Outer Measure Research Articles

The observational evidence that all microflares that accelerate electrons to high energies are rooted in sunspots

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. 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." 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. 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<!PCT!> 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<!PCT!> 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<!PCT!> exhibit a relatively flat flare loop geometry. 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

Tracking multiple space objects using multiple surveillance sensors is a critical approach in many Space Situation Awareness (SSA) applications. In this process, the uncertainties of targets, dynamics, and observations are usually represented by the probability distributions. However, precise characterization of uncertainty becomes challenging due to imperfect knowledge about some key aspects, such as birth targets and sensor detection profiles. To overcome this challenge, this paper proposes a multi-sensor possibility PHD filter based on the theory of outer probability measures. An effective compensation method is introduced to tackle variations in the fields of view of SSA sensors or instances of missed detections, aiming to mitigate the inconsistency in localized information. The proposed method is adapted to centralized and distributed sensor networks, offering effective solutions for multi-sensor multi-target tracking. The major innovation of the proposed method compared with typical methods is the proper description of epistemic uncertainty, which yields more robust performance in the scenarios of lacking some information about the system. The effectiveness of the multi-sensor possibility PHD filter is demonstrated by a comparison with conventional methods in two simulated scenarios.

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

We consider an inverse problem of determining the coefficients of a fractional p -Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.

Merging of coherent upper conditional probabilities defined by Hausdorff outer measures

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 .

Common extension of two sigma-finite measures

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.

Realization of reproductive qualities of cows and productivity of young stock.

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).

Solving stationary inverse heat conduction in a thin plate

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.

Multipoint formulas in inverse problems and their numerical implementation

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).

Multipliers of Variationally Mcshane Integrable Functions in Locally Convex Space

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.

Inverse Problem for a Planar Conductivity Inclusion

This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.

Coherent conditional previsions with respect to inner and outer Hausdorff measures to represent conscious and unconscious human brain activity

Coherent upper and lower conditional probabilities defined by Hausdorff outer and inner measures are proposed to represent respectively the unconscious and conscious activities of the human brain when information is given. In the model uncertainty measures are defined according to the complexity of the conditioning event that represents the given information. The model is applied to explain mathematically the conjunction fallacy in Linda's Problem and the bias of selective attention described in the so-called invisible Gorilla experiment, that is often taken as a characteristic example of the inescapable limitations of human perception. Once people are concentrated on doing a specific action, they do not notice unexpected events (having 0 probability) occurring in the meantime. When applying the model, selective attention is no longer a bias since it is able to explain this function of the human brain mathematically. Moreover different reactions of people to unexpected events can be represented in different metric spaces with metrics which are not bi-Lipschitz. In these metric spaces coherent upper conditional probabilities defined by Hausdorff outer measures are not mutually absolutely continuous and so they do not share the same null events.

An equivalent source method for holographic visualization of acoustic sources in an apertured enclosure based on exterior measurements

Most acoustical holography techniques are developed to visualize sound sources or reconstruct sound field in a free-space environment, where no boundary reflected or scattered sound are present. However, in some engineering applications, a sound scatter is an inseparable part of source mechanism. For example, for a centrifugal fan, the aerodynamic noise sources are the turbulent pressure inside the fan shell, but the source distribution is significantly altered if the fan shell is removed. Thus, in order to visualize the sources inside the fan shell by holography methods based on exterior sound measurements, the measurements need to be performed with the presence of the shell (which is a sound scatter). This obviously violates the free-space assumption for most acoustical holography methods. In this work, traditional free-space equivalent source methods are extended to visualize acoustic sources situated in an enclosure with openings based on sound measurements exterior to the enclosure. The key step is to replace the free-space Green’s function used in traditional equivalent source methods by a Green’s function that satisfies the boundary condition of the apertured enclosure. Simulations are performed to prove the concept of the proposed acoustical holography method.

