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We show that magnetic reconnection and topology-change can be understood, and distinguished, in terms of trajectories of Alfv\'enic wave-packets ${\bf x}(t)$ moving along the magnetic field ${\bf B(x}, t)$ with Alfv\'en velocity $\dot{\bf x}(t)={\bf V}_A({\bf x},t)$, i.e. adopting a Lagrangian formalism for virtual particles. A considerable simplification is attained, in fact, by directly employing elementary concepts from hydrodynamic turbulence without appealing to the fictitious and complicated notion of magnetic field lines moving through plasma. In incompressible flows, Alfv\'enic trajectories correspond to the dynamical system $\dot{\bf x}(t)={\bf B}$, where $\bf B$ solves the induction equation, with phase space $(\bf x,B)$. Metric topology of this phase space, at any time $t$, captures the intuitive notion that nearby wave-packets should remain nearby at a slightly different time $t\pm \delta t$, unless topology changes e.g., by dissipation. In fact, continuity conditions for magnetic field allow rapid but continuous divergence of these trajectories, i.e., reconnection, but not discontinuous divergence which would change magnetic topology. Thus topology can change only due to time-reversal symmetry breaking e.g., by dissipation or turbulence. In laminar and even chaotic flows, the separation of Alfv\'enic trajectories at all times remains proportional to their initial separation, i.e., slow reconnection, and topology changes by dissipation with a rate proportional to resistivity. In turbulence, trajectories diverge super-linearly with time independent of their initial separation, i.e., fast reconnection, and magnetic topology changes by turbulent diffusion with a rate independent of small-scale plasma effects. Our results strongly support the Lazarian-Vishniac model of stochastic reconnection and its reformulation by Eyink in terms of stochastic flux-freezing.

In this study, we introduce a novel Mylar-based pouch motor design that leverages the reversible actuation capabilities of Peltier junctions to enable agonist-antagonist muscle mimicry in soft robotics. Addressing the limitations of traditional silicone-based materials, such as leakage and phase-change fluid degradation, our pouch motors filled with Novec 7000 provide a durable and leak-proof solution for geometric modeling. The integration of flexible Peltier junctions offers a significant advantage over conventional Joule heating methods by allowing active and reversible heating and cooling cycles. This innovation not only enhances the reliability and longevity of soft robotic applications but also broadens the scope of design possibilities, including the development of agonist-antagonist artificial muscles, grippers with can manipulate through flexion and extension, and an anchor-slip style simple crawler design. Our findings indicate that this approach could lead to more efficient, versatile, and durable robotic systems, marking a significant advancement in the field of soft robotics., Comment: submitted to IROS 2024, 7 pages, 9 figures

In this work, we introduce an advanced thermo-active variable impedance module which builds upon our previous innovation in thermal-based impedance adjustment for actuation systems. Our initial design harnessed the temperature-responsive, viscoelastic properties of Polycaprolactone (PCL) to modulate stiffness and damping, facilitated by integrated flexible Peltier elements. While effective, the reliance on compressing and the inherent stress relaxation characteristics of PCL led to suboptimal response times in impedance adjustments. Addressing these limitations, the current iteration of our module pivots to a novel 'shear-mode' operation. By conducting comprehensive shear rheology analyses on PCL, we have identified a configuration that eliminates the viscoelastic delay, offering a faster response with improved heat transfer efficiency. A key advantage of our module lies in its scalability and elimination of additional mechanical actuators for impedance adjustment. The compactness and efficiency of thermal actuation through Peltier elements allow for significant downsizing, making these thermal, variable impedance modules exceptionally well-suited for applications where space constraints and actuator weight are critical considerations. This development represents a significant leap forward in the design of variable impedance actuators, offering a more versatile, responsive, and compact solution for a wide range of robotic and biomechanical applications., Comment: Submitted to IROS 2024, 7 pages, 10 figures

