Differentials on graph complexes II: hairy graphs

We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many nonzero hairy graph cohomology classes out of (known) non-hairy classes by studying the cancellations in those sequences. This provide a first glimpse at the tentative global structure of the hairy graph cohomology.     

Tripartite connection condition for a quantum graph vertex

We discuss formulations of boundary conditions in a quantum graph vertex and demonstrate that the so-called ST-form can be further reduced up to a form more effective in certain applications: In particular, in identifying the number of independent parameters for given ranks of two connection matrices, or in calculating the scattering matrix when both matrices are singular. The new form of boundary conditions, called the PQRS-form, also gives a natural scheme to design generalized low and high pass quantum filters.     

Symmetric vertex models on planar random graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular (Φ³) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent – a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models).The relations between the vertex weights and Ising model parameters in the 4-vertex model on Φ³ graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model.Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.     

Approximation of a general singular vertex coupling in quantum graphs

The longstanding open problem of approximating all singular vertex couplings in a quantum graph is solved. We present a construction in which the edges are decoupled; an each pair of their endpoints is joined by an edge carrying a δ potential and a vector potential coupled to the “loose” edges by a δ coupling. It is shown that if the lengths of the connecting edges shrink to zero and the potentials are properly scaled, the limit can yield any prescribed singular vertex coupling, and moreover, that such an approximation converges in the norm-resolvent sense.     

Fourier could be a data scientist: From graph Fourier transform to signal processing on graphs

The legacy of Joseph Fourier in science is vast, especially thanks to the essential tool that the Fourier transform is. The flexibility of this analysis, its computational efficiency and the physical interpretation it offers makes it a cornerstone in many scientific domains. With the explosion of digital data, both in quantity and diversity, the generalization of the tools based on Fourier transform is mandatory. In data science, new problems arose for the processing of irregular data such as social networks, biological networks or other data on networks. Graph signal processing is a promising approach to deal with those. The present text is an overview of the state of the art in graph signal processing, focusing on how to define a Fourier transform for data on graphs, how to interpret it and how to use it to process such data. It closes showing some examples of use. Along the way, the review reveals how Fourier's work remains modern and universal, and how his concepts, coming from physics and blended with mathematics, computer science, and signal processing, play a key role in answering the modern challenges in data science.     

Supersymmetry and cohomology of graph complexes

Letters in Mathematical Physics - I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves $$widehat{Z}_{I}in H^{*}left( bar{mathcal...     

Quantum deleting and cloning in a pseudo-unitary system

In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.     

Magnetoresonances on a lasso graph

We consider a charged spinless quantum particle confined to a graph consisting of a loop to which a halfline lead is attached; this system is placed into a homogeneous magnetic field perpendicular to the loop plane. We derive the reflection amplitude and show that there is an infinite ladder of resonances; analyzing the resonance pole trajectories, we show that half of them turn into true embedded eigenvalues provided the flux through the loop is an integer or half-integer multiple of the flux unit hc/e. We also describe a general method to solve the scattering problem on graphs of which the present model is a simple particular case. Finally, we discuss ways in which a state localized initially at the loop decays.     

Sum rules for the vertex function from local commutativity (III) Relations applicable to the time-ordered and anti-time-ordered boundary values of the vertex function

A set of sum rules, independent of the ones presented in two earlier papers [1, 2], are derived from the analyticity properties of the vertex function proved by Källén and Wightman [3]. The sum rules are applicable to the time-ordered and anti-time-ordered boundary values. In an explicit example it is shown how the relations can be combined to re-derive an earlier result [2] by different methods.     

Estimates of general Mayer graphs. I: Construction of upper bounds for a given graph by means of sets of subgraphs

A large number of physical quantities (thermodynamic and correlation functions, scattering amplitudes, intermolecular potentials, etc. ...) can be expressed as sums of an infinite number of multiple integrals of the following type: $$\Gamma \left( {x_1 ,. . . , x_n } \right) = \int {\prod {f_L \left( {x_{i,} x_j } \right)dx_{n + 1} . . . dx_{n + k} } }$$ These are called Mayer graphs in statistical mechanics, Feynman graphs in quantum field theory, and multicenter integrals in quantum chemistry. We call themn-graphs here. In a preceding note [Physics Letters 62A:295 (1977)], we have proposed a new estimation method which provides upper bounds for arbitraryn-graphs. This article is devoted to developing in detail the foundations of this method. As a first application, we prove that all virial coefficients of polar systems are finite. More generally, we show on some examples that our estimation method can givefinite upper bounds forn-graphs occurring in the perturbative developments of thermodynamic functions of neutral, polar, and ionized gases and of Green's functions of Euclidean quantum field theories (thus improving Weinberg's theorem), as also in variational approximations of intermolecular potentials. Our estimation method is based on the Hölder inequality which is an improvement over the mean value estimation method, employed by Riddell, Uhlenbeck, and Groeneveld, except for the hard-sphere gas, where both methods coincide. The method is applied so far only to individual graphs and not to thermodynamic functions.     

