Complex-valued (p,q)-harmonic morphisms defined and studied.
problem Defining and studying (p,q)-harmonic morphisms between Riemannian manifolds.
method Introducing and characterizing complex-valued (p,q)-harmonic morphisms.
result Characterization and new non-trivial examples of complex-valued (p,q)-harmonic morphisms.
New solutions found for complex-valued harmonic morphisms with rational exponents.
problem Finding new solutions to complex-valued harmonic morphisms.
method Generalizing from polynomial solutions to rational exponents.
result Many new solutions to the problem.
We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the exis…
New solutions found using rational exponents in complex-valued geometry.
problem Difficult non-linear problems in differential geometry.
method Generalizing from polynomial solutions to rational exponents.
result Many new solutions found.
We describe the relationship between complex-valued harmonic morphisms from Minkowski 4-space} and the shear-free ray congruences of mathematical physics. Then we show how a horizontally conformal submersion on a domain of Euclidean 3-space gives the boundary values at infinity of a complex-valued harmonic morphism on …
New p-harmonic and harmonic morphisms found on Lie groups.
problem Constructing explicit p-harmonic and harmonic morphisms on Lie groups.
method Using the method of eigenfamilies to construct explicit complex-valued p-harmonic functions and harmonic morphisms.
result Explicit complex-valued p-harmonic functions and harmonic morphisms constructed on non-compact classical Lie groups.
We construct new complex-valued harmonic morphisms from Euclidean spaces from functions which are holomorphic with respect to Hermitian structures. In particular, we give the first global examples of complex-valued harmonic morphisms from Rn for each n>4 which do not arise from a Kähler structure; it is know…
New complex-valued maps found on complex geometries.
problem Finding new maps on complex geometries.
method Solving non-linear PDEs based on manifold geometry.
result Constructed new proper biharmonic and (2,1)-harmonic maps.
In this paper we give a positive answer to the open existence problem for complex-valued harmonic morphisms from the non-compact irreducible Riemannian symmetric spaces SLn(R)/SO(n), SU∗(2n)/Sp(n) and their compact duals SU(n)/SO(n) and SU(2n)/Sp(n). Furthermore we prove the existence of globally defined, com…
New submersion proves complex-valued harmonic map existence.
problem Existence of non-constant harmonic morphisms.
method Constructing harmonic Riemannian submersions from symmetric spaces.
result Existence of non-constant, globally defined complex-valued harmonic morphism.
We prove local existence of complex-valued harmonic morphisms from any Riemannian homogeneous spaces of positive curvature, except the Berger space Sp(2)/SU(2).
We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic morphisms from Euclidean spaces and spheres.
We construct the first known complex valued harmonic morphisms from the non-compact Lie groups SL(n,R), SU*(2n) and Sp(n,R) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO*(2n), SO(p…
We consider four dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension two. These foliations produce local complex-valued harmonic morphisms.
We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harm…
We consider 5-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we give a partial classification of left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.
Study conformal foliations on Lie groups, finding new families and harmonic morphisms.
problem Classifying conformal foliations on Lie groups with minimal leaves.
method Analyzing left-invariant foliations generated by specific subgroups.
result New multi-dimensional families of Lie groups with conformal foliations.
Equivalences between conformal foliations on Euclidean 3-space, Hermitian structures on Euclidean 4-space, shear-free ray congruences on Minkowski 4-space, and holomorphic foliations on complex 4-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued …
In this paper we give a unified framework for the construction of complex valued harmonic morphisms from the real, complex and quaternionic Grassmannians and their non-compact duals. This gives a positive answer to the corresponding open existence problem in the real and quaternionic cases.
Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.
problem Creating explicit solutions for p-harmonic functions and harmonic morphisms. method Using joint eigenfunctions of the Laplace-Beltrami and conformality operators.
result Induces solutions on dual non-compact Riemannian symmetric spaces.
New minimal surfaces found in spheres and hyperbolic spaces.
problem Constructing minimal submanifolds in even-dimensional spheres and hyperbolic spaces.
method Using complex-valued harmonic morphisms.
result Explicit examples of minimal submanifolds in S4 and H4. In this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r≥3. The second method provides us with global solutio…
Minimal conformal foliations on Lie groups are shown to be fibres of harmonic morphisms.
problem Characterizing minimal conformal foliations on Lie groups.
method Analyzing left-invariant semi-Riemannian metrics and harmonic morphisms.
result Minimal conformal foliations of codimension two are fibres of complex-valued harmonic morphisms.
We construct 4-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension 2. We show that these foliations are holomorphic with respect to an (integrable) Hermitian structure which is not K\" ahler. We then prove that the Riemannian Lie groups constructed are {\it …
The study constructs minimal submanifolds in complex and quaternionic projective spaces.
problem Finding minimal submanifolds in complex and quaternionic projective spaces.
method Using complex-valued harmonic morphisms.
result Complete minimal submanifolds of odd-dimensional complex projective spaces and their dual hyperbolic spaces are constructed.
New biharmonic functions created on Lie groups.
problem Constructing explicit biharmonic functions on Lie groups.
method Developed a new scheme for constructing complex-valued biharmonic functions on Riemannian Lie groups.
result Manufactured infinite series of new solutions on SU(n) and showed applicability to SO(n) and Sp(n). New families of Lie groups with special foliations discovered.
problem Characterizing left-invariant foliations on semi-Riemannian Lie groups.
method Classifying foliations generated by specific subgroups.
result Constructing new families of Lie groups with conformal minimal foliations.
