The Boltzmann Medal is a prestigious award presented in the field of statistical mechanics and thermodynamics. It is named after the Austrian physicist Ludwig Boltzmann, who made significant contributions to the understanding of statistical mechanics and kinetic theory. The medal is awarded to scientists who have made outstanding contributions to the development of statistical mechanics, thermodynamics, and related areas of physics. Recipients of the Boltzmann Medal are recognized for their innovative research and advancements that have had a lasting impact on the field.

The Boltzmann constant, denoted as \( k_B \) or simply \( k \), is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It plays a crucial role in statistical mechanics and thermodynamics. The Boltzmann constant is defined as: \[ k_B = 1.

The Course of Theoretical Physics typically refers to an academic program or series of courses focused on the theoretical aspects of physics. This field involves the formulation of physical principles and laws using mathematical models and abstract concepts, seeking to explain and predict various physical phenomena. Key components of a theoretical physics course might include: 1. **Classical Mechanics:** Explores the motion of bodies under the influence of forces, including Newton's laws, energy conservation, and oscillations.

Fermi–Dirac statistics is a quantum statistical framework that describes the distribution of particles, specifically fermions, which are particles that obey the Pauli exclusion principle. Fermions include particles like electrons, protons, and neutrons, and they have half-integer spin (e.g., 1/2, 3/2). In systems of indistinguishable fermions, no two particles can occupy the same quantum state simultaneously.

The EPS Statistical and Nonlinear Physics Prize is an award given by the European Physical Society (EPS) to recognize outstanding contributions in the fields of statistical physics and nonlinear phenomena. This prize honors researchers who have made significant advancements or discoveries in these areas, which encompass a wide range of topics including complex systems, phase transitions, and nonlinear dynamics. The award aims to celebrate the important role of statistical mechanics and nonlinear science in understanding and modeling physical systems.

Entropy of network ensembles refers to a concept in statistical physics and network theory that quantifies the amount of uncertainty or disorder in a particular ensemble of networks. In this context, a "network ensemble" is a collection of networks that share certain properties or constraints, such as degree distribution, clustering coefficient, or overall connectivity structure. ### Key Concepts: 1. **Network Ensembles**: - These are groups of networks that are generated under specific statistical rules.

Gibbs' paradox highlights an apparent contradiction in statistical mechanics regarding the entropy of mixing identical particles or gases. It arises when considering the entropy change associated with mixing two gases or ensembles of particles that are indistinguishable. In classical thermodynamics, when two different gases are mixed, the entropy of the system increases due to the increased number of available microstates.

Cluster expansion is a mathematical and computational technique used to analyze and represent complex systems, particularly in statistical mechanics, statistical physics, and combinatorial optimization. The method involves expressing a system's properties or behavior in terms of sums over clusters, or groups of interacting components. This approach can simplify the study of many-particle systems by allowing one to break down the interactions into manageable parts.

Configuration entropy refers to the measure of the number of microstates (specific arrangements) corresponding to a given macrostate (overall state) of a system. In other words, it quantifies the degree of disorder or randomness associated with a particular arrangement of particles in a system. In thermodynamics and statistical mechanics, entropy is often associated with the level of uncertainty or disorder within a system. Specifically, configuration entropy appears in contexts where the arrangement of particles or components influences the system's properties.

The Coulomb gap refers to an energy gap that arises in disordered electronic systems, particularly in granular or amorphous materials where localized charge carriers interact weakly with one another. This concept is often discussed in the context of insulating materials and systems near the metal-insulator transition.

Mean-field theory (MFT) is a statistical physics and mathematical physics approach that simplifies complex many-body systems by averaging the effects of all individual particles or entities on one another. In this framework, instead of dealing with the complicated interactions of every particle in a system, the average effect of all particles is considered to define a "mean field" that influences each particle.

Fick's laws of diffusion describe how substances diffuse, providing a quantitative framework for understanding the movement of particles within a medium. There are two main laws: ### Fick's First Law: This law states that the flux of a substance (the amount of substance passing through a unit area per unit time) is proportional to the concentration gradient.

