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## Post-edited  Random subgroups, totally non free actions and factor representations Grigorchuk, Rostislav (Auteur de la Conférence) | CIRM (Editeur )

I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely many factor representations of type $II_1$. I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely ...

## Multi angle  On some operator-theoretic aspects of ergodic theory Haase, Markus (Auteur de la Conférence) | CIRM (Editeur )

I will describe the main features and methods of a strictly operator-theoretic/functional-analytic perspective on structural ergodic theory in the spirit and in continuation of a recent book project (with T.Eisner, B.Farkas and R.Nagel). The approach is illustrated by a review of some classical results by Abramov on systems with quasi-discrete spectrum and by Veech on compact group extensions (joint work with N.Moriakov).

## Multi angle  The unsolved problems of Halmos Weiss, Benjamin (Auteur de la Conférence) | CIRM (Editeur )

Sixty years ago Paul Halmos concluded his Lectures on Ergodic Theory with a chapter Unsolved Problems which contained a list of ten problems. I will discuss some of these and some of the work that has been done on them. He considered actions of $\mathbb{Z}$ but I will also widen the scope to actions of general countable groups.

##### Codes MSC

Nuage de mots clefs ici

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Computer Architecture, Memory Interleaving. Decoding step can be interleaved?

A CPU has a cache with block size $$64$$ bytes. The main memory has $$k$$ banks, each bank being $$c$$ bytes wide. Consecutive $$c$$ − byte chunks are mapped on consecutive banks with wrap-around. All the $$k$$ banks can be accessed in parallel, but two accesses to the same bank must be serialized. A cache block access may involve multiple iterations of parallel bank accesses depending on the amount of data obtained by accessing all the k banks in parallel. Each iteration requires decoding the bank numbers to be accessed in parallel and this takes $$k/2ns$$.The latency of one bank access is $$80ns$$. If $$c=2$$ and $$k=24$$, the latency of retrieving a cache block starting at address zero from main memory is:

One of explanation I found is:

Cache block = 64 bytes.

Each bank in memory is 2 bytes and there are 24 such banks. So, in 1 iteration we can get 2*24 = 48 bytes and getting 64 bytes requires 2 iterations.

So, latency = k/2 + 80 + k/2 + 80 (since in each iteration we need to select the banks and the bank decoding time (k/2) is independent of the number of banks we are going to access)

= 12 + 80 + 12 + 80 = 184 ns

But total memory capacity here is 2*24 = 48 bytes

What I'm unable to understand is, when 1st memory access is going on, can't we decode the 2nd bank request.

So, that would make it, $$12 + 80 + 80$$ as in the 1st memory access of $$80$$, we have decoded the 2nd bank req and as soon as 1st memory access is completed, we will start the 2nd memory access which is decoded.

Are my thoughts conceptually correct? Or am I missing something?

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# Physics Study Guide/Fluids

## Buoyancy

Buoyancy is the force due to pressure differences on the top and bottom of an object under a fluid (gas or liquid).

Net force = buoyant force - force due to gravity on the object

## Bernoulli's Principle

Fluid flow is a complex phenomenon. An ideal fluid may be described as:

• The fluid flow is steady i.e its velocity at each point is constant with time.
• The fluid is incompressible. This condition applies well to liquids and in certain circumstances to gases.
• The fluid flow is non-viscous. Internal friction is neglected. An object moving through this fluid does not experience a retarding force. We relax this condition in the discussion of Stokes' Law.
• The fluid flow is irrotational. There is no angular momentum of the fluid about any point. A very small wheel placed at an arbitary point in the fluid does not rotate about its center. Note that if turbulence is present, the wheel would most likely rotate and its flow is then not irrotational.

As the fluid moves through a pipe of varying cross-section and elevation, the pressure will change along the pipe. The Swiss physicist Daniel Bernoulli (1700-1782) first derived an expression relating the pressure to fluid speed and height. This result is a consequence of conservation of energy and applies to ideal fluids as described above.

Consider an ideal fluid flowing in a pipe of varying cross-section. A fluid in a section of length Δx1 moves to the section of length Δx2 in time Δt. The relation given by Bernoulli is:

 ${\displaystyle P+{\frac {1}{2}}\rho v^{2}+\rho gh=K}$

[where: P is pressure at cross-section, K is a constant, h is height of cross-section, ρ is density, and v is velocity of fluid at cross-section.]

In words, the Bernoulli relation may be stated as: As we move along a streamline the sum of the pressure (P), the kinetic energy per unit volume and the potential energy per unit volume remains a constant.

(To be concluded)


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# A hackable, multi-functional, and modular extrusion 3D printer for soft materials

## Abstract

Three-dimensional (3D) printing has emerged as a powerful tool for material, food, and life science research and development, where the technology’s democratization necessitates the advancement of open-source platforms. Herein, we developed a hackable, multi-functional, and modular extrusion 3D printer for soft materials, nicknamed Printer.HM. Multi-printhead modules are established based on a robotic arm for heterogeneous construct creation, where ink printability can be tuned by accessories such as heating and UV modules. Software associated with Printer.HM were designed to accept geometry inputs including computer-aided design models, coordinates, equations, and pictures, to create prints of distinct characteristics. Printer.HM could further perform versatile operations, such as liquid dispensing, non-planar printing, and pick-and-place of meso-objects. By ‘mix-and-match’ software and hardware settings, Printer.HM demonstrated printing of pH-responsive soft actuators, plant-based functional hydrogels, and organ macro-anatomical models. Integrating affordability and open design, Printer.HM is envisaged to democratize 3D printing for soft, biological, and sustainable material architectures.