Soft robotics is a rapidly growing field that spans the fields of chemistry, materials science, and engineering. Due to the diverse background of the field, there have been contrasting naming schemes such as 'intelligent', 'smart' and 'adaptive' materials which add vagueness to the broad innovation among literature. Therefore, a clear, functional and descriptive naming scheme is proposed in which a previously vague name -- Soft Material for Soft Actuators -- can remain clear and concise -- Phase-Change Elastomers for Artificial Muscles. By synthesizing the working principle, material, and application into a naming scheme, the searchability of soft robotics can be enhanced and applied to other fields. The field of thermo-active soft actuators spans multiple domains and requires added clarity. Thermo-active actuators have potential for a variety of applications spanning virtual reality haptics to assistive devices. This review offers a comprehensive guide to selecting the type of thermo-active actuator when one has an application in mind. Additionally, it discusses future directions and improvements that are necessary for implementation., Comment: 16 pages, 10 figures, accepted to Robotics Reports

Unsupervised Machine Learning techniques have been applied to Natural Language Processing tasks and surpasses the benchmarks such as GLUE with great success. Building language models approach achieves good results in one language and it can be applied to multiple NLP task such as classification, summarization, generation and etc as an out of box model. Among all the of the classical approaches used in NLP, the masked language modeling is the most used. In general, the only requirement to build a language model is presence of the large corpus of textual data. Text classification engines uses a variety of models from classical and state of art transformer models to classify texts for in order to save costs. Sequence Classifiers are mostly used in the domain of text classification. However Token classifiers also are viable candidate models as well. Sequence Classifiers and Token Classifier both tend to improve the classification predictions due to the capturing the context information differently. This work aims to compare the performance of Sequence Classifier and Token Classifiers and evaluate each model on the same set of data. In this work, we are using a pre-trained model as the base model and Token Classifier and Sequence Classier heads results of these two scoring paradigms with be compared.., Comment: 11 Pages, 3, figures

In this article, we propose using deep learning and transformer architectures combined with classical machine learning algorithms to detect and identify text anomalies in texts. Deep learning model provides a very crucial context information about the textual data which all textual context are converted to a numerical representation. We used multiple machine learning methods such as Sentence Transformers, Auto Encoders, Logistic Regression and Distance calculation methods to predict anomalies. The method are tested on the texts data and we used syntactic data from different source injected into the original text as anomalies or use them as target. Different methods and algorithm are explained in the field of outlier detection and the results of the best technique is presented. These results suggest that our algorithm could potentially reduce false positive rates compared with other anomaly detection methods that we are testing., Comment: 8 Pages, 4 Figures

We discuss application of methods from the Kraichnan model of turbulent advection to the study of non-equilibrium concentration fluctuations arising during diffusion in liquid mixtures at high Schmidt numbers. This approach treats nonlinear advection of concentration fluctuations exactly, without linearization. Remarkably, we find that static and dynamic structure functions obtained by this method reproduce precisely the predictions of linearized fluctuating hydrodynamics. It is argued that this agreement is an analogue of anomaly non-renormalization which does not, however, protect higher-order multi-point correlations. The latter should thus yield non-vanishing cumulants, unlike those for the Gaussian concentration fluctuations predicted by linearized theory., Comment: 22 pages, 2 figures

The Frobenius coin problem in three variables, for three positive relatively prime integers $a_1< a_2< a_3$ asks to find the largest number not representable as $a_1x_1+a_2x_2+a_3x_3$ with non-negative integer coefficients $x_1$, $x_2$ and $x_3$. In this article, we present a new algorithm to solve this problem that is faster and in our belief simpler than all existing algorithms and runs in $\mbox{O}(\log a_1)$ steps.

