Geometric Index Research Articles

Dynamics of coastal land-based aquaculture pond in China and Southeast Asia from 1990 to 2020

The extensive expansion of coastal land-based aquaculture ponds (CLAP) has resulted in substantial wetland loss. To effectively monitor the dynamics of CLAP on a large scale, critical spatial information derived from continuous datasets is essential. Here, we generate the inaugural map depicting the changes in aquaculture pond coverage across the coastal regions both of China and Southeast Asia from 1990 to 2020 by utilizing the long time-series Landsat archive with an object-oriented hierarchical classification method. We extracted coastal water bodies using the Otsu algorithm and established a water inundation frequency index to eliminate interference from temporary water facilities. This process allowed us to identify potential distribution areas of aquaculture ponds. To precisely map the spatial distribution of CLAP within these identified areas, we employed an object-oriented K-nearest neighbor approach with spectral, geometric, and water index features as classification features. The cumulative coverage of CLAP in China and Southeast Asia reached 23,121 km2 by 2020, marking a net increase of 12,700 km2 since 1990. Notably, China, Indonesia, and Vietnam emerged as the primary contributors to these changes within our study area. Regions exhibiting dense aquaculture activities are predominantly concentrated in the northern region above 10 degrees latitude and along the eastern coastlines of various countries. This aquaculture pond map reserves fine spatial detail by effectively distinguishing drainage channels, rice seedling fields, and embankments surrounding aquaculture ponds. It serves as a valuable resource for guiding coastal resource management and environmental protection initiatives.

Single isocenter dynamic conformal arcs-based radiosurgery for brain metastases: Dosimetric comparison with Cyberknife and clinical investigation

PurposeTo compare the dosimetric quality of automatic multiple brain metastases planning (MBM) with that of Cyberknife (CK) based on the clinical tumor condition, such as the tumor number, size, and location. Methods76 treatment plans for 46 patients treated with CK were recalculated with the MBM treatment planning system. Conformity index (CI), homogeneity index (HI), gradient index (GI), lesion underdosage volume factor (LUF), healthy tissue overdose volume factor (HTOF), geometric conformity index (g) and mean dose to normal organs were compared between CK and MBM for tumor number, size, shape and distance from the brainstem or chiasm. ResultsThe results showed that the mean brain dose was significantly smaller in MBM than CK. CI did not differ between MBM and CK; however, HI was significantly more ideal in CK (p = 0.000), and GI was significantly smaller in MBM (P = 0.000). LUF was larger in CK (p = 0.000) and HTOF and g was larger in MBM (p = 0.003, and 0.012). For single metastases, CK had significantly better HTOF (p = 0.000) and g (p = 0.002), but there were no differences for multiple tumors. Brain dose in MBM was significantly lower and CI was higher for tumors < 30 mm (p = 0.000 and 0.000), whereas HTOF and g for tumors < 10 mm were significantly smaller in CK (p = 0.041 and p = 0.016). Among oval tumors, brain dose, GI and LUF were smaller in MBM, but HTOF and g were smaller in CK. There were no particular trends for tumors close to the brainstem, but HTOF tended to be smaller in CK (0.03 vs. 0.29, p = 0.068) for tumors inside the brainstem. ConclusionsMBM can reduce the brain dose while achieving a dose distribution quality equivalent to that with CK.

Automated construction of multi-leaf walls with natural stones

This paper introduces an innovative strategy for the construction of multi-leaf walls utilizing raw stones, a significant step forward in sustainable building practices. Our approach leverages advanced computational algorithms and image processing techniques to automatically build walls with irregular stones. By focusing on the physical assembly of these materials, the objective is to optimize their use in creating structurally robust walls. The core of our methodology lies in the efficient arrangement of irregular objects based on image-convolution techniques. The proposed packing algorithm is proven to be efficient and versatile in stacking stones without restriction in the shape and size of units. Evaluations of constructed walls based on geometric indices show that the stone layout meets masons’ rules-of-art. This work presents a scalable wall construction solution that could revolutionize the use of natural stones in construction.

