Entire Maps Research Articles

Spheres over fields, their entire rational maps and applications

The paper summarizes some results on algebraic geometry presence in the homotopy theory. For the homotopy group πm(Sn), denote by πalgm(Sn) its subset of homotopy classes represented by ℝ-entire rational maps Sm→Sn of spheres. The main result of this paper concerns to the study of πn+kalg(Sn) for k=0,...,7.

Dynamic Probabilistic Pruning: A General Framework for Hardware-Constrained Pruning at Different Granularities.

Unstructured neural network pruning algorithms have achieved impressive compression ratios. However, the resulting-typically irregular-sparse matrices hamper efficient hardware implementations, leading to additional memory usage and complex control logic that diminishes the benefits of unstructured pruning. This has spurred structured coarse-grained pruning solutions that prune entire feature maps or even layers, enabling efficient implementation at the expense of reduced flexibility. Here, we propose a flexible new pruning mechanism that facilitates pruning at different granularities (weights, kernels, and feature maps) while retaining efficient memory organization (e.g., pruning exactly k -out-of- n weights for every output neuron or pruning exactly k -out-of- n kernels for every feature map). We refer to this algorithm as dynamic probabilistic pruning (DPP). DPP leverages the Gumbel-softmax relaxation for differentiable k -out-of- n sampling, facilitating end-to-end optimization. We show that DPP achieves competitive compression ratios and classification accuracy when pruning common deep learning models trained on different benchmark datasets for image classification. Relevantly, the dynamic masking of DPP facilitates for joint optimization of pruning and weight quantization in order to even further compress the network, which we show as well. Finally, we propose novel information-theoretic metrics that show the confidence and pruning diversity of pruning masks within a layer.

The relationship between high dominant frequency and low voltage areas in patients with atrial fibrillation: digital twin study

Abstract Funding Acknowledgements Type of funding sources: Public grant(s) – National budget only. Main funding source(s): a grant [2022R1I1A1A0106342211] from the Basic Science Research Program run by the National Research Foundation of Korea (NRF) which is funded by the Ministry of Science, ICT & Future Planning (MSIP) Background It has been suggested that the high dominant frequency (DF) site plays an atrial fibrillation (AF) driver, but its relationship with the atrial substrate voltage map (Vm) in the entire chamber AF map has yet to be studied. We explored the relationship between the DF site during AF and Vm in the entire chamber maps by applying the human left atrial (LA) electroanatomical map (EAM) to the digital twin AF map. Methods We acquired LA substrate Vm (at sinus rhythm, >800 points bipolar electrograms) of 110 patients with persistent AF (male 72.7%, 58.7±9.8 years old) who underwent AF catheter ablation. We applied EAM to digital twins (0.4 million nodes), reflecting anatomy, histology, and electrophysiology. We evaluated the highest 10% DF areas for 6 sec after inducing virtual AF. The correlation between the highest 10% DF and the low Vm areas (<0.2mV) was evaluated in the LA region of 10 segments. Results The mean voltage of the entire LA area was 1.96±0.85 mV (2.19±0.82 mV extra-PV area, 1.33±0.93 mV PV area) in 110 patients. The correlation between low Vm and high DF areas was not significant in the entire LA (r=0.106, p<0.001), extra-PV (r=0.157, p<0.001), and PV (r=0.079, p=0.097) areas. In the LA 10 segment regional analysis, 72 patients (65.5%) showed the co-localization of the highest DF in the region with the lowest Vm, and 14 patients (12.7%) had the lowest Vm segment in the region with the highest DF. The mean voltage where high DF appeared was 0.69±1.35 mV, and the mean voltage of the region where high DF did not appear was 0.59±1.15 mV (p=0.171). Conclusion High DF site, a parameter representing a localized focal source, had statistically significant but very poor correlations with LA extra-PV voltage substrate in patients with persistent AF.

A Class of Potentials for Hyperbolic Transcendental Entire Maps

A Class of Potentials for Hyperbolic Transcendental Entire Maps

Oscillating wandering domains for p$p$‐adic transcendental entire maps

We give examples of transcendental entire maps over C p $\mathbb {C}_p$ having an oscillating wandering Fatou component.

