Quantitative Evaluation of the Soil Signal Effect on the Correlation between Sentinel-1 Cross Ratio and Snow Depth

High-resolution Synthetic Aperture Radar (SAR), as an efficient Earth observation technology, can be used as a complementary means of observation for snow depth (SD) and can address the spatial heterogeneity of mountain snow. However, there is still uncertainty in the SD retrieval algorithm based on SAR data, due to soil surface scattering. The aim of this study is to quantify the impact of soil signals on the SD retrieval method based on the cross-ratio (CR) of high-spatial resolution SAR images. Utilizing ascending Sentinel-1 observation data during the period from November 2016 to March 2020 and a CR method based on VH- and VV-polarization, we quantitatively analyzed the CR variability characteristics of rock and soil areas within typical thick snow study areas in the Northern Hemisphere from temporal and spatial perspectives. The correlation analysis demonstrated that the CR signal in rock areas at a daily timescale shows a strong correlation (mean value > 0.60) with snow depth. Furthermore, the soil areas are more influenced by freeze-thaw cycles, such that the monthly CR changes showed no or negative trend during the snow accumulation period. This study highlights the complexity of the physical mechanisms of snow scattering during winter processes and the influencing factors that cause uncertainty in the SD retrieval, which help to promote the development of high-spatial resolution C-band data for snow characterization applications.

Conformal bootstrap near the edge

We propose a bootstrap program for CFTs near intersecting boundaries which form a co-dimension 2 edge. We describe the kinematical setup and show that bulk 1-pt functions and bulk-edge 2-pt functions depend on a non-trivial cross-ratio and on the angle between the boundaries. Using the boundary OPE (BOE) with respect to each boundary, we derive two independent conformal block expansions for these correlators. The matching of the two BOE expansions leads to a crossing equation. We analytically solve this equation in several simple cases, notably for a free bulk field, where we recover Feynman-diagrammatic results by Cardy.

NORMALIZED PARAMETERS OF A MAGNETORESISTIVE SENSOR IN BRIDGE CIRCUITS

Magnetoresistive sensors are considered as part of bridge circuits for measuring magnetic field strength and electric current value. Normalized or relative expressions are introduced to change the resistance of the sensor and the measured bridge voltage to increase the information content of the regime to provide the possibility of comparing the regimes of different sensors. To justify these expressions, a geometric interpretation of the bridge regimes, which leads to hyperbolic straight line geometry and a cross ratio of four points, is given. Upon a change in the sensor resistance, the bridge regime is quantified by the value of the cross ratio of four samples (three characteristic values and the current or real value) of voltage and resistance. The cross ratio, as a dimensionless value, is taken as a normalized expression for the bridge voltage and sensor resistance. Moreover, the cross ratio value is an invariant for voltage and resistance. The proposed approach considers linear and nonlinear dependences of measured voltage on sensor resistance from general positions.

Counting tropical rational space curves with cross-ratio constraints

This is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. The multiplicities of such cross-ratio floor diagrams can be calculated by enumerating certain rational tropical curves in the plane.

Discrete curvature and torsion from cross-ratios

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature.

Non-Euclidean Biosymmetries and Algebraic Harmony in Genomes of Higher and Lower Organisms

The article is devoted to the study of the relationship of non-Euclidean symmetries in inherited biostructures with algebraic features of information nucleotide sequences in DNA molecules in the genomes of eukaryotes and prokaryotes. These genomic sequences obey the universal hyperbolic rules of the oligomer cooperative organization, which are associated with the harmonic progression 1/1, 1/2, 1/3,.., 1/n. The progression has long been known and studied in various branches of mathematics and physics. Now it has manifested itself in genetic informatics. The performed analysis of the harmonic progression revealed its connection with the cross-ratio, which is the main invariant of projective geometry. This connection consists in the fact that the magnitude of the cross-ratio is the same and is equal to 4/3 for any four adjacent members of this progression. The long DNA nucleotide sequences have fractal-like structure with so called epi-chains, whose structures are also related to the harmonic progression and the projective-geometrical symmetries. The received results are related additionally to a consideration of DNA double helix as helical antenna. This fact of the connection of genetic informatics with the main invariant of projective geometry can be used to explain the implementation of some non-Euclidean symmetries in genetically inherited structures of living bodies.

Cross-ratio Dynamics on Ideal Polygons

Two ideal polygons, $(p_1,\ldots ,p_n)$ and $(q_1,\ldots ,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\alpha $-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha $ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is generically a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures and show that these relations, with different values of the constants $\alpha $, commute, in an appropriate sense. We investigate the case of small-gons and describe the exceptional ideal pentagons and hexagons that possess infinitely many $\alpha $-related polygons.