ADAPTIVE STEPSIZE RUNGE-KUTTA ALGORITHM We implemented the 4th order Runge-Kutta method to solve chaotic ODE system and it turned out to be a power solver. However, one of its drawback is that the step size is fixed. When the solution happens to be smooth and change slightly, the number of steps could be reduced for efficiency. On the other hand, if the solution is changing sharply, fixed step size could fail to account for the abrupt variations and the errors might accumulate over time. As a result, for the purpose of higher efficiency and accuracy, adaptive step size Runge-Kutta method would be a better ODE solver and could handle more equations at the same time. As mentioned by the author in the Numerical Recipes for c++, “many small steps should tiptoe through treacherous terrain while a few great strides through smooth uninteresting countryside.” The step size is controlled by the truncation errors between the fifth order and the fourth order values. If the error exceeds the tolerance, we should reduce the step size until the error is below the tolerance. On the other hand, if the error is small, it will be more efficient to increase the step size so that the error is about the same order of magnitude as the tolerance. The general form of the adaptive step size Runge-Kutta method is as follows. The tn and yn are the time and the vector containing N ODE values at the nth step. k_1 = hf(t_n, y_n)\\ k2 = hf(x_n+c_2h, y_n+a_{21}k_1)\\ ...\\ k6 = hf(t_n+c_6h, y_n+a_{61}h+...+a_{65}k5)\\ y_{n+1} = y_n+^{6} b_ik_i \\ y^*_{n+1} = y_n+^{6} b^*_ik_i where a’s, b′s, b*’s and c’s are the various constants found by Dormand and Prince and all the values are tabulated in Chapter 17.2 of Numerical Recipes. The yn + 1* is alculated to the fifth order so the error term can be written as \Delta = \left| y_{n+1} - y^*_{n+1} \right| Note that here the || is the absolute value of each element rather than the norm of the N-element vector. By picking the maximum value in the Δ vector, _err_, we can compare _err_ to our tolerance _err0_. First of all, if _err_ is larger than _err0_, we have to scale down the step size and recalculate again until the err < err0. Secondly, for scaling the step size, whichever the case is, we can always apply the same scaling factor, scale = \left( {err} \right)^{1/4} \times S The S here is the safe factor with the requirement |S|<1. We chose S = 0.8 in the program. The ${4}$ power was used instead of ${5}$ since it scales up/down the step size more efficiently and the function _sqrt_ in c++ is easier and more accurate. Thirdly, we cannot scale the step size abruptly, like from h₀ to 0.01h₀ or 50h₀. In such case, the adaptive step size Runge-Kutta method would have ignored some features in the 50h₀ step or become inefficient using 0.01h₀ step size. We have to put extreme scaling values _maxscale_ and _minscale_. The re-scaled step size would be passed to the next run and so on. Finally, in order to avoid infinite _while_ loops, we added a _counter_ to keep track of the number of loops. If _counter_ exceeds certain number (in the program 100), we will break the while loop and print the message that the err was never reduced below err0 at this step. However, this didn’t happen even though we’ve tried tens of cases with billions of steps. Usually, it only took less than 5 loops to reduce the error within our tolerance. Let’s talk about the specified error tolerance err0 and the number of steps to achieve certain final t value. The initial step size doesn’t make a difference because the step size will get scaled up/down depending upon the solution and system. Given the same number of steps, the final t value reached depends mainly on err0. The smaller the tolerance is, the smaller the step size would be for each step. Therefore, the final t would end up far below the expected value. We can either increase the number of steps or the error tolerance which is a more efficient way than the former one. In the following testing and computations, this issue took place most of the time and it’s necessary to choose a good and aacceptable err0 and number of steps to solve the ODE system. We found that err0 = 10−5 of the initial value would work well for most of the cases.

Welcome to Authorea!

PLOTTING OF BESSEL FUNCTIONS The Bessel Functions of the first kind Jν(x) and the second kind Yν(x) are two functions of interest in this program. The program incorporates two header files “NR3.H” and “BESSEL.H” and outputs the function values between x = 0 to x = 20. The first two Bessel functions J₀, J₁, Y₀ and Y₁ are built in within “BESSEL.H” and we can use the recurrence formula to construct higher order Bessel functions. J_{\nu+1}(x) = {x}J_\nu(x) - J_{\nu-1}(x) Y_{\nu+1}(x) = {x}Y_\nu(x) - Y_{\nu-1}(x) The first three Jν and Yν are plotted by MATLAB as shown in Fig. 1 and Fig. 2.

