contribution to the spectrum by using second-order Rayleigh-Schr¨ odinger perturba- tion theory (there is an intermediate state sum involved, but since we are doing the calculation numerically, this is not a serious problem). There is also the issue of degen- eracy but the existence of a higher conserved charge once again renders the problem effectively non-degenerate. The resulting three-loop data for large-K was fit in Chap- ter 2 to a power series in K −1 to read off the expansion coefficients E su(2) 3,n . It turns out that, to numerical precision, the coefficients are non-vanishing only for n > **5** (as re- quired by BMN scaling). The results of this program are reproduced for convenience from Chapter 2 in table 4.17, where they are compared with string theory predictions derived (in the manner described in previous paragraphs) from eqn. (4.2.10). (The accuracy of the match is displayed in the last column of table 4.17.) The important point is that there is substantial disagreement with string results at O(λ 3 ) for all energy levels: the low-lying states exhibit a mismatch ranging from roughly 19% to 34%, and there is no evidence that this can be repaired by taking data on a larger range of lattice sizes. There is apparently a general breakdown of the correspondence between string theory and gauge theory anomalous dimensions at three loops, despite the precise and impressive agreement at first and second order. This disagreement was first demonstrated in the two-impurity regime [26]. It is perhaps not surprising that the three-loop disagreement is reproduced in the three-impurity regime, but it provides us with more information that may help to clarify this puzzling phenomenon.

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In the previous section we have shown that string equations of motion admit the Lax representation provided the parameter κ in the Lagrangian takes values ± 1. It is for these values of κ that the model exhibits the local fermionic symmetry. In addition to the κ-symmetry, the string sigma model has the usual reparametrization invariance. Due to these local symmetries not all degrees of freedom appearing in the Lagrangian (1.35) are physical. Thus, ultimately we would like to understand if and how **integrability** is inherited by the physical subspace which is obtained by making a gauge choice and imposing the Virasoro constraints. In this section we will make a first step in this direction by analyzing in detail the transformation properties of the Lax connection under the κ-symmetry and diffeomorphism transformations. We also indicate a relation between the Lax connection and the global psu(2, 2 | 4) symmetry of the model.

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[23] at the classical level. One certainly hopes **integrability** to persist also in the quantum theory, although it is unclear at present how this could be precisely imple- mented. Inspired by the all-loop Bethe ansatz conjectures on the gauge theory side and aware of the obtained data in the plane wave, flat space and spinning strings lim- its Arutyunov, Staudacher and one of the present authors were able to write down a set of quantum string Bethe equations [24] which are structurally very similar to the gauge theory equations of [15], differing by a so-called dressing factor which depends on (an infinite set of) undetermined functions of λ and thus taking into account the three loop discrepancies. These functions should be determined by comparison with quantum string data. First steps in this direction have been performed in [22, 25, 26]. The quantum string Bethe equations of [24] have been also generalized to the full P SU(2, 2 | 4) setting in [16] 3 .

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initio calculations within it is to define it on a discretized spacetime or lattice. Lattice field theory methods have been recently become a subject of study also in the framework of worldsheet string models [10–12]. This approach bypasses the subtleties of realizing supersymmetry on the lattice - which characterise the lattice approach to the duality from the gauge theory side [13] - in that the Green-Schwarz **superstring** formulation that we use displays supersymmetry only in the target space. In the two-dimensional string world-sheet model under analysis supersymmetry appears as a flavour symmetry. Importantly, local symmetries (diffeomorphism and fermionic kappa-symmetry) are all fixed, and only scalar fields (some of which anti-commuting) appear, assigned to sites. This rather simplified setting - useful to have at most quartic fermionic interactions - still retains the sophisticated dynamics of relevant observables in this framework.

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In measuring the action at small values of the coupling g, we observe a divergence com- patible with a quadratic behavior ∼ a −2 . It is certainly possible that the reasoning leading to the line of constant physics (4.1) might be subject to change once all fields correlators are investigated – something which we leave for the future. However, in the lattice regu- larization performed here such divergences are expected. In continuum perturbation theory, power-divergences arising in this [8] and analogue models [11] are set to zero using dimen- sional regularization. From the perspective of a hard cut-off regularization like the lattice one, this is related to the emergence in the continuum limit of power divergences – quadratic, in the present two-dimensional case – induced by mixing of the (scalar) Lagrangian with the identity operator under UV renormalization. The problem of renormalization in presence of power divergences is in general non trivial, and one of the ways to proceed – which is our way here – is via non-perturbative subtractions of those divergences. While with the present data we are able to reliably and non-perturbatively subtract them, in general this procedure leads to potentially severe ambiguities, with errors diverging in the continuum limit. In the future it may be therefore worthwhile to explore whether other schemes – e.g. the Schr¨ odinger functional scheme [52] – could be used as a proper definition of the effective action under in- vestigation. We remark however that for the other physical observable here investigated, the h **x** **x** ∗ i correlator, we encountered no problems in proceeding to the continuum limit.

