The point-to-set principle [J. H. Lutz and N. Lutz, ACM Trans. Comput. Theory, 10(2), 7 (2018).] characterizes the Hausdorff dimension of a subset [Formula: see text] by the effective (or algorithmic) dimension of its individual points. This characterization has been used to prove several results in classical, i.e. without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand’s projection theorem, a fundamental result in fractal geometry. In this paper, we introduce the notion of optimal oracles for subsets [Formula: see text]. One of the primary motivations of this definition is that, if [Formula: see text] has optimal oracles, then the conclusion of Marstrand’s projection theorem holds for [Formula: see text]. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of [Formula: see text] agree, then [Formula: see text] has optimal oracles. Moreover, we show that the existence of sufficiently nice outer measures on [Formula: see text] implies the existence of optimal Hausdorff oracles. In particular, the existence of exact gauge functions for a set [Formula: see text] is sufficient for the existence of optimal Hausdorff oracles, and is therefore sufficient for Marstrand’s theorem. Thus, the existence of optimal oracles extends the currently known sufficient conditions for Marstrand’s theorem to hold. Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a generalization of Davies theorem on projections.

This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect to a monotone set function of bounded variation. In this work, we prove that an integral representation with respect to a countably additive measure is also possible, on the Borel σ-algebra, even when the coherent upper prevision is defined by the Choquet integral with respect to a Hausdorff measure, which is not of bounded variation. It occurs since Hausdorff outer measures are metric measures, and therefore every Borel set is measurable with respect to them. Furthermore, when the conditioning event has a Hausdorff measure in its own Hausdorff dimension equal to zero or infinity, coherent conditional probability is defined via the countably additive Möbius transform of a monotone set function of bounded variation. The paper demonstrates the continuity of coherent conditional previsions induced by Hausdorff measures.

Against the backdrop of the deep integration of the global digital economy and cross-border e-commerce, and addressing the lack of measure theory and quantitative analysis challenges for high-dimensional dynamic data in this field, this study constructs a measure theory system for cross-border e-commerce markets that combines mathematical rigor and economic interpretability based on Carathéodory’s extension theorem. Based on functional analysis and measure theory, the study defines the market fundamental set as the topological product space of a time index set and a multi-dimensional transaction state space. By constructing a combined structure of a left-open right-closed interval semiring and a power set semiring that satisfies the closure of Boolean algebra operations, an algebraic framework is established for the unified measurement of continuous and discrete variables. On the semiring structure, a σ-finite premeasure integrating Lebesgue measure and counting measure is defined. With the help of the countable covering mechanism generated by outer measure and the measure screening rules of Carathéodory’s measurability condition, the axiomatic extension from premeasure to complete measure on the σ-algebra is completed. Through the verification of Carathéodory’s condition for subsets of null sets and the transmission of outer measure monotonicity, the completeness of the measure space is strictly proved, and the core property that “subsets of null sets must be measurable” is established, providing a solid measure-theoretic foundation for mathematical modeling of cross-border e-commerce markets. At the empirical analysis level, the study uses micro-panel data on global cross-border e-commerce transactions from 2018 to 2024. Through the Kolmogorov-Smirnov test in non-parametric hypothesis testing, the distribution isomorphism between the theoretical measure and empirical data is verified. Based on the measure space theory, a Generalized Method of Moments (GMM) panel regression model is constructed. System GMM and Difference GMM estimation methods are used to handle endogeneity issues. Combined with instrumental variable methods and lag variable techniques, key parameters such as the logarithmic elasticity of economic scale between importing and exporting countries, the spatial decay effect of geographical distance, and the asymmetric inhibitory effect of tariff policies are quantitatively analyzed. A graph neural network model integrating measure theory is innovatively designed. By introducing a completeness regular term, the measure constraints on null sets and their subsets are achieved. Combined with the SHAP value interpretability analysis method, the marginal contribution of each characteristic variable in model decision-making is revealed. The study finds that the constructed measure space not only satisfies the axiomatic requirements of modern measure theory such as completeness and σ-finiteness, but also through the empirical tests of the GMM model and graph neural network, it is confirmed that it can effectively characterize the economic scale effect, spatial distance decay law, and policy sensitivity characteristics in cross-border e-commerce transactions, providing a methodological innovation paradigm based on measure theory for quantitative analysis in the field of international business in the digital economy era.