## Results and discussion

### A modular extrusion printer based on a robotic arm

Figure 1 highlights the key features of ‘Printer.HM’, an extrusion-based 3D printer as an affordable and hackable alternative to commercial bioprinters (see Supplementary Table 2 for a comparison of their specifications). The system is built on a hackable robotic arm (see Supplementary Fig. 2 for a real-life setup), which does not rely on repurposing an existing fused deposition modelling printer, as their proprietary firmware may still limit its customisability and the number of utilities (e.g. printheads and UV module) that can be fitted into the system18,20. Housed in an enclosure and built on an aluminium breadboard for ease of reconfiguring different modules, the core part of the printer consists of dispensing modules (i.e. four custom-built piston-driven printheads); and a stage, of which motion control is afforded by the robotic arm. The use of a moveable built plate that is controlled by a robotic arm here, instead of a set of 3-axis Gantry linear stages, offers advantages of compactness and ease of assembly. The estimated build volume of Printer.HM is ~ 490 cm3, limited by the maximum payload capacity of the robotic arm.

As the printing competency of soft materials crucially depends on the rheology and crosslinking of the inks3, ‘Printer.HM’ is equipped with add-on auxiliary tools, including a syringe heater, a stage heater, and a UV module for assisting printing of hydrogels (Supplementary Fig. 2). The heaters are capable of controlling the temperatures of the stage and the syringe from room temperatures to ~ 60 °C, which is sufficient for most types of hydrogels and elastomeric materials. Various stages were custom-designed to fit different sizes of receiving substrates or reservoirs, including standard glass slides, petri dishes (90, 55 and 35 mm) and rectangular containers (40 and 30 mm) (Supplementary Fig. 2b). The printheads of ‘Printer.HM’ were built from simple mechanical components, such as lead screw, micro-stepper motors with dispensing resolution of 0.8 µm per step (see Supplementary Note V) and linear rails for increased stability and compactness. The syringe holders of the printheads were 3D printed, enabling customisation for fitting different sizes of dispensing utilities specific to the users’ experiments. As proof-of-concept, we customised the printheads to accommodate 3 ml or 1 ml syringes that are compatible with most laboratory-based applications.

Being able to freely tailor the print path is important as it directly controls the properties of the printed constructs, such as their mechanical properties21, stimuli-responsive morphing behaviour12,22 and the cell orientation in the constructs23. Commercial and existing custom-made extrusion-based printers generally employ CAD models/G-code as the only option to describe printing designs1,3,7 (Supplementary Tables 1 and 2). The lack of geometry input options offered in these systems could restrict the design freedom and the customisability of the print path especially for actuator structures24. Thus, printer.HM was designed to accept four different geometry inputs for creating prints of distinct characteristics. They are coordinates, equations, G-codes and pictures.

Overall, the modular design of ‘Printer.HM’ allows users to reconfigure the setup based on their experimental requirements and resource limitation, as well as encouraging the research community to expand the functionalities of the system via designing new modules. As a proof-of-concept, four printheads were built here. The total cost of this fully equipped and four-printhead system is around ~ £1900, while a single printhead system costs ~ £900. This offers significant cost saving compared to commercial bioprinters3. The installation time of ‘Printer.HM’ is around 2–4 h, excluding the time required for 3D printing parts. A step-by-step instruction of the printer assembly is provided in Supplementary Note III, to promote the reproducibility of the system.

### Printing competency

Compared to the existing open-source printers, the broader set of auxiliary tools associated with ‘Printer.HM’ greatly enhances its ability to construct different materials and geometry combinations (see Supplementary Fig. 1 and Supplementary Table 1). To exemplify, Printer.HM was used to perform extrusion printing of soft materials that require different gelation mechanisms, including thermally-induced gelation and photo-induced crosslinking, as shown in Fig. 2a. Thermal-responsive hydrogels self-assemble and undergo phase transitions at their critical temperatures that are defined by lower critical solution temperature (LCST) or upper critical solution temperature (UCST)25,26. UCST hydrogels, including gelatin and agarose, undergo gel formation upon cooling at a temperature below their UCST27,28. On the contrary, gelling of LCST (e.g. Pluronic F127) hydrogels occurs when increasing the temperature above their LCST28. With Printer.HM, the syringe heater assists the printing of UCST hydrogels (i.e. gelatin) via heating the inks during extrusion, while the stage heater helps preserving the printed shape of LCST hydrogels via enhancing its rheology at elevated temperature at the built plate. The UV module allows in-situ crosslinking of photo-polymerizable hydrogels (i.e. methacrylate hydroxypropyl cellulose29) during printing. In addition, as shown in Fig. 2b, the printer is capable of printing a wide variety of biomaterials, from Poly(ethylene glycol) diacrylate (PEGDA, a low viscosity ink with viscosity 20 mPa s30), collagen, silicone elastomer (see Supplementary Fig. 8) to a highly viscous solution of sodium carboxyl methyl cellulose (1500 Pa s).

We further demonstrate the print resolution of ‘Printer.HM’ by printing a line pattern with Pluronic F127. The test was performed with Pluronic F127 as it is commonly used in literature. Using a non-optimised setting of the operation parameters, the median Pluronic F127 feature achieved with ‘Printer.HM’ was around ~ 150 µm (Fig. 2c), which is comparable with the resolution typically attained in extrusion-based bioprinting7,31. However, it should be noted that the resolution of the printed features is predominantly determined by the ink properties. To further evaluate the performance of Printer.HM, we measured the line widths of the filament resulting from different extrusion flowrate ($$Q$$) and stage speed ($$v_{stage}$$) setting, and compared the measured line width with the theoretical line width (~ $$\sqrt {\frac{4Q}{{\pi \cdot v_{stage} }}}$$, see Supplementary Note VII for further information). As shown in Fig. 2d, a value of r2 close to 1 is found, indicating a good agreement between the measured and theoretical line widths and hence suggesting the satisfactory performance of Printer.HM.