FINITE DIFFERENCE ALGORITHM We have used Runge-Kutta method to solve ODEs with initial conditions; however, many of the physical systems cannot be described by ODEs. The partial differential equations with boundary conditions are inevitable for physical systems such as time-independent Schrodinger equations and electromagnetic fields. Those equations involve partial differential in spatial coordinates with specified boundaries and/or sources. When trying to solve such system numerically, the finite difference algorithm is a straightforward and easy-to-program method. We will use the 2D Laplace equation as an example to demonstrate the algorithm. \nabla^2\phi(x,y) = 0 The finite difference algorithm requires to mesh-grid the space of interest and the value on each grid point (i, j) represents the ϕ(x = ih₁, y = jh₂)=ϕij where h₁ and h₂ are the spacing in x and y axes. It’s always required to set up another “flag” grid which determines whether certain combination of (i, j) is on the boundary or source (electrodes). Next, based on the finite difference of adjacent values to ϕij and spacing, the second partial differential can be approximated. We can solve for ϕij in terms of its neighbors for the first iteration. ^{FD} = {4}\left( +++ \right) The flag grid is like a mask, leaving all the specified boundary values unchanged during iterations. Similar to Runge-Kutta method, we evaluate the maximum error in each iteration. \Delta = max | ^{FD}- | If the maximum error is larger than the specified tolerance Δϕij > ϵmax, we should pass the new values of ϕij to Eq. (2) and repeat the calculation until the error is below the tolerance or the number of iterations exceeds certain upper bound. The new ϕijnew can be the following form, which is the weighed average of ϕij and ϕijFD. ^{new} = (1-\omega) + \omega^{FD} where 0 < ω < 2. If 1 < ω < 2, the algorithm converges more quickly and this is known as the successive over-relaxation algorithm. The regime where 0 < ω < 1 corresponds to the under-relaxation where the solution converges relatively slowly. Here, we will solve the Laplace equation in a cylinder with three disks of inner and outer radius. Instead of utilizing Eq. (2), we will use the finite difference in cylindrical coordinates and assume azimuthal symmetry. ^{FD} = {4}\left( +++ \right)+{8j}\left( - \right) ,where z = ih and r = jh. The above equation applies to points where r ≠ 0. On the central axis where r = 0, we should have ^{FD} = {6}\left( 4++\right) In order to use finite difference algorithm, we need to set up matrices in c++. We used _arrayt.h_ to generate and do the calculations for matrices. OVER-RELAXATION First, we investigated the powerful role of ω here. We made the grid spacing h = 0.1 and error tolerance ϵmax = 2 and calculated the number of iterations for the error to go below the tolerance. The results are shown in the following table. ω 1.0 1.2 1.4 1.6 1.8 1.9 ---------------------- ----- ----- ----- ----- ----- ------ Number of iterations 655 511 431 371 359 1077 The best choice of ω seems to be around ω = 1.8 and we would use this over-relaxation value for the following calculation. SOLUTIONS For solving the electric potential inside the cylinder with three disk electrodes and boundaries, the grid spacing is the same h = 0.1 while the error tolerance is much smaller, ϵmax = 0.01. This one percent error would increase the number of iterations significantly to N = 4231. The contour plot of the numerical solution in space is shown in Fig. 1, where the red line represents the middle disk electrode held at ψ = 2000V and the black line and rectangle are the two grounded electrodes ψ = 0V. The electric potential contour is distorted by the two grounded electrodes, compressing the potential lines in the z direction. The potential at about r = 10mm is symmetric while that closer to the central axis is slightly asymmetric due to the mismatch in inner radii of the two grounded electrodes. The electric potential is mainly restricted among the disks and near the origin. Outside of those disks, the potential is negligibly zero everywhere. The configuration is seen in the electron guns of the electron microscopes to focus electrons by adjusting the fixed voltage of the center electrode.

Ergodicity The system of classical particles bouncing back and forth is not ergodic since each particles can only occupy certain close volume in phase space, with fixed x, y position, Px, Py momenta and ±pz. Therefore, the given volume in phase space of the ensemble will not evolve with time. Heat engine (a) True. (b) True, since the work in done on the gas. (c) False, it’s a refrigerator. (d) True. W = (P_0)(3V_0) + ^{V_0} {V} dV = 3P_0V_0 - 4P_0V_0\log4 (e) True, a part does no work so ΔU = Q. \Delta U = {2} nRT_c - {2} nRT_h = -{2} (3P_0V_0) = -{2}P_0V_0 where n is the number of particles in mole and R = 8.317 is the ideal gas constant. (f) True. The whole cycle ΔU = 0, the heat input and the net work done _onto_ the gas must equal the heat output. Q_h + W = Q_c

CALCULATION OF I/I₀ WITH VARYING n The analytical Fresnel Diffraction Theory gives the light intensity at x₀ on a viewing plane with distance z behind the edge the following form. I(x_0) = {\lambda z} \left| \int^{x_0}_{-\infty} \exp \left( {\lambda z} \right) dx \right| ^2 where λ and I₀ are the light wavelength and intensity respectively. It can be transformed into the following form by substituting $u_0 = x_0 {\lambda z}}$: {I_0} = {2}\left[ (C(u_0)-C(-\infty))^2 + (S(u_0)-S(-\infty))^2 \right] where C(u_0) = \int^{u_0}_0 \cos \left( {2} \right) du\\ S(u_0) = \int^{u_0}_0 \sin \left( {2} \right) du\\ As a result, instead of calculating the complex integral in Eq. (1), we simplify the task by evaluating C(u₀) and S(u₀). First, I’ll start with calculating both C(u₀) and S(u₀) by the trapezoid rule. The inputs of the program are the number of iterations n and the integral upper limit u₀. For mth iteration, the interval will be cut into 2m equally spaced trapezoids and the intensity ratio in Eq. (2) as well as the errors in C(u₀) and S(u₀) will be printed out. The output is shown in the following tables. As n increases, the errors in both functions are reduced dramatically, giving the convergence of the intensity ratio, shown as bold face numbers.

