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# Generating clean fuel with individual gold atoms

A July 22, 2016 news item on Nanowerk highlights an international collaboration focused on producing clean fuel,

A combined experimental and theoretical study comprising researchers from the Chemistry Department and LCN [London Centre for Nanotechnology], along with groups in Argentina, China, Spain and Germany, has shed new light on the behaviour of individual gold atoms supported on defective thin cerium dioxide films – an important system for catalysis and the generation of clean hydrogen for fuel.

A July ??, 2016 LCN press release, which originated the news item, expands on the theme of catalysts, the research into individual gold atoms, and how all this could result in clean fuel,

Catalysis plays a vital role in our world; an estimated 80% of all chemical and materials are made via processes which involve catalysts, which are commonly a mixture of metals and oxides. The standard motif for these heterogeneous catalysts (where the catalysts are solid and the reactants are in the gas phase) is of a high surface area oxide support that is decorated with metal nanoparticles a few nanometres in diameter. Cerium dioxide (ceria, CeO2) is a widely used support material for many important industrial processes; metal nanoparticles supported on ceria have displayed high activities for applications including car catalytic converters, alcohol synthesis, and for hydrogen production. There are two key attributes of ceria which make it an excellent active support material: its oxygen storage and release ability, and its ability to stabilise small metal particles under reaction conditions. A recent system that has been the focus of much interest has been that of gold nanoparticles and single atoms with ceria, which has demonstrated high activity towards the water-gas-shift reaction, (CO + H2O —> CO2 + H2) a key stage in the generation of clean hydrogen for use in fuel cells.

The nature of the active sites of these catalysts and the role that defects play are still relatively poorly understood; in order to study them in a systematic fashion, the researchers prepared model systems which can be characterised on the atomic scale with a scanning tunnelling microscope.

Figure: STM images of CeO2-x(111) ultrathin films before and after the deposition of Au single atoms at 300 K. The bright lattice is from the oxygen atoms at the surface – vacancies appear as dark spots

These model systems comprised well-ordered, epitaxial ceria films less than 2 nm thick, prepared on a metal single crystal, upon which single atoms and small clusters of gold were evaporated onto under ultra-high-vacuum (essential to prevent contamination of the surfaces). Oxygen vacancy defects – missing oxygen atoms in the top layer of the ceria – are relatively common at the surface and appear as dark spots in the STM images. By mapping the surface before and after the deposition of gold, it is possible to analyse the binding of the metal atoms, in particular there does not appear to be any preference for binding in the vacancy sites at 300 K.

Publishing their results in Physical Review Letters, the researchers combined these experimental results with theoretical studies of the binding energies and diffusion rates across the surface. They showed that kinetic effects governed the behaviour of the gold atoms, prohibiting the expected occupation of the thermodynamically more stable oxygen vacancy sites. They also identified electron transfer between the gold atoms and the ceria, leading to a better understanding of the diffusion phenomena that occur at this scale, and demonstrated that the effect of individual surface defects may be more minor than is normally imagined.

Here’s a link to and a citation for the paper,

Diffusion Barriers Block Defect Occupation on Reduced CeO2(111) by P.G. Lustemberg, Y. Pan, B.-J. Shaw, D. Grinter, Chi Pang, G. Thornton, Rubén Pérez, M. V. Ganduglia-Pirovano, and N. Nilius. Phys. Rev. Lett. Vol. 116, Iss. 23 — 10 June 2016 2016DOI:http://dx.doi.org/10.1103/PhysRevLett.116.236101 Published 9 June 2016

This paper is behind a paywall.

# Gold nanoparticles and two different collective oscillations

An April 27, 2016 news item on phys.org describes research into gold nanoparticles and Surface Plasmon Resonance at Hokkaido University and the University of Tsukuba (Japan),

The research group of Professor Hiroaki Misawa of Research Institute for Electronic Science, Hokkaido University and Assistant Professor Atsushi Kubo of the Faculty of Pure and Applied Sciences, University of Tsukuba, have successfully observed the dephasing time of the two different types of collective motions of electrons generated on the surface of a gold nanoparticle for the first time in the world, by combining a laser that emits ultrashort light pulses with a photoemission electron microscope.

An April 26, 2016 Hokkaido University press release, which originated the news item, explains further,

When gold is reduced to the size in nanometer scale, its color is red instead of gold. When gold nanoparticles are exposed to light, the collective oscillations of electrons existing on the localized surface of the gold causes red light to be strongly absorbed and dispersed.

This phenomenon is called Surface Plasmon Resonance. The red color of stained glass is also a result of this phenomenon. Recently, gold nanoparticles have been widely used in various fields, such as application in pregnancy tests.

This collective oscillations of electrons on the surface of gold nanoparticles caused by light was considered to be a phenomenon that sustained only for an extremely short time, and difficult to measure due to this shortness.

Our research group developed a methodology to measure the dephasing time of the collective oscillations of electrons occurring on the surface of gold nanoparticles by combining a laser that emits ultrashort light pulses of a few femtoseconds (1 femtosecond: 1´10-15 seconds), and a photoemission electron microscope in high spatial resolution.

When measured by this technique, the different dephasing times of the two different collective oscillations, namely dipole and quadrupole surface plasmon modes, could be resolved and identified as 5 femtoseconds and 9 femtoseconds, respectively.

