The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta…
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Study of a functional equation associated to the Kummer's equation of the trilogarihtm. Applicationsmath.CV
In this paper we study a functional equation associated to the Kummer's equation (K) of the trilogarithm. Then we apply our results to web geometry and to characterize the functions solution of (K).
We study the spectral functions, and in particular the zeta function, associated to a class of sequences of complex numbers, called of spectral type. We investigate the decomposability of the zeta function associated to a double sequence with respect to some simple sequence, and we provide a technique for obtaining the…
Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian groupmath.DG
We prove Patterson's conjecture about the singularities of the Selberg zeta function associated to a convex-cocompact, torsion free group acting on a hyperbolic space.
Defines spectral Einstein functional for manifolds with boundary.
problem Calculating the spectral Einstein functional for manifolds with boundaries.
method Defined spectral Einstein functional associated with the Dirac operator and proved a theorem for 4D manifolds.
result Proof of Kastler-Kalau-Walze type theorem for spectral Einstein functional.
Proves Kastler-Kalau-Walze theorem for spectral Einstein functional on low-dimensional manifolds.
problem Proving Kastler-Kalau-Walze type theorems for spectral Einstein functional.
method Defining spectral Einstein functional associated with Dirac operator and proving theorem for low-dimensional manifolds.
result Proves Kastler-Kalau-Walze type theorem for spectral Einstein functional on low-dimensional manifolds with boundary.
Study on likelihood functions, associative equations, and Frobenius manifolds.
problem Maximum likelihood estimation and associativity equations in statistical models.
method Analyzes the cone of concentration matrices, log-likelihood function, and Frobenius manifolds.
result Maximum likelihood degree is indexed by components of Frobenius residuals.
Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.
problem Calculating Einstein-like functionals for sub-Dirac operators.
method Introduced spectral Einstein functional for sub-Dirac operators on manifolds with boundary.
result Proved a theorem for spectral Einstein functions on four-dimensional manifolds.
Study on new Monge-Ampère functionals and their variational problems.
problem Existence and uniqueness of solutions for nonlinear eigenvalue problems.
method Introduction of a family of real Monge-Ampère functionals and proving Sobolev type inequalities.
result Existence of solutions for a nonlinear eigenvalue problem.
We propose the use of the functional determinant of geometric operators in constructing an entropy functional associated to geometric flows. Our approach is based on the direct computation of the partition function, with a well-defined set of microstates and macrostates in the canonical ensemble. The approach is motiva…
Introduces Fitzpatrick losses, tighter than Fenchel-Young losses.
problem Improving loss functions for machine learning.
method Introduces Fitzpatrick losses based on the Fitzpatrick function.
result Fitzpatrick losses are tighter than Fenchel-Young losses.
Analytic torsions on contact spheres are calculated using Rumin complex.
problem Calculating analytic torsions for contact spheres.
method Explicitly wrote down eigenvalues of Rumin Laplacian and expressed analytic torsion functions in terms of Riemann zeta function.
result Functions of analytic torsions vanish at the origin and were determined.
Study of twisted Ruelle zeta function on hyperbolic manifolds and its relation to analytic torsion.
problem Analyzing the twisted Ruelle zeta function on hyperbolic manifolds.
method Investigating the twisted Ruelle zeta function associated with geodesic flow and acyclic representations.
result The twisted Ruelle zeta function equals the square of the refined analytic torsion multiplied by an exponential involving the eta invariant.
Model stores many more patterns than neurons, improving pattern recognition.
problem Storing and retrieving many more patterns than neurons in a network.
method Constructs a family of models interpolating between feature-matching and prototype modes, corresponding to neural networks with various activation functions.
result Higher rectified polynomials can be used in neural networks for improved pattern recognition.
The paper defines differential invariants for -structures and calculates their number.
problem Understanding scalar differential invariants for -structures.
method Analyzing the action of diffeomorphisms on jet bundles and quotient spaces.
result Calculates the number of scalar differential invariants for -structures.
We describe convolutional networks using harmonic functions.
problem Understanding the function space and smoothness of convolutional networks.
method Using reproducing kernel Hilbert spaces and functional ANOVA decomposition.
result Convolutional networks can be decomposed into a sum of elementary functions.
