34,175 research outputs found
Recommended from our members
Ensuring Access to Safe and Nutritious Food for All Through the Transformation of Food Systems
Quantum Mechanics Lecture Notes. Selected Chapters
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
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
For each integer partition we give a simple combinatorial
expression for the sum of the Jack character over the
integer partitions of with no singleton parts. For 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 where 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 --process in classical invariant theory,
which we call the principal lower hook product
Tonelli Approach to Lebesgue Integration
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
Nonparametric Two-Sample Test for Networks Using Joint Graphon Estimation
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
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
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
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
- …