The fundamental thermodynamic relation is a central concept in thermodynamics that relates changes in internal energy to changes in entropy and volume. It is derived from the first and second laws of thermodynamics and describes the changes in a system’s state as it exchanges heat and work with its surroundings.

2026-02-07: might have lost a the left North Face etip recycled gloves black with white signs, rebuying without too much thinking, redundancy is fine now that I'm old and don't have time: www.amazon.co.uk/dp/B08BDPDNH7

2025-12-28: after losing the Karrimor X Lite Run Black Headband for good going to try a few ones online:

2024-11: OK, I give up, going to try and find the Extremities XDRY again, as these are the perfect gloves for the UK. Buying www.amazon.co.uk/dp/B00OISFUQ4. Their marketing is very confusing and there are several different names for the product, but the label reads "extremities insulated waterpoof sticky power liner glove", black, large. This is the official page: www.terra-nova.co.uk/products/insulated-waterproof-sticky-power-liner-gloves X-Dry Stretch, 100% polyester.

2024-10: going to try Endura Men's Strike Glove - Hi-Viz Yellow, Large www.endurasport.com/strike-glove-hi-viz-yellow/12929542.html | www.wiggle.com/endura-strike-gloves-930526#colcode=93052613. These gloves are high quality, but they are too hot and fleecy for the UK, unless it is an exceptionally cold winter day around 0. This means that they get quite wet from sweat, and they are not that breathable. Also they are very hard to take off and put on, which is painful if you are like me who uses your cell phone for navigation, and while it works with touchscreens, sometimes it's just too annoying and you have to take them off.

2024-05: going to try www.sealskinz.com/products/water-repellent-all-weather-glove Sealskinz Tasburgh Water Repellent All Weather love black large because Extremities XDRY has a hole on right thumb due to gear shifting. This glove is very similar to the Nike academy hyperwam, not really a winter glove feels like. Weird branding...

2023-11: Endura Windchill Bibtights XL - Black: www.endurasport.com/windchill-bibtights-black/12926004.html because knees of dhb one open. I'm an idiot for getting XL... it's just that L felt too short on the top. But ultimately XL was too baggy, this became quite apparent on first usage, L would likely have been better, just would have needed to pull it a bit more. Quality is very good, very water repellent, kept me warm on 2C under very light rain. Also tried www.endurasport.com/xtract-bibtight-black/12926142.html but the sizing was completely different. L on that one as obviously too large, M felt correct.

2023-07: lost my waterproof cap, not cycling specialized, but was useful. Was by itself in crappy backpocket of cheap muddy fox jersey, lots of wind, some fence jumping... Hadn't noted know the brand/model... sad. Let's try Castelli A/C 2 Cycling Cap? www.castelli-cycling.com/US/en/Men/Collection/Hot-Weather-Kit/A-C-2-CYCLING-CAP/p/4523032_008 Single size. Slightly small, but hard to be sure.

2023-07: www.dhbsport.com/products/aeron-short-sleeve-jersey-2-0 dhb Aeron Short Sleeve Jersey 2.0 Colour Ink Blue Size (Standard) L. To prevent crap from falling again hopefully. Felt too small. Material: 100% polyester, inserts, 64% polyester, 6% elastane. Compared to the non 2.0 which I have, the material feels more glossy/slippery. Slightly weird.

2023-05: www.wiggle.com/p/dhb-classic-shorts?color=black&sizeStandard=M dhb Classic Shorts, medium.

2022-05: buying another "dhb Lightweight Mesh Long Sleeve Base Layer", small www.wiggle.com/p/dhb-lightweight-mesh-long-sleeve-base-layer. I notice that it is actually a bit not long enough. Doesn't make a huge difference, but maybe medium or large next time.

2022-12 Rohan Polar Gloves, black, large, 45 pounds. Shell: 85% polyamide, 15% elastane. Insulation: 100% polyester. 100g insulation on back of the hand. www.rohan.co.uk/mens/accessories-polar-gloves Let's see how they perform after hands were very cold on a bike ride with the Extremities XDRY.

