arXiv research

The heights of Alexandroff square transformation groups are computed and proven.

Novel approach integrates Multivariate Square-root Lasso into Synthetic Control for high-dimensional data.

It is proved that for every fractal continuous mapping F: I\to I^2 of the unit interval onto the unit square there is a pair of points x,y\in I, such that |F(x)-F(y)|^2\ge 5|x-y|.

Study on minimal hypersurfaces in a unit sphere, proving specific isometries.

We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…

A new approach to kernel adaptive filters reduces sparsity for monotonic signals.

This paper controls the capacity of weight-normalized deep neural networks using rectified linear units.

Estimates the first eigenvalue of a Schrödinger operator on minimal submanifolds.

An isometric immersion $x:M^n\rightarrow S^{n+p}$ is called Willmore if it is an extremal submanifold of the Willmore functional: $W(x)=\int_{M^n} (S-nH^2)^{\frac{n}{2}}dv$, where $S$ is the norm square of the second fundamental form and $H$ is the mean curvature. Examples of Willmore submanifolds in the unit sphere ar…

Study non-asymptotic estimation bounds for LTI models with Gaussian noise.

Introduces BMF for efficient matrix factorization of large data.

Totally geodesic submanifolds in spheres have restricted curvature properties.

The paper improves bounds on how many squares can fit in a rectangle and still have stable homology.

In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f…

Optimizes embedding accuracy for data variance and error.

Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.

ADMM algorithm solves nonlinear matrix decompositions efficiently.

In this paper we propose and investigate a novel nonlinear unit, called $L_p$ unit, for deep neural networks. The proposed $L_p$ unit receives signals from several projections of a subset of units in the layer below and computes a normalized $L_p$ norm. We notice two interesting interpretations of the $L_p$ unit. First…

Supermixed labyrinth fractals extend mixed fractals by using multiple patterns.

Characterizes algebraic squares of irreducible complex spinors in various dimensions.

If F is a family of mod 2 flat k-cycles in the unit n-ball, we lower bound the maximal volume of any cycle in F in terms of the homology class of F in the space of all cycles. We give examples to show that these lower bounds are fairly sharp.

Let $L^2$ be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of $L^2$. We also investigate this connection in the context of restricted Grassmann manifolds associated to $p…

Paper proves uniqueness of minimal hypersurfaces in specific domains.

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $Σ_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $Σ_g$. For …

New bounds on homological eigenvalues relate to Weil-Petersson length.

Optimizes experimental design using synthetic controls for better outcomes.

Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrighta…

Paper studies rigid properties of closed CSL submanifolds in unit spheres.

Paper proves Simon's third gap conjecture for minimal surfaces in spheres.

We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of unit ball in $\mathbb R^n$ that have square integrable mean curvature. As one consequence we obtain a Li-Yau type inequality in this setting, thereby generalizing results of Oliveira and Soret, and Fraser and Schoen. In th…

Paper proves a conjecture about minimal hypersurfaces in spheres.

Paper introduces quadrilateral labyrinth fractals and their properties.

FF algorithm uses goodness as a measure of input quality, derived from likelihood-ratio tests.

Study curvature inequalities for real hypersurfaces in complex space forms.

We establish $L^{\infty}$ and $L^2$ error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with $\ell^1$ and $\ell^0$ controls on their inner and outer parameters. With the squared ReLU ridge function, we show th…

Study calculates liquidity costs for delta hedging of European options.

We study the relations between the quaternion $H$-type group and the boundary of the unit ball on two dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion $H$-type group into its subspace of boundary values of $q$-holomorphic functions is consider. …

Study measures volume of foliations on surfaces, finding integrability range.

Enhances spoken speech quality using EEG signals.

Infinite width ReLU networks can approximate functions with bounded Euclidean norm.

Paper classifies hypersurfaces in a sphere with specific curvature properties.

Deep neural networks often avoid suboptimal points due to close proximity of local minima to global optima.

Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventu…

The disbalance of Supply and Demand is typically considered as the driving force of the markets. However, the measurement or estimation of Supply and Demand at price different from the execution price is not possible even after the transaction. An approach in which Supply and Demand are always matched, but the rate $I=…

Study shows GRU model with dropout outperforms in Bitcoin price prediction.

Using an analog of the boundary element method in engineering and science, we analyze and model unemployment rate in Austria, Italy, the Netherlands, Sweden, Switzerland, and the United States as a function of inflation and the change in labor force. Originally, the model linking unemployment to inflation and labor for…

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

Proves properties of minimal hypersurfaces with free boundary in a ball.