4,4-Bis(isopropylthio)-1,1-diphenyl-2-azabuta-1,3-diene Adducts with Cadmium(II), Mercury(II) and Copper(I) Iodides: Crystal, Molecular and Electronic Structures of d10 Transition Metal Chelate Complexes

The thioether-functionalized 2-azabutadiene (iPrS)2C=C(H)-N=CPh2 L ligates to CdI2 and HgI2 to form the chelate compounds [CdI2{(iPrS)2C=C(H)-N=CPh2] (1) and [HgI2(iPrS)2C=C(H)-N=CPh2] (2). Their crystal structures were solved via X-ray diffraction. Both crystallize in the non-centrosymmetric space groups: monoclinic P21 (1) and orthorhombic P212121 (2), respectively. The closed-shell d10 metal centers are four-coordinated (two iodides and S and N coordinating atoms from the ligand L) in both complexes. The geometrical indexes τ indicate that a highly distorted trigonal pyramidal is adopted for 1 and a seesaw geometry for 2. The comparative nature of metal–ligand bonds is discussed on the basis of metric parameters and of QT-AIM (quantum theory of atoms in molecules) calculations. L was also treated with CuI to obtain the dinuclear species [LCu(μ2-I2)CuL] (3), in which the two Cu(I) centers are linked by a short metal–metal bond. The geometric and electronic properties of 3 are compared with those of 1 and 2.

Morphology of Abdominal Aortic Aneurysms and Correlation with Biomechanical Tests of Aneurysmal Wall Fragments

Evaluate how specific morphologic aspects of abdominal aortic aneurysms (AAAs), including asymmetries, curvatures, tortuosities, and angulations, among others can influence the intrinsic biomechanical properties of the AAA's wall. This study analyzed the correlation of geometric measurements (1-dimensional, 2-dimensional, and 3-dimensional) of preoperative tomographic images of AAA with uniaxial biomechanical tests of the arterial wall fragments of these AAA obtained in open surgical repair of aneurysms. It was a multicenter, experimental, and observational study, and initially 54 individuals were selected who underwent open surgical of AAA, with valid biomechanical tests of the anterior wall of the AAA. Seven individuals were excluded because they had poor preoperative quality computed tomography scans and/or artifacts that impeded image segmentation and extraction of AAA geometric indices. The aortic fragments were subjected to uniaxial biomechanical destructive tests to obtain the following data: maximum load, failure stress, failure tension, failure strain energy, strain, and fragment thickness. In the same patients, preoperative computed tomography scans were performed with the extraction of 26 geometric indices, subdivided into 9 1-dimensional indices, 6 2-dimensional indices, and 11 3-dimensional indices. Data were subjected to statistical analysis using SPSS version28. Comparing ruptured and unruptured AAA, no statistical difference was observed between the biomechanical and geometric parameters. The fragment thickness of the ruptured AAA was lower than that of the unruptured AAA (P<0.05). By comparing tomographic geometric indices and biomechanical parameters of the aortic fragments using Pearson's coefficient, positive and linear correlations (P<0.05) were observed between the geometric variable maximum diameter (Dmax) of the AAA with maximum load (r=0.408), failure tension (r=0.372), and failure stress (r=0.360). Positive and linear correlations were also observed between the variable diameter/height ratio (DHr) and the maximum load (r=0.360), failure tension (r=0.354), and failure stress (r=0.289). The geometric variable DHr was dependent and correlated with Dmax. Simple regression analysis showed that R2 varied between 8.3% and 16.7%, and all models were significant (P<0.05). Dmax and DHr were linearly and positively correlated with the resistance parameters (maximum load, failure tension, and failure stress) of the AAA fragments. The DHr variable is dependent and correlated with Dmax. There was no correlation between the other geometric indices and the biomechanical parameters of the AAA wall. The asymmetries did not globally influence the biomechanics of AAA wall; however, they may influence regionally. Larger AAAs were stronger than smaller ones. Therefore, such findings may point toward Dmax is still the main geometric parameter, which influences the anterior wall, and possibly globally in the AAA.

Sum Geometric Harmonic Means Index of Graphs

In this paper, the concept of sum geometric harmonic means index of a graph G, denoted by SGhm(G) is introduced and sum geometric harmonic means index SGhm(G) of few families of graphs is computed. Further, we establish the bounds for sum geometric harmonic means index. General Terms AMS, subject Classification 05C07; 92E10. Keywords Graph, Molecular graph, Sum geometric harmonic means index .

