A black hole is a solution to Einstein’s **equations**. Classically it does not have many physical quantities for us to study; however quantum-mechanically, it is a thermal dynamical system. An analogy between the laws of black hole dynamics and the laws of thermodynamics was discovered more than 30 years ago. In particular, Bekenstein and Hawking showed that a black hole’s entropy is one-quarter the area of its event horizon in gravitational units. In 1995 Strominger and Vafa gave a microscopic description of black hole entropy [26] in terms of D-brane bound states, where D-branes are the sources of the black hole. Later on, it was found that the partition function of a 4-dimensional black hole is related to the partition function of a closed **topological** **string**. In the limit of a large black hole, that is, small curvature of the event horizon, this beautiful relation is known as the Ooguri, Strominger, and Vafa (OSV) conjecture [25]. On one side, the black hole is a solution to Einstein’s **equations** in 4 dimensions which is obtained by type II **string** compactification on Calabi-Yau 3-folds; on the other side, the closed **topological** **string** is evaluated at the attractor point of moduli space of the same Calabi-Yau 3-fold associated to the black hole charge.

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The extended **holomorphic** **anomaly** **equations** of the previous chapter express the anti- **holomorphic** derivatives of amplitudes at worldsheet genus g and boundary number h in terms of amplitudes with lower genus and/or boundary number, plus insertions corresponding to insertions of marginal closed **string** states, or equivalently covariant derivatives. Integrating these **equations** thus allows the recursive solving of amplitudes, up to a **holomorphic** function (integration constant) at each order. This chapter will develop and prove a technique for doing so, using a set of rules which take the form of Feynman rules. Note that we assume the vanishing of disk one-point functions throughout, so the new anomalies identified in the last chapter are not present — indeed, as they do not admit a recursive structure, the approaches in this chapter require their absence.

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There is at present no doubt that some mathematical concepts of ﬁbre bundle the- ory have become an established part of mathematical physics because ﬁbre bundles provide a natural and very deep framework for discussing the concepts of relativity and invariance, describing gravitation and other gauge ﬁelds, and giving a geometrical interpretation to quantization and the canonical formalism of particles and ﬁelds. Fibre bundles provide the language which is needed for dealing with local problems of dif- ferential geometry and ﬁeld **theory**. They are necessary to understand and solve global, **topological** problems, such as those arising in connection with magnetic monopoles and instantons. For example, in an attempt to understand the properties of Donaldson invariants of four manifolds, E. Witten presented a new approach to using physics to illuminate Donaldson **theory** (Witten [64] and Donaldson [19]). (For a very illuminating survey of the Seiberg-Witten **equations** and their relation to **topological** invariants of four-manifolds, see [19]. Donaldson has stressed the importance of these **equations** by the following words: “Since 1982 the use of gauge **theory**, in the shape of the Yang- Mills instanton **equations**, has permeated research in 4-manifold topology. (. . .) A body of techniques has built up through the eﬀorts of many mathematicians, producing re- sults which have uncovered some of the mysteries of 4-manifold **theory**, and leading to substantial internal conundrums within the ﬁeld itself. In the last three months of 1994 a remarkable thing happened: this research was turned on its head by the introduction of a new kind of diﬀerential-geometric **equations** by Seiberg and Witten: in the space of a few weeks, long-standing problems were solved, new and unexpected results were found, along with simpler new proofs of existing ones, and new vistas for research opened up” [19, page 45].) He suggested that, instead of computing the Donaldson in- variants by counting SU (2) instanton solutions, one can obtain the same invariants by cutting the solutions of the dual **equations**, which involve U (1) gauge ﬁelds and monopoles. From a physical point of view, the dual description via monopoles and Abelian gauge ﬁelds should be simpler than the microscopic SU (2) description since in the renormalization group sense it arises by “integrating out the irrelevant degrees of freedom.”