We consider the effects of thermal noise on the Batchelor-Kraichnan theory of high Schmidt-number mixing in the viscous-dissipation range of turbulent flows. Using fluctuating hydrodynamics for a binary fluid mixture at low Mach numbers, we justify linearization around the deterministic Navier-Stokes solution in the dissipation range. For the latter solution we adopt the standard Kraichnan model and derive asymptotic high-Schmidt limiting equations for the concentration field, in which the thermal velocity fluctuations are exactly represented by a Gaussian random velocity which is white in time. We obtain the exact solution for the concentration spectrum in this high-Schmidt limiting model, showing that the Batchelor prediction in the viscous-convective range is unaltered. Thermal noise dramatically renormalizes the bare diffusivity in this range, but the effect is the same as in laminar flow and thus hidden phenomenologically. However, in the viscous-diffusive range at scales below the Batchelor length (typically micron scales) the predictions based on deterministic Navier-Stokes equations are drastically altered by thermal noise. Whereas the classical theories predict rapidly decaying spectra in the viscous-diffusive range, we obtain a $k^{-2}$ power-law spectrum starting just below the Batchelor length. This spectrum corresponds to non-equilibrium giant concentration fluctuations, due to the imposed concentration variations advected by thermal velocity fluctuations which are experimentally well-observed in quiescent fluids. At higher wavenumbers, the concentration spectrum instead goes to a $k^2$ equipartition spectrum due to equilibrium molecular fluctuations. We work out detailed predictions for water-glycerol and water-fluorescein mixtures. Finally, we discuss broad implications for turbulent flows and novel applications of our methods to experimentally accessible laminar flows., Comment: There was a rather subtle issue of defining meaningful concentration gradients, poorly explained in our original submission, which is discussed and explained in the second submission

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.

We study the effects of data size and quality on the performance on Automated Essay Scoring (AES) engines that are designed in accordance with three different paradigms; A frequency and hand-crafted feature-based model, a recurrent neural network model, and a pretrained transformer-based language model that is fine-tuned for classification. We expect that each type of model benefits from the size and the quality of the training data in very different ways. Standard practices for developing training data for AES engines were established with feature-based methods in mind, however, since neural networks are increasingly being considered in a production setting, this work seeks to inform us as to how to establish better training data for neural networks that will be used in production., Comment: 14 pages, 3 figures, 5 tables

These days different platforms such as social media provide their clients from different backgrounds and languages the possibility to connect and exchange information. It is not surprising anymore to see comments from different languages in posts published by international celebrities or data providers. In this era, understanding cross languages content and multilingualism in natural language processing (NLP) are hot topics, and multiple efforts have tried to leverage existing technologies in NLP to tackle this challenging research problem. In this survey, we provide a comprehensive overview of the existing literature with a focus on transfer learning techniques in multilingual tasks. We also identify potential opportunities for further research in this domain.

Automated Essay Scoring (AES) is a cross-disciplinary effort involving Education, Linguistics, and Natural Language Processing (NLP). The efficacy of an NLP model in AES tests it ability to evaluate long-term dependencies and extrapolate meaning even when text is poorly written. Large pretrained transformer-based language models have dominated the current state-of-the-art in many NLP tasks, however, the computational requirements of these models make them expensive to deploy in practice. The goal of this paper is to challenge the paradigm in NLP that bigger is better when it comes to AES. To do this, we evaluate the performance of several fine-tuned pretrained NLP models with a modest number of parameters on an AES dataset. By ensembling our models, we achieve excellent results with fewer parameters than most pretrained transformer-based models., Comment: 11 pages, 1 figure, 3 tables

Generative adversarial networks (GANs) are one of the most powerful generative models, but always require a large and balanced dataset to train. Traditional GANs are not applicable to generate minority-class images in a highly imbalanced dataset. Balancing GAN (BAGAN) is proposed to mitigate this problem, but it is unstable when images in different classes look similar, e.g. flowers and cells. In this work, we propose a supervised autoencoder with an intermediate embedding model to disperse the labeled latent vectors. With the improved autoencoder initialization, we also build an architecture of BAGAN with gradient penalty (BAGAN-GP). Our proposed model overcomes the unstable issue in original BAGAN and converges faster to high quality generations. Our model achieves high performance on the imbalanced scale-down version of MNIST Fashion, CIFAR-10, and one small-scale medical image dataset.

