Euclidean Distance Research Articles

Abstract Kaolin, as a versatile non-metallic mineral resource, is widely used in ceramics, casting, construction, and other fields. To rapidly and accurately identify and classify three polymorphs of kaolin, while addressing the challenge of aligning resource characteristics with application requirements, a novel method for kaolin identification and classification has been proposed. This method employs terahertz time-domain spectroscopy (THz-TDS) in conjunction with cluster analysis (CA) and principal component analysis (PCA). It facilitates the extraction of spectral data, the calculation of Euclidean distances, and effective dimensionality reduction of the dataset. Terahertz spectroscopy, a non-destructive analytical technique, was employed to obtain terahertz spectra from three kaolin specimens using a transmissive terahertz spectrometer. The refractive index, dielectric constant, and absorption coefficient of the samples within the frequency range of 0.5 THz to 2.75 THz were calculated from the terahertz time-domain spectroscopy data, utilizing established equations, including the fast Fourier transform. Using all available refractive index and absorption coefficient data within this frequency band as input variables, cluster analysis and principal component analysis were conducted separately to determine the Euclidean distance and the first principal component (PC1) for the corresponding samples. The findings reveal significant variations in the refractive index and absorption coefficient among the analyzed samples. Principal component analysis (PCA) identified three primary components for both the refractive index and absorption coefficient, which accounted for cumulative contribution rates of 99.99% and 97.26%, respectively. Cluster analysis (CA) categorized the samples into three distinct groups based on Euclidean distance metrics, with the CA clusters closely aligning with the PCA results. The sandy kaolin (SZGLT-GX325) exhibited the highest PC1 score and the greatest Euclidean distance from the second group, measuring 156.16. Utilizing the CA-PCA model, the three types of kaolin were accurately identified and classified based on inter-sample variations, achieving a classification accuracy of 99.99%. This study underscores the utility of terahertz time-domain spectroscopy combined with chemometric techniques, providing a robust framework for the rapid, precise, and non-destructive analysis of kaolin, thereby demonstrating significant potential for practical applications.

Coffea racemosa and C. zanguebariae show promising characteristics for cultivation under stress conditions. However, their potential for breeding programs requires further characterization, especially regarding fruit attributes. This study aimed to characterize the bean/husk ratio and the nutrient content in bean and husks from 22 accessions of Coffea racemosa and another 22 of C. zanguebariae cultivated in Mozambique. Ripe fruits were collected, dried, and manually peeled to evaluate the percentage of bean and husk. The nutrient content (N, P, K, Ca, Mg, S, Fe, Zn, Cu, Mn, and B) was quantified separately by standard methodology. The data were summarized in scatter plots, and differences among accessions were analyzed using Euclidean distance and grouped following the Unweighted Pair Group Method with Arithmetic Mean. On average, beans accounted for 54.4% (ranging from 34.5% to 66.5%) of the fruit mass in C. racemosa and 60.4% (38.8% to 81.4%) in C. zanguebariae. Macronutrient content in beans followed the order N > K > Mg > P > S > Ca (average N = 19.98 kg ton−1 of beans) in C. racemosa and N > K > Ca > Mg > S > P (average N = 25.42 kg ton−1 of beans) in C. zanguebariae. Micronutrient content in beans followed the order Fe > B > Mn > Cu > Zn in both species, with average Fe content of 325.8 and 473.72 g ton−1 of beans for C. racemosa and C. zanguebariae, respectively. No correspondence occurred between the bean and husk nutrient content. Coffea racemosa and C. zanguebariae exhibit bean proportions and nutritional profiles comparable to those of commercial species, highlighting their high potential for coffee diversification and genetic breeding. These results provide new evidence supporting the inclusion of C. racemosa and C. zanguebariae in breeding programs aimed at climate-resilient and nutritionally distinct coffee varieties.