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In paper [7], it design a redundancy **S**-Box circuit in ASIC. Because of the redundancy, it cost more area, power and increase the path delay. We have implemented that design in FPGA to compare that design with our design. After implementation, the comparison of our **S**-Box design and others design is shown in Table VI.

The equivalence postulate point of view is that the fundamental equa- tion is the QHJE, which is a third–order non–linear differential equation. It is equivalent to the Schr¨odinger equation (in the sense discussed above eq. (**5**)), but requires specifying more initial conditions than for the Schr¨odinger equation. We have that there is a moduli space of solutions of the QHJE, which corresponds to the same wave function. That is, there are hidden variables which depend on the Planck length and are not detected in the solutions of the Schr¨odinger equation. This means that the Schr¨odinger equation with its related apparatus provides an effective description, albeit an extremely successful one from the experimental point of view. Now, the vacuum energy in conventional quantum mechanics is an artifact of the Hilbert space construction, i.e. it is an artifact of the effective descrip- tion. But from the point of view of the equivalence postulate the more complete solution is given by the QHJE, which admits a non–trivial solu- tion also for the state with vanishing energy and vanishing potential. The existence of such a specialized state already indicates that it may have something to do with the vacuum energy, as according to the equivalence postulate all other states are connected to this special state by coordinate transformations. This leads to the existence of a fundamental length scale with all the expected implications of modifications of the uncertainty rela- tions, and space–time uncertainty relations, etc. However, the important fact is the existence of the additional term in the quantum HJ equation, Q(q) = (¯ h/2m) {**S** 0 , q } , which is never vanishing. This term can be in-

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Starting with the classical algebraic curve describing a particular solution one can develop a semiclassical quantization [16, 44] by deforming the cuts definining the algebraic curve (adding extra roots) [45, 46]. Fluctuations are then perturbations of the cuts, and the one-loop correction to the energy is given as usual by the sum of the energy shifts (or characteristic frequencies) due to these fluctuations. Alternatively, one may try to guess the quantum extension of the classical finite gap integral equations, having as guiding principle the gauge theory information implying a description in terms of an asymptotic Bethe Ansatz [43]. Improved by the phase [47, 48] extracted from the 1-loop string data of [49], the Bethe Ansatz result for the 1-loop correction to string energy was shown [50] to agree, for a generic classical **superstring** solution, with the approach based on extracting the characteristic frequencies by perturbing the algebraic curve. This general equivalence was recently extended to include also the exponentially suppressed finite size effects with the asymptotic Bethe Ansatz starting point replaced by an appropriate Thermodynamic Bethe Ansatz (see [17] and references therein).

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Localization has been proven to be one of the most powerful tools in obtaining non pertur- bative results in quantum supersymmetric gauge theories [3]: An impressive number of new exact results have been derived in different dimensions, mainly when formulated on spheres or products thereof [3, 16]. In order to gain further intuition on the relation between localization and sigma-model perturbation theory in different and more general settings, we re-examine this issue addressing as follows the problem of how to possibly eliminate the ambiguity related to the partition function measure. We consider the string dual to a non-maximal circular Wil- son loop - the family of 1/4-BPS operators with path corresponding to a latitude in **S** 2 ∈ **S** **5** parameterized by an angle θ 0 and studied at length in [15, 17, 18] - and evaluate the corre-

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In Ref. [24] we address the following question: Given a supersymmetric string vacuum at the Planck scale, is it possible to obtain hierarchical supersymmetry breaking in the observable sector? A supersymmetric string vacuum is obtained by finding solutions to the cubic level F and D constraints. We take a gauge coupling in agreement with gauge coupling unification, thus taking a fixed value for the dilaton VEV. We then investigate the role of nonrenormalizable terms and strong hidden sector dynamics. The hidden sector contains two non–Abelian hidden gauge groups, SU(**5**) × SU (3), with matter in vector–like representations. The hidden SU (3) group is broken near the Planck scale. We analyze the dynamics of the hidden SU(**5**) group. The SU (**5**) hidden matter mass matrix is given by

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scaling dimension has been computed for small ’t Hooft coupling λ 1 in field perturbation theory up to an impressive **5**-loop order [1]. The motivation for going to such high orders arose from the enormous progress in understanding the hidden integrable system behind the spectral problem of this AdS/CFT duality pair (for a recent review see [2]). Here, the computation of the Konishi scaling dimension has become something of a testing ground for the application of **integrability** techniques going beyond the asymptotic Bethe ansatz [3] in the form of the thermodynamic Bethe ansatz for the mirror model [4] or the Y-system [**5**]. The assumption of **integrability** is powerful enough to evaluate the Konishi scaling dimension to even higher orders [6], with the present record being set at eight [7] or even nine loops [8]. The **integrability** based results in principle can yield the scaling dimensions of short operators dual to the spectrum of short excited AdS **5** × **S** **5** strings for any value of the coupling λ and in particular also at