For an algebraically closed field $K$ of characteristic zero and a non-singular matrix $A\in \mbox{GL}_n(K)$, a semi-invariant polynomial of $A$ is defined to be a polynomial $p(x)=p(x_1,\dots,x_n)$ with coefficients in $K$ such that $p(xA)=\lambda p(x)$ for some $\lambda\in K$. In this article, we classify all semi-invariant polynomials of $A$ in terms of a canonically constructed basis that will be made precise in the text., Comment: A major revise, with a lot of new material

Let $n\ge 1$ and $s\ge 1$ be integers. An almost $s$-stable subset $A$ of $[n]=\{1,\dots,n\}$ is a subset such that for any two distinct elements $i, j\in A$, one has $|i-j|\ge s$. For a family $\cal F$ of non-empty subsets of $[n]$ and an integer $r\ge 2$, the chromatic number of the $r$-uniform Kneser hypergraph $\mbox{KG}^r({\cal F})$, whose vertex set is $\cal F$ and whose edge set is the set of $\{A_1,\dots, A_r\}$ of pairwise disjoint elements in $\cal F$, has been studied extensively in the literature and Abyazi Sani and Alishahi were able to give a lower bound for it in terms of the equatable $r$-colorability defect, $\mbox{ecd}^r({\cal F})$. In this article, the methods of Chen for the special family of all $k$-subsets of $[n]$, are modified to give lower bounds for the chromatic number of almost stable general Kneser hypergraph $\mbox{KG}^r({\cal F}_s)$ in terms of $\mbox{ecd}^s({\cal F})$. Here ${\cal F}_s$ is the collection of almost $s$-stable elements of $\cal F$. We also propose a generalization of a conjecture of Meunier., Comment: Minor typos and inaccuracies are fixed

Let $n\ge 1$, $r\ge 2$, and $s\ge 0$ be integers and ${\cal P}=\{P_1,\dots, P_l\}$ be a partition of $[n]=\{1,\dots, n\}$ with $|P_i|\le r$ for $i=1,\dots, l$. Also, let $\cal F$ be a family of non-empty subsets of $[n]$. The $r$-uniform Kneser-type hypergraph $\mbox{KG}^r({\cal F}, {\cal P},s)$ is the hypergraph with the vertex set of all $\cal P$-admissible elements $F\in {\cal F}$, that is $|F\cap P_i|\le 1$ for $i=1,\dots, l$ and the edge set of all $r$-subsets $\{F_1,\dots, F_r\}$ of the vertex set that $|F_i\cap F_j|\le s$ for all $1\le i

Local magnetic reversals are an inseparable part of magnetohydrodynamic (MHD) turbulence whose collective outcome on an arbitrary scale in the inertial range may lead to a global stochastic reconnection event with a rate independent of small scale physics. We show that this picture is intimately related to the nanoflare theory of the solar corona. First, we argue that due to stochastic flux freezing, a generalized version of flux freezing in turbulence, the magnetic field follows the turbulent flow in a statistical sense. Bending and stretching an initially smooth field, therefore, the turbulence generally increases the magnetic spatial complexity. Strong magnetic shears associated with such a highly tangled field can trigger local reversals and field annihilations that convert magnetic energy into kinetic and thermal energy respectively. The former maintains the turbulence, which incidentally continues to entangle the field completing the cycle, while the latter enhances the heat generation in the dissipative range. We support this theoretical picture invoking recent analytical and numerical studies which suggest a correlation between magnetic complexity and magnetic energy dissipation. The amplification of multiple local, in-phase reversals by super-linear Richardson diffusion may initiate a global reconnection at larger scales, however, even in the absence of such a global stochastic reconnection, the small scale reversals will continue to interact with the turbulence. We employ conventional scaling laws of MHD turbulence to illustrate that these local events are indeed efficient in both enhancing the turbulence and generating heat. Finally, using an MHD numerical simulation, we show that the time evolution of the magnetic complexity is statistically correlated with the kinetic energy injection rate and/or magnetic-to-thermal energy conversion rate.

