Search Results (885)

Background/Objectives: Large language models have created new pathways for patients to access health information, yet little is known about how the general population uses these conversational artificial intelligence (AI) tools for oral health concerns. This study investigated patterns of AI use as a source of oral health information, the nature of data shared with these systems, users’ perceptions, and the impact on dental care-seeking behavior. Methods: A cross-sectional study was conducted among adults from Bihor County, Romania, using a structured 16-item online questionnaire distributed via social media, with eligibility restricted to individuals who had previously used conversational AI for oral health information. The final sample comprised 393 valid responses from this self-selected group of users. Fisher’s exact test and Z-tests with Bonferroni correction were applied (α = 0.05). Results: Most participants were female (68.2%), university-educated (52.9%), and lived in an urban setting (88.3%). Significant differences in patterns of AI use for oral health information were identified according to age, sex, and living environment (p < 0.001). Younger participants used AI more frequently, while older individuals perceived the information as less clear. The vast majority used AI for informational or preliminary guidance purposes, with very few treating it as a substitute for professional opinion. A relevant subset shared visual data (intraoral photographs or radiographs) with AI systems, raising data privacy concerns. Rural participants more frequently delayed dental visits and less often discussed AI-derived information with their dentist compared to urban participants. Conclusions: When used by the public as a source of oral health information, AI is increasingly adopted, with adoption shaped by age, sex, and socioeconomic context. These findings concern only this informational use and do not extend to other applications of AI in dentistry, such as diagnostic support, image analysis, or clinical decision-making. Dental professionals should proactively engage patients about their use of AI for oral health information to ensure that digitally obtained content is appropriately contextualized. Full article

Deaf, deafblind, and hard-of-hearing (DDBHH) American Sign Language (ASL) users assigned female at birth experience disparities in cervical cancer screening. There is a need to further clarify the roles of education status and English proficiency in adherence to the U.S. Preventive Services Task Force (USPSTF) screening guidelines. Using the NCI Health Information National Trends Survey in ASL, we surveyed participants; those who reported being female and within the USPSTF age-eligible range (21–65) were included in analyses. Associations were determined using Pearson Chi-squared tests. A two-sided p ≤ 0.05 was considered significant. Six hundred and forty-three participants, recruited between July 2023 and October 2025, answered questions in ASL and English, with 65% self-reporting screening adherence. Chi-square results indicated that more years of education were significantly associated with screening adherence [X2 (2) = 9.72, p = 0.008]. Consistent with the results, English proficiency was not a significant predictor of screening adherence [Fisher’s exact p = 0.267]. Those who self-report as HS graduates and/or have left college demonstrate lower adherence as well. These findings suggest that educational attainment may be an important factor associated with screening adherence among DDBHH ASL users and warrants further investigation in efforts to improve adherence toward Healthy People 2030 targets. Full article

Objectives: This retrospective study investigated the incidence and risk factors of dry socket following mandibular third molar extraction. Methods: This retrospective analysis utilized clinical information and diagnostic X-ray images of patients who underwent tooth extraction by a single oral and maxillofacial surgeon between June 2020 and March 2024. Dry socket was diagnosed according to established clinical criteria. Panoramic radiographs were used to assess angulation (Winter’s classification), depth and gingival coverage. Statistical analyses were conducted to examine differences in dry socket incidence. Categorical comparisons were performed using chi-square or Fisher’s exact tests. Multivariate logistic regression was used to identify independent risk factors for dry socket. Odds ratios with 95% confidence intervals were calculated. Results: A total of 3033 mandibular third molars were extracted from 2384 patients (859 males and 1525 females). The overall incidence of dry socket was 2.6% (80 cases), with a significantly higher rate in female patients (3.3%) than in male patients (1.4%) (p < 0.01). Increased risk was observed in age ≥ 30 years compared to early 20 s (p < 0.01), and the use of oral contraceptives (p < 0.01). Multivariate logistic regression also identified these three factors as significant independent risk factors. Conclusions: These findings suggest that patient-related factors influencing healing capacity contribute to dry socket development. Recognizing these associations may assist clinicians in identifying high-risk cases and implementing preventive strategies to reduce postoperative complications in third molar surgery. Full article

