Formula For Volume Research Articles

On $q$-analogues of Zeta Functions of Root Systems II

In a previous paper, we introduced functions $\zeta_r(s,a,\Delta;q)$, which are $q$-analogues of zeta functions associated with root systems $\Delta$, and their $p$-deformations $\zeta_r(s,a,\beta,\Delta;p,q)$. It is known that zeta functions of root systems satisfy certain functional relations including Witten's volume formula. In this paper, we consider $q$-extensions of these functional relations for $\Delta=\Delta(A_2), \ \Delta(A_3), \ \Delta(B_2), \ \Delta(G_2)$. We also generalize expressions of ``Weyl group symmetric'' linear combination of $\zeta_r(s,a,\beta,\Delta;p,q)$ for $\Delta=\Delta(A_1), \ \Delta(A_2), \ \Delta(A_3)$, which were obtained in the previous paper, to arbitrary root systems $\Delta$ by using an identity due to Macdonald.

An introduction to gradient solitons on statistical Poisson manifold and perfect fluid Poisson manifold

In this article, we introduce the idea of statistical Poisson manifold and perfect fluid Poisson manifold. Then, we investigate the well known almost solitons like Ricci soliton, Yamabe solitons or Einstein solitons on Poisson manifold and discuss the statistical structures. Next, we find a volume formula of the Poisson manifold and discuss the bounds of |Ric|2 for gradient solitons. Afterward, we study statistical structure on a perfect fluid Poisson manifold. Then, we obtain the volume formula and bounds of |Ric|2 for perfect fluid Poisson manifold.

Experimental research on wear mechanism of diamond wheels for grinding Cf/SiC composites grooves

Aiming at the problems of low processing efficiency, low groove shape accuracy and poor surface quality caused by grinding wheel wear in the grinding process of Cf/SiC composite grooves. In this paper, based on the motion mode of grinding wheel grinding, the wear volume formula of grinding wheels was analyzed, and the grinding wheel wear comparison experiment of different bond diamond grinding wheels grinding Cf/SiC composite grooves was carried out, and the more suitable kind of diamond grinding wheel for grinding Cf/SiC composite grooves was selected. From the experimental results, compared with the ceramic-bonded diamond wheel and resin-bonded diamond wheel, the grinding times of electroplated diamond wheel grinding Cf/SiC composites were increased by 20 and 104, respectively, and the surface roughness of the groove was most stable and the processing damage was smallest. Therefore, the best quality of grinding Cf/SiC composites groove was obtained by the electroplated diamond grinding wheel, and its life in grinding Cf/SiC composites was the longest. At the same time, the wear mechanism of the electroplated grinding wheel was explored, the wear on the bottom surface of the electroplated grinding wheel was more serious than that on the circumferential surface. The wear on the bottom surface was mainly wear and tear, and it gradually diffused along the inner diameter of the grinding wheel with the increasing grinding times.

A Phase II Randomized Clinical Trial to Assess Toxicity and Quality of Life of Breast Cancer Patients with Hypofractionated vs. Conventional Fractionation Radiotherapy with Regional Nodal Irradiation