### Printing with versatile geometry inputs

Figure 3 illustrates the wide variety of constructs fabricated using different geometry inputs, each with their own strengths depending on the architecture requirement. Being the most ordinary form of the geometry inputs, coordinates is particularly useful for creating simple linear or regular patterns, such as one-dimensional channels (Fig. 3a, Supplementary Video 1). Meanwhile, equation input enables the creation of seamless one-stroke curvy patterns, however it is not suitable for complex patterns that are not describable by equations (Fig. 3b, Supplementary Video 2). Figure 3b demonstrates that simple tubular constructs can be readily produced via an equation of circle, without the need to prepare CAD files. On the other hand, three-dimensional intricate objects can be well-described by 3D CAD models, the standard geometry format used in 3D printing (Fig. 3c, Supplementary Video 3). Lastly, picture input enables the creation of customised motifs via photos of hand-drawn patterns or pictures created by any drawing software. By leveraging the picture input option, user-designed patterns, for example circuit- and vascular-like patterns, can be readily fabricated (Fig. 3d.i–d.ii).

To further illustrate the benefits of having print path customisability, in particular for soft robotics applications, we demonstrated the creation of a soft morphing system made of a pH-responsive hydrogel by leveraging an anisotropic print path. As showed in Fig. 3d.iii and Supplementary Video 4, the 2D construct created with a heterogeneous print path exhibited an anisotropic swelling response and morphed into a flower shape. It is worthwhile mentioning that the operation flexibility does not limit to the four geometry inputs provided here. As the control programme is entirely hackable, users can freely amend the programme for unprecedented designs.

### Multi-functionalities in one platform

By virtue of the customisable control programme, operations with ‘Printer.HM’ are user-amendable and multi-functional. We demonstrate that operations, such as automated dispensing, printing with variable speed and non-planar printing, can be carried out with ‘Printer.HM’. To exemplify, liquid handling always plays an indispensable role in life science experiments. Thus, we transformed ‘Printer.HM’ into a dispenser by modifying the control programme. Figure 4a and Supplementary Video 5 show that droplets of cell suspension were automatically dispensed on a petri dish. The dispensed volume of the droplets is controllable by the extrusion flowrate and the dispensing duration. By simply setting different duration of the dispensing time, droplets with various sizes were obtained. This capability might be useful for automating the hanging drop method to produce cell spheroids and dispensing active ingredients within a printed object.

Features with continuously narrowing width can be easily generated with variable speed of the stage, as demonstrated in Fig. 4b, which can assist in creating hierarchical vascular network. Further, we demonstrate the capability of performing non-planar printing using the platform. As opposed to conventional commercial 3D printers that rely on plane-by-plane slicing, non-planar printing requires the ink to be printed on a freeform surface by moving the motion part of the printer in all 3 axes at the same time. Figure 4c and Supplementary Video 6 show a line pattern was printed on a 3D target nose model. The 3D surface of the target object was assessed using a 3D scanner and the line pattern was projected according to the non-planar geometry using a custom-written code (further described in Materials and Methods). With the ability to deposit inks directly onto variable object surfaces, novel applications of 3D extrusion printing technology could become possible, such as depositing freeform circuits32 and functional materials. Lastly, we demonstrate nozzle-based ‘pick-and-place’ of meso-objects in Fig. 5a and Supplementary Video 7. Such an operation enables applications for locating cell spheroids between different environments.

The multiple printheads equipped in ‘Printer.HM’ facilitate the fabrication of multi-material constructs. As a demonstration, Fig. 5b.ii shows a four-layer construct composed of Pluronic F127 inks coloured with different dyes printed in air, and a model of the respiratory system with lungs and trachea made of alginate inks printed inside a support bath (Fig. 5c.ii–c.iii). This capability opens up future potential in generating sophisticated tissue anatomy, which are usually multi-component and spatially heterogeneous.

## Conclusion

Notwithstanding, several limitations are noted in ‘Printer.HM’. First, the design of a moving stage system used in ‘Printer.HM’ might potentially compromise the fidelity of low viscosity objects that are printed in-air. To reduce the potential impact, a very slow speed of the stage can be used when printing delicate structures of low viscosity materials. Second, ‘Printer.HM’ does not encompass cooling systems that assist printing of protein-based inks and an ink retraction mechanism that prevents inks from unintentional oozing. A retraction mechanism will be of interest in future work, which can be achieved by setting a reverse rotation of the mechanically driven printhead motor, to further enhance the print fidelity. It is also worth mentioning that printing soft materials, such as hydrogels, in general does not require a high heating temperature (i.e. > 60 °C), and the feature resolution of extrusion-based printing is typical > 100 μm7. Therefore, although Printer.HM has a lower mechanical resolution and a narrower heating range than commercial systems (see Supplementary Table 2), a satisfactory printing performance is achieved. Furthermore, the modular design of the system enables easy reconfiguration and the expandability of the system. Users can incorporate new functionalities, such as microfluidic printheads, coolers etc., to ‘Printer.HM’ in future development. In summary, our work established an affordable 3D extrusion printer with improved customisability and all-in-one functionalities, benefitting the do-it-yourself research community and potentially facilitating the development of open and innovative fabrication strategies in diverse fields, such as tissue engineering, soft robotics, food, and eco-friendly material processing.