Background/Objectives: Root canal treatment (RCT) may be required after full-coverage crown placement in initially vital teeth due to biological and restorative factors affecting pulp vitality. However, the timing and distribution of clinical variables associated with teeth requiring RCT after crown placement remain insufficiently described. The aim of this study was to evaluate this timing and distribution using retrospective data collected between 2011 and 2024. Methods: This study included 588 vital teeth treated with metal–ceramic or zirconia crowns between 2011 and 2024. Recorded variables were age, gender, tooth type and location, crown material, bruxism, number of abutment teeth, and time to RCT. Statistical analyses were performed using SPSS v25.0, and as data were not normally distributed (Shapiro–Wilk test), Mann–Whitney U, Kruskal–Wallis, and chi-square or Fisher’s exact tests were used (p < 0.05). Results: Of the 588 teeth, 36.1% were from male and 63.9% from female patients, and most crowns were metal–ceramic (80.8%), followed by zirconia (19.2%). Significant associations were observed between tooth group and jaw location (p < 0.001), number of abutment teeth (p < 0.001), and crown material type (p < 0.001). RCT was most frequently recorded during the fifth year after crown placement (52.6% of cases), with a mean time of 4.13 ± 1.79 years. Tooth extraction was observed in 5.3% of teeth. Conclusions: This study provides descriptive information regarding the distribution and timing of RCT in initially vital teeth following fixed prosthodontic restorations. The findings should be interpreted as a characterization of affected cases rather than as evidence of risk factors or predictors of RCT occurrence. Full article

Quantum gate estimation and tomography pipelines routinely combine intrinsically defined likelihoods with priors or regularization terms specified in local Euclidean coordinates. This practice implicitly replaces the Haar reference measure on $SU\left(2\right)$ with Lebesgue measure, specifying a different statistical model rather than a reparametrization of the intended one. We show that omitting the associated chart-volume factor alters the optimization objective itself, modifying its gradient field and stationary-point structure. The gradient discrepancy $\nabla L_G-\nabla L_E=-\nabla \log J_{\exp }$ is nonzero for all $v\ne 0$ so that flat-coordinate surrogate objectives can converge to points that are non-stationary for the corresponding Haar-consistent objective even in regimes where local Gaussian approximations are assumed valid. We prove a formal non-equivalence proposition and validate a leading-order Fisher-information correction analytically and numerically. Large-scale multi-start optimization experiments ($N=\mathrm{11,900}$ runs) demonstrate that the discrepancy is regime dependent and most pronounced under moderate-to-strong regularization or limited data. The fix requires a single-line modification to any gradient-based optimizer. These results identify reference-measure selection as an explicit modeling decision with direct consequences for optimization and inference in gate-set tomography, randomized benchmarking, and Bayesian gate estimation on curved parameter manifolds; quantitative validation is restricted to single-qubit systems, though the mechanism extends to any regularized optimization on a curved parameter manifold. Full article

Carbonation in CO₂-storage solid-waste backfill materials couples CO₂ transport, mineral reaction, and strength evolution, making carbonation-front prediction and transport–reaction inference important for evaluating sequestration performance. This study proposes an evidence-ranked, physics-guided inverse-learning framework for carbonation-front prediction, auxiliary strength reconstruction, PDE-residual assessment, and practical-identifiability analysis. The framework represents carbonation using group-conditioned latent fields of effective CO₂ concentration and remaining reactive capacity, maps latent carbonation degree to measured depth through a differentiable front operator, and reconstructs unconfined compressive strength through a supervised auxiliary head. Empirical front laws and reaction–diffusion physics-informed neural-network variants were evaluated using held-out ranking, repeated stratified splits, residual-weight sweeps, front-operator threshold and smoothing-coefficient sensitivity checks, profile-likelihood and Fisher-information diagnostics, and controlled synthetic tests. Results show that the grouped Weibull front law achieved the best short-range carbonation-depth interpolation, while the retained constant-diffusion PINN was used as a diagnostic formulation within the physics-guided family to improve auxiliary strength reconstruction and to evaluate residual consistency, front-threshold selection, parameter sharing, and inverse-parameter behavior rather than to replace the empirical depth regressor. Increasing the PDE-residual weight substantially reduced residual magnitudes, but profile-likelihood and Fisher-information diagnostics indicated strong parameter trade-offs; the fitted diffusion, reaction, depletion, and diffusion–decay quantities are therefore interpreted as effective, observation-conditional parameters rather than unique material constants. The proposed framework provides a prediction-first and attribution-aware approach for analyzing carbonation evolution in solid-waste backfill materials and supports coordinated assessment of front advancement, strength response, and transport–reaction behavior, while explicitly delimiting the generalization and physical interpretation that can be supported by sparse literature-derived observations. Full article