On a New Measure on the Levi-Civita Field $$ \mathcal{R} $$

The Levi-Civita field $$ \mathcal{R} $$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on $$ \mathcal{R} $$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over $$ \mathcal{R} $$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on $$ \mathcal{R} $$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on $$ \mathcal{R} $$ that proves to be a better generalization of the Lebesgue measure from $$ \mathbb{R} $$ to $$ \mathcal{R} $$ and that leads to a family of measurable sets in $$ \mathcal{R} $$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in $$ \mathbb{R} $$ hold.

A finitary outer measure logic

Abstract We extend classical first-order logic with a family of weak probability quantifiers, which we call submeasure quantifiers. Formulas are finitary, but infinitary deduction rules are needed. We consider first-order structures that are equipped with a countable family of submeasures (hence the name of the new quantifiers). We prove that every consistent set of sentences in the resulting logic is satisfiable in some structure as above. Then we restrict the set of formulas by requiring that no submeasure quantifier occurs within the scope of some classical quantifier. By suitably extending the deduction rules, we prove that every consistent set of sentences from the restricted class of formulas is satisfiable in some structure whose submeasures are actually outer measures. To perform the last step, we apply nonstandard techniques à la A. Robinson.

BDG inequalities and their applications for model-free continuous price paths with instant enforcement

Shafer and Vovk introduce in their book [8] the notion of instant enforcement and instantly blockable properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of “typical” price paths. In this paper an outer measure on the space $[0,+\infty )\times \Omega $ is introduced, which assigns zero value exactly to those sets (properties) of pairs of time t and an elementary event ω which are instantly blockable. Next, for a slightly modified measure, Itô’s isometry and BDG inequalities are proved, and then they are used to define an Itô-type integral. Additionally, few properties are proved for the quadratic variation of model-free continuous martingales, which hold with instant enforcement.

Duality for double iterated outer $L^p$ spaces

We study the double iterated outer $L^p$ spaces, that is, the outer $L^p$ spaces associated with three exponents and defined on sets endowed with a measure and two outer measures. We prove that in the case of finite sets, under certain relations between t

Measure defined on Γ–algebra and some of their generalizations

This work aims to introduce concepts about Γ–algebra calculate relative to Γ–algebra and outer measure in relation to Γ–algebra are examples. After that, we establish a relationship between them. The notions of null-additive and weakly null-additive as two generalizations of measure and extreme measure are introduced and studied too.

On SDEs with Lipschitz coefficients, driven by continuous, model-free martingales

We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard’s iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingales. We work with a new outer measure which assigns zero value exactly to those properties which are instantly blockable.

О взаимоотношении движений динамических систем в сепарабельном локально компактном метрическом пространствес инвариантной мерой

In this paper, we study the interrelation between recurrent and outgoing motions of dynamical systems. An outgoing motion is a motion whose α- and ω-limit sets are either empty or non-compact. It is shown that in a separable locally compact metric space Σ with invariant Caratheodory measure, almost all points lie on trajectories of motions that are either recurrent or outgoing, i. e. in the space Σ, the set of points Γ lying on the trajectories of nonoutgoing and non-recurrent motions has measure zero. Moreover, any motion located in Γ is both positively and negatively asymptotic with respect to the corresponding compact minimal sets. The proof of this assertion essentially relies on the classical Poincare-Caratheodory and Hopf recurrence theorems. From this proof and Hopf’s theorem, it follows that in a separable locally compact metric space, there can exist non-recurrent Poisson-stable motions, but all these motions must necessarily be outgoing. At the same time, in the compact space Σ any Poisson-stable motion is recurrent.

Soft Carathéodory extension theorem and soft outer measure

Molodsov defined the concept of a soft set, which is widely used when drawing conclusionsif we have unclear, incomplete and imprecise information. Soft set theory became an interest of many researchers who defined new concepts such as soft measure, soft ?-algebra, soft premeasure, soft outer measure, soft content, etc. In this paper, it was shown that with a certain extension, we can construct a soft measure on the obtained soft ?-algebra starting from a soft premeasure on a soft semiring.