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The idea of ports can be (and is) extended to waveguide devices, but a port can no longer be defined in terms of circuit poles because in waveguides the electromagnetic waves are not guided by electrical conductors. They are, instead guided by the walls of the waveguide. Thus, the concept of a circuit conductor pole does not exist in this format. Ports in waveguides consist of an aperture or break in the waveguide through which the electromagnetic waves can pass. The bounded plane through which the wave passes is the definition of the port. |
Waveguides have an additional complication in port analysis in that it is possible (and sometimes desirable) for more than one waveguide mode to exist at the same time. In such cases, for each physical port, a separate port must be added to the analysis model for each of the modes present at that physical port. |
The concept of ports can be extended into other energy domains. The generalised definition of a port is a place where energy can flow from one element or subsystem to another element or subsystem. This generalised view of the port concept helps to explain why the port condition is so defined in electrical analysis. If the algebraic sum of the currents is not zero, such as in example diagram (c), then the energy delivered from an external generator is not equal to the energy entering the pair of circuit poles. The energy transfer at that place is thus more complex than a simple flow from one subsystem to another and does not meet the generalised definition of a port. |
An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits. |
There are a number of very well known and often used equivalent circuits in linear network analysis. These include resistors in series, resistors in parallel and the extension to series and parallel circuits for capacitors, inductors and general impedances. Also well known are the Norton and Thévenin equivalent current generator and voltage generator circuits respectively, as is the Y-Δ transform. None of these are discussed in detail here; the individual linked articles should be consulted. |
One-element networks are trivial and two-element, two-terminal networks are either two elements in series or two elements in parallel, also trivial. The smallest number of elements that is non-trivial is three, and there are two 2-element-kind non-trivial transformations possible, one being both the reverse transformation and the topological dual, of the other. |
Example 3 shows the result is a Π-network rather than an L-network. The reason for this is that the shunt element has more capacitance than is required by the transform so some is still left over after applying the transform. If the excess were instead, in the element nearest the transformer, this could be dealt with by first shifting the excess to the other side of the transformer before carrying out the transform. |
"spooky action at a distance" on the assumption of QM's completeness). |
The no-communication theorem states that, within the context of quantum mechanics, it is not possible to transmit classical bits of information by means of carefully prepared mixed or pure states, whether entangled or not. The theorem disallows all communication, not just faster-than-light communication, by means of shared quantum states. The theorem disallows not only the communication of whole bits, but even fractions of a bit. This is important to take note of, as there are many classical radio communications encoding techniques that can send arbitrarily small fractions of a bit across arbitrarily narrow, noisy communications channels. In particular, one may imagine that there is some ensemble that can be prepared, with small portions of the ensemble communicating a fraction of a bit; this, too, is not possible. |
The theorem is built on the basic presumption that the laws of quantum mechanics hold. Similar theorems may or may not hold for other related theories, such as hidden variable theories. The no-communication theorem is not meant to constrain other, non-quantum-mechanical theories. |
The basic assumption entering into the theorem is that a quantum-mechanical system is prepared in an initial state, and that this initial state is describable as a mixed or pure state in a Hilbert space "H". The system then evolves over time in such a way that there are two spatially distinct parts, "A" and "B", sent to two distinct observers, Alice and Bob, who are free to perform quantum mechanical measurements on their portion of the total system (viz, A and B). The question is: is there any action that Alice can perform on A that would be detectable by Bob making an observation of B? The theorem replies 'no'. |
An important assumption going into the theorem is that neither Alice nor Bob is allowed, in any way, to affect the preparation of the initial state. If Alice were allowed to take part in the preparation of the initial state, it would be trivially easy for her to encode a message into it; thus neither Alice nor Bob participates in the preparation of the initial state. The theorem does not require that the initial state be somehow 'random' or 'balanced' or 'uniform': indeed, a third party preparing the initial state could easily encode messages in it, received by Alice and Bob. Simply, the theorem states that, given some initial state, prepared in some way, there is no action that Alice can take that would be detectable by Bob. |
