5,979 research outputs found
Opening the system to the environment: new theories and tools in classical and quantum settings
The thesis is organized as follows. Section 2 is a first, unconventional, approach to the topic of EPs. Having grown interest in the topic of combinatorics and graph theory, I wanted to exploit its very abstract and mathematical tools to reinterpret something very physical, that is, the EPs in wave scattering. To do this, I build the interpretation of scattering events from a graph theory perspective and show how EPs can be understood within this interpretation. In Section 3, I move from a completely classical treatment to a purely quantum one. In this section, I consider two quantum resonators coupled to two baths and study their dynamics with local and global master equations. Here, the EPs are the key physical features used as a witness of validity of the master equation. Choosing the wrong master equation in the regime of interest can indeed mask physical and fundamental features of the system. In Section 4, there are no EPs. However I transition towards a classical/quantum framework via the topic of open systems. My main contribution in this work is the classical stochastic treatment and simulation of a spin coupled to a bath. In this work, I show how a natural quantum--to--classical transition occurs at all coupling strengths when certain limits of spin length are taken. As a key result, I also show how the coupling to the environment in this stochastic framework induces a classical counterpart to quantum coherences in equilibrium. After this last topic, in Section 5, I briefly present the key features of the code I built (and later extended) for the latter project. This, in the form of a Julia registry package named SpiDy.jl, has seen further applications in branching projects and allows for further exploration of the theoretical framework. Finally, I conclude with a discussion section (see Sec. 5) where I recap the different conclusions gathered in the previous sections and propose several possible directions.Engineering and Physical Sciences Research Council (EPSRC
UMSL Bulletin 2023-2024
The 2023-2024 Bulletin and Course Catalog for the University of Missouri St. Louis.https://irl.umsl.edu/bulletin/1088/thumbnail.jp
New techniques for integrable spin chains and their application to gauge theories
In this thesis we study integrable systems known as spin chains and their applications to the study of the AdS/CFT duality, and in particular to N “ 4 supersymmetric Yang-Mills theory (SYM) in four dimensions.First, we introduce the necessary tools for the study of integrable periodic spin chains, which are based on algebraic and functional relations. From these tools, we derive in detail a technique that can be used to compute all the observables in these spin chains, known as Functional Separation of Variables. Then, we generalise our methods and results to a class of integrable spin chains with more general boundary conditions, known as open integrable spin chains.In the second part, we study a cusped Maldacena-Wilson line in N “ 4 SYM with insertions of scalar fields at the cusp, in a simplifying limit called the ladders limit. We derive a rigorous duality between this observable and an open integrable spin chain, the open Fishchain. We solve the Baxter TQ relation for the spin chain to obtain the exact spectrum of scaling dimensions of this observable involving cusped Maldacena-Wilson line.The open Fishchain and the application of Functional Separation of Variables to it form a very promising road for the study of the three-point functions of non-local operators in N “ 4 SYM via integrability
A Local-to-Global Theorem for Congested Shortest Paths
Amiri and Wargalla (2020) proved the following local-to-global theorem in
directed acyclic graphs (DAGs): if is a weighted DAG such that for each
subset of 3 nodes there is a shortest path containing every node in ,
then there exists a pair of nodes such that there is a shortest
-path containing every node in .
We extend this theorem to general graphs. For undirected graphs, we prove
that the same theorem holds (up to a difference in the constant 3). For
directed graphs, we provide a counterexample to the theorem (for any constant),
and prove a roundtrip analogue of the theorem which shows there exists a pair
of nodes such that every node in is contained in the union of a
shortest -path and a shortest -path.
The original theorem for DAGs has an application to the -Shortest Paths
with Congestion (()-SPC) problem. In this problem, we are given a
weighted graph , together with node pairs ,
and a positive integer . We are tasked with finding paths such that each is a shortest path from to , and every
node in the graph is on at most paths , or reporting that no such
collection of paths exists.
When the problem is easily solved by finding shortest paths for each
pair independently. When , the -SPC problem recovers
the -Disjoint Shortest Paths (-DSP) problem, where the collection of
shortest paths must be node-disjoint. For fixed , -DSP can be solved in
polynomial time on DAGs and undirected graphs. Previous work shows that the
local-to-global theorem for DAGs implies that -SPC on DAGs whenever
is constant. In the same way, our work implies that -SPC can be
solved in polynomial time on undirected graphs whenever is constant.Comment: Updated to reflect reviewer comment
2023-2024 Boise State University Undergraduate Catalog
This catalog is primarily for and directed at students. However, it serves many audiences, such as high school counselors, academic advisors, and the public. In this catalog you will find an overview of Boise State University and information on admission, registration, grades, tuition and fees, financial aid, housing, student services, and other important policies and procedures. However, most of this catalog is devoted to describing the various programs and courses offered at Boise State
On Hypergraph Supports
Let be a hypergraph. A support is a graph
on such that for each , the subgraph of induced on the
elements in is connected. In this paper, we consider hypergraphs defined on
a host graph. Given a graph , with ,
and a collection of connected subgraphs of , a primal support
is a graph on such that for each , the
induced subgraph on vertices is connected. A \emph{dual support} is a graph on
s.t. for each , the induced subgraph
is connected, where . We present
sufficient conditions on the host graph and hyperedges so that the resulting
support comes from a restricted family.