To compare acute toxicity between conventional fractionated radiation therapy (CFRT) and hypofractionated radiation therapy (HFRT) for patients undergoing breast-conserving surgery or mastectomy with breast or chest wall and regional nodal irradiation (RNI). The secondary endpoints were acute and subacute toxicity, cosmesis, quality of life, and lymphedema features. Unblinded randomized trial with a total of 86 patients treated with CFRT (n = 33; 50 Gy/25 fractions ± sequential boost [10 Gy/5 fractions]) versus HFRT (n = 53; 40 Gy/15 fractions ± concomitant boost [8 Gy/15 fractions]). Women included for enrollment were 18 years or older and diagnosed with breast carcinoma, staging T1-4, with at least one positive lymph node (American Joint Committee on Cancer - AJCC 8th). The surgery performed was mastectomy or breast-conservative surgery with the investigation of sentinel lymph node or axillary dissection. Toxic effects and cosmesis evaluation used the Common Terminology Criteria for Adverse Events version 4.03 (CTCAE) and the Harvard/NSABP/RTOG scale. Patients-reported QoL was determined using the European Organization for Research and Treatment of Cancer quality of life questionnaire (EORTC QLQ-C30) and the breast cancer-specific supplementary questionnaire (QLQ-BR23). The comparative analysis of categorical variables between the control and experimental groups used the Chi-square test and Fisher's exact test, while numerical variables basing on the Mann-Whitney U test. The intergroup comparison of the EORTC QLQ-C30 used the Generalized Equations Estimating method. All analyses used 2-sided α = 0.05 with R software version 4.1.2. Lymphedema determination compared volume differences between the affected and contralateral arms using the Casley-Smith volume formula. Grade 2 and grade 3 skin rash dermatitis were lower with HFRT than with CF-RT (28% vs. 52%, 0% vs. 6%; p = 0.022, respectively). HFRT vs. CFRT had lower rate of grade 2 hyperpigmentation (23% vs. 55%; p = 0.005). Overall rates of any physician-assessed grade ≥ 2 and grade ≥ 3 acute toxicities were no different between HFRT vs. CFRT arms. There was no statistical difference between the CFRT and the HFRT regarding cosmesis, lymphedema rate (13% vs. 12% HFRT vs. CFRT; p = 1.000, respectively), and functional and symptom scales during irradiation and after six months of treatment. HF-RT showed lower rates of acute toxicity, with no changes in quality-of-life outcomes.

Experimental trajectories in Euclidean and hyperbolic geometry

The Euler tetrahedron volume formula is used to define dimensionality of space and the Euclidean straight line and area. The double disk modul (DDM) is used to define trajectories on the metrical surfaces. The geodetical paths on the planet Mars can be realized by the Mars double disk modul (MDDM). The relations of generalized Lobacevskii geometry are derived and related to the Einstein equations. It is considered spherical and pseudospherical geometry, Riemann geometry, Lobacevskii and generalized Lobacevskii geometry, Poincare model, Beltrami model, gravity as deformation of space and Schwinger theory of gravity.

All-Factor Average Method of Reserve Parameters with Crude Oil Volume and Mass Constraints

In order to make the average value of each reserve parameter of a set of oil reserves more representative, this paper puts forward the all-factor average method of reserve parameters with crude oil volume and mass constraints. In the first step, the two constraint methods of crude oil volume and mass are adopted to calculate the average value of various parameters of the total items. The weight coefficient when the parameter is averaged is, respectively, the partial derivative of the volume or mass reserve calculation formula with respect to the parameter. Compared with the original calculation results, the average of their parameters all show a shift towards values with a significant share of reserves, especially effective porosity, oil saturation, and crude oil volume factor. The all-factor average method considers a more comprehensive set of factors than the original method. Therefore, each new average parameter should also be much more representative. Since the current reserve specification stipulates that each parameter needs to retain a certain number of decimal places, there inevitably is some carry error between the reserve results calculated by the parameters of the total items and the accumulated reserves of each unit. The second step is to select the optimal average value of each reserve parameter by using the full permutation combination selection method to reduce the carry error. A set of parameters that minimizes the sum of squared relative errors of crude oil volume and mass reserves is selected using the full permutation combination selection method, which is the optimal selection of the average value of the total set of items. Compared with the original method, the full permutation combination selection method can effectively reduce the carry error.

