Up until now, with the exception of pure Chern-Simons theory, we have considered only **theories** with extended supersymmetry, i.e., at least N = 3. There is no inherent reason why the local- ization calculation requires this, since in our derivation of the matrix model we used only N = 2 supersymmetry. However, we wrote a specific form for the **superconformal** transformations of the matter fields, and although this form is necessary in **theories** with extended supersymmetry, it is not the only one compatible with N = 2 supersymmetry. Generically, the matter may come in some other representation of the algebra – specifically, with an R-charge other than than 1 2 – and then the δ-exact terms above cannot be used. In this chapter, following [7, 5], we attempt to find these more general representations, and use them to localize arbitrary N = 2 **theories**. We will see that in addition to opening up the interesting realm of N = 2 **theories** to investigation via **localization**, one is led to insights into RG flow that apply even to nonsupersymmetric **three**-**dimensional** **theories**.

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Besides these, there can be mixed branches where scalars from both types of multiplets acquire expectation values. The points at which different classical branches meet are often RG fixed points with interesting interacting conformal **field** **theories**. Quantum effects are especially important at these fixed points, which are often strongly interacting. In some situations, IR **dualities** relate different UV Lagrangian descriptions of the same fixed point. The **theories** are said to be in the same universality class. We will study various examples of this phenomenon in 3.1. The fact that the two types of branches meet can constrain the metric on the Coulomb branch. An example is SQED (see also 3.1.3) (U (1) gauge theory with charge 1 and charge -1 chirals) in which the one complex **dimensional** Coulomb branch pinches off when it intersects the (one complex **dimensional**) Higgs branch. The effect can be argued to exist on the basis of the U (1) J symmetry (see 1.1.6)

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diﬀerent constraint, for example of the form x + 2y = const. The ray of maximal entropy is again given by the union of the midpoints of the line segments x + 2y = c contained in the dual toric cone, and we can read oﬀ k((1, 2)) = (2, 1). This is illustrated in red in Figure 9. As a next example, consider the simplest **three**-**dimensional** toric variety. In its simplest presentation, where both the toric and the dual toric cone are simply the ﬁrst octant of the lattice Z 3 , the symmetry relating the **three** generators is evident and trivially k( ~b) = (1, 1, 1). For a less trivial example, consider the presentation of the toric data of C 3 in the conventions of Section 2, where we choose the SL(3, Z ) frame so that the generators of the toric cone reside in the z = 1 plane. It is straightforward to compute the generators of the dual toric cone and the Reeb vector using the results from Section 2, and one gets

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The symmetries of Maxwell’s classical equations have played a defining role in modern physics. Their importance for relating observers which moved at constant velocity with respect to each other was realised by Lorentz, prior to the development of special relativity by Einstein in 1905. Their gauge sym- metry was discovered by Weyl in the 1920’s and was extended to construct the standard model in 1967, However, it is only more recently that the true importance of electromagnetic duality and conformal invariance has become apparent. In fact, the conformal invariance of the classical Maxwell equa- tions was realised as long ago as 1909 [1]. Unfortunately, quantum effects in Maxwell’s theory coupled to electrons and in all other four-**dimensional** the- ories which were subsequently studied for many years, lead to violations of their conformal symmetries. The corresponding anomaly is directly related to the appearance of infinities in quantum **field** theory. Despite this, in the 1960’s and 1970’s there was a revival of interest in four **dimensional** confor- mal symmetry [2] and it was found that the two and **three** point Green’s functions could be determined up to constants by conformal symmetry [3]. With the discovery of supersymmetry, examples of conformally invariant four **dimensional** quantum **field** **theories** were found. The first such example [4] being the N = 4 Yang-Mills theory. Subsequently, it was realised that there were an infinite number of N = 2 **theories** [5] and even some N = 1 **theories** [6]. More recently other examples of conformally invariant supersymmetric **theories** have been found [7].

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A non-perturbative duality in **three** dimensions, known as 3-D mirror symmetry, has been proposed by K. Intriligator and N. Seiberg [9], and later studied by a num- ber of authors [10]-[27]. The mirror symmetry predicts quantum equivalence of two different **theories** in the IR limit. In this regime a supersymmetric gauge theory is described by a strongly coupled **superconformal** **field** theory. The duality exchanges masses and Fayet-Iliopoulos terms as well as the Coulomb and Higgs branches imply- ing that electrically charged particles in one theory correspond to the magnetically charged objects (monopoles) in the other. Also, since the Higgs branch does not re- ceive quantum corrections and the Coulomb branch does, mirror symmetry exchanges classical effects in one theory with quantum effects in the dual theory. Many aspects of the **three**-**dimensional** mirror symmetry have a string theory origin.

