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Causal Inference by String Diagram SurgeryNov 20 2018Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear ... More

Tinker, tailor, solver, proofOct 30 2014We introduce Tinker, a tool for designing and evaluating proof strategies based on proof-strategy graphs, a formalism previously introduced by the authors. We represent proof strategies as open-graphs, which are directed graphs with additional input/output ... More

Synthesising Graphical TheoriesFeb 27 2012Apr 17 2012In recent years, diagrammatic languages have been shown to be a powerful and expressive tool for reasoning about physical, logical, and semantic processes represented as morphisms in a monoidal category. In particular, categorical quantum mechanics, or ... More

Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum ComputingMar 01 2012Mar 22 2012This work is about diagrammatic languages, how they can be represented, and what they in turn can be used to represent. More specifically, it focuses on representations and applications of string diagrams. String diagrams are used to represent a collection ... More

Finite matrices are complete for (dagger-)hypergraph categoriesJun 23 2014Aug 19 2015Hypergraph categories are symmetric monoidal categories where each object is equipped with a special commutative Frobenius algebra (SCFA). Dagger-hypergraph categories are the same, but with dagger-symmetric monoidal categories and dagger-SCFAs. In this ... More

Abstract Tensor Systems as Monoidal CategoriesAug 16 2013The primary contribution of this paper is to give a formal, categorical treatment to Penrose's abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system ... More

Generalised Compositional Theories and Diagrammatic ReasoningJun 11 2015This chapter provides an introduction to the use of diagrammatic language, or perhaps more accurately, diagrammatic calculus, in quantum information and quantum foundations. We illustrate the use of diagrammatic calculus in one particular case, namely ... More

Open Graphs and Monoidal TheoriesNov 18 2010String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ... More

!-Graphs with Trivial Overlap are Context-FreeJan 24 2015Apr 10 2015String diagrams are a powerful tool for reasoning about composite structures in symmetric monoidal categories. By representing string diagrams as graphs, equational reasoning can be done automatically by double-pushout rewriting. !-graphs give us the ... More

Tensors, !-graphs, and non-commutative quantum structures (extended version)Mar 04 2015!-graphs provide a means of reasoning about infinite families of string diagrams and have proven useful in manipulation of (co)algebraic structures like Hopf algebras, Frobenius algebras, and compositions thereof. However, they have previously been limited ... More

Equational reasoning with context-free families of string diagramsApr 10 2015Oct 13 2015String diagrams provide an intuitive language for expressing networks of interacting processes graphically. A discrete representation of string diagrams, called string graphs, allows for mechanised equational reasoning by double-pushout rewriting. However, ... More

A first-order logic for string diagramsMay 02 2015Equational reasoning with string diagrams provides an intuitive means of proving equations between morphisms in a symmetric monoidal category. This can be extended to proofs of infinite families of equations using a simple graphical syntax called !-box ... More

The compositional structure of multipartite quantum entanglementFeb 12 2010Aug 17 2010While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a long-standing open problem in quantum ... More

Categorical Quantum Mechanics II: Classical-Quantum InteractionMay 27 2016This is the second part of a three-part overview, in which we derive the category-theoretic backbone of quantum theory from a process ontology, treating quantum theory as a theory of systems, processes and their interactions. In this part we focus on ... More

Can quantum theory be characterized in information-theoretic terms?Apr 20 2016Apr 21 2016Does information play a significant role in the foundations of physics? We investigate whether information-theoretic constraints characterize quantum theory. In a C*-algebraic framework, this is known to hold via three equivalences: no broadcasting and ... More

Tensors, !-graphs, and non-commutative quantum structuresDec 30 2014Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques. Diagrams allow us ... More

Categorical Quantum Mechanics I: Causal Quantum ProcessesOct 19 2015May 27 2016We derive the category-theoretic backbone of quantum theory from a process ontology. More specifically, we treat quantum theory as a theory of systems, processes and their interactions. In this first part of a three-part overview, we first present a general ... More

Quantomatic: A Proof Assistant for Diagrammatic ReasoningMar 03 2015Oct 13 2015Monoidal algebraic structures consist of operations that can have multiple outputs as well as multiple inputs, which have applications in many areas including categorical algebra, programming language semantics, representation theory, algebraic quantum ... More

Categories of Quantum and Classical Channels (extended abstract)Aug 01 2014We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive ... More

A Graphical Language for Proof StrategiesFeb 27 2013Oct 07 2013Complex automated proof strategies are often difficult to extract, visualise, modify, and debug. Traditional tactic languages, often based on stack-based goal propagation, make it easy to write proofs that obscure the flow of goals between tactics and ... More

