# Selective Self-Assembly using Re-Programmable Magnetic Pixels

Martin Nisser1Yashaswini Makaram 1Authors are with MIT CSAIL<EMAIL_ADDRESS>Martin Nisser1 Yashaswini Makaram1 Faraz Faruqi1 Ryo Suzuki2 Stefanie Mueller1
1Authors are with MIT CSAIL<EMAIL_ADDRESS>is with University of
Calgary

###### Abstract

This paper introduces a method to generate highly selective encodings that can
be magnetically ”programmed” onto physical modules to enable them to self-
assemble in chosen configurations. We generate these encodings based on
Hadamard matrices, and show how to design the faces of modules to be maximally
attractive to their intended mate, while remaining maximally agnostic to other
faces. We derive guarantees on these bounds, and verify their attraction and
agnosticism experimentally. Using cubic modules whose faces have been covered
in soft magnetic material, we show how inexpensive, passive modules with
planar faces can be used to selectively self-assemble into target shapes
without geometric guides. We show that these modules can be easily re-
programmed for new target shapes using a CNC-based magnetic plotter, and
demonstrate self-assembly of 8 cubes in a water tank.

##  I Introduction

Modular self-assembly is a parallel process for the bottom-up fabrication of
arbitrarily complex geometries from a set of disjoint modules. At the
molecular scale, self-assembly is effectively used in nature to assemble
complex biological structures [1], and formal models evaluating the power of
self-assembly have proven it to be Turing-universal. Seeking to harness its
capabilities, engineers have used self-assembly across scales to create
microstructured materials with designed optical and mechanical properties at
the $\mu$m-scale [2], to self-assembled structures at the mesoscale [3] up to
aerial self-assembly of structures at the m-scale [4]. As an alternative to
automating the assembly of robots in a top-down manner [5], reconfigurable
robotics has explored bottom-up self-assembly in two forms, active and
passive.

Active self-assembly by modular self-reconfigurable robots (MSRR) involves
modules that modulate their behavior online in order to locate, position and
bond themselves to their neighbors, for which each module requires embedding
with computation, sensing and actuation. Because most systems choose cubic
modules to facilitate tessellation, a key discriminator between systems is
often the actuation, which include electropermanent magnets [6],
electromagnets [7, 8], momentum wheels [9, 10], and COTs servos or motors
[11]. While active assembly has been used to successfully reconfigure a
variety of robotic systems, it is these actuators that are typically the most
significant challenge to scaling systems up in number and down in size due to
the cost and complexity of embedding them into individual modules [12].

Figure 1: (Left) We enable stochastic self assembly using inexpensive ($0.23)
cubic modules. (Above, right) We accomplish this by magnetically programming
module faces with uniquely mating pairs of encodings based on Hadamard
matrices, and show bounds on their performance. (Below, right) Key to the
modules’ success is their ability to attract strongly to their mates, while
remaining agnostic in all other translations, rotations, and to non-mating
modules. Figure 2: Overview of our stochastic self assembly approach. (Left)
Magnetic Pixels: Cubic modules are programmed with matrices of magnetic
pixels. These permit module faces to mate selectively to assemble target
geometries. 3x3 Matrices shown for clarity; our modules utilize 8x8. (Center
left) Scalability: Our binary-valued, 8x8 matrices can encode 264 module faces
with unique permutations, and modules are inexpensive ($0.23). (Center right)
Selectivity Guarantees: We leverage Hadamard matrices to encode magnetic
pixels on faces with 2 criteria. ”Locally”, mating faces attract in only one
configuration; ”globally”, non-mating faces attract in no configuration.
(Right) Reconfigurability: Encodings are ”programmed” as magnetic pixels using
magnets installed on a 3-axis CNC. Modules can be re-programmed to self
assemble into new target geometries.

In contrast, passive self-assembly obviates the need for emdedded acuation and
control. Instead, system actuation is outsourced to an external excitation,
and in the case of stochastic self-assembly [13], this excitation requires no
local control and is governed instead only by global parameters such as
excitation magnitude. Stochastic assembly sacrifices efficiency and
predictability for advantages in cost, complexity and scale; by enabling the
environment to actuate reconfiguration, it trades off deterministic assembly
times of individual modules for statistical assembly rates of the collective.
To assemble stochastically, modules require pre-programming to enforce correct
mating during random collisions with their intended mate. This programmed
specificity between pairs of mating faces is typically achieved via
minimization of free surface energy via topology[14], wettability [15],
magnetic forces [16] or electrostatic [2] interaction. While the wider
scientific community has often been interested in constraining the self-
assembly problem to 2D, for instance by using a shaker table [17], roboticists
have leveraged liquid tanks to study assembly in 3D. Fluidic assembly at the
mesoscale has become a particularly widely studied problem in robotics [18,
19, 20, 21, 22, 12]. Existing stochastically self-assembling modules typically
include two features to enable assembly: first, embedded magnets that generate
near-field forces to bring modules close, and second, selective geometry on
module faces that encodes the specificity to only permit bonds between mating
pairs [23, 14, 17, 24]. However, three key challenges remain for the
development of stochastic self-assembling systems: (1) scalability that shows
how modules can be made both numerous and small; (2) selectivity guarantees
that help bound module misassembly; and (3) reconfigurability that let modules
acquire different target shapes.

(1) Scalability: To assemble arbitrarily complex geometries, encodings for 3D
modules must support selectivity great enough to permit uniquely mating pairs
of modules in the hundreds or even thousands. In addition, modules must be
inexpensive and simple enough to be fabricated in these quantities. Due to
this dual problem, a significant corpus of previous research demonstrates the
stochastic self-assembly for tiled 2D arrays, such as chessboards[2, 23, 25],
with only two module types where each module is selective to entirely half of
all modules in the set. On the other hand, the individual fabrication of
heterogeneous module topologies with manually embedded permanent magnets poses
a significant challenge to scalable fabrication.

(2) Selectivity guarantees: Because magnet arrangements typically used to
generate near-field forces are poorly discriminating to each other, this
framework often leads to misassemblies, because near field forces between both
mating and non-mating face magnets are equally strong. In addition, protruding
geometrical features used for selectivity can lead to ”jamming” by obstructing
assembly paths [23], and bounds on the expected misassembly rate between
geometrically dissimilar modules may be difficult to compute.

(3) Reconfigurability: To date, structures self-assembled at the mesoscale are
not reconfigurable. Because module selectivity is achieved by fabricating
individualized geometries, any set of fabricated modules encode only a single
target shape (or single set, for non-deterministic encodings). Such modules
are therefore unable to be ”re-programmed” to self-assemble new target shapes:
new shapes require a unique batch of modules to be fabricated from scratch,
inhibiting their utility and increasing their potential unit cost.

In this paper, we introduce a method to design and ”program” selective
encodings [26] onto cubic modules in a way that addresses all three challenges
above (Fig. 2). We program module faces with patterns of magnetic pixels which
can attract or repulse pixels of another face (Fig. 2 left), and if the number
of pixels in attraction match those in repulsion, the faces are agnostic to
each other. Using this observation, we show how to program modules with
encodings that allow them to selectively mate with other cubes to self
assemble in a unique target structure. Formulated as matrices, we demonstrate
the number of unique encodings that can be programmed given criteria on
attraction and agnosticism.

Our modules consist of PLA cubes, with squares of soft magnetic material
(inexpensive COTS fridge magnets) bonded to their sides (Fig. 2 center left).
The encodings on these faces generate both near-field forces and selectivity.
This selectivity lets us use homogeneous cubic modules with planar faces,
making modules both non-jamming and inexpensive to mass fabricate across
scales. Key to our approach is the generation of selective encodings, for
which we leverage Hadamard matrices (Fig. 2 center right), and similar
procedures may have been used to create industrial-grade magnets with tailored
selectivity properties in industry [27]. The two polarities of the magnetic
pixels we encode onto faces correspond to elements of these binary-valued
matrices. Our matrices enforce two key criteria. A local criterion dictates
behavior between faces intended to mate. If every pixel on one face, face A,
is magnetically opposite to pixels on another, then we call that face its
mate, A’. These faces thus form a maximally attractive mating pair. Our matrix
pairs are designed to exhibit maximal attraction in this one configuration,
while placing guarantees on agnosticism in all other translations and
rotations. A global criterion dictates behavior between two faces not intended
to mate. For this, we ensure that any given matrix pair, A and A’, cannot mate
with any other faces B, B’, C, C’ etc., in any configuration, and we place
guarantees on this agnosticism. We further demonstrate how to ”program” module
faces using a simple magnetic plotter consisting of two oppositely oriented
permanent magnets affixed to a 3-axis CNC (Fig. 2 right). Crucially, these
soft magnetic faces are re-programmable, and thus modules can be repeatedly
re-programmed with new encodings in order to self-assemble into new target
geometries in 3 dimensions. Finally, we design and fabricate a set of 8
modules and demonstrate their stochastic self-assembly in a water tank.

The paper is structured as follows. We begin by introducing a procedure to
generate selective magnetic encodings, and derive bounds on the number of
modules that can be utilized given a threshold on agnosticism between
programmed encodings. We describe the physical modules themselves, and the
magnetic plotting technique used to program them. We demonstrate our ability
to make predictions with regard to the attraction and agnosticism between
various magnetic encodings, and verify these experimentally. Finally, we
demonstrate stochastic self-assembly of our system using 8 modules in a water
tank.

##  II Encoding generation

This section describes how we generate encodings that satisfy the global and
local criteria given above. Our encodings are based on Hadamard matrices,
whose unique properties have lent their use to applications including Code
Division Multiple Access and error correcting code. The Hadamard matrix is a
square matrix whose rows are all mutually orthogonal and whose elements are
either $1$ or $-1$ (here representing magnetic pixel polarization). As a
consequence of its row orthogonality, it follows that its columns are mutually
orthogonal too. As a result, the dot product of any pair of rows, or any pair
of columns, is equal to 0. In addition, if a single row or column is
multiplied by -1 before taking the dot product, the product remains 0.

Defining the mate of matrix A as A’$=$-A, the Hadamard product (1) between a
matrix A of order N and its mate, normalized by the number of elements N2, is
-1. This implies maximum magnetic attraction. Conversely, the normalized
Hadamard product between A and itself is +1, connoting maximal repulsion. Let
$S_{G}$ be a global score indicating the maximum attraction between non-mating
pairs A and B in any configuration, and let $S_{L}$ be the local score
indicating maximum attraction between mating pairs A and A’ in all wrong
configurations. To permit self assembly, satisfying the global and local
agnosticism criteria requires the force enacted by our fluid $F_{f}$ to
satisfy $-1<F_{f}<min(S_{L},S_{G})$ in order to both break apart unintended
misassemblies and allow correct assemblies to survive. Because controlling the
fluidic force is challenging, the goal is then to maximize $min(S_{L},S_{G})$,
making them maximally agnostic, in order to place $F_{f}$ between these
values.