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These images are compared with those reconstructed from holograms of the full 4 π sphere. Figures 7(a) and (b) show the reconstructed intensities of 1/2 1/2 0 and 1/4 1/4 1/4 par- allel to [100], [010] and [001] axes. The FWHMs of 1/2 1/2 0 and 1/4 1/4 1/4 parallel to [100] and [010] axes are about 0.03 nm, which is almost identical to the values in the case of the full sphere. The difference in the peak positions is within 0.004 nm. On the other hand, along the [001] axis, the peak of the limited sphere is broader than that of the full sphere. The FWHMs of the peaks of 1/2 1/2 0 and 1/4 1/4 1/4 increase from 0.032 nm to 0.079 nm and from 0.029 nm to 0.098 nm, respectively. This broadening prevents accurate determina- tion of the atomic position along the depth of the film. This broadening problem is solved by increasing the Q-range. One positive solution is to detect fluorescent **X**-rays excited by the incident beams passing through the substrate. This enlarges the angular range of the experimental hologram close to that of the full 4 π sphere. Consequently, it will provide us with the three-dimensional atomic positions around a specific atom in a thin film.

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Following the idea of Johnson (2002), we can express the variables u, W , p as double-asymptotic expansion (in ε and δ) with terms, depending only on η(**x**, t) and explicitly on z. As a result, a single nonlinear equation for η will be obtained, and thus all variables will be expressed through the solution of this equation.

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As an optic technique, CH is affected by factors such as surface slope and its position within the working range. In order to avoid these factors of influence, the roughness specimens were located on a test bench taking care that the test surfaces stayed parallel to the XY plane of the Conoscan 4000 scanner at distance equal to the sensor stand-off. The main directions of the test specimens were also aligned with the machine **X**-Y axes.

by means of constructing the Lax (zero-curvature) representation for the **superstring** equations of motion. In [28] we have shown that this connection admits a consistent reduction to the fields describing excitations from the su (1 | 1) sector. Thus, the non- trivial interacting Dirac Hamiltonian [28] which governs the dynamics in this sector is integrable, but its integrable properties are not transparent rather they are hidden in the highly non-trivial Lax pair. This pair can be formulated in terms of two 4 × 4 matrices, L σ and L τ , depending on a spectral parameter z and satisfying the condition of zero curvature

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If sufficient data are available, the model can be extended to consider situations where different stellar populations are in sub-disks with different radial scale lengths and scale heights This is relevant because analysis of the Milky Way [**5**] found different star populations in sub-disks with scale height inversely related to sub-disk radial scale lengths.

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The use of 3D technology in the field of **holography** opens up new opportunities in creating models in space. Modern information technology software significantly accelerates the process of creating holograms and improves the result. 3D technology helps to reduce the size of devices to reproduce holograms and increase their productivity. The disadvantage is that to model and reproduces holograms with a 3D image, computers with greater productivity than for 2D should be used, but the use of productive devices is a necessary step for the transition to the era of **holography** and the replacement of conventional projectors and showcases with modern and more productive ones. Information technology software for reproducing 3D images must be multipurpose and work on any device. It will create a fundamentally new situation in presentations and advertising.

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Alaska Department of Education and Early Development, Division of Teaching and Learning Support, Assessment, Accountability and Student Information, Standards Based Assessments (SBAS), [r]

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be shown that the superpotential terms of such a state with the Standard Model states vanish to all orders of nonrenormalizable terms. In this case the exotic states can interact with the Standard Model states only via the gauge interactions and cannot decay into them . In such a model therefore an exotic state will be stable and one has to check that its mass density does not over close the universe. These constraints were investigated in detail in ref. [39]. Several general remarks however are in order. Exotic states that do not have GUT origin are generic in **superstring** models. They arise due to the breaking of the non-Abelian gauge symmetries at the string rather than in the effective field theory level. Such states are therefore a generic signature of **superstring** compactification. Thus, they may lead to possible observable experimental signatures. For example, all the level one models predict the existence of fractionally charged states at least with Planck scale masses. Specific free fermionic models also predict the existence of Standard Standard Model states which are exotic from the point of view of the underlying SO(10). These may be color triplets, electroweak dou- blets or Standard Model singlets. Such states may provide an experimental signature of specific classes of **superstring** compactifications.

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Yes **X** Resources are being developed that support the standards. The resources will include content support and lesson models around the domains. In addition, examples of differentiated lessons and assessment supports will also be developed. Professional development will focus on standards orientation and best practices. Workshops will be in person at various locations around the state and virtual.

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