## Materials and methods

### Mechanical design

‘Printer.HM’ is an open-source extrusion 3D printer that consists of a commercially available open-source robotic arm (uArm Swift Pro Desktop Robotic Arm) and a dispensing module as the core part, and heating systems, a UV module and an inspection camera as optional utilities. The robotic arm controlled the x, y and z axis motion of the 3D printed stage. Various stages were custom-designed to fit different sizes of receiving substrates or reservoirs, including standard glass slides, petri dishes (90, 55 and 35 mm) and rectangular containers (40 and 30 mm) (Supplementary Fig. 2b). The dispensing module is composed of do-it-yourself (DIY) piston-driven printheads that were built from simple mechanical components (i.e. stepper motor, linear rail and ball bearing) and custom-designed 3D printed parts. All CAD files of the 3D printed parts of ‘Printer.HM’ are accessible and are available on our Github repository, thus users can freely amend the parts to better tailor to their applications if needed. The 3D printed parts were printed with polylactic acid (PLA) or acrylonitrile butadiene styrene (ABS) using an Ultimaker S3 3D printer. As a proof-of-concept, four printheads were built here and they were designed to accommodate 1 ml or 3 ml syringes, but users can adjust the number of printheads or amend the design of the syringe holder to fit other sizes of dispensing tools in accordance with their experimental need.

The stage and syringe heating systems in ‘Printer.HM’ are composed of a custom-made aluminium holder that was wrapped with nichrome wires (UMNICWIRE2, Ultimachine) as the heating element and a K-type thermocouple (Z2-K-1M, Labfacility) as the temperature sensor. A UV LED light source (5 W, 365 nm, NSUV365, Nightsearcher) was employed here and was mounted onto the aluminium breadboard of ‘Printer.HM’. Meanwhile, users can select different light sources based on the choice of the photo-initiators. An inspection camera unit was mounted onto the aluminium breadboard for in-situ monitoring and recording the printing process. The dispensing module and the heating systems were connected to Arduino boards, while the robotic arm has a built-in Arduino for controlling. Assembly instruction of the printer and the electrical circuit of ‘Printer.HM’ is described in Supplementary Note III.

### Programme description

The printing operation was implemented by a custom-written Python programme that synchronously communicates with the Arduino boards of the robotic arm and the dispensing module, whereas the heating modules were independently controlled by graphical user interfaces (GUI) that communicate with the Arduino boards of the heaters which users can freely customise the programme for their needs. All the operation programme used in this study are available on Github.

### Printing operation

Prior to printing, the ink was centrifuged at 1000 g for 3 min to remove bubbles. The ink was drawn into a 1 ml or 3 ml syringe, and the syringe was loaded to the syringe holder of the setup. A collecting reservoir, such as petri dish or glass slides, was loaded to the 3D printed custom-made stage. Four Python control programmes were written for importing different types of geometry inputs—coordinates, equation, CAD model and picture inputs. Printing parameters, such as printing speed, offset position, extrusion flowrate and initial z-position, are user-adjustable and can be defined in the control programme. By default, the constructs were printed at the centre of the collecting reservoir, unless an offset position was defined.

#### Printing with coordinate input

A list of coordinates (x = [x1, x2, …, xn], y = [y1, y2, …, yn]) was directly loaded to the programme, where xn and yn denote the x and y coordinates of the nth point (see Supplementary Fig. 10).

#### Printing with equation input

A set of cartesian or parametric equations together with the defined range of the independent variable was inputted in the control programme (see Supplementary Fig. 11). The curve was discretised by at least 100 evenly spaced points, depending on the length of the curve. The constructs shown in Fig. 3b were fabricated using equations of sine wave, butterfly curve and circle. 3D features were produced by printing stacked layers of the 2D curve according to the defined object and layer heights.

#### Printing with CAD model input

3D CAD models were either designed using Autodesk Inventor or downloaded from GradCAD (https://grabcad.com/library/software/stl) or Thingiverse (www.thingiverse.com). Prior to the printing process, the CAD model was converted to a G-code file using Slic3r (https://slic3r.org/) with the user-defined slicing parameters (i.e. fill pattern, fill density, extrusion width and layer height). The G-code file was then imported to the Python control programme (see Supplementary Fig. 12).

#### Printing with picture input

Pictures of the printing design or photos of the hand-drawn sketches were imported to Inkscape. They were converted to G-code using the ‘Gcodetools’ extension on Inkscape (https://inkscape.org/), which was an extension designed for CNC machines. Step-by-step procedure of the conversion can be found in Supplementary Note IX. The generated G-code was then imported to the control programme for picture input, which was written to accept the G-code generated by this extension (see Supplementary Fig. 13).

### Heating operation

Syringe heating and stage heating were applied when required. They were controlled by a custom-written graphical user interface (GUI), where users can directly specify the desired set-point temperature. The acceptable deviation from the desired set-point temperature was defaulted to ± 0.5 °C here. The control programme for the heating operation are available on Github.

### Non-planar printing

A 2D line pattern for printing was designed on Inkscape and was converted to a G-code file. The 3D shape of the target object (a nose model made of Ecoflex, Supplementary Fig. 8) was captured using a 3D scanner (EinScan H, SHINING 3D®) and was saved as a STL file. To analyse the surface of the target object, the STL file of the nose model was converted into a G-code file using Slic3R with the following slicing settings (fill pattern = ‘Hilbert curve’, extrusion width = 0.2 mm, fill density = 100% and layer height = 0.2 mm). A dense infill setting and a Hilbert curve infill pattern were used here for precisely describing the target object. The G-codes of the target object (the nose model) and the printing pattern (a line pattern) were then imported to a custom-written path planning Python programme. In the programme, the z-position of the printing pattern was projected in accordance with the z-position of the target object at the similar x, y positions. By default, the programme assumes that the pattern is printed around the centre of the target object, but an offset position can be used if needed. The programme outputs a text file of the projected coordinate array, which was then imported to the control programme used for Picture input to implement the printing.

