Canonical Map Research Articles

Deep Generative Medical Image Harmonization for Improving Cross-Site Generalization in Deep Learning Predictors.

In the medical imaging domain, deep learning-based methods have yet to see widespread clinical adoption, in part due to limited generalization performance across different imaging devices and acquisition protocols. The deviation between estimated brain age and biological age is an established biomarker of brain health and such models may benefit from increased cross-site generalizability. To develop and evaluate a deep learning-based image harmonization method to improve cross-site generalizability of deep learning age prediction. Retrospective. Eight thousand eight hundred and seventy-six subjects from six sites. Harmonization models were trained using all subjects. Age prediction models were trained using 2739 subjects from a single site and tested using the remaining 6137 subjects from various other sites. Brain imaging with magnetization prepared rapid acquisition with gradient echo or spoiled gradient echo sequences at 1.5 T and 3 T. StarGAN v2, was used to perform a canonical mapping from diverse datasets to a reference domain to reduce site-based variation while preserving semantic information. Generalization performance of deep learning age prediction was evaluated using harmonized, histogram matched, and unharmonized data. Mean absolute error (MAE) and Pearson correlation between estimated age and biological age quantified the performance of the age prediction model. Our results indicated a substantial improvement in age prediction in out-of-sample data, with the overall MAE improving from 15.81 (±0.21) years to 11.86 (±0.11) with histogram matching to 7.21 (±0.22) years with generative adversarial network (GAN)-based harmonization. In the multisite case, across the 5 out-of-sample sites, MAE improved from 9.78 (±6.69) years to 7.74 (±3.03) years with histogram normalization to 5.32 (±4.07) years with GAN-based harmonization. While further research is needed, GAN-based medical image harmonization appears to be a promising tool for improving cross-site deep learning generalization. 4 TECHNICAL EFFICACY: Stage 1.

Surfaces with canonical map of maximum degree

We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the Z / 3 \mathbb {Z}/3 -invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth.

SURREAL ORDERED EXPONENTIAL FIELDS

AbstractIn 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$ -vector space) to be isomorphic to an initial subfield ( $K$ -subspace) of ${\mathbf {No}}$ , i.e. a subfield ( $K$ -subspace) of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$ . In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $({\mathbf {No}}, \exp )$ . These include all models of $T({\mathbb R}_W, e^x)$ , where ${\mathbb R}_W$ is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of ${\mathbf {No}}$ , which includes ${\mathbf {No}}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field ${\mathbb T}^{LE}$ of logarithmic-exponential transseries into ${\mathbf {No}}$ is shown to be initial, as are the ordered exponential fields ${\mathbb R}((\omega ))^{EL}$ and ${\mathbb R}\langle \langle \omega \rangle \rangle $ .

Actor of a crossed module of dialgebras via tetramultipliers

We study the representability of actions in the category of crossed modules of dialgebras via tetramultipliers. We deduce a pair of dialgebras in order to construct an object which, under certain circumstances, is the actor (also known as the split extension classifier). Moreover, we give give a full description of actions in terms of equations. Finally, we check that under the aforementioned circumstances, the center coincides with the kernel of the canonical map from a crossed module to its actor.

Examples of Surfaces with Canonical Map of Maximal Degree

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois étale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois étale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

AbstractLet$\pi$be a group satisfying the Farrell–Jones conjecture and assume that$B\pi$is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group$\pi$whose canonical map to$B\pi$has degree 1, and show that two such manifolds ares-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If$\pi$is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.
 

Irreducible cone spherical metrics and stable extensions of two line bundles

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in U(1). By using the theory of indigenous bundles, we construct on a compact Riemann surface X of genus gX≥1 a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in 2πZ>1, which is generically injective in the algebro-geometric sense as gX≥2. As an application, we prove the following two results about irreducible metrics:• as gX≥2 and d is even and greater than 12gX−7, the effective divisors of degree d which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension ≥2(d+3−3gX) in Symd(X);• as gX≥1, for almost every effective divisor D of degree odd and greater than 2gX−2 on X, there exist finitely many cone spherical metrics representing D.