We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measure κ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

The aim of the study was to obtain phenotypic characteristics of the resource population chickens characterizing the development of the musculoskeletal system and body build for the subsequent study of the structural variability of the chicken genome and the search for gene variants that determine the manifestation of meat productivity phenotypes, including the development of the musculoskeletal system. Materials and methods. The object of the study were chickens of the F0 generation of the Uzbek fighting breed (n = 65) and Cornish (n = 79); hybrids F1 (♂ Cornish x ♀ Uzbek fighting; n = 117) and F2 (♂F1 x ♀ F1; n = 133). At the age of one day, 2 and 4 weeks, the weight (g) of the chickens was determined. Exterior measurements were taken at the age of 4 weeks. The following indicators were estimated (cm): chest girth; metatarsus girth; keel length; metatarsus length. The body conformation indices were calculated. Results. The chickens of the original F0 breeds and F1, F2 hybrids differed significantly in live weight at the age of 2 and 4 weeks. The weight of the Uzbek Fighting breed gene pool chickens was 3,5–4,2 times lower than that of the Cornish breed; the F1 and F2 hybrids had intermediate indicators. In Cornish breed chickens (F0), the coefficient of variability in live weight at the age of 4 weeks was 13,8 %; in Uzbek Fighting breed chickens (F0) – 15,0 %; in F1 and F2 hybrids – 20,5 and 17,6 %, respectively. The maximum values of the massiveness, eirisomy and shortness indices were obtained on Cornish breed chickens, while Uzbek Fighting breed chickens had low indicators; F1 and F2 hybrids are intermediate. Heterosis was established for the relative length of the metatarsus in F1 and F2 hybrids compared to the original breeds. The data obtained proved the feasibility of using the Cornish and Uzbek fighting breeds to create a resource population for the purpose of conducting research to find gene variants that determine the manifestation of meat productivity phenotypes.

Let \mathcal{I} be a surjective closed operator ideal with associated outer measure \gamma_{{\mathcal{I}}} , and let \gamma_{{\mathfrak{I}}} be the extension of \gamma_{{\mathcal{I}}} to bilinear operators. We establish a convexity inequality for the measure \gamma_{{\mathfrak{I}}} of a bilinear operator interpolated by the real method. Our arguments do not require that \mathcal{I} be symmetric.

The measure-preserving, but not necessarily invertible, ergodic transformations of the compact metric space with the Caratheodory measure are considered. The behavior of the Birkhoff sums for integrable and almost everywhere bounded functions with zero mean value in terms of the Caratheodory measure is studied. It is shown that for almost all points of the metric space there is an infinite sequence of "moments of time"; along which the Birkhoff sums tend to zero and at the same moments the trajectory points approach their initial position as close as possible (as in the Poincare return theorem). As an example, we consider the transformation ; of the single segment; closely related to Bernoulli tests.

Relevance. Accounting for live weight in reindeer husbandry is an important indicator for breeding, but due to the difficulty of determining it, there is a need to develop a predictive matrix of animal weight through biometric exterior measurements. The purpose of the study is to develop a model for predicting the live weight of reindeer using regression analysis. Methods. The research was carried out using exterior measurements and weighing results of Nenets reindeer (males n = 48, females n = 50) from the Tazovsky district on the Yamal Peninsula at the age of 2 to 9 years. Calculation of statistical parameters, visualization of correlation data and regression analysis using the least squares method were carried out in MS Excel and R-studio.Results. The greatest variability among exterior measurements was noted in the indicators “chest width CW” (9.6%) and “live weight LW” (9.4%). When conducting a correlation analysis, multicollinearity was revealed between the height at the withers HW and the height at the elbow HE r = 0.824 (p ≤ 0.001). High and significant correlations of live weight LW with chest depth CD and chest girth CG r = 0.651 and r = 0.687 (p ≤ 0.001), head length HL r = 0.678 (p ≤ 0.001), height at withers HW r = 0.663 (p ≤ 0.001) and body length BL r = 0.639 (p ≤ 0.001). The most effective model m2 was determined, including chest girth and body length in its structure, the coefficient of determination of which was R2 = 0.70, with multiple R = 0.83, which reflects 70% of the explained variable in the model, with an approximation of 4.2%. As a result, a table was created for predicting the live weight of reindeer using biometric data, which will help simplify selection and breeding work in future populations of hard-to-reach areas.