New method uses Brownian motion to estimate Kleinian group orbital functions.
problem Estimating orbital functions of Kleinian groups.
method Developed a new method using Brownian motion.
result Gave estimates of orbital functions for nilpotent covers.
Study convexity of Mabuchi functional in big cohomology classes.
problem Convexity of Mabuchi functional in big cohomology classes.
method Defined an invariant related to transcendental Fujita approximations and established convexity under vanishing of this invariant.
result Established almost convexity along weak geodesics in big cohomology classes.
Proposes AAA for efficient association estimation with confounders.
problem Summarizing log odds ratio as a function of confounders.
method Develops efficient DML estimators for AAA.
result Demonstrates practicality and effectiveness of AAA estimators.
This paper shows how to calculate risk measures for sums of two counter-monotonic risks.
problem Calculating risk measures for sums of two counter-monotonic risks.
method Using a fixed distortion function and expressing the risk measure of a sum as the sum of two related measures of the marginals.
result The risk measure of a sum of two counter-monotonic risks can be expressed as the sum of two related distortion risk measures of the marginals.
New families of complete hyperbolic affine spheres are constructed from Hildebrand's semi-homogeneous cones.
problem Constructing families of complete hyperbolic affine spheres from semi-homogeneous cones.
method Computed isothermal parametrizations, affine metrics, and cubic forms for Hildebrand's examples; constructed associated families using Weierstrass functions.
result Generic members of the constructed affine spheres are given by Weierstrass functions.
Proves new theorems for manifolds with boundary.
problem No boundary conditions for manifolds.
method Generalizes Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
result Proof of new theorems for even and odd dimensional manifolds with boundary.
The study examines Poincaré series for links of normal surface singularities, proving their quasipolynomiality and polynomial generalization.
problem Counting function of Poincaré series for rational homology sphere plumbed 3-manifolds.
method Interpreting Poincaré series as an alternating sum of coefficient functions, using Jeffrey--Kirwan residues and Némethi's surgery formulas.
result Proves quasipolynomiality and polynomial generalization of the Seiberg-Witten invariant.
Investigates webs related to cluster algebras and polylogarithms.
problem Understanding webs associated with cluster algebras and polylogarithms.
method Introducing AMP webs and analyzing their properties, proving results and conjectures.
result Many webs associated with polylogarithms and cluster algebras are AMP webs.
The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex. The relationship ``curvature-associator'' is investigated. Based on this generic example, we extend the geometric language of vector fields to a purely algebraic setting, similar to the context of Gerstenhaber algebras. We int…
We introduce here a natural functional associated to any : \emph{spectral length functional}, on the space of "generalized paths" in , closely related to both the Hofer length functional and spectral invariants and establish some of its properties. This functional is smooth on its…
New proof of energy functional monotonicity via geodesics in measure space.
problem Proving monotonicity of energy functional in generalized Ricci flow.
method Defining adapted cost functional, geodesics, and entropy functional.
result Monotonicity of cost along backwards heat flow and energy functional along generalized Ricci flow.
New method approximates partition function of graphical models using gauge functions and polynomials.
problem Computing the partition function of graphical models is computationally challenging.
method Combines gauge function technique with real stable polynomials to approximate partition function.
result Belief Propagation estimations in the sequence do not decrease and low-bound the partition function.
Crowdsourced gene set queries reveal protein-protein interactions and gene-gene associations.
problem Lack of integrative analysis of diverse gene set queries.
method Harnessed thousands of user-submitted gene sets to construct a global gene-gene association network.
result The constructed network recapitulates known protein-protein interactions and gene-gene functional associations.
The geometric complex of a Morse-Bott-Smale pair and an extension of a theorem by Bismut-Zhangmath.GT
We define the geometric complex associated to a Morse-Bott-Smale vector field, cf. [Austin-Braam, 1995], and its associated spectral sequence. We prove an extension of the Bismut-Zhang theorem to Morse-Bott-Smale functions. The proof is based on the Bismut-Zhang theorem for Morse-Smale functions, see [Bismut-Zhang, 199…
Witten deformation connects manifold spectra to Morse functions.
problem Understanding spectral properties of Riemannian manifolds.
method Rellich-Kato theorem applied to Witten deformation.
result Relates spectral package to Morse complex and harmonic oscillators.