Undistinguished particles and the Gibbs factor (a) \Omega = {N!} S = k_B \space \log(\Omega) = k_B(N\log(V) - B\log(N) + N) = Nk_B\left(1+\log\left({N}\right) \right) The average volume of a region for single particle is v = V/N with the entropy of s = kBlog(v). Therefore, the configuration entropy is related to N times the entropy of a particle in its region with average volume. (b) S = Nk_B\left({2}-\log(\rho\lambda^3)\right) = Nk_B\left({2}+\log\left({N}\lambda^3\right)\right) The configuration entropy for an ideal gas particle is also related to the small average volume v. S_c = Nk_B\log(v) But the ideal gas particles also occupy the momentum space and each particle has the average length of λ in each dimension. So S_p = 3Nk_B\log(\lambda) = Nk_B\log(\lambda^3) Combine the configuration and momentum entropy for one particle, S = S_c+S_p = Nk_B\log(v*\lambda^3) = Nk_B\log\left({N}*\lambda^3\right) The total entropy is related to the N times the space occupied both in configuration and momentum space.

This is the first project report for ECE 5390/MSE 5472, focusing on the models and analyzing the conductance of 1D charge density waves (CDW). For normal conductors, the charge density in the conducting band is uniform (Fig.1(a)) but for certain materials, the charge density fluctuates with position. The CDW is the ground state of many quasi-1D materials, such as NbSe₃ when cooled down below Peierls temperature TP and undergoing Peierls Transition . Below TP, the distortion of lattice minimizes the total energy of conducting electrons and the elastic energy of the distortion, rendering it the energetically favorable state. The electrons pile up around the periodic distortion potentials and the charge density in the conducting band varies with the same period 2kF (Fig.1(b)).

