Hasil untuk "Orthopedic surgery"

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arXiv Open Access 2026
TopoLS: Lattice Surgery Compilation via Topological Program Transformations

Junyu Zhou, Yuhao Liu, Ethan Decker et al.

Lattice surgery is a leading approach for implementing fault-tolerant logical operations in surface code quantum computing, but compiling efficient lattice surgery layouts remains challenging. Existing compilers are largely circuit-centric and operate directly on gate sequences, limiting their ability to exploit the topological flexibility of merge-split operations and minimize space--time volume. We present TopoLS, a topology-centric compiler that uses ZX diagrams as an intermediate representation for lattice surgery compilation. TopoLS combines semantic-preserving ZX-level program transformations, including spider fusion and topology-aware slicing, with a Monte Carlo Tree Search (MCTS)-based synthesis procedure that constructs pipe-diagram embeddings by jointly optimizing placement and routing in 3D space--time. To scale to large circuits, TopoLS further introduces topology-aware partitioning that decomposes the compilation task into bounded subproblems and limits the routing frontier during embedding. Across evaluated benchmarks, TopoLS achieves an average $46\%$ reduction in space--time volume over prior circuit-centric compilers, with improvements ranging from $25\%$ to $90\%$, and exhibits strong empirical scalability on large benchmark families. Compared with SAT-based formulations that become intractable on larger instances, TopoLS offers a practical end-to-end solution for optimized lattice surgery compilation. TopoLS has been integrated into the TQEC ecosystem, enabling downstream circuit-level simulation and resource estimation workflows.

en quant-ph
arXiv Open Access 2025
Flat connections at infinity on knot surgery manifolds

Aditya Dwivedi, Archana Maji, Dmitry Noshchenko et al.

$\rm SL(2,\mathbb{C})$ Chern-Simons theory on a closed 3-manifold is one of the most interesting, yet tractable examples of a QFT. On one hand, its non-perturbative structure is not yet fully understood; on the other, the mathematical structure turns out to be very rich. In this work we explore the new phenomenon of flat connections at infinity on various knot surgery manifolds. Such flat connections can be understood as asymptotic ends in the non-compact moduli space of flat $\rm SL(2,\mathbb{C})$ connections. We focus on the examples of $\pm 1/r$-surgeries on torus, twist and some double twist knot complements in $S^3$. Surprisingly, our findings suggest that flat connections at infinity are abundant even for simple low-crossing knot surgeries. We therefore believe that their presence would shed light on the resurgent nature of the path integral.

en hep-th, math-ph
arXiv Open Access 2025
Endoscopic fiber-coupled diamond magnetometer for cancer surgery

A. J. Newman, S. M. Graham, C. J. Stephen et al.

Interoperative measurements using magnetic sensors is a valuable technique in cancer surgery for finding magnetic tracers. Here we present a fiber-coupled nitrogen-vacancy (N-V) center magnetometer capable of detecting iron oxide suspension (MagTrace from Endomagnetics Ltd.) used in breast cancer surgeries. Detection of an iron mass as low as 0.56~mg has been demonstrated, 100 times less than that of a recommended dose at a maximum distance of 5.8~mm. Detection of an iron concentration as low as 2.8 mg/ml has also been demonstrated, 20 times less than a recommended dose. The maximum working distance from the sensor can be as large as 14.6~mm for higher concentrations. The sensor head has a maximum diameter of 10~mm which would allow it to be used for endoscopy, laparoscopy and interoperative surgery.

en physics.med-ph, cond-mat.mtrl-sci
DOAJ Open Access 2025
Artificial intelligence for pediatric height prediction using large-scale longitudinal body composition data

Dohyun Chun, Hae Woon Jung, Jongho Kang et al.

