15,504 research outputs found
Detecting the prime divisors of the character degrees and the class sizes by a subgroup generated with few elements
We prove that every finite group G contains a three-generated subgroup H with the following property: a prime p divides the degree of an irreducible character of G if and only if it divides the degree of an irreducible character of H: There is no analogous result for the prime divisors of the sizes of the conjugacy classes
The independence graph of a finite group
Given a finite group we denote by the graph whose vertices
are the elements and where two vertices and are adjacent if there
exists a minimal generating set of containing and We prove that
is connected and classify the groups for which is a
planar graph
Freely Scalable Quantum Technologies using Cells of 5-to-50 Qubits with Very Lossy and Noisy Photonic Links
Exquisite quantum control has now been achieved in small ion traps, in
nitrogen-vacancy centres and in superconducting qubit clusters. We can regard
such a system as a universal cell with diverse technological uses from
communication to large-scale computing, provided that the cell is able to
network with others and overcome any noise in the interlinks. Here we show that
loss-tolerant entanglement purification makes quantum computing feasible with
the noisy and lossy links that are realistic today: With a modestly complex
cell design, and using a surface code protocol with a network noise threshold
of 13.3%, we find that interlinks which attempt entanglement at a rate of 2MHz
but suffer 98% photon loss can result in kilohertz computer clock speeds (i.e.
rate of high fidelity stabilizer measurements). Improved links would
dramatically increase the clock speed. Our simulations employed local gates of
a fidelity already achieved in ion trap devices.Comment: corrected typos, additional references, additional figur
Decomposition, Reformulation, and Diving in University Course Timetabling
In many real-life optimisation problems, there are multiple interacting
components in a solution. For example, different components might specify
assignments to different kinds of resource. Often, each component is associated
with different sets of soft constraints, and so with different measures of soft
constraint violation. The goal is then to minimise a linear combination of such
measures. This paper studies an approach to such problems, which can be thought
of as multiphase exploitation of multiple objective-/value-restricted
submodels. In this approach, only one computationally difficult component of a
problem and the associated subset of objectives is considered at first. This
produces partial solutions, which define interesting neighbourhoods in the
search space of the complete problem. Often, it is possible to pick the initial
component so that variable aggregation can be performed at the first stage, and
the neighbourhoods to be explored next are guaranteed to contain feasible
solutions. Using integer programming, it is then easy to implement heuristics
producing solutions with bounds on their quality.
Our study is performed on a university course timetabling problem used in the
2007 International Timetabling Competition, also known as the Udine Course
Timetabling Problem. In the proposed heuristic, an objective-restricted
neighbourhood generator produces assignments of periods to events, with
decreasing numbers of violations of two period-related soft constraints. Those
are relaxed into assignments of events to days, which define neighbourhoods
that are easier to search with respect to all four soft constraints. Integer
programming formulations for all subproblems are given and evaluated using ILOG
CPLEX 11. The wider applicability of this approach is analysed and discussed.Comment: 45 pages, 7 figures. Improved typesetting of figures and table
Design and fabrication of chemically robust three-dimensional microfluidic valves
A current problem in microfluidics is that poly(dimethylsiloxane) (PDMS), used to fabricate many microfluidic devices, is not compatible with most organic solvents. Fluorinated compounds are more chemically robust than PDMS but, historically, it has been nearly impossible to construct valves out of them by multilayer soft lithography (MSL) due to the difficulty of bonding layers made of non-stick fluoropolymers necessary to create traditional microfluidic valves. With our new three-dimensional (3D) valve design we can fabricate microfluidic devices from fluorinated compounds in a single monolithic layer that is resistant to most organic solvents with minimal swelling. This paper describes the design and development of 3D microfluidic valves by molding of a perfluoropolyether, termed Sifel, onto printed wax molds. The fabrication of Sifel-based microfluidic devices using this technique has great potential in chemical synthesis and analysis
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Directed Placement for mVLSI Devices
Continuous-flow microfluidic devices based on integrated channel networks are becoming increasingly prevalent in research in the biological sciences. At present, these devices are physically laid out by hand by domain experts who understand both the underlying technology and the biological functions that will execute on fabricated devices. The lack of a design science that is specific to microfluidic technology creates a substantial barrier to entry. To address this concern, this article introduces Directed Placement, a physical design algorithm that leverages the natural "directedness" in most modern microfluidic designs: fluid enters at designated inputs, flows through a linear or tree-based network of channels and fluidic components, and exits the device at dedicated outputs. Directed placement creates physical layouts that share many principle similarities to those created by domain experts. Directed placement allows components to be placed closer to their neighbors compared to existing layout algorithms based on planar graph embedding or simulated annealing, leading to an average reduction in laid-out fluid channel length of 91% while improving area utilization by 8% on average. Directed placement is compatible with both passive and active microfluidic devices and is compatible with a variety of mainstream manufacturing technologies
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