We primarily study two classes of graphs: If the host graph has genus
and the hypergraphs satisfy a topological condition of being
\emph{cross-free}, then there is a primal and a dual support of genus at most
. If the host graph has treewidth and the hyperedges satisfy a
combinatorial condition of being \emph{non-piercing}, then there exist primal
and dual supports of treewidth . We show that this exponential blow-up
is sometimes necessary. As an intermediate case, we also study the case when
the host graph is outerplanar. Finally, we show applications of our results to
packing and covering, and coloring problems on geometric hypergraphs
Fixed-Parameter Algorithms for Computing RAC Drawings of Graphs
In a right-angle crossing (RAC) drawing of a graph, each edge is represented
as a polyline and edge crossings must occur at an angle of exactly ,
where the number of bends on such polylines is typically restricted in some
way. While structural and topological properties of RAC drawings have been the
focus of extensive research, little was known about the boundaries of
tractability for computing such drawings. In this paper, we initiate the study
of RAC drawings from the viewpoint of parameterized complexity. In particular,
we establish that computing a RAC drawing of an input graph with at most
bends (or determining that none exists) is fixed-parameter tractable
parameterized by either the feedback edge number of , or plus the vertex
cover number of .Comment: Accepted at GD 202
Near-optimal quantum strategies for nonlocal games, approximate representations, and BCS algebras
Quantum correlations can be viewed as particular abstract states on the tensor product of
operator systems which model quantum measurement scenarios. In the paradigm of nonlocal games,
this perspective illustrates a connection between optimal strategies and certain
representations of a finitely presented -algebra affiliated with the nonlocal game.
This algebraic interpretation of quantum correlations arising from nonlocal games has been
valuable in recent years. In particular, the connection between representations and
strategies has been useful for investigating and separating the various frameworks for
quantum correlation as well as in developing cryptographic primitives for untrusted
quantum devices. However to make use of this correspondence in a realistic setting one
needs mathematical guarantees that this correspondence is robust to noise.
We address this issue by considering the situation where the correlations are not ideal.
We show that near-optimal finite-dimensional quantum strategies using arbitrary quantum
states are approximate representations of the affiliated nonlocal game algebra for
synchronous, boolean constraint systems (BCS), and XOR nonlocal games. This result
robustly extends the correspondence between optimal strategies and finite-dimensional
representations of the nonlocal game algebras for these prominent classes of nonlocal
games. We also show that finite-dimensional approximate representations of these nonlocal
game algebras are close to near-optimal strategies employing a maximally entangled state.
As a corollary, we deduce that near-optimal quantum strategies are close to a near-optimal
quantum strategy using a maximally entangled state.
A boolean constraint system is -definable from another boolean constraint system
if there is a -formula defining over . There is such a -formula if all the constraints in can be defined via conjunctions of relations in using additional boolean variables if needed. We associate a finitely presented -algebra, called a BCS algebra, to each boolean constraint system . We show that -definability can be interpreted algebraically as -homomorphisms between BCS algebras. This allows us to classify boolean constraint languages and separations between various generalized notions of satisfiability. These types of satisfiability
are motivated by nonlocal games and the various frameworks for quantum correlations and
state-independent contextuality. As an example, we construct a BCS that is -satisfiable in the sense that it has a representation on a Hilbert space but has no tracial
representations, and thus no interpretation in terms of commuting operator correlations
The separating variety for 2x2 matrix invariants
We study the action of the group G = GL_2(C) of invertible matrices over the complex numbers on the complex vector space V of n-tuples of 2x2 matrices. The algebra of invariants C[V]^G for this action is well-known, and has dimension 4n-3 and minimum generating set E_n with cardinality 1/6(n^3+11n). In recent work, Kaygorodov, Lopatin and Popov showed that this generating set is also a minimal separating set by inclusion, i.e. no proper subset is a separating set. This does not mean it has smallest possible cardinality among all separating sets. We show that if S is a separating set for C[V]^G then |S| is at least 5n-5. In particular for n=3, the set E_n is indeed of minimal cardinality, but for n>3 may not be so. We then show that a smaller separating set does in fact exist for n>4, We also prove similar results for the left-right action of SL_2(C)xSL_2(C) on V
Hyperkähler structure of bow varieties
In this thesis we study Cherkis bow varieties and its description in terms of linear flows
on the Jacobian variety of certain spectral curve. We describe explicitly the bow variety
of a deformed instanton moduli space over Taub-NUT, i.e. the bow variety consisting
of one arrow and interval with r l-points, and find a spectral description in terms of
conditions on certain divisors. We find an asymptotic metric for the bow variety by
constructing a model space using twistor methods and showing that the corresponding
metric is asymptotically close to the one of the bow variety.In dieser Dissertation untersuchen wir Cherkis Bogenvarietäten und deren Beschreibung
als lineare Flüsse auf der Jacobischen Varietät einer bestimmten Spektralkurve.
Wir beschreiben explizit die Bogenvarietät eines deformierten Instanton-Modulraums
über der Taub-NUT-Mannigfaltigkeit, das heißt wir beschreiben die Bogenvarietät, die
aus einem Pfeil und einem Intervall mit r l-Punkte besteht, und finden eine spektrale
Darstellung in Form von Bedingungen an spezielle Divisoren. Wir finden eine
asymptotische Metrik für diese Bogenvarietät, indem wir mittels Methoden aus der
Twistortheorie einen Modellraum konstruieren und zeigen, dass die zugehörige Metrik
asymptotisch nah an der eigentlichen Metrik der Bogenvarietät liegt
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