Isotropic geometry of graph surfaces associated with product production functions in economicsmath.DG
The paper classifies graph surfaces of product production functions in isotropic 3-space.
problem Understanding the geometry of production functions in economics.
method Analysis of graph surfaces in isotropic 3-space with constant curvature.
result Several classification results for product production functions in isotropic 3-space.
Sub-signature operators and the Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundarymath.DG
The paper defines a new functional and proves related theorems for manifolds with boundary.
problem Defining and proving theorems for manifolds with boundary.
method Defining the spectral Einstein functional and relating it to the noncommutative residue.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for spectral Einstein functional on 4D manifolds with boundary.
Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.
problem Characterizing the behavior of volume functions associated with quadratic differentials.
method Analyzing the Thurston volume of unit balls in measured lamination spaces.
result The volume function is not proper and is -integrable for any .
We investigate classification results for general quadratic functions on torsion abelian groups. Unlike the previously studied situations, general quadratic functions are allowed to be inhomogeneous or degenerate. We study the discriminant construction which assigns, to an integral lattice with a distinguished characte…
Study on hypermaps and KP hierarchy, proving tau function and enumerative meaning.
problem Understanding the partition function of meromorphic functions on the Riemann sphere.
method Analysis of Hurwitz Dubrovin--Frobenius manifold structure and rational reductions of the KP hierarchy.
result The all genera partition function is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy.
Study provides bounds for longest curves with intersections on Teichmüller space.
problem Finding the longest curve with a given number of intersections on Teichmüller space.
method Uses lower bounds for the infimum of length functions associated with curve collections and dual cube complexes.
result Obtains estimates for the longest curve with k self-intersections.
Energy Transformercs.LG
Energy Transformer integrates attention, energy models, and associative memory.
problem Lack of clear theoretical foundations in attention mechanisms and straightforward design of energy functions in energy-based models.
method Proposes Energy Transformer, a sequence of attention layers with a specifically engineered energy function.
result Obtained strong results on graph anomaly detection and classification tasks.
Study Kleinian groups using orbital functions and heat kernels.
problem Understanding Kleinian groups through orbital functions and heat kernels.
method Using the heat kernel approach developed in \cite{artmoiheatcounting1}.
result Developed a method to study Kleinian groups using orbital functions and heat kernels.
We show how the space of complex spin structures of a closed oriented three-manifold embeds naturally into a space of quadratic functions associated to its linking pairing. Besides, we extend the Goussarov-Habiro theory of finite type invariants to the realm of compact oriented three-manifolds equipped with a complex s…
Study critical exponent for geodesic currents using quasi-metric spaces.
problem Understanding the critical exponent for geodesic currents.
method Associated a quasi-metric space to geodesic currents and defined a metric for filling currents, studying the critical exponent and its relation to curve intersection growth.
result The critical exponent equals the exponential growth rate of the intersection function for closed curves.
We study the asymptotic behaviour of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor . Assuming the data in question is invariant under an -action (locally around ) we prove that this density function has a distri…
Study CR geometry surface area elements and singular Yamabe problem solutions.
problem Solving the singular CR Yamabe problem in 3D CR geometry.
method Expressed CR invariant surface area elements, deduced Euler-Lagrange equations, provided solutions.
result One energy functional coefficient is shown to be proportional to the log term in volume renormalization.
Spectral Theory of Almost Periodic Banach--Malcev Algebras and Applications to Moufang Dynamicsmath.DG
New spectral theory for non-associative algebras with applications to Moufang dynamics.
problem Spectral theory of non-associative algebras and their applications.
method Introducing almost periodic Banach--Malcev algebras and analyzing their spectral properties.
result Spectral characterization and continuous functional calculus for almost periodic derivations.
Explicit computation of Drinfeld associator in the case of the fundamental representation of gl(N)hep-th
We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of . Several tests of the results are presented. It can be explicitly seen that components of this solution for the associator coincide w…
Mathematical construction of Chern-Simons partition function using reflection positivity.
problem Constructing a mathematical framework for Chern-Simons functional integrals.
method Reflection positive functional on Banach space of connections, unitary operators, weak limit.
result Nonperturbative construction of partition function without renormalization.
Study L-functions for knot group deformations, proving torsion and zero simplicity.
problem Investigate L-functions for knot group deformations.
method Analyze twisted knot modules and associated L-functions.
result Show torsion property and verify zero simplicity of L-functions.
We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.