Mar Cortesa,b, Alejandra Climenta, Laura Dubreuil Vallc, Giulio Ruffinic  Douglas Labara, Dylan Edwardsa a. Non invasive Brain Stimulation and Human Motor Control Laboratory, Burke Medical Research Institute, Weill Medical College of Cornell University, 785 Mamaroneck Avenue, 10605, White Plains, NY, USA.b. Universitat de Barcelona, Barcelona, Spain. c. Neuroelectrics Corporation, Cambridge (MA), USA. Corresponding author:Mar CortesNon invasive Brain Stimulation and Human Motor Control Laboratory, Burke Medical Research Institute, Weill Medical College of Cornell University, 785 Mamaroneck Avenue, 10605, White Plains, NY, [email protected]: +1 914 368 3181 Abstract: (250)[MC1]  Background: Existing strategies to enhance motor function following Spinal Cord Injury (SCI) are suboptimal leaving patients with considerable disability. Available evidence suggests that transcranial direct current stimulation (tDCS) is a promising method to improve motor dysfunction. How tDCS affects resting brain activity monitored by EEG is little explored. Objective: Investigate the effects of anodal tDCS on brain signaling (EEG) and neurophysiology (TMS) when targeting forearm muscles below the level of the lesion in chronic SCI subjects. Methods: We conducted a randomized, single blind, sham-controlled, cross-over study in seven chronic SCI subjects with cervical lesions. We investigated the effects of 20-minute anodal tDCS applied over the left primary motor cortex (M1) on electroencephalography (EEG) power spectrum density, coherence and frequency band power. Subjects were randomized to receive either 1mA or sham stimulation.  The EEG data acquisition pre and post stimulation comprised 5-minute takes of 24 bit, 500 S/s 8-channel EEG using StarStim Ag/AgCl EEG electrodes (at F3, F4, Cz, C4, P3 and P4; and Pi Ag/AgCl electrodes at C3, anode, AF8, return). Results: In comparison to sham stimulation, 20-minutes of active 1mA tDCS induced a pattern of faster activity around the anodal stimulating electrode, and slowing activity near the return electrode in the frequency (full band) and mean power domain (gamma band). In addition, tDCS increased coherence in the fastest bands (gamma, beta 2) and decreased coherence in slower frequency bands (theta, SMR), with no relation with brain topography or the stimulation electrode polarity. Conclusions: These findings show that tDCS is capable of inducing modulation of ongoing oscillatory brain rhythms captured by EEG, in spinal cord injury patients. The combined use of EEG and tDCS sets the stage for optimizing tDCS protocols targeting motor cortex and may have application in treatment of motor dysfunction and chronic pain.  Key words: spinal cord injury, EEG, tDCS, motor cortex, TMS Abbreviations:    Introduction[JC2]  Transcranial direct current stimulation (TDCS) delivered over primary motor cortex (M1) can increase or decrease corticomotor excitability as determined by the amplitude of the motor evoked potential (MEP) from stimulating M1 with supra-threshold transcranial magnetic stimulation (TMS). The effect on MEP amplitude depends on the duration and intensity of stimulation and the polarity and spatial arrangement of the TDCS electrodes, and these effects can persist for up to an hour after TDCS is ceased. It is assumed that the TDCS after-effect results from some form(s) of neuromodulation at the cortical level such as synaptic potentiation or depression, or effects on ion channels. However, further insight into the cortical mechanisms in the human is limited experimentally. The electroencephalogram (EEG) measures far-field potentials from synchronized neural populations over cortical regions that extend beyond M1, and so offers a method for measuring the effect of TDCS on neural activity across the cortex.  In particular, such measurements do not depend on spinal cord conduction and alpha-motoneuron activation, which is the case for MEP recordings. This could have advantages for determining the effect of TDCS at the cortical level in spinal cord injury (SCI). While most SCI studies focus on the level of injury, there is increasing interest in the compensatory changes that might occur in the cortex, and whether they might facilitate or impede recovery. In the present study we measured the effect of EEG after 20 min of anodal or sham TDCS applied to the dominant M1 in chronic cervical SCI. We show that there are changes in EEG power, frequency and coherence that are spatially-related to the TDCS electrode configuration, but that spread widely across the hemispheres. [MC3]  Materials and methods Participants and study designThe randomized, single blind, sham controlled, cross over study was conducted on seven chronic SCI subjects, all males with an average age of 51.14 ± 10.57 years (mean ±SD, range 34 to 65 years). Participants presented traumatic SCI at the cervical level (C4-C8); some degree of motor function in wrist extension (score 1-5 over 5 on the Medical Research Council (MRC) scale for motor strength in the right extensor carpi radialis (ECR) muscle); chronic injury (>9 months after injury); and tolerance to sitting upright for at least one hour (see Table 1 for baseline characteristics). Patients were excluded if they presented with: progressive neurodegenerative disorder; concomitant traumatic brain injury or stroke; clinically significant cognitive impairment; medically unstable; change in medication during the study; or presented contraindications to brain stimulation (history of seizures/epilepsy, presence of metallic implants in the brain, pacemaker, pregnancy). The subjects were randomized in two groups depending on whether they were receiving stimulation or not:  a) Sham or control group, b) 1mA or active group. We investigated the effects of 20-minute anodal t-DCS applied over the primary motor cortex (M1) on: a) quantitative electroencephalography (EEG) power spectrum density, coherence and frequency band power. At Baseline we collected: clinical and functional evaluations were performed prior the brain stimulation intervention, and included the upper extremity motor score (UEMS), American Spinal Injury Association impairment scale (AIS), spinal cord independence measure (SCIM III) and visual analog scale (VAS) pain questionnaires. [DE4]  The study was approved by the Burke Medical Rehabilitation Institutional Review Board and conformed to the standards set out by the 1964 Declaration of Helsinki.                  AddTable 1 Transcranial direct current stimulation (t-DCS) interventionParticipants remained seated in their own wheelchair or were provided with a comfortable chair. The StarStimNE noninvasive wireless t-DCS/EEG neurostimulator (NE Neuroelectrics®, Barcelona, Spain) was used to both record EEG data and deliver the direct current sequentially. The StarStimNE neurostimulator included a wireless neoprene cap, based on the International 10-20 system, which was placed on the participants’ heads by aligning the central CZ electrode position with the vertex (intersection of nasion-inion and inter-aural line mid-point).Small Ag/AgCl gelled electrodes, with a surface contact area of 3.14 cm2 specific to the StarStimNE device (Pi electrodes, Neuroelectrics®), were placed over the left M1 at C3 (anode) and contralateral supraorbital area, (AF8; cathode) (Figure 1). The electrodes were connected to a control box device, which was wirelessly connected to a computer and communicated with the NIC software (version 1.2, Neuroelectrics®).  Add Figure 1 During anodal stimulation, direct current was delivered from a current-control circuit in the battery-driven stimulator within the control box device. The current was set at 1mA intensity and applied for 20 minutes. [DE5] For the sham stimulation, electrodes were placed in the same position and participants received a short ramp (30sec total up / down[DE6] ) at the beginning and end of the stimulation period.  