Using airborne vector magnetic data to calculate the projection of magnetic anomaly vector onto normal geomagnetic field

The total-field magnetic anomaly [Formula: see text] is an approximation of the projection [Formula: see text] of the magnetic anomaly vector [Formula: see text] onto the normal geomagnetic field [Formula: see text]. However, for highly magnetic sources, the approximation error of [Formula: see text] cannot be ignored. To reduce the error, we have developed a method for calculating [Formula: see text] by using airborne vector magnetic data based on the vector relationship of geomagnetic field [Formula: see text]. The calculation uses the magnitude of the vectors [Formula: see text], [Formula: see text], and [Formula: see text] through a simple approach. To ensure that each magnitude has the same level, we normalize the magnitude of [Formula: see text] using the total-field magnetic data measured by the scalar magnetic sensor. The method is applied to the measured airborne vector magnetic data at the Qixin area of the East Tianshan Mountains in China. The results indicate that the calculated [Formula: see text] has high precision and can distinguish the approximation error less than 3.5 nT. We also analyze the characteristics of the approximation error that are caused by the effects of different total magnetization inclinations. These error characteristics are used to predict the total magnetization inclination of a 2D magnetic source based on the measured airborne vector magnetic data.

Adaptive Kalman Filter for Linear Systems with Additive and Multiplicative Noises

This manuscript investigates adaptive Kalman filter problem of of linear systems with multiplicative and additive noises. The main contributions are stated in two aspects. Firstly, compared with the estimation problem of linear systems with additive noises, we propose an algorithm that is applicable to the linear systems with both additive and multiplicative noises. To solve the technical issue raised by the multiplicative noise, a variational Bayesian approach is proposed. Moreover, the proposed approach is capable of estimating the multiplicative and additive measurement noise covariances as a whole, while the existing algorithms often operate in a separate way. Secondly, in contrast with existing literature, where the covariance of the multiplicative noise is assumed to be fixed and known, we focus on the situation that the covariances of both additive and multiplicative noises are time-varying and unknown. Towards this end, a novel adaptive Kalman filter is proposed to jointly estimate the covariances of multiplicative and additive noises based on projection formula and a VB approach.

Adaptive Kalman Filter for Linear Systems with Additive and Multiplicative Noises

This manuscript investigates adaptive Kalman filter problem of of linear systems with multiplicative and additive noises. The main contributions are stated in two aspects. Firstly, compared with the estimation problem of linear systems with additive noises, we propose an algorithm that is applicable to the linear systems with both additive and multiplicative noises. To solve the technical issue raised by the multiplicative noise, a variational Bayesian approach is proposed. Moreover, the proposed approach is capable of estimating the multiplicative and additive measurement noise covariances as a whole, while the existing algorithms often operate in a separate way. Secondly, in contrast with existing literature, where the covariance of the multiplicative noise is assumed to be fixed and known, we focus on the situation that the covariances of both additive and multiplicative noises are time-varying and unknown. Towards this end, a novel adaptive Kalman filter is proposed to jointly estimate the covariances of multiplicative and additive noises based on projection formula and a VB approach.

A remark on convolution products for quiver Hecke algebras

In this paper, we investigate a connection between convolution products for quiver Hecke algebras and tensor products for quantum groups. We give a categorification of the natural projection [Formula: see text] sending the tensor product of the highest weight vectors to the highest weight vector in terms of convolution products. When the quiver Hecke algebra is symmetric and the base field is of characteristic [Formula: see text], we obtain a positivity condition on some coefficients associated with the projection [Formula: see text] and the upper global basis, and prove several results related to the crystal bases. We then apply our results to finite type [Formula: see text] using the homogeneous simple modules [Formula: see text] indexed by one-column tableaux [Formula: see text].

Max-projective modules

Weakening the notion of [Formula: see text]-projectivity, a right [Formula: see text]-module [Formula: see text] is called max-projective provided that each homomorphism [Formula: see text], where [Formula: see text] is any maximal right ideal, factors through the canonical projection [Formula: see text]. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are [Formula: see text]-projective (respectively, max-projective) are characterized. For a commutative Noetherian ring [Formula: see text], we prove that injective modules are [Formula: see text]-projective if and only if [Formula: see text], where [Formula: see text] is [Formula: see text] and [Formula: see text] is a small ring. If [Formula: see text] is right hereditary and right Noetherian then, injective right modules are max-projective if and only if [Formula: see text], where [Formula: see text] is a semisimple Artinian and [Formula: see text] is a right small ring. If [Formula: see text] is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective.