### Dispensing of cell suspension

3T3 mouse embryo fibroblast cell line was cultured in a 25 cm2 flask and was passaged using standard protocol. Cell culture media used here were 10 v/v% fetal bovine serum (F0804, Sigma) and 1 v/v% penicillin–streptomycin (P43333, Sigma) in DMEM (31885023, Life technologies). A cell suspension with 2 × 106 cells/ml was used in the dispensing experiments, with the cells stained with Calcein AM (C3099, Fisher Scientific) at 2 μM working concentration for live cell staining. To prevent cell sedimentation, immediately after resuspension, the cell ink was drawn into a 1 ml luer-lok syringe and was loaded into the syringe holder of the printer for dispensing operation. The control programme for dispensing operation are available on Github.

### Ink preparation

Supplementary Table 6 summarises the inks and the support baths used for fabricating the constructs demonstrated in this work. The inks used here were SE1700 (Dow), 30 w/v% and 40 w/v% Pluronic F127 (P2443, Sigma), a pre-crosslinked alginate ink, a pre-crosslinked hydroxyapatite-alginate ink, 10 w/v% carboxymethyl cellulose sodium salt (21902, Sigma), 10% gelatin (G1890, Sigma), 25% polyacrylic acid (450 kDa, 181285, Merck Life), collagen (50201, Ibidi), a PEGDA solution, 68 wt% methacrylate hydroxypropyl cellulose and 3 w/v% sodium hyaluronate (251770250, Fisher Scientific). Some of the inks were stained with sodium fluorescein (46960, Sigma) or Rhodamine B (A13572.18, Alfa Aesar). Unless further specified, the inks were prepared by dissolving the desired concentration of the chemical powder in deionised water. The methacrylate hydroxypropyl cellulose ink was prepared following the method described in our previous study29. The SE1700 ink was produced by mixing the base precursor and the catalyst at a weight ratio of 10:1. The alginate ink was prepared by pre-crosslinking a 10 w/v% alginate (W201502, Sigma) solution with a 200 mM CaCl2 (C5670, Sigma) solution at a 10:3 volumetric ratio. The hydroxyapatite-alginate ink was made of 15 w/v% hydroxyapatite (21223, Sigma) dispersed in a 5 w/v% alginate solution, which was then pre-crosslinked with a 200 mM CaCl2 solution at a 10:1 volumetric ratio. The PEGDA ink was prepared by mixing PEGDA (Mn 700, 455008, Merck Life), deionised water and a 10 w/v% Irgacure 2959 (g/100 ml ethanol, 410869, Sigma) at a 2:8:1 volumetric ratio. Ecoflex ink (Smooth-On Inc.) was prepared following a similar formulation reported in literature34, where Part A Ecoflex 00-30 was mixed with Part B Ecoflex 00-30 (with 1.2w/v% Slo-jo and 1.2 w/v% Thivex) with the addition of a drop of acrylic light orange color ink for visualisation. The supportive baths used here were 1.3% xanthan gum (G1253, Sigma), 1 w/v% Carbopol ETD 2020 (Lubrizol), 4.5 w/v% gelatin slurry and 6 w/v% fumed silica (S5130, Merck Life) in mineral oil (330760, Merck Life). The Carbopol, gelatin slurry and fumed silica-mineral oil supportive baths were prepared following the protocols described in previous studies35,36,37.

## Data and code availability

All data generated or analysed during this study are included in this published article and its supplementary information files. The code and the CAD designs of the 3D printed parts are available on Github (https://github.com/iekmanlei/Printer.HM) and through Zenodo (https://doi.org/10.5281/zenodo.5353394).

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## Acknowledgements

This work was supported by the European Research Council (ERC-StG, 758865) and UKRI – EPSRC (EP/S009000/1). I.M.L. acknowledges the financial support from the W.D. Armstrong Trust and the Macao Postgraduate Scholarship Fund. C.L.L. acknowledges the support from the University of Macau via a UM Macao Fellowship and the Clarendon Scholarship Fund. Y.S. acknowledges the financial support from the Chinese Scholarship Council. The authors thank Dr Yang Cao for her assistance with the cell dispensing experiment, Abby Thompson for her help with the 3D scanner, Ian Ganney for his help with the machining work, and Dr Clement Chun Lam Chan and Prof Silvia Vignolini (Department of Chemistry, University of Cambridge) for providing the methacrylate hydroxypropyl cellulose solution.

## Author information

Authors

### Contributions

All authors conceived the experiments and reviewed the manuscript.

### Corresponding author

Correspondence to Yan Yan Shery Huang.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Lei, I.M., Sheng, Y., Lei, C.L. et al. A hackable, multi-functional, and modular extrusion 3D printer for soft materials. Sci Rep 12, 12294 (2022). https://doi.org/10.1038/s41598-022-16008-6

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• DOI: https://doi.org/10.1038/s41598-022-16008-6


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Now showing items 1-1 of 1

• #### Rate of Convergence of the Density Estimation of Regression Residual

[OWP-2012-08] (Mathematisches Forschungsinstitut Oberwolfach, 2012)
Consider the regression problem with a response variable $Y$ and with a $d$-dimensional feature vector $X$. For the regression function $m(x) = \mathbb{E}\{Y|X = x\}$, this paper investigates methods for estimating the ...