Globular: an online proof assistant for higher-dimensional rewritingDec 04 2016This article introduces Globular, an online proof assistant for the formalization and verification of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their construction ... More

PyZX: Large Scale Automated Diagrammatic ReasoningApr 09 2019The ZX-calculus is a graphical language for reasoning about ZX-diagrams, a tensor network-like language that can represent arbitrary linear maps between qubits. Using the ZX-calculus, we can intuitively reason about quantum mechanics, and optimise and ... More

Compositional Quantum LogicFeb 20 2013Quantum logic aims to capture essential quantum mechanical structure in order-theoretic terms. The Achilles' heel of quantum logic is the absence of a canonical description of composite systems, given descriptions of their components. We introduce a framework ... More

Proceedings Second Graphs as Models WorkshopDec 04 2016Graphs are used as models in all areas of computer science: examples are state space graphs, control flow graphs, syntax graphs, UML-type models of all kinds, network layouts, social networks, dependency graphs, and so forth. Once such graphical models ... More

Globular: an online proof assistant for higher-dimensional rewritingDec 04 2016Jan 22 2018This article introduces Globular, an online proof assistant for the formalization and verification of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their construction ... More

CNOT circuit extraction for topologically-constrained quantum memoriesApr 01 2019Many physical implementations of quantum computers impose stringent memory constraints in which 2-qubit operations can only be performed between qubits which are nearest neighbours in a lattice or graph structure. Hence, before a computation can be run ... More

Pattern graph rewrite systemsApr 30 2012Apr 01 2014String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric representations of string ... More

Open Graphs and Computational ReasoningJul 22 2010We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of half-edges (edges ... More

Completely positive projections and biproductsAug 21 2013Dec 30 2014The recently introduced CP*-construction unites quantum channels and classical systems, subsuming the earlier CPM-construction in categorical quantum mechanics. We compare this construction to two earlier attempts at solving this problem: freely adding ... More

Reducing T-count with the ZX-calculusMar 25 2019Reducing the number of non-Clifford quantum gates present in a circuit is an important task for efficiently implementing quantum computations, especially in the fault-tolerant regime. We present a new method for reducing the number of T-gates in a quantum ... More

Reducing T-count with the ZX-calculusMar 25 2019Mar 27 2019Reducing the number of non-Clifford quantum gates present in a circuit is an important task for efficiently implementing quantum computations, especially in the fault-tolerant regime. We present a new method for reducing the number of T-gates in a quantum ... More

Categories of Quantum and Classical ChannelsMay 16 2013Sep 16 2014We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators. For example, it turns the category of finite-dimensional Hilbert spaces into the category of finite-dimensional ... More

Coherent Parity Check Construction for Quantum Error CorrectionNov 23 2016We present a framework for constructing and analysing quantum error correction codes that gives simple and intuitive tools for designing codes based on device specifications. Built from a direct analog of classical parity checking, these coherent parity ... More

Graphical Structures for Design and Verification of Quantum Error CorrectionNov 23 2016Jan 12 2018We introduce a high-level graphical framework for the design, analysis, and verification of quantum error correcting codes. The coherent parity check construction for stabilizer codes allows us to construct a broad range of quantum codes based on classical ... More

Rewriting modulo symmetric monoidal structureFeb 22 2016Feb 23 2016String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory. ... More

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculusFeb 08 2019We present a new circuit-to-circuit optimisation routine based on an equational theory called the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, a flexible, lower-level language for describing quantum computations graphically. Then, ... More

Graphical Fourier Theory and the Cost of Quantum AdditionApr 16 2019The ZX-calculus is a convenient formalism for expressing and reasoning about quantum circuits at a low level, whereas the recently-proposed ZH-calculus yields convenient expressions of mid-level quantum gates such as Toffoli and CCZ. In this paper, we ... More

Strong Complementarity and Non-locality in Categorical Quantum MechanicsMar 22 2012Apr 27 2012Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, ... More

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculusFeb 08 2019Feb 26 2019We present the theoretical foundations for a new quantum circuit optimisation technique based on an equational theory called the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing ... More

The GHZ/W-calculus contains rational arithmeticMar 14 2011Graphical calculi for representing interacting quantum systems serve a number of purposes: compositionally, intuitive graphical reasoning, and a logical underpinning for automation. The power of these calculi stems from the fact that they embody generalized ... More

The Matrix of Linear MappingsJan 27 2010On the set of mappings of the given set, we define the product of mappings. If A is associative algebra, then we consider the set of matrices, whose elements are linear mappings of algebra A. In algebra of matrices of linear mappings we define the operation ... More

