Mathematical Physics

arXiv:math-ph

Articles in mathematical physics which are of interest to both mathematicians and physicists.

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arXiv:math-ph

Articles in mathematical physics which are of interest to both mathematicians and physicists.

Looking for a broader view? This category is part of:

Mathematical Physics

Trending in Mathematical Physics

2603.28975

Remarks on "Further comments on "Rebuttal of "Refutation of "Comment on "Reply to "Comments on "A genuinely natural information measure" " " " " " "

It's a bit tedious, but as John Doe and Jean Roe have insisted on offering further comments on our comprehensive refutation of the former's already tiringly obstinate advances, we feel compelled to review their not even wrong opinions once again, hoping to put some sense back into the discourse.

2603.28975

Mar 2026Mathematical Physics

3,164 papers

From BV-BFV Quantization to Reshetikhin-Turaev Invariants

We propose a program for bridging the gap between the perturbative BV-BFV quantization of Chern-Simons theory and the non-perturbative Reshetikhin-Turaev (RT) invariants of 3-manifolds, passing through factorization homology of $\mathbb{E}_n$-algebras and the derived algebraic geometry of character stacks. We conjecture that the modular tensor category underlying the RT construction arises as the $\mathbb{E}_2$-category from BV-BFV quantization of Chern-Simons theory on the disk, with the derived character stack $\mathrm{Loc}_G(Σ)$ and its shifted symplectic structure mediating the proposed identification. We formulate seven conjectures, including a main conjecture asserting natural equivalence of the BV-BFV and RT constructions as (3-2-1)-extended topological quantum field theories, develop a proof strategy via deformation quantization of shifted symplectic stacks, and clarify the role of $\mathbb{E}_n$-Koszul duality in translating between perturbative and non-perturbative data. Supporting evidence is examined in the abelian, low-genus, and Seifert fibered cases. Connections to resurgence, categorification, and the geometric Langlands program are discussed as further motivation, though significant technical gaps remain open.

2604.04556Apr 2026

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Mathematical and numerical studies on ground states of the extended Gross-Pitaevskii equation with the Lee-Huang-Yang correction

We study the ground states of the extended Gross--Pitaevskii equation with the Lee--Huang--Yang correction from both theoretical and numerical perspectives. Starting from the three-dimensional model, we derive reduced one- and two-dimensional equations through nondimensionalization and dimensional reduction. We establish existence and nonexistence results for ground states in different spatial dimensions, both in free space and under confining external potentials. For the numerical computation of ground states, we propose a normalized gradient flow method with a Lagrange multiplier. The numerical results show how the model parameters affect the ground-state profiles, and reveal different regimes in the free-space parameter plane, including no-ground-state, soliton-like, and droplet-like regions. We also introduce a simple flat-top approximation for the droplet regime and present two- and three-dimensional computations to illustrate more general localized structures.

2604.04460Apr 2026

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2604.04173

Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part II: A Model from Local QFT

This paper completes a previous work by constructing a class of positive-energy relativistic spatial localization observables in Minkowski spacetime within quantum field theory, using the stress-energy-momentum tensor smeared with suitable test functions. For each timelike direction, the construction yields a family of positive operator-valued measures (POVMs) on spacelike hypersurfaces, well defined on every n-particle sector and satisfying a natural relativistic causality condition excluding superluminal propagation of detection probabilities. These observables arise from local or quasi-local field-theoretic quantities and provide a rigorous version of earlier heuristic proposals. In the one-particle sector, the construction reduces to the observable introduced previously, and its first moment reproduces the Newton-Wigner position operator under suitable normalization conditions. Because the normally ordered stress-energy-momentum tensor is not positive on the full Fock space, as implied by the Reeh-Schlieder theorem, we study quantum energy inequalities and derive lower bounds controlling deviations from positivity. This leads to regularized families of positive operators approximating the localization effects. We also construct conditional localization observables for finite laboratories using modified local energy operators and their Friedrichs self-adjoint extensions. Using Haag duality and Kadison's result on affiliation, we show that the resulting conditional POVMs belong to local von Neumann algebras and therefore commute for causally separated regions, in agreement with the Araki-Haag-Kastler framework. These results support the view that commutativity of localization observables is recovered at the level of conditional measurements in finite spacetime regions.