Study on the law and mechanism of anisotropic conductivity of carbon nanotubes film prepared by floating catalytic chemical vapor deposition method

Carbon nanotubes (CNTs) film has attracted extensive attention in the field of electronics, sensors, and other potential applications due to their excellent electrical conductivity. The conductivity of CNTs film is one of the most important aspects of engineering applications. This study investigates the conductivity of CNTs film with varying areal density, prepared using the floating catalyst chemical vapor deposition (FCCVD) method and densified by rolling. The square resistance of pre- and post-rolling films was measured to characterize the electrical properties. Experimental results indicate that square resistance decreases with increasing areal density and stabilizes eventually. A mathematical formula was derived to explain the relationship between areal density and square resistance, incorporating volume and areal density formulas. Experimental data curves of pre- and post-rolling films were fitted, yielding mathematical relations consistent with the derived formulas. The electrical conductivity of post-rolling CNTs film was superior to pre-rolling in experiment and calculation. The charge carrier transport mechanism in CNTs film was studied by analyzing its internal structure and electrical properties. Surface conductivity was over 1000 times higher than volume conductivity, attributed to the distribution of CNTs bundles in collection and thickness directions. Charge carrier transport capacity decreased with increasing layers in thickness direction due to contact resistance and large resistance at tube–tube junctions. Layers of CNTs near the current application surface significantly contribute to charge carrier transport at high areal density levels.

Investigation of Secondary School Students' Processes of Constructing Area and Volume Relations of Rectangular Prisms

The purpose of this study is to examine the knowledge construction processes of 5th grade students, who do not have any knowledge about the surface area and volume of the rectangular prism, within the framework of the observable epistemic actions of the RBC (Recognizing, Building-with, Constructing) abstraction model. This study was carried out according to the case study pattern, which is one of the qualitative research types. The study was carried out in the 2021-2022 academic year. For the research, six students were selected by the maximum diversity sampling method. Three homogeneous groups were formed and semi-structured interviews were conducted. An activity paper consisting of eight problems was used to obtain data. The obtained data were evaluated in the context of observable epistemic actions of RBC. Only a student with a low level of mathematics achievement couldn’t construct knowledge of the surface area of the prism. All students have found the number of unit cubes that can fit inside the rectangular prism. But three students with a high level of success were able to construct and use volume formulas. In the research, it has been seen that recognition and building-with actions are easier to perform than construction action. As the level of mathematics achievement decreased, the speed and success of abstraction decreased.

Evidence that Chinese white olive (Canarium album (Lour.) DC.) fruits are solids of revolution

ABSTRACT Although many fruit geometries resemble a solid of revolution, this assumption has rarely been rigorously examined. To test this assumption, 574 fruits of Canarium album (Lour.) DC. which appear to have an ellipsoidal shape, were examined to determine the validity of a general avian-based egg-shape equation, referred to as the explicit Preston equation (EPE). The assumption that the C. album fruit geometry is a solid of revolution is tested by applying the volume formula for a solid of revolution using the EPE. The goodness of fit of the EPE was assessed using the adjusted root-mean-square error (RMSEadj). The relationship between the observed volume (V obs) of each fruit, as measured by water displacement in a graduated cylinder, and the predicted volumes (V pre) based on the EPE was also evaluated using the equation V pre = slope * V obs. All the RMSEadj values were smaller than 0.05, which demonstrated the validity of the EPE based on C. album fruit profiles. The 95% confidence interval of the slope of V pre vs. V obs included 1.0, indicating that there was no significant difference between V pre and V obs. The data confirm that C. album fruits are solids of revolution. This study provides a new approach for calculating the volume and surface area of geometrically similar fruits, which can be extended to other species with similar fruit geometries to further explore the ontogeny and evolution of angiosperm reproductive organs.

Volume formula and growth rates of the balls of strings under the edit distances

Analogs of balls in n-dimensional Euclidean space are defined in the monoid of strings composed of letters of an alphabet if a metric is defined on the monoid. These analogs are not only balls in the monoid of strings but also regular languages. Since the Levenshtein distance, a reasonable metric between strings, was introduced in 1965, researchers have aimed to determine a formula for computing the volumes of balls of strings under this distance and their growth rate. However, this problem remains open. In this paper, we first derive a formula for the volumes of balls of strings under a metric obtained by extending the Hamming distance as a metric on the whole monoid and subsequently reveal the growth rate of balls of strings under the Levenshtein distance using this formula. Furthermore, we describe a conjecture on the formula for the volumes of spheres of strings under the Levenshtein distance. Next, we construct an efficient randomized algorithm that computes estimates of the volumes of the balls whose estimation errors are less than a given value with a probability greater than a specified value. Finally, we examine the validity of the proposed conjecture, using the (true) volumes obtained by exhaustive searches for small spheres and estimates of the volumes obtained by this algorithm for large ones.