A data science approach for analysis and reconstruction of spinodal-like composition fields in irradiated FeCrAl alloys

A statistical method for the analysis of continuously distributed data representative of composition fluctuations in irradiated FeCrAl alloys acquired using Energy Dispersive X-ray Spectroscopy (EDS) method is presented. Using probability distribution functions, direct and cross-covariances between the elemental compositions, the effects of alloy composition and irradiation dose were investigated on the spatial distribution and length scale of composition fluctuations at the nanoscale. It was observed that, for neutron-irradiated FeCrAl alloys, the distribution of Fe and Cr followed a left-skewed and right-skewed distribution, respectively for all (average) alloy compositions and irradiation doses. The analysis also revealed enhanced spatial gradients in the elemental compositions at higher irradiation dose. Direct and cross-covariance estimates of the experimental data were also utilized for reconstruction of composition data through fitting it to a parametric form of the covariance functions. Linear Model of Coregionalization was used to determine the parameters of the covariance functions. Subsequently, a spectral method was utilized for simulating a realization of the alloy compositions. Close correspondence was observed between the experimental and the reconstructed data which was analyzed using probability distribution functions and covariance functions. Composition space of the experimental and reconstructed data and dislocation velocities as a function of applied stress and line directions over the entire composition maps were also examined.

Pyramidal and conditional convolution attention network for hyperspectral image classification using limited training samples

ABSTRACT Hyperspectral image classification methods based on convolutional neural network (CNN) and attention mechanism have obtained satisfactory results. However, the currently applied convolution operations treat all inputs equally, and the multiscale information extraction of hyperspectral data is not sufficient. The used attention mechanism is also one-dimensional, two-dimensional, or a simple combination of them. Alongside these, the high-precision classification models rely on many training samples. Based on these, we propose a pyramidal and conditional convolution attention network (PCAN), which can fully and effectively capture the information of hyperspectral input data through convolution kernels with different weights and different scales. And for the first time, the attention of three-dimensional weights is applied to hyperspectral data, and satisfactory results are achieved on minimal training sets. Specifically, PCAN is composed of spectral branch and spatial branch. The spectral branch uses conditionally parameterized convolutions (CondConv) to customize the convolution kernels according to the input features. It directly generates three-dimensional weights to quantify the importance of the entire feature maps. The spatial branch uses the pyramidal convolution (PyConv), which includes convolution kernels of different spatial sizes and depths and can capture different levels of detailed information in hyperspectral images. At the same time, competition and cooperation between different channels are captured through gating adaptation and channel normalization. Experiments on three well-known hyperspectral data sets show that our model achieves higher accuracy and less training time than other advanced methods, especially with fewer training samples.

Disconnected Julia set of Halley's method for exponential maps

We investigate the Halley method of exponential maps. Our main result is that, unlike Newton's method, the Julia set of Halley's method may be disconnected when applied to entire maps of form where p and q are polynomials and q is non-constant. We also describe the nature of the fixed points and classify rational Halley's maps of entire functions.

On the Dynamics of Entire Maps with Symmetry

On the Dynamics of Entire Maps with Symmetry

Certain dynamical aspects of a family fλ(z) =λ ez/z+1 for z ∈ C when λ < 0

We study the change of dynamics of transcendental meromorphic functions f? = ? ez/z+1 for z ? C when ? varies on the negative real axis. It is shown that there is a ?^ such that the Fatou set of f? is empty for ? < ?^ whereas the Fatou set is an invariant parabolic basin corresponding to a real rationally indifferent fixed point x^ if ? = ?^ . In fact, the Fatou set is an invariant attracting basin of a real negative fixed point ?? if ?^ < ? < 0. Also the dynamics of fn? for n ? 2 at the fixed points is investigated for different values of ?. As a generalization of f?, we observed some dynamical issues for the class of entire maps F?,a,m(z) = ?(z+a)m exp(z) where ?, a ? C and m ? N.

Criniferous entire maps with absorbing Cantor bouquets

<p style='text-indent:20px;'>It is known that, for many transcendental entire functions in the Eremenko-Lyubich class <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are <i>criniferous</i>. In this paper, we extend this result to a new class of maps in <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>. Furthermore, we show that if a map belongs to this class, then its Julia set contains a <i>Cantor bouquet</i>; in other words, it is a subset of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{C} $\end{document}</tex-math></inline-formula> ambiently homeomorphic to a straight brush.</p>

A converse landing theorem in parameter spaces

In this article, we prove that for several one-dimensional holomorphic families of holomorphic maps, in the parameter plane, there exists a local piece of a curve that lands at a given parabolic parameter, in the spirit of well-known results about the quadratic and the exponential families. We also show that, under some assumptions, this general result partially answers the existence and landing questions of ray structures in the parameter planes for holomorphic families of transcendental entire maps.