2604.04173Apr 2026

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2604.04010

Sharp upper bounds for the density of relativistic atoms: Noninteracting case

We prove an optimal upper bound for the density of electrons of an infinite Bohr atom (no electron-electron interactions) described by the relativistic operators of Chandrasekhar and Dirac. We also consider densities in each angular momentum channel separately.

2604.04010Apr 2026

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2604.03963

Two Approximate Solutions of the Ornstein-Zernike (OZ) Integral Equation

This thesis explores the evolution of liquid-state theories based on the Ornstein-Zernike (OZ) equation, summarizing the foundational methods developed by Baxter, Lebowitz, Wertheim, and others. A unifying feature of these approaches is their shared analytical strategy: by introducing an intermediate function with specific mathematical properties, they effectively decouple the total correlation function and the direct correlation function. This allows the OZ equation to be solved within specific spatial intervals by exploiting regions where either the total or direct correlation function is known. Furthermore, this work presents a comprehensive derivation of analytical solutions to the OZ integral equation under the hard-sphere model. This includes applications of the Percus-Yevick (PY) approximation for both single- and multi-component systems, as well as the Mean Spherical Approximation (MSA) for systems of charged hard spheres. Building upon these analytical solutions, explicit expressions for macroscopic thermodynamic properties, such as the equation of state and activity coefficients, are rigorously derived. These derivations extensively employ advanced mathematical techniques, including Fourier transforms, complex analysis, and integral equation theory. Notably, many of the intermediate analytical steps and thermodynamic derivations presented herein offer a level of clarity and completeness previously absent from the existing literature.

2604.03963Apr 2026

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Ergodic Schrodinger operators on the Bethe lattice and a modified Thouless formula

The main result of this paper is a modified Thouless formula relating the density of states for ergodic Schrodinger operators on the Bethe lattice to the Lyapunov exponent. The modified Thouless formula consists of a Thouless-like term, involving the density of states, and a remainder term. The remainder term vanishes when the connectivity $κ$ equals one, yielding the usual Thouless formula for ergodic Schrodinger operators on $\mathbb{Z}$. We prove the remainder term is nontrivial for $κ\geq 2$. We also discuss the automorphism group of the Bethe lattice and its relation to ergodic Schrodinger operators. In particular, we clarify the use of the multiparameter noncommutative ergodic theorem in evaluating the limit of Green's functions along certain paths.

2604.03880Apr 2026

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2604.03801

Variational formulation of a general dissipative fluid system with differential forms

This work is devoted to the study of dissipative fluid systems, through the lens of a geometric variational formulation. Building upon previous works extending Hamilton's principle to non-equilibrium thermodynamics, the present method incorporates an arbitrary number of additional variables expressed as differential forms. Dissipation sources, thermodynamic flux closures, and their associated boundary conditions are also all expressed in this differential-form framework. The resulting equations are consistent with the fundamental laws of thermodynamics, namely conservation of total energy and positive entropy production. Onsager's principle is also given a simple formulation, while Curie's principle is revisited within this geometric setting through the lens of representation theory. It is shown that this general framework encompasses physically relevant models, such as multi-species magnetohydrodynamics (MHD) equations with intricate dissipation mechanisms.