Context. In general, large solar flares are more efficient at accelerating high-energy electrons than microflares. Nonetheless, sometimes microflares that accelerate electrons to high energies are observed. Their origin is unclear. Aims. We statistically characterized microflares with strikingly hard spectra in the hard X-ray (HXR) range, which means that they are efficient at accelerating high-energy electrons. We refer to these events as “hard microflares”. Methods. We selected 39 hard microflares, based on their spectral hardness estimated from the Solar Orbiter/STIX HXR quicklook light curves in two energy bands. The statistical analysis is built on spectral and imaging information from STIX combined with extreme ultraviolet (EUV) and magnetic field maps from SDO/AIA and SDO/HMI. Results. The key observational result is that all hard microflares in this dataset have one of the footpoint rooted directly within a sunspot (either in the umbra or the penumbra). This clearly indicates that the underlying magnetic flux densities are large. For the events with the classic two-footpoints morphology, the absolute value of the mean line-of-sight magnetic flux density (and vector magnetic field strength) at the footpoint rooted within the sunspot ranges from 600 to 1800 G (1500 to 2500 G), whereas the outer footpoint measures from 10 to 200 G (100 to 400 G), therefore about ten times weaker. In addition, approximately 78% of the hard microflares, which exhibited two HXR footpoints, have similar or even stronger HXR flux from the footpoint rooted within the sunspot. This contradicts the magnetic mirroring scenario. The median footpoint separation, measured through HXR observations, is approximately 24 Mm, which aligns with regular events of similar GOES classes. In addition, about 74% of the events could be approximated by a single-loop geometry, demonstrating that hard microflares typically have a relatively simple morphology. Out of these events, around 54% exhibit a relatively flat flare loop geometry. Conclusions. We conclude that all hard microflares are rooted in sunspots, which implies that the magnetic field strength plays a key role in efficiently accelerating high-energy electrons, with hard HXR spectra associated with strong fields. This key result will allow us to further constrain our understanding of the electron acceleration mechanisms in flares and space plasmas.

Multi-sensor possibility PHD filter for space situational awareness

Determining coefficients for a fractional p-Laplace equation from exterior measurements

Within the Memphis region of Egypt, the Old Kingdom period (approximately 2686–2181 BCE) saw the construction of over 70 pyramids. The extant survey data for these pyramids, derived mainly from architectural studies, have not been substantially updated. The information from this period predominantly encompasses descriptive accounts and schematic representations, focusing on the dimensions, internal chambers, and passageways of the pyramids. Notably absent are detailed documentations that provide insights into the individual stone blocks and the intricacies of their masonry. Consequently, the current understanding of the pyramid construction methodologies of this era remains largely hypothetical. Since 2005, I have been actively engaged in an interdisciplinary 3D survey initiative, focusing on acquiring laser- scanned 3D data of Memphite pyramids. This endeavor aims to analyze their structural forms to elucidate the progression in construction techniques during this epoch. Beginning in 2013, the “Giza 3D Survey” project was initiated as a collaboration between academia and industry. This project sought to comprehensively update the exterior measurement data of the Giza pyramid complex, employing “Structure from Motion/Multi-View Stereo (SfM/MVS)” technologies to generate 3D representations from photographic imagery. This comprehensive approach seeks to unravel the construction methodologies of these ancient structures. As a result of a decade-long survey, it has been discovered that the pyramids of Giza are not merely accumulations of limestone blocks. Instead, they contain a central “core structure” in a step-like formation, a framework made up of “backing stones,” and areas filled with debris and sand, referred to as “chamber method.”

Coherent upper conditional probabilities defined by Hausdorff measures on metric spaces are proven to represent merging opinions with increasing information when the metrics are bi-Lipschitz equivalent .

A σ – finite measure is obtained as the unique common extension of two σ – finite measures defined on two disjoint rings. This is done by first considering an outer measure on a σ – ring containing both disjoint rings. This outer measure then gives rise to the unique extension in question.

For the first time, on the basis of complex research, scientifically based and experimentally proven zootechnical feasibility of using the Prevention-N-A biologics developed by us based on the polysaccharide complex of yeast cells Saccharomyces cerevisiae and the aminoglycoside group bactericidal preparation in the technology of obtaining and growing calves in comparison with the previously approved PS-2 preparation. It is proved that three-time intramuscular injection of PS-2 and Prevention-N-A biologics to down-calvers 45-40, 25-20 and 15-10 days before calving in a dose of 10 ml prevents gynecological diseases and increases reproductive function. Under the influence of preparations in cows, the time of separation of the fetal membranes was reduced by 6.0 and 6.4 hours, retention of placenta, post-parturient complications and breast diseases were prevented. The risk of subinvolution of the uterus and endometritis in the first case decreased by 3.0 and 2.0 times, respectively, and in the second - was excluded (P<0.05). Against the background of immunocorrection in cows, the time of onset of heat by 11.6 and 14.2 days, the insemination index by 1.6 and 1.8 times, the service period by 22.4 and 28.4 days, and fertilization in 1 oestrus increased by 2.5 and 3.0 times (P<0.05-0.01). It was found that twice intramuscular injection of PS-2 and Prevention-N-A to calves on 2...3rd and 7... 9th day of life in a dose of 3 ml stimulates their growth and development, reduces the incidence. By the end of the growing period, the animals of the 1st and 2nd experimental groups were superior in live weight to control peers by 4.6 and 7.0 kg, rearing - 13.8 and 17.0 kg, and fattening - by 19.4 and 24.2 kg, respectively (P<0.05-0.01). A similar pattern occurred in the nature of changes in exterior measurements and the growth coefficient of animals of the compared groups. In calves of the experimental groups, the incidence of respiratory and digestive organs decreased by 2.3 and 7.0 times, the recovery time - by 1.3 and 4.3 days, respectively, compared to the control (P<0.05).