Electroencephalography (EEG)The StarStimNE multichannel wireless device (NE Neuroelectrics®, Barcelona, Spain), which allows for simultaneous electroencephalography (EEG) and tDCS[DE7] , was used to record EEG data. The EEG data consisted of 5-minute takes (pre and post stimulation) with 24 bit resolution, 500 S/s 8-channel EEG collected with StarStimNE Ag/AgCl EEG electrodes (geltrodes, NE022; positions at F3, F4, Cz, C4, P3 and P4; Pi stimulating electrodes at C3 and AF8), based on the International 10-20 system, with the cap aligned to the central CZ electrode position (vertex).[DE8]  The electrodes were connected to a control box device, which was wirelessly connected to a computer and communicated with the NIC software (version 1.2, Neuroelectrics®). Transcranial magnetic stimulation (TMS) Electromyography (EMG): A bipolar surface EMG electrode (1 cm diameter, 2 cm inter-pole distance; Biometrics Ltd, UK) was placed over the right ECR muscle, with the forearm relaxed in a pronated position and supported by a cushion. The EMG activity was amplified and filtered on site (x1000 gain, band-pass filter 20-460 Hz; SX230-1000), digitized at 2 kHz (CED 1401, Cambridge Electronic Design, Cambridge, UK) and stored for offline analysis using Spike 2.6 software. Measurements were performed at rest. During the experiment, free running EMG was continuously monitored with visual feedback of EMG to ensure complete muscle relaxation.The right ECR muscle was selected for clinical relevance; where restoration of motor function in this muscle can help increase independence in quadriplegic subjects with activities of daily living.  Transcranial Magnetic Stimulation (TMS): A figure-of-eight coil (Model DB-80, Tonika Elektronik A/S, Farum, Denmark), connected to a MagPro X100 series (MagVenture A/S, Farum, Denmark) magnetic stimulator, was placed congruent with the head and the handle rotated 45° lateral from mid-sagittal so as to induce currents in the brain approximately perpendicular to the central sulcus. Resting motor threshold (RMT) was established at C3, and was defined as the minimum TMS intensity required to elicit a reliable MEP in the contralateral ECR amplitude of >50 µV in at least 50% of consecutive trials.[DE9]  EEG data processing/statistical analysisResting quantitative EEG and corticospinal excitability (transcranial magnetic stimulation; MEP) were recorded before (PRE) and at the end (POST) of each intervention. Quantitative EEG measures included: mean power, mean frequency and mean coherence. The three measures were normalized as the percentage change between PRE and POST stimulation, for each frequency band and electrode. The frequency bands were defined as: Theta= [4 8] Hz, Alpha-1=[8 10] Hz, Alpha-2=[10 12] Hz, SMR=[12 16] Hz, Beta-1=[16 25] Hz, Beta-2=[25 35] Hz, Gamma=[35 40] Hz.The mean power is defined as the average power in uV2 in a given frequency band. The mean frequency is defined as the average frequency in a given band, weighting each frequency value by their corresponding power at that frequency. The mean coherence of a given electrode is defined as the average coherence of this electrode with all the other ones (i.e., the similarity of that signal with all the other electrodes).--> Giulio to revise - DONEThe EEG data were referenced to the average of all 8 electrodes. An automated quality check of the data was then carried out using 8 sec epochs. Epochs have been rejected if they do not meet quality criteria (too high mean power at full band or line noise, or motion artifacts as detected by the built in accelerometer). PSD changes and coherence analysis was carried out for each quality passed epoch. Average PSD’s and coherence have then been computed for each subject.EEG data were discarded from analysis for channels presenting bad signal quality during the entire 5-minute recording. Those subjects that did not have more than 5 valid channels were completely discarded. After the discarding process, in order to have the same subjects in all the conditions, 7 subjects were analyzed for Sham and 1mA.Resting MEP amplitude (peak-to-peak; ECR muscle) was measured following single-pulse TMS (10/12) set at 120% [DE12] of the RMT over the C3[DE13] . Raw and normalized values were used for analysis[DE14] . Results are presented as mean ± standard deviation (SD), and standard error of the mean (SEM)[DE15] .After normalizing the EEG features as the percentage change between PRE and POST stimulation, statistical comparison between Sham and Active (1mA) stimulation has been done. The p value in the T tests has been calculated with a Wilcoxon one-tail T test with the assumption of paired samples. Changes are considered statistically significant when p =< 0.05. PLEASE ADD THE STUDY DATA ANALYSIS – GIULIO / LAURA --> Described above Results Effects of tDCS on EEGNormalized pre-post EEG power, frequency and coherence for those subjects who received 1mA tDCS compared to those who received sham stimulation shows: a) significant increase of the mean power in the 1mA group in the Gamma frequency band under C3 (p=0.0035), the anodal stimulating electrode; b) a significant increment in the active group of mean frequency around the anode (stimulating) electrode (C3, F3) (p=0.0047) with a decreased mean frequency in the Alpha band near the return electrode (P4) (p=0.035); and c) significantly increased mean coherence in the active group in the fastest frequency bands Beta2 under Cz and C4 (p=0.007), Gamma band under Cz electrode (p=0.0023), C3 electrode (p=0.0006) and C4 electrode (p=0.0006), and SMR band under C3 electrode (p=0.05). LAURA – ADD THE TYPE OF TEST USED FOR THE RESULTS AND THE P VALUES OBTAINED --> Described  above Effects of tDCS on Corticospinal excitabilityBaseline values for resting MEP amplitude were similar between interventions (average: 0.37±0.05 mV; mean±SEM). No significant changes were observed for 1 mA a-tDCS or sham. The stimulation intensities used to obtain the RMT were not significantly different between 1 mA (64±17% MSO) and sham (66±16% MSO). All participants presented MEP responses.   Discussion1. tdcs does modulate EEG in SCI population. 2. The effect of the TDCS in EEG is location specific and band specific. 3. The fastest bands are more affected. Augmented power and frequency of fast bands (Gamma and B2) under the anode electrode - there is more fast activity under the anodal electrode.  Why is this?3.1. Anodal stimulation leads to increased spontaneous neuronal firing, increase motor cortex excitability. So we would expect more fast activity and higher amplitudes under the anode electrode. 3.2. Brain activity changes (increase fast band) are related to motor learning – changes in Mu rhythm (preparation for movement, getting ready for movement). This is shown in the coherence increase of SMR band under the stimulating electrode.  MAR – ALL OF THE ABOVE 3.2.1. Is there are connection between C3 and P4 in resting state brain? Why is alpha decreased significantly in P4 (cortex parietal) in the contralateral hemisphere.  CONECTIONS BETWEEN MOTOR CORTEX AND SENSORY MOTOR CORTEX?? – JANA 3.3. Anodal tDCS makes C3 to be more related to the rest of the head, by increasing the connectivity of the SMR under the stimulating electrode. INTERPRETANTION OF THIS POINT TO GIULIO/LAURA, ANYTHING PUBLISHED BEFORE?Here you have a reference using EEG coherence: Rafael Polanıia, Michael A. Nitsche, and Walter Paulus. "Modulating Functional Connectivity Patterns and Topological Functional Organization of the Human Brain with Transcranial Direct Current Stimulation". Human Brain Mapping (2010). 3.4. This is an excitability independent change (no TMS change associated). Can EEG pick up changes in the brain than TMS cannot?                                                       LABAR? Conclusion These findings show that t-DCS is capable of inducing modulation of ongoing oscillatory brain rhythms captured by EEG, in spinal cord injury patients. The combined use of EEG and t-DCS sets the stage for optimizing t-DCS protocols targeting motor cortex and may have application in treatment of motor dysfunction and chronic painFigures 