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There are two different **topological** gauge theories in 3d which can be obtained by twisting N = 4 d = 3 super-Yang-Mills **theory**. The first one is the dimensional reduction of the Donaldson-Witten twist of N = 2 d = 4 super-Yang-Mills **theory**. The second one is intrinsic to 3d and has been first discussed by Blau and Thompson [3]. We will refer to them as A-type and B-type **topological** gauge theories respectively. The reason for this terminology is that the BPS **equations** in the former **theory** are elliptic, as in the usual A-model, while in the latter **theory** they are overdetermined, as in the usual B-model. The definition and some properties of the B-type 3d gauge **theory** (for a general gauge group) are described in the Section 5.3. In this subsection we only deal with the abelian case. Consider the 3d bosonic fields A, φ and the fermionic fields ψ + i ψ, χ, η ˜ − i˜ η, ψ 4 − i ψ ˜ 4 . It is

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We derive an approximate analytic relation between the number of consistent heterotic Calabi-Yau compactiﬁcations of **string** **theory** with the exact charged matter content of the standard model of particle physics and the **topological** data of the internal manifold: the former scaling exponentially with the number of Kähler parameters. This is done by an estimate of the number of solutions to a set of Diophantine **equations** representing constraints satisﬁed by any consistent heterotic **string** vacuum with three chiral massless families, and has been computationally checked to hold for complete intersection Calabi-Yau threefolds (CICYs) with up to seven Kähler parameters. When extrapolated to the entire CICY list, the relation gives ∼ 10 23 **string** **theory** standard models; for the class of Calabi-Yau hypersurfaces in toric varieties, it gives ∼ 10 723 standard models.

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a behavior that is drastically different from the one showed in the Boundary formulation of SFT [49] (as well in frameworks such CFT, boundary state approach, RG flow analysis, etc.), in which the tachyon monotonically rolls from the perturbative unstable vacuum towards the true vacuum. The Cubic SFT tachyon starts from the maximum of the po- tential at t = −∞, rolls down past the minimum and undergoes evergrowing oscillations, still the energy being conserved. The pressure oscillates similarly, thus not representing tachyon matter. In [56] an argument was done to reconcile this discrepancy, based on the field redefinition we derived in [13], that would map the CSFT oscillating solution to the BSFT “well-behaved” one. As we explained at the end of Chapter 3, it seems to us that some more informations about this mapping should come at least from the knowledge of a further order - the third - of this field redefinition. This would be an important step forward a meaningful interpretation of the puzzling behavior of the Cubic SFT solution. Another point of view about this discrepancy is related to an important general is- sue of Cubic open **string** field **theory**, the gauge fixing. To perform explicit calculations in **string** field **theory**, the gauge symmetry of the cubic SFT action must be fixed. Al- most all the studies in Cubic SFT - included the results quoted above - are conducted in Feynman–Siegel gauge. This gauge, however, is known to show a pathological behav- ior in the effective tachyon potential [28]. Namely, branch points appear in both side of larger and smaller field values of the tachyon so that one could not go beyond this small region 1 . These branch points arise because the trajectory in field space associ- 1 It is worth mentioning again here that in the Boundary SFT approach the tachyon potential can be

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When explaining cognitive control in either mode, the brain is represented as a network of brain regions connected through white matter tracts. In the vertical mode, the explanans is the global **topological** property such as small-worldliness and the explanandum is the global physical property such as global cognitive control (whether global cognitive control is theoretically possible). The most basic way to illustrate small-worldliness as a global **topological** property is through Watts and Strogatz (1998) model. Amongst the most established ways to quantify networks are the average path length L(p) and clustering coefficient C(p). The L(p) measures the average number of edges that have to be traversed in order to reach one node from the other. Clustering is understood as a tendency of a small group of nodes to form connected triangles around one central node, which indicates that the connected neighboring nodes are also neighbors among themselves; hence they form a cluster or a clique. The clustering coefficient is a measure of this tendency, which characterizes a value for all nodes in a network (Kaiser and Hilgetag 2004, p. 312; van den Heuvel and Sporns 2013, p. 683). Networks that have high clustering coefficients and low path lengths are called small-world networks (Watts and Strogatz 1998). Small-worldliness as a global **topological** property indicates that almost any two nodes in the network will be connected either directly or through a minimal number of indirect connections, which shortens the distance between the nodes within a neighborhood of nodes as well as between neighborhoods of nodes, and neighborhoods of neighborhoods, which further ensures that the energy requirements for changing any of the trajectories will be minimal, and thus explains why the network is globally or in principle controllable.