The level of spatial complexity associated with a given vector field on an arbitrary range of scales \iffalse ${\bf F(x}, t)$ can be quantified by a simple, time-dependent function $S(t)={1\over 2}(1-\hat{\bf F}_l.\hat{\bf F}_L)_{rms}$ with ${\bf F}_l$ (${\bf F}_L$) being the average field in a volume of scale $l$ ($L>l$) and the unit vector defined as $\hat{\bf F}={\bf F}/|{\bf F}|$. Thus,\fi can be quantified by a simple, scale-dependent function of time; $0\leq S(t)\leq 1$. Previous work has invoked kinetic and magnetic complexities, associated with velocity and magnetic fields ${\bf u(x}, t)$ and ${\bf B(x}, t)$, to study magnetic reconnection and diffusion in turbulent and magnetized fluids. In this paper, using the coarse-grained momentum equation, we argue that the fluid jets associated with magnetic reconnection events at an arbitrary scale $l$ in the turbulence inertial range are predominantly driven by the Lorentz force ${\bf{N}}_l=({\bf j\times B})_l-{\bf j}_l\times {\bf B}_l$. This force, is induced by the subscale currents and is analogous to the turbulent electromotive force ${\cal E}_l=({\bf u\times B})_l-{\bf u}_l\times {\bf B}_l$ in dynamo theories. Typically, high (low) magnetic complexities during reconnection imply large (small) spatial gradients for the magnetic field, i.e., strong (weak) Lorentz forces ${\bf N}_l$. Reconnection launches jets of fluid, hence the rate of change of kinetic complexity is expected to strongly correlate with the power injected by the Lorentz force ${\bf N}_l$. We test this prediction using an incompressible, homogeneous magnetohydrodynamic (MHD) simulation and associate it with previous results. It follows that the stronger (weaker) the turbulence, the more (less) complex the magnetic field and the stronger (weaker) the reconnection field and thus the ensuing reconnection.

In this paper, we prove a generalization of a conjecture of Erd\"{o}s, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph $\mbox{KG}^r_s(n,k)$, has all $k$-subsets of $\{1,\dots,n\}$ as the vertex set and all multi-sets $\{A_1,\dots, A_r\}$ of $k$-subsets with $s$-wise empty intersections as the edge set. The case $r=s=2$, was considers by Kneser \cite{K} in 1955, where he conjectured that its chromatic number is $n-2(k-1)$. This was finally proved by Lov\'asz \cite{L} in 1978. The case $r>2$ and $s=2$, was considered by Erd\"os in 1973, and he conjectured that its chromatic number is $\left\lceil\frac{n-r(k-1)}{r-1}\right\rceil$. This conjecture was proved by Alon, Frankl and Lov\'asz \cite{AFL} in 1986. The case where $s>2$, was considered by Sarkaria \cite{S} in 1990, where he claimed to prove a lower bound for its chromatic number which generalized all previous results. Unfortunately, an error was found by Lange and Ziegler \cite{Z'} in 2006 in the induction method of Sarkaria on the number of prime factors of $r$, and Sarkaria's proof only worked when $s$ is less than the smallest prime factor of $r$ or $s=2$. In this paper, by applying the $\mathbb Z_p$-Tucker lemma of Ziegler \cite{Z} and Meunier \cite{M}, we finally prove the general Erd\"{o}s conjecture and prove the claimed result of Sarkaria for any $2\le s\le r$. We also provide another proof of a special case of this result, using methods similar to those of Alon, Frankl, and Lov\'asz \cite{AFL} and compute the connectivity of certain simplicial complexes that might be of interest in their own right., Comment: Minor edit and correcting few typos