Quaternion RhapsodySep 04 2009Apr 04 2011In this paper I explore the set of quaternion algebras over field. Quaternion algebra E(C,-1,-1) is isomorphic to tensor product of complex field C and quaternion algebra H=E(R,-1,-1). Considered two sets of quaternion functions, which satisfy to equation ... More

Linear Map of $D$-AlgebraFeb 12 2015May 08 2015Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more general point of ... More

Differential Equation over Banach AlgebraJan 05 2018In the book, I considered differential equations of order $1$ over Banach $D$-algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. In noncommutative Banach algebra, initial value ... More

A note on a multiplicity result for the mean field equation on compact surfacesNov 11 2014Mar 06 2017We are concerned with a Liouville-type equation with exponential nonlinearities on a compact surface which describes the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We provide the first multiplicity result for this ... More

Mappings of Conjugation of Quaternion AlgebraFeb 23 2012Jun 10 2012In the paper I considered mappings of conjugation of quaternion algebra. I proved the theorem that there is unique expansion of R-linear mapping of quaternion algebra relative to the given set of mappings of conjugation.

Quadratic Equation over Associative D-AlgebraMay 30 2015Jan 17 2016In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $a\in R$, $a<0$, then the equation has infinitely many roots. Otherwise, the ... More

Derivative of Map of Banach algebraMay 14 2015Let $A$ be Banach algebra over commutative ring $D$. The map $f:A\rightarrow A\ $ is called differentiable in the Gateaux sense, if $$f(x+a)-f(x)=\partial f(x)\circ a+o(a)$$ where the Gateaux derivative $\partial f(x)$ of map $f$ is linear map of increment ... More

Representation of Universal AlgebraDec 17 2009Feb 07 2015Theory of representations of universal algebra is a natural development of the theory of universal algebra. Morphism of the representation is the map that conserve the structure of the representation. Exploring of morphisms of the representation leads ... More

The rotation of brown dwarfsOct 20 2016One of the characteristic features of low-mass stars is their propensity to shed large amounts of angular momentum throughout their evolution. This distinguishs them from brown dwarfs which remain fast rotators over timescales of gigayears. Brown dwarfs ... More

Vector Space Over Division RingJan 16 2005Jul 16 2010A system of linear equations over a skew field has properties similar to properties of a system of linear equations over a field. Even noncommutativity of a product creates a new picture the properties of system of linear equations and of vector space ... More

Division in Associative $D$-AlgebraDec 13 2014From the symmetry between definitions of left and right divisors in associative $D$-algebra $A$, the possibility to define quotient as $A\otimes A$-number follows. In the paper, I considered division and division with remainder. I considered also definition ... More

Integral of Map into Abelian $Ω$-groupOct 11 2013Mar 12 2014The common in ring, module and algebra is that they are Abelian group with respect to addition. This property is enough to study integration. I treat integral of measurable map into normed Abelian $\Omega$-group. Theory of integration of maps into $\Omega$-group ... More

Linear Equation in Finite Dimensional AlgebraDec 21 2009Apr 30 2012In the paper I considered methods for solving equations of the form axb+cxd=e in the algebra which is finite dimensional over the field.

The frequency of large variations in the near-infrared fluxes of T Tauri starsNov 08 2011Variability is a characteristic feature of young stellar objects (YSOs) and could contribute to the large scatter observed in HR diagrams for star forming regions. For typical YSOs, however, the long-term effects of variability are poorly constrained. ... More

Multiplicity results for the mean field equation on compact surfacesNov 11 2014Dec 10 2014We are concerned with a Liouville-type equation with exponential nonlinearities on a compact surface which describes the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We provide the first multiplicity result for this ... More

Free Algebra with Countable BasisNov 27 2012In this book I treat the structure of D-module which has countable basis. If we do not care for topology of D-module, then we consider Hamel basis. If norm is defined in D-module, then we consider Schauder basis. In case of Schauder basis, we consider ... More

The Gateaux Derivative of Map over Division RingAug 23 2009I consider differential of mapping $f$ of continuous division ring as linear mapping the most close to mapping $f$. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher ... More

Introduction into Calculus over Division RingDec 30 2008Feb 26 2013Based on twin representations of division ring in an Abelian group I consider $D$\Hyph vector spaces over division ring. Morphism of $D$\Hyph vector spaces is linear map of $D$\Hyph vector spaces. I consider derivative of function $f$ of continuous division ... More

Introduction into Calculus over Banach algebraJan 11 2016Let $A$ be Banach $D$-algebra wit norm $|a|$. The map $$f:A\rightarrow A$$ is called differentiable on the set $U\subset A$, if at every point $x\in U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =\partial_x f(x)\circ dx +o(dx) =\frac{\partial ... More