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Introduction

Benefit from the natural features of quantum superposition and quantum entanglement, quantum computing (QC) has significant potential for solving classically intractable computational problems in areas such as machine learning1, cryptography2, chemistry3,4 and others. There are various quantum computing physical implementation systems up to now. Superconducting quantum computing has steady progress in gate and measurement fidelity5,6,7,8 and IBM’s roadmap for scaling quantum technology9 has been verified solidly so far10 based on this technical route. However, the implementation of a practical architecture that can scale to millions of qubits and solve large-scale practical problems remains a challenging milestone11. Actually, superconducting quantum processors must be placed in a cryogenic environment to be initialized close to their quantum ground state. To control its quantum state evolution, each qubit is individually addressed with corresponding microwave signal lines connected to room temperature control devices. The control costs and complexity related to the multiple coaxial lines per qubit limit the possible scalable size of a superconducting quantum chip, whose upper limit is about a few thousand qubits subjected to current techniques11.

Currently, physical operations, such as pulsed excitation, are used to conduct quantum computation in the hardware level. These physical operations are normally described by unitary operators that show the state evolution of qubits. Intuitively, quantum computation can be regarded as circuits of quantum gates with a series of ordered sequences of unitary operators12. However, gate-model quantum computers have inherent constraints in their architecture including coupling graph, primitive gates supported13,14,15, etc. Quantum circuits cannot be directly executed on these computers. According to the Solovay Kitaev theorem, we can approximate any quantum unitary operations as quantum circuits based on a finite set of operators within an arbitrary tolerance. However, finding an optimal strategy to build and compute such a sequence remains in suspense. The problem is defined as quantum compilation and optimization to rebuild the approximating optimal circuits suitable for different platforms. Quantum circuit compilation (QCC) takes a nonconforming quantum circuit as input and generates a circuit that can be executed on the target platform according to its constraints, including coupling graph, primitive gate set and so on. A quantum compiler is a toolchain between quantum program and pulse scheduling, with which programmers do not need to consider the physical constraints of a particular quantum processor. Every quantum compiler has its own trade-off between the number of qubits, the length of the sequences, the compilation time and the final execution time.

Generally, the quantum hardware can be modeled in four abstract layers: the quantum data plane where the qubits reside, the control and measurement plane, the control processor plane, and the host processor16. Challenges at the quantum-classical interface are examined with the goal of architecting a scaled-up quantum computer comprising millions of qubits. The quantum-classical interface, consisting of control and readout sub-systems, connects platform-specific operation to high-level software tools. These sub-systems include digital logic for generating and detecting analog waveforms, signal sources, data converters, amplifiers and so on. Separating the distinct sub-systems of the interface that perform readout and control, general arguments are given for why distributing the components of these sub-systems over significant distances and across large temperature gradients presents a major challenge to scaling-up the technology17. Largely addressing these issues, an architecture for the interface that leverages cryo-CMOS circuits proximal to the quantum plane is motivated, leveraging protocols that enable massively-parallel readout of qubits via frequency multiplexing17,18.

The novelty of this paper is performing a significant control optimization at software level in quantum computing. Our work targets the multiplexing quantum control architecture with M control channels and N qubits, where $$2\le M\le N$$. Theoretically, all high-level quantum logic gates can be decomposed into the combination of basic single qubit operations and two qubits controlled operations. Consequently, the extreme case of our quantum control architecture has two centralized control channels, controlling each qubit through a multiplexer. The experiment results show that our work can reduce the hardware cost of control system greatly and improve its utilization efficiency significantly while introducing limited circuit depth expansion.

Results

High-efficiency control architecture

Considering the parallelism and quantum circuit depth, it is better to use one control channel to control a qubit respectively. However, the utilization efficiency of the quantum control system is limited when running quantum benchmarks. We do experiments on several quantum platforms of IBM with one-to-one control system. Based on the experimental results, we calculate the utilization efficiency of control systems during execution. The utilization efficiency is the ratio of the time when control channels are not idle to the whole execution time period. The average utilization efficiency of the benchmarks is about 60% in our experiments. To further improve the utilization efficiency and reduce the number and costs of control channels, we use M control channels to control N qubits, losing a little parallelism accordingly, as shown in Fig. 1. Obviously, $$M \le N$$ and the extreme case is $$M = 2$$ because of the existence of two-qubits operations.

Dependence analysis

Some of the control operations are independent such as different single qubit operations applied to independent qubits. In addition, there exists a number of operations which have dependences to each other. Here, we define the dependences in quantum control system as two main kinds, sequence dependence and coherence dependence.

Sequence dependence is defined as the dependence between two control operations with timing sequence. These two operations are used on the same single qubit or two qubits with orders of priority. For example, a controlled-NOT (CNOT) gate on two qubits $$q_i$$ and $$q_j$$, denoted as $$CNOT(q_i,q_j)$$, can be executed only when all its predecessor gates on $$q_i$$ and $$q_j$$ have been executed already.

Coherence dependence is defined as the dependence between two qubits with controlled operations, such as CNOT. These two qubits are tightly coupled and should be controlled in the same time slots.

The two kinds of dependences above are both true dependences and cannot be eliminated at the compiler level. Related quantum operations should be executed orderly or simultaneously. Consequently, our high-efficiency quantum control architecture has at least two control channels to deal with the coherence dependence.

High-performance compilation strategy

To fully exploit the advantages of the multiplexing control architecture, we optimize the quantum circuit at basis gates level. Considering a Toffoli gate, it will be decomposed into a sequential circuit consisting of a series of single-qubit and two-qubit operations. Figure 2 gives a simple example with two Hadamard gates and a Toffoli gate and Fig. 2a shows the decomposed circuit. Obviously, traditional one-to-one control method has a higher efficiency when considering circuit depth. However, the overall utilization efficiency of the control system is dissatisfactory. The cost of control system plays an important role in NISQ era and quantum error correction and should be taken into consideration carefully when balancing the budget and overall performance. Figure 2b shows the control process and results.