Magnetic reconnection, topological change in magnetic fields, is a fundamental process in magnetized plasmas. It is associated with energy release in regions of magnetic field annihilation, but this is only one facet of this process. Astrophysical flows normally have very large Reynolds numbers and are expected to be turbulent, in agreement with observations. In strong turbulence magnetic lines constantly reconnect everywhere at all scales, making magnetic reconnection an intrinsic part of turbulent cascade. We note that this is inconsistent with the usual practice of regarding magnetic lines as persistent dynamical elements. A number of theoretical, numerical, and observational studies, starting with Lazarian & Vishniac (1999), demonstrated that 3D turbulence makes magnetic reconnection fast and that these two processes are intrinsically connected. We discuss the dramatic violation of the textbook concept of magnetic flux-freezing in the presence of turbulence and demonstrate that in the presence of turbulence the plasma effects are subdominant to turbulence as far as the magnetic reconnection is concerned. This justifies an MHD-like treatment of magnetic reconnection at scales much larger than the relevant plasma scales. We discuss numerical and observational evidences supporting the turbulent reconnection model. In particular, we show that tearing reconnection is suppressed in 3D and, unlike the 2D case, the 3D reconnection induces turbulence that makes reconnection independent of resistivity. We show that turbulent reconnection dramatically affects the key astrophysical processes, e.g., star formation, turbulent dynamo, acceleration of cosmic rays. We provide criticism of the concept of "reconnection-mediated turbulence" and explain why turbulent reconnection is very different from enhanced turbulent resistivity and hyper-resistivity, and why the latter has fatal conceptual flaws., Comment: 94 pages, 39 figures

In this paper, we present a new comparative study on automatic essay scoring (AES). The current state-of-the-art natural language processing (NLP) neural network architectures are used in this work to achieve above human-level accuracy on the publicly available Kaggle AES dataset. We compare two powerful language models, BERT and XLNet, and describe all the layers and network architectures in these models. We elucidate the network architectures of BERT and XLNet using clear notation and diagrams and explain the advantages of transformer architectures over traditional recurrent neural network architectures. Linear algebra notation is used to clarify the functions of transformers and attention mechanisms. We compare the results with more traditional methods, such as bag of words (BOW) and long short term memory (LSTM) networks., Comment: 17 pages, 7 figures

We study the evolution of turbulent magnetic fields from a topological point of view, invoking commonplace mathematical tools from general topology and dynamical systems theory which connect magnetic field evolution to time reversal invariance, entropy increase and the second law of thermodynamics. We show that in fact magnetic topology is well-defined only in the phase space corresponding to a dynamical system governed by the induction equation. Hence the field's topology and stochasticity can be studied in terms of the corresponding phase space trajectories rather than the field lines in real Euclidean space. In fact, our results suggest that magnetic field lines should not be taken too literally because their existence and uniqueness and more importantly continuity in time require strong mathematical conditions, hardly satisfied in astrophysical systems. As for magnetic topology change, it is shown that the phase space topology is preserved in time for a magnetic field which, besides satisfying few continuity conditions, solves a time reversal invariant induction equation. What breaks the time symmetry in the induction equation is the presence of non-ideal plasma effects at small scales such as resistivity, which results from random collisions between diffusing electrons and other particles. The small scale, stochastic disturbances produced thereby are super-linearly amplified by Richardson diffusion in the turbulent cascade, which are eventually manifested as large scale reconnection events, somehow similar to stretching quantum fluctuations during inflation to seed large scale cosmological structures. This suggests that reconnection is rooted in the second law of thermodynamics that dictates entropy increase which in turn breaks the time symmetry.

