arXiv research

New examples show non-integer Hausdorff dimensions in collapsing spaces.

Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.

Infinite knots have non-integer trace values.

New framework constructs holographic tensor networks using hyperbolic buildings.

Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.

Conjecturally, the only knots in $S^3$ with non-integer surgeries producing Seifert fibered spaces are torus knots and cables of torus knots. In this paper, we make progress on the associated realization problem. Let $Y$ be a small Seifert fibered space arising by $p/q$-surgery on a knot in $S^3$, where $p/q$ is positi…

The study confirms conjectures about slopes of knots using knot Floer homology.

We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution …

The paper studies non-integer curvature flows and proves convergence to spheres under specific conditions.

A slope $p/q$ is a characterizing slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established s…

The Wodzicki residue and the cut-off integral extend to classical symbol-valued forms. We show that they obey a Stokes' type property and that the extended Wodzicki residue can be interpreted as a complex residue like the ordinary one. In the case of cut-off integrals, Stokes' property (i.e. vanishing on exact forms) o…

We study the fractional gravity for spacetimes with non-integer dimensions. Our constructions are based on a geometric formalism with the fractional Caputo derivative and integral calculus adapted to nonolonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical …

We show that if the branched double cover of an alternating link arises as $p/q \in \mathbb{Q} \setminus \mathbb{Z}$ surgery on a knot in $S^3$, then this is exhibited by a rational tangle replacement in an alternating diagram.

The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in t…

Effects of randomness on non-integer power law tails in multiplicatively interacting stochastic processes are investigated theoretically. Generally, randomness causes decrease of the exponent of tails and the growth rate of processes. Explicit calculations are performed for two examples: uniformly distributed and two p…

Advances in fractional analysis suggest a new way for the physics understanding of Riemann's conjecture. It asserts that, if s is a complex number, the non trivial zeros of zeta function in the gap [0,1], is characterized by . This conjecture can be understood as a consequence of 1/2-order fractional differential chara…

Existing approaches to combine both additive and multiplicative neural units either use a fixed assignment of operations or require discrete optimization to determine what function a neuron should perform. However, this leads to an extensive increase in the computational complexity of the training procedure. We present…

We present power low rank ensembles (PLRE), a flexible framework for n-gram language modeling where ensembles of low rank matrices and tensors are used to obtain smoothed probability estimates of words in context. Our method can be understood as a generalization of n-gram modeling to non-integer n, and includes standar…

Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.

A slope $p/q$ is a characterising slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that when $K$ is a hyperbolic knot its set of characterising slopes contains all but finitely many slopes $p/q$ with $q \geq 3$. We prove stronger results for hyper…

Existing approaches to combine both additive and multiplicative neural units either use a fixed assignment of operations or require discrete optimization to determine what function a neuron should perform. This leads either to an inefficient distribution of computational resources or an extensive increase in the comput…

The paper calculates Bachelier option prices using Taylor expansions and applies it as a variance reduction technique.

Solves constant mean curvature Dirichlet problem on catenoids with improved estimates.

This paper is essentially a short version of hep-th/9404046. We compute multiplicative anomaly det(AB)/(detA detB) =F(A,B) for elliptic pseudo-differential operators (PDOs) A, B on a closed manifold M in terms of their symbols. We prove that F(A,B)=1 for elliptic differential operators close to positive-definite ones o…

QPCA improves PCA for cyclostationary data.

Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(…

This paper proves a theorem about Dehn surgery using a new theorem about PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictl…

The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete descrip…

We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with non-empty boundary which has a quadrilateral in each tetrahedron. While this condition is far from being necessary, it is powerful enough to give two new results: the e…

Proposes continuous convolution layers for flexible feature map resizing.

We study three fundamental statistical-learning problems: distribution estimation, property estimation, and property testing. We establish the profile maximum likelihood (PML) estimator as the first unified sample-optimal approach to a wide range of learning tasks. In particular, for every alphabet size $k$ and desired…

This paper shows universality in spectrum behavior for random inner-product kernel matrices in polynomial regime.

This paper optimizes matrix-based Renyi's entropy computation for large datasets.

This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Heisenberg PsiDOs introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Heisenberg PsiDOs…

This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in $S^3$. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsv{á}th and S…

New initialization schemes preserve fractional moments of weights in deep networks, improving training and test performance.

Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.

Paper relates asymptotic dimension to cofinal dimension using coarse proximities.

We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the …

Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation dimension. There are two major difficulties in estimation based on this method. …

We establish cohomological and extension dimension versions of the Hurewicz dimension-raising theorem

Study on CR structures in 7D, proving maximal symmetry dimension.

We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^…

In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo \cite{Buyalo1}) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: micr…

This paper studies three aspects around dimension datum: (1), a generalization of the dimension datum, which we call the tau-dimension datum; (2), dimension data of disconnected subgroups; (3), compactness of isospectral sets of normal homogeneous spaces.

Thurston's spine dimension exceeds virtual cohomological dimension.

Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.

Short note shows unbounded dimensions in Fano K-moduli spaces.