Background: Impacted maxillary canines frequently cause external root resorption of adjacent lateral incisors, a complication that may compromise tooth survival and orthodontic outcomes. Cone-beam computed tomography (CBCT) enables precise three-dimensional assessment of canine position and resorption severity, yet the quantitative relationship between canine angulation and resorption remains incompletely characterized. Objectives: The objectives were to quantify the correlation between impacted maxillary canine angulation relative to the vertical reference line and the severity of adjacent lateral incisor root resorption using CBCT and to evaluate the influence of crown position on resorption prevalence. Materials and Methods: This retrospective cross-sectional study was conducted at the Department of Oral Radiology, King Abdulaziz University Dental Hospital (KAUDH), Jeddah, Saudi Arabia, using CBCT scans acquired between January 2018 and December 2023. CBCT scans of 262 patients with impacted maxillary canines were analyzed. A total of 108 impacted canines from 88 patients were evaluated. Canine angulation was measured as the acute angle between the canine long axis and a vertical reference line on coronal CBCT sections. Crown position was classified as palatal, buccal, or center. Lateral incisor root resorption was graded using the modified Kaley and Phillips classification (Grade 0–4), where Grade 4 represents side resorption. Statistical analyses included Spearman rank correlation, Mann–Whitney U test, chi-square test, and Kruskal–Wallis test with post hoc comparisons (α = 0.05). Results: Lateral incisor root resorption was detected in 17 of 108 canines (15.7%). Mean canine angulation was significantly higher in the resorption group (45.5° ± 21.4°) compared with the no-resorption group (32.7° ± 18.0°; Mann–Whitney U = 809.5, p = 0.019). Spearman correlation revealed a significant positive association between angulation and resorption grade (ρ = +0.243, p = 0.019). Canines angled ≥ 45° exhibited a 3.5-fold higher resorption rate (32.1% vs. 9.2%; OR = 4.66, Fisher’s Exact p = 0.012). Palatal crown position was associated with significantly higher resorption prevalence (25.5%) compared with center (5.1%) and buccal (11.1%) positions (χ² = 7.258, p = 0.027). Unilateral impaction was associated with significantly higher resorption prevalence than bilateral (22.4% vs. 4.9%; p = 0.031). Conclusions: Larger canine angulation from the vertical reference line is significantly correlated with increased severity of adjacent lateral incisor root resorption. Palatal crown position, angulation ≥ 45°, and unilateral impaction pattern are important risk indicators. CBCT-based angulation measurement provides clinically relevant quantitative data to inform early intervention and treatment planning. Full article

Background: The utilization of robot-assisted surgery in pediatric patients is increasing, with particularly notable advantages in complex reconstructive procedures. This study aims to evaluate the safety and efficacy of robotic-assisted resection of choledochal cysts in infants aged less than 3 months. Methods: A total of 73 infants with choledochal cysts who were admitted to the Department of General Surgery, Children’s Hospital of Zhejiang University School of Medicine, between April 2019 and December 2025 were included. The patients were divided into a robotic-assisted surgery (RAS) group (n = 39) and a laparoscopic-assisted surgery (LAS) group (n = 34). Clinical data, including demographic information, laboratory indexes, surgical data, and prognostic data, were retrospectively reviewed, and the Mann–Whitney U test, independent-samples t-test, and Fisher’s exact test were used for statistical analysis. Results: The groups were comparable in terms of age, sex, weight, pre- and postoperative biochemical markers, fasting time, cyst diameter, and operative time. Overall, 80.8% of cases were prenatally detected. The RAS group had a significantly shorter postoperative hospital stay (p = 0.004, Z = −2.864), drainage tube duration (p = 0.002, Z = −3.100), and hepaticojejunostomy time (p < 0.0001, df = 71, 95%CI (−5.70, −3.04)) compared to the LAS group. In the LAS group, three patients developed anastomotic fistulas, all of whom required reoperation, and one patient developed adhesive bowel obstruction, whereas in the RAS group, one patient developed incision infection, one developed cholangitis, one developed adhesive bowel obstruction, and one presented with postoperative liver function abnormalities. The hospitalization cost in the LAS group was significantly lower than that in the RAS group (p < 0.0001, Z = −5.468). Conclusions: In experienced pediatric centers, robotic-assisted resection of choledochal cysts is safe and effective for infants aged less than 3 months and deserves further exploration. Full article