2604.03801Apr 2026

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2604.03729

Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part I: A General Analysis

We investigate whether commutativity is necessary to represent relativistic locality for localization observables of relativistic quantum systems in Minkowski spacetime. A well known no-go theorem by Halvorson and Clifton shows that commutativity of localization effects for causally separated regions is incompatible with other seemingly natural assumptions about spatial localization. Since commutativity is taken to represent locality in the Araki-Haag-Kastler framework of QFT, this prompts the question whether it follows from more elementary locality principles of quantum theory. Using Busch's operational analysis in terms of no-signaling and relativistic consistency, we argue that for particle-like systems commutativity is not implied by these principles. Assuming a natural local detectability principle, elementary localization observables are not localized in arbitrarily small spacetime neighborhoods of the relevant spatial regions, but rather in regions containing the entire rest space (a Cauchy surface) on which the measurement is performed. This reflects the particle picture itself, where localization occurs at a unique place on a rest space filled with ideal detectors, and therefore does not directly conflict with the Araki-Haag-Kastler notion of locality. We also show that commutativity and localization can coexist for less idealized localization procedures. To this end, we introduce conditional localization POVMs associated with bounded spatial regions interpreted as laboratories. By the gentle measurement lemma, these observables describe conditional localization probabilities and can, in principle, satisfy commutativity for causally separated laboratories. They may therefore be represented by local observables in the Araki-Haag-Kastler sense. Explicit examples will be presented in forthcoming work within local QFT.

2604.03729Apr 2026

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Perfect fluid equations with nonrelativistic conformal symmetry: Exact solutions

The group-theoretic approach is used to construct exact solutions to perfect fluid equations invariant under the Schrodinger group, or the l-conformal Galilei group, or the Lifshitz group. In each respective case, the velocity vector field looks similar to the Bjorken flow. It is shown that one can reach an arbitrarily high density (and hence pressure) for a short period of time by adjusting the value of l and other free parameters available.

2604.03621Apr 2026

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Scattering of TE and TM waves by inhomogeneities of a 2D material, low-frequency behavior of the scattering amplitude, and low-frequency invisibility

The propagation of the transverse electric (TE) and transverse magnetic (TM) waves in an effectively two-dimensional (2D) isotropic medium is described by Bergmann's equation of acoustics. We develop a dynamical formulation of the stationary scattering of these waves and explore its application in the study of the low-frequency behavior of the scattering data. Specifically, we introduce a suitable notion of fundamental transfer matrix for TE and TM waves in 2D. This is an integral operator $\widehat{\mathbf{M}}$ that carries the information about the scattering properties of the medium and admits a Dyson series expansion involving a non-Hermitian Hamiltonian operator. For situations where the inhomogeneities of the medium are confined to a layer of thickness $\ell$, we use the Dyson series for $\widehat{\mathbf{M}}$ to construct the series expansion of the scattering amplitude in powers of $k\ell$, where $k$ is the incident wavenumber. We derive analytic expressions for the leading- and next-to-leading-order terms of this series, verify the effectiveness of their application to a class of exactly solvable models, and use them to study low-frequency invisibility. In particular, we develop a low-frequency cloaking scheme which is applicable for both TE and TM waves. Our results have immediate applications in the study of low-frequency scattering of acoustic waves in a 2D fluid as these waves are also described by Bergmann's equation.

2604.03149Apr 2026

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2604.03051

Higher order derivative moments of CUE characteristic polynomials and the Riemann zeta function

We use random matrix theory for the Circular Unitary Ensemble (CUE) to study moments of derivatives of the Riemann zeta function shifted a small distance from the critical line. The corresponding CUE moments are studied in the limit of large matrix size in two regimes: when the spectral parameter is (1) suitably far inside the unit disc, and (2) at a small distance from the unit circle. In case (1), we obtain an asymptotic formula as a combinatorial sum over contingency tables, while in case (2) we obtain a sum over certain determinants with multiplicative coefficients given by Kostka numbers. The latter result is also valid exactly on the unit circle. Then, we consider the analogous problem for mean values of derivatives of the zeta function with suitable shifts. Assuming the Lindelöf hypothesis, we show that this mean value gives rise to the same sum over contingency tables obtained in the CUE. For sufficiently low-order moments, we establish this result unconditionally.

2604.03051Apr 2026

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2604.03012

Cartan connections for an infinite family of integrable vortices

An infinite family of integrable vortex equations is studied and related to the Cartan geometry of the underlying Riemann surfaces. This Cartan picture gives an interpretation of the vortex equations as the flatness of a non-Abelian connection. Solutions of the vortex equations also give rise to magnetic zero-modes for a certain Dirac operator on the lifted geometry. The family of integrable vortex equations is parametrised by a positive number $n$, that is equal to unity in the standard case and an integer in the case of polynomial vortex equations; finally, it may be extended to any positive real number.