A scalable quantum device with a multiplexing control architecture could be a solution to save the cost of control system. As shown in Fig. 1, there are limited control channels connected to each qubit through a multiplexer. Due to the limited control channels, some of the gates cannot be executed in parallel as in traditional compilation. The quantum circuit should be rescheduled during compilation to guarantee all the gates being executed in a right order. Here we propose two scheduling methods for multiplexing quantum control devices. The static scheduling method simply gives a control order of gates that could have been executed in parallel in traditional one-to-one control system. To further optimize the circuit, a dynamic scheduling method is given based on dependency analysis.

Static scheduling. A basic quantum circuit is obtained after transpilation and optimization. Without loss of generality, we assume a single-qubit quantum operation is executed in one time slot and a two-qubits quantum operation is executed in two time slots. Based on the assumption, we can get a basic quantum circuit with timestamps. Taking a quantum control system with two control channels as an example, we use $$c_0$$ and $$c_1$$ to indicate the control channels and $$t_0, t_1, ..., t_s$$ to represent the timestamps. For static scheduling, we use the decomposed circuit directly to map the control channels without any timing optimization, as shown in Fig. 2c. This method is simple and can guarantee the validity of the compiler easily. To further improve the performance, we bring about a dynamic scheduling scheme with timing optimization.

Dynamic scheduling. To further improve the utilization efficiency of quantum control channels and reduce the circuit depth, we propose a dynamic scheduling algorithm based on dependence analysis. A timing optimization is applied first and the independent quantum operations are reordered to maximize the utilization efficiency of control channels. The independent control operations can be reordered dynamically and spatiotemporally. We assume the control channels as hardwares of a pipelining and control operation to qubits as instructions. Our dynamic scheduling method is similar to the out-of-order launch and sequential execution process in classical computing. The result of dynamic scheduling is shown in Fig. 2d. The practically executed circuit depth can be further reduced by 4.1% and the utilization efficiency improved by 4.2%.

Experimental results evaluation. To validate the advantages of our scheme, we performed experiments on five IBM quantum computers with various benchmarks, including the Bernstein–Vazirani algorithm (BV)19, hidden shift algorithm (HS)20,21, quantum Fourier transform (QFT)22 and multi-qubit QC gates such as Toffoli and Fredkin23. We use these benchmarks because they constitute an essential part of many large QC applications that have been used in prior work on NISQ system evaluation24,25,26,27,28. The details of the benchmark are listed in Table 1. According to the experimental results shown in Figs. 3 and 4, we can improve the utilization efficiency significantly while only introducing a little circuit depth expansion. In spite of the hardware differences, the trends of optimization and extra cost are similar. The utilization efficiencies of traditional quantum control systems based on various benchmarks are ranging from around 50–80%. Our static scheduling method can help to improve the range from 90 to 100%. Compared with static scheduling, our dynamic scheduling method can bring about an extra 4% performance improvement. Our high-performance compilation strategy for multiplexing control quantum architecture has significant potential to implement quantum algorithms which need relatively fewer qubits while deeper circuit depth, such as Grover algorithm. The circuit depth expansion and utilization efficiency of control channels after scheduling depend on the number of channels and the quantum circuits. Before scheduling, the circuit needs to be compiled adapting to the hardware constraints. Due to different hardware constraints including coupling graph, the transformed circuit before scheduling varies on different platforms, which leads to different results on different devices. On IBM Belem, the utilization efficiency of QFT5 increased from 42.02 to 90.87% (increased by 116.25%). The utilization efficiency of the nine benchmarks increased by 60.16% on average on IBM Belem. On IBM Bogota, the average utilization efficiency increased by 42.85%. The values were 44.06%, 55.48% and 44.67% on the other three platforms. In summary, the utilization efficiency increased by 49.44% on average. Some of the benchmarks like QFT3 has a good result with high utilization efficiency and low circuit depth expansion. Restricted to the dependence among gates, few gates in circuit QFT3 can be executed in parallel even in one-to-one control system. When we decrease the number of control channels, the parallelism of the circuit will not be affected much. So we can get a low circuit depth expansion after scheduling in QFT3. The results show a trade-off between depth expansion and utilization efficiency. In general, with the increase of M, both depth expansion and utilization efficiency will decline.

Discussion

In this paper, we proposed a high-performance compilation strategy for multiplexing quantum control architecture. The experimental results showed the efficiency of our strategy. The quantum circuits of complex quantum algorithms can be decomposed and transpiled into basic quantum circuits matching target quantum instruction sets. For future work, we would focus more on the trade-off between depth expansion and utilization efficiency when M changes and study how to choose M to get a satisfactory result. We will also try to apply our strategy to quantum algorithms solving practical problems based on larger-scale quantum devices. In addition, we are going to pay more attention to new mapping and compilation methods to further improve the performance of our scheduling schemes.

Methods

In this section, we introduce our optimization strategies for multiplexing quantum control architecture. We first give preprocessing steps in the "Preprocessing" section. The static scheduling and dynamic scheduling methods are introduced in the "Static scheduling and dynamic scheduling" section. Then we analyse complexity of our methods in the "Complexity analysis" section. We summarize the notations we use in Table 2.

Preprocessing

Before applying our methods, some preprocessing steps need to be done to generate required data including circuit DAG, initialized circuit DAGT and initialized front layer.