When some treatment labels are missing, the number of the observed treated group is much smaller than the control group. The imbalanced dataset can make treatment effect estimation more challenging. To address this issue, we develop an inference framework that integrates an Absent Data Generating (ADG) algorithm with kernel matching. The proposed method incorporates a modified Kernel Fisher Discriminant (KFD) into the causal inference framework to recover missing treatment assignments. SMOTE and Borderline-SMOTE are mainly designed for imbalanced classification problems. These methods generate synthetic observations through local interpolation. In contrast, the ADG algorithm learns the main patterns in the observed treated units. The algorithm then generates the unobserved samples by updating the data structure step by step and drawing samples based on the learned relationships. Consequently, it provides a distribution-based mechanism for recovering missing treatment information rather than relying solely on neighborhood-based interpolation. The recovered treatment labels are subsequently incorporated into kernel matching to improve ATT estimation and interval inference. Monte Carlo simulations and a real-data application demonstrate that the proposed method provides more accurate ATT estimates and more reliable confidence intervals under severe treatment-group imbalance. Full article

There is a lack of information about the shark fisheries in Tamaulipas, a coastal state of the Gulf of Mexico, and the species that sustain them. This challenges the development of management and conservation strategies for this group. This study evaluates the ecological vulnerability of elasmobranch species under fishing pressure using a Productivity and Susceptibility Analysis (PSA). PSA was conducted using the literature biological data of the organisms caught in La Pesca, Tamaulipas, Mexico, and semi-structured surveys applied to artisanal fishers. Results indicate that eleven shark species and four ray species sustain this fishery, with biological productivity values ranging from 1.15 to 2.23 and susceptibility values from 1.91 to 2.09. The minimum ecological risk value was observed in Rhizoprionodon terranovae (v = 1.26), and the highest in Gymnura lessae (v = 2.14). Secondary, non-local biological productivity data limit the quality score but not the overall study validity, highlighting the need for primary regional data on Tamaulipas elasmobranchs. Our results rank elasmobranch species by ecological risk in La Pesca, Tamaulipas, helping policymakers prioritize species for research and conservation and determine whether current management matches local artisanal fishery realities or requires regional adjustments. Further regional studies are required to improve biological productivity data for elasmobranchs supporting artisanal fisheries in Tamaulipas. Full article

La Paz Bay is one of the most important destinations for recreational fishing events in Baja California Sur, Mexico, with significant tourist activity that supports ecosystem services and provides economic benefits to participants, benefits that have not yet been economically measured. The aim of this study was to estimate the willingness to pay (WTP) of anglers who participated in sport-fishing events and to propose tourism promotion and recreational fisheries management strategies. Applying 184 face-to-face surveys at sport fishing events in LPB held in 2022 and 2023 to collect information regarding fishers’ and fishing trips characteristics, and using the individual travel cost method to estimate the individual WTP per angler, which ranges from USD 50.96 to 625.63, and the recreational economic value of fishing events in La Paz was estimated at USD 0.89 to 1.11 million. Strategies for conserving species reserved for sport fishing and promoting tourism are discussed, which could help improve tournament activity and promote the rational use of natural resources. This study represents the first effort aiming to value sport fishing events in Mexico. Also, demonstrates the economic relevance of sport fishing events for recreational fisheries management and tourism-promoting policies in LPB. It provides evidence that conserving sport-fishing species could enhance management strategies and sustainable tourism promotion policies for this recreational activity. Full article