Screening Potato Landraces to Cope with Climate Change in the Central Andes of Peru

Agriculture in the Andean region is mainly small-scale and rainfed, especially in Peru where almost 80% of its population depends on agriculture-related activities. Climate change in addition to social factors threatens the food security of this region. The forecast of more frequent dry spells would especially affect potato crops, domesticated centuries ago in the Andes, where there remains a great genetic diversity. This study aimed to characterize the response to drought stress of 79 potato landraces traditionally grown in the Central Andes of Peru (Pasco region) as a first selection for future breeding evaluations. The used indices were mean productivity (MP), geometric mean productivity (GMP), stress tolerance (STI), stress susceptibility (SSI), and tolerance index (TOL), and a scoring methodology that integrates all of them into a single descriptor in a simple and fast way. The varied responses showed a wide genetic diversity within the assessed landraces, where at least nine of them own high resilience and productivity qualities, and many others are highly vulnerable to drought. It is recommended to complement these studies with physiological and molecular evaluations in stress situations, especially in those with tolerance qualities highlighted in this study, and thus promote the conservation of the biodiversity of this region.

Clinical evaluation of deep learning and atlas-based auto-segmentation for organs at risk delineation

Manual delineation of organs at risk and clinical target volumes is essential in radiotherapy planning. Atlas-based auto-segmentation (ABAS) algorithms have become available and been shown to provide accurate contouring for various anatomical sites. Recently, deep learning auto-segmentation (DL-AS) algorithms have emerged as the state-of-the-art in medical image segmentation. This study aimed to evaluate the effect of auto-segmentation on the clinical workflow for contouring different anatomical sites of cancer, such as head and neck (H&N), breast, abdominal region, and prostate. Patients with H&N, breast, abdominal, and prostate cancer (n = 30 each) were enrolled in the study. Twenty-seven different organs at four sites were evaluated. RayStation was used to apply the ABAS. Siemens AI-Rad Companion Organs RT was used to apply the DL-AS. Evaluations were performed with similarity indices using geometric methods, time-evaluation, and qualitative scoring visual evaluations by radiation oncologists. The DL-AS algorithm was more accurate than ABAS algorithm on geometric indices for half of the structures. The qualitative scoring results of the two algorithms were significantly different, and DL-AS was more accurate on many contours. DL-AS had 41%, 29%, 86%, and 15% shorter edit times in the HnN, breast, abdomen, and prostate groups, respectively, than ABAS. There were no correlations between the geometric indices and visual assessments. The time required to edit the contours was considerably shorter for DL-AS than for ABAS. Auto-segmentation with deep learning could be the first step for clinical workflow optimization in radiotherapy.

Connection number topological aspect for backbone DNA networks.

The present study investigates the complex topological characteristics of DNA networks, with a specific emphasis on the innovative metric known as Connection Number (CN) as a key factor in determining network structure. The Connection Number, represented as CN(v) for a vertex v, measures the count of unique paths that link v to every other vertex in the network. By employing rigorous mathematical modeling and analysis techniques, we are able to reveal the profound implications of CN (complex networks) in characterizing the structural robustness and dynamics of information flow within DNA networks. The study of how the theory of graphs and chemicals interact is known as chemical graph theory. This paper, computing the hyper Zagreb connection index, augmented connection index, inverse sum connection index, harmonic connection index, symmetric division connection index, geometric arithmetic connection index, and atom bond connectivity connection index, of two significant types of backbone DNA and Barycentric subdivision of backbone DNA networks. Direct method computation is used to produce these Connection-based topological descriptors.

The Geometric–Arithmetic index of trees with a given total domination number

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. In Vukičević and Furtula (2009), Vukičević defined a new topological index, named geometric–arithmetic index of a graph G and denoted by GA(G), as GAG=∑uv∈E2dudvdu+dv,where du and dv denote the degrees of vertices u and v, respectively. We obtain an upper bound for the geometric–arithmetic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound. Additionally, by a known relation between the geometric–arithmetic and arithmetic–geometric indices, we get a new lower bound for the arithmetic–geometric index.

Effect of Selected Music Soundtracks on Cardiac Vagal Control and Complexity Assessed by Heart Rate Variability.