Aplikasi Pembelajaran Bangun Ruang dengan Augmented Reality (Studi Kasus : Sdn Rorotan 07 Jakarta)

The use of technology and reliance on the internet has altered students' learning practices, especially how they approach mathematics. As a result, a novel method of teaching mathematics is required, particularly for geometric concepts that are challenging for children to grasp. For students to better comprehend geometric forms and volume formulas, this study intends to develop spatial learning applications incorporating augmented reality (AR). The method employed in this study is called the Multimedia Development Live Cycle (MDLC), and it includes the steps of ideation, design, material collecting, assembly, testing, and distribution. The results showed that this application succeeded in increasing students' understanding in the form of geometric shapes through the use of spatial learning applications with augmented reality. This was effective in increasing students' understanding of using 3D objects and interactions through applications made students more interested and motivated to learn mathematics.

Large-scale experimental study on scour around offshore monopile under combined wave and current condition

This work conducts large scale (1:13) experiments to study local scour around a monopile under combined irregular wave and current condition. The Keulegan–Carpenter (KC) numbers range from 1.1 to 6.73, and the relative current strength parameter Ucw numbers range from 0.17 to 0.77. The large-scale experiment indicates the 3D shape of scour holes is always an inverted cone under wave-current condition which has nothing to do with wave irregularity and Ucw. The changes of maximum scour location are associated with KC and Ucw numbers. The location of maximum scour depth is shifted from upstream to downstream of the monopile with large KC number under current-dominated condition. The characteristic scour timescale is closely related to KC, Ucw and Shields parameter θw when Ucw<0.3 and may be only related to θw when Ucw>0.7. The equilibrium scour depth is a function of vortex scale parameter referring to KC and Ucw numbers. A change in the definition of KC can improve the accuracy of existing formula under large-scale experimental condition, and scour volume formulae are further proposed from the relation with maximum scour depth. Finally, a scour volume formula of KC and Ucw numbers is suggested to estimate the protection material amount around monopile.

Differential Analysis and Prediction of Planar Shape at the Head and Tail Ends of Medium-Thickness Plate Rolling

This paper aims to improve planar shape prediction accuracy in the rolling process of medium and thick plates. We present a model based on the strip method that addresses limitations in predicting planar shape variations at the head and tail ends of rolled pieces. By analysing the rolling process, we introduce the concept of an imaginary strip longitudinal length difference to represent planar shape characteristics effectively. By analysing the change in metal shape in the rolling deformation zone, the calculation formula for metal volume in the deformation zone is derived. This establishes a relationship between the longitudinal length difference at the rolled piece ends and the metal volume in the deformation zone. The prediction of plane shape difference between the end and the head of medium and medium-thickness plate is realized. The experimental results confirm the feasibility and effectiveness of the proposed method.

A polyhedral approach to the arithmetic and geometry of hyperbolic link complements

We provide an elementary polyhedral approach to study and deduce results about the arithmeticity and commensurability of an infinite family of hyperbolic link complements [Formula: see text] for [Formula: see text]. The manifold [Formula: see text] is the complement of [Formula: see text] by the ([Formula: see text])-link chain [Formula: see text] and has [Formula: see text] cusps. We show that [Formula: see text] is closely related to a hyperbolic Coxeter orbifold that is commensurable to an orbifold with a single cusp. Vinberg’s arithmeticity criterion and certain cusp density and volume computations allow us to reproduce some of the main results in [20] and [18] about [Formula: see text] in a comparatively elementary and direct way. As a by-product, we give a rigorous proof of Thurston’s volume formula for [Formula: see text] and deduce that, for [Formula: see text], the volume of [Formula: see text] is strictly bigger than the volume of the ([Formula: see text])-cyclic cover over one component of the Whitehead link.