When developing new apple varieties (Malus domestica Borkh.), selection breeders should take into account not only the specifi c features of the cultivation area but also the desired set of economically valuable characteristics, including fruit storability with minimal quality loss. Apple fruits should meet a number of essential criteria, important to both producers and consumers. New genotypes should maximally conform to the characteristics of a commercial “model” variety developed and adapted to particular weather conditions. The parameters of storability under low storage temperatures should be accompanied by minimal quality loss and a high resistance to physiological disorders and microbiological diseases during storage. This study was aimed at identifying the most promising apple varieties in terms of an integrated set of storability criteria using “simulation” and “optimal” variety models. A simulation model of an “ideal” apple variety was developed, taking into account the criteria of fruit storability and durability during storage. On this basis, an “optimal” apple variety model for the central part of the Russian Federation (Central Region) was developed. The “ideal” and “optimal” models were described using 16 post-harvest criteria for 35 apple varieties studied over a 20-year period. The data were processed using a hierarchical clustering algorithm (HCA) with the Ward method and factor analysis with a multivariate principal component analysis to identify the most promising apple genotypes based on their maximum similarity to the model in terms of a set of storability traits and fruit resistance to physiological disorders and microbiological diseases during refrigeration at various temperatures. The degree of similarity between the developed models and the studied genotypes was determined based on the Euclidean distance. The Svezhest and Orlovsky Partizan varieties were closest to the “ideal” model. The criteria for the Svezhest variety were used as the basis for developing an “optimal” model. The Yubiley Moskvy, Orlovsky Partizan, Start, and Turgenevskoye varieties were closest to the “optimal” model.

Hyperspectral imaging (HSI) has become a powerful remote sensing and material analysis tool because it can capture detailed spectral information in hundreds of adjacent bands. Nevertheless, the high dimensionality and redundancy in HSI data make precise and efficient classification challenging. This paper presents an extensive comparative study of both traditional and state-of-the-art Machine Learning algorithms for HSI classification. Classical classifiers like Support Vector Machines (SVM) and K-Nearest Neighbors (KNN) are compared with state-of-the-art methods like Collaborative and Sparse Representation-based approaches, Convolutional Recurrent Neural Networks (CRNN), Classification and Regression Trees (CRT), and Local Fisher Discriminant Analysis with Class Mean Modeling (LFDA-CMM). The specific emphasis is given to the Nearest Regularized Subspace (NRS) classifier family, which utilizes different distance measures—i.e., Spectral Angle Mapper, Manhattan, Euclidean, Cosine, and Chi-square distances—to achieve improved classification accuracy. Experimental results on two benchmark datasets, Indian Pines and University of Pavia, show that the proposed NRS-MD method consistently achieves better performance in accuracy, Kappa coefficient, and computational complexity. These results emphasize the capability of subspace models with regularization to meet the challenges of hyperspectral image classification and present valuable information for choosing appropriate methods in practical applications.

Application of Euclidean distance for multi-trait selection in in vitro–derived sugarcane mutant candidates

This paper analyzes how to improve the performance of luminescent solar concentrators based on dye-doped POFs stacked in layers by adjusting the geometric distances between them and choosing appropriate dyes. Improvements in the external photon efficiency greater than 10% could be achieved in this way. These improvements complement other reported ones, including a significant boost in output power when a layer of 10 POFs is used instead of a single POF, far exceeding the tenfold increase in the number of fibers, or an additional large enhancement by adding just one more layer. Our study's results are related to the degree of matching between the spectral absorption band of the dye used in each layer and the incident power spectrum that reaches the layer. This study focuses on the promising dyes Lumogen Red and Lumogen Yellow, using Monte Carlo simulations and experimental results. It opens the door to new analyses using stacked polymer fibers with improved light-trapping properties.