The aim of our study was to investigate the effect of selected passive music therapy frequencies without vocals on selected cardio-vagal and complexity indices of short-term heart rate variability (HRV) in healthy youth, in terms of calming the human. 30 probands (15 male, averaged age: 19.7+/-1.4 years, BMI: 23.3+/-3.8 kg/m2) were examined during protocol (Silence baseline, Music 1 (20-1000 Hz), Silence 1, Music 2 (250-2000 Hz), Silence 2, Music 3 (1000-16000 Hz), and Silence 3). Evaluated HRV parameters in time, spectral, and geometrical domains represent indices of cardio-vagal and emotional regulation. Additionally, HRV complexity was calculated by approximate entropy and sample entropy (SampEn) and subjective characteristics of each phase by Likert scale. the distance between subsequent R-waves in the electrocardiogram (RR intervals [ms]) and SampEn were significantly higher during Music 3 compared to Silence 3 (p=0.015, p=0.021, respectively). Geometrical cardio-vagal index was significantly higher during Music 2 than during Silence 2 (p=0.006). In the subjective perception of the healthy youths evaluated statistically through a Likert scale, the phases of music were perceived significantly more pleasant than the silent phases (p<0.001, p=0.008, p=0.003, respectively). Our findings revealed a rise of cardio-vagal modulation and higher complexity assessed by short-term HRV indices suggesting positive relaxing effect music especially of higher frequency on human organism.

Mapping Connectivity Patterns: Degree-Based Topological Indices of Corona Product Graphs

Graph theory (GT) is a mathematical field that involves the study of graphs or diagrams that contain points and lines to represent the representation of mathematical truth in a diagrammatic format. From simple graphs, complex network architectures can be built using graph operations. Topological indices (TI) are graph invariants that correlate the physicochemical and interesting properties of different graphs. TI deal with many properties of molecular structure as well. It is important to compute the TI of complex structures. The corona product (CP) of two graphs G and H gives us a new graph obtained by taking one copy of G and V G copies of H and joining the i th vertex of G to every vertex in the i th copy of H . In this paper, based on various CP graphs composed of paths, cycles, and complete graphs, the geometric index (GA) and atom bond connectivity (ABC) index are investigated. Particularly, we discussed the corona products P s ⨀ P t , C t ⨀ C s , K t ⊙ K s , K t ⊙ P s , and P s ⊙ K t and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical findings supported by proofs.

Topological Descriptors of Molecular Networks via Reverse Degree

Porphyrazine and tetrakis porphyrazine are organic compounds with intricate ring structures. They are made up of four indole rings that fuse together to form a larger cyclic structure. These structures have been extensively studied in physics and chemistry. They exhibit unique electronic and optical properties due to their highly conjugated systems. In several scientific domains, they are used as the building blocks for functional materials, molecular electronics, sensors, and catalysis. In order to derive a topological index from data, molecules are represented as graphs, with atoms acting as nodes and bonds acting as edges. There are many different techniques and algorithms that can be used to determine the topological index. We have used the reverse degree-based methodology for the mentioned structures. Reverse degree-based topological indices evaluate the structural complexity and stability of chemical compounds by adding the reversed atom degree sequence to their molecular networks. In this study, we calculated the reverse degree-based topological indices like the reverse redefined Zagreb indices, the reverse forgotten index, the reverse hyper-Zagreb index, the reverse Balaban index, the reverse atom bond connectivity index, the reverse geometric arithmetic index, and the reverse general Randić index for porphyrazine and tetrakis porphyrazine networks. This research advances our knowledge of the fundamental laws of physics and chemistry by illuminating the complex interrelationships between molecular structure, biological interactions, and chemical reactivity.

Morphological indexes to describe snow-cover patterns in a high-alpine area

Abstract The spatiotemporal distribution of snow affects hydrological and climatological processes at different scales. Accordingly, quantifying geometric features of snow-cover patterns is important, providing a valuable complement for snow water equivalent (SWE) modelling. This study on satellite-based morphological analysis originally uses two types of geometric indexes: (1) MN, Minkowski numbers (area (MN1), perimeter (MN2), Euler number (MN3)), and (2) CL, average chord length, to describe the morphology of Sentinel-2-derived snow-covered areas (SCAs), within the high-alpine site Zugspitze for a 5 year period. Results indicate that they capture the seasonal variability of snow-cover patterns, particularly during accumulation and ablation. Being to some degree independent from each other, MN2, MN3 and CL provide additional information upon shape, connectivity and length scale of snow cover, compared to most used indexes (e.g. fractional SCA). Correlation values up to +0.7 for MN2, +0.58 for MN3 and +0.46 for CL were observed with selected topographic characteristics, suggesting a close connection between geometric features of snow cover and ground features. Comparing in situ SWE measurements with MN and CL shows a correlation between −0.5 and +0.5. These indexes can hence be applied in combination with in situ data and/or modelling approaches to improve spatially distributed SWE in high-alpine catchments.