Objective We developed a precise, reliable artificial intelligence (AI) model for predicting the future height of children and adolescents based on anthropometric and body composition data. Materials and Methods We used an extensive longitudinal dataset from a large-scale Korean cohort study, which included 588,546 measurements from 96,485 children and adolescents aged 7–18. We developed a prediction model using the light gradient boosting method and integrated anthropometric and body composition metrics along with their standard deviation scores (SDSs) and velocity parameters. Model performance was assessed through root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). We employed Shapley additive explanations (SHAP) for model interpretability. Results The model accurately predicted future heights. For males, the average RMSE, MAE, and MAPE were 2.51 cm, 1.74 cm, and 1.14%, respectively, with female prediction results showing comparable accuracy (2.28 cm, 1.68 cm, and 1.13%, respectively). Shapley additive explanations analysis revealed that the SDS of height, height velocity, and soft lean mass velocity were key predictors of future height. The model created personalized growth curves through estimation of individual-specific height trajectories, comparison with actual measurements, and identification of key variables using local SHAP values. Conclusion Our model produces accurate and personalized growth curves, incorporating explainable AI techniques for enhanced clinical understanding. This method advances pediatric growth assessment and provides robust clinical decision support. Despite limitations including the absence of handwrist radiography comparison and Korean population specificity, our approach demonstrates significant potential for early identification of growth disorders and optimization of growth outcomes.

Computer applications to medicine. Medical informatics
DOAJ Open Access 2025
Condromalacia patella

Ozer Ulku

Abstract Condromalacia-patella which is seen on the patellafemoral face is the degeneration of the patella articular cartilage, and generally missed since it doesnt give clinical symptom and can t be observed in rontgenogram. Here is given some information for the diagnosis and treatment of this disease releated to a patient in our clinic. ÖzetKondromalasia patetla, patellanın patella-femoral yüzünde görülen, çoğu zaman klinik ve rontgen belirtisi vermeden gözden kaçan, patella eklem kartilajının dejenerasyonudur. Kliniğimizde rastladığımız bir vak a sebebiyle bu hastalık hakkında kısaca teşhis kriterleri ve tedavi hakkında bilgi verilmiştir.

Orthopedic surgery
DOAJ Open Access 2025
Case Report: CD34-Negative, S100-Positive Spindle Cell Tumor With a MYH10-RET Fusion

Yifei Wang, Xiaowei Li, Chunfang Yao et al.

With the application of next-generation sequencing (NGS) in soft tissue tumors, NTRK gene rearrangement is becoming known to pathologists as a molecular hallmark of spindle cell tumors, and other spindle cell tumors with related kinases are being reported. However, cases of RET rearranged spindle cell tumors not associated with NTRK rearrangements are rarely reported. Here, we describe a case of RET rearranged spindle cell tumor in a 3-year-old girl who presented with swelling and pain in the arm. Histologically, the tumor consisted of a fascicular arrangement of monomorphic spindle cells infiltrating and growing in adipose and muscle tissue with visible mitotic activity. Immunohistochemistry showed spindle cells negative for CD34, positive for S100 protein, and focal staining for Pan-TRK. MYH10-RET fusion was identified by NGS. Fluorescence in situ hybridization (FISH) analysis confirmed the RET gene rearrangement but did not detect the NTRK1/2/3 gene rearrangement. In conclusion, we describe a rare case of CD34-negative, S100-positive spindle cell tumor with MYH10-RET fusion.

arXiv Open Access 2024
EgoSurgery-Phase: A Dataset of Surgical Phase Recognition from Egocentric Open Surgery Videos

Ryo Fujii, Masashi Hatano, Hideo Saito et al.

Surgical phase recognition has gained significant attention due to its potential to offer solutions to numerous demands of the modern operating room. However, most existing methods concentrate on minimally invasive surgery (MIS), leaving surgical phase recognition for open surgery understudied. This discrepancy is primarily attributed to the scarcity of publicly available open surgery video datasets for surgical phase recognition. To address this issue, we introduce a new egocentric open surgery video dataset for phase recognition, named EgoSurgery-Phase. This dataset comprises 15 hours of real open surgery videos spanning 9 distinct surgical phases all captured using an egocentric camera attached to the surgeon's head. In addition to video, the EgoSurgery-Phase offers eye gaze. As far as we know, it is the first real open surgery video dataset for surgical phase recognition publicly available. Furthermore, inspired by the notable success of masked autoencoders (MAEs) in video understanding tasks (e.g., action recognition), we propose a gaze-guided masked autoencoder (GGMAE). Considering the regions where surgeons' gaze focuses are often critical for surgical phase recognition (e.g., surgical field), in our GGMAE, the gaze information acts as an empirical semantic richness prior to guiding the masking process, promoting better attention to semantically rich spatial regions. GGMAE significantly improves the previous state-of-the-art recognition method (6.4% in Jaccard) and the masked autoencoder-based method (3.1% in Jaccard) on EgoSurgery-Phase. The dataset is released at https://github.com/Fujiry0/EgoSurgery.

en cs.CV, cs.AI
DOAJ Open Access 2024
Autologous Costal Chondral/Osteochondral Transplantation and Costa‐Derived Chondrocyte Implantation for Articular Cartilage Repair: Basic Science and Clinical Applications

Yuxuan Wei, Hao Guo, Zhuhong Chen et al.