We construct a family of self-adjoint operators on the prime numbers whose entries depend on pairwise arithmetic divergences, replacing geometric distance with number-theoretic dissimilarity. The resulting spectra encode how coherence propagates through the prime sequence and define an emergent arithmetic geometry. From these spectra we extract observables such as the heat trace, entropy, and eigenvalue growth, which reveal persistent spectral compression): eigenvalues grow sublinearly, entropy scales slowly, and the inferred dimension remains strictly below one. This rigidity appears across logarithmic, entropic, and fractal-type kernels, reflecting intrinsic arithmetic constraints. Analytically, we show that for the unnormalized Laplacian, the continuum limit of its squared Hamiltonian corresponds to the one-dimensional bi-Laplacian, whose heat trace follows a short-time scaling proportional to t−1/4. Under the spectral dimension convention ds=−2dlogΘ/dlogt, this result produces ds=1/2 directly from first principles, without fitting or external hypotheses. This value signifies maximal spectral compression and the absence of classical diffusion, indicating that arithmetic sparsity enforces a coherence-limited, non-Euclidean geometry linking spectral and number-theoretic structure.

. The mixed geographically weighted regression model has been extensively studied both in its estimation and applications. However, in most existing research results of the model, the spatial weights are commonly computed based on the Euclidean distance, which is a straight-line distance metric. Considering the diversity of the sample data and the complexity of the geography, the Euclidean distance may be inappropriate. Thus, it is necessary to define an appropriate distance metric when calibrating a space model. In this article, a class of non Euclidean distance metrics – Minkowski distance, by varying its exponent parameter and the coordinate rotation, is used to compute the spatial weights in calibrating the mixed geographically regression model. Considering the fact that the two-step estimation procedure is less time-consuming and can obtain explicit expressions of the coefficients, we mainly focus on the performance of the two-step estimation method with Minkowski distance. Simulation results demonstrate that the proposed method has more robustness to the sample data and tends to obtain more accurate constant and varying coefficient estimates. A real-world dataset is analyzed to show the application of the proposed estimation method and the article is ended with a conclusion.

This study aimed to develop a portable mixed reality navigation (PMRN) system for neurosurgery and assess its feasibility, providing design insights for the optimization and wider adoption of mixed reality navigation. The PMRN system linked a head-mounted display (HMD) and a portable laptop via a router, forming a local network. Active infrared tracking enabled real-time localization of a custom surgical probe. Accuracy was first tested in a laboratory using simulation models, then evaluated in a prospective clinical study of 42 patients with intracranial lesions. Clinically, the probe of a traditional optical navigation (TON) system was placed at PMRN-localized targets, and the Euclidean distance between the two systems was measured to quantify localization error. Doctors also subjectively assessed the alignment of mixed reality holograms with patient anatomy to evaluate reliability. In laboratory testing, two doctors each performed 10 simulated head model localizations, with an average time of under 5 min. The times decreased with practice and stabilized at approximately 3.5 min. The maximum fiducial registration error (FRE) and target registration error (TRE) were 2.5 mm and 2.3 mm, respectively, with no significant difference between doctors. In the clinical study (n = 42 patients), the mean localization time was 4.3 ± 1.0 min, the FRE was 2.2 ± 0.7 mm, and the TRE was 1.7 ± 0.5 mm. Doctors rated the hologram–patient alignment as satisfactory in 40 cases (95%); the remaining two cases showed a 3 mm deviation, which was attributed to slight patient movement or reduced device performance in low-battery mode. The PMRN system is portable, easy to use, and achieves millimeter-level accuracy in neurosurgical procedures with high surgeon satisfaction. It meets the needs of most routine operations but requires further refinement for submillimeter-precision procedures. These results confirm its clinical feasibility and support its further development and adoption in mixed reality neurosurgical navigation.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-24603-6.