Degree-based topological indices of boron nanotubes

In the past two decades, boron nanotubes have received significant attention from researchers and scientists due to their wide-ranging applications in electronics, nanodevices, optical engineering, nanobiotechnology, and cosmetics. These nanotubular structures composed of boron present exceptional electrical and mechanical properties, making them highly potential nanomaterials. In this article, we study the molecular structure of significant classes of boron nanotubes, namely, trihexagonal boron nanotubes, triangular boron nanotubes, and boron-α nanotubes. Furthermore, we calculate various topological indices for these nanotubes, including the augmented Zagreb index, Sombor index, reduced Sombor index, sum-connectivity index, and arithmetic–geometric index. These indices hold substantial importance in assessing the physical, chemical, and biological characteristics of boron nanotubes.

Capsule production promotes Group B Streptococcus intestinal colonization.

Late-onset disease is the most common clinical presentation of Group B Streptococcus (GBS) infection during infancy, and gastrointestinal (GI) colonization is an important precursor. Previously, we described a murine model of postnatal GBS GI colonization that resulted in sustained colonization and progression to invasive disease. Capsular polysaccharide is an important GBS virulence factor. Vaccines based on a subset of capsular serotypes are in clinical trials. However, little is known regarding the role of specific GBS capsular serotypes in GI colonization. We examined the role of GBS capsule in GI colonization using capsule-producing and acapsular strains derived from GBS strain A909 (serotype Ia) in a murine model. Using isogenic GBS strains differing only in capsular serotypes, we explored the role of specific serotypes in GI colonization by determining competitive indices during cocolonization. We found that GBS A909 colonizes the murine GI tract without causing invasive disease. In monocolonization experiments, there was colonization persistence with the capsule-producing strain (100%) compared to the acapsular mutant strain (13%). In cocolonization experiments, the capsule-producing strain outcompeted its isogenic acapsular mutant, with a geometric mean competitive index of 8, 95% confidence interval (CI) [1.7, 38.9] in the colon at 7 days post-colonization. A909 expressing its native serotype Ia capsule outcompeted an isogenic mutant that expresses serotype III capsule, with a geometric mean competitive index of 2.5, 95% CI [1.2, 5.1] in the colon at 7 days post-colonization. Thus, polysaccharide capsule production enhances GBS GI colonization in vivo. In an A909 genetic background, the production of a serotype Ia capsule provides a competitive advantage over an isogenic strain producing type III capsule. The murine model is a valuable tool to understand the role of GBS capsule types in GI colonization. IMPORTANCE The establishment of GBS intestinal colonization is believed to be a critical precursor to late-onset disease in neonates, which has a significant impact on neurodevelopment outcomes in this population. Our prior work described a murine model of postnatal Group B Streptococcus (GBS) acquisition and invasive disease. Using this model, we explored the importance of GBS polysaccharide capsule production on gastrointestinal colonization. We found that the expression of capsule (compared to isogenic acapsular strains) provides an advantage in intestinal colonization and, importantly, that capsule type Ia has an advantage over capsule type III in a GBS A909 strain background. We speculate that specific serotypes may differ in colonization fitness, which may play a role in serotype distribution in neonatal disease.

Analysis of genetic diversity and genome-wide association study for drought tolerance related traits in Iranian bread wheat