Now, the row and column orthogonality described above yields that taking the
Hadamard product between A and A’ becomes 0 if one matrix is translated in
only X or only Y, yielding maximal agnosticism. However, agnosticism is not
guaranteed for matrices translated in both X and Y, or if they are rotated. To
find matrices that maximize agnosticism in these configurations, we perform
two searches. The first search shows that Hadamard matrices perform optimally
among the set of all square matrices. The second search identifies the number
of matrices that can be generated for a given bound on agnosticism
performance.

$(A\odot B)_{ij}=(A)_{ij}(B)_{ij}$ (1)

###  II-A Matrix search

We generate every binary-valued (values 1 and -1) matrix A of order N=4
(yielding $2^{16}$ matrices), and compute the cross-correlation between A and
A’, which is equivalent to their Hadamard product in every translation ($2N-1$
values) and also compute their Hadamard product in rotations of 0∘, 90∘, 180∘,
and 270∘ (4 values, upper-bounded by computational resources). These values
are used to assess the local criterion. We then check which of these are
Hadamard matrices by evaluating which matrices A satisfy $AA^{T}=N(I_{N})$,
finding 768 matrices (1.2%) out of the total 65,536 ($2^{16}$). We then rank
every matrix by its most attractive (most negative) value for all translations
and rotations (excluding its one mating configuration of -1) to give its local
criterion score $S_{L}$. An optimal $S_{L}$ score is 0, implying perfect
agnosticism in all configurations besides its mating configuration.

Assessing a set of matrices’ global criterion score, $S_{G}$, requires finding
the worst (most negative) Hadamard product in translation and rotation of
every combination of matrices in the set. This is the power set of matrices,
$2^{65,536}$ combinations for the matrices above, a computationally
intractable search; searching for all possible cliques (maximal complete
subgraphs) in a graph is an NP-complete problem. However, any guarantee of a
matrix’ performance in agnosticism is given by $min(S_{L},S_{G})$. Therefore,
to narrow our search, we build sets by searching those with best $S_{L}$
first, stopping our search once $S_{G}=S_{L}$. We consider a graph $G$ where
vertices represent these matrices, and where two vertices ($V_{1},V_{2}$) are
adjacent if their Hadamard product (in translation and rotation) is above some
$S_{G}$. Hence we reduce the problem of searching for mutually compatible sets
in the power set to a search for cliques in G. We determine $S_{G}$ for
progressively larger clique sizes of the 65,536 matrices using a variation of
the algorithm by Bron and Kerbosch [28, 29], and find that given a $S_{G}$ and
$S_{L}$, the set of Hadamard matrices produce a clique size equivalent to the
set of all possible matrices. Thus, the maximal clique search, which exhibits
exponential complexity, is reduced to the set of Hadamard matrices (a small
fraction, 1.2%, of the total matrices possible) and still finds maximally
sized cliques given some $S_{G}$ and $S_{L}$.

Initial experiments revealed that matrices of order N=4 were unable to produce
adequately agnostic values for $S_{G}$ and $S_{L}$ for clique sizes greater
than 5 such that our stochastic fluid could discriminate between correct and
incorrect mates. However, with a goal to assemble an 8-module system into a
”meta cube” (Fig. 13), we require a minimum clique size of 12. We therefore
use our result regarding maximal cliques to search across Hadamard matrices of
order N=8.

###  II-B Generating Hadamard Matrices

We generate a normalized Hadamard matrix of order N=8 using the recursive
procedure described below: let $H$ be a Hadamard matrix of order N. We can use
this to create the partitioned matrix of order $2N$ shown in (2).

Figure 3: (Above) Normalized, naturally ordered Hadamard matrices generated in
a procession of orders N of 2, 4, 8 and 16. Binary values of 1 and -1 are
represented as dark and light pixels, respectively. (Below) Their mates.

$\begin{bmatrix}H&H\\\ H&-H\end{bmatrix}$ (2)

We can apply this rule generally for higher orders using:

$H_{2^{k}}=\begin{bmatrix}H_{2^{k-1}}&H_{2^{k-1}}\\\
H_{2^{k-1}}&-H_{2^{k-1}}\end{bmatrix}$ (3)

Initializing $H_{1}$ as 1, we generate $H_{8}$ and see that the Hadamards
produced in this construction are fractal (Fig. 3). We then generate all $8!$
matrices that are permutations of its rows. As the rows of $H_{8}$ are
orthogonal, the rows of all its permutations remain orthogonal and Hadamard.

To meet our goal of finding a clique of size 12, we seed a threshhold for both
$S_{L}$ and $S_{G}$ of -0.2, and iteratively search for maximal cliques. We
continue to lower the threshold of $S_{G}$ by 0.02 until we grow the maximal
clique size to 12. Fig. 4 illustrates the relationship between maximum clique
size, its mutual $S_{G}$ score, and the number of cliques of that size. We
select the 12 matrices from one of the 4 size-12 cliques to program and
assemble a meta cube, exhibiting a combined agnosticism score of
$min(S_{G},S_{L})=S_{G}=-0.36$.

Figure 4: The size of a clique (the number of mutually compatible encodings)
related to both 1) the number of such cliques, and 2) its global agnosticism
score $S_{G}$

##  III Magnetically programmed modules

Figure 5: (Above) Magnetic plotter programming a module. (Middle) Simulated
and plotted matrices viewed through magnetic viewing film; (Left) A normalized
order-8 Hadamard; (Right) A checkerboard. Dark and light pixels in simulation
represent opposite magnetic polarities. The same material was reprogrammed to
produce these patterns in turn. (Below) Module fabrication. (Left) a white PLA
cube is covered with (middle) squares of black soft magnetic material; (Right)
once all six squares are bonded, it is ready for programming.

Soft magnetic material is material that can be easily coerced to become
magnetized when placed in an external magnetic field. When removed from the
field, they nonetheless retain a significant fraction of their magnetization,
letting them function as magnets. This process is repeatable, allowing modules
with soft magnetic faces to be re-programmed with new encodings that encode
different target structures.

We build a magnetic plotter to stamp our cubes’ soft magnetic faces with
magnetic encodings (Fig. 5, above). The plotter consists of a pair of
oppositely polarized permanent magnets (3mm diameter, 6mm length) installed in
a housing mounted onto a 3-axis CNC (SnapMaker 3-in-1). Each magnet thereby
plots opposite pixel values, where binary-valued pixels correspond to
oppositely polarized regions—magnetic pixels—of soft magnetic material. A
script translates these matrices into G-code, allowing the plotter to program
faces without manual intervention. Once plotted, encodings can be viewed using
magnetic viewing film (Fig. 5, middle).

We 3D print 25mm cubes from PLA and bond square faces of 26-mil thick soft
magnetic material to its 6 faces (Fig. 5, below). The cubes are printed with
internal cubic cavities of side length 18mm to neutrally buoy them in tap
water. Cubes are placed in the CNC platform and programmed with encodings that
produce the desired target configuration once mated. Each module costs $0.23
in materials ($0.19 PLA, $0.04 for 6 soft magnetic squares), and requires 12
minutes to program all 6 faces with our un-optimized G-code. Taken together,
cubes programmed with these encodings are therefore inexpensive, easy to
manufacture, physically re-programmable, attractive over distances in contrast
to contact adhesives (such as glue), and generate forces selectively without
consuming power during operation.

##  IV Results

In this section, we measure the magnetic force of the programmed encodings in
attraction and repulsion, and compare this to predictions made by taking
Hadamard products of the associated matrices. We then evaluate the performance
of the matrices in terms of the global and local agnosticism criteria.

Figure 6: (Above) Predicted vs (below) measured magnetic force for a
checkerboard matrix translated with its mate.

###  IV-A Empirical validation

We place one programmed cube on a scale (KUBEI pocket, 0.1mN accuracy)
installed on the CNC platform, and mount a second cube above it on the CNC end
effector, aligning the mating faces to be planar at 0.5mm distance in Z. We
use the CNC to translate the second cube one pixel at a time, in X and Y, and
measure the force generated at each increment. To maximise signal to noise,
the cubes are programmed with mating checkerboard patterns of size N=8 (Fig.
5, middle right). This is done because unlike the encoding matrices which are
largely agnostic, checkerboard patterns generate large cyclic forces in
attraction and repulsion as like and unlike pixels align and misalign with
each translation. The predicted results are generated by taking the cross-
correlation between mating checkerboard matrices, i.e. the matrices are
translated pixel-wise while taking their Hadamard product (1) at each
increment. The correlations in this and subsequent figures are plotted
normalized by the peak attractive (negative) force in order to facilitate
comparison between predicted and measured results.

Fig. 6 shows the predicted (above) and measured (below) results, which match
well visually. However the measured repulsive values are weaker than those
predicted by correlating the matrix values. We calibrated the scale to rule
out ascribing this result to anisotropic measurement sensitivity. Rather, this
is likely an effect of coercivity of the pixels on each other; if two
attractive magnetic dipoles are brought into contact, they reinforce their
attractive alignments. However, repulsive dipoles will realign to an
attractive equilibrium if free to rotate; a condition which the low coercivity
of the soft magnetic faces may support. Accounting for this, we implement a
scaling factor of 0.09 to the repulsive forces predicted by correlation that
normalizes the magnitude of the repulsive pixels to those in attraction. We
compute a normalized sum of squared differences of 0.014 between the measured
and predicted results using this scaling factor, supporting our model as an
accurate predictor of force between magnetically programmed faces.

###  IV-B Local agnosticism criterion

In this section, we use our model to evaluate the success of the local
agnosticism criterion; the forces between our generated matrices and their
mates. The figures are emblematic of the performance of all matrices in our
clique.

Figure 7: Translational agnosticism of a matrix A with its mate A’. They
exhibit maximal attraction (-1) when centered, remaining largely agnostic (0)
elsewhere.

To evaluate selectivity in translation, we correlate the matrices in X and Y,
taking the Hadamard product at each pixel increment (Fig. 7). A peak
normalized attractive force of -1 is produced with the matrices
translationally centered at (0,0); this corresponds to 256 Pascals, or 160 mN
between square faces of side 25mm. Elsewhere, the correlation is dominantly
agnostic (centered about 0) or repulsive (positive), bounded in attraction by
-0.25.

To illustrate how the Hadamard product produces a dominantly agnostic
interaction between the matrices besides their mating configuration, Fig. 8
visualizes the pixel-wise attraction and repulsion during the correlation of a
normalized Hadamard with its inverse. Here, red pixels indicate repulsion
(+1), green attraction (-1), and yellow non-overlap (0). Summing the pixels
over a square produces the Hadamard product, or the net force, that is plotted
in each data point in Fig. 7. The translationally centered position (0,0) in
Fig. 8 produces uniform attraction, whereas other positions produce an exactly
or largely agnostic interaction due to equal numbers of attractive or
repulsive pixels cancelling out.