Harmonic morphisms and p-harmonic functions constructed on symmetric spaces.
problem Constructing harmonic morphisms and p-harmonic functions on symmetric spaces.
method Using Cartan embedding and related maps to relate tension field and conformality operator.
result Simple formulae relating tension field and conformality operator on symmetric spaces to those on their images.
New method creates minimal submanifolds using complex-valued eigenfunctions.
problem Creating minimal submanifolds in compact Riemannian manifolds.
method Employing complex-valued eigenfunctions.
result Manufactured minimal submanifolds in compact Riemannian manifolds.
Complex-valued neural networks avoid spurious local minima.
problem Finding spurious local minima in neural networks.
method Proved no spurious local minima for shallow complex neural networks with quadratic activations.
result Complex-valued weights eliminate spurious local minima in neural networks.
A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with complex-valued vectors followed by (2) taking the absolute value of every entry…
Complex-valued Gaussian process improves time series analysis of brain oscillations.
problem Improving time series analysis of complex-valued signals, especially brain oscillations.
method Modeling real-valued signals as the real part of a latent complex-valued Gaussian process with new covariance functions.
result Complex-valued Gaussian process provides better estimates of amplitude and frequency than existing methods.
Paper introduces new complex-valued kernel for improved regression.
problem Limited solution for complex-valued regression using real-valued kernels.
method Derives widely RKHS (WRKHS) and pseudo-kernel to address limitations.
result WRKHS yields better performance in complex-valued regression.
Complex-valued neural networks perform similarly to real-valued models for real-valued classification tasks.
problem Comparing real-valued and complex-valued neural networks for real-valued classification tasks.
method Comparison of neural networks with similar capacity sizes, using various activation functions and weight initialisation strategies.
result Complex-valued neural networks perform equal to or slightly worse than real-valued models for real-valued classification tasks.
Proposes random Euler filters for efficient complex-valued nonlinear signal processing.
problem Efficiently processing complex-valued nonlinear signals with reduced computational cost.
method Introduces linear and widely-linear random Euler complex-valued filters with fixed network structures.
result Analytical minimum mean square error and optimum step-size derived for transient and steady-state performances.
The paper uses complex-valued functions to simplify plane differential geometry and kinematics.
problem Simplifying complex problems in plane differential geometry and kinematics.
method Consistent use of complex-valued functions of a real variable.
result Derives results in a particularly simple, uniform, and transparent way.
In the spirit of Ray and Singer we define a complex valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex valued Reidemeister torsion, including its phase. …
Bayesian sparsification improves complex-valued neural networks by 50-100x with minimal performance loss.
problem Efficiently compressing complex-valued neural networks for embedded systems.
method Extending Sparse Variational Dropout to complex-valued networks and conducting a numerical study.
result Achieved state-of-the-art performance on MusicNet with 50-100x compression.
Study complex-valued VAEs for radar OOD detection.
problem Detecting out-of-distribution signals in complex radar environments.
method Proposed and compared several detection metrics for CVAE.
result CVAE-MSE and latent-based scores outperform ANMF-Tyler.
CRBM extracts speech features from complex spectra directly.
problem Speech coding ignores phase information in complex spectra.
method CRBM learns relationships between visible and hidden units from complex-valued spectra.
result CRBM outperforms conventional methods in speech coding.
Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.
problem Finding a unified scheme for complex-valued eigenfunctions on Riemannian symmetric spaces.
method Employing the Cartan embedding for classical compact Riemannian symmetric spaces and quaternionic Grassmannians.
result Construction of new eigenfunctions on quaternionic Grassmannians.
A new method for complex-valued signals improves convergence and performance.
problem Nonlinear channel equalization and complex-valued signals with different properties.
method Generalized complex kernel least-mean-square (gCKLMS) algorithm.
result The gCKLMS algorithm converges faster and performs better than previous methods.
C-SURE improves complex-valued deep learning models by shrinking estimates, outperforming MLE and SurReal.
problem Improving accuracy and robustness of complex-valued deep learning models.
method Proposes a Stein's unbiased risk estimate (SURE) for complex-valued data and integrates it into a prototype CNN classifier.
result C-SURE outperforms SurReal and MLE in accuracy and robustness on complex-valued datasets.
New method constructs complex-valued r-harmonic functions on Riemannian manifolds.
problem Constructing complex-valued r-harmonic functions on Riemannian manifolds.
method Introducing a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds and applying it to specific semisimple Lie groups.
result The method successfully constructs complex-valued r-harmonic functions on various Riemannian manifolds, including specific Lie groups.
Algorithm simulates complex-valued Gaussian processes efficiently.
problem Simulating noncircular or improper complex-valued stationary Gaussian processes.
method Circulant embedding method for multivariate Gaussian processes.
result Exact simulation possible except for negative eigenvalues.
This paper proves a generalization bound for complex-valued neural networks scaling with spectral complexity.
problem Ensuring the performance of complex-valued neural networks on unseen data.
method Theoretical derivation using Maurey Sparsification Lemma and Dudley Entropy Integral, empirical validation on various datasets.
result The spectral complexity of weight matrices is a significant factor in the generalization ability of complex-valued neural networks.
This research explores complex-valued neural networks and their implementation.
problem The challenges of implementing complex-valued neural networks and their potential for non-complex data.
method Detailed theory and implementation of CVNN, including Wirtinger calculus, complex backpropagation, and modules like complex layers and activation functions. Python implementation using cvnn toolbox.
result Demonstrates the potential of CVNN for non-complex data through simulations.
CVNN outperforms RVNN on non-circular data.
problem Classifying complex-valued data with statistical dependence.
method Comparison of CVNN and RVNN on non-circular data.
result CVNN outperforms RVNN in accuracy and generalization.