Problem Statement A random walk on the 2-dimensional integer lattice begins at the origin. At each step, the walker moves one unit either left, right, or up, each with probability $\frac13$. (No downward steps ever.) A walk is a success if it reaches the point (1, 1). What is the probability of success? Note: One can vary the problem by varying the target point. Eg., use (1, 0) or (0, 1) instead. Perhaps there is a good method to resolve the general case of target (a, b). Source: Bruce Torrence, Randolph-Macon College

INTRODUCTION As available datasets increase in size, machine learning models can successfully use more and more parameters. In applications such as computer vision, models with up to 144 million  parameters are not uncommon and reach state-of-the-art performance. Experts can train and deploy such models on large machines, but effective use of lower-resource hardware such as commodity laptops or even mobile phones remains a challenge. One way to address the challenge of large models is through model compression using hashing . In general, this amounts to reducing a parameter set S = {s₀, s₁, ...sD} to a greatly reduced set R = {r₀, r₁, ..., rd} with d ≪ D by randomly tying parameters to hash buckets (si = rh(i)). This turned out to perform very well for neural networks, leading to the so-called HashedNets. Many machine learning models involve several linear projections representable by matrix-vector products W ⋅ x where x is input data and W consists of model parameters. In most such models, this linear algebra operation is the performance bottleneck; neural networks, in particular, chain a large number of matrix-vector products, intertwined with non-linearities. In terms of dimensionality, modern systems deal with millions training samples xi lying in possibly high-dimensional spaces. The shape of W, (dout, din), depends on how deep a layer is in a network: at the first layer, din depends on the data being processed, while dout at the final layer depends on the desired system output (i.e., dout = 1 for binary classification, and dout = p if the output can fall in p classes). In middle layers, dimensionality is up to the model designer, and increasing it can make the model more powerful but bigger and slower. Notably, middle layers often have square Wh. When W is stored in a reduced hashed format Wh, many common trade-offs may change. The goal of our project is to explore the performance bottlenecks of the Wh ⋅ x operation where Wh is a hashed representation of an array that stays constant for many inputs xi. Since neural networks are typically trained with batches of input vectors x concatenated into an input matrix X, we will look at the general case of matrix-matrix multiplication, where the left matrix is in a reduced hashed format Wh ⋅ X. Taking advantage of massively parallel GPU architecture can be important even when dealing with smaller models. In March 2015, Nvidia announced a SoC for mobile devices with a GPU performance of 1 teraflop, the Tegra X1 ; we foresee future mobile devices to have stronger and stronger GPUs. The objectives of our project are to: 1. Investigate fast applications of Wh ⋅ X when Wh is small enough to be fully loaded into memory. In this case, is it faster to first materialize the hashed array and use existing fast linear algebra routines? Can the product be computed faster on a GPU with minimal memory overhead? This can lead to highly efficient deployment of powerful models on commodity hardware or phones. 2. Analyze performance when even after hashing Wh is too big. In the seminal work that popularized the usage large scale deep convolutional neural networks and training using the GPU , Krizhevsky predicts that GPUs with more memory can lead to bigger networks with better performance. Hashing-based compression can help practitioners prototype very large models on their laptops before deciding which configuration to spend cloud computing resources on. Can we make HashedNets training on GPUs efficient? This may involve forcing a predictable split on the hash function to allow for independent division of work.