The perturbation of a semiconductor from the thermodynamic equilibrium often leads to the display of nonlinear dynamics and formation of spatiotemporal patterns due to the spontaneous generation of competing processes. Here, we describe the ultrafast imaging of nonlinear carrier transport in silicon, excited by an intense femtosecond laser pulse. We use scanning ultrafast electron microscopy (SUEM) to show that, at a sufficiently high excitation fluence, the transport of photoexcited carriers slows down by turning into an oscillatory process. We attribute this nonlinear response to the electric field, generated by the spatial separation of these carriers under intrinsic and photo-induced fields; we then provide an advection-diffusion model that mimics the experimental observation. Our finding provides a direct imaging evidence for the electrostatic oscillation of hot carriers in highly excited semiconductors and offers new insights into their spatiotemporal evolution as the equilibrium is recovered.

We study the topology associated with physical vector and scalar fields. A mathematical object, e.g., a ball, can be continuously deformed, without tearing or gluing, to make other topologically equivalent objects, e.g., a cube or a solid disk. If tearing or gluing get involved, i.e., the deformation is not continuous anymore, the initial topology will consequently change giving rise to a topologically distinct object, e.g., a torus. This simple concept in general topology may be employed in the study of physical systems described by fields. Instead of continuously deforming objects, we can take a continuously evolving field, with an appropriately defined topology, such that the topology remains unchanged in time unless the system undergoes an important physical change, e.g., a transition to a different energy state. For instance, a sudden change in the magnetic topology in an energetically relaxing plasma, a process called reconnection, strongly affects the dynamics, e.g., it is involved in launching solar flares and generating large scale magnetic fields in astrophysical objects. In this topological formalism, the magnetic topology in a plasma can spontaneously change due to the presence of dissipative terms in the induction equation which break its time symmetry. We define a topology for the vector field $\bf F$ in the phase space $(\bf x, F)$. As for scalar fields represented by a perfect fluid, e.g., the inhomogeneous inflaton or Higgs fields, the fluid velocity $\bf u$ defines the corresponding topology. The vector field topology in its corresponding phase space $(\bf x, F)$ will be preserved in time if certain conditions including time reversal invariance are satisfied by the field and its governing differential equation.

Previous work has introduced scale-split energy density \psi for a given vector field B in order to quantify the field stochasticity S(t). Application to turbulent magnetic fields leads to the prediction that tangling magnetic field by turbulence increases magnetic stochasticity. An increasing stochasticity in turn leads to disalignments of the coarse-grained fields at smaller scales thus they average to weaker fields at larger scales upon coarse-graining. The field's resistance against tanglement by the turbulence may lead at some point to its sudden slippage through the fluid, decreasing the stochasticity and increasing the energy density. Thus the maxima (minima) of magnetic stochasticity are expected to approximately coincide with the minima (maxima) of energy density, occurrence of which corresponds to slippage of the magnetic field through the fluid. Field-fluid slippage, on the other hand, has been already found to be intimately related to magnetic reconnection. In this paper, we test these theoretical predictions numerically using a homogeneous, incompressible magnetohydrodynamic (MHD) simulation. Apart from expected small scale deviations, possibly due to e.g., intermittency and strong field annihilation, the theoretically predicted global relationship between stochasticity and magnetic energy is observed in different sub-volumes of the simulation box. This may indicate ubiquitous local field-fluid slippage and small scale reconnection events in MHD turbulence. We also show that the maximum magnetic stochasticity, i.e., \partial_t S(t)=0 & \partial^2_t S(t)<0, leads to sudden increases in kinetic stochasticity level which may correspond to fluid jets driven by the reconnecting field lines, i.e., reconnection. This suggests a new mathematical approach to the reconnection problem.