The proof proceeds by defining how the total Hilbert space "H" can be split into two parts, "H""A" and "H""B", describing the subspaces accessible to Alice and Bob. The total state of the system is assumed to be described by a density matrix σ. This appears to be a reasonable assumption, as a density matrix is sufficient to describe both pure and mixed states in quantum mechanics. Another important part of the theorem is that measurement is performed by applying a generalized projection operator "P" to the state σ. This again is reasonable, as projection operators give the appropriate mathematical description of quantum measurements. After a measurement by Alice, the state of the total system is said to have "collapsed" to a state "P"(σ). |
The proof of the theorem is commonly illustrated for the setup of Bell tests in which two observers Alice and Bob perform local observations on a common bipartite system, and uses the statistical machinery of quantum mechanics, namely density states and quantum operations. |
Alice and Bob perform measurements on system S whose underlying Hilbert space is |
It is also assumed that everything is finite-dimensional to avoid convergence issues. The state of the composite system is given by a density operator on "H". Any density operator σ on "H" is a sum of the form: |
where "Ti" and "Si" are operators on "H""A" and "H""B" respectively. For the following, it is not required to assume that "Ti" and "Si" are state projection operators: "i.e." they need not necessarily be non-negative, nor have a trace of one. That is, σ can have a definition somewhat broader than that of a density matrix; the theorem still holds. Note that the theorem holds trivially for separable states. If the shared state σ is separable, it is clear that any local operation by Alice will leave Bob's system intact. Thus the point of the theorem is no communication can be achieved via a shared entangled state. |
Alice performs a local measurement on her subsystem. In general, this is described by a quantum operation, on the system state, of the following kind |
where "V""k" are called Kraus matrices which satisfy |
means that Alice's measurement apparatus does not interact with Bob's subsystem. |
Supposing the combined system is prepared in state σ and assuming, for purposes of argument, a non-relativistic situation, immediately (with no time delay) after Alice performs her measurement, the relative state of Bob's system is given by the partial trace of the overall state with respect to Alice's system. In symbols, the relative state of Bob's system after Alice's operation is |
where formula_8 is the partial trace mapping with respect to Alice's system. |
From this it is argued that, statistically, Bob cannot tell the difference between what Alice did and a random measurement (or whether she did anything at all). |
The scallop theorem is a consequence of the subsequent forces applied to the organism as it swims from the surrounding fluid. For an incompressible Newtonian fluid with density formula_1 and viscosity formula_2, the flow satisfies the Navier–Stokes equations |
where formula_4 denotes the velocity of the swimmer. However, at the low Reynolds number limit, the inertial terms of the Navier-Stokes equation on the left-hand side tend to zero. This is made more apparent by nondimensionalizing the Navier–Stokes equation. By defining a characteristic velocity and length, formula_5 and formula_6, we can cast our variables to dimensionless form: |
By plugging back into the Navier-Stokes equation and performing some algebra, we arrive at a dimensionless form: |
where formula_9 is the Reynolds number, formula_10. In the low Reynolds number limit (as formula_11), the LHS tends to zero and we arrive at a dimensionless form of Stokes equations. Redimensionalizing yields |
The proof of the scallop theorem can be represented in a mathematically elegant way. To do this, we must first understand the mathematical consequences of the linearity of Stokes equations. To summarize, the linearity of Stokes equations allows us to use the reciprocal theorem to relate the swimming velocity of the swimmer to the velocity field of the fluid around its surface (known as the swimming gait), which changes according to the periodic motion it exhibits. This relation allows us to conclude that locomotion is independent of swimming rate. Subsequently, this leads to the discovery that reversal of periodic motion is identical to the forward motion due to symmetry, allowing us to conclude that there can be no net displacement. |
The reciprocal theorem describes the relationship between two flows in the same geometry where inertial effects are insignificant compared to viscous effects. Consider a fluid filled region formula_13 bounded by surface formula_14 with a unit normal formula_15. Suppose we have solutions to Stokes equations in the domain formula_13 possessing the form of the velocity fields formula_4 and formula_18. The velocity fields harbor corresponding stress fields formula_19 and formula_20 respectively. Then the following equality holds: |
The reciprocal theorem allows us to obtain information about a certain flow by using information from another flow. This is preferable to solving Stokes equations, which is difficult due to not having a known boundary condition. This particularly useful if one wants to understand flow from a complicated problem by studying the flow of a simpler problem in the same geometry. |
One can use the reciprocal theorem to relate the swimming velocity, formula_22, of a swimmer subject to a force formula_23 to its swimming gait formula_24: |
Now that we have established that the relationship between the instantaneous swimming velocity in the direction of the force acting on the body and its swimming gate follow the general form |
where formula_27 and formula_28 denote the positions of points |
on the surface of the swimmer, we can establish that locomotion is independent of rate. Consider a swimmer that deforms in a periodic fashion through a sequence of motions between the times formula_29 and formula_30. The net displacement of the swimmer is |