Figure 8: Pixel-wise attraction and repulsion of a normalized order-8 Hadamard
matrix during translation with its mate. These are maximally attractive with
the matrix pair translationally centered at (0,0), during full overlap.
Elsewhere, attractive and repulsive forces largely cancel.

$\begin{bmatrix}x^{\prime}\\\
y^{\prime}\end{bmatrix}=\begin{bmatrix}cos(\theta)&-sin(\theta)\\\
sin(\theta)&cos(\theta)\end{bmatrix}\begin{bmatrix}x\\\ y\end{bmatrix}$ (4)

To evaluate selectivity in rotation, Hadamard products can be evaluated
between the matrices for orientations of 90${}^{\circ}n$, where n is an
integer. Elsewhere, the matrices do not superimpose. Therefore to evaluate
these products at arbitrary angles, we first discretize the order-8 matrices
by a factor of 10 to produce 80x80 grids. Each matrix element is then said to
be at location (x,y) with respect to an origin placed at the center of the
matrix, and we use the rotation matrix (4) to compute its new location (x’,y’)
after an arbitrary rotation of $\theta$. We smooth the result using a 3x3
averaging kernel to remove artefacts from the discretization, then evaluate
the Hadamard product to evaluate the net force, assigning a value of 0 (non-
overlap, or agnosticism) where pixels do not overlap. Fig. 9 plots this force
for 10 matrices from -180∘ to 180∘ in 10∘ increments. While our search
procedure optimized only for rotations of multiples $90^{\circ}$, this
procedure verifies performance for arbitrary rotations. The attractive force
is bounded by -0.25 in all orientations besides the intended mating
configuration (0∘). Pure agnosticism is further enforced at 90${}^{\circ}n$,
where the square matrices line up.

Figure 9: Rotational agnosticism of a matrix A with its mate A’. They exhibit
maximal attraction (-1) when centered, remaining largely agnostic (0)
elsewhere.

Fig. 10 visualizes the pixel-wise attraction and repulsion during the rotation
of a normalized Hadamard with its inverse. At 0∘, 90∘, 180∘ and 270∘, locally
attractive and repulsive pixels sum to 0, producing agnosticism. Elsewhere,
attractive and repulsive pixel interactions cancel to within an attractive
bound of -0.2.

Figure 10: Pixel-wise attraction and repulsion of a normalized order-8
Hadamard matrix during rotation with its mate. For rotations of 90n∘,
attraction and repulsion cancel exactly; elsewhere, approximately.

###  IV-C Global agnosticism criterion

In this section, we evaluate the global agnosticism criterion between two
emblematic matrices from our clique. Using the same tools used in the local
case above, Fig. 11 shows the correlation between two matrices, illustrating
their agnosticism over all translations. Equivalently, Fig. 12 illustrates
their agnosticism in rotation, with a negative bound of -0.36 that indicates
that attraction between non-mating faces is never greater than 36% of the
attractive force between mating faces in alignment, as derived in our search.

Figure 11: Translational agnosticism between non-mating matrices; agnosticism
dominates for all configurations. Figure 12: Rotational agnosticism between
non-mating matrices; agnosticism dominates for all configurations.

###  IV-D Stochastic self-assembly

We built and programmed 8 cubes to stochastically self-assemble into a ”meta
cube” (Fig. 13), to suggest how cubes could be programmed to assemble
structures into recursively larger cubes. In this configuration, each cube
occupies a vertex in an octree and connects to 3 other cubes, yielding an
assembly that requires 12 pairs of mating encodings. To do this, we selected
one of the 4 maximal cliques (size 12) of mutually compatible encodings (Fig.
1 above, right). We wrote a script to translate these encodings into G-code
and deployed this on our magnetic plotter, programming each module face in 2
minutes per face. We released all cubes into a glass container (cubic, 200mm
side length) filled with tap water, that was stochastically perturbed by a
hydraulic pump (Hygger WaveMaker 1600gph). The pump was programmed to produce
stochastic flows of random magnitude and frequency to stimulate brownian
motion of the cubes. We inserted a laser-cut mesh between the cubes and the
pump to promote turbulent flow and to prevent cubes from becoming drawn into
the pump inlet. We experimentally calibrated the force of our stochastic
disturbance to exceed the attractive force (-0.36) of misassemblies until no
permanent misassemblies were observed. Following this procedure, the cubes
acquired their correct positions to self-assemble the structure in 32 hours
(see supplemental video). After assembly, we re-programmed all faces with new
encodings to acquire different final target shapes and measured individual
mate forces, observing no difference in the strength of individual mates after
reprogramming.

Figure 13: Stochastic self-assembly from (top left) a random arrangement of 8
cubes to a (bottom right) meta cube.

##  V Discussion

In this paper, we introduced a method to build and program modules capable of
selective self-assembly. These modules are scalable; they are passive,
inexpensive ($0.23/module) and consist of COTS bulk material. We have
introduced a method of generating encodings that are provably selective. We
demonstrate a method of generating highly selective cliques of mutually
compatible face encodings for modules, and place guarantees on agnosticism for
non-mating configurations in translation and rotation, verifying these results
experimentally by instantiating encodings as magnetic polarities. We have
shown that these modules can be programmed with encodings that result in
specified self-assembled geometries using a simple magnetic plotter, and
demonstrated that modules can be re-programmed for new target shapes. Finally,
we demonstrated self-assembly using 8 modules in tap water. While our
technique was successful at self-assembling conservatively sized systems, a
number of limitations and avenues for future work present themselves.

The high selectivity of these encodings—agnosticism in translation and
rotation—result in significant assembly times, as it constrains the influence
of its near field force to areas directly above module faces. This diminishes
the ability of programmed faces to easily attract their mates over distances,
requiring them to enter each others’ narrow volumes of magnetic influence
before mutual attraction results in a bond. First, the water chamber size and
flow rate could be better calibrated. Future work could investigate adapting
the matrices to exhibit larger volumes of influence in the form of magnetic
potential wells, with gradients of attraction that help direct mates to their
correct orientation. In addition, these modules may be a promising candidate
to be used in conjunction with semi-directed stochastic assembly methods [18,
20, 19], which manipulate the fluid to guide mating modules into their basins
of attraction. This may also help address the current system’s inability to
control assembly order, which would be crucial for complex target geometries.

Using COTS fridge magnet, the attractive pressure between our mating faces is
relatively conservative (250 Pa). Future work could investigate the use of
more strongly magnetizable materials to increase mating strength. These
encodings could equivalently be used to build strong interfaces by replacing
programmed pixels with arrangements of permanent magnets, and for active self-
assembly using electromagnets [30] or electropermanent magnets whose encodings
could be changed online. In addition, the programming step may be rapidly
accelerated using parallel arrays of electromagnets to program every pixel
simultaneously.

Our search procedure successfully founded matrices that permitted self-
assembly for matrices of order N=8, however future work could improve our
search to extend to larger cliques and matrix orders in shorter time. While we
have demonstrated the implementation of selectively mating encodings
magnetically, future work could also investigate the encoding of these
matrices in other binary media, such as in the patterning of electrostatic
charge, wettability or chemical bonding, that may permit self-assembly at
smaller scales.

## References

  * [1] G. M. Whitesides, J. P. Mathias, and C. T. Seto, “Molecular self-assembly and nanochemistry: a chemical strategy for the synthesis of nanostructures,” _Science_ , vol. 254, no. 5036, pp. 1312–1319, 1991.
  * [2] B. A. Grzybowski, A. Winkleman, J. A. Wiles, Y. Brumer, and G. M. Whitesides, “Electrostatic self-assembly of macroscopic crystals using contact electrification,” _Nature materials_ , vol. 2, no. 4, pp. 241–245, 2003\.
  * [3] M. E. Nisser, S. M. Felton, M. T. Tolley, M. Rubenstein, and R. J. Wood, “Feedback-controlled self-folding of autonomous robot collectives,” in _2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2016, pp. 1254–1261.
  * [4] A. Papadopoulou, J. Laucks, and S. Tibbits, “From self-assembly to evolutionary structures,” _Architectural Design_ , vol. 87, no. 4, pp. 28–37, 2017.
  * [5] M. Nisser, C. C. Liao, Y. Chai, A. Adhikari, S. Hodges, and S. Mueller, “Laserfactory: A laser cutter-based electromechanical assembly and fabrication platform to make functional devices & robots,” in _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_ , 2021, pp. 1–15.
  * [6] S. Hauser, M. Mutlu, and A. J. Ijspeert, “Kubits: solid-state self-reconfiguration with programmable magnets,” _IEEE Robotics and Automation Letters_ , vol. 5, no. 4, pp. 6443–6450, 2020.
  * [7] M. Nisser, D. Izzo, and A. Borggraefe, “An electromagnetically actuated, self-reconfigurable space structure,” 2017.
  * [8] M. Nisser, L. Cheng, Y. Makaram, R. Suzuki, and S. Mueller, “Electrovoxel: Electromagnetically actuated pivoting for scalable modular self-reconfigurable robots,” in _2022 International Conference on Robotics and Automation (ICRA)_. IEEE, 2022, pp. 4254–4260.
  * [9] J. W. Romanishin, K. Gilpin, and D. Rus, “M-blocks: Momentum-driven, magnetic modular robots,” in _2013 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2013, pp. 4288–4295.
  * [10] J. W. Romanishin, K. Gilpin, S. Claici, and D. Rus, “3d m-blocks: Self-reconfiguring robots capable of locomotion via pivoting in three dimensions,” in _Robotics and Automation (ICRA), 2015 IEEE International Conference on_. IEEE, 2015, pp. 1925–1932.
  * [11] J. Baca, S. Hossain, P. Dasgupta, C. A. Nelson, and A. Dutta, “Modred: Hardware design and reconfiguration planning for a high dexterity modular self-reconfigurable robot for extra-terrestrial exploration,” _Robotics and Autonomous Systems_ , vol. 62, no. 7, pp. 1002–1015, 2014.
  * [12] V. Zykov, H. Lipson _et al._ , “Experiment design for stochastic three-dimensional reconfiguration of modular robots,” in _IEEE Int. Conf. Intell. Robots Syst., Self-Reconfigurable Robot. Workshop, San Diego, CA_. Citeseer, 2007.
  * [13] S. Tibbits, “The self-assembly line,” 2012.
  * [14] A. Hacohen, I. Hanniel, Y. Nikulshin, S. Wolfus, A. Abu-Horowitz, and I. Bachelet, “Meshing complex macro-scale objects into self-assembling bricks,” _Scientific reports_ , vol. 5, no. 1, pp. 1–8, 2015.
  * [15] N. Bowden, A. Terfort, J. Carbeck, and G. M. Whitesides, “Self-assembly of mesoscale objects into ordered two-dimensional arrays,” _Science_ , vol. 276, no. 5310, pp. 233–235, 1997.
  * [16] Y. Lu, A. Bhattacharjee, D. Biediger, M. Kim, and A. T. Becker, “Enumeration of polyominoes & polycubes composed of magnetic cubes,” in _2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2021, pp. 6977–6982.
  * [17] M. Jılek, M. Kulich, and L. Preucil, “Centimeter-scaled self-assembly: A preliminary study,” in _Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO). ScitePress_ , 2020, pp. 438–445.
  * [18] M. T. Tolley, M. Krishnan, D. Erickson, and H. Lipson, “Dynamically programmable fluidic assembly,” _Applied Physics Letters_ , vol. 93, no. 25, p. 254105, 2008.
  * [19] M. T. Tolley and H. Lipson, “Fluidic manipulation for scalable stochastic 3d assembly of modular robots,” in _2010 IEEE international conference on robotics and automation_. IEEE, 2010, pp. 2473–2478.
  * [20] M. Tolley and H. Lipson, “Programmable 3d stochastic fluidic assembly of cm-scale modules,” in _2011 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2011, pp. 4366–4371.
  * [21] M. Krishnan, M. T. Tolley, H. Lipson, and D. Erickson, “Increased robustness for fluidic self-assembly,” _Physics of Fluids_ , vol. 20, no. 7, p. 073304, 2008.
  * [22] M. Kalontarov, M. T. Tolley, H. Lipson, and D. Erickson, “Hydrodynamically driven docking of blocks for 3d fluidic assembly,” _Microfluidics and Nanofluidics_ , vol. 9, no. 2, pp. 551–558, 2010.
  * [23] M. Jílek, M. Somr, M. Kulich, J. Zeman, and L. Přeučil, “Towards a passive self-assembling macroscale multi-robot system,” _IEEE Robotics and Automation Letters_ , vol. 6, no. 4, pp. 7293–7300, 2021\.
  * [24] D. Tsutsumi and S. Murata, “Multistate part for mesoscale self-assembly,” in _SICE Annual Conference 2007_. IEEE, 2007, pp. 890–895.
  * [25] S. Miyashita, Z. Nagy, B. J. Nelson, and R. Pfeifer, “The influence of shape on parallel self-assembly,” _Entropy_ , vol. 11, no. 4, pp. 643–666, 2009\.
  * [26] M. Nisser, Y. Makaram, and S. Mueller, “Stochastic self-assembly with magnetically re-programmable voxels,” 2022.
  * [27] “Correlated magnetics,” http://www.polymagnet.com/, accessed: 2021-11-25.
  * [28] C. Bron and J. Kerbosch, “Algorithm 457: finding all cliques of an undirected graph,” _Communications of the ACM_ , vol. 16, no. 9, pp. 575–577, 1973\.
  * [29] A. Hagberg, P. Swart, and D. S Chult, “Exploring network structure, dynamics, and function using networkx,” Los Alamos National Lab.(LANL), Los Alamos, NM (United States), Tech. Rep., 2008.
  * [30] M. Nisser, L. Cheng, Y. Makaram, R. Suzuki, and S. Mueller, “Programmable polarities: Actuating interactive prototypes with programmable electromagnets,” in _The Adjunct Publication of the 34th Annual ACM Symposium on User Interface Software and Technology_ , 2021, pp. 121–123.


