Conditional probabilities via line arrangements and point configurations

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterisation of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables, however, in the presence of hidden variables, they contain higher degree minors. This leads to the study of the structure of determinantal hypergraph ideals whose decompositions can be understood in terms of point and line configurations in the projective space.Comment: 24 pages, 5 figure

Hitting time for quantum walks on the hypercube

Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as well as unitary evolution. We derive an expression for hitting time using superoperators, and numerically evaluate it for the discrete walk on the hypercube. The values found are compared to other analogues of hitting time suggested in earlier work. The dependence of hitting times on the type of unitary ``coin'' is examined, and we give an example of an initial state and coin which gives an infinite hitting time for a quantum walk. Such infinite hitting times require destructive interference, and are not observed classically. Finally, we look at distortions of the hypercube, and observe that a loss of symmetry in the hypercube increases the hitting time. Symmetry seems to play an important role in both dramatic speed-ups and slow-downs of quantum walks.Comment: 8 pages in RevTeX format, four figures in EPS forma

Error tolerance of two-basis quantum key-distribution protocols using qudits and two-way classical communication

We investigate the error tolerance of quantum cryptographic protocols using $d$-level systems. In particular, we focus on prepare-and-measure schemes that use two mutually unbiased bases and a key-distillation procedure with two-way classical communication. For arbitrary quantum channels, we obtain a sufficient condition for secret-key distillation which, in the case of isotropic quantum channels, yields an analytic expression for the maximally tolerable error rate of the cryptographic protocols under consideration. The difference between the tolerable error rate and its theoretical upper bound tends slowly to zero for sufficiently large dimensions of the information carriers.Comment: 10 pages, 1 figur

Tripartite to Bipartite Entanglement Transformations and Polynomial Identity Testing

We consider the problem of deciding if a given three-party entangled pure state can be converted, with a non-zero success probability, into a given two-party pure state through local quantum operations and classical communication. We show that this question is equivalent to the well-known computational problem of deciding if a multivariate polynomial is identically zero. Efficient randomized algorithms developed to study the latter can thus be applied to the question of tripartite to bipartite entanglement transformations

The Routing of Complex Contagion in Kleinberg's Small-World Networks

In Kleinberg's small-world network model, strong ties are modeled as deterministic edges in the underlying base grid and weak ties are modeled as random edges connecting remote nodes. The probability of connecting a node $u$ with node $v$ through a weak tie is proportional to $1/|uv|^\alpha$, where $|uv|$ is the grid distance between $u$ and $v$ and $\alpha\ge 0$ is the parameter of the model. Complex contagion refers to the propagation mechanism in a network where each node is activated only after $k \ge 2$ neighbors of the node are activated. In this paper, we propose the concept of routing of complex contagion (or complex routing), where we can activate one node at one time step with the goal of activating the targeted node in the end. We consider decentralized routing scheme where only the weak ties from the activated nodes are revealed. We study the routing time of complex contagion and compare the result with simple routing and complex diffusion (the diffusion of complex contagion, where all nodes that could be activated are activated immediately in the same step with the goal of activating all nodes in the end). We show that for decentralized complex routing, the routing time is lower bounded by a polynomial in $n$ (the number of nodes in the network) for all range of $\alpha$ both in expectation and with high probability (in particular, $\Omega(n^{\frac{1}{\alpha+2}})$ for $\alpha \le 2$ and $\Omega(n^{\frac{\alpha}{2(\alpha+2)}})$ for $\alpha > 2$ in expectation), while the routing time of simple contagion has polylogarithmic upper bound when $\alpha = 2$. Our results indicate that complex routing is harder than complex diffusion and the routing time of complex contagion differs exponentially compared to simple contagion at sweetspot.Comment: Conference version will appear in COCOON 201

Quantum walks with infinite hitting times

Hitting times are the average time it takes a walk to reach a given final vertex from a given starting vertex. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition, which for discrete time quantum walks is that the degeneracy of the evolution operator be greater than the degree of the graph. The set of initial states which give an infinite hitting time form a subspace. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. In the case of the discrete walk, if this condition is satisfied the walk will have infinite hitting times for any choice of a coin operator, and we give a class of graphs with infinite hitting times for any choice of coin. Hitting times are not very well-defined for continuous time quantum walks, but we show that the idea of infinite hitting-time walks naturally extends to the continuous time case as well.Comment: 28 pages, 3 figures in EPS forma

Quantum walks on quotient graphs

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup used to construct it. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A {\bf 74}, 042334 (2006)] that the hitting time can be infinite. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with fast hitting times correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.Comment: 18 pages, 7 figures in EPS forma

