop Space

DNA Framework Programmed Conformational Reconstruction of Antibody Complementary Determining Region.

The conformation of complementary determining region (CDR) is crucial in dictating its specificity and affinity for binding with an antigen, making it a focal point in artificial antibody engineering. Although desirable, programmable scaffolds that can regulate the conformation of individual CDRs with nanometer precision are still lacking. Here, we devise a strategy to program the CDR conformation by anchoring both ends of a free CDR loop to specific sites of a DNA framework structure. This method allows us to define the span of a single CDR loop with an ∼2 nm resolution. Using this approach, we create a series of DNA framework based artificial antibodies (DNFbodies) with varied CDR loop spans, leading to different antibody-antigen binding affinities. We find that an optimized single CDR loop (∼2.3 nm span) exhibits ∼3-fold improved affinity relative to natural antibodies, confirming the critical role of the CDR conformation. This study may inspire the rational design of artificial antibodies.

Reflexive homology

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

Color Doppler, HDlive, and HDlive Silhouette features of an umbilical cord cyst before 11 weeks of gestation: Diagnostic value.

We describe transvaginal color Doppler, HDlive, and HDlive Silhouette features of an umbilical cord cyst (UCC) before 11 weeks of gestation. In this cohort study, 135 transvaginal dating scans were performed at 7 to 10 + 6 weeks of gestation, and 17 UCCs were identified (12.6%). UCC was evaluated using color Doppler, HDlive, and HDlive Silhouette. The clinical characteristics, pregnancy courses, and outcomes were also investigated. UCC location was on the fetal side in six cases, at the free loop in 10 cases, and on the placental side in one case. There were seven single and 10 multiple cysts. Cyst diameters ranged from 3.3 to 11.3 mm (mean, 5.6; standard deviation, ±2.1). Blood flow inside the cyst was noted in three cases (17.6%). HDlive clearly showed the spatial relationships among UCC, the umbilical cord, midgut herniation, yolk sac, and embryo. Location of UCC could be clearly identified with HDlive. HDlive Silhouette showed central cysts inside UCCs in seven cases (41.2%). HDlive Silhouette also clearly demonstrated the sac of midgut herniation in the umbilical cord in 12 cases (70.6%). All UCCs resolved before 15 weeks (mean, 11.1 weeks; standard deviation, ±1.5). All fetuses with UCCs showed good neonatal outcomes. The incidence of UCC was high compared with that in previous reports. Color Doppler, HDlive, and HDlive Silhouette may provide information on the nature and origin of UCCs before 11 weeks of gestation. UCC before 11 weeks of gestation may be a common, transient, and benign finding.

Free Bol loops of exponent two

A Bol loop is a loop that satisfies the identity x((yz)y)=((xy)z)y. In this paper, we give a construction of the free Bol loops of exponent two. We define a canonical form of all their elements and describe their multiplication law based on this form.

Level structures on p-divisible groups from the Morava E-theory of abelian groups

The close relationship between the scheme of level structures on the universal deformation of a formal group and the Morava E-cohomology of finite abelian groups has played an important role in the study of power operations for Morava E-theory. The goal of this paper is to explore the relationship between level structures on the p-divisible group given by the trivial extension of the universal deformation by a constant p-divisible group and the Morava E-cohomology of the iterated free loop space of the classifying space of a finite abelian group.

Homology transfer products on free loop spaces: orientation reversal on spheres

Homology transfer products on free loop spaces: orientation reversal on spheres

The cohomology of free loop spaces of rank $2$ flag manifolds

The cohomology of free loop spaces of rank $2$ flag manifolds

Differences in umbilical vein predict placental sharing and outcome in monochorionic twins with selective growth-restriction

To examine the association between umbilical venous diameters and flow with placental sharing and adverse outcomes in monochorionic twin pregnancies diagnosed with selective fetal growth restriction (sFGR). The venous placental territory determines the oxygenated blood flow to the fetuses. Unequal placental sharing is considered the main contributor but is not yet quantifiable during pregnancy. Prospective cohort study of monochorionic diamniotic twin pregnancies diagnosed with sFGR. The following ultrasound variables were measured: intra-abdominal and free loop diameter of the umbilical vein (UV), estimated fetal weight (EFW), and cord insertions. The time-averaged maximum velocity in the intra-abdominal UV was measured and used to calculate UV-flow. Ratios (larger/smaller twin) were used for analysis. Adverse outcome was defined as the occurrence of fetal demise or birth before 34 weeks. We included 102 consecutive sFGR pregnancies, of which 88 placentas (86%) underwent injection studies with barium sulfate. The overall survival was 89%: 7 cases with single fetal demise (7%), 5 with double demise (5%), and 5 neonatal deaths (5%) occurred. Univariable linear regression analysis showed a significant association for all of the variables with placental sharing. In multivariable analysis, the UV-variables and a discordant cord insertion remained significantly associated with placental sharing. EFW differences and a concordant or intermediate cord insertion were not significantly associated with placental sharing. Logistic regression and receiver operating characteristics curve analysis demonstrated that EFW discordance and the cord insertion site had a poor predictive accuracy for an unequally shared placenta or an adverse outcome. In contrast, the predictive accuracy of the UV-variables was fair. Differences in UV-diameter and flow between the twins at the time of sFGR diagnosis inform us of the placental sharing, more so than differences in EFW. Differences in UV-variables between the twins also seem to increase the risk of fetal demise and birth before 34 weeks.View Large Image Figure ViewerDownload Hi-res image Download (PPT)