# Varying activity and the bursts properties of FRB 20240114A probed with GMRT
down to 300 MHz

Ajay Kumar<EMAIL_ADDRESS>National Centre for Radio Astrophysics
(NCRA - TIFR), Pune - 411007, India  Yogesh Maan<EMAIL_ADDRESS>National Centre for Radio Astrophysics (NCRA - TIFR), Pune - 411007, India
Yash Bhusare<EMAIL_ADDRESS>National Centre for Radio Astrophysics
(NCRA - TIFR), Pune - 411007, India

###### Abstract

Repeating fast radio bursts can exhibit a wide range of burst repetition
rates, from none to hundreds of bursts per hour. Here, we report the detection
and characteristics of 57 bursts from the recently discovered FRB 20240114A,
observed with GMRT in the frequency ranges 300-500 MHz and 550-750 MHz.
Majority of the bursts show narrow emission-bandwidth with $\Delta\nu/\nu\sim$
around 10%. All of the bursts we detect are faint ($<$10 Jy ms), and thus
probe the lower end of the energy distribution. We determine the rate function
for FRB 20240114Aat 400 MHz, and downward drift rates at 400 and 650 MHz, and
discuss our measurements in the context of the repeating FRB population. We
observe sudden variations in the burst activity of FRB 20240114A over time.
Our data as well as the other publicly available information on other
observations of FRB 20240114A so far, there is an indication that FRB
20240114A potentially exhibit a chromaticity in its burst activity. While the
burst properties of FRB 20240114A are similar to other repeating FRBs, the
frequency-dependent activity, if established, could provide crucial clues to
the origin of repeating FRBs. We also place the most stringent 5$\sigma$ upper
limits of 600 $\mu$Jy and 89 $\mu$Jy on any persistent radio source (PRS)
associated with FRB 20240114A at 400 MHz and 650 MHz, respectively, and
compare these with the luminosity of the known PRSs associated with FRB121102A
and FRB190520B.

Fast Radio Bursts (2008) — FRBs (2008) — Radio Transients (1868)

††facilities: Giant Meterwave Radio Telescope, Khodad, Pune ††software:
astropy (Astropy Collaboration et al., 2013), matplotlib (Hunter, 2007), scipy
(Virtanen et al., 2020), your (Aggarwal et al., 2020), RFIClean (Maan et al.,
2021), PRESTO (Ransom, 2011)

##  1 Introduction

Fast radio bursts (FRB) are coherent bright radio flashes of cosmological
origin with duration ranging from a few micro to several milliseconds. Their
exceptional high dispersion measure (DM) along the line of sight can not be
explained by the Galactic electron content, thus predicting their
extragalactic origin. This is now confirmed with several repeating FRBs
localized to a variety of galaxies (Chatterjee et al., 2017a; Bannister et
al., 2019; Marcote et al., 2020; Bhandari et al., 2020; Heintz et al., 2020).
The origin and emission mechanism of FRBs still remain an open question.
Currently there are $\sim$800 published FRBs, of which some are repeating in
nature. Discovery of the first repeating FRB, FRB 20121102A (Spitler et al.,
2014), ruled out cataclysmic events as the source for repeating FRBs.
Currently there are close to sixty repeaters known 111https://www.chime-
frb.ca/repeaters but only a few are known to be extremely active occasionally.
A few repeating FRBs show periodicity in their activity, e.g., FRB180916 and
FRB121102A exhibit periodicities of 16.35 and about 157 days (Chime/Frb
Collaboration et al., 2020; Rajwade et al., 2020a). Some FRBs also show very
high bursts rates ($>$100 bursts/hr) occasionally, e.g., FRB201124A and
FRB220912A (Zhou et al., 2022; Feng et al., 2023). Several repeaters have been
followed up or monitored over a long duration of time and their properties
very well studied but we still we do not have source models that can explain
all their observed properties. Discovery of highly active repeaters can
provide more detailed insights into the observed spectro-temporal properties
like narrow-band emission-behaviour and downward drifting pattern that has
been seen across all repeaters.

Recently CHIME reported a discovery of a highly active repeating FRB 20240114A
(Shin & CHIME/FRB Collaboration, 2024). Starting on January 14, 2024,
CHIME/FRB detected three bursts from FRB 20240114A within a week, and at a DM
of 527.7 pc cm-3 consistent for all the bursts. The CHIME/FRB daily exposure
time of only 4 minutes for this source suggested that a high burst rate above
the fluence thresholds of $\sim 1$ Jy ms (Shin & CHIME/FRB Collaboration,
2024). There were prompt follow ups from various telescopes including Parkes,
FAST and ours using various frequency bands of GMRT (300-500 MHz, 550-750 MHz
and 1060-1460 MHz), which confirmed the high activity of FRB 20240114A
(Uttarkar et al., 2024; Pelliciari et al., 2024a; Kumar et al., 2024).

Subsequently, FRB was localized to an arc-second precision by the MeerKAT
telescope (Tian et al., 2024). Following this localization, we focused on low
frequency study of FRB 20240114A using band-3 (300-500 MHz) of uGMRT. One of
our observations was on 05 March when there was a burst storm as reported by
FAST (Zhang et al., 2024a). Since then, several other detections have been
reported from P-band to frequencies up to 6 GHz (Uttarkar et al., 2024; Ould-
Boukattine et al., 2024a; Zhang et al., 2024b; Pelliciari et al., 2024a; Kumar
et al., 2024; Panda et al., 2024a; Zhang et al., 2024a; Pelliciari et al.,
2024b; Snelders et al., 2024; Tian et al., 2024; Ould-Boukattine et al.,
2024b; Hewitt et al., 2024; Joshi et al., 2024; Shin & CHIME/FRB
Collaboration, 2024).

Low frequency study of active repeating FRBs provide constraints to various
proposed progenitors models for repeating FRBs (e.g., Pastor-Marazuela et al.,
2021a). They also help in studying some of the propagation effects and the
underlying emission mechanism (Gopinath et al., 2023; Pleunis et al., 2021a).
Narrow-band behaviour has been reported for repeating FRBs (Kumar et al.,
2021a) and it has been also been linked to giant pulses from pulsars and
magnetars (Thulasiram & Lin, 2021; Maan et al., 2019a). Magnetars appear to be
the plausible originators of some of the repeating FRBs, especially since the
detection of a millisecond duration burst from the Galactic magnetar SGR
1935+2154 with an energy comparable to the faintest FRBs (CHIME/FRB
Collaboration et al., 2020; Bochenek et al., 2020). The low-frequency
observations further help in studying the wide-band spectral behaviour of FRBs
and probe any potential chromatic evolution of the activity (Pastor-Marazuela
et al., 2021a; Pleunis et al., 2021a).

In this paper, we report detection of 57 bursts at low radio frequencies and
narrow-band emission from FRB 20240114A. We present the results from the
observations done in band-4 (550-750 MHz), band-3 (300-500 MHz) and band-5
(1060-1460 MHz) of the upgraded GMRT. We describe the observation and the
search procedure in section 2, the rate function at 400 MHz and extreme
narrow-band emission in section 3. Burst properties are detailed in section 3,
we discuss the implications of our results for observed properties of the
repeating FRBs and their proposed formation channels in section 5. Finally, we
conclude in the Section 6.

##  2 Observations and search procedure

Following a possible high level of activity from the repeating FRB 20240114A
reported by (Shin & CHIME/FRB Collaboration, 2024), we conducted prompt
follow-up observations using the upgraded giant metrewave radio telescope
(uGMRT; Gupta et al., 2017). We utilized the wide frequency coverage of uGMRT,
spanning 300 MHz to 1460 MHz. Specifically, we observed the source for 114
minutes in band-4 (550–750 MHz) on 01 February 2023, 104 and 71 minutes in
band-3 (300–500 MHz) on 08 and 12 February 2023, respectively, and 65 minutes
in band-5 (1060–1460 MHz) on 13 February 2023. These observations utilized
recording in three different beams: phased-array (PA) beam and the PA spectral
voltage (PASV; spectral voltage recording at the Nyquist rate) beam utilizing
antennas in compact configuration, and the incoherent array (IA) beam. We
simultaneously recorded the interferometric visibility data with a sampling
time of 0.67 seconds to potentially localize the source within a few
arcseconds using a burst with adequate signal-to-noise ratio (S/N). On 12 Feb
2024, we formed four PA beams with different antenna configurations to
potentially obtain a localization better than the position known at that time
(about 1.5 arcmin uncertainty, Shin & CHIME/FRB Collaboration, 2024).