Circuit DAG. The sequence dependences of gates can be abstracted by a circuit Directed Acyclic Graph (DAG), where the gates are the vertices, and the edges are the execution dependences of gates. A two-qubit gate $$CNOT(q_i,q_j)$$ can be executed only when all its predecessors on $$q_i$$ and $$q_j$$ have been executed already. A DAG can be constructed by traversing the entire quantum circuit with complexity O(g), where g is the number of gates in circuit.

Circuit DAGT. We define circuit directed acyclic graph with timestamps (DAGT) by assigning each vertex of circuit DAG with a timestamp. The timestamp of a gate is the earliest time it can be executed. Gates with the same timestamp can be executed at the same time. The gates in circuit are classified in single-qubit gates and two-qubit gates. Without loss of generality, here we assume the duration time of single-qubit gate is 1 timeslot while two-qubit gate is 2 timeslots.

Front layer. A front layer is a set of gates whose predecessors have all been executed. It can be initialized by choosing the vertices in circuit DAGT whose indegree is zero. These vertices represent for the gates that do not need to be executed after any other gates in quantum circuit.

Algorithm  1 shows the pseudo code of our algorithm for generating a DAGT for one-to-one control system by scanning through the entire DAG. In the beginning, the circuit DAGT is initialized with the circuit DAG by assigning each vertex with timestamp 0. Front layer F needs to be initialized, too. In each iteration, the algorithm will first check whether all the gates in F are executable (whether the timestamp of gate is equal to the current timestamp ts). For each executable gate, we obtain its successor gates and update their timestamps. A successor gate’s timestamp is the maximum of the sum of its predecessor’s timestamp and the duration time. If a successor gate’s dependencies are all resolved (all its predecessors are removed from F), it can be added to F. The algorithm terminates when all gates are executed.

Static scheduling and dynamic scheduling

In the "Preprocessing" section, the timestamps of gates in circuit DAGT are computed considering the sequence dependence constraint and different duration time of gates. However, on the platform with multiplexing control architecture, there are also control constraints. As shown in Fig. 1, there are limited control channels connected to each qubit through a multiplexer with which qubits can be controlled through time division multiplexing. Due to the limited control channels, some of the gates cannot be executed in parallel. The quantum circuit should be rescheduled to guarantee all the gates being executed in a right order.

Algorithm 4 shows the pseudocode of a static scheduling method to generate a circuit DAGT for multiplexing control architecture. Algorithm 4 first initializes DAGT, F and successors. The variable ts is the current timestamp. To represent the information about control channels, we use $$channels\_unoccupied$$ to represent the number of idle channels and a channelList to record the remaining time to be occupied of each channel. A gate is executable only when it is in F and there are enough idle channels left. In the iteration, the algorithm will check whether all the gates in F are executable. If a gate is executable, it will find idle channels for it and record the remaining occupied time of the channel. Then the algorithm will obtain successors of this gate in DAGT and add them to successors. If there are not enough idle channels, it will increase timestamp and update the remaining occupied time of channels. When all the gates in F are executed, the algorithm will assign F with successors and start iteration again until there are no successors. Algorithms 2 and 3 give the pseudocode of allocating and updating control channels.

Although Algorithm 4 gives a scheduling method for multiplexing control architecture, there still remains room to reduce the circuit depth and improve the utilization efficiency of control system. In Algorithm  4, only when all gates in F have been executed completely can the successors be added to F. However, some of the successors whose dependences are all resolved can be executed once there are enough idle control channels. In this way, the utilization efficiency of control system can be improved. The circuit depth may also be reduced than static scheduling. Thus, we propose a dynamic scheduling method based on the static scheduling. Algorithm 5 shows the pseudocode. Different from static scheduling, Algorithm 5 will add new gates to F dynamically if possible.

Complexity analysis

In Algorithm 1, the circuit DAGT can be generated by traveling the circuit DAG. The complexity of Algorithm 1 is $$O(g+e)$$ where g is the number of gates and e is the number of edges of circuit DAG.

Algorithm 4 can be divided into four steps: find executable gates in F and allocate channels, assign timestamps for executable gates and increase timestamps by one, remove gates that finish executing and add its successors whose dependencies are all resolved, update channels. Each gate in the circuit will be executed once in the algorithm. Both Allocating channels (Algorithm 2) and Updating channels (Algorithm 3) have a complexity of O(M). The worst case is that only one gate can be executed in each iteration. In this case, the complexity of Algorithm 4 is $$O(g\times M+e)$$, which has more relations with circuit size and channel numbers. M is much less than the number of qubits in multiplexing control quantum architecture. Algorithm 5 has the same complexity as Algorithm 4, which is acceptable in compilation.


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Find length of the curve help

• November 4th 2008, 09:31 PM
khuezy
Find length of the curve help
find the length of the curve r(t) = <2t, t^2, 1/3t^3>

I did this
r'(t) = <2, 2t, t^2>

|r'(t)|

When I take the integral, I don't know how to solve for it.
• November 4th 2008, 10:15 PM
Chris L T521
Quote:

Originally Posted by khuezy
find the length of the curve r(t) = <2t, t^2, 1/3t^3>

I did this
r'(t) = <2, 2t, t^2>

|r'(t)|

When I take the integral, I don't know how to solve for it.
Second, the integral is $\int_{t_0}^{t_f}\sqrt{\left(\frac{\,dx}{\,dt}\righ t)^2+\left(\frac{\,dy}{\,dt}\right)^2+\left(\frac{ \,dz}{\,dt}\right)^2}\,dt=\int_{t_0}^{t_f}\sqrt{4+ 4t^2+t^4}\,dt=\int_{t_0}^{t_f}\sqrt{(t^2+2)^2}\,dt$ $=\int_{t_0}^{t_f}\left(t^2+2\right)\,dt$, for $t_0\leq t\leq t_f$