There has been increasing application of autologous costal chondral/osteochondral transplantation (ACCT/ACOT) and costa‐derived chondrocyte implantation (ACCI) for articular cartilage repair over the past three decades. This review presents the major evidence on the properties of costal cartilage and bone and their qualifications as grafts for articular cartilage repair, the major clinical applications, and the risks and strategies for costal chondral/osteochondral graft(s) harvest. First, costal cartilage has many specific properties that help restore the articular surface. Costa, which can provide abundant cartilage and cylindrical corticocancellous bone, preserves permanent chondrocyte and is the largest source of hyaline cartilage. Second, in the past three decades, autologous costal cartilage‐derived grafts, including cartilage, osteochondral graft(s), and chondrocyte, have expanded their indications in trauma and orthopaedic therapy from small to large joints, from the upper to lower limbs, and from non‐weight‐bearing to weight‐bearing joints. Third, the rate of donor‐site complications of ACCT or ACOT is low, acceptable, and controllable, and some skills and accumulated experience can help reduce the risks of ACCT and ACOT. Costal cartilage‐derived autografting is a promising technique and could be an ideal option for articular chondral lesions with or without subchondral cysts. More high‐quality clinical studies are urgently needed to help us further understand the clinical value of such technologies.

Orthopedic surgery
arXiv Open Access 2023
Interactive Shape Sonification for Tumor Localization in Breast Cancer Surgery

Laura Schütz, Trishia El Chemaly, Emmanuelle Weber et al.

About 20 percent of patients undergoing breast-conserving surgery require reoperation due to cancerous tissue remaining inside the breast. Breast cancer localization systems utilize auditory feedback to convey the distance between a localization probe and a small marker (seed) implanted into the breast tumor prior to surgery. However, no information on the location of the tumor margin is provided. To reduce the reoperation rate by improving the usability and accuracy of the surgical task, we developed an auditory display using shape sonification to assist with tumor margin localization. Accuracy and usability of the interactive shape sonification were determined on models of the female breast in three user studies with both breast surgeons and non-clinical participants. The comparative studies showed a significant increase in usability (p<0.05) and localization accuracy (p<0.001) of the shape sonification over the auditory feedback currently used in surgery.

DOAJ Open Access 2023
Automated Optimization-Based Deep Learning Models for Image Classification Tasks

Daudi Mashauri Migayo, Shubi Kaijage, Stephen Swetala et al.

Applying deep learning models requires design and optimization when solving multifaceted artificial intelligence tasks. Optimization relies on human expertise and is achieved only with great exertion. The current literature concentrates on automating design; optimization needs more attention. Similarly, most existing optimization libraries focus on other machine learning tasks rather than image classification. For this reason, an automated optimization scheme of deep learning models for image classification tasks is proposed in this paper. A sequential-model-based optimization algorithm was used to implement the proposed method. Four deep learning models, a transformer-based model, and standard datasets for image classification challenges were employed in the experiments. Through empirical evaluations, this paper demonstrates that the proposed scheme improves the performance of deep learning models. Specifically, for a Virtual Geometry Group (VGG-16), accuracy was heightened from 0.937 to 0.983, signifying a 73% relative error rate drop within an hour of automated optimization. Similarly, training-related parameter values are proposed to improve the performance of deep learning models. The scheme can be extended to automate the optimization of transformer-based models. The insights from this study may assist efforts to provide full access to the building and optimization of DL models, even for amateurs.

Electronic computers. Computer science
DOAJ Open Access 2023
Outcomes of total hip and knee arthroplasty in special populations: a synopsis and critical appraisal of systematic reviews

Dimitris Challoumas, David Munn, Henrietta Stockdale et al.