Banana (Musa spp.) is an important fruit production in Brazil, but crop productivity is still too low. The ‘Nanica’ cultivar and fertigation have been introduced, but more accurate guidelines are needed to support fertilization decisions at the orchard scale. This study aimed to develop customized nutrient standards for fertigated ‘Nanica’. A commercial ‘Nanica’ orchard provided 129 observations on yield and foliar nutrient concentrations from 2010 to 2017 in eight groves of 3.26 ha each. Plant density averaged 1479 plants ha−1. The diagnostic leaf was analyzed for 13 elements. Concentration values were transformed into centered log ratios (clr), weighted log ratios (wlr), and isometric log ratios (ilr) to account for nutrient interactions and normalize the data. Yield cutoff between low- and high yielders was set at 27 t ha−1 semester−1. The XGBoost classification models relating yield to tissue composition returned an area under curve averaging 0.715 for log ratio expressions. Nutrient standards were expressed as clr, wlr, and raw concentration means and standard deviations of performing specimens. The clr and wlr diagnoses of a low-yielding and imbalanced specimen against a benchmark specimen (Euclidean distance = 2.5) or the performing subpopulation (Mahalanobis distance = 37.6, p < 0.01) indicated Mn shortage and Na excess. Sufficiency concentration ranges may not agree with log ratio diagnoses, especially for Mn. The clr and wlr nutrient standards were site-specific, supporting precision farming. The concept developed in this paper is applicable to endogenous research conducted by stakeholders in orchards worldwide.

BackgroundMild traumatic brain injury (mTBI) affects millions worldwide and frequently leads to secondary depression. Early identification of high-risk individuals is critical for targeted mental-health screening in this population. Data-driven phenotyping offers a promising avenue to unmask hidden symptom patterns, but few studies have combined unsupervised clustering of post-concussion profiles with established clinical and psychosocial metrics. We aimed to classify post-concussion symptom-severity profiles in adults with mTBI and to evaluate their association with secondary depression risk, adjusting for Glasgow Coma Scale (GCS) score, psychological resilience, age, sex, and time since injury.MethodsIn this cross-sectional analysis, 249 adults with mTBI (GCS 13–15) were recruited from a tertiary hospital in northern Taiwan. We performed hierarchical clustering using Ward’s method with Euclidean distance (with BIC support) to derive three symptom-severity phenotypes from the Rivermead Post-Concussion Questionnaire items, then used k-means clustering to assign individuals to these by minizing within-cluster variance. Depression, defined as a Beck Depression Inventory-II ≥ 13, was modeled as an outcome in generalized linear models, adjusting for GCS and psychological resilience. Model discrimination was evaluated via area under the receiver operating characteristic curve (AUC).ResultsThree distinct symptom clusters (mild, moderate, severe) were identified. The severe cluster was characterized by prominent visual symptoms, including light sensitivity and double vision. Compared with the mild cluster, the moderate cluster had 5.06-fold higher depression odds (95% CI [2.08–12.31]; p < .001) and the severe cluster 17.17-fold higher odds (95% CI [5.66–52.14]; p < .001). Higher resilience was independently protective (OR = 0.95, 95% CI [0.93–0.96]; p < .001), as was each additional GCS score (OR = 0.20, 95% CI [0.06–0.62]; p = .005). The full model showed excellent discrimination with an AUC of 88%, 95% CI [0.83–0.92].ConclusionsOur data-driven approach shows that distinct post-concussion symptom-severity phenotypes, when integrated with GCS and resilience metrics, yields a robust tool for identifying mTBI survivors at high risks of depression. These findings support early, targeted mental-health screening and lay the groundwork for prospective validation and personalized intervention strategies.Clinical trial numberNCT04243226. Registered on January 20. 2020.Supplementary InformationThe online version contains supplementary material available at 10.1186/s12888-025-07512-w.

This study presents an improved MOPSO algorithm, ATP-QL-MOPSO, for lightweight and crashworthiness optimization of automotive battery pack systems (BPS). Traditional MOPSO struggles with hyperparameter tuning and local optima. The proposed method integrates Q-learning (QL) and adaptive t-distribution perturbation (ATP) to address these issues. In QL, particles act as agents with independent velocity updates, using Euclidean distance and three velocity parameters as state and action spaces. ATP dynamically adjusts the t-distribution shape to avoid local optima. ATP-QL-MOPSO showed improved performance in ZDT tests with reduced Inverted Generational Distance (In ZDT1, IGD is reduced by 55.6% compared to standard MOPSO). Applied to BPS, it achieved higher hypervolume values, increased X -direction displacement by 6.044%, decreased Y -direction displacement by 2.787%, and reduced mass by 3.273%. This demonstrates that QL automates hyperparameter tuning, while ATP improves convergence, making it superior to MOPSO in optimizing BPS for weight reduction and crash resistance, with potential for broader applications.