In this final report for ECE 5390/ MSE 5472, the tunneling model for charge density waves (CDW) in the quasi-one-dimensional material will be used to derive both DC and AC conductivity. The DC conductivity in the absence of AC signals has a threshold voltage, above which the conductivity is non-zero. The AC conductivity follows the same behavior due to the photon-assisted tunneling of CDW but there is an additional resonance contribution. Finally, the conductivity for mixed DC/AC signals is shown with the photon-assisted conductivity at low field and resonant AC conductivity.

OVERVIEW This document presents a parametric model to help design an Interferometric Array. It focuses in the value vs. cost trade-off inherent to many of its architecture definitions. This is a Multiple Objective problem. This document describes design parameters to consider in § [sec:var] and a set of equations for research and cost objectives in § [sec:obj]. A spreadsheet that uses these design parameters and produces a CSV file for analysis of the emerging Pareto Front is introduced in § [sec:spreadsheet]. This output enables the of Multiple Objective Visual Analytics (MOVA) for complex engineered systems as proposed in . PARAMETERS This section presents selected design parameters that influence selected objectives in § [sec:obj]. We will select design parameters that are specification agnostic. As an example of this, the parameters will be relevant to multiple antenna specifications, including offset Gregorian and symmetric Cassegrain. Antenna Parameters Antenna Collecting Area We will use A in this document as each array element collecting area (thus we could also write π ⋅ D², with D being the dish diameter). Antenna Efficiency We will use ηa in this document as the antenna efficiency with \eta_a = } \cdot } \cdot } \cdot } as defined in . Antenna Quantity We will use N in this document as the number of array elements. Antenna Pad Parameters Pad Quantity We will use P in this document as the number of pad built for the array. In case re-configuration of the array is envisioned, there might be a bigger number of pads ready for aperture connection to the system. Pad Position We will use the geographic latitudes and longitudes to establish pad location in this document. We will calculate the length of the possible baselines using pad positions. We will also calculate length and complexity of the roads, fiber and power networks needed using pad positions. We will use B as the maximum array element separation in any single configuration. Reciever System Parameters Number of Receivers per Array Element We will use R as the number of frequency bands, being Ri the different frequency bands. If the array bandwidth is λmax − λmin, it is useful for our analysis to use wavelength λ = λmin. Notes: high bandwidth ration: up to 7 might be practical, but could compromise Ae/Tsys. High absolute bandwidth is challenging for digitalization. up to 20GHz might be practical. Receivers Efficiency Signal Processing and Transmision Parameters Notes: directly at RF (no reference), single sideband down conversion (LO and timing reference), double sideband (IQ) down conversion (two LO, two references, LO tunable. Bits per sample (dynamic range) Correlator Aspects Position We will geographic latitude and longitude to establish correlator location in this document. We will calculate fiber, power and road network aspects based in this information. Efficiency We will use ηc as correlator efficiency in this document, with \eta_c(t_{int}) = }{ t_{int}} as defined in . OBJECTIVES This section aims to include array performance objectives that might be influenced by design variables in § [sec:var]. Fourier Plane Coverage As derived in , the antenna diameter determines its beam size $ \approx {D}$. If the plane area ${\lambda}$ is divided in cells of size ${\lambda}$ then N_{occ} \leqslant \pi ({D})^2 Point Source Sensitivity An overall measure of performance is the System Equivalent Flux Density, SEFD, defined in as the flux density of a source that would deliver the same amount of power: SEFD = {}{{2k_B}}} in units of Janskys where Tsys is the system temperature including contributions from receiver noise, feed losses, spillover, atmospheric emission, galactic background and cosmic background, and kB = 1.380 × 10−23 Joule K−1 is the Boltzmann constant. According to , if we assume N apertures with the same SEFD, observing the same bandwidth Δν, during the same integration time tint, then weak-source limit in the sensitivity of a synthesis image of a single polarization is \Delta I_m = {{\eta_s }}{{}}} in units of Janskys per synthesized beam area, with ηs most important factor being correlator efficiency ηc. Surface Brightness Sensitivity Operations Costs Components reliability Maintenance complexity Calibration Software Costs Calibration Hardware Costs Power Consumption Cost Re-configuration Systems Operation Cost Up-front Costs Cost of Antennas Construction According to , a commonly used rule of thumb for the cost of an antenna is that it is proportional to Dα, where α ≈ 2.7 for values of D from a few meters to tens of meters. For N antennas of diameter D meters with accuracy ${16}$, where λ is in millimeters we could use as an upper limit for Antenna construction cost. = {10})^{2.7}}{(\lambda^{0.7})} + 500 in K$. Cost of Front-end system For M frequency bands, each 30% wide, and dual polarization we could use as an upper limit for Front-End System Cost: = 45MN + 200M in K$. Cost of LO system We could use as an upper limit for LO System Cost: = 80N+100 in K$. IF Transmission Cost We could use as an upper limit for IF Transmission Cost: = 8BN + 30N + 400 in K$. Correlator Cost We could use Correlator Cost as an upper limit: = 2N^2 + 112N +1360 in K$. Cost of Re-configuration Systems Construction DATA FOR VISUAL ANALYTICS - SPREADSHEET IMPLEMENTATION This section presents a spreadsheet that produces data in the right format for performing visual analytics, consistent with variables in § [sec:var] and objectives in § [sec:obj]. VISUALIZATION TOOL NOTES CONVERSATION NOTES Engineering cost vs. Calibration cost Tricky because you can compensate antenna quality with software. So the equations must capture this trade off.