A phase II randomized clinical trial to assess toxicity and quality of life of patients with breast cancer with hypofractionated versus conventional fractionation radiotherapy with regional nodal irradiation.

588 Background: The primary aim was comparing acute toxicity between conventional fractionated radiation therapy (CF-RT) and hypofractionated radiation therapy (HF-RT) for patients undergoing breast-conserving surgery or mastectomy with breast or chest wall and regional nodal irradiation (RNI). The secondary endpoints were acute and subacute toxicity, cosmesis, quality of life, and lymphedema features. Methods: Unblinded randomized trial with total of 86 patients treated with CF-RT (n = 33; 50 Gy/25 fractions ± sequential boost [10 Gy/5 fractions]) versus HF-RT (n = 53; 40 Gy/15 fractions ± concomitant boost [8 Gy/15 fractions]). Toxic effects and cosmesis evaluation used the Common Terminology Criteria for Adverse Events version 4.03 (CTCAE) and the Harvard/NSABP/RTOG scale. Patients-reported QoL was determined using European Organization for Research and Treatment of Cancer quality of life questionnaire (EORTC QLQ-C30) and the breast cancer-specific supplementary questionnaire (QLQ-BR23). Lymphedema determination compared volume differences between the affected and contralateral arms using the Casley-Smith volume formula. Results: Grade 2 and grade 3 skin rash dermatitis were lower with HF-RT than with CF-RT (28% vs. 52%, 0% vs. 6%; p = 0.022, respectively). HF-RT vs. CF-RT had lower rate of grade 2 hyperpigmentation (23% vs. 55%; p = 0.005). No differences in overall rates of any physician-assessed grade 2 or higher and grade 3 or higher acute toxicity between HF-RT vs. CF-RT were registered. There was no statistical difference between the CF-RT and the HF-RT regarding cosmesis, lymphedema rate (13% vs. 12% HF-RT vs. CF-RT; p = 1.000, respectively), and functional and symptom scales during irradiation and after six months of treatment. Conclusions: HF-RT showed lower rates of acute toxicity, with no changes in quality-of-life outcomes. Clinical trial information: NCT04015531 .

Predicting live weight using body volume formula in lactating water buffalo

Live weight (LW) is an important piece of information within production systems, as it is related to several other economic characteristics. However, in the main buffalo-producing regions in the world, it is not common to periodically weigh the animals. We develop and evaluate linear, quadratic, and allometric mathematical models to predict LW using the body volume (BV) formula in lactating water buffalo (Bubalus bubalis) reared in southeastern Mexico. The LW (391.5 ± 138.9 kg) and BV (333.62 ± 58.51 dm3) were measured in 165 lactating Murrah buffalo aged between 3 and 10 years. The goodness-of-fit of the models was evaluated using the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), coefficient of determination (R2), mean-squared error (MSE) and root MSE (RMSE). In addition, the developed models were evaluated through cross-validation (k-folds). The ability of the fitted models to predict the observed values was evaluated based on the RMSEP, R2, and mean absolute error (MAE). LW and BV were significantly positively and strongly correlated (r = 0.81; P < 0.001). The quadratic model had the lowest values of MSE (2788.12) and RMSE (52.80). On the other hand, the allometric model showed the lowest values of BIC (1319.24) and AIC (1313.07). The Quadratic and allometric models had lower values of MSEP and MAE. We recommend the quadratic and allometric models to predict the LW of lactating Murrah buffalo using BV as a predictor.

Comparison of conventional Doppler imaging techniques and superb microvascular imaging in determination of vascularization in undescended testes.