∗-Strongly regular rings

Let [Formula: see text] be a ring with involution ∗. An element [Formula: see text] is called ∗-strongly regular if there exists a projection [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] is invertible, and [Formula: see text] is said to be ∗-strongly regular if every element of [Formula: see text] is ∗-strongly regular. We discuss the relations among strongly regular rings, ∗-strongly regular rings, regular rings and ∗-regular rings. Also, we show that an element [Formula: see text] of a ∗-ring [Formula: see text] is ∗-strongly regular if and only if [Formula: see text] is EP. Hence we finally give some characterizations of EP elements.

The Casimir effect in string theory

We discuss the Casimir effect in heterotic string theory. This is done by considering a [Formula: see text] twist acting on one external compact direction and three internal coordinates. The hyperplanes fixed by the orbifold generator [Formula: see text] realize the two infinite parallel plates. For the latter to behave as “conducting material,” we implement in a modular invariant way the projection [Formula: see text] on the spectrum running in the vacuum-to-vacuum amplitude at one-loop. Hence, the relevant projector to account for the Casimir effect is orthogonal to that commonly used in string orbifold models which is [Formula: see text]. We find that this setup yields the same net force acting on the plates in the context of quantum field theory and string theory. However, when supersymmetry is not present from the onset, finiteness of the resultant force in field theory is reached by adding formally infinite forces acting on either side of each plate, while in string theory, both contributions are finite. On the contrary, when supersymmetry is spontaneously broken à la Scherk–Schwarz, finiteness of each contribution is fulfilled in field and string theory.

Development in understanding of Gauss-Krüger projection and its outcomes

The role of Gauss and Krüger is made clear in developing Gauss-Krüger projection. Gauss developed the projection and Krüger had brought Gauss’s posthumous work into the open. From studying such historical issues, useful projection formula was found and this is now implemented for actual usage in surveying in Japan.

Toward an explicit eigencurve for GL(3)

Starting with a numerically noncritical (at [Formula: see text]) Hecke eigenclass [Formula: see text] in the homology of a congruence subgroup [Formula: see text] of [Formula: see text] (where [Formula: see text] divides the level of [Formula: see text]) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of [Formula: see text] to a Hecke eigenclass [Formula: see text] with coefficients in a module of [Formula: see text]-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection [Formula: see text] to weight space of the eigencurve around the point [Formula: see text] corresponding to the system of Hecke eigenvalues of [Formula: see text]. We do this under the conjecturally mild hypothesis that [Formula: see text] is smooth at [Formula: see text].

Complete hypersurfaces with constant Möbius scalar curvature

Let [Formula: see text] be an umbilical free hypersurface in the unit sphere [Formula: see text]. Four basic invariants of [Formula: see text], under the Möbius transformation group of [Formula: see text] are the Möbius metric [Formula: see text], the Möbius second fundamental form [Formula: see text], the Blaschke tensor [Formula: see text] and the Möbius form [Formula: see text]. In this paper, we study complete hypersurfaces with constant normalized Möbius scalar curvature [Formula: see text] and vanishing Möbius form [Formula: see text]. By computing the Laplacian of the funtion [Formula: see text], where the trace-free Blaschke tensor [Formula: see text], and applying the well known generalized maximum principle of Omori–Yau, we obtain the following result: [Formula: see text] must be either Möbius equivalent to a minimal hypersurface with constant Möbius scalar curvature, when [Formula: see text]; [Formula: see text] in [Formula: see text], when [Formula: see text]; the pre-image of the stereographic projection [Formula: see text] of the circular cylinder [Formula: see text] in [Formula: see text], when [Formula: see text]; or the pre-image of the projection [Formula: see text] of the hypersurface [Formula: see text] in [Formula: see text], when [Formula: see text].