Grid-Obstacle Representations with Connections to Staircase Guarding

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Alaraajojen lihasten spastisuus ennen ja jälkeen avustetun polkuharjoittelun

Opinnäytetyön tavoitteena oli kerätä tietoa aivoverenkiertohäiriötä, selkäydinvauriota sekä CP-vammaa sairastavien neurologisten asiakkaiden spastisten alaraajojen lihasten spastisuuden aiheuttaman lihasaktivaation mahdollisesta muutoksesta ennen ja jälkeen avustetulla polkulaitteella suoritetun polkuharjoituksen. Tarkoituksena oli tuottaa tutkittua tietoa kyseisen terapiamuodon vaikutuksesta edellä mainittuja oireyhtymiä sairastavien kuntoutuksessa. Toimeksiantaja voi hyödyntää tuloksia suunnitellessaan ja arvioidessaan neurologisten asiakkaiden kuntoutuksessa käytettäviä terapiamuotoja. Lisäksi tarkoituksena oli tuottaa fysioterapia-alalle tietoa terapiamuodon vaikutuksesta alaraajojen spastisuuteen. Työn tekijät syvensivät työn kautta omaa ammattitaitoaan tulevaa ammattia varten. Opinnäytetyömme tutkimusongelmana oli miten polkulaitteella suoritettu 20 minuutin avustettu polkuliike vaikuttaa aivoverenkiertohäiriötä, selkäydinvauriota sekä CP-vammaa sairastavien neurologisten asiakkaiden spastisuuden aiheuttamaan alaraajojen lihasaktivaatioon. Opinnäytetyö toteutettiin tapaustutkimuksena, johon osallistui viisi tutkimushenkilöä. Tutkimuksen aineisto kerättiin määrällisin menetelmin, joita olivat elektromyografia (EMG), Modified Modified Ashworth Scale (MMAS) sekä kysymyslomake. EMG ja MMAS mittaukset suoritettiin yhtäaikaisesta ennen polkuharjoitusta ja sen jälkeen. Mittareilla saadut tulokset analysoitiin MegaWin-ohjelmalla ja Microsoft Excel-taulukkolaskentaohjelmalla. Tulokset on esitetty numeerisessa ja graafisessa muodossa. Tutkimuksesta saatujen tulosten mukaan spastisuuden aiheuttama lihasaktivaatio väheni polkuharjoittelun jälkeen jokaisessa mitatussa lihaksessa EMG- ja MMAS -mittareilla mitattuna. Myös kysymyslomakkeella saatujen tulosten mukaan polkuharjoittelun vaikutukset spastisuuteen ovat positiivisia. Näin ollen tutkimustulosten perusteella avustetulla polkuharjoittelulla oli lihasten spastisuutta alentava vaikutus. Pienen tutkimusjoukon johdosta tuloksia ei voi kuitenkaan yleistää, mutta ne ovat suuntaa-antavia.The aim of this thesis is to gather information on possible changes in the spasticity of the lower limb muscles before and after assisted cycling exercise in clients with stroke, spinal cord injury and cerebral palsy. The purpose of this thesis is to produce information about the effects of the assisted cycling exercise in rehabilitation with clients suffering from the above mentioned injuries. The commissioner, Kemijärven Fysikaalinen Hoitolaitos Ky, can benefit from the achieved results while planning the rehabilitation of neurological clients. The authors’ purpose is to generate knowledge on the effects of assisted cycling exercise in spasticity of the lower limb muscles for physiotherapy field to use. The authors benefit from the thesis by obtaining their own expertise for the upcoming profession. The research problem of this thesis was to discover how the 20-minute assisted cycling exercise affects the spasticity of the lower limbs muscles in clients with stroke, spinal cord injury and cerebral palsy. This thesis is a case study in which participated five study subjects. The research data was gathered with the following quantitative methods: Electromyography (EMG), Modified Modified Ashworth Scale (MMAS) and questionnaire. EMG and MMAS were administrated simultaneously before and after assisted cycling exercise. The results were analysed with MegaWin-program and Microsoft Excel Spreadsheet. The results are displayed in numerical and graphical form. The results of this thesis show that after the assisted cycling exercise the muscle activation caused by spasticity, previously measured by EMG and MMAS, was reduced in every tested muscle. According to results from the questionnaire the effects of assisted cycling exercise was also positive. Therefore, it could be said that assisted cycling exercise reduces the spasticity in lower limb muscles. Due to the limited amount of participant in the study group, the results cannot be generalised, nevertheless, they can be used as directional information

Thresholded Covering Algorithms for Robust and Max-Min Optimization

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".Comment: 24 page