Background: Chronic inflammation is a key driver of cardiovascular disease progression and may impact transcatheter aortic valve replacement (TAVR) outcomes. Higher periaortic adipose tissue (PAAT) attenuation reflects aortic wall inflammation and can be measured on routine preprocedural CT. Yet, its prognostic value in TAVR patients remains unclear. Aim: To explore whether PAAT attenuation predicts long-term mortality in TAVR patients beyond traditional risk factors. Methods: We retrospectively analyzed preprocedural CT scans from consecutive TAVR patients treated at a single tertiary center between 2013 and 2023. The aorta was automatically segmented using a deep learning-based segmentation tool (TotalSegmentator) from the sinotubular junction to the distal infrarenal segment. PAAT attenuation was defined as the mean attenuation (Hounsfield units, HU) of all voxels within a 10mm radial cylinder around the aortic wall and an attenuation range of -190 to -30 HU. PAAT attenuation was associated with 5-year all-cause mortality using Cox regression models, adjusting for technical parameters (tube voltage, signal-to-noise ratio, BSA-indexed PAAT volume) and clinical covariates (age, sex, BMI, and Society of Thoracic Surgeons [STS] risk score). Incremental predictive value of PAAT attenuation was evaluated using Harrell’s C-statistic, and a high-attenuation threshold was derived by Euclidean distance within a receiver operating characteristic framework. Results: The study included 1,003 patients (51.5 % male, mean age 80±8y, BMI 28.6±6.0 kg/m 2 , median STS score 4.3 [2.5–7.0]%), followed for a median 22 (14–37) months; 5-year mortality rate was 23.6% (n=237). Mean PAAT attenuation was -77.3±7.3 HU. Non-survivors had higher mean PAAT attenuation than survivors (-75.9 vs -77.8 HU, P&lt;0.001), Figure 1 . Those with high PAAT attenuation (&gt;-77 HU) were slightly older, and more often female (both P≤0.05). PAAT attenuation independently predicted mortality (aHR [per 10 HU] 1.71, 95%-CI: 1.29–2.26; P&lt;0.001) after adjustment. Adding PAAT attenuation to the clinical model (age, sex, BMI, STS score) improved discrimination for 5-year death (Harrell’s C from 0.69 to 0.70; P&lt;0.001). Conclusions: High periaortic fat attenuation on preprocedural CT independently predicts long-term mortality in patients undergoing TAVR. Quantifying PAAT inflammation may offer additional prognostic value beyond established clinical risk factors and refine preprocedural risk stratification.

To accurately predict short-term traffic flow, considering the time characteristics of traffic flow and combining the characteristics of DFT, KNN and LSTM models, a DFT-KNN-LSTM hybrid model is proposed. The model first uses the DFT method to decompose the traffic flow data into trend term and residual term data to remove the influence of residual term data on traffic flow prediction. Secondly, the KNN algorithm based on Euclidean distance is used to screen the traffic flow data with high similarity between K days and target forecast days in the data. Furthermore, the filtered data are used as the training set, and the target day's data are used as the test set, which is substituted into the LSTM model for prediction; Finally, the mean absolute error ( MAE ), mean square error ( MSE ) and root mean square error ( RMSE ) were used as evaluation indexes to analyze and evaluate the prediction results. Taking the traffic flow data collected from Xizhaosi Street in Dongcheng District of Beijing as an example, the prediction performance of the combined model is analyzed. The results show that compared with other commonly used prediction models, the MSE of the DFT-KNN-LSTM combined model is improved by 3.24 % ~ 19.05 %, RMSE is improved by 1.54 % ~ 9.98 %, and MAE is improved by 3.05 % ~ 8.97 %. It can be seen that the combined model has better prediction performance than the traditional single model and other combined models, and can be better applied to short-term traffic flow prediction.