Our aim was to gain an idea about testicular injury by comparing the reduced volume, which is one of the indirect indicators of testicular damage in undescended testes, and by evaluating the reduced microvascular blood flow by superb microvascular imaging, and also to determine whether superb microvascular imaging modes could detect microvascular blood flow in more detail in the decreased volume of undescended testes. We compared testicular blood flow in undescended testes via conventional Doppler imaging, color superb microvascular imaging, and monochrome superb microvascular imaging techniques with contralateral normally located testis and normal control group. Each sample of testicular tissue was evaluated using a qualitative method. Spot color encoding and linear flow color encoding counts determined in testicular parenchyma were counted separately and expressed as numerical data. The localization of the examined testes in the grayscale was noted (proximal inguinal canal, medial inguinal canal, distal inguinal canal, and scrotal). The volume of undescended testes was calculated automatically via a formula for volume. Monochrome superb microvascular imaging is significantly superior in visualizing the vascularity of undescended testes compared with color Doppler, power Doppler and color superb microvascular imaging (p = 0.001). Also, undescended testes have a significantly lower blood flow compared with contralateral normal testes (p = 0.001). The volume of undescended testes was significantly lower than the contralateral normal testes. The volume, structure and blood flow are indirect signs of testicular damage in undescended testes. Monochrome superb microvascular imaging can detect vascularity in undescended testes better than the conventional Doppler imaging technique and color superb microvascular imaging. Based on our findings, we can report that monochrome superb microvascular imaging can be used to evaluate testicular injury and vascularity of undescended testes.

MP24-02 CLUSTER ANALYSIS IDENTIFIES BENIGN PROSTATIC HYPERTROPHY CLINICAL PHENOTYPES

You have accessJournal of UrologyCME1 Apr 2023MP24-02 CLUSTER ANALYSIS IDENTIFIES BENIGN PROSTATIC HYPERTROPHY CLINICAL PHENOTYPES Matthew Simmons, Nathaniel Taylor, W Carter Reed, Pablo Santamaria, Zachary Klaassen, Sherita King, and Martha Terris Matthew SimmonsMatthew Simmons More articles by this author , Nathaniel TaylorNathaniel Taylor More articles by this author , W Carter ReedW Carter Reed More articles by this author , Pablo SantamariaPablo Santamaria More articles by this author , Zachary KlaassenZachary Klaassen More articles by this author , Sherita KingSherita King More articles by this author , and Martha TerrisMartha Terris More articles by this author View All Author Informationhttps://doi.org/10.1097/JU.0000000000003249.02AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookLinked InTwitterEmail Abstract INTRODUCTION AND OBJECTIVE: Benign prostatic hypertrophy (BPH) is a heterogeneous disease that is affected by multiple variables including infection, genetics and inflammation. This study assessed whether hierarchical cluster analysis could better characterize BPH patients. METHODS: Patients who underwent TURP or simple prostatectomy between January 2017 to August 2022 were reviewed. 92 patients who had CT imaging within 18m of surgery were selected. CT-based prostate volumes (PV) were calculated using the ellipsoid volume formula (L*W*H*(π/6)). Metabolic syndrome score (MetX) was calculated by adding normalized BMI score to the number of metabolic syndrome-related diseases present (i.e., hypertension, hyperlipidemia, type II diabetes mellitus each disease given a score of 1). Data were scaled using Z transformation. Hierarchical clustering was conducted with Heatmapper software using average linkage and Pearson distance measurement methodology. RESULTS: Mean age of the cohort was 68 (SD=8.5), and 40% were African American. Hypertension, hyperlipidemia and type II diabetes mellitus were present in 73%, 58% and 32% of patients, respectively. Mean prostate volume was 84 ml (SD=56). Median lobes (ML) were present in 26%. Intraprostatic calcifications (IPCs) were present in 37%. Clustering revealed three clear groupings. Group 1 cnsisted of men with IPCs. These men had lower than average age, PV, and MetX scores. Group 2 consisted of men with either MLs or high PV without concomitant high MetX score. Two categories of men were included in Group 2 - younger than average men with MLs; and older than average men men with high PVs. Group 3 consisted of men with high MetX score. Age and PV was lower than average in this group. CONCLUSIONS: Cluster analysis identified three distinct clinical phenotypes of BPH. Classification of BPH subtypes may allow for improved diagnosis and therapy. Future studies will correlate molecular subtyping data and clinical outcomes data to further validate findings. Source of Funding: Medical College of Georgia © 2023 by American Urological Association Education and Research, Inc.FiguresReferencesRelatedDetails Volume 209Issue Supplement 4April 2023Page: e317 Advertisement Copyright & Permissions© 2023 by American Urological Association Education and Research, Inc.MetricsAuthor Information Matthew Simmons More articles by this author Nathaniel Taylor More articles by this author W Carter Reed More articles by this author Pablo Santamaria More articles by this author Zachary Klaassen More articles by this author Sherita King More articles by this author Martha Terris More articles by this author Expand All Advertisement PDF downloadLoading ...