We develop a quantitative relationship between magnetic diffusion and the level of randomness, or stochasticity, of the diffusing magnetic field in a magnetized medium. A general mathematical formulation of magnetic stochasticity in turbulence has been developed in previous work in terms of the ${\cal L}_p$-norm $S_p(t)={1\over 2}|| 1-\hat{\bf B}_l.\hat{\bf B}_L||_p$, $p$th order magnetic stochasticity of the stochastic field ${\bf B}({\bf x}, t)$, based on the coarse-grained fields, ${\bf B}_l$ and ${\bf B}_L$, at different scales, $l\neq L$. For laminar flows, stochasticity level becomes the level of field self-entanglement or spatial complexity. In this paper, we establish a connection between magnetic stochasticity $S_p(t)$ and magnetic diffusion in magnetohydrodynamic (MHD) turbulence and use a homogeneous, incompressible MHD simulation to test this prediction. Our results agree with the well-known fact that magnetic diffusion in turbulent media follows the super-linear Richardson dispersion scheme. This is intimately related to stochastic magnetic reconnection in which super-linear Richardson diffusion broadens the matter outflow width and accelerates the reconnection process.

This pedagogical note revisits the concept of electromagnetic helicity in classical systems. In particular, magnetic helicity and its role in mean field dynamo theories is briefly discussed highlighting the major mathematical inconsistency in most of these theories---violation of magnetic helicity conservation. A short review of kinematic dynamo theory and its classic theorems is also presented in the Appendix.

This short note serves as an introduction to gravitational radiation through reviewing the inspiral-plunge transition phase in extreme mass ratio binaries. We study the relativistic motion of a compact object (CO) of mass $m$ around a massive black hole of mass $M\gg m$. The Kerr-Newman metric, effective potential for the general case of elliptical orbits, gravitational radiation, orbital energy and angular momentum of a coalescing CO in Kerr spacetime and gravitational wave frequency and signal to noise ratio are briefly reviewed. The main focus is on the transition from inspiral to plunge for a CO assuming that a test particle approach is plausible in the regime $m\ll M$ without appealing to a perturbative analysis. The effective potential is used to obtain the properties of the Innermost Stable Circular Orbit (ISCO) near which the adiabatic inspiral phase ends abruptly and the CO enters the plunge phase. For the transition phase, the effective potential is expanded in terms of parameters such as the radial (coordinate) distance from the ISCO and the deviation of particle's angular momentum from its value at the ISCO to obtain the equation of motion. The equations of motion, during the inspiral and transition phases, are joined numerically and the gravitational wave frequency, number of wave cycles and signal to noise ratio (SN) during the transition is obtained for circular/inclined as well as elliptical/inclined orbits. The limitations and inaccuracies of the current methods used to approach this problem is discussed. A short introduction to the fundamental concepts of General Relativity, in particular Einstein Field Equations is also provided in the Appendix.

We review the current theoretical models of the inward advection of the large scale external magnetic fields in accretion discs. The most plausible theories for launching astrophysical jets rely on strong magnetic fields at the inner parts of the host accretion disks. An internal dynamo can in principle generate small scale magnetic fields in situ but generating a large scale field in a disk seems a difficult task in the dynamo theories. In fact, as far as numerous numerical experiments indicate, a dynamo-generated field in general would not be coherent enough over the large length scales of order the disk's radius. Instead, a large scale poloidal field dragged in from the environment, and compressed by the accretion, provides a more promising possibility. The difficulty in the latter picture, however, arises from the reconnection of the radial field component across the mid-plane which annihilates the field faster than it is dragged inward by the accretion. We review the different mechanisms proposed to overcome these theoretical difficulties. In fact, it turns out, that a combination of different effects, including magnetic buoyancy and turbulent pumping, is responsible for the vertical transport of the field lines toward the surface of the disk. The radial component of the poloidal field vanishes at the mid-plane, which efficiently impedes reconnection, and grows exponentially toward the surface where it can become much larger than the vertical field component. This allows the poloidal field to be efficiently advected to small radii until the allowed bending angle drops to of order unity, and the field can drive a strong outflow.