Abstract Accurate State of Health (SOH) estimation for Lithium Batteries (LIBs) is essential for reliable energy management. However, complex nonlinear degradation and noise-sensitive data make it difficult. To address these limitations, this study proposes an advanced framework integrating the improved Dandelion Optimization (IDO) algorithm with a Gated Recurrent Unit (GRU) neural network. The proposed method first extracts critical health indicators from raw battery cycling data. To mitigate the adverse effects of noise during data acquisition, a Gaussian filtering technique is applied to preprocess the extracted features. The core innovation lies in the IDO-GRU architecture, where the GRU network’s hyperparameters are optimized by an improved Dandelion Optimization algorithm. The IDO incorporates three key improvements: a Euclidean distance strategy to enhance population diversity, a golden sine search mechanism to improve local exploitation, and adaptive inertia weights to balance global exploration. Experiments on NASA and CALCE datasets show remarkable accuracy. The results show that the maximum RMSE is 0.0048 and MAPE is 0.50% on the CALCE dataset, while on the NASA dataset, RMSE is 0.0013 and MAPE is 0.11%. The proposed framework demonstrates robust adaptability across diverse battery chemistries, thereby offering a novel and scalable solution for real-world SOH monitoring in electric vehicles systems.

We propose a geometry topological framework to analyze storm dynamics by coupling persistent homology with Anti-de Sitter (AdS)-inspired metrics. On radar images of a bow echo event, we compare Euclidean distance with three compressive AdS metrics (α = 0.01, 0.1, 0.3) via time-resolved H1 persistence diagrams for the arc and its internal cells. The moderate curvature setting (α=0.1) offers the best trade-off: it suppresses spurious cycles, preserves salient features, and stabilizes lifetime distributions. Consistently, the arc exhibits longer, more dispersed cycles (large-scale organizer), while cells show shorter, localized patterns (confined convection). Cross-correlations of H1 lifetimes reveal a temporal asymmetry: arc activation precedes cell activation. A differential indicator Δ(t) based on Wasserstein distances quantifies this divergence and aligns with the visual onset in radar, suggesting early warning potential. Results are demonstrated on a rapid Corsica bow echo; broader validation remains future work.

Introduction: Type B aortic dissection (TBAD) is a serious condition that may benefit from early endovascular repair. In the acute phase (aTBAD; &lt;14 days onset), the high compliance of the dissection flap improves success of endovascular repair. In the chronic phase (cTBAD; &gt;90 days onset), the flap stiffens and makes endovascular repair less effective. This study aimed to better characterize dissection flap behavior and understand the cause of stiffening. Hypothesis: We hypothesize that remodeling of the dissection flap in TBAD leads to decreased dynamic displacement over time, driven by structural thickening and stiffening. Methods: We combined in vivo imaging and ex vivo tissue analyses. A total of 15 2D phase-contrast magnetic resonance imaging (PC-MRI) datasets were analyzed (4 cTBAD; 11 aTBAD). For dissection flap thickness, a separate cohort of 41 tissue samples were used (20 cTBAD; 10 aTBAD, 11 control). We used an in-house MATLAB code to analyze flap motion in the 2D PC-MRI data. We extracted maximum displacement by calculating the Euclidean distance between corresponding points on the flap at systole and diastole. After surgical excision, dissection flap samples were placed into cryopreservation medium (10% DMSO + 90% RPMI) and stored in a -80 °C freezer. Prior to measurement, tissue was thawed at around 37 °C, and tissue thickness was measured by a micrometer. Control dissection flaps were created by peeling healthy descending aortic tissue at the level of the media and combined intimal and partial medial layers were measured together as control flap thickness. Group differences were assessed using the Mann-Whitney U test. Results: 2D PC-MRI revealed large differences in dissection flap displacement between cTBAD and aTBAD (Fig. 1A,B). Mean maximum displacements in the acute group (5.4±1.9 mm) were significantly higher than chronic (1.5±1.1 mm; p &lt; 0.01). Thickness measurements of the dissection flaps varied greatly: cTBAD = 1.8±0.4, aTBAD = 1.3±0.4, control = 0.8±0.1 mm. All group comparisons revealed significant differences (Fig. 1C). Since bending stiffness is proportional to the cube of thickness, we can also expect a significant increase in chronic dissection flap stiffness. Conclusions: Dissection flaps in chronic TBAD exhibit reduced motion and increased stiffness compared to the acute phase. These properties are related to structural thickening which highlights the importance of early intervention before adverse remodeling occurs.

