arXiv research

The h-principle helps solve complex geometric problems.

Solves open problem on simple surfaces with novel twistor correspondence.

In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves i…

Smooth structures on diffeological spaces and sheaves, resolving conjectures.

Proves divisibility relations for symplectic curve polynomials.

In the work we discuss two invariants of conjugacy classes of braids. The first invariant is the conformal module which occurred in connection with the interest in the 13th Hilbert Problem. The second is a popular dynamical invariant, the entropy. It occurred in connection with Thurston's theory of surface homeomorphis…

Establishes jet transversality for regular maps from flexible manifolds.

The abstract discusses families of holomorphic maps to Oka manifolds with approximation theorems.

Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ o…

Every nonflat conformal minimal surface is homotopic to a proper one.

By attaching a Lie algebra of germs of analytic vector fields to every point of a (real or complex) analytic variety V we construct the Nagano foliation of the variety. We prove that the Nagano foliation of V is a stratification. The treatment of the subject is totally coordinate free but relies on the Oka-Cartan-Serre…

Milnor fibrations were extended by Mutsuo Oka for certain mixed polynomial. In this paper, we study singular points of differentiable maps into the 2-dimensional torus, called Milnor fibration product maps, obtained by several Milnor fibrations for mixed polynomial. We give a characterization of singular points of such…

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immer…

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Rieman…

The Mittag-Leffler theorem is extended to meromorphic curves and minimal surfaces.

Unified classification of equivariant principal bundles using higher homotopy theory.

The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.

Islamic banks being commercial entities strive to earn profit within shariah ambit. Therefore, they seem to be basing themselves upon two knowledge streams namely i) Islamic jurisprudence principles, and ii) banking principles. Islamic jurisprudence principles primarily aim at bringing shariah compliance while banking …

The study establishes uncertainty principles on harmonic manifolds of rank one.

Shows flexible sheaves as fibrant objects for Gromov's h-principle.

A new method to break down insurance costs into risk and uncertainty.

Study on maximum principles for nonlinear equations on Riemannian manifolds.

Derives Fredholm criteria for isotypical components from a Simonenko principle.

Establishes a boundary maximum principle for varifolds with fixed contact angle.

Study proves Maximum Principles for unbounded Riemannian domains.

A pricing principle is introduced for non-attainable claims in incomplete markets.

We show that a well known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group.

Proves a principle for one-phase Bernoulli problem minimizers.

New method proves $h$-principles for stable forms on manifolds.

We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle

In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is o…

Note establishes a local maximum principle for Ricci flow under curvature conditions.

Along with fruitful applications of Deep Neural Networks (DNNs) to realistic problems, recently, some empirical studies of DNNs reported a universal phenomenon of Frequency Principle (F-Principle): a DNN tends to learn a target function from low to high frequencies during the training. The F-Principle has been very use…

Derives time-averaged active inference from control principles.

New principle for harmonic maps helps study higher-dimensional submanifolds.

In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contra…

In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to …

We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation.We demonstrate the LME principle BY deriving …

3-manifolds study Hasse norm principle, akin to number fields.

New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.

Maps to manifolds transverse to certain distributions satisfy an $h$-principle.

Paper proves h-principles for symplectic structures and foliations.

Previous studies have shown that deep neural networks (DNNs) with common settings often capture target functions from low to high frequency, which is called Frequency Principle (F-Principle). It has also been shown that F-Principle can provide an understanding to the often observed good generalization ability of DNNs. …

Study shows strong min-max principle for phase transitions.

We prove that for an algebraic curvature tensor on a pseudo-Euclidean space, the Jordan-Osserman condition implies the Rakić duality principle, and that the Osserman condition and the duality principle are equivalent in the diagonalisable case.

Why deep neural networks (DNNs) capable of overfitting often generalize well in practice is a mystery [#zhang2016understanding]. To find a potential mechanism, we focus on the study of implicit biases underlying the training process of DNNs. In this work, for both real and synthetic datasets, we empirically find that a…