Angular momentum and disk evolution in very low mass systemsJun 24 2013This review summarises recent observational results regarding the evolution of angular momentum and disks in brown dwarfs. The observations clearly show that brown dwarfs beyond ages of 10 Myr are exclusively fast rotators and do not spin down with age. ... More

New existence results for the mean field equation on compact surfaces via degree theorySep 27 2014Dec 10 2014We consider a class of equations with exponential non-linearities on a compact surface which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We prove an existence result via degree theory. This yields ... More

Blow-up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensitiesSep 17 2016A class of equations with exponential nonlinearities on a compact Riemannian surface is considered. More precisely, we study an asymmetric sinh-Gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with variable intensities. ... More

Algebra of Fractions of Algebra with ConjugationMay 09 2012In the paper, I considered construction of algebra of fractions of algebra with conjugation. I also considered algebra of polynomials and algebra of rational mappings over algebra with conjugation.

Orthogonal Basis and Motion in Finsler GeometryJul 24 2011Aug 12 2011Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this ... More

Introduction into Geometry over Division RingMay 31 2009Feb 12 2011Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the notion of tower ... More

Normed Ω-GroupMay 15 2013Oct 17 2013Since sum which is not necessarily commutative is defined in \Omega-algebra A, then \Omega-algebra A is called \Omega-group. I also considered representation of \Omega-group. Norm defined in \Omega-group allows us to consider continuity of operations ... More

Polynomial over Associative D-AlgebraFeb 25 2013Apr 12 2015In the paper I considered algebra of polynomials over associative D-algebra with unit. Using the tensor notation allows to simplify the representation of polynomial. I considered questions related to divisibility of polynomial of any power over polynomial ... More

An Existence Result for the Mean Field Equation on Compact Surfaces in a Doubly Supercritical RegimeApr 15 2012Mar 16 2014We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean field equation of the equilibrium turbulance with arbitrarily signed vortices. For the first time, we consider the problem ... More

Correspondence between Row-Column Determinants and Quasideterminants of Matrices over Quaternion AlgebraFeb 09 2011In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row ... More

Orthonormal Basis in Minkowski SpaceJan 19 2012Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this ... More

DNA brick self-assembly with an off-lattice potentialJun 07 2016We report Monte Carlo simulations of a simple off-lattice patchy-particle model for DNA `bricks'. We relate the parameters that characterise this model with the binding free energy of pairs of single-stranded DNA molecules. We verify that an off-lattice ... More

Exact solution of pulled, directed vesicles with sticky walls in two dimensionsNov 28 2018We analyse a directed lattice vesicle model incorporating both the binding-unbinding transition and the vesicle inflation-deflation transition. From the exact solution we derive the phase diagram for this model and elucidate scaling properties around ... More

Distinguishing Thermal Fluctuations from Instrumental Error for High Pressure Charged GasJun 05 2016Feb 26 2017Thermodynamic parameters such as temperature and pressure can be defined from the statistical behavior of a system. Therefore, thermal fluctuation is an inseparable characteristic of these parameters which eventually finds its way into experimental data. ... More

Numerical evidence for nucleated self-assembly of DNA brick structuresFeb 25 2014Apr 25 2014The observation by Ke et al. [Science 338, 1177 (2012)] that large numbers of short, pre-designed DNA strands can assemble into three-dimensional target structures came as a great surprise, as no colloidal self-assembling system has ever achieved the ... More

Dusty disks at the bottom of the IMFNov 15 2007'Isolated planetary mass objects' (IPMOs) have masses close to or below the Deuterium-burning mass limit (~15 Jupiter masses) -- at the bottom of the stellar initial mass function. We present an exploratory survey for disks in this mass regime, based ... More

Discrete Component AnalysisApr 18 2006This article presents a unified theory for analysis of components in discrete data, and compares the methods with techniques such as independent component analysis, non-negative matrix factorisation and latent Dirichlet allocation. The main families of ... More

Distinguishing Thermal Fluctuations from Instrumental Error for High Pressure Charged GasJun 05 2016Thermodynamic parameters such as temperature and pressure could be defined from the statistical behavior of the system. Therefore, always there exists a natural thermal fluctuations in these parameters which leads to fluctuations in experimental data. ... More

Counting occurences of 132 in a permutationMay 09 2001Aug 02 2001We study the generating function for the number of permutations on n letters containing exactly $r\gs0$ occurences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in $S_{2r}$.