Abstract Introduction This study aimed to present and critically appraise the best available evidence investigating associations between some pre-defined patient-related characteristics and perioperative complications or other outcomes in THA and TKA. Methods Electronic databases were searched (Medline, EMBASE, Scopus, CENTRAL) for systematic reviews assessing the following pre-defined patient-related characteristics as possible risk factors for worse peri-operative outcomes in THA and TKA: smoking, alcohol excess, rheumatoid arthritis, human immunodeficiency virus infection, hepatitis C virus infection, mental health conditions, and solid organ transplantation. Our primary outcome was periprosthetic joint infection. Results were analysed separately for THA, TKA and THA/TKA (mixed data). Results Based on at least two systematic reviews being in agreement, the following patient-related characteristics were associated with increased incidence of complications as follows: a) Smoking for all-cause revision in THA, for periprosthetic joint infection in TKA and THA/TKA; b) alcohol excess for periprosthetic joint infection in THA/TKA; c) human immunodeficiency virus for periprosthetic joint infection in THA/TKA; d) hepatitis C virus for overall complications, periprosthetic joint infection and all-cause revision in THA and THA/TKA, and for overall complications in TKA. Our study found conflicting evidence for a) smoking as a risk factor for periprosthetic joint infection and aseptic loosening in THA; b) human immunodeficiency virus as a risk factor for all-cause revision for THA/TKA; c) hepatitis C virus as a risk factor for periprosthetic joint infection and all-cause revision in TKA. No certainty of evidence was assigned to these results as this was not assessed by the authors of the majority of the included systematic reviews. Conclusion We found that smoking, excess alcohol consumption, RA, and infection with HIV and HCV were associated with a higher incidence of periprosthetic joint infection in one or both of THA and TKA or mixed THA/TKA data. All our results should be interpreted and communicated to patients with caution as the quality of the included systematic reviews was generally poor.

Orthopedic surgery
DOAJ Open Access 2023
Evaluation of Chronic Lateral Ankle Instability with a Sprain Stimulator: A Controlled Study in Physically Active Subjects

Matthieu Lalevee MD, PhD, Eli Schmidt, Nacime Salomao Barbachan Mansur MD, PhD et al.

Category: Ankle; Basic Sciences/Biologics Introduction/Purpose: Chronic Lateral Ankle Instability (CLAI) represents a significant socioeconomic burden. Paradoxically, its management has changed little over the years, notably because research is divided into functional and mechanical instability, whereas CLAI encompasses both. Sprain simulators can encompass both functional and mechanical instability by assessing the maximal ankle inversion velocity (MIV) during a simulated inversion trauma. We built a sprain simulator capable of producing a sudden ankle inversion motion during walking. We aimed to differentiate subjects with chronic lateral ankle instability (CLAI) from controls and quantify functional CLAI as well as impairments in activities of daily living and sports using a sprain simulator. Methods: Forty-five physically active subjects were included and assigned to a CLAI group, a control group, or excluded according to the International Ankle Consortium selection criteria. Each subject walked on a treadmill with instability boots after completing the Identification of Functional Ankle Instability (IdFAI) and the Foot Ankle Ability Measurement (FAAM) questionnaires. A simulated trauma was unexpectedly triggered by the observer. Maximal inversion velocities (MIV) were measured at this very moment using inertial moment units. We normalized these values by the average MIV of the 5 stance phases of the same foot preceding the simulated trauma (Ratio MIV). Normality of data were assessed with the Shapiro-Wilk test. The groups were compared using Student T test for normal and Mann-Whitney U test for nonnormal variables. Multivariate linear regressions were performed to assess the relation between, the IdFAI, the FAAM Activities of Daily Living Subscale, the FAAM Sports Subscale and the explanatory variables. Results: Twenty-six ankles were excluded, 32 composed the CLAI group and 32 the control group. Mean MIV were 213.5+/-54.7°/s and 177+/-64.2°/s (p=0.02), and mean Ratio MIV were 1.22+/-0.13 and 1.08+/-0.08 (p < 0.001) in the CLAI and Control groups respectively. In multivariate analysis, Ratio MIV was associated with higher values of IdFAI (β=42.8 [12.9;72.8],p=0.006), lower values of FAAM Activities of Daily Living Subscale (β=-14.1 [-27.8;-0.5],p=0.04) and lower values of FAAM Sports Subscale (β=-7.2 [-13.7;- 0.6],p=0.03) whereas MIV was not. Conclusion: Inversion velocities caused by a sprain simulator clearly differentiated CLAI from controls in our study. Ratio MIV showed good ability to quantify functional CLAI as well as impairments in activities of daily living and sports. This tool should be used in future studies in an attempt to provide a complete picture of CLAI encompassing its functional and mechanical aspects which may lead to improved LAS and CLAI management.