Abstract Number ‐ 60: Influence of Primary Wind Diameter and Length on Coil Volume: Comparing Stryker and CEREPAK coils

Introduction Coiling procedures for intracranial aneurysms (IAs) performed in the US have steadily increaseda, and volumetric efficiency may become an important consideration for hospitals. Coil volume depends on length and primary wind diameter (PWD), and volumetric efficiency can be achieved by either inserting more coils or using coils with larger volumetric filling. This scenario analysis evaluates the influence of PWD and length on total coil volume by comparing four similar Stryker and CEREPAK coils. Methods In this analysis, we test the influence of PWD (D2) and length (L) on the Coil Volume Formula: [Π(D2/2)2(L)]c. In two hypothetical scenarios, relative percent difference in total coil volume was calculated and compared between four Stryker Target 360 coils and four similar CEREPAK coils that were matched based on closest PWD and L. In scenario I, L remained constant and D2 varied. In scenario II, D2 and L were inversely varied. In both scenarios, coil volume achieved by CEREPAK and Stryker coils was calculated and descriptively compared. Results Scenario I: (a) Stryker Target XL 360 10mm x 40cm (D2 = 0.014”, length = 40cm) with volume of 39.726mm3 was matched to the CEREPAK Heliform XL 10mm x 40cm (D2 = 0.015”, length = 40cm) with 15% greater volume of 45.604mm3. (b)Stryker Target 360 3mm x 6cm (D2 = 0.010”, length = 6cm) with volume 3.040mm3 was matched to the CEREPAK Freeform 3mm x 6cm (D2 = 0.012”, length = 6cm) with 44% greater volume of 4.378mm3. Scenario II: (a) Stryker Target XL 360 12mm x 45cm (D2 = 0.014”, length = 45cm) with volume 44.692mm3 was matched to the CEREPAK Heliform XL 12mm x 42cm (D2 = 0.015”, length = 42cm) with 7% greater volume of 47.884mm3. (b)Stryker Target 360 5mm x 20cm (D2 = 0.010”, length = 20cm) with volume 10.134mm3 was matched to the CEREPAK Freeform 5mm x 15cm (D2 = 0.012”, length = 15cm) with 8% greater volume of 10.945mm3. Conclusions The four selected CEREPAK coils compared to four equivalent Stryker coils achieved higher total coil volume in all scenarios. An increase in PWD by as little as 0.001”‐0.002”, improved relative coil volume by 15%‐44% (with equivalent coil length) and 7–8% (with varied coil length by 3–5cm). Clinicians evaluate many coil characteristics when treating an IA, however, PWD might be given consideration over length due to its impact on total coil volume.

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

The volume of the unit sphere in every dimension is given an interpretation as a product of special values of the zeta function of $\mathbb {Z}$, akin to volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A product formula is foun