Background: Complementary Medicine and Therapies (CMTs) are increasingly used worldwide, particularly in Saudi Arabia, due to growing interest in holistic and integrative healthcare. However, concerns regarding safety, regulatory awareness and evidence-based use persist, while comparative knowledge data between family physicians and the general population remain limited. This study aimed to compare knowledge of CMTs between family physicians and the general population in Saudi Arabia. Methods: A comparative cross-sectional analysis was conducted using two independent datasets collected in Saudi Arabia from a total of 1307 participants. Variables related to the participants’ demographic characteristics, and their knowledge of and information sources on CMTs, were harmonised prior to analysis. Between-group comparisons were performed using Chi-square and Fisher’s exact tests, with statistical significance set at p < 0.05. Results: The family physicians demonstrated significantly higher levels of knowledge of CMTs than the general population. While 35.5% of the physicians demonstrated good knowledge levels, 82.0% of the public participants exhibited poor or low knowledge. However, knowledge of regulatory and quality control measures governing CMTs remained limited among both the physicians (21.0%) and the public (24.6%). Both the physicians (66.1%) and the members of the public (34.8%) primarily relied on healthcare professionals as major sources of information on CMTs. Significant associations were observed between the participant group and the knowledge-related variables, overall knowledge levels and information sources (p < 0.001). Conclusions: Strengthening public education, physician training and regulatory awareness may support safer and more evidence-based integration of CMTs within healthcare systems. Full article

Objective: This study evaluated hepatitis B vaccine (HepB) uptake, associated influencing factors, and post-vaccination immune responses among preterm infants residing in Fujian Province. The findings can support targeted improvements in hepatitis B prevention and control strategies tailored for this high-risk neonatal population. Methods: We conducted a multicenter cross-sectional study combined with short-term prospective serological follow-up across five counties, cities and districts of Fujian Province between 2022 and 2023. A total of 779 eligible preterm infants were enrolled in this study. We collected demographic information of participating mothers and infants, as well as complete HepB vaccination records throughout the study period. For 363 enrolled infants, we performed serological tests to detect hepatitis B surface antigen (HBsAg) and hepatitis B surface antibody (HBsAb) at 1–2 months after they completed the full HepB vaccination series. To explore factors linked to timely administration of the first HepB dose (HepB1), completion of the full vaccination course and HBsAb serostatus, we adopted a set of statistical approaches including descriptive statistics, the chi-square test (Fisher’s exact test was used for groups with small sample sizes) and binary logistic regression. Results: The timely HepB1 vaccination rate among all preterm infants was 78.18%, while 63.80% completed the full vaccination schedule as required. In the serology cohort, the HBsAb positive rate was 90.91%, and 8.82% of infants showed double-negative HBsAg and HBsAb results, indicating susceptibility to HBV infection. Multivariate analysis identified multiple risk factors for delayed vaccination. Preterm infants were more likely to receive vaccinations late if their mothers tested HBsAg-negative (HepB1: OR = 25.231, 95%CI: 4.997–127.406; full-course HepB: OR = 2.440, 95%CI: 1.395–4.269), were delivered in county-level or lower-tier medical facilities (HepB1: OR = 3.724, 95%CI: 2.107–6.580), or were born via cesarean section (HepB1: OR = 3.460, 95%CI: 2.169–5.520; full-course HepB: OR = 1.954, 95%CI: 1.411–2.704). Additional risk factors included a gestational age below 34 weeks (HepB1: OR = 4.369, 95%CI: 1.894–10.081; full-course HepB: OR = 2.237, 95%CI: 1.148–4.359) and a birth weight less than 2500 g (HepB1: OR = 2.251, 95%CI: 1.397–3.629; full-course HepB: OR = 1.513, 95%CI: 1.065–2.150). Conclusions: Preterm infants enrolled from five regions in Fujian Province achieved robust immune protection following standard HepB vaccination. However, timely first-dose coverage and on-schedule full-course vaccination remain suboptimal in this cohort. Observed gaps in routine vaccination management at primary care settings highlight a key area for improvement in local hepatitis B prevention. Targeted standardized training for maternity care staff at county-level facilities, paired with a full-cycle follow-up system for preterm infant vaccination, may further strengthen hepatitis B mother-to-child transmission (MTCT) interruption in the study regions. Full article