Sensitivity of our above observations was limited as the actual source was
about 1.2 arcmin away from the best estimated position at that time. After FRB
20240114A was localized by MeerKAT to RA = 21h27m39.83s and Dec = 04d19’46.02”
(Tian et al., 2024) with about an arcsec precision, we followed it for six
more hours at band-3, now utilizing the full sensitivity of the PA beams.
Observations were done in three epochs on 05, 08, and 24 March 2024 with the
cumulative on-source times of 55, 85, and 75 minutes, respectively. More
recently, we also had two observations at band-4, on 29 May 2024 and 15 June
2024, with cumulative on-source times of 126 minutes in each. In each of these
sessions, we recorded the PA beam with 4096 frequency channels and a time
resolution of 655 $\mu s$. PASV data were also recorded simultaneously, which
were coherently dedispersed and converted to filterbank format with 1024
frequency channels and a time resolution of 81.92 $\mu s$ in the offline
processing.

In all the above sessions, we use the following reduction and search
procedures. The raw data are recorded with 16384, 4096, 4096 frequency
channels and with time resolutions of $655.36\mu s$, $327.28\mu s$, $163.84\mu
s$ in bands 3, 4 and 5, respectively. For the band-3 observations conducted in
March, we convert the PASV data to filterbank format with a sampling time of
$81.92\mu s$ and 1024 frequency channels. All these raw data are passed
through a number of RFI excision rounds as follows. First,
RFIClean222https://github.com/ymaan4/RFIClean (Maan et al., 2021) is used to
mitigate periodic, broadband and spiky RFI as well as convert the data to
SIGPROC filterbank format. Subsequently, dedispersion from SIGPROC is used to
subband the data to 1024 frequency channels, followed by one more round of RFI
mitigation using RFIClean. Finally, we execute rfifind from PRESTO (Ransom,
2011) to compute a mask of further identified RFI-contaminated segments in the
data.

We use prepdata from PRESTO to dedisperse over a range of DMs from 522 to 532
pc cm-3, with a step of 0.1 pc cm-3 for band-3 and 0.2 pc cm-3 for band-4. The
dedispersed time-series are searched for single pulses using
single_pulse_search.py above S/N of 7 and with a maximum boxcar width of 0.5
sec. The candidates are clustered in arrival times and DM, and then the
candidate corresponding to highest S/N in the individual clusters is verified
visually using the waterfall plot prepared using the package your (Aggarwal et
al., 2020).

For the sub-band search, we divide the 200 MHz bandwidth in band-4 as well as
band-3 into 7 sub-bands. Each sub-band has a bandwidth of 50 MHz and their the
central frequency varies from 325 to 475 MHz for band-3, and 575 to 725 MHz
for band-4, in steps of 25 MHz. Thus, there is 50% overlap between successive
sub-bands. The dedispersion for these sub-bands is realized by ignoring the
frequency channels not desired for a particular sub-band. The remaining part
of the search procedure is same as described earlier.

In the Figure 1 top panel, we show the detection and non-detection of bursts
reported at different frequencies in four different ranges: 0.3 to 0.8 GHz,
0.8 to 2.2 GHz, 2.2 to 4 GHz and 4 to 6 GHz, over the duration from 14/01/24
to 09/05/24 roughly every ten days. Here, we use the limited information that
is available from various Atels published during this period.

Figure 1: Top: A summary of the observations and detection/non-detection of
bursts reported at different frequencies, ranging from P-band to C-band, over
the duration from 14/01/24 to 09/05/24. The data used here are from various
Atels published between 28/01/24 and 09/05/24 (Shin & CHIME/FRB Collaboration,
2024; Uttarkar et al., 2024; Ould-Boukattine et al., 2024a; Zhang et al.,
2024b; Pelliciari et al., 2024a; Tian et al., 2024; Kumar et al., 2024; Panda
et al., 2024a; Zhang et al., 2024a; Snelders et al., 2024; Pelliciari et al.,
2024b; Ould-Boukattine et al., 2024b; Hewitt et al., 2024; Joshi et al., 2024;
Limaye & Spitler, 2024). Bottom: Dynamic Spectra and time series of some of
the bursts detected at band-3 (300-500 MHz).

##  3 Burst Analysis and Results

We detected 10 bursts from our band-3 and band-4 observations conducted before
the source well localized (Tian et al., 2024). In these observations, the
pointing center was offset from the actual source position by 1.2 arcminutes
(Tian et al., 2024). Of the 10 bursts, 5 were detected in band-4 (550-750 MHz)
with an on-source time of 114 minutes, but only in the IA beam indicating that
the actual source position was offset by more than the PA beam width (1.1
arcmin). Additionally, we detected 5 bursts in band-3 in the PA beam despite
the offset as the PA beam width was adequately larger at these lower
frequencies. We corrected the measured fluences for these observations to
account for the offset from the actual position. Consequently, we estimate the
burst rate to be 2.6 hr-1 at 650 MHz above the fluence of 0.64 Jy ms, seven
days after CHIME detected two bright bursts from FRB 20240114A (Shin &
CHIME/FRB Collaboration, 2024).

From our later observations in March 2024, conducted in band-3 (300-500 MHz)
and now using the improved position estimate, we detected significant
activity. On 5th March 2024, approximately 3 hours and 15 minutes after a FAST
observation wherein a sudden extremely high activity (burst rate of 500 hr-1
above a fluence of 0.015 Jy ms Zhang et al., 2024a) was noticed, we detected
about 27 bursts within 52 minutes, implying a burst rate of 31 hr-1 at a
fluence threshold of 0.2 Jy ms. The rate varies significantly over short
durations, as we detected only 5 bursts on 8th March 2024, and all within the
first 10 minutes out of nearly 75 minutes of the total on-source time. The
rate seemed to have decreased by a factor of 8 in three days. Also, there were
several bursts reported at P and L-band after this session (Snelders et al.,
2024; Ould-Boukattine et al., 2024b; Hewitt et al., 2024; Joshi et al., 2024)
implying varying activity level. On 24th March, we detected 15 bursts from an
on-source time of 75 minutes. We did not detect any burst from our band-4
session on 29 May 2024.

In total, we detected 52 bursts in band-3 (300-500 MHz) and 5 bursts in band-4
(550-750 MHz). Some of these bursts exhibit a downward drifting pattern and
majority are narrow-band in nature, as shown in Figure 1. Many bursts are
primarily centered in the higher part of band-3, i.e., above 400 MHz, as
evident from Figure 2, however, it is most likely caused by typical more RFI
contamination in the central and lower half of band-3.

Figure 2: Left: Variation of the measured peak frequency with the emission
bandwidth. The red line denotes the Gaussian kernel density function for the
distribution of the fractional emission bandwidth ($\Delta\nu/\nu$) at top and
for the peak frequency at right. Right: Example of a burst showing downward
drifting pattern and Gaussian fits to the spectra for each component. In the
right panel, spectra obtained from averaging along time axis for the regions
marked in left panel under each component are shown. The red and green
horizontal lines indicate the peak frequency for each of the component.

###  3.1  Rate Function

For each of the bursts, we use the filterbank files and the RFI mask described
in Section 2, to prepare a time series dedispersed at the detection DM (i.e.,
the DM which maximizes the S/N). We extract the time series around the burst’s
arrival time and normalize the time series in units of the root mean square
(RMS) noise measured from off-pulse. The time series is then calibrated to Jy
units using the radiometer equation. To obtain the fluence for each burst, we
sum the calibrated time series around the pulse region. After measuring the
fluence for each pulse, we characterize the rate function of FRB 20240114A at
400 MHz. Figure 3 shows the log N - log S plot. We fit a power law using the
optimization function scipy.optimize.minimize, taking into account Poissonian
errors as described by (Cash, 1979). All bursts have fluences below 10 Jy ms,
suggesting that we are probing the lower end of the energy distribution.
However, there have also been bright burst reported with fluences in excess of
10 Jy ms at P-band (Ould-Boukattine et al., 2024b; Pelliciari et al., 2024b).
We do not include fluences measured for 5 bursts detected in band-4 since they
were detected in IA beam.

To measure the pulse widths, we use the scipy module curve_fit to fit a
Gaussian profile to each of the burst (component). The pulse widths range from
0.6 ms to 17 ms. The median pulse width of FRB 20240114A at 400 MHz is 4.3 ms,
which is broader compared to 1.4 ms at 650 MHz(Panda et al., 2024b). The
median for the SNR-optimised DM is 528.1 pc cm-3.

Figure 3: Cumulative distribution of the fluences of the bursts detected at
band-3 (300-500 MHz). The green dashed line indicates the fluence threshold
for a 2 ms burst detected at a S/N of 10. The red line indicates the fitted
power-law.

###  3.2 Emission bandwidths

To characterize the emission bandwidth, we first down-sample the data from
1024 to 128 frequency channels and then average along the time axis around the
burst region to estimate the average burst spectrum, as illustrated in the
right panel of Figure 2. The average spectrum is the fitted with a Gaussian
function to estimate the peak frequency as well as the emission bandwidth
(full width at half maximum). Most of the bursts exhibit narrow-band emission,
as also apparent in Figure 1.

In Figure 2, two major trends are apparent. First, the peak emission frequency
seems to decrease (roughly from 480 to 440 MHz) as the emission bandwidth
increases from about 20 to 60 MHz. It is most likely caused by the band-edge
and search sensitivity effects — near the top band-edge, we might be seeing
only parts of the burst emission otherwise continuing outside our observing
bandwidth. A similar but inverted trend would be expected at the lower edge of
the band, and we see some hints of it. However, several parts of the spectrum
in the lower half of the band (300-400 MHz) are typically contaminated by RFI,
which effectively degrade the sensitivity in this part of the band.
Consequently, we see this inverted trend effectively starting from roughly
around the middle of the band. The distribution of the peak frequencies
peaking around 450 MHz is also a consequence of the above band-edge effects.
Figure 2 demonstrates that the emission is narrow-band for nearly all the
bursts. The fractional bandwidth distribution peaks around 0.12.