Combinatorics of criniferous entire maps with escaping critical values

A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its Julia set. In particular, our description precisely reflects the splitting phenomenon. This combinatorial structure provides the foundation for further study of this class of functions. For example, we use these results in another paper to give the first full description of the topological dynamics of a class of transcendental entire maps with unbounded postsingular set.

The maximum modulus set of a quasiregular map

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are either the whole plane, or a countable union of analytic curves. We show that in the quasiregular case, by way of contrast, any closed set containing at least one point of each modulus can be attained as the maximum modulus set of a quasiregular map. These examples are all of polynomial type. We also show that, subject to an additional constraint, such sets can even be attained by quasiregular maps of transcendental type.

Dynamics of transcendental Hénon maps III: Infinite entropy

<p style='text-indent:20px;'>Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.</p>

SCGA‐Net: Skip Connections Global Attention Network for Image Restoration

AbstractDeep convolutional neural networks (DCNN) have shown their advantages in the image restoration tasks. But most existing DCNN‐based methods still suffer from the residual corruptions and coarse textures. In this paper, we propose a general framework “Skip Connections Global Attention Network” to focus on the semantics delivery from shallow layers to deep layers for low‐level vision tasks including image dehazing, image denoising, and low‐light image enhancement. First of all, by applying dense dilated convolution and multi‐scale feature fusion mechanism, we establish a novel encoder‐decoder network framework to aggregate large‐scale spatial context and enhance feature reuse. Secondly, the solution we proposed for skipping connection uses attention mechanism to constraint information, thereby enhancing the high‐frequency details of feature maps and suppressing the output of corruptions. Finally, we also present a novel attention module dubbed global constraint attention, which could effectively captures the relationship between pixels on the entire feature maps, to obtain the subtle differences among pixels and produce an overall optimal 3D attention maps. Extensive experiments demonstrate that the proposed method achieves significant improvements over the state‐of‐the‐art methods in image dehazing, image denoising, and low‐light image enhancement.

Accurate On-Chip Temperature Sensing for Multicore Processors Using Embedded Thermal Sensors

Thermal issues seriously restrict the quality, reliability, and lifetime of semiconductor chips. To prevent thermal runaway situations, modern processors deploy numerous on-die thermal sensors to collect temperature information, which is then used to guide dynamic thermal management (DTM) mechanisms. Accurate on-chip temperature information is critical for DTM as temperature overestimation will degrade the performance by an unnecessary invocation of thermal control mechanisms, and underestimation will result in the reliability issues of the systems. In this article, two effective techniques are proposed to achieve an accurate on-chip temperature sensing. The first technique is a synergistic calibration method for forecasting the actual temperatures of noisy thermal sensors. Second, we propose a full thermal characterization technique based on convolutional neural networks (CNNs) to accurately recover the entire thermal maps by using a limited number of thermal sensors. In a realistic scenario, these two techniques can be used in combination to provide a more accurate thermal monitoring. By utilizing the sophisticated infrared imaging setup, the effectiveness of the proposed techniques is validated on a real 45-nm AMD quad-core chip. The simulation results show that the methods achieve significant improvements compared with existing techniques in the literature. The successful implementation of the proposed methods will significantly improve the efficiency of DTM.

Stable Components in the Parameter Plane of Transcendental Functions Of Finite Type

We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call shell components, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the virtual center, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. Our dynamical plane results apply without the restriction of finite type.

Two-dimensionally controllable DSR generation from dumbbell-shaped mode-locked all-fiber laser**Project supported by the National Natural Science Foundation of China (Grant Nos. 61435009, 61235008, and 61405254).

An all-fiber dumbbell-shaped dual-amplifier mode-locked Er-doped laser that can function in dissipative soliton resonance (DSR) regime is demonstrated. A nonlinear optical loop mirror (NOLM) and a nonlinear amplifying loop mirror (NALM) are employed to initiate the mode-locking pulses. Unlike conventional single-amplifier structure, the output peak power of which remains unchanged when pump power is varied, the proposed structure allows its output peak power to be tuned by changing the pump power of the two amplifiers while the pulse duration is directly determined by the amplifier of nonlinear amplifying loop mirror. The entire distribution maps of peak power and pulse duration clearly demonstrate that the two amplifiers are related to each other, and they supply directly a guideline for designing tunable peak power DSR fiber laser. Pulse width can change from 800 ps to 2.6 ns and peak power varies from 13 W to 27 W. To the best of our knowledge, the peak power tunable DSR pulse is observed for the first time in dumbbell-shaped Er-doped all-fiber mode-locked lasers.

On the distribution of Julia sets of holomorphic maps

In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of C* can neither contain a free Jordan arc nor be contained in any finite set of straight lines.