Orthopedic surgery
DOAJ Open Access 2022
Are outcomes after fixation of distal humerus coronal shear fractures affected by surgical approach? A systematic review and meta-analysis

Kalin J. Fisher, MD, Michael G. Livesey, MD, Oliver C. Sax, DO, MS et al.

Background: Surgical management of coronal shear fractures of the distal humerus is associated with a high rate of complications. Several surgical approaches have been described to address these fractures. The complication profiles associated with each approach have not previously been compared, and that is the aim of the present study. Methods: A systematic review of the literature was performed to identify all studies addressing coronal shear fractures of the distal humerus published between 2001 and January 2022. Of the 189 articles identified, 45 met the criteria for inclusion. Summaries of continuous data were calculated using the inverse variance method for pooling with random effects models. Fixed effects model estimates were reported unless significant heterogeneity was observed between studies. A subset of 6 studies reported the surgical approach and complications associated with the operative management of capitellar shear fractures without posterior comminution. The complication profiles of the extended lateral and anterolateral approaches were compared. Results: The 45 studies included yielded 899 patients. The average age was 44.9 years (95% confidence interval [CI]: 39.7 to 50.2). The fracture type was Dubberley A in 38% (n = 342), Dubberley B in 33% (n = 300), and not reported in the remainder. The reoperation rate was 13.8% (95% CI: 9.6% to 19.5%). Pooled complication rates included post-traumatic arthritis in 21.2% (95% CI: 18.0% to 24.9%), heterotopic ossification in 12.0% (95% CI: 9.2% to 15.6%), nerve injury in 7.8% (95% CI: 5.6% to 10.9%), and avascular necrosis in 7.4% (95% CI: 5.3% to 10.2%). The complication rate in noncomparative studies was 25.8% following the lateral approach and 16.7% following the anterolateral approach. Reported complications following the anterolateral approach were pain (9.5%) and nerve injury (7.1%). Reported complications following the lateral approach included arthritis (9.1%), heterotopic ossification (6.1%), avascular necrosis (4.5%), instability (3.0%), nerve injury (1.5%), and wound issues (1.5%). Discussion and Conclusion: Complications are common following operative management of capitellar shear fractures. In noncomparative studies, the complication rate was higher following the extended lateral compared to the anterolateral approach for Dubberley A fractures. Additionally, the reported complications following the extended lateral approach may impact long-term outcomes. Insufficient comparative evidence currently exists to recommend one approach over the other. High-quality comparative studies are needed.

Orthopedic surgery, Diseases of the musculoskeletal system
DOAJ Open Access 2022
Bilateral Posterior Fracture-Dislocation of the Shoulder after Electrical Shock Treated with Bilateral Hemiarthroplasty: A Case Report

Kechagias VA, Katounis CA, Badras SL et al.

This is a case report of a bilateral posterior fracture- dislocation of the shoulder after electrical shock and presents the first such patient treated with bilateral shoulder hemiarthroplasty. At first presentation, the upper limbs of the patient were in a position of internal rotation, and passive and active external rotations were painful. Radiographs and computed tomography of both shoulders showed bilateral posterior fracture-dislocation. Defects over 50% of the head articular surfaces led to unstable and unsuccessful initial closed and open reductions. The patient was then treated with cemented hemiarthroplasties with very good functional results two years post-operatively. This case presentation underlines the critical value of systematic clinical and radiographic evaluation of severe bilateral shoulder fracture- dislocations, prior to the ultimate proper treatment with cemented humeral shoulder hemiarthroplasties, followed by appropriate rehabilitation programme, for successful patient outcomes.