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We study Seiberg-like **dualities** for **three**-**dimensional** N ¼ 2 **theories** with flavors in fundamental and adjoint representations. The recent results of Intriligator and Seiberg provide a derivation of an Aharony duality from a Giveon–Kutasov duality. We extend their result to the case of more general **theories** involving various masses for fundamental quarks and adjoint fields. By fine-tuning the vacuum expectation value of a scalar **field** and using various identifications between gauge groups and their singlet duals, we derive several examples of Aharony **dualities**. For **theories** with an adjoint **field**, we discuss the connection between the Aharony **dualities** proposed by Kim and Park for **theories** with multiple Coulomb branches and Giveon–Kutasov–Niarchos **dualities**.

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One of the reasons that make the M-brane **dualities** more challenging than the D3-brane duality is that the M-theory background does not contain a dilaton eld, and therefore there is no weak- coupling limit. Also, it is not obvious that a classical action describing the conformal eld theory that is dual to the M-theory solution needs to exist. For example, in the case of the M2 duality, we can consider the maximally supersymmetric SU (N ) Yang-Mills theory that describes the world- volume theory on a collection of N coincident D2-branes as a weak coupling description in the UV of the desired SCFT. To be specic, this **three**-**dimensional** SU (N ) Yang-Mills theory while maximally supersymmetric it is not conformal, i.e., it has a dimensionfull coupling. But if we ow to the infrared of this gauge theory, the coupling becomes innite and one reaches the conformally invariant xed point of the theory. Although, there is no guarantee that this xed point has a dual Lagrangian description. The M5 case is even worst since there is not even a weak coupling description in the ultraviolet of the required SCFT because the theory is six-**dimensional**.

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importance to this text. One of the greatest obstacles faced by **field** **theories** is that of their strongly coupled, non-perturbative regimes. Since the methods of perturbation theory cannot be employed, such regimes are notoriously difficult to investigate. An interesting new method for investigating the strongly coupled regimes comes in the form of strong-weak duality. Certain classes of theory have been discovered whose strongly coupled regime is physically equivalent to the weakly coupled regime of another ‘dual’ theory. The **theories** give rise to the same observable phenomena. Importantly the two **theories** are also related by a ‘duality’ transformation. Such **theories** are interesting as the behaviour of the strongly coupled regime of one theory can be ascertained by perturbatively investigating the weakly coupled regime of its dual, and then making a duality transformation. Specifically, for effective **three** **dimensional** N = 2 **theories** with zero Chern-Simons level, Aharony duality was theorised [2], and for effective **three** **dimensional** N = 2 **theories** with non-zero

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time, as mentioned in section 3, the non-compact quantum dilogarithm function, which is closely related to sine double function, appeared in a partition function of SL(2, R ) Chern-Simons theory. From this observation, one can read off **field** contents and Chern-Simons coupling corresponding to a single tetrahedron by matching variables between two such as a certain quantity of a chiral multiplet and an edge parameter. At the same time, one can also check with moduli space of suerpsymmet- ric vacua (or supersymmetric parameter space) of 3d N = 2 **theories** and the moduli space of flat SL(2, C) connection of Chern-Simons theory on a single tetrahedron.

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and the Spin(2) × Spin(4) subgroup of the Spin(6) R-symmetry. The translational and conformal symmetries are broken. This is the same deformation as that used by S. Kim to compute the **superconformal** index of the ABJM theory [16]. We will see that the same kind of deformation can be used to study any **three**-**dimensional** gauge theory with enough supersymmetry. One big advantage of this method is that we have control over the conformal dimensions of an important class of monopole operators. We show that for k = 1, 2 the ABJM theory has monopole operators which are conformal primaries of dimension 2 and transform as vectors under Lorenz transformations. Such operators must be conserved currents, which enables us to conclude that the R-symmetry and consequently supersymmetry are enhanced. The other model we consider is an N = 4 d = 3 U (N ) gauge theory with an adjoint and a fundamental hypermultiplet. This theory has no Chern-Simons term and is not conformal but flows to a nontrivial IR fixed point. String theory arguments show that it must be IR dual to N = 8 super-Yang-Mills theory with gauge group U (N ). This implies that it must have enhanced supersymmetry in the infrared, and we show that this is indeed the case. There are several important differences compared to the case of the ABJM theory. In particular, we find that some currents predicted by the duality are realized by monopole operators with a vanishing topological charge (but nonzero GNO charges). This is a nice illustration of the importance of nontopological disorder operators in quantum **field** theory. We also show that for N > 1 the U (N ) × U (N ) k = 1 ABJM theory as well as the IR limit of N = 4 U (N ) theory with an adjoint and a fundamental hypermultiplet have a free sector (also with N = 8 supersymmetry). This decoupled sector is not visible on the perturbative level, but its existence is predicted by the conjecture [11] that both **theories** are dual to the IR limit of N = 8 U (N ) super-Yang-Mills theory. 2