###  3.3 Frequency drift rates

The sad-trombone effect or the downward-drifting pattern, is characteristic of
repeating FRBs, with different repeating FRBs showing a wide variety of drift
rates ranging from $-1$ to $-60$ MHz/ms (Pleunis et al., 2021b). Many of the
bursts exhibit this downward drifting with single and multiple sub-bursts. The
drift rates are typically estimated by first determining the structure-
optimized DM and then exploiting 2-dimensional correlation. Due to the low S/N
of our bursts, we could not utilise
DM_phase333https://github.com/danielemichilli/DM_phase (Seymour et al., 2019)
for determining the structure optimized DM. We also could not use
frbgui444https://github.com/mef51/frbgui (Chamma et al., 2023) for the same
reason. So, we use the S/N optimized DM to measure the relative TOAs and the
central frequencies of the successive components in multi-component bursts to
estimate the drift rate. The peak frequency of the sub-bursts are estimated as
described in Section 3.2. The relative TOAs are determined by fitting a
Gaussian function to each component in the band-averaged time profile. There
are three bursts for which we could measure drift rates. One example of a
burst for which we measured the drift rate is shown in right panel in Figure
4. The drift rate for this burst is estimated to be $-0.35\pm 0.07$ MHz/ms.
For the other bursts, the drift rate is estimated to be $-11.7\pm 0.8$ MHz/ms
and $-2.07\pm 0.14$ MHz/ms. These measurements are comparable to the drift
rates obtained for other repeaters at similar frequencies, as shown in Figure
4.

Using the data made publicly available by Panda et al. (2024b), we also
measure drift rates for 4 bursts detected at band-4 which show well-separated,
drifting components. The arrival times of the components are provided by the
authors and we estimate the corresponding peak frequencies visually. We
liberally assign large uncertainties to these visual estimates of peak
frequencies, which often cover at least half of the total emission bandwidth
for the respective components. The drift rates thus estimated are also shown
in Figure 4.

Figure 4: Drift rates as function of frequency, with both the quantities in
the rest frame of the respective host galaxy, for FRB 20240114A and other
repeaters (e.g. FRB121102A, FRB180916B). The dashed blue line and the region
shaded around it indicate the power-law fit with its uncertainty, fitted by
(Wang et al., 2022). References for measurements on other FRBs: Zhou et al.
(2022); Kumar et al. (2023); CHIME/FRB Collaboration et al. (2019); Hessels et
al. (2019); Pastor-Marazuela et al. (2021b); Marthi et al. (2020).

##  4 Imaging analysis and results

Our observations in February and early March (at band-3 as well as band-4)
often utilised recording the interferometric visibilities at 0.67 s
integration time with the aim to localize the source to arcsecond precision
(the position uncertainty at that time was about 1.5 arcmin Shin & CHIME/FRB
Collaboration, 2024). However, these observation did not result in detection
of a burst with adequately high S/N for such a localization. Once the source
was localized to arcsec precision by (Tian et al., 2024), our remaining
observations (on 5, 8 and 24 March 2024) in band-3, used the new position as
the pointing center with an integration time of 10.7 s. For band 3
observations, we used 3C48 as the flux calibrator, and for band 4, we used
3C147 and 3C48 in different sessions, and with J2130+050 as the phase
calibrator in all the observations. We used
CAPTURE555https://github.com/ruta-k/CAPTURE-CASA6 (Kale & Ishwara-Chandra,
2021) for RFI flagging and calibration of the data.

After calibrating the data, we use CASA task tclean for imaging, with the
deconvolution algorithm set to MTMFS and the Briggs weighting scheme (robust =
0). Self-calibration was performed until convergence to an image was achieved.
We used uGMRTprimarybeam for primary beam correction for the band-4 image
obtained from the 01 Feb 2024 session, since our pointing was slightly offset
from the source position. From this observation, we place an upper limit of
140 $\mu$Jy (at 5$\sigma$ level) on any continuum emission associated with the
FRB (Kumar et al., 2024). From our more recent observation on 15 June 2024, we
place the most stringent 5$\sigma$ upper limit of 89 $\mu$Jy for continuum
emission from any associated PRS or the host galaxy at 650 MHz (Panda et al.,
2024b, placed a 5$\sigma$ upper limit of 124 $\mu Jy$ at the same frequency).
Similarly, at band-3, we obtained the final images individually for the
datasets from 5, 8, and 24 March observations. In the best image, we do not
detect any continuum emission up to 600 $\mu$Jy (at 5$\sigma$ level) with an
on-source time of 78 minutes. We note that there is a bright (325 mJy) source
between the FWHM and first null of the uGMRT primary beam. Thus, deconvolution
errors limits the RMS noise achievable for the band-3 image despite the longer
on-source time. We provide the upper limits for spectral luminosity at 650 MHz
and 400 MHz using the redshift of 0.13 (Bhardwaj et al., 2024). We also obtain
image in band-5 with on source time of 65 min but the observations were
heavily affected by RFI. Nevertheless, we provide an upper limit of 1 mJy for
the PRS at 1.26 GHz using these band-5 data. Figure 5666L$\nu$ is calculated
assuming a flat spectrum. shows the comparison of our upper limits on emission
from a persistent radio source (PRS) associated with FRB 20240114A, along with
that for PRSs associated with FRB121102A and FRB190520B, and upper limits on
similar emission from various repeating and non-repeating FRBs, as well as SGR
1935+2154.

Figure 5: Top: Images of the sky regions around the sky-position of FRB
20240114A, at 650 MHz and 400 MHz. The MeerKAT localisation of FRB 20240114A
is marked with a circle at the center of the images. Bottom: Comparison of
upper limits and measurements of spectral luminosity of PRS, associated with
repeating and non-repeating FRBs, and SGR 1935+2154, along with our upper
limits for FRB 20240114A at different frequencies. References: Law et al.
(2022); Resmi, L. et al. (2021); Chatterjee et al. (2017b); Zhang et al.
(2023); Niu et al. (2022). For the original references for upper limits on
various repeaters, non-repeating FRBs and SGR 1935+2154, please see Law et al.
(2022).

##  5 Discussion

Our efforts to understand the nature of FRB 20240114A included a prompt
follow-up in February and March 2024, at band-3 as well as band-4 of uGMRT
right after its discovery. From the observation on 1 February at band-4, we
detected bursts in the IA beam but not in the PA beam, suggesting that the
position reported by CHIME was offset from the actual source position by at
least 1.1 arcmin (i.e., half-power beam width), a finding later confirmed by
the MeerKAT localization (Tian et al., 2024). Subsequent observations in
band-3 and band-5 also did not result in bursts adequately bright enough to
localize the source. In these early observations, we detected five bursts in
Band-3 in the PA beam (but not in the IA beam, as these were all faint
bursts), and none at band-5. With these detections, we confirmed the high
activity phase of FRB 20240114A in the 300-750 MHz frequency range (Kumar et
al., 2024). Using the subsequent observations in March, now with adequately
precise position estimate from MeerKAT, we measure significant variations in
the activity levels of FRB 20240114A which could be related to the underlying
emission mechanism or propagation effects. While a few repeating FRBs show
periodic modulation in their activity (Chime/Frb Collaboration et al., 2020;
Rajwade et al., 2020b), a few others show high activity phases (e.g., burst
storms) followed by a long quiescence (Lanman et al., 2022). Currently there
is no clear picture what causes such modulations, periodic or otherwise, in
the bursting activity.

###  5.1 Frequency-dependent activity

FRB 20240114A has been followed up by various telescopes at a variety of
observing frequencies covering the range 0.3$-$6 GHz. We compiled a summary of
these observations and burst detections reported by various groups through
ATels and show it in Figure 1. Limited by the details provided in various
ATels as well as for the convenience of displaying, we quantize the observing
frequency into four ranges: 0.3-0.8, 0.8-2, 2-4, and 4-6 GHz. Similarly, the
time span between the discovery and first week of May is divided into 10
parts. We note that at L-band and lower frequencies (i.e., the first two
frequency ranges), bursts from FRB 20240114A have been detected regularly.
However, at the frequencies higher than 2 GHz, bursts have been detected only
in the later half of April and beginning of May, despite regular observations
earlier too. So, there is an indication of a potential chromaticity in the
activity, where FRB 20240114A is becoming active at higher frequencies at the
later epochs. Non-detection of any burst from our latest band-4 observation on
29 May 2024 also indicate towards quenching of the burst activity at lower
frequencies.

FRB 20180916B is the only FRB for which evidence of chromaticity in its
activity has been seen (Pastor-Marazuela et al., 2021b; Pleunis et al.,
2021c). Despite having only a moderate burst rate, the discovery of the
chromaticity was aided by the knowledge of the underlying modulation period of
the FRB 20180916B. For FRB 20240114A, an underlying periodicity, if any, in
its activity is yet to be detected. Moreover, the frequency chromaticity in
FRB 20240114A appears to be opposite to that in FRB 20180916B, i.e., the burst
activity at higher frequencies seem to be delayed in FRB 20240114A while the
opposite is true for FRB 20180916B.

We note that the currently available information on FRB 20240114A is not
sufficient to establish or rule out above potential chromaticity. For example,
bursts at in the 2$-$6 GHz frequency range have been detected by sensitive
telescopes like Effelsberg and Nancay (Limaye & Spitler, 2024; Hewitt et al.,
2024; Joshi et al., 2024). However, the observations displayed for the 4$-$6
GHz range in Figure 1 are mostly by much smaller telescopes and different
setups, e.g., with smaller bandwidths (Ould-Boukattine et al., 2024b). Any
lack of sub-band searches might also have contributed to the sensitivity
differences. Furthermore, the narrow-band emission of the bursts combined with
narrow observing bandwidths might also have been decisive. With a detailed
information of various observing setups, it might be possible to conduct
simulations to further probe if there is indeed chromaticity in the FRB
20240114A’s activity, however, that is beyond the scope of this work.
Nevertheless, a systematic, multi-frequency monitoring of the source is needed
to better probe this potential chromaticity as well as any underlying
periodicity.

Furthermore, Zhang et al. (2024b) conducted observations on 28 and 29 January
2024, and 01 and 04 February 2024, detecting 38 bursts with a total on-source
time of 2 hours. The inferred burst rate at approximately 1.1 GHz is 20 hr-1
above the fluence threshold of around 0.015 Jy ms. Assuming a mean power law
index of -1.5, the burst rate would be only 0.072 hr-1 at a fluence threshold
of 0.64 Jy ms. However, we estimate the burst rate to be 2.6 hr-1 at 400 MHz
from our first band-3 observation on 08 February 2024, indicating a sharp
evolution of burst rate with observing frequency and/or time.