Orthopedic surgery
DOAJ Open Access 2022
Percutaneous two unilateral iliosacral S1 screw fixation for pelvic ring injuries: a retrospective review of 38 patients

Whee Sung Son, Jae-Woo Cho, Nam-Ryeol Kim et al.

Purpose Percutaneous iliosacral (IS) screw fixation for pelvic ring injuries is a minimally invasive technique that reduces the amount of blood loss and shortens the procedure time. Moreover, two unilateral IS S1 screws exhibit superior stability to a single IS screw and are also safer for neurological injuries than an S2 screw. Therefore, this study aimed to evaluate fixation using percutaneous two unilateral IS S1 screws for pelvic ring injuries and its subsequent clinical outcomes. Methods We retrospectively reviewed 38 patients who underwent percutaneous two unilateral IS S1 screw fixation for pelvic ring injuries. The procedure time, blood loss, achievement of bone union, radiological outcomes (Matta and Tornetta grade), and postoperative complications were evaluated. Results The mean procedure time, hemoglobin loss, bone union rate, and time to union were 40.1 minutes (range, 18–102 minutes), 0.6 g/dL (range, 0.3–1.0 g/dL), 100%, and 153.2 days (range, 61–327 days), respectively. The Matta and Tornetta grades were excellent, good, and fair in 24 (63.1%), 11 (28.9%), and three patients (7.9%), respectively, and the postoperative complications were S1 screw loosening, widening of the symphysis pubis (2.3 and 2.5 mm), lumbosacral plexopathy, and S1 radiculopathy in one (2.6%), two (5.3%), one (2.6%), and one patient (2.6%), respectively. However, all neurological complications recovered spontaneously. Conclusions Percutaneous two unilateral IS S1 screw fixation was useful for treating pelvic ring injuries. In particular, it involved a short procedure time with little blood loss and also led to 100% bone union and good radiological outcomes.

Medical emergencies. Critical care. Intensive care. First aid
arXiv Open Access 2021
Approximation of mean curvature flow with generic singularities by smooth flows with surgery

Joshua Daniels-Holgate

We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We conclude by improving the entropy bound on the low-entropy Schoenflies conjecture.

en math.DG, math.AP
DOAJ Open Access 2021
Predictors of In-Hospital Mortality in Older Adults Undergoing Hip Fracture Surgery: A Case–Control Study

Ming-Hsiu Chiang, Huan-Ju Lee, Yi-Jie Kuo et al.

Introduction: Although surgery has been proven to improve the long-term survival of older adults with hip fracture, in-hospital mortality directly resulting from repair of hip fracture is undesirable. This study aimed to identify potential prognostic factors that predict in-hospital mortality risk in elderly patients following hip fracture surgery. Materials and Methods: This case–control study comprehensively collected data from older adults with hip fracture admitted to a single medical centre. Age was selected as the cross-matching factor. Univariate and binary multivariate logistic regression models were used to estimate the odds ratios with 95% confidence intervals. A receiver operating characteristic curve was constructed to quantify the discrimination power of the model. Results: Among a total of 841 older adults who received hip fracture surgery, 17 died during hospitalisation, yielding a 2.0% in-hospital mortality rate. Using a binary multivariate logistic regression model to perform a comparison with 51 age-matched patients in survival groups, the model revealed that estimated glomerular filtration rate (eGFR) and malignant cancer history were the only 2 factors significantly correlated with in-hospital mortality. The prognostic values for the eGFR and malignant cancer history were acceptable, with areas under the curve of .76 and .67, respectively. Conclusion: The prevalence of in-hospital mortality following hip fracture is low. After adjustment for age, eGFR and malignant cancer history were identified as factors significantly correlated with in-hospital mortality. The findings of this study could assist in the early screening and detection of patients with high in-hospital mortality risks.

Orthopedic surgery, Geriatrics
arXiv Open Access 2020
Surgery Approach to Rudyak's Conjecture

Alexander Dranishnikov, Jamie Scott

Using the surgery we prove the following: THEOREM. Let $f:M \to N$ be a normal map of degree one between closed manifolds with $N$ being $(r-1)$-connected, $r\ge 1$. If $N$ satisfies the inequality $\dim N \leq 2r \cat N - 3$, then for the Lusternik-Schnirelmann category $\cat M \geq \cat N$ .

en math.AT, math.GT

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