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2), the amount of energy in the non-axisymmetric part of the **field** increases exponentially. The growth rate is approximately propor- tional to the strength of the magnetic **field** (see S.I. Figure 1), i.e. for the case depicted in Figs. 1 and 2 the non-axisymmetric energy growth timescale is ∼ 1.5kyr. The exponential growth stops if one of the following two conditions is fulfilled: either 0.5−0.9 of the total magnetic energy has decayed, or the non-axisymmetric **field** contains about 50% of the total energy. While initially the m = 0 mode was dominant, once the instability develops energy is transferred and a local maximum appears around 10 . m . 20 (see S.I. Figure 3), with some further local maxima at higher har- monics, leading to a characteristic wavelength for the instability of about 5km at the equator, with finer structure appearing later giving kilometre-sized features. These features are well above the resolu- tion of the simulation and appear both in the low and high resolu- tion runs. This behaviour is indicative of the density-shear instabil- ity in Hall-MHD, where, based on analytical arguments (21; 22), the wavelength of the dominant mode is 2πL where L is the scale height of the electron density and the magnetic **field**, which can be approximated by the thickness of the crust.

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When taking RSS experience attenuation model to estimate the range, [23] points out that the attenuation follows the lognormality experience model, but there are many factors infacting the parameters in the model. It proposed a curve fitting method to work out the function between distance and receiving power. Through a lot of experiments, it reached a conclusion that the range measurement accuracy is higher when distance is shorter. WSN is essentially a multi-hop network. MDS-based **localization** algorithm utilizes RSS experience attenuation model to estimate the range between nodes. It is inevitable to estimate multi-hop distance to be measured. With the coverage area and the denstiy of nodes getting change, it cannot solve the problems radically to increast the send power of nodes. Thus, the normal solution is to search the shortest path between two nodes. All of one hop distance of the path is accumulated as the estimation of multi-hop distance. Therefore the measurment error of the multi-hop distance is inversely proportional to the density of nodes and the connectivity of network. It is also found that the more hops between nodes, the more error to the measurment range. Employing culster to divide the network can not only reduce communication costs, but also decreace the complexity of the computation. Besides, the structure of cluster can also dramatically reduce the multi-hop in dissimilarity matrix, and improve the overall range measurment accuracy.

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E is minimised using a Levenberg-Marquardt solver to perform the nonlinear op- timisation, where the initial values may be estimated using standard geometry. The solver was as implemented within the Matlab function lsqnonlin [8]. Finally, to provide useful **dimensional** units to the returned coordinates, a scaling factor for the system is calculated based on known distances between the physical points used for calibration. Note that in this simple model, focal length f is effectively an arbitrary scaling factor, which could vary inﬁnitely if {t} varied accordingly, so it is held con- stant during optimisation.

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We can reveals that the detection range with the L-shaped antenna array is significantly greater than that with the inverted T-shaped antenna array when the number of effective digits or antenna spacing are the same through comparative analysis of results in Figure 3 and Figure 4. This is due to the fact that the L-shaped antenna array consists of two ellipsoid equations and one spherical equation while the inverted T-shaped antenna array consists of **three** ellipsoid equations. An ellipsoid equation contains a large number of divisions besides secondary operations, but spherical equation only contains secondary operations. So the required number for the computation of spherical equation is fewer than that of the ellipsoid equation. Therefore, the detection range of the L-shaped is significantly greater than that of the inverted T-shaped when the number of effective digits or antenna spacing is same.