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We think this consequence is an intuitively undesirable property. It generally prevents any act of learning (updating with) the actual world. Indeed, the main problem of Formal Learning **Theory** (learning the true world, or the correct possibility, from a given set of possibilities) becomes automatically unattainable. Similarly, the physicist’s dream of finding a true “**theory** of everything” is declared impossible by fiat, as a matter of logic. More importantly, even if necessity of error might seem realistic within a Lewisian “large- world interpretation” of possible-world semantics (in which each world must really come with a full description of all the myriad of ontic facts of the world), this property seems completely unrealistic when we adopt the more down-to-earth “small-world” models that are common in Computer Science, Game **theory** and other applications. In these fields, the “worlds” in any usable model come only with the description of the facts that are relevant for the problem at hand: e.g. in a scenario involving the throwing of a fair coin, the relevant fact is the upper face of the coin. A model for this scenario will involve typically only two possible worlds: Head and Tail. Requiring that the agent must always have a false belief means in this context that the agent can never find out which of the coin’s faces is the upper one: an obviously absurd conclusion!

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solution with respect to the formal solution depending on the geometry of the problem. The ﬁnal main result states the splitting of both formal and analytic solutions to the prob- lem under study as a sum of three terms. More precisely, if F denotes the Banach space of **holomorphic** and bounded functions deﬁned in T × H β , and u(t, ˆ z, ) stands for the

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Construction of automorphic forms via Poincar´ e series is a classical technique, and there is a significant amount of literature on this subject. For example, it is well- known that any **holomorphic** k-differential (k is an integer, k ≥ 2) on a compact Riemann surface of genus g ≥ 2 is obtained from the Poincar´ e series of a polynomial in z of degree not higher than k(2g − 2). There are various descriptions of the kernel of the Poincar´ e series operator: See, in particular, [25], [26], [30], [31], [33].

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principles within a gauge field **theory** framework, but at the same time also explains the scale hierarchy between the Planck scale and the abundance of black hole entropy. Dawid claims that it is unlikely that there is more than one **theory** that explains all of those phenomena, rather than just one (in an unexpected way). Thus UEA ‘constrains’ underdetermination. But why is it unlikely? Although it seems plausible that a **theory** is likely to fit the facts when it was constructed to fit the facts, the reverse does not follow. Moreover, Dawid should be aware that the use-novelty account he appeals to has been shown to be highly problematic (Musgrave 1974). Suppose scientist S1, due to her ignorance of the **theory** itself or her ignorance of the extant phenomena, did not expect **theory** T to explain phenomenon P, whereas S2 did. Then, absurdly, T would receive more confirmation from P, when considered by S1, than when considered by S2. Furthermore, why can we not appreciate **string** **theory** simply for its unifying power? For reasons that escape me, Dawid is unwilling to base his account on the latter.

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A robot can be moved by a user or for itself using exploration techniques such as frontier-based exploration [13]. While a robot is roaming around the environment, it records RSSIs at each sampling time. In addition, it records the position of itself to display the node map and to cluster anomalous positions if the number of the generated nodes is too many. Note that the positions are not input to the HTM. The position of the robot can be estimated by using GMapping [14] that showed good localization performance on mobile robots. The HTM for the **topological** node map takes only the RSSI as an input to compute the **anomaly** score. The position information will not be used to recognize a node since the purpose of the recognition is to globally localize a robot using only WiFi signals.