###  5.2 Spectro-temporal Properties

The width distribution (not shown but assessed separately) of the bursts we
detected shows a median width of 4.3 ms at 400 MHz which is larger than the
median width of 1.4 ms reported at 650 MHz (Panda et al., 2024b). In the left
panel in Figure 2, we see that the peak frequency of majority of the bursts is
above 400 MHz, potentially suggesting a preference for the emission
frequencies to be above 400 MHz. However, there could also be detection
biases, e.g., lower sensitivity in the lower half of band-3 due to presence of
RFI and scatter broadening. Most of the bursts in our sample show narrow-band
emission. Similar behavior has been observed at 600 MHz with CHIME and at
L-band with FAST (Zhang et al., 2024b; Shin & CHIME/FRB Collaboration, 2024).
Narrow-band emission has also been observed in the giant pulses of the Crab
pulsar and FRB 20190711A (Thulasiram & Lin, 2021; Kumar et al., 2021b) with
similar values of $\Delta\nu/\nu$ as for FRB 20240114A. There is the
possibility of bursts being narrow due to emission being below our detection
threshold at other frequencies and brightest at frequencies where we detect
them, which effectively implies steep modulations in the burst spectra. Cordes
et al. (2017) shows that caustics can produce strong magnifications ($\leq
10^{2}$) in plasma and present as spectral peaks in the frequency range 0.1-1
GHz. Plasma lensing has been invoked to explain narrow-band and high-energy
bursts for repeating FRBs, e.g., FRB121102A, FRB201124A (Chen et al., 2024).
Metzger et al. (2019) invokes synchrotron maser emission in ultra-relativistic
magnetized shocks to produce the downward drifting pattern. Under certain
conditions, it predicts a simultaneous narrow-band emission that is not
intrinsic but most likely caused by the propagation effects, e.g. plasma
lensing.

Wang et al. (2022) modeled the drift rates of various repeating FRBs as a
function of frequency (in the rest frame of the host galaxy) with a power-law.
They suggest that the downward drift can be explained through curvature
radiation in magnetars originating from different heights. Figure 4 shows that
the drift rates we measured for FRB 20240114A also agree with the relationship
modeled by Wang et al. (2022). In the lower frequency regime, FRB180916B has
been the only other FRB with such measurements. Our measurements for FRB
20240114A further populate this part of the phase space.

###  5.3 Energetics

Several bursts have now been reported from FRB 20240114A that have fluence in
excess of 50 Jy ms (Ould-Boukattine et al., 2024a; Pelliciari et al., 2024b;
Ould-Boukattine et al., 2024b; Shin & CHIME/FRB Collaboration, 2024), however,
all the bursts except one we detected have fluences below 10 Jy ms (twice of
the CHIME/FRB’s typical detection threshold). This might suggest that most of
the time the source might be emitting relatively low energy bursts, also seen
by Zhang et al. (2024b), and high energy bursts might be following a different
mean burst rate and power-law statistics. If this is really the case, it would
remain to be seen whether the two population of bursts are caused by different
emission mechanisms or propagation effects (e.g., plasma lensing).
Particularly for FRB20201124A, Kirsten et al. (2023) argue that the higher
energy burst population cannot be explained by lensing effects and the
increased brightness is intrinsic.

For FRB 20240114A, We obtain a power-law index of $-1.24\pm 0.11$ for the
bursts we detected at 400MHz. We note that, much like several other works
following the same approach, our fluences are estimated from the band-averaged
bursts, even for the bursts clearly having narrow bandwidths. Our measured
power-law index of $-1.24\pm 0.11$ reconciles with that for several other
repeaters for which a wide range from relatively flat index of -0.5 to as
steep as -4.9 has been reported (Kirsten et al., 2023). This index also varies
with time and the range of observed energies of the bursts (Kirsten et al.,
2023). For FRB 20240114A, Panda et al. (2024b) fit a broken power law for
bursts detected at 650 MHz with index at lower end to be -0.7 which is flatter
than what we measure at 400 MHz. While this could suggest a dynamic or
frequency-dependent change in the energy distribution, it could also result
from incompleteness in search for lower fluence bursts.

A similar power-law index has been seen for several other repeating FRBs as
well as the giant pulses from the Crab pulsar and the magnetar XTE 1810$-$197
(Maan et al., 2019b; Serylak et al., 2009). However, different measurements
have resulted into different power-law indices for the same sources,
indicating temporal evolution or a more complex distribution of the bursts.

###  5.4 Persistent Radio Source

Persistent radio sources (PRS) have been associated with two repeating FRBs so
far, FRB 20190520B and FRB 20121124A (Chatterjee et al., 2017b; Niu et al.,
2022). For both FRBs, we see that there is temporal evolution of the
associated PRS flux densities (Rhodes et al., 2023; Chen et al., 2023). Figure
5 shows the spectral luminosity of the PRS associated with the above two
repeating FRBs at different frequencies, along with the upper limits for a few
repeating and one-off FRBs as well as our upper limits on the spectral
luminosity of any PRS associated with FRB 20240114A at 400 and 650 MHz.
Specially at 650 MHz, our upper limit is almost an order of magnitude below
the spectral luminosity of the two known PRS.

Two models have been invoked to explain the persistent radio emission — one
involves magnetized ion-electron nebula (Margalit & Metzger, 2018) and other
involves a hypernebula i.e hyperacrreting compact object (Sridhar & Metzger,
2022). While out limits are quite stringent at lower frequencies, a spectral
turn-over around or above 1 GHz can not be ruled out. In that case, it might
still be possible to detect the PRS at higher frequencies. Furthermore, the
turn-over is expected to shift to lower radio frequencies with time (Margalit
& Metzger, 2018), and hence the PRS could also become detectable even at 650
MHz at a later epoch. In such a case, our current limits would be useful in
constraining the temporal evolution. Overall, a regular sensitive observations
at a few different frequencies would be useful to probe any associated PRS.

Law et al. (2022) studied any potential correlations between various burst
properties, e.g., the repetition rate, and the associated PRS luminosity. An
interesting idea is that the excess DM caused by the host galaxy, DMhost,
might be causally connected to the PRS (Law et al., 2022). For FRB 20240114A,
DMhost could be around 330pc cm-3 (Bhardwaj et al., 2024), which is larger
than the typical contribution from the ISM in the host galaxies. When compared
to the DMhost and repetition rate of FRBs with and without an associated PRS,
FRB 20240114A’s DMhost and repetition rate might favor the presence of a PRS.
However, DMhost normalized by the stellar mass of the galaxy could be more
informative (Law et al., 2022, e.g., their figure 4).

##  6 Conclusions

We report detection of 57 bursts from FRB 20240114A at band-3 (300-500 MHz)
and band-4 (550-750 MHz) of uGMRT. We also confirm the lower frequency
counterpart of the burst storm that was reported at L-band with a rate of 500
hr-1 using FAST on 5th Mar 2024, but with a bursts rate of 31 hr-1 above a
fluence threshold of 0.22 Jy ms for 2 ms wide bursts at 400 MHz. From a
detailed analysis of our low-energy burst sample as well as considering the
reported detection of FRB 20240114A from other telescopes, we conclude the
following.

  1. 1.

The fluence distribution of the FRB 20240114A bursts at 400 MHz in the range
0.1$-$10 Jy ms is well fit with a single power-law index of -1.24$\pm$0.11.

  2. 2.

The bursts from FRB 20240114A exhibit narrow-band (about 10% fractional
bandwidth) behaviour and downward drifting pattern similar to many other
repeating FRBs. Our drift-rate estimates for FRB 20240114A are such
measurements at the second lowest frequency for any FRB so far. The drift
rates are consistent with the power-law fit for such estimates from various
FRBs at their rest frame frequencies, and might indicate curvature radiation
at different heights as the origin of the observed downward drifting emission.

  3. 3.

We place the most stringent upper limits at 400 and 650 MHz on the spectral
luminosity of any underlying PRS associated with FRB 20240114A. The upper
limit at 650 MHz is nearly an order of magnitude below the luminosity of the
two known PRS at this frequency. However, a spectral turnover at higher
frequencies can not be ruled out as the cause of the non-detection, and
continued monitoring would be helpful.

  4. 4.

Finally, FRB 20240114A shows frequency dependent burst rate. By combining the
information available from follow-ups at other telescopes and frequencies, we
also show that the burst activity of FRB 20240114A is potentially frequency
dependent. Our latest observation at band-4 also indicate towards a potential
quenching of the activity at low frequencies.

Overall, the similarity of various properties of FRB 20240114A to other
repeating FRBs suggest a common intrinsic nature of emission. Our results also
warrant for a multi-frequency monitoring of FRB 20240114A to establish the
potential frequency dependent activity and probe a spectral turn-over
evolution of any associated PRS.

##  7 Software and third party data repository citations

## Acknowledgments

YM acknowledges support from the Department of Science and Technology via the
Science and Engineering Research Board Startup Research Grant
(SRG/2023/002657). We would like to thank Ramananda Santra for useful
discussions regarding interferometric data analysis. AK would like to thank
Arvind Balasubramanian for useful discussions. We would like to thank the
Centre Director and the observatory for the prompt time-allocation and
scheduling of our observations. GMRT is run by the National Centre for Radio
Astrophysics of the Tata Institute of Fundamental Research.