Now consider the swimmer deforming in the same manner but at a different rate. We describe this with the mapping |
This result means that the net distance traveled by the swimmer does not depend on the rate |
at which it is being deformed, but only on the geometrical sequence of shape. This is the first key result. |
If a swimmer is moving in a periodic fashion that is time invariant, we know that the average displacement during one period must be zero. To illustrate the proof, let us consider a swimmer deforming during one period that starts and ends at times formula_29 and formula_30. That means its shape at the start and end are the same, i.e. formula_37. Next, we consider motion obtained by time-reversal |
symmetry of the first motion that occurs during the period starting and ending at times formula_38 and formula_39. using a similar mapping as in the previous section, we define formula_40 and formula_41 and define the shape in the reverse motion to be the same as the shape in the forward motion, formula_42. Now we find the relationship between the net displacements in these two cases: |
This is the second key result. Combining with our first key result from the previous section, we see that formula_44. We see that a swimmer that reverses its motion by reversing its sequence of shape changes leads to the opposite distance traveled. In addition, since the swimmer exhibits reciprocal body deformation, the sequence of motion is the same between formula_38 and formula_39 and formula_29 and formula_30. Thus, the distance traveled should |
be the same independently of the direction of time, meaning that reciprocal motion cannot be used for net motion in low Reynolds number environments. |
The scallop theorem holds if we assume that a swimmer undergoes reciprocal motion in an infinite quiescent Newtonian fluid in the absence of inertia and external body forces. However, there are instances where the assumptions for the scallop theorem are violated. In one case, successful swimmers in viscous environments must display non-reciprocal body kinematics. In another case, if a swimmer is in a non-Newtonian fluid, locomotion can be achieved as well. |
In his original paper, Purcell proposed a simple example of non-reciprocal body deformation, now commonly known as the Purcell swimmer. This simple swimmer possess two degrees of freedom for motion: a two-hinged body composed of three rigid links rotating out-of-phase with each other. However, any body with more than one degree of freedom of motion can achieve locomotion as well. |
In general, microscopic organisms like bacteria have evolved different mechanisms to perform non-reciprocal motion: |
The assumption of a Newtonian fluid is essential since Stokes equations will not remain linear and time-independent in an environment that possesses complex mechanical and rheological properties. It is also common knowledge that many living microorganisms live in complex non-Newtonian fluids, which are common in biologically relevant environments. For instance, crawling cells often |
Non-Newtonian fluids have several properties that can be manipulated to produce small scale locomotion. |
First, one such exploitable property is normal |
stress differences. These differences will arise from the stretching of the fluid by the flow of |
The Clausius theorem (1855) states that for a thermodynamic system (e.g. heat engine or heat pump) exchanging heat with external reservoirs and undergoing a thermodynamic cycle, |
where formula_2 is the infinitesimal amount of heat absorbed by the system from the reservoir and formula_3 is the temperature of the external reservoir (surroundings) at a particular instant in time. The closed integral is carried out along a thermodynamic process path from the initial/final state to the same initial/final state. In principle, the closed integral can start and end at an arbitrary point along the path. |
If there are multiple reservoirs with different temperatures formula_4, then Clausius inequality reads: |
In the special case of a reversible process, the equality holds. The reversible case is used to introduce the state function known as entropy. This is because in a cyclic process the variation of a state function is zero. In other words, the Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around. |
for an infinitesimal change in entropy "S" applies not only to cyclic processes, but to any process that occurs in a closed system. |
The Clausius theorem is a mathematical explanation of the second law of thermodynamics. It was developed by Rudolf Clausius who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius theorem provides a quantitative formula for understanding the second law. |
Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius theorem was first published in 1862 in Clausius' sixth memoir, "On the Application of the Theorem of the Equivalence of Transformations to Interior Work". Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δ"Q") into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation |
with 𝛿"Q" being energy flow into the system due to heating and "T" being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not. This relation is the "Clausius inequality". |
Now that this is known, there must be a relation developed between the Clausius inequality and entropy. The amount of entropy "S" added to the system during the cycle is defined as |
If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality. |
The temperature that enters in the denominator of the integrand in the Clausius inequality is actually the temperature of the external reservoir with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir. |
Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoir, the net change in entropy of the "universe", so to say, is formula_12. |
When the system takes in heat by an infinitesimal amount formula_13(formula_14), for the net change in entropy formula_15 in this step to be positive, the temperature of the "hot" reservoir formula_16 needs to be slightly greater than the temperature of the system at that instant. If the temperature of the system is given by formula_17 at that instant, then formula_18, and formula_19 forces us to have: |
This means the magnitude of the entropy "loss" from the reservoir, formula_21 is less than the magnitude of the entropy gain formula_22(formula_14) by the system: |
Similarly, when the system at temperature formula_24 expels heat in magnitude formula_25 (formula_26) into a colder reservoir (at temperature formula_27) in an infinitesimal step, then again, for the Second Law of Thermodynamics to hold, one would have, in an exactly similar manner: |
Here, the amount of heat 'absorbed' by the system is given by formula_29(formula_30), signifying that heat is transferring from the system to the reservoir, with formula_31. The magnitude of the entropy gained by the reservoir, formula_32 is greater than the magnitude of the entropy loss of the system formula_33 |
Since the total change in entropy for the system is 0 in a cyclic process, if one adds all the infinitesimal steps of heat intake and heat expulsion from the reservoir, signified by the previous two equations, with the temperature of the reservoir at each instant given by formula_34, one gets, |
In summary, (the inequality in the third statement below, being obviously guaranteed by the second law of thermodynamics, which is the basis of our calculation), |
For a reversible cyclic process, there is no generation of entropy in each of the infinitesimal heat transfer processes, so the following equality holds, |
Thus, the Clausius inequality is a consequence of applying the second law of thermodynamics at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself. |
Adiabatic quantum computation (AQC) is a form of quantum computing which relies on the adiabatic theorem to do calculations and is closely related to quantum annealing. |
First, a (potentially complicated) Hamiltonian is found whose ground state describes the solution to the problem of interest. Next, a system with a simple Hamiltonian is prepared and initialized to the ground state. Finally, the simple Hamiltonian is adiabatically evolved to the desired complicated Hamiltonian. By the adiabatic theorem, the system remains in the ground state, so at the end the state of the system describes the solution to the problem. Adiabatic quantum computing has been shown to be polynomially equivalent to conventional quantum computing in the circuit model. |
The time complexity for an adiabatic algorithm is the time taken to complete the adiabatic evolution which is dependent on the gap in the energy eigenvalues (spectral gap) of the Hamiltonian. Specifically, if the system is to be kept in the ground state, the energy gap between the ground state and the first excited state of formula_1 provides an upper bound on the rate at which the Hamiltonian can be evolved at time When the spectral gap is small, the Hamiltonian has to be evolved slowly. The runtime for the entire algorithm can be bounded by: |
where formula_3 is the minimum spectral gap for |
AQC is a possible method to get around the problem of energy relaxation. Since the quantum system is in the ground state, interference with the outside world cannot make it move to a lower state. If the energy of the outside world (that is, the "temperature of the bath") is kept lower than the energy gap between the ground state and the next higher energy state, the system has a proportionally lower probability of going to a higher energy state. Thus the system can stay in a single system eigenstate as long as needed. |
Universality results in the adiabatic model are tied to quantum complexity and QMA-hard problems. The k-local Hamiltonian is QMA-complete for k ≥ 2. QMA-hardness results are known for physically realistic lattice models of qubits such as |
where formula_5represent the Pauli matrices Such models are used for universal adiabatic quantum computation. The Hamiltonians for the QMA-complete problem can also be restricted to act on a two dimensional grid of qubits or a line of quantum particles with 12 states per particle. If such models were found to be physically realisable, they too could be used to form the building blocks of a universal adiabatic quantum computer. |
In practice, there are problems during a computation. As the Hamiltonian is gradually changed, the interesting parts (quantum behaviour as opposed to classical) occur when multiple qubits are close to a tipping point. It is exactly at this point when the ground state (one set of qubit orientations) gets very close to a first energy state (a different arrangement of orientations). Adding a slight amount of energy (from the external bath, or as a result of slowly changing the Hamiltonian) could take the system out of the ground state, and ruin the calculation. Trying to perform the calculation more quickly increases the external energy; scaling the number of qubits makes the energy gap at the tipping points smaller. |
Adiabatic quantum computation solves satisfiability problems and other combinatorial search problems. Specifically, these kind of problems seek a state that satisfies |
This expression contains the satisfiability of M clauses, for which clause formula_7 has the value True or False, and can involve n bits. Each bit is a variable formula_8 such that formula_7 is a Boolean value function of formula_10. QAA solves this kind of problem using quantum adiabatic evolution. It starts with an Initial Hamiltonian formula_11: |
