arXiv research

We prove the $3$-manifold $\RP^3 \# \RP^3$ is of $\Z_{2}$-coefficient homology $(1, 2)$-systolic freedom. Given a Riemannian metric on $\RP^{3}\# \RP^{3}$, we define $\Z_{2}$-coefficient homology $1$-systole as the infimum of lengths of all nonseparating geodesic loops representing nontrivial classes in $H_{1}(\RP^3\#\…

Khovanov homology invariant proved for links in $\mathbb{RP}^3$.

For a given real generic curve $\ga: S^1\to \Bbb {RP}^n$ let $D_\ga$ denote the ruled hypersurface in $\Bbb {RP}^n$ consisting of all osculating subspaces to $\ga$ of codimension 2. A curve $\ga: S^1\to \Bbb {RP}^n$ is called convex if the total number of its intersection points (counted with multiplicities) with any h…

We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake RP^{6}s of Hirsch and Milnor and on the analogous fake RP^{14}s.

New proof of Yamabe invariant for RP^3 using harmonic functions.

We show that the manifold *CP^2 # *RP^4, which is homotopy equivalent but not homeomorphic to CP^2 # RP^4, is in fact smoothable.

The study connects cubic differentials to convex RP^2-structures and their ends.

A new method, Residual-Permuted Sums, improves confidence region construction for linear regression models.

New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.

Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal $\mathbb{Z}_2^2$-equivariant triangulations of $2$-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{R…

Study instanton Floer homology for links in RP^3 and use it to detect knots.

Improves cancer classification accuracy using Random Projection combined with other methods.

We study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \to P_m(RP^2 {x_1,...,x_n} \to P_{n+m}(RP^2) \stackrel{p_{\ast}}{\to} P_n(RP^2) \to 1$, where $n\geq 2$ and $m\geq 1$, and $p_{\ast}$ is the homomorphis…

In this paper we give examples of closed smooth submanifolds of RP^n which are isotopic to nonsingular projective subvarieties of RP^n but they can not be isotopic to the real parts of nonsingular complex projective subvarieties of CP^n.

P. Arnoux and A. Marin showed that any triangulation of $\mathbb{RP}^n$ contains more than $\frac{(n+1)(n+2)}{2}$ vertices if $n \geq 3$. We construct some natural triangulation of $\mathbb{RP}^n$ with $\frac{n(n+5)}{2}-1$ vertices for all $n \geq 3$. Previously, it was known that $\mathbb{RP}^n$ has $\mathbb{Z}_2^n$-e…

Defines a Rasmussen invariant for links in RP^3.

We define deformations of $RP^2$-manifolds.

New algorithm LSTD($λ$)-RP uses random projections and eligibility traces for efficient reinforcement learning.

Let S be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural correspondence between convex RP^2 structures on S and pairs (Σ,U) consisting of a conformal structure Σon S and a holomorphic cubic differential U over Σ. …

Investigates MAD-RP portfolios for asset allocation.

Estimates eigenvalues for surfaces in RP^3, proving a strict inequality.

Paper introduces Bar-Natan homology for special links in a modified space.

New method uses data characteristics for better random projections.

New homologies defined for null homologous links in RP^3, linking to Heegaard Floer homology.

The paper extends a connected-sum inequality to calculate λ-Yamabe invariants of certain manifolds.

Links with isotopic preimages in $S^3$ are isotopic in $RP^3$.

RP-WNO extends WNO with uncertainty quantification, useful for scientists and engineers.

Classifies stable hypersurfaces in real projective spaces and confirms the isoperimetric conjecture.

Random projection improves deep learning performance on high-dimensional data.

There is an elegant relation found by Fabricius-Bjerre [Math. Scand 40 (1977) 20--24] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in RP^2. We note that the quantities in the formula are naturally dual to each …

We determine the lower central and derived series of the n-string braid groups B_n(RP^2) of the real projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. For n=1,2, B_n(RP^2) is finite and its lower central and derived series …

Novel method learns time series dynamics without reconstruction.

RPS uses graph-based representation learning for better portfolio optimization.

Enhances graph neural networks with Relational Pooling for better graph classification.

RP method improves noisy label classification with CNN, achieving 0.46 error across all MNIST digits.

Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_…

New findings on stable commutator lengths in recursively presented groups.

The purpose of this short note is to prove the uniqueness of Hamiltonian volume minimizing Lagrangian submanifolds which are Hamiltonian isotopic to RP^n in CP^n modulo isometric group actions.

Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.

Using basic properties of one-sided Heegaard splittings, a direct proof that geometrically compressible one-sided splittings of RP^3 are stabilised is given. The argument is modelled on that used by Waldhausen to show that two-sided splittings of S^3 are standard.

Non-orientable 4-manifolds are simple branched coverings of RP^4 and twisted S^3-bundles.

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and to $\Z_{4}$ if $n\geq 4$. Further, for all $n\geq 3$, up to isomorphism, the foll…

We prove the projective plane $\rp^2$ is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by A.N.Dranishnikov) in extension theory. One of the main tools is the computation …

Unified view of LR and RP gradients explained via divergence theorem.

Information theoretical measures, such as entropy, mutual information, and various divergences, exhibit robust characteristics in image registration applications. However, the estimation of these quantities is computationally intensive in high dimensions. On the other hand, consistent estimation from pairwise distances…

The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …

We show that the loop spaces of real projective spaces are topologically approximated by the spaces of rational maps from RP(1) to RP(n). As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces.

The study constructs metrics with positive 2nd Ricci curvature on various manifolds.