2022-08 dhb Aeron Short Finger Gel Gloves 2.0 medium white www.wiggle.co.uk/dhb-aeron-short-finger-gel-gloves-20. For sun/fall protection. In theory should be small according to size guide, but small was over, and I have my doubts it is correct.

2021-12: bought a North Face etip recycled gloves black with white signs, large about 40 dollars: www.thenorthface.co.uk/shop/en-gb/tnf-gb/mens-etip-gloves-4sha?variationId=KY4 let's see how that goes. These ended up being very good. They are not very warm, good for 10-15 C maybe, which is a important range. Water performance is very good

2021-12:

2021-06: Castelli UPF 50+ Light Arm Sleeves (Skins) Large White, 29 pounds www.castelli-cycling.com/bg/upf-50-+-light-arm-sleeves/p/451603621P-001. Felt really good. Really does not trap much heat, and completely blocks off sun. It makes you develop a small layer of sweat that keeps you cool, cooler than without the sleeves.

2021-02 Castelli PRO THERMAL SKULLY red www.wiggle.co.uk/castelli-pro-thermal-skully www.castelli-cycling.com/gb/pro-thermal-skully/p/452054220A to help with cold, the X Lite Run Black Headband is just not made for winter.

2021-01: Madison Cycle Everywear Sportive Toe covers. L/XL. www.madison.cc/shop/sportive-thermal-toe-covers/VARCLA121/CLA12104

2021-01: "Endura FS260-NEMO Glove II" Neoprene Winter Handwarmer Large. www.endurasport.com/Gants-Nemo-FS260-Pro-II/p/E1216-Black Let's see how neoprene feels like. Basically it becomes wet with your transpiration, but is still relatively warm, would take it well down to 5C. Likely would work well with heavy rain. Not bad. But does feel a bit cooler than the Extremities XDRY, especially after you walk for a bit, and your hand becomes cold, and it is very hard to warm it because it is also wet.

2020-12-21: "dhb Neoprene Nylon Overshoes" www.wiggle.co.uk/dhb-neoprene-nylon-overshoes large black. After 2 rides, noticed that the front lower part of left foot (the one I put on and off the most) had a hole in it, not sure where it was made. Edit: I later noticed that it is because I have the reflex of braking slightly with my shoes in certain conditions, e.g. at lower speeds in close proximity to pedestrians, the sounds of which also serve as a way to alert them without the need for a loud bell. But it destroys the overshoe, so I have to undo that reflex.

2020-11-06 www.alpinetrek.co.uk/castelli-perfetto-ros-long-sleeve-cycling-jersey/ "Castelli Perfetto RoS Long Sleeve - Cycling jersey" size L Large. On Wiggle: www.wiggle.co.uk/castelli-perfetto-ros-long-sleeve-jersey Feels great. Together with the dhb Merino Long Sleeve Base Layer, I can gown down to 0 Celcius, no problem, and up to 14 with the "Mesh Long Sleeve Base Layer". Breathes great, blocks wind and rain reasonably, and can holds mild rain out on a short 1.5h ride without problem. I used this so much it is unbelievable.

2020-10-17:

For sizing see also: Ciro Santilli's body.

2020-11: Castelli Thermoflex 2 Arm Warmers, Large, 28 pounds: www.wiggle.co.uk/castelli-thermoflex-2-arm-warmers | www.castelli-cycling.com/gb/men/accessories/cycling-leg-and-arm-warmers/p/451953020A-010 advertized 8°-20°C. Was going to buy the DHB ones for 10 pounds less, but always out of so Castelli it is.

2020-08: started getting getting cold, so let's do it:

2020-05 www.wiggle.co.uk/dhb-aeron-short-sleeve-jersey-4/ dhb Aeron Short Sleeve Jersey 100% polyester, Red Large. This one fit OK. Length feels right. When on cycling position, a bit baggy under arms though, but guessing M would be too short? Still already feels less baggy than the cheap muddy fox jersey. Aeron is the second lowest level of DHB top, after the super basic 25 pound one. Material is thin polyester, could be better, but OK.