where formula_13 shows the Hamiltonian corresponding to the clause formula_7. Usually, the choice of formula_13 won't depend on different clauses, so only the total number of times each bit is involved in all clauses matters. Next, it goes through an adiabatic evolution, ending in the Problem Hamiltonian formula_16: |
where formula_18 is the satisfying Hamiltonian of clause C. |
For a simple path of adiabatic evolution with run time T, consider: |
which is the adiabatic evolution Hamiltonian of our algorithm. |
According to the adiabatic theorem, we start from the ground state of Hamiltonian formula_11 at the beginning, proceed through an adiabatic process, and end in the ground state of problem Hamiltonian formula_16. |
We then measure the z-component of each of the n spins in the final state. This will produce a string formula_25 which is highly likely to be the result of our satisfiability problem. The run time T must be sufficiently long to assure correctness of the result. According to adiabatic theorem, T is about formula_26, where |
is the minimum energy gap between ground state and first excited state. |
Adiabatic quantum computing is equivalent in power to standard gate-based quantum computing that implements arbitrary unitary operations. However, the mapping challenge on gate-based quantum devices differs substantially from quantum annealers as logical variables are mapped only to single qubits and not to chains. |
The D-Wave One is a device made by Canadian company D-Wave Systems, which claims that it uses quantum annealing to solve optimization problems. On 25 May 2011, Lockheed-Martin purchased a D-Wave One for about US$10 million. In May 2013, Google purchased a 512 qubit D-Wave Two. |
The question of whether the D-Wave processors offer a speedup over a classical processor is still unanswered. Tests performed by researchers at Quantum Artificial Intelligence Lab (NASA), USC, ETH Zurich, and Google show that as of 2015, there is no evidence of a quantum advantage. |
The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an isolated system is nonnegative, and can only be zero when the system has no gravitating objects. Although these statements are often thought of as being primarily physical in nature, they can be formalized as mathematical theorems which can be proven using techniques of differential geometry, partial differential equations, and geometric measure theory. |
Richard Schoen and Shing-Tung Yau, in 1979 and 1981, were the first to give proofs of the positive mass theorem. Edward Witten, in 1982, gave the outlines of an alternative proof, which were later filled in rigorously by mathematicians. Witten and Yau were awarded the Fields medal in mathematics in part for their work on this topic. |
An imprecise formulation of the Schoen-Yau / Witten positive energy theorem states the following: |
The meaning of these terms is discussed below. There are alternative and non-equivalent formulations for different notions of energy-momentum and for different classes of initial data sets. Not all of these formulations have been rigorously proven, and it is currently an open problem whether the above formulation holds for initial data sets of arbitrary dimension. |
The original proof of the theorem for ADM mass was provided by Richard Schoen and Shing-Tung Yau in 1979 using variational methods and minimal surfaces. Edward Witten gave another proof in 1981 based on the use of spinors, inspired by positive energy theorems in the context of supergravity. An extension of the theorem for the Bondi mass was given by Ludvigsen and James Vickers, Gary Horowitz and Malcolm Perry, and Schoen and Yau. |
Gary Gibbons, Stephen Hawking, Horowitz and Perry proved extensions of the theorem to asymptotically anti-de Sitter spacetimes and to Einstein–Maxwell theory. The mass of an asymptotically anti-de Sitter spacetime is non-negative and only equal to zero for anti-de Sitter spacetime. In Einstein–Maxwell theory, for a spacetime with electric charge formula_1 and magnetic charge formula_2, the mass of the spacetime satisfies (in Gaussian units) |
with equality for the Majumdar–Papapetrou extremal black hole solutions. |
An initial data set consists of a Riemannian manifold and a symmetric 2-tensor field on . One says that an initial data set : |
Note that a time-symmetric initial data set satisfies the dominant energy condition if and only if the scalar curvature of is nonnegative. One says that a Lorentzian manifold is a development of an initial data set if there is a (necessarily spacelike) hypersurface embedding of into , together with a continuous unit normal vector field, such that the induced metric is and the second fundamental form with respect to the given unit normal is . |
This definition is motivated from Lorentzian geometry. Given a Lorentzian manifold of dimension and a spacelike immersion from a connected -dimensional manifold into which has a trivial normal bundle, one may consider the induced Riemannian metric as well as the second fundamental form of with respect to either of the two choices of continuous unit normal vector field along . The triple is an initial data set. According to the Gauss-Codazzi equations, one has |
where denotes the Einstein tensor of and denotes the continuous unit normal vector field along used to define . So the dominant energy condition as given above is, in this Lorentzian context, identical to the assertion that , when viewed as a vector field along , is timelike or null and is oriented in the same direction as . |
The ends of asymptotically flat initial data sets. |
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