The aim of the study was to study the genetic structure of the reindeer population of the native Nenets breed and to evaluate the morphometric parameters in the resulting genetic groups. Pearson's Chi-square (?2) test was used to assess the statistical significance of differences between the empirical and theoretical genotype frequencies. Theoretical occurrence frequencies were calculated based on the Hardy-Weinberg law. A z-test was used to identify partial differences between the empirical and theoretical distributions. The software Python 3.9.7, GenAlEx 6.51 was used for data processing and statistical analysis. The population studied according to STR profiles is split into three independent groups (I, II, III) with sizes of 33, 53, 16 individuals, respectively. Genetic differentiation between the identified subpopulations was weak, but statistically significant (Fst = 0.022, p = 0.01). The greatest differentiation is observed between subpopulations I and III (Fst = 0.025, p = 0.03), while the differences between I - II and II - III are less pronounced (Fst = 0.017, Fst = 0.015). In subpopulations I and II, all loci are polymorphic; in subpopulation III, only 93.75% of the loci are polymorphic. The average number of alleles per locus is 6.44?0.63, 6.56?0.60, 5.25?0.54, respectively, for subpopulations I, II, III. The effective number of alleles for subpopulations is 3.73?0.44, 3.55?0.38, 3.40?0.40, respectively. The Shannon information index for the three subpopulations is at the same level and amounts to 1.38?0.13, 1.36?0.12, 1.27?0.13. The distribution according to the Hardy-Weinberg law and the empirical distribution of heads according to the genotypes of the growth hormone gene AA, and BB are statistically significantly different and are 36.603:47.795:15.603 and 22:77:1, respectively. Therefore, this equilibrium is absent in the studied population and a shift in the distribution of allele frequencies is observed. Genetic drift, migration processes or selection pressure may mediate such a shift. For the myostatin gene, no statistically significant differences were found between the theoretically expected and empirical distributions of genotypes. Females with the GHAA genotype had a higher live weight than their peers with the GHgenotype. The study of reindeer exterior measurements and physique indices in conjunction with somatotropin gene polymorphism found that individuals with the GHAA genotype outperform their peers with the GHgenotype, while low variability of indicators for these traits was observed. In the studied sample of both sex and age groups, a rather high value of the Gregory index (0.86...0.99) was found, which is typical for beef cattle. The obtained results can serve as the basis for the development of a system of genetic assessment in reindeer breeding and indicate the priority groups of animals for further breeding.

We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.

We present the first numerical study of multipoint formulas for finding leading coefficients in asymptotic expansions arising in potential and scattering theories. In particular, we implement different formulas for finding the Fourier transform of potential from the scattering amplitude at several high energies. We show that the aforementioned approach can be used for essential numerical improvements of classical results including the slowly convergent Born–Faddeev formula for inverse scattering at high energies. The approach of multipoint formulas can be also used for recovering the x-ray transform of potential from boundary values of the scattering wave functions at several high energies. Determination of total charge (electric or gravitational) from several exterior measurements is also considered. In addition, we show that the aforementioned multipoint formulas admit an efficient regularization for the case of random noise. In particular, we proceed from theoretical works (Novikov 2020 Inverse Problems 36 095001; 2021 Russ. Math. Surv. 76 723–5).

The study was conducted to see and collect data on how leadership style, work discipline, and job satisfaction affect employee performance. The research I conducted used quantitative methods and descriptive analysis. For this study, data were collected online through a randomized method through a likert scale questionnaire 1-5 to 190 respondents who met the following criteria: they are over 18 years old and have worked or participated in an organization, team, or company. For data analysis, the SmartPLS application is used to perform inner measurements, reliability values, discriminant validity, Cronbach's Alpha, and exterior measurements. The outer measurements also use T-test calculations, determinacy coefficient, f-square, and VIF. The main purpose of this study is to prove that leadership style contributes positively to employee performance, work discipline has a positive influence on employee performance, and job satisfaction has a positive impact on employee performance.

Abstract We study measurable real valued multipliers of variationally McShane (resp. McShane) integrable functions defined on a σ-finite outer regular quasi-Radon measure space and taking values in a complete locally convex topological vector space X. We also show: in case X is representable by semi-norm then essentially bounded real measurable functions are multipliers of functions which are Pettis integrable as well as integrable by semi-norm. The space of real valued measurable and essentially bounded functions turn out to be precisely the multipliers of variationally McShane (resp. McShane) integrable functions in case X is representable by semi-norm.