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The accuracy of **localization** with six randomly situated detectors is greatly improved in figure 15 due to cross-pairing the detectors. Cross-pairing allowed for the analysis of fifteen conic regions instead of just **three**. The results seen in figure 15 are much more precise and accurate than in figure 14 even though the intersecting region is still slightly under the source location. Figure 16 and figure 17 show the results of having six detectors in a fixed array instead of randomly situated. The array layout is the same as basic configuration as previously used in the other examples, and the new detector pair was added as to make a triangular array of detector pairs that are paired vertically. As a side note, this array most closely emulates the hydrophone array used for the underwater acoustics experiments mentioned in chapter 1. The triangular array produces **three** conic regions with the same orientation in the 𝑧̂ direction. The intersection of these **three** conic regions is usually scattered in several separate volumes, one of which contains the signal source location. These scattered results occur quite frequently when only **three** conic regions are being intersected; even if they are randomly situated. Figure 14’s lack of scattering is coincidental from randomization of detector locations. Furthermore, cross-pairing the six

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or underexposure. The flame radiant existence **field** is reconstructed using the flame image, and temperature **field** is further calculated via the lookup table between radiant existence and temperature. These researches proved that the radiative imaging technique is efficient for the 3-D temperature **field** reconstruction in large scale flames. However, up to date the conventional cameras are used for these radiative imaging techniques to record the radiation intensity where the direction of each ray cannot be recorded simultaneously. So the multi-cameras are needed in radiative imaging system for the measurement of flame temperature **field**. This leads to some issues such as high degree of coupling and synchronization of the multi-cameras, making the operation and mounting of the system costly and inconvenient. A single camera system has also been employed for the radiative imaging technique [13] to reconstruct the 3-D temperature **field**. In this model, the beam of rays detected by each pixel is regarded as the principal ray for simplification. However, the simplification is based on the fact that the distance between the camera and the flame is far enough. The farther distance will result in the smaller image of the flame with certain size. The smaller image implies that the pixels of the CCD sensor are not employed to capture the flame image at the utmost extent.

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Location information is prerequisite for wireless sensor network monitoring and control applications. In this paper, we have provided a tutorial and survey of enhanced DV-Hop **localization** algorithms under 3D WSNs since most previous works concen- trated on 2D space. We summarized representative survey of range-free **localization** algorithms that concentrated both under 2D and 3D WSNs. Moreover, improved DV- Hop algorithms proposed in literature are investigated, with comparing in terms of **localization** error, computational complexity, energy consumption, node type and coplanar issue. We divided enhanced 3D DV-Hop **localization** into four types based on location processing characteristic. Each type of enhanced method is needed to balance location performance and algorithm cost, like nature-inspired based algorithm is greatly boosted **localization** accuracy, but computational complexity is high. Future probably hot topics of 3D DV-Hop **localization** are mobility network, irregular topol- ogy networks and multiple-objective optimization model-based **localization** schemes.

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The zeta function regularization technique is used to study the Casimir effect for a scalar **field** of mass m satisfying Dirichlet boundary conditions on a spherical surface of radius a. In the case of large scalar mass, ma 1 , simple analytic expressions are obtained for the zeta function and Ca- simir energy of the scalar **field** when it is confined inside the spherical surface, and when it is con- fined outside the spherical surface. In both cases the Casimir energy is exact up to order a m − 2 − 1 and contains the expected divergencies, which can be eliminated using the well established re- normalization procedure for the spherical Casimir effect. The case of a scalar **field** present in both the interior and exterior region is also examined and, for ma 1 , the zeta function, the Casimir energy, and the Casimir force are obtained. The obtained Casimir energy and force are exact up to order a m − 2 − 1 and a m − 3 − 1 respectively. In this scenario both energy and force are finite and do not need to be renormalized, and the force is found to produce an outward pressure on the spher- ical surface.

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On the basis of the research of DV-Hop **localization** algorithm, this paper im- proved the shortcomings of large algorithm **localization** error to better reflect the actual average hop distance and effectively reduced the **localization** error. In order to further improve the **localization** accuracy of the algorithm, without additional hard- ware and communication overhead, adaptive cuckoo algorithm is introduced to opti- mize the improved DV-Hop **localization** algorithm on position estimation. The simu- lation results show that the proposed algorithm is significantly superior to the im- proved DV-Hop **localization** algorithm and the traditional DV-Hop algorithm. There- fore, the introduction of adaptive CS algorithm, to some extent, improved the toler- ance of DV-Hop **localization** algorithm and made it a better applicability. In the fol- lowing work, how to improve the **localization** accuracy in the case of uneven distribu- tion of the anchor nodes and the invalidation of some nodes will need to be further studied.

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