34,175 research outputs found

    Quantum Mechanics Lecture Notes. Selected Chapters

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    These are extended lecture notes of the quantum mechanics course which I am teaching in the Weizmann Institute of Science graduate physics program. They cover the topics listed below. The first four chapter are posted here. Their content is detailed on the next page. The other chapters are planned to be added in the coming months. 1. Motion in External Electromagnetic Field. Gauge Fields in Quantum Mechanics. 2. Quantum Mechanics of Electromagnetic Field 3. Photon-Matter Interactions 4. Quantization of the Schr\"odinger Field (The Second Quantization) 5. Open Systems. Density Matrix 6. Adiabatic Theory. The Berry Phase. The Born-Oppenheimer Approximation 7. Mean Field Approaches for Many Body Systems -- Fermions and Boson

    Positive Geometries of S-matrix without Color

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    In this note, we prove that the realization of associahedron discovered by Arkani-Hamed, Bai, He, and Yun (ABHY) is a positive geometry for tree-level S-matrix of scalars which have no color and which interact via cubic coupling. More in detail, we consider diffeomorphic images of the ABHY associahedron. The diffeomorphisms are linear maps parametrized by the right cosets of the Dihedral group on n elements. The set of all the boundaries associated with these copies of ABHY associahedron exhaust all the simple poles. We prove that the sum over the diffeomorphic copies of ABHY associahedron is a positive geometry and the total volume obtained by summing over all the dual associahedra is proportional to the tree-level S matrix of (massive or massless) scalar particles with cubic coupling. We then provide non-trivial evidence that the projection of the planar scattering forms parametrized by the Stokes polytope on these realizations of the associahedron leads to the tree-level amplitudes of scalar particles, which interact via quartic coupling. Our results build on ideas laid out in our previous works, leading to further evidence that a large class of positive geometries which are diffeomorphic to the ABHY associahedron defines an ``amplituhedron" for a tree-level S matrix of some local and unitary scalar theory. We also highlight a fundamental obstruction in applying these ideas to discover positive geometry for the one loop integrand when propagating states have no color.Comment: 33 Pages, 4 Figure

    Jack Derangements

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    For each integer partition λ⊢n\lambda \vdash n we give a simple combinatorial expression for the sum of the Jack character θαλ\theta^\lambda_\alpha over the integer partitions of nn with no singleton parts. For α=1,2\alpha = 1,2 this gives closed forms for the eigenvalues of the permutation and perfect matching derangement graphs, resolving an open question in algebraic graph theory. A byproduct of the latter is a simple combinatorial formula for the immanants of the matrix J−IJ-I where JJ is the all-ones matrix, which might be of independent interest. Our proofs center around a Jack analogue of a hook product related to Cayley's Ω\Omega--process in classical invariant theory, which we call the principal lower hook product

    Tonelli Approach to Lebesgue Integration

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    Leonida Tonelli devised an interesting and efficient method to introduce the Lebesgue integral. The details of this method can only be found in the original Tonelli paper and in an old italian course and solely for the case of the functions of one variable. We believe that it is woth knowing this method and here we present a complete account for functions of every number of variables

    Finding and Counting Patterns in Sparse Graphs

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    Nonparametric Two-Sample Test for Networks Using Joint Graphon Estimation

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    This paper focuses on the comparison of networks on the basis of statistical inference. For that purpose, we rely on smooth graphon models as a nonparametric modeling strategy that is able to capture complex structural patterns. The graphon itself can be viewed more broadly as density or intensity function on networks, making the model a natural choice for comparison purposes. Extending graphon estimation towards modeling multiple networks simultaneously consequently provides substantial information about the (dis-)similarity between networks. Fitting such a joint model - which can be accomplished by applying an EM-type algorithm - provides a joint graphon estimate plus a corresponding prediction of the node positions for each network. In particular, it entails a generalized network alignment, where nearby nodes play similar structural roles in their respective domains. Given that, we construct a chi-squared test on equivalence of network structures. Simulation studies and real-world examples support the applicability of our network comparison strategy.Comment: 25 pages, 6 figure

    Ideograph: A Language for Expressing and Manipulating Structured Data

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    We introduce Ideograph, a language for expressing and manipulating structured data. Its types describe kinds of structures, such as natural numbers, lists, multisets, binary trees, syntax trees with variable binding, directed multigraphs, and relational databases. Fully normalized terms of a type correspond exactly to members of the structure, analogous to a Church-encoding. Moreover, definable operations over these structures are guaranteed to respect the structures' equivalences. In this paper, we give the syntax and semantics of the non-polymorphic subset of Ideograph, and we demonstrate how it can represent and manipulate several interesting structures.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421

    A simplified lower bound for implicational logic

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    We present a streamlined and simplified exponential lower bound on the length of proofs in intuitionistic implicational logic, adapted to Gordeev and Haeusler's dag-like natural deduction.Comment: 31 page

    Maxflow-Based Bounds for Low-Rate Information Propagation over Noisy Networks

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    We study error exponents for the problem of low-rate communication over a directed graph, where each edge in the graph represents a noisy communication channel, and there is a single source and destination. We derive maxflow-based achievability and converse bounds on the error exponent that match when there are two messages and all channels satisfy a symmetry condition called pairwise reversibility. More generally, we show that the upper and lower bounds match to within a factor of 4. We also show that with three messages there are cases where the maxflow-based error exponent is strictly suboptimal, thus showing that our tightness result cannot be extended beyond two messages without further assumptions
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