In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations whi…
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48 results for “mixed type equations”
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
We prove the existence of C^{\infty} local solutions to a class of mixed type Monge-Ampere equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point. Existence of C^{\infty} global solutions to a corresponding class of linear mixed ty…
Study improves Poisson equation solutions on various manifolds.
problem Improving solutions to Poisson equation on different types of manifolds.
method Established estimates for mixed boundary conditions on manifolds with specific curvature properties.
result Generalized existing theorems to broader Riemannian settings.
The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a h…
Study solves complex Hessian equation on Hermitian manifolds.
problem Solving Hessian equations on Hermitian manifolds with mixed structure.
method Derive a priori estimates and solve Dirichlet problem under conditions.
result Solvability of the Dirichlet problem for mixed Hessian equations.
Remarks on the Hölder-continuity of solutions to parabolic equations with conic singularitiesmath.AP
Study proves Hölder continuity for solutions to parabolic equations with conic singularities.
problem Proving Hölder continuity of solutions to parabolic equations with conic singularities.
method Using the results from previous works \cite{LSU} and \cite{FS}, the Hölder continuity is proven for linear parabolic equations of mixed type.
result Hölder continuity of solutions to parabolic equations with conic singularities is established.
The paper studies Einstein-Hilbert action on complex manifolds.
problem Deriving equations for the Einstein-Hilbert action on almost -product manifolds.
method Adapted variations of metric, deriving Euler-Lagrange equations.
result Presented a nice form of Einstein equation.
The paper derives an integral formula for mixed scalar curvature of singular distributions.
problem Differential geometry of singular distributions on Riemannian manifolds.
method Proves divergence theorem and Codazzi equation for singular distributions.
result Derives an integral formula for mixed scalar curvature of singular distributions.
Researchers create solutions for naked singularities in Einstein vacuum equations.
problem Constructing solutions for the interior region of naked singularities in Einstein vacuum equations.
method Novel self-similarity and study of mixed degenerate elliptic-hyperbolic PDE's.
result Gluing together interior and exterior solutions produces a naked singularity.
In this paper, we present smooth examples of degenerate hyperbolic and mixed type Monge-Ampere equations in the plane, which do not admit a local C^3 solution.
Paper proves inequalities on Hermitian manifolds with applications to bounded solutions.
problem Establishing mixed Hessian inequalities on Hermitian manifolds.
method Weak convergence theorem of complex Hessian operators and general mixed Hessian inequality.
result Existence of bounded solutions of complex Hessian equations.
In the thesis at hand we give a comprehensive discussion of basic problems for generalized Maxwell equations with mixed boundary conditions using the calculus of alternating differential forms on Riemannian manifolds of arbitrary dimension. We prove compactness results, Hodge decompositions and Poincare type estimates.…
Researchers prove constant solutions for a specific Finslerian equation.
problem Investigating exponentially harmonic functions on Finslerian spaces.
method Analyzing the exponential energy functional and using nonnegative Ricci curvature conditions.
result Any bounded solution to the Finslerian equation is constant.
New methods solve complex PDEs with mixed boundary conditions.
problem Solving inhomogeneous Robin type boundary value problems for linear PDEs.
method Odd and even Hilbert transforms.
result Non-standard solutions to various PDEs in finance, stochastic analysis, etc.
Some recent developments on isometric immersions via compensated compactness and gauge transformsmath.AP
Survey on recent developments in isometric immersions using PDE techniques.
problem Analyzing isometric immersions with low Sobolev regularity.
method Compensated compactness and Coulomb-Uhlenbeck gauges.
result Weak continuity and stability of Gauss-Codazzi-Ricci equations.
Study of mixed equation combining gauge theory and symplectic geometry.
problem Regularity and compactness of solutions to the mixed equation.
method Combining Uhlenbeck and Gormov compactness theorems.
result Moduli spaces of solutions to the mixed equation satisfy compactness properties.
