We find a Polyakov-type action for strings moving in a torsional Newton-Cartan geometry. This is obtained by starting with the relativistic Polyakov action and fixing the momentum of the string along a non-compact null isometry. For a flat target space, we show that the world-sheet theory becomes the Gomis-Ooguri action. From a target space perspective these strings are non-relativistic but their world-sheet theories are still relativistic. We show that one can take a scaling limit in which also the world-sheet theory becomes non-relativistic with an infinite-dimensional symmetry algebra given by the Galilean conformal algebra. This scaling limit can be taken in the context of the AdS/CFT correspondence and we show that it is realized by the 'Spin Matrix Theory' limits of strings on AdS 5 × S 5 . Spin Matrix theory arises as non-relativistic limits of the AdS/CFT correspondence close to BPS bounds. The duality between non-relativistic strings and Spin Matrix theory provides a holographic duality of its own and points towards a framework for more tractable holographic dualities whereby non-relativistic strings are dual to near BPS limits of the dual field theory.2 As will be clear in Sec. 2.1 we find in this paper that the TNC geometry is extended with a periodic target space direction.3 This Nambu-Goto form was also obtained in [20]. 4 The GCA was also observed in earlier work on non-relativistic limits of AdS/CFT [21]. See also Ref. [22] for useful work on representations of the GCA and aspects of non-relativistic conformal two-dimensional field theories.
Non-relativistic string theories promise to provide simpler theories of quantum gravity as well as tractable limits of the AdS/CFT correspondence. However, several apparently distinct non-relativistic string theories have been constructed. In particular, one approach is to reduce a relativistic string along a null isometry in target space. Another method is to perform an appropriate large speed of light expansion of a relativistic string. Both of the resulting non-relativistic string theories only have a well-defined spectrum if they have nonzero winding along a longitudinal spatial direction. In the presence of a Kalb-Ramond field, we show that these theories are equivalent provided the latter direction is an isometry. Finally, we consider a further limit of non-relativistic string theory that has proven useful in the context of AdS/CFT (related to Spin Matrix Theory). In that case, the worldsheet theory itself becomes non-relativistic and the dilaton coupling vanishes.
We study the (2+1) dimensional Dirac equation in an homogeneous magnetic field (relativistic Landau problem) within a minimal length, or generalized uncertainty principle -GUP-, scenario. We derive exact solutions for a given explicit representation of the GUP and provide expressions of the wave functions in the momentum representation. We find that in the minimal length case the degeneracy of the states is modified and that there are states that do not exist in the ordinary quantum mechanics limit (β → 0). We also discuss the massless case which may find application in describing the behavior of charged fermions in new materials like Graphene.
We obtain exact solutions of the (2+1) dimensional Dirac oscillator in a homogeneous magnetic field within a minimal length (∆x0 = √ β), or generalized uncertainty principle (GUP) scenario. This system in ordinary quantum mechanics has a single left-right chiral quantum phase transition (QPT). We show that a non zero minimal length turns on a infinite number of quantum phase transitions which accumulate towards the known QPT when β → 0. It is also shown that the presence of the minimal length modifies the degeneracy of the states and that in this case there exist a new class of states which do not survive in the ordinary quantum mechanics limit β → 0.
Setting sale prices correctly is of great importance for firms, and the study and forecast of prices time series is therefore a relevant topic not only from a data science perspective but also from an economic and applicative one. In this paper, we examine different techniques to forecast sale prices applied by an Italian food wholesaler, as a step towards the automation of pricing tasks usually taken care by human workforce. We consider ARIMA models and compare them to Prophet, a scalable forecasting tool by Facebook based on a generalized additive model, and to deep learning models exploiting Long Short-Term Memory (LSTM) and Convolutional Neural Networks (CNNs). ARIMA models are frequently used in econometric analyses, providing a good benchmark for the problem under study. Our results indicate that ARIMA models and LSTM neural networks perform similarly for the forecasting task under consideration, while the combination of CNNs and LSTMs attains the best overall accuracy, but requires more time to be tuned. On the contrary, Prophet is quick and easy to use, but considerably less accurate.
This paper presents a comparison of different methodologies for monitoring the plants growth in a greenhouse. A 2D measurement based on Computer Vision algorithms and 3D shape measurements techniques (Structured light, LIDAR and photogrammetry) are compared. From the joined 2D and 3D data, an analysis was performed considering health plant indicators. The methodologies are compared among each other. The acquired data are then fed into Deep Learning algorithms in order to detect anomalies in plant growth. The final aim is to give an assessment on the image acquisition methodologies, selecting the most suitable to be used to create the Deep Learning model inputs saving time and resources.
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