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence resultsMay 06 2016We are concerned with the following class of equations with exponential nonlinearities: $$ \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, $$ which is related to the Tzitz\'eica equation. Here $h_1, h_2$ are two smooth positive ... More

Quantitative Measurements of Electromechanical Response with a Metrological Atomic Force MicroscopeMay 21 2015An ongoing challenge in atomic force microscope (AFM) experiments is the quantitative measurement of cantilever motion. The vast majority of AFMs use the optical beam deflection (OBD) method to infer the deflection of the cantilever. The OBD method is ... More

Avoiding maximal parabolic subgroups of S_kJun 21 2000We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook ... More

A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspectsMay 27 2016We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper we describe the blow-up phenomenon and highlight the differences from the ... More

On the global bifurcation diagram of the Gel'fand problemJan 20 2019For domains of first kind [7,13] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel'fand problem crossing the origin. At least to our knowledge this is the first result about the exact ... More

Nonequilibrium wetting of finite samplesMar 23 2005Mar 15 2006As a canonical model for wetting far from thermal equilibrium we study a Kardar-Parisi-Zhang interface growing on top of a hard-core substrate. Depending on the average growth velocity the model exhibits a non-equilibrium wetting transition which is characterized ... More

Asynchronous Graph Pattern Matching on Multiprocessor SystemsJun 13 2017Jun 14 2017Pattern matching on large graphs is the foundation for a variety of application domains. Strict latency requirements and continuously increasing graph sizes demand the usage of highly parallel in-memory graph processing engines that need to consider non-uniform ... More

Exotic cluster structures on $SL_n$: the Cremmer-Gervais caseJul 03 2013This is the second paper in the series of papers dedicated to the study of natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our ... More

Topological classification of generic real rational functionsOct 21 2001To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to a face of ... More

On double Hurwitz numbers in genus 0Nov 14 2006We study double Hurwitz numbers in genus zero counting the number of covers $\CP^1\to\CP^1$ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the ... More

Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domainsApr 19 2018The aim of this paper is to complete the program initiated in [50], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general ... More

Classification of blow-up limits for the sinh-Gordon equationFeb 07 2016The aim of this paper is to use a selection process and a careful study of the interaction of bubbling solutions to show a classification result for the blow-up values of the elliptic sinh-Gordon equation $$\Delta u+h_1e^u-h_2e^{-u}=0 \quad \mathrm{in}~B_1\subset\mathbb{R}^2.$$ ... More

On the Topological degree of the Mean field equation with two parametersFeb 10 2016We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $$ - \Delta u = \rho_1 \left( \frac{h_1 \,e^{u}}{\int_M h_1 \,e^{u} } - \frac{1}{|M|} \right) - \rho_2 \left( \frac{h_2 \,e^{-u}}{\int_M h_2 \,e^{-u} ... More

Poisson Geometry of Directed Networks in an AnnulusDec 31 2008Jul 11 2010As a generalization of Postnikov's construction (see arXiv: math/0609764), we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson brackets introduced ... More

Cremmer--Gervais cluster structure on $SL_n$Aug 12 2013We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification ... More

Homogeneous TIP4P/2005 ice nucleation at low supercoolingJun 21 2013We present a partial free energy profile for the homogeneous nucleation of ice using an all-atom model of water at low supercooling, at which ice growth dynamics are reasonably accessible to simulation. We demonstrate that the free energy profile is well ... More

Effects of surface interactions on heterogeneous ice nucleation for a monatomic water modelApr 22 2014Despite its importance in atmospheric science, much remains unknown about the microscopic mechanism of heterogeneous ice nucleation. In this work, we perform hybrid Monte Carlo simulations of the heterogeneous nucleation of ice on a range of generic surfaces, ... More

The Rotation-Disk Connection in Young Brown Dwarfs: Strong Evidence for Early Rotational BrakingJan 16 2019Jan 24 2019We use Kepler/K2 lightcurves to measure rotation periods of brown dwarfs and very low mass stars in the Upper Scorpius star-forming region. Our sample comprises a total of 104 periods. Depending on the assumed age of Upper Scorpius, about a third of them ... More

On the properties of the exchange graph of a cluster algebraMar 06 2007May 11 2015We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra $\A$ in two cases: when $\A$ is of geometric type and when $\A$ is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that ... More

Symmetry and uniqueness of solutions to some Liouville-type equations and systemsMar 07 2017We prove symmetry and uniqueness results for three classes of Liouville-type problems arising in geometry and mathematical physics: asymmetric Sinh-Gordon equation, cosmic string equation and Toda system, under certain assumptions on the mass associated ... More

Drinfeld double of $GL_n$ and generalized cluster structuresMay 18 2016Oct 24 2017We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the Poisson ... More