Background/Objectives: Hallux valgus (HV), a common foot deformity, is difficult to quantify beyond traditional angular measurements. This study introduces a novel dynamic distance mapping technique to visualize HV progression and identify spatial features linked to severity. Methods: A retrospective analysis of 335 feet from 178 patients undergoing HV surgery at Hasharon Hospital, Israel (2014–2024), utilized custom Python software to annotate 24 landmarks on preoperative standing anteroposterior radiographs. This generated 276 normalized Euclidean distances, analyzed via Pearson correlation against HV angles (HVA, IMA, DMAA, HIA). Results: Seven distances correlated negatively (r &gt; 0.4, p &lt; 0.05) and seven positively with HVA, involving the distal phalanx, sesamoids, and second metatarsal. Eleven distances showed strong positive correlation (r &gt; 0.4, p &lt; 0.05) with IMA, reflecting displacement patterns. Moderate correlations were observed with DMAA (six negative, r −0.3 to −0.4; two positive, r 0.3 to 0.4, p &lt; 0.05) and HIA (two negative, r −0.3 to −0.4, p &lt; 0.05). Visualizations highlighted progressive spatial changes. Conclusions: Dynamic distance mapping provides valuable insights into hallux valgus (HV) progression, as evidenced by significant correlations with HVA and IMA, supporting its potential role in surgical planning. However, its ability to capture 3D deformities requires validation against weightbearing computed tomography (WBCT). Future research should explore correlations with specific indications for corrective osteotomies to enhance clinical applicability.

Azacytidine is the only approved treatment for patients with higher-risk myelodysplastic syndromes (MDS); yet less than half of the patients will achieve a response, whereas the duration of response is highly heterogeneous and there are no predictors for response duration. The aim of this study is to estimate the patient’s time to loss of response (LoR) to azacytidine based on clinical measurements during treatment. To this end, a personalized prediction framework is proposed that estimates the LoR of a new patient using a patient similarity-based approach. Namely, the new patient’s clinical data—represented as a multivariate time series—are compared to a reference set of patients. The comparison uses distance metrics that quantify how similar two patients’ time series are, assuming patients with similar trajectories tend to have similar LoR. Then, the LoR of the new patient is predicted by averaging the outcomes of the most similar reference patients. The pipeline includes a data normalization strategy that centers each feature on its baseline value and scales it to highlight relative changes and distance metrics to quantify similarity. Both real-world and simulated data were utilized to evaluate the proposed methodology, employing the leave-one-out validation and the Mean Absolute Percentage Error (MAPE) to assess accuracy. The estimated MAPE was found to be 30.52% and 11.82% in the real-world and simulated dataset, respectively. The best and most robust predictions were achieved using the Euclidean distance metric and setting the number of most similar patients around three to five. This study proposes a personalized predictive approach for the LoR to azacitidine in the MDS clinical setting, demonstrating potential for a serviceable prediction of LoR and forming the foundation for further research.

We construct a set of $2^n$ points in $\mathbb{R}^n$ such that all pairwise Manhattan distances are odd integers, which improves the recent linear lower bound of Golovanov, Kupavskii and Sagdeev. In contrast to the Euclidean and maximum metrics, this shows that the odd-distance set problem behaves very differently to the equilateral set problem under the Manhattan metric. Moreover, all coordinates of the points in our construction are integers or half-integers, and we show that our construction is optimal under this additional restriction.