arXiv research

We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the…

String similarity models are vital for record linkage, entity resolution, and search. In this work, we present STANCE --a learned model for computing the similarity of two strings. Our approach encodes the characters of each string, aligns the encodings using Sinkhorn Iteration (alignment is posed as an instance of opt…

In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to …

A virtual $n$-string is a chord diagram with $n$ core circles and a collection of arrows between core circles. We consider virtual $n$-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string $α$, Turaev associated a based matrix that encodes invariants…

We extend knot Floer homology to string links in D^{2} \times I and to d-based links in arbitrary three manifolds, without any hypothesis on the null-homology of the components. As for knot Floer homology we obtain a description of the Euler characteristic of the resulting homology groups (in D^{2} \times I) in terms o…

Chemical autoencoders are attractive models as they combine chemical space navigation with possibilities for de-novo molecule generation in areas of interest. This enables them to produce focused chemical libraries around a single lead compound for employment early in a drug discovery project. Here it is shown that the…

We explore the use of random forest and gradient boosting, two powerful tree-based machine learning algorithms, for the detection of cosmic strings in maps of the cosmic microwave background (CMB), through their unique Gott-Kaiser-Stebbins effect on the temperature anisotropies.The information in the maps is compressed…

Recently, randomly mapping vectorial data to strings of discrete symbols (i.e., sketches) for fast and space-efficient similarity searches has become popular. Such random mapping is called similarity-preserving hashing and approximates a similarity metric by using the Hamming distance. Although many efficient similarit…

In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…

Analysis of large-scale sequential data has been one of the most crucial tasks in areas such as bioinformatics, text, and audio mining. Existing string kernels, however, either (i) rely on local features of short substructures in the string, which hardly capture long discriminative patterns, (ii) sum over too many subs…

The discovery of novel materials and functional molecules can help to solve some of society's most urgent challenges, ranging from efficient energy harvesting and storage to uncovering novel pharmaceutical drug candidates. Traditionally matter engineering -- generally denoted as inverse design -- was based massively on…

Unified description of string and brane worldvolumes using auto-parallel vector fields.

Statistical models usually require vector representations of categorical variables, using for instance one-hot encoding. This strategy breaks down when the number of categories grows, as it creates high-dimensional feature vectors. Additionally, for string entries, one-hot encoding does not capture information in their…

Machine learning improves invoice matching speed and accuracy.

New research shows larger language models improve data processing for diverse entries.

Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology classes in spaces of knots and links in higher-dimensional Euclidean spaces. In previo…

Synthetic reference strings are as effective as real ones for training citation parsing models.

We propose a formulation of the term structure of interest rates in which the forward curve is seen as the deformation of a string. We derive the general condition that the partial differential equations governing the motion of such string must obey in order to account for the condition of absence of arbitrage opportun…

We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant ver…

String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signat…

We employ Hirzebruch-type invariants obtained from iterated p-covers to investigate concordance of links and string links. We show that the invariants naturally give various group homomorphisms of the string link concordance group into L-groups over number fields. We also obtain homomorphisms of successive quotients of…

Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial $n$ may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial m…

New theory connects string theory to swampland distance conjecture.

Derives path integrals for perturbative strings on various backgrounds.

Identifies all perturbative vacua in bosonic string theory.

String geometry theory connects strings to space-time and finds string vacua.

The crucial importance of metrics in machine learning algorithms has led to an increasing interest in optimizing distance and similarity functions, an area of research known as metric learning. When data consist of feature vectors, a large body of work has focused on learning a Mahalanobis distance. Less work has been …

Derives path-integrals for superstrings on curved backgrounds using string geometry theory.

Chiral string integrands simplify to ambitwistor string integrands in the tensionless limit.

Perturbative string amplitudes are correctly derived from the string geometry theory, which is one of the candidates of a non-perturbative formulation of string theory. In order to derive non-perturbative effects rather easily, we formulate topological string geometry theory. We derive the perturbative partition functi…

Black holes offer insights into machine learning's loss landscapes.

Machine learning refactors knowledge to improve learning efficiency.

A virtual string is a scheme of self-intersections of a closed curve on a surface. We study algebraic invariants of strings as well as two equivalence relations on the set of strings: homotopy and cobordism. We show that the homotopy invariants of strings form an infinite dimensional Lie group. We also discuss connecti…

A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string …

The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial evaluations cannot produce exponential simplifications of link diagrams. In this pap…

There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field theory combined.

New formulas link string bordism to integers.

Extended Alexander groups are used to define an invariant for open virtual strings. Examples of non-commuting open strings and a ribbon-concordance obstruction are given. An example is given of a slice virtual open string that is not ribbon. Definitions are extended to open n-strings.

BOSS optimizes string inputs using string kernels and genetic algorithms.

String topology coproduct and Turaev cobracket computed for surfaces.

We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is particularly suitable to deal with string connections: it enables us to prove that every string structure admits a string connect…

Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also shows that virtual string links up to unwelded equivalence are classified by those…

In the 1920's Artin defined the braid group in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of th…

Paper shows string cobordism at 24 dims can be determined by elliptic genus.

Introduces string structures linking to loop spaces.

The actions, anomalies, and quantization conditions allow the M2-brane and the M5-brane to support, in a natural way, structures beyond Spin on their worldvolumes. The main examples are twisted String structures. This also extends to twisted String^c structures, which we introduce and relate to twisted String structure…

A virtual string is a scheme of self-intersections of a closed curve on a surface. We introduce virtual strings and study their geometric properties and homotopy invariants. We also discuss connections between virtual strings, Gauss words, and virtual knots.

Study topological G₂ and Spin(7) strings at 1-loop using double complexes.