Background: Pneumothorax is a clinically heterogeneous condition with a substantial nursing-care burden; yet, contemporary descriptive data that combine medical and nursing variables remain scarce in the Croatian setting. Our aim was to describe the demographic profile, treatment patterns, and nursing-care requirements of patients treated for pneumothorax at a single regional hospital. Moreover, we explored factors associated with chest drainage and with adverse nursing-sensitive outcomes. Methods: We conducted a retrospective, single-centre, cross-sectional analysis of adult and adolescent patients consecutively admitted due to pneumothorax to Varaždin General Hospital between 1 January 2019 and 31 December 2021. Sociodemographic, clinical, and nursing-care variables were extracted from the hospital information system and the electronic nursing documentation. Nursing diagnoses were classified using NANDA International (NANDA-I) terminology. Proportions are reported with 95% Wilson score confidence intervals. Bivariate associations between categorical variables were assessed using the Fisher exact test with Haldane–Anscombe-corrected odds ratios; the Kruskal–Wallis test with Bonferroni-corrected pairwise comparisons were used for continuous distributions. Independent associations with the chest drainage placement, prolonged length of stay (>14 days), and worsening of dependency category were assessed with L1-penalised logistic regression (α = 0.5), with 1000-iteration non-parametric bootstrap 95% CIs and p-values. Results: Of 60 patients included, 39 (65.0%; 95% CI 52.4–75.8) were male and 33 (55.0%; 42.5–66.9) were aged 60 years or older. Spontaneous pneumothorax accounted for 27 cases (45.0%; 33.1–57.5), traumatic for 23 (38.3%; 27.1–51.0), and iatrogenic for 10 (16.7%; 9.3–28.0). Chest drainage was used in 44 patients (73.3%; 61.0–82.9), universally in iatrogenic cases. After adjustment, age ≥ 60 years was independently associated with the receipt of chest drainage (adjusted OR 3.67; 95% CI 1.21–13.56; p = 0.026), with a prolonged length of stay (adjusted OR 3.69; 95% CI 1.02–21.00; p = 0.042) and with functional deterioration (adjusted OR 4.29; 95% CI 1.21–22.62; p = 0.028). Risk for falls (58.3%) and Bathing self-care deficit (26.7%) were the most frequent NANDA-I diagnoses; 14 patients (23.3%) deteriorated by at least one dependency category by discharge. Conclusions: Patients hospitalised with pneumothorax at our centre were predominantly older men with a substantial nursing-care workload. An older age was the most consistent independent correlate of both invasive treatment and adverse nursing-sensitive outcomes. The findings provide a descriptive baseline for the Croatian setting and should be interpreted as hypothesis-generating, given the modest sample size and the single-centre retrospective design. Full article

Graph-based narrative extraction relies on a coherence function to score transitions between events, but the coherence metrics in current use are defined operationally and lack an information-theoretic foundation. We study the composite metric $C=\sqrt{A·T}$, where A is the angular similarity of document embeddings and $T=1-d_{\mathrm{JS}}$ is the topic proximity through the Jensen–Shannon distance of soft cluster memberships, and we provide an information-geometric reading of this metric together with an axiomatic characterization of the geometric-mean combinator. On the product manifold ${\mathbb{S}}^{d-1}\times {\Delta }_{+}^{K-1}$, the negative log-coherence decomposes additively into an angular and a topic cost. Because the Riemannian metric tensor induced by the Jensen–Shannon distance on the simplex is proportional to the Fisher information matrix, the topic component is locally consistent with the Fisher–Rao metric singled out by Chentsov’s theorem. Within a parametric family of combinators (the compensability spectrum), the geometric mean is the unique combinator consistent with four natural axioms (a boundary/veto condition, symmetry, log-additivity, normalization), and the construction also motivates a proper product metric $d_{\times }$ that we use as a reference distance. Experiments on four corpora spanning news and academic domains (40 to 6000 documents), three general-purpose embedding families (GPT-4/ada-002, MPNet, MiniLM-L6) plus citation-aware SPECTER2, and three alternative topic models (LDA, soft k-means, GMM) are consistent with the framework: the Fisher identity holds with $R\ge 0.99$, the geometric mean tracks $d_{\times }$ closely ($\rho =0.999$), and a downstream LLM-as-judge consistency check shows that the geometric mean is not empirically dominated by any alternative combinator or single-channel baseline. Sweeping the compensability spectrum, the bottleneck-coherence gap between extracted storylines and random sequences splits into a symmetric component—maximized at the geometric mean on the four corpora above and a fifth, human-navigation corpus—and a displacement term; a cross-modal case study on a human-curated image narrative reproduces the same effect in a second modality. Together, these results provide an information-geometric justification for the composite coherence metric and articulate the conditions under which the geometric mean is the natural choice. Full article