2604.03012Apr 2026

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2604.02957

On a stability of time-optimal version of the Boundary Control method

Let $Ω$ be a Riemannian manifold with boundary. The time-optimal version of the BC-method determines the parameters in the $T$-neigh\-bor\-hood $Ω^T$ of $\partialΩ$ from the boundary observations (response operator) $R^{2T}$ on the time segment $[0,2T]$. It visualizes the invisible waves supported in $Ω^T$, by reconstructing the operator $W^T$ that creates these waves. The visualization is based on the triangular factorization of the operator $C^T:=W^{T\,*}W^T$ in the form $C^T:=F^{T\,*}F^T$ with a factor $F^T=U^{T}W^T$, where $U^T$ is a unitary operator. The factorization $C^T\mapsto F^T$ has certain continuity properties, due to which the time-optimal reconstruction $R^{2T}\mapsto C^T\mapsto F^T\mapsto W^T$ turns out to be continuous (stable) in the sense of relevant operator topologies (convergences). As an example, determination of the potential $q$ in the wave equation $u_{tt}-Δu+qu=0$ from $R^{2T}$ is considered. We show that $R^{2T}_j\to R^{2T}$ implies $q_j\to q$ in $H^{-2}(Ω^T)$. However, the question of quantitative estimates of stability (the rate of convergence) remains open.

2604.02957Apr 2026

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Dimensional consistency in fractional differential equations with non singular kernels

The purpose of this article is to address the issues of dimensional consistency that arise in the process of replacing the ordinary time derivative operator by a fractional derivative operator in order to write a fractional differential equation. We show that by performing a simple change of variables fulfilling certain conditions ensures the consistency in physical dimensions for fractional differential equations with non singular kernels. An example of the proposed method is given.

2604.02432Apr 2026

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2604.02297

Commutator Estimates for Low-Temperature Fermi Gases

We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.

2604.02297Apr 2026

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Tensor invariants for multipartite entanglement classification

Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.

2604.02269Apr 2026

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2604.02013

A Rigorous Functional-Integral Construction of Toral Chern-Simons Theory

We construct the functional integral of Abelian Chern-Simons theory with toral gauge group $\mathbb T=\mathfrak t/Λ\cong U(1)^n$ at level $K$, where $K:Λ\timesΛ\to\mathbb Z$ is an even, integral, nondegenerate symmetric bilinear form, by exact zeta-regularized Gaussian evaluation of the formal quotient integral over connections modulo gauge. For closed $3$-manifolds, this yields a topological invariant; for manifolds with boundary, the relative functional integral produces the canonical boundary state. The resulting theory satisfies the required axioms of a $(2+1)$-dimensional TQFT.

2604.02013Apr 2026

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Functional relations in renormalization group methods for a class of ordinary differential equations

We develop a renormalization group (RG)-based perturbation scheme for a class of ordinary differential equations, including first-order systems with semisimple or nilpotent linear parts, as well as scalar higher-order equations. The key observation is that the secular coefficients arising in naive perturbation theory satisfy an exact functional relation. This yields, in a unified manner, several fundamental features of the RG method: the renormalized amplitudes satisfy a closed functional relation with a group-like structure, the RG equation governing their slow dynamics is obtained directly, the absence of secular terms is ensured to all orders, and the relation between bare and renormalized amplitudes admits an explicit inversion. The results extend earlier ones for second-order scalar equations.

2604.01744Apr 2026

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Edge localization and Lifshitz tails for graphs with Ahlfors regular volume growth

In this work, we study the Anderson model on graphs with Ahlfors $α$-regular volume growth. We show that, under mild regularity assumptions of the random distribution, Lifshitz-tail type estimates near the bottom of the spectrum lead to exponential decay of fractional moments of the Green's function and thus spectral and dynamical localization at low energies. This generalizes the result of [4] from the lattice $\mathbb{Z}^d$ to Ahlfors $α$-regular graphs. In addition, we establish Lifshitz tail estimates for the integrated density of states, with the Lifshitz exponent determined by the ratio of the volume growth rate and the random walk dimension of the underlying graphs, under certain assumptions on low lying eigenvalues of the Dirichlet and Neumann Laplacian on the graph. As an application, we verify all conditions on the Sierpinski gasket graph and obtain that, under mild regularity assumptions of the random distribution, for any fixed disorder, the Anderson model on the Sierpinski gasket graph has pure point spectrum and exhibits strong dynamical localization near the bottom of the spectrum.