Estimates of general Mayer graphs III: Upper bounds obtained by means of spanningn-trees

We obtain computable upper bounds for any given Mayer graph withn root-points (orn-graph). These are products of integrals of the type \(\left( {\int {\left| {f_L } \right|^{z_{iL} y_i^{ - 1} } dx} } \right)^{yi} \) , where thez _iL andy _i are nonnegative real numbers whose sum overi is equal to 1. As a particular case, we obtain the canonical bounds (see their definition in Section 2.2): $$\left| {\int {\prod\limits_L {f_L \left( {x_i ,x_j } \right)dx_{n + 1} \cdot \cdot \cdot dx_{n + k} } } } \right| \leqslant \prod\limits_L {\left( {\int {\left| {f_L } \right|^{\alpha _L } dx} } \right)^{\alpha _L^{ - 1} } } $$ where theα _L 's satisfy the conditionα _L ≥1 for anyL, and ∑ _L α _L ^?1 =k (k is the number of variables that are integrated over). These bounds are finite for alln-graphs of neutral systems. We obtain also finite bounds for all irreduciblen-graphs of polar systems, and for certainn-graphs occurring in the theory of ionized systems. Finally, we give a sufficient condition for an arbitraryn-graph to be finite.     

The Hubbard model on a complete graph: exact analytical results

We derive the analytical expression of the ground state of the Hubbard model with unconstrained hopping at half filling and for arbitrary lattice sites.     

The Hubbard model on a complete graph: exact analytical results

We derive the analytical expression of the ground state of the Hubbard model with unconstrained hopping at half filling and for arbitrary lattice sites.     

Luminescent europium(III) and terbium(III) nicotinate complexes covalently linked to a 1,10-phenanthroline functionalised sol-gel glass

Sol-gel glasses with covalently linked lanthanide complexes are luminescent materials which can be processed at ambient temperatures, which have a good solubility and uniform distribution of the complexes in the host matrix. In this study, a luminescent terbium(III) complex was covalently coupled to an organic-inorganic hybrid material prepared by the sol-gel process. This was realised by use of nicotinate groups as the ligands for the terbium(III) ion. The [Tb(C₅H₄NCO₂)₃(phen)(H₂O)₂] complex was immobilised on the sol-gel glass matrix and showed a green photoluminescence upon irradiation with ultraviolet light. The nicotinate groups act as an antenna to absorb the incident light and channel the excitation energy to the terbium(III) ion. The sol-gel glass was also prepared for the corresponding europium(III) complex. In this case, excitation of the europium(III) ion was possible via both the nicotinate ligands and the 1,10-phenanthroline ligands. High-resolution luminescence and excitation spectra were recorded and the radiative lifetimes were measured.     

Intermolecular energy transfer between lanthanide complexes: 6. Influence of metal-to-ligand ratio on the transfer from terbium (III) to europium (III) complexes of L-histidine

The nature of the complexes formed between Tb³⁺ and L-histidine was investigated by making use of the fact that electronic energy can be absorbed by Tb³⁺/HIS complexes and subsequently transferred in a non-radiative manner to Eu³⁺/HIS complexes. The transfer was found to depend on the possible association of donor and quencher, and this information was used to outline the conditions under which the histidine complexes were found to self-associate. Supplementary information was obtained from measurements of the pH dependence of Tb³⁺ luminescence and differential absorbance spectra of Ho³⁺/HIS complexes. At low pH, the energy transfer occurred primarily by collisional means (implying the presence of monomeric complexes), whereas the transfer was governed by the formation of associated complexes at high pH.     

Intermolecular interaction and a concentration-quenching mechanism of phosphorescent Ir(III) complexes in a solid film

Solid-state self-quenching processes of highly efficient Ir(III) phosphorescent emitters are investigated by the measurement of thin film photoluminescence quantum efficiency and transient lifetime as a function of doping concentration in a host matrix. The radiative decay rate constant is found to be independent from the average distance between dopant molecules (R), and the concentration-quenching rate constant is shown to be dependent on R(-6). The quenching dependence on R strongly suggests that luminescent concentration quenching in a phosphorescent Ir(III) complex:host film is controlled by dipole-dipole deactivating interactions as described by the F?rster energy transfer model.