Local conservation a. The density of tin atoms ρSn(X) is locally conserved since there is no tin atom created or annihilated. b. It depends. Under the constraints that one cannot change his/her political belief, yes, the density of democrats is locally conserved. However, if changing of political belief is allowed, the density of democrats is not locally conserved. There might be democrats changing to republicans or vise versa, which implies that the democrats or republicans can teleport, created or annihilated. Density-dependent diffusion Suppose there is no external force. We can write the current as J = -D(\rho)\nabla\rho And apply {\partial t} = -\nabla\cdot J {\partial t} = {\partial\rho} \nabla\rho\cdot\nabla\rho + D(\rho)\nabla^2\rho {\partial t} = {\partial\rho} (\nabla\rho)^2 + D(\rho)\nabla^2\rho This is the diffusion equation with density-dependent diffusion constant.

Absorption boundary conditions First observe that the probability density at the origin at any time is zero. \rho(0,t) = 0 And the probability density on x = x′ at t = 0 is \rho(x,0) = \delta(x-x') We can assume that there is a image probability density at t = 0 on x = −x′. Therefore the total probability at t = 0 should be \rho(x,0) = \delta(x-x')-\delta(x+x') The Green’s function for each delta function is a time-dependent Gaussian distribution, G(x,t) = {} e^{-x^2/4Dt} with the particle originally at the origin. For this case, both delta functions diffuse, yielding the same Gaussian distribution. G(x,t) = {} \left( e^{-(x-x')^2/4Dt} - e^{-(x+x')^2/4Dt}\right) The time evolution probability distribution is shown in fig.1.

Weirdness in high dimensions Most of the volume of high dimensional spheres is near the surface. As we can approximate the volume as V ≈ r3N in 3N dimensions, near the center, r ≈ 0 and V ≈ 0. Near the surface, where r becomes larger and contributes to most of the volume. For statistical mechanics, instead of taking the whole E sphere into account, which might involve high-dimensional integral, it’s easier to focus on the shell of E sphere. Then the momenta can be considered to be $p = $ instead of doing \int_0^{} (....)dp

Undistinguished particles For N distinguishable particles, if at certain instant they occupy N states in the phase space, there are N! different choices of such configuration. The total volume for this system in phase space is the product of the volume for each particle in its own sub phase space. \Omega_D = N! \space \omega_1 \omega_2 \omega_3 ... \omega_N If the particles are now undistinguished, there is only one choice of such configuration without permutation. \Omega_U = \omega_1 \omega_2 \omega_3 ... \omega_N Therefore, $\Omega_D = N!\space \Omega_U $. For the mixing problem of black and white particles of the same pressure on both sides, if you cannot tell the difference between black and white ones, you cannot extract work from the mixing. Actully, you won’t know if they’ve mixed already or not. However, suppose the black and white particles are of different pressure, and you still cannot tell them apart. If there were a membrane which allows particles of lower pressure to penetrate through, this membrane then can tell the difference between the two sides without telling the particles apart. Then work can be extracted. Now if a door can distinguish the black from the white particles and let through white ones, after very long time, the white particles will distribute themselves equally in the two sides, i.e. there will be ${4} N\space white + 0 \space black$ in the left and ${4} N \space white + {2} N \space black$ in the right. And we can extract work from it due to the pressure difference between the two sides.

Gallium Nitride, Band-to-band Tunneling, PN diode