2604.01584Apr 2026

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2604.01424

A Note on the Resolvent Algebra and Functional Integral Approach to the Free Bose Einstein Condensation

In this paper, we present a systematic description of the structure of Bose-Einstein condensation (BEC) in the free Bose gas from the viewpoint of the correspondence between the operator-algebraic formulation based on the resolvent algebra and the functional integral representation. By clarifying the representation-theoretic structure of finite-temperature BEC states and rigorously analyzing the correspondence between their direct integral decomposition and the ergodic decomposition of the associated probability measures, we provide a framework in which general features of phase transitions-such as the emergence of order parameters, the decomposition of states, and clustering properties-are explicitly described using BEC in the free Bose gas as a concrete example. Furthermore, we construct in detail the correspondence between the decomposition of measures in the functional integral approach and that of operator-algebraic representations, thereby establishing the equivalence between the probabilistic and algebraic aspects, and providing a guiding principle for isolating the essential structures by disentangling the additional mathematical complications arising from the treatment of infrared singularities in interacting systems. These results lay a foundation for the rigorous analysis of phase transitions in non-relativistic constructive quantum field theory and quantum statistical mechanics, and serve as a starting point for extensions to interacting models.

2604.01424Apr 2026

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Page 1 of 159

Mathematical Physics Papers (math-ph) - ScienceStack

3,164 papers

From BV-BFV Quantization to Reshetikhin-Turaev Invariants

2604.04556Apr 2026

View

Mathematical and numerical studies on ground states of the extended Gross-Pitaevskii equation with the Lee-Huang-Yang correction

2604.04460Apr 2026

View

2604.04173

Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part II: A Model from Local QFT

2604.04173Apr 2026

View

2604.04010

Sharp upper bounds for the density of relativistic atoms: Noninteracting case

2604.04010Apr 2026

View

2604.03963

Two Approximate Solutions of the Ornstein-Zernike (OZ) Integral Equation

2604.03963Apr 2026

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Ergodic Schrodinger operators on the Bethe lattice and a modified Thouless formula

2604.03880Apr 2026

View

2604.03801

Variational formulation of a general dissipative fluid system with differential forms

2604.03801Apr 2026

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2604.03729

Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part I: A General Analysis

2604.03729Apr 2026

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Perfect fluid equations with nonrelativistic conformal symmetry: Exact solutions

2604.03621Apr 2026

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Scattering of TE and TM waves by inhomogeneities of a 2D material, low-frequency behavior of the scattering amplitude, and low-frequency invisibility

2604.03149Apr 2026

View

2604.03051

Higher order derivative moments of CUE characteristic polynomials and the Riemann zeta function

2604.03051Apr 2026

View

2604.03012

Cartan connections for an infinite family of integrable vortices

2604.03012Apr 2026

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2604.02957

On a stability of time-optimal version of the Boundary Control method

2604.02957Apr 2026

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Dimensional consistency in fractional differential equations with non singular kernels

2604.02432Apr 2026

View

2604.02297

Commutator Estimates for Low-Temperature Fermi Gases

2604.02297Apr 2026

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Tensor invariants for multipartite entanglement classification

2604.02269Apr 2026

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2604.02013

A Rigorous Functional-Integral Construction of Toral Chern-Simons Theory

2604.02013Apr 2026

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Functional relations in renormalization group methods for a class of ordinary differential equations

2604.01744Apr 2026

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Edge localization and Lifshitz tails for graphs with Ahlfors regular volume growth

2604.01584Apr 2026

View

2604.01424

A Note on the Resolvent Algebra and Functional Integral Approach to the Free Bose Einstein Condensation

2604.01424Apr 2026

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Page 1 of 159