2020-05 www.wiggle.co.uk/castelli-perfetto-light-2-short-sleeve-jersey/ Castelli Perfetto Light 2 Short Sleeve Jersey 84% Nylon, 16% Lycra. When it arrived, was WAY too small, so returned. Material looked and felt amazing.

2019-12 www.wiggle.co.uk/dhb-merino-long-sleeve-base-layer-1/?sku=5360752872 "dhb Merino Long Sleeve Base Layer", black, small, was right size, wool is a bit itchy, but does feel like it dries off more quickly on long rides than the cheap Muddy Fox polyester cycling Jersey. Pretty warm, too much for 15C.

2019-12 www.wiggle.co.uk/dhb-aeron-winter-weight-merino-sock/?sku=100335580 dhb Aeron Winter Weight Merino Sock, blue, UK 9.5-12. Did not seem to make much of a difference, feet still cold, not wet.

2019-11 Nike academy hyperwarm gloves: www.nike.com/gb/t/hyperwarm-academy-football-gloves-s9Dd8D/GS0373-013 (archive) for 17.99 Pounds from physical retailer. Good intermediate between the Extremities gloves when it gets a bit warmer. 60% polyester/27% nylon/5% rubber/4% elastane.

2019-10 XGC Men's Cycling Shorts/Bike Shorts And Cycling Underwear With High-Density High-Elasticity And Highly Breathable 4D Sponge Padded www.amazon.co.uk/dp/B07BDJP64W (archive). Grey color, red inner butt padding. 2020-08: after using it a lot, noticed that the padding stiches started to come off a bit.

2018 (?) Extremities XDRY gloves Looks like: www.sportsdirect.com/extremities-wp-p/line-glove-91-907293 (archive) Gets wet after a few minutes of medium rain. But does dry quickly. Baggy, multilayer. Says 40 pounds, but always at 20 pound discount, so it is just a marketing trick. On website: Outershell 96% Polyester/4% Lycra (Spandex)Palm 45% Polyester/55% PolyurethaneLining 100% Polyester

2017-09 Trespass Cruzado Male Gloves size S. www.trespass.com/cruzado-unisex-gloves (archive) Markings: crossover gloves. Technical Performance TP75: www.trespass.com/advice/trespass-tp-ratings-guide/ Lost right hand around 2018/2019.

2017 Muddy fox cycling Jersey, orange, polyester, Large, cheap. Works well enough I guess. Could be a bit more tight fitting.

2017 www.amazon.co.uk/Nevica-Unisex-Reversible-Skuff/dp/B07GSD4R7Y Nevica Skuff black

2017 www.ebay.co.uk/itm/254646634761 Karrimor X Lite Run Black Headband. Had a hole in the ear for a long time. Miraculously surived being almost lost on several occasions, including once it fell off from bike and a dude coming behind pointed it to me. But then alas on 2025-12-28 I lost it, noticed next day wasn't there, by chance found it on neighboring park, but it had been torn very badly, a dog must have done that, so I had to throw it away!

2015 UNIQLO tights 53% cotton, 32% polyester, 15% elastane. 2020-08: after riding it a lot, it ripped a bit near upper inner tights.

2015 Odlo top and tights, polyester. Not sure exact model, not necessarily meant for cycling. Top does not fit tightly, did not feel like it was removing sweat effectively.

You just map the value (1, 1) $C_m\times C_n$ to the value 1 of $C_{mn}$, and it works out. E.g. for $C_2\times C_3$, the group generated by of (1, 1) is:

A concise to describe a specific permutation.

A permutation group can then be described in terms of the generating set of a group of specific elements given in cycle notation.

E.g. en.wikipedia.org/w/index.php?title=Mathieu_group&oldid=1034060469#Permutation_groups mentions that the Mathieu group $M_{11}$ is generated by three elements:which feels quite compact for a simple group with 95040 elements, doesn't it!