The tangent complex of K-theory

We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is cyclic homology (over k). This equivalence is compatible with the $\lambda$-operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_\infty \to K$. The proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory a la Lurie-Pridham.

Assessment of River-Sea Interaction in the Danube Nearshore Area (Ukraine) by Bioindicators and Statistical Mapping

There is a lack of understanding of the main drivers that form the picture of biological communities of transitional waters in deltaic ecosystems under the influence of terrestrial sources. Analysis of hydrochemical parameters in relation to phytoplankton communities in the Ukrainian part of the Danube coastal zone (in August 2018) is the focus of current work. The goal was to identify patterns in the distribution of environmental parameters (salinity and nutrients) in the area of the shipping channel through the Bystry arm, as well as to assess the state of water quality. The ecological bioindicators approach using modern statistical methods, and ecological mapping shows sufficient achievements in interpreting the results. The indicators of salinity (mesohalobes) had better describe the character of the transportation of fresh riverine waters than salinity gradient. The composition of 35 indicator phytoplankton species corresponds to 3 and 4 water quality classes in the coastal zone. High N:P ratios showed an imbalance in the ecosystem as an indicator of production and destruction processes. Statistical maps of the indicator species distribution revealed the river current’s influence on the nearshore water mass. Ecological maps of surface and bottom variables show various environmental impacts resulting from dredging in the shipping channel and excavated soil dumping. Canonical correspondence analysis (CCA) and statistical maps revealed two pools of factors with oppositely directed effects on phytoplankton: salinity, on one hand, and nutrients, on the other. Miozoa and Chlorophyta have an opposite interaction with salinity and oxygen and can be ecosystem change indicators in further analysis

Quantum Unique Ergodicity for Eisenstein Series in the Level Aspect

We prove a variety of quantum unique ergodicity (QUE) results for Eisenstein series in the level aspect. A new feature of this variant of QUE is that the main term involves the logarithmic derivative of a Dirichlet L-function on the 1-line. A zero of this L-function near the 1-line can thus have a distorting effect on the main term. We obtain quantitative control on the test function and thereby prove an asymptotic formula in the level aspect version of the problem with test functions of shrinking support. Surprisingly, this asymptotic formula shows some obstruction to equidistribution that may retrospectively be interpreted as being caused by the growth of Eisenstein series in the cusps. We also make some coarse descriptions on the unevenness of the mass distribution of level N Eisenstein series on the fibers of the canonical projection map from \(Y_0(N)\) to \(Y_0(1)\).

Some infinite sequences of canonical covers of degree 2

Abstract In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.

The Segal conjecture for topological Hochschild homology of Ravenel spectra

In the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing the algebraic K-theory of X(n) using trace methods, which approximates the algebraic K-theory of the sphere spectrum in a precise sense. We solve the homotopy limit problem for topological Hochschild homology of T(n) under the assumption that the canonical map \(T(n)\rightarrow BP\) of homotopy commutative ring spectra can be rigidified to map of \(E_2\) ring spectra. We show that the obstruction to our assumption holding can be described in terms of an explicit class in an Atiyah-Hirzebruch spectral sequence.

On an open problem of characterizing the birationality of $4K$

We answer an open problem raised by Chen and Zhang in 2008 and prove that, for any minimal projective 3-fold $X$ of general type with the geometric genus $\geq 5$, $X$ is birationally fibred by a pencil of $(1,2)$-surfaces (i.e. $c_1^2=1$, $p_g=2$) if and only if the $4$-canonical map $\varphi_{4,X}$ is non-birational. The statement does not hold for those with the geometric genus $\leq 4$ according to our examples.

The influence of air temperature diversity in Central Europe on the occurrence of very strong and extreme cold stress in Poland in winter months

The aim of this research was to illustrate the relationship between the occurrence of very strong and extreme cold stress among human beings under the Universal Thermal Climate Index (UTCI) across Poland and largescale near-surface air temperature over the central part of Europe. The statistical downscaling procedure of canonical correlation analysis (CCA) with respect to the period between 1971 and 2000 was applied to extract the main modes of large-scale factors and their local responses. The greatest frequency of occurrence of the discussed cold stress exceeded 35% in January. Most of the variance (39-44%) of the local field in each winter month is explained by the first coupled canonical maps. T he main patterns of large-scale field show negative anomalies of monthly mean air temperature in central Europe from -1°C up to -3°C. It can indicate an increase in frequency of the occurrence of analysed cold stress categories throughout the entire area of Poland, by 2% to even over 10%, depending on the region. The best quality model was obtained in eastern Poland, especially in the north-east and south-west of Poland.