BackgroundDrought is most likely the most significant abiotic stress affecting wheat yield. The discovery of drought-tolerant genotypes is a promising strategy for dealing with the world’s rapidly diminishing water resources and growing population. A genome-wide association study (GWAS) was conducted on 298 Iranian bread wheat landraces and cultivars to investigate the genetic basis of yield, yield components, and drought tolerance indices in two cropping seasons (2018–2019 and 2019–2020) under rainfed and well-watered environments.ResultsA heatmap display of hierarchical clustering divided cultivars and landraces into four categories, with high-yielding and drought-tolerant genotypes clustering in the same group. The results of the principal component analysis (PCA) demonstrated that selecting genotypes based on the mean productivity (MP), geometric mean productivity (GMP), harmonic mean (HM), and stress tolerance index (STI) can help achieve high-yield genotypes in the environment. Genome B had the highest number of significant marker pairs in linkage disequilibrium (LD) for both landraces (427,017) and cultivars (370,359). Similar to cultivars, marker pairs on chromosome 4A represented the strongest LD (r2 = 0.32). However, the genomes D, A, and B have the highest LD, respectively. The single-locus mixed linear model (MLM) and multi-locus random-SNP-effect mixed linear model (mrMLM) identified 1711 and 1254 significant marker-trait association (MTAs) (-log10 P > 3) for all traits, respectively. A total of 874 common quantitative trait nucleotides (QTNs) were simultaneously discovered by both MLM and mrMLM methods. Gene ontology revealed that 11, 18, 6, and 11 MTAs were found in protein-coding regions (PCRs) for spike weight (SW), thousand kernel weight (TKW), grain number per spike (GN), and grain yield (GY), respectively.ConclusionThe results identified rich regions of quantitative trait loci (QTL) on Ch. 4A and 5A suggest that these chromosomes are important for drought tolerance and could be used in wheat breeding programs. Furthermore, the findings indicated that landraces studied in Iranian bread wheat germplasm possess valuable alleles, that are responsive to water-limited conditions. This GWAS experiment is one of the few types of research conducted on drought tolerance that can be exploited in the genome-mediated development of novel varieties of wheat.

Evaluation of Four Maize (Zea mays L.) Genotypes Using Drought Tolerance Indices

Drought tolerance is not often considered as an independent trait by breeders. The objective of this study was to evaluate and identify drought tolerant genotypes using eight drought tolerance indices namely the Stress Susceptibility Index (SSI), the YSI, the YR (Yr), yield index (YI), tolerance index (TOL), average productivity (MP), mean geometric productivity (GMP) and stress tolerance index (STI) of maize genotypes (Zea mays L.). A field trial was conducted to evaluate four genotypes during the hot dry season of 2016 and 2017 at the irrigated perimeter of Djirataoua. Drought tolerance indices were calculated on the basis of yield under optimal and stressed conditions. The comparison of the means of drought tolerance demonstrated the effects of drought on yield and showed significant differences between genotypes. The correlation coefficient and principal component analysis showed that the GMP, MP and STI indices were able to discriminate drought-sensitive and tolerant genotypes. Two genotypes CZH131001 and CZH142013 produced high grain yield under both optimal and stressed conditions. Overall, GMP, MP, and STI indices can be used as effectively drought tolerance screening indices and able to identify better genotypes, suitable for both optimal and stress conditions.

Geometric versus Hemodynamic Indexes for Rupture-Destined Aneurysms: A Retrospective Cohort and a Repeated-Measures Study

Plain Language SummaryThe incidence of intracranial aneurysm (IA) is 2–3% in the general population, and with the widespread use of neurovascular imaging techniques, more IAs are being diagnosed. IA management is crucial as it can lead to fatal subarachnoid hemorrhage, causing high mortality and neurological issues. About 1% of patients with IA experience severe hemorrhage annually, which raises questions about the best method to assess rupture risk. Past studies used geometric indexes and computational fluid dynamics to differentiate between ruptured and unruptured aneurysms. However, these studies mostly focused on cross-sectional data and did not analyze the evolutionary changes of ruptured aneurysms. In this study, we call such aneurysms rupture-destined aneurysms (RDAs) to differentiate from stable unruptured intracranial aneurysm (UIA). In the present study, the key findings could be summarized as below: (1) the RDA group showed a significantly higher hemodynamic instability than the UIA group; (2) the UIA group showed features of hemodynamic stability, even with geometric growth; (3) aneurysmal rupture risk should be assessed using both geometric and hemodynamic information.

On the coincidence of optimal completions for small pairwise comparison matrices with missing entries

AbstractIncomplete pairwise comparison matrices contain some missing judgements. A natural approach to estimate these values is provided by minimising a reasonable measure of inconsistency after unknown entries are replaced by variables. Two widely used inconsistency indices for this purpose are Saaty’s inconsistency index and the geometric inconsistency index, which are closely related to the eigenvector and the logarithmic least squares priority deriving methods, respectively. The two measures are proven to imply the same optimal filling for incomplete pairwise comparison matrices up to order four but not necessarily for order at least five.