## References

  * Aggarwal et al. (2020) Aggarwal, K., Agarwal, D., Kania, J. W., et al. 2020, Journal of Open Source Software, 5, 2750, doi: 10.21105/joss.02750
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Bannister et al. (2019) Bannister, K. W., Deller, A. T., Phillips, C., et al. 2019, Science, 365, 565, doi: 10.1126/science.aaw5903
  * Bhandari et al. (2020) Bhandari, S., Sadler, E. M., Prochaska, J. X., et al. 2020, ApJ, 895, L37, doi: 10.3847/2041-8213/ab672e
  * Bhardwaj et al. (2024) Bhardwaj, M., Kirichenko, A., & Gil de Paz, A. 2024, The Astronomer’s Telegram, 16613, 1
  * Bochenek et al. (2020) Bochenek, C. D., Ravi, V., Belov, K. V., et al. 2020, Nature, 587, 59, doi: 10.1038/s41586-020-2872-x
  * Cash (1979) Cash, W. 1979, ApJ, 228, 939, doi: 10.1086/156922
  * Chamma et al. (2023) Chamma, M. A., Rajabi, F., Kumar, A., & Houde, M. 2023, Monthly Notices of the Royal Astronomical Society, 522, 3036–3048, doi: 10.1093/mnras/stad1108
  * Chatterjee et al. (2017a) Chatterjee, S., Law, C. J., Wharton, R. S., et al. 2017a, Nature, 541, 58, doi: 10.1038/nature20797
  * Chatterjee et al. (2017b) —. 2017b, Nature, 541, 58, doi: 10.1038/nature20797
  * Chen et al. (2023) Chen, G., Ravi, V., & Hallinan, G. W. 2023, The Astrophysical Journal, 958, 185, doi: 10.3847/1538-4357/ad02f3
  * Chen et al. (2024) Chen, X., Hu, B., Wang, P., et al. 2024, Plasma lensing interpretation of FRB 20201124A bursts at the end of September 2021.  https://arxiv.org/abs/2401.18036
  * CHIME/FRB Collaboration et al. (2019) CHIME/FRB Collaboration, Andersen, B. C., Bandura, K., et al. 2019, ApJ, 885, L24, doi: 10.3847/2041-8213/ab4a80
  * Chime/Frb Collaboration et al. (2020) Chime/Frb Collaboration, Amiri, M., Andersen, B. C., et al. 2020, Nature, 582, 351, doi: 10.1038/s41586-020-2398-2
  * CHIME/FRB Collaboration et al. (2020) CHIME/FRB Collaboration, Andersen, B. C., Bandura, K. M., et al. 2020, Nature, 587, 54, doi: 10.1038/s41586-020-2863-y
  * Chime/Frb Collaboration et al. (2020) Chime/Frb Collaboration, Amiri, M., Andersen, B. C., et al. 2020, Nature, 582, 351, doi: 10.1038/s41586-020-2398-2
  * Cordes et al. (2017) Cordes, J. M., Wasserman, I., Hessels, J. W. T., et al. 2017, ApJ, 842, 35, doi: 10.3847/1538-4357/aa74da
  * Feng et al. (2023) Feng, Y., Li, D., Zhang, Y.-K., et al. 2023, arXiv e-prints, arXiv:2304.14671, doi: 10.48550/arXiv.2304.14671
  * Gopinath et al. (2023) Gopinath, A., Bassa, C. G., Pleunis, Z., et al. 2023, Monthly Notices of the Royal Astronomical Society, 527, 9872, doi: 10.1093/mnras/stad3856
  * Gupta et al. (2017) Gupta, Y., Ajithkumar, B., Kale, H. S., et al. 2017, Current Science, 113, 707, doi: 10.18520/cs/v113/i04/707-714
  * Heintz et al. (2020) Heintz, K. E., Prochaska, J. X., Simha, S., et al. 2020, ApJ, 903, 152, doi: 10.3847/1538-4357/abb6fb
  * Hessels et al. (2019) Hessels, J. W. T., Spitler, L. G., Seymour, A. D., et al. 2019, ApJ, 876, L23, doi: 10.3847/2041-8213/ab13ae
  * Hewitt et al. (2024) Hewitt, D. M., Huang, J., Hessels, J. W. T., et al. 2024, The Astronomer’s Telegram, 16597, 1
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Joshi et al. (2024) Joshi, P., Medina, A., Earwicker, J. T., et al. 2024, The Astronomer’s Telegram, 16599, 1
  * Kale & Ishwara-Chandra (2021) Kale, R., & Ishwara-Chandra, C. H. 2021, Experimental Astronomy, 51, 95, doi: 10.1007/s10686-020-09677-6
  * Kirsten et al. (2023) Kirsten, F., Ould-Boukattine, O., Herrmann, W., et al. 2023, Connecting repeating and non-repeating fast radio bursts via their energy distributions.  https://arxiv.org/abs/2306.15505
  * Kumar et al. (2024) Kumar, A., Maan, Y., & Bhusare, Y. 2024, The Astronomer’s Telegram, 16452, 1
  * Kumar et al. (2021a) Kumar, P., Shannon, R. M., Flynn, C., et al. 2021a, MNRAS, 500, 2525, doi: 10.1093/mnras/staa3436
  * Kumar et al. (2021b) —. 2021b, MNRAS, 500, 2525, doi: 10.1093/mnras/staa3436
  * Kumar et al. (2023) Kumar, P., Luo, R., Price, D. C., et al. 2023, MNRAS, 526, 3652, doi: 10.1093/mnras/stad2969
  * Lanman et al. (2022) Lanman, A. E., Andersen, B. C., Chawla, P., et al. 2022, The Astrophysical Journal, 927, 59
  * Law et al. (2022) Law, C. J., Connor, L., & Aggarwal, K. 2022, ApJ, 927, 55, doi: 10.3847/1538-4357/ac4c42
  * Limaye & Spitler (2024) Limaye, P., & Spitler, L. 2024, The Astronomer’s Telegram, 16620, 1
  * Maan et al. (2019a) Maan, Y., Joshi, B. C., Surnis, M. P., Bagchi, M., & Manoharan, P. K. 2019a, ApJ, 882, L9, doi: 10.3847/2041-8213/ab3a47
  * Maan et al. (2019b) —. 2019b, ApJ, 882, L9, doi: 10.3847/2041-8213/ab3a47
  * Maan et al. (2021) Maan, Y., van Leeuwen, J., & Vohl, D. 2021, Astronomy & Astrophysics, 650, A80, doi: 10.1051/0004-6361/202040164
  * Marcote et al. (2020) Marcote, B., Nimmo, K., Hessels, J. W. T., et al. 2020, Nature, 577, 190, doi: 10.1038/s41586-019-1866-z
  * Margalit & Metzger (2018) Margalit, B., & Metzger, B. D. 2018, The Astrophysical Journal Letters, 868, L4, doi: 10.3847/2041-8213/aaedad
  * Marthi et al. (2020) Marthi, V. R., Gautam, T., Li, D. Z., et al. 2020, MNRAS, 499, L16, doi: 10.1093/mnrasl/slaa148
  * Metzger et al. (2019) Metzger, B. D., Margalit, B., & Sironi, L. 2019, MNRAS, 485, 4091, doi: 10.1093/mnras/stz700
  * Niu et al. (2022) Niu, C. H., Aggarwal, K., Li, D., et al. 2022, Nature, 606, 873, doi: 10.1038/s41586-022-04755-5
  * Ould-Boukattine et al. (2024a) Ould-Boukattine, O. S., Hessels, J. W. T., Kirsten, F., et al. 2024a, The Astronomer’s Telegram, 16432, 1
  * Ould-Boukattine et al. (2024b) Ould-Boukattine, O. S., Dijkema, T. J., Gawronski, M., et al. 2024b, The Astronomer’s Telegram, 16565, 1
  * Panda et al. (2024a) Panda, U., Bhattacharyya, S., Dudeja, C., Kudale, S., & Roy, J. 2024a, The Astronomer’s Telegram, 16494, 1
  * Panda et al. (2024b) Panda, U., Roy, J., Bhattacharyya, S., Dudeja, C., & Kudale, S. 2024b, arXiv e-prints, arXiv:2405.09749, doi: 10.48550/arXiv.2405.09749
  * Pastor-Marazuela et al. (2021a) Pastor-Marazuela, I., Connor, L., van Leeuwen, J., et al. 2021a, Nature, 596, 505, doi: 10.1038/s41586-021-03724-8
  * Pastor-Marazuela et al. (2021b) —. 2021b, Nature, 596, 505, doi: 10.1038/s41586-021-03724-8
  * Pelliciari et al. (2024a) Pelliciari, D., Geminardi, A., Bernardi, G., et al. 2024a, The Astronomer’s Telegram, 16434, 1
  * Pelliciari et al. (2024b) —. 2024b, The Astronomer’s Telegram, 16547, 1
  * Pleunis et al. (2021a) Pleunis, Z., Michilli, D., Bassa, C. G., et al. 2021a, ApJ, 911, L3, doi: 10.3847/2041-8213/abec72
  * Pleunis et al. (2021b) Pleunis, Z., Good, D. C., Kaspi, V. M., et al. 2021b, ApJ, 923, 1, doi: 10.3847/1538-4357/ac33ac
  * Pleunis et al. (2021c) Pleunis, Z., Michilli, D., Bassa, C. G., et al. 2021c, ApJ, 911, L3, doi: 10.3847/2041-8213/abec72
  * Rajwade et al. (2020a) Rajwade, K. M., Mickaliger, M. B., Stappers, B. W., et al. 2020a, MNRAS, 495, 3551, doi: 10.1093/mnras/staa1237
  * Rajwade et al. (2020b) —. 2020b, MNRAS, 495, 3551, doi: 10.1093/mnras/staa1237
  * Ransom (2011) Ransom, S. 2011, PRESTO: PulsaR Exploration and Search TOolkit, Astrophysics Source Code Library, record ascl:1107.017
  * Resmi, L. et al. (2021) Resmi, L., Vink, J., & Ishwara-Chandra, C. H. 2021, A&A, 655, A102, doi: 10.1051/0004-6361/202039771
  * Rhodes et al. (2023) Rhodes, L., Caleb, M., Stappers, B. W., et al. 2023, MNRAS, 525, 3626, doi: 10.1093/mnras/stad2438
  * Serylak et al. (2009) Serylak, M., Stappers, B. W., Weltevrede, P., et al. 2009, MNRAS, 394, 295, doi: 10.1111/j.1365-2966.2008.14260.x
  * Seymour et al. (2019) Seymour, A., Michilli, D., & Pleunis, Z. 2019, DM_phase: Algorithm for correcting dispersion of radio signals, Astrophysics Source Code Library, record ascl:1910.004
  * Shin & CHIME/FRB Collaboration (2024) Shin, K., & CHIME/FRB Collaboration. 2024, The Astronomer’s Telegram, 16420, 1
  * Snelders et al. (2024) Snelders, M. P., Bhandari, S., Kirsten, F., et al. 2024, The Astronomer’s Telegram, 16542, 1
  * Spitler et al. (2014) Spitler, L. G., Cordes, J. M., Hessels, J. W. T., et al. 2014, ApJ, 790, 101, doi: 10.1088/0004-637X/790/2/101
  * Sridhar & Metzger (2022) Sridhar, N., & Metzger, B. D. 2022, ApJ, 937, 5, doi: 10.3847/1538-4357/ac8a4a
  * Thulasiram & Lin (2021) Thulasiram, P., & Lin, H.-H. 2021, MNRAS, 508, 1947, doi: 10.1093/mnras/stab2692
  * Thulasiram & Lin (2021) Thulasiram, P., & Lin, H.-H. 2021, Monthly Notices of the Royal Astronomical Society, 508, doi: 10.1093/mnras/stab2692
  * Tian et al. (2024) Tian, J., Pastor-Marazuela, I., Stappers, B., et al. 2024, The Astronomer’s Telegram, 16446, 1
  * Uttarkar et al. (2024) Uttarkar, P. A., Kumar, P., Lower, M. E., & Shannon, R. M. 2024, The Astronomer’s Telegram, 16430, 1
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Wang et al. (2022) Wang, W.-Y., Yang, Y.-P., Niu, C.-H., Xu, R., & Zhang, B. 2022, ApJ, 927, 105, doi: 10.3847/1538-4357/ac4097
  * Zhang et al. (2024a) Zhang, J., Wu, Q., Cao, S., et al. 2024a, The Astronomer’s Telegram, 16505, 1
  * Zhang et al. (2024b) Zhang, J., Zhu, Y., Cao, S., et al. 2024b, The Astronomer’s Telegram, 16433, 1
  * Zhang et al. (2023) Zhang, X., Yu, W., Law, C., et al. 2023, ApJ, 959, 89, doi: 10.3847/1538-4357/ad0545
  * Zhou et al. (2022) Zhou, D. J., Han, J. L., Zhang, B., et al. 2022, Research in Astronomy and Astrophysics, 22, 124001, doi: 10.1088/1674-4527/ac98f8