Positive Scalar Curvature due to the Cokernel of the Classifying Map

This paper contributes to the classification of positive scalar curvature metrics up to bordism and up to concordance. Let $M$ be a closed spin manifold of dimension $\ge 5$ which admits a metric with positive scalar curvature. We give lower bounds on the rank of the group of psc metrics over $M$ up to bordism in terms of the corank of the canonical map $KO_*(M)\to KO_*(B\pi_1(M))$, provided the rational analytic Novikov conjecture is true for $\pi_1(M)$.

Generalized Cartan-Behnke-Stein's theorem and $q$-pseudoconvexity in a Stein manifold

Let $D$ be an open subset of an $n$-dimensional Stein manifold, where $n\geq 2$. Assume that the canonical map $H^{n-1}(D,\mathcal{O})\to H^{n-1}(D,\mathcal{M})$ is injective. Then, we prove that $D$ is pseudoconvex of order 1, which generalizes the well-known theorem of Cartan-Behnke-Stein.

Self-dual codes over a family of local rings

We construct an infinite family of commutative rings $${R_{q,\varDelta }}$$ and we study codes over these rings as well as the structure of the rings. We define a canonical Gray map from $${R_{q,\varDelta }}$$ to vectors over the residue finite field of q elements and use it to relate codes over $${R_{q,\varDelta }}$$ to codes over the finite field $${\mathbb {F}}_q$$ . Finally, we determine the parameters for when self-dual codes exist and give various constructions for self-dual codes over $${R_{q,\varDelta }}$$ .

EEG microstate periodicity explained by rotating phase patterns of resting-state alpha oscillations

Spatio-temporal patterns in electroencephalography (EEG) can be described by microstate analysis, a discrete approximation of the continuous electric field patterns produced by the cerebral cortex. Resting-state EEG microstates are largely determined by alpha frequencies (8-12 Hz) and we recently demonstrated that microstates occur periodically with twice the alpha frequency.To understand the origin of microstate periodicity, we analyzed the analytic amplitude and the analytic phase of resting-state alpha oscillations independently. In continuous EEG data we found rotating phase patterns organized around a small number of phase singularities which varied in number and location. The spatial rotation of phase patterns occurred with the underlying alpha frequency. Phase rotors coincided with periodic microstate motifs involving the four canonical microstate maps. The analytic amplitude showed no oscillatory behaviour and was almost static across time intervals of 1-2 alpha cycles, resulting in the global pattern of a standing wave.In n=23 healthy adults, time-lagged mutual information analysis of microstate sequences derived from amplitude and phase signals of awake eyes-closed EEG records showed that only the phase component contributed to the periodicity of microstate sequences. Phase sequences showed mutual information peaks at multiples of 50 ms and the group average had a main peak at 100 ms (10 Hz), whereas amplitude sequences had a slow and monotonous information decay. This result was confirmed by an independent approach combining temporal principal component analysis (tPCA) and autocorrelation analysis.We reproduced our observations in a generic model of EEG oscillations composed of coupled non-linear oscillators (Stuart-Landau model). Phase-amplitude dynamics similar to experimental EEG occurred when the oscillators underwent a supercritical Hopf bifurcation, a common feature of many computational models of the alpha rhythm.These findings explain our previous description of periodic microstate recurrence and its relation to the time scale of alpha oscillations. Moreover, our results corroborate the predictions of computational models and connect experimentally observed EEG patterns to properties of critical oscillator networks.

The generalized inverses of the products of two elements in a ring

In this paper, new extensions of Jacobson's lemma and Cline's formula for Drazin inverses, generalized Drazin inverses, and ($m,n$)-pseudo-inverses are given. As applications, we provide a formula for the powers of the products of two elements and establish connections between two B-Fredholm elements through the canonical map.