We construct and study a family of double-periodic almost entire solutions of the maximal surface equation. The solutions are parameterized by a submanifold of -matrices (the so-called generating matrices). We show that the constructed solutions are either space-like or of mixed type with the light-cone type…
Study of Einstein-Hilbert action on metric-affine spaces with connections.
problem Formulating and solving variational problems for mixed Einstein-Hilbert action.
method Developed variational formulas for extrinsic geometry, derived Euler-Lagrange equations, and characterized critical points.
result Derived new equations analogous to Einstein and Cartan equations, with a new Ricci type tensor.
Paper develops methods for estimating gradients of Finslerian Schrödinger equations.
problem Estimating gradients of Finslerian Schrödinger equations.
method Develops new Laplacian comparison theorem and applies it to Finslerian Schrödinger equation.
result Global and local Li-Yau type gradient estimates for positive solutions.
In this paper, we study a class of fully nonlinear metric flow on Kähler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the result of Song-Weinkove. As a consequence, under the given condition, we solved the…
Starshaped compact hypersurfaces in warped product manifolds II: a class of Hessian type equationsmath.DG
The study finds starshaped compact hypersurfaces in warped products with curvature estimates.
problem Finding starshaped compact hypersurfaces in warped product manifolds.
method Deriving global curvature estimates and interior second order a priori estimates for solutions to associated equations.
result Existence of starshaped compact hypersurfaces in warped product manifolds.
Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.
problem Derives gradient estimate for a nonlinear parabolic equation on Finsler manifolds.
method Leverages a new Laplacian comparison theorem to derive a Li-Yau type gradient estimate.
result Establishes a Li-Yau type gradient estimate for the Finslerian logarithmic Schrödinger equation.
We introduce and study the flow of metrics on a foliated Riemannian manifold , whose velocity along the orthogonal distribution is proportional to the mixed scalar curvature, $\Sc_{\,\rm mix}$. The flow is used to examine the question: When a foliation admits a metric with a given property of $\Sc_{\,\rm mix}$ (…
Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functionsstat.ML
We extend neural networks with fractional and mixed activation functions for better function approximation.
problem Limitations in approximating higher-order smooth functions in complex spaces.
method Incorporating fractional exponents in activation functions and defining new density functions.
result Improved accuracy and broader applicability of neural network approximation theory.
Local solubility of Bao--Ratiu equations proven for surfaces with specific curvature conditions.
problem Existence of asymptotic directions for volume-preserving diffeomorphisms on surfaces.
method Analysis of degenerate Monge--Ampère equation following Han's work.
result Asymptotic directions always exist locally about a point on surfaces with specific curvature conditions.
The paper studies deformations of mixed type surfaces in Lorentz-Minkowski space.
problem Deformations of mixed type surfaces with singular points.
method Introduced L-Gauss map around non-degenerate lightlike points to prove fundamental theorem of surface theory.
result Real analytic mixed type surfaces admit non-trivial isometric deformations around generic lightlike points.
Proves regularity of geodesic equation on Hermitian manifolds.
problem Regularity of geodesic equation in mixed volume forms space.
method Ellipticity conditions, uniform Laplacian estimates, explicit subsolutions.
result Existence of unique solution to Donaldson equation.
New examples of mixed-type zero-curvature graphs found.
problem Finding new examples of zero-curvature graphs in Lorentz-Minkowski space.
method Using Konderak's representation formula to construct entire zero-curvature graphs over specific planes.
result Existence of new types of entire zero-curvature graphs in mixed-type in Lorentz-Minkowski space.
The Index of Some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fieldsmath.FA
Computes indices of mixed order Dirac-type operators and related tensor fields.
problem Computing indices of mixed order Dirac-type operators and tensor fields.
method Using Hilbert complexes and differential operators of mixed order, computing indices with cohomology groups of tensor fields.
result Computation of indices for elasticity and biharmonic complexes.
Solves a mix of many random linear equations using tensor decomposition and alternating minimization.
problem Estimating multiple linear models from mixed samples with unknown labels.
method Combination of tensor decomposition and alternating minimization.
result Guaranteed exact solution with linear sample complexity in dimension and polynomial in .
Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifoldsmath.DG
Study on bounded Gaussian curvature in mixed type surfaces of Lorentzian manifolds.
problem Characterizing the behavior of Gaussian curvature at non-degenerate lightlike points of mixed type surfaces.
method Introducing invariants and using Gauss-Bonnet type formula results.
result Gauss-Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.
Formulas for heat equation solutions on manifolds with boundary.
problem Solving heat equations on manifolds with boundaries.
method Path integral formulas for approximating solutions.
result Approximations of solutions by integrals over path spaces.
Bayesian test assesses dependence between mixed data types.
problem Assessing dependence between text, image, and sound data.
method Bayesian kernelised correlation test using Dirichlet process model.
result Demonstrated effectiveness compared to other methods.
New method solves complex curvature equations.
problem Solving semilinear scalar curvature equations.
method Mixed convex integration method.
result New proof of scalar curvature result.
Study exhaustions for complex orbits in almost homogeneous manifolds.
problem Complex orbits in almost homogeneous manifolds.
method Complex homogeneous Monge-Ampère equations.
result Rigidity results on complex spaces.
A novel graph spectral method for mixed categorical and numerical data.
problem Feature learning for mixed data types (numerical and categorical).
method Graph spectral decomposition of the graph Laplacian to model probabilistic dependence structure.
result Increased separability and clusterability of observations in the transformed feature space.
The paper studies variations of metrics on foliated manifolds and finds solutions to specific actions.
problem Variations of metrics on foliated pseudo-Riemannian manifolds.
method Developed variation formulas and applied to Einstein-Hilbert type actions.
result Found solutions like twisted products, conformal submersions, and isoparametric foliations.
New model clusters mixed-type data with missing values, improving air quality analysis.
problem Clustering mixed-type data with missing values and regime persistence.
method Statistical jump model incorporating regime persistence and handling missing data.
result Superior performance in inferring persistent air quality regimes compared to traditional methods.
We study the geometry of type II supergravity compactifications in terms of an oriented vector bundle , endowed with a bundle metric of split signature and further datum. The geometric structure is associated with a so-called generalised -structure and characterised by an -spinor , which we can regard as a …
Mixed-platonic 3-manifoldsmath.GT
New mixed-platonic 3-manifolds from different polyhedra types.
problem Finding new hyperbolic knot complements with hidden symmetries.
method Defined mixed-platonic 3-manifolds and studied their properties.
result No mixed-platonic hyperbolic knot complement has hidden symmetries.
Study geodesic equation on mixed-volume forms on balanced manifolds, proving existence of solutions.
problem Existence of solutions to the Calabi-Yau equation for balanced metrics.
method Introduced a metric space of mixed-volume forms and derived a geodesic equation.
result Existence of solutions to the Calabi-Yau equation on all balanced manifolds.
It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space is called of mixed type if it changes causal type from space-like to time-like. In , Osamu Kobayashi found …
Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.
problem Christoffel-Minkowski problem and Hessian equations under rotational symmetries.
method Constructing explicit convex solutions to mixed Monge-Ampère equations on \(\mathbb{R}^n\) under radial symmetry.
result Explicit representation formula for the support function of the resulting convex body.
The paper improves ODE solvers by integrating diverse information types.
problem Improving accuracy and physical meaningfulness of ODE solutions.
method Leveraging probabilistic solvers to include second-order information and physical conservation laws.
result Solutions become more accurate and physically meaningful with additional information.
Large deviation principles for multivariate stochastic volatility models.
problem Understanding the behavior of log-processes in multivariate stochastic volatility models.
method Establishing a comprehensive sample path large deviation principle for log-processes.
result Asymptotic formulas for first exit times and barrier option prices derived from the LDP.
Proposes CDTD, a diffusion model for mixed-type tabular data.
problem Adapting diffusion models to mixed-type tabular data.
method Score matching and score interpolation for continuous features, adaptive noise schedules for categorical features.
result Consistently outperforms state-of-the-art models in mixed-type tabular data.
We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\…