5,979 research outputs found

    Opening the system to the environment: new theories and tools in classical and quantum settings

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    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

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    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

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    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

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    Amiri and Wargalla (2020) proved the following local-to-global theorem in directed acyclic graphs (DAGs): if GG is a weighted DAG such that for each subset SS of 3 nodes there is a shortest path containing every node in SS, then there exists a pair (s,t)(s,t) of nodes such that there is a shortest stst-path containing every node in GG. 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 (s,t)(s,t) of nodes such that every node in GG is contained in the union of a shortest stst-path and a shortest tsts-path. The original theorem for DAGs has an application to the kk-Shortest Paths with Congestion cc ((k,ck,c)-SPC) problem. In this problem, we are given a weighted graph GG, together with kk node pairs (s1,t1),…,(sk,tk)(s_1,t_1),\dots,(s_k,t_k), and a positive integer c≤kc\leq k. We are tasked with finding paths P1,…,PkP_1,\dots, P_k such that each PiP_i is a shortest path from sis_i to tit_i, and every node in the graph is on at most cc paths PiP_i, or reporting that no such collection of paths exists. When c=kc=k the problem is easily solved by finding shortest paths for each pair (si,ti)(s_i,t_i) independently. When c=1c=1, the (k,c)(k,c)-SPC problem recovers the kk-Disjoint Shortest Paths (kk-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed kk, kk-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 (k,c)(k,c)-SPC on DAGs whenever k−ck-c is constant. In the same way, our work implies that (k,c)(k,c)-SPC can be solved in polynomial time on undirected graphs whenever k−ck-c is constant.Comment: Updated to reflect reviewer comment

    2023-2024 Boise State University Undergraduate Catalog

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    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

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    Let H=(X,E)\mathcal{H}=(X,\mathcal{E}) be a hypergraph. A support is a graph QQ on XX such that for each E∈EE\in\mathcal{E}, the subgraph of QQ induced on the elements in EE is connected. In this paper, we consider hypergraphs defined on a host graph. Given a graph G=(V,E)G=(V,E), with c:V→{r,b}c:V\to\{\mathbf{r},\mathbf{b}\}, and a collection of connected subgraphs H\mathcal{H} of GG, a primal support is a graph QQ on b(V)\mathbf{b}(V) such that for each H∈HH\in \mathcal{H}, the induced subgraph Q[b(H)]Q[\mathbf{b}(H)] on vertices b(H)=H∩c−1(b)\mathbf{b}(H)=H\cap c^{-1}(\mathbf{b}) is connected. A \emph{dual support} is a graph Q∗Q^* on H\mathcal{H} s.t. for each v∈Xv\in X, the induced subgraph Q∗[Hv]Q^*[\mathcal{H}_v] is connected, where Hv={H∈H:v∈H}\mathcal{H}_v=\{H\in\mathcal{H}: v\in H\}. 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: (1)(1) If the host graph has genus gg 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 gg. (2)(2) If the host graph has treewidth tt and the hyperedges satisfy a combinatorial condition of being \emph{non-piercing}, then there exist primal and dual supports of treewidth O(2t)O(2^t). 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

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    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 90∘90^\circ, 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 GG with at most bb bends (or determining that none exists) is fixed-parameter tractable parameterized by either the feedback edge number of GG, or bb plus the vertex cover number of GG.Comment: Accepted at GD 202

    Near-optimal quantum strategies for nonlocal games, approximate representations, and BCS algebras

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    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 BB is pppp-definable from another boolean constraint system B′B' if there is a pppp-formula defining BB over B′B'. There is such a pppp-formula if all the constraints in BB can be defined via conjunctions of relations in B′B' using additional boolean variables if needed. We associate a finitely presented ∗*-algebra, called a BCS algebra, to each boolean constraint system BB. We show that pppp-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 C∗C^*-satisfiable in the sense that it has a representation on a Hilbert space HH but has no tracial representations, and thus no interpretation in terms of commuting operator correlations

    The separating variety for 2x2 matrix invariants

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    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

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    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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