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The laws of quantum mechanics were found to be successful in describing single particles and atoms. However, when particles and atoms accumulate them self to from a bulk matter, quantum laws suffers from some long standing problems [5,6].One of them is known as many body problem[7].The behavior of superconductors(Sc) at high temperature, till know, have no well defined simple full quantum **theory** to describe them [8,9]. **Theory** that can describe the early universe and unify gravity with other forces [10, 11].This may be attributed to the fact that, when particles ender a medium, quantum **equations** accounts for the effect of potential only, does not account for the effect of friction anther particles. Some attempts were made by the first authore do consider the friction effect. This work is concerned with alternative derivations based on Maxwell's and classical oscillator **equations**. The derivations were done in sections 2,3 and 4.The new equation is applied to oscillators in sec 5. Discussion and conclusion. The new equation is applied to oscillators in sections 6 and 7.

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Our cyclic universe started from the zero-energy universe through the four-stage cyclic transformation. Emer- ging from the zero-energy universe, the four-stage transformation consists of the 11D positive-negative energy dual membrane universe, the 10D positive-negative energy dual **string** universe, the 10D positive-negative ener- gy dual particle universe, and the 4D (light)-variable D (dark) positive-negative energy dual particle asymme- trical universe. The transformation can then be reversed back to the zero-energy universe through the reverse four-stage transformation. The light universe is an observable four-dimensional universe started with the infla- tion and the Big Bang, and the dark universe is a variable dimensional universe from 10D to 4D. The dark uni- verse could be observed as dark energy only when the dark universe turned into a four-dimensional universe. The four-stage transformation explains the four force fields in our universe. The theoretical calculated percen- tages of dark energy, dark matter, and baryonic matter are 68.3, 26.4, and 5.3, respectively, in agreement with observed 68.3, 26.8, and 4.9, respectively. According to the calculation, dark energy started in 4.28 billion years ago in agreement with the observed 4.71 ± 0.98 billion years ago.

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We obtain eﬃcient conditions under which some or all solutions of a nonlinear equation in a **topological** vector space preordered by a closed wedge are comparable with respect to the corresponding preordering. Conditions suﬃcient for the equivalence of comparable solutions are also given. The wedge under consideration is not assumed to be a cone, nor any continuity conditions are imposed on the mappings considered.

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There is an important question about choice of the additional conditions, additional requirements, that should be added to **equations** (4), (5), (6) in order to provide the uniqueness of the solution. These conditions should allow to evaluate the functions and to plot the ﬁgures. I suggest hints for these additional conditions and I illustrate these with evaluation of the Fourth ackermann, which already has special name, tetration:

Abstract In this paper, we obtained the 1-soliton solutions of the symmetric regularized long wave (SRLW) equation and the (3+1)-dimensional shallow water wave **equations**. Solitary wave ansatz method is used to carry out the integration of the **equations** and obtain **topological** soliton solutions The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Note that, this method is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.

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The main goal of this paper is the initial value problem for a system of autonomous differential **equations**. Let us consider a nonempty set Φ in the phase space G of such system. If a trajectory of the system crosses the set Φ then the starting point of the trajectory is called a starting (initial) point of reachability and Φ is called a set of reachability. Some basic properties of the starting points set of reachability are studied in the paper.

From a different perspective, one would like to close this logical circle and check that the same result is obtained in the framework of the quantum **string** Bethe **equations** pro- posed originally in [17]. Indeed, it would be very nice to show that these **equations** repro- duce the scaling function in the suitable long **string** limit. Also, one expects to find some simplifications due to the fact that only the first terms in the strong coupling expression of the dressing must be dealt with. On the other side, the BES equation certainly requires all the weak-coupling terms if it has to be extrapolated at large coupling.

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In Section 2, **string** **theory** with oscillating space-time dimension number is derived from varying speed of light in the framework of special relativity. In Section 3, the digital space structure consists of attachment space (denoted as 1) for rest mass and reversible movement and detachment space (denoted as 0) for irreversible ki- netic energy. In Section 4, the proposed cosmology based on **string** **theory** and the digital space structure leads to the orbital structure for the periodic table of elementary particles. Section 5 discusses the periodic table of ele- mentary particles.

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