Research using gold nanoparticles as optical antennae to harvest light for photovoltaic cell and an artificial photosynthesis system that can split water to obtain hydrogen is progressing. The successful measurement of the dephasing time of the collective oscillations of electrons is considered to be a useful guideline in developing these systems.

Here’s a link to and a citation for the paper,

Dissecting the Few-Femtosecond Dephasing Time of Dipole and Quadrupole Modes in Gold Nanoparticles Using Polarized Photoemission Electron Microscopy by Quan Sun†, Han Yu, Kosei Ueno, Atsushi Kubo, Yasutaka Matsuo, and Hiroaki Misawa. ACS Nano, 2016, 10 (3), pp 3835–3842 DOI: 10.1021/acsnano.6b00715Publication Date (Web): February 15, 2016

This paper appears to be open access.

# Black gold: ultralight, high density nanoporous gold

South Korean researchers have found a way to fabricate a new kind of gold nanoparticle according to a March 28, 2016 news item on ScienceDaily,

A new material is more solid and 30 percent lighter than standard gold, scientists report. In their study, the team investigated grain boundaries in nanocrystalline np-Au and found a way to overcome the weakening mechanisms of this material, thereby suggesting its usefulness.

A March 28, 2016 Ulsan National Institute of Science and Technology (UNIST) press release (also on EurekAlert) by Chorok Oh, which originated the news item, provides more information,

A team of Korean research team, led by Professor Ju-Young Kim (School of Materials Science and Engineering) of Ulsan National Institute of Science and Technology (UNIST), South Korea has recently announced that they have successfully developed a way to fabricate an ultralight, high-dense nanoporous gold (np-Au).

In a new paper, published in Nano Letters on March 22, the team reported that this newly developed material, which they have dubbed “Black Gold” is twice more solid and 30% lighter than standard gold.

According to Prof. Kim, “This particular nanoporous gold has a 100,000 times wider surface when compared to standard gold. Moreover, due to its chemically stablity, it is also harmless to humans.”

The surfaces of np-Au are rough and the metal loses its shine and eventually turns black when they are at sizes less than 100 nanometres (nm). This is the reason that they are called “Black Gold”.

In their study, the team investigated grain boundaries in nanocrystalline np-Au and found a way to overcome the weakening mechanisms of this material, thereby suggesting its usefulness.

The team used a ball milling technique to increase the flexural strength of the three gold-silver precursor alloys. Then, using free corrosion dealloying of silver from gold-silver alloys, they were able to achieve the nanoporous surface. According to the team, “The size of the pores can be controlled by the temperature and concentration of nitrate.” Moreover, they also note that this crack-free nanoporous gold samples are reported to exhibit excellent durability in three-point bending tests.

Prof. Kim’s team notes, “Ball-milled np-Au has a much greater density of two-dimensional defects than annealed and prestrained np-Au, where intergranular fracture is preferred.” They continue, “Therefore, the probable existence of grain boundary opening in the highest tensile region is attributed to the flexural strength of np-Au.”

They suggest that this newly developed technique can be also applied to many other metal, as the np-Au produced by this technique have shown increased strength and durability while still maintaining the good qualities of standard gold.

This means that this technique can be also used in other technologies, like catalytic-converting as observed by platinum, the automobile catalyst and palladium, the hydrogen sensor catalyst.

Here’s a link to and a citation for the paper,

Weakened Flexural Strength of Nanocrystalline Nanoporous Gold by Grain Refinement by Eun-Ji Gwak and Ju-Young Kim. Nano Lett., Article ASAP DOI: 10.1021/acs.nanolett.6b00062 Publication Date (Web): March 16, 2016

This paper is behind a paywall.

# Mass production of nanoparticles?

With all the years of nanotechnology and nanomaterials research it seems strange that mass production of nanoparticles is still very much in the early stages as a Feb. 24, 2016 news item on phys.org points out,

Nanoparticles – tiny particles 100,000 times smaller than the width of a strand of hair – can be found in everything from drug delivery formulations to pollution controls on cars to HD TV sets. With special properties derived from their tiny size and subsequently increased surface area, they’re critical to industry and scientific research.

They’re also expensive and tricky to make.

Now, researchers at USC [University of Southern California] have created a new way to manufacture nanoparticles that will transform the process from a painstaking, batch-by-batch drudgery into a large-scale, automated assembly line.

A Feb. 24, 2016 USC news release (also on EurekAlert) by Robert Perkins, which originated the news item, offers additional insight,

Consider, for example, gold nanoparticles. They have been shown to easily penetrate cell membranes without causing any damage — an unusual feat given that most penetrations of cell membranes by foreign objects can damage or kill the cell. Their ability to slip through the cell’s membrane makes gold nanoparticles ideal delivery devices for medications to healthy cells or fatal doses of radiation to cancer cells.

However, a single milligram of gold nanoparticles currently costs about $80 (depending on the size of the nanoparticles). That places the price of gold nanoparticles at$80,000 per gram while a gram of pure, raw gold goes for about $50. “It’s not the gold that’s making it expensive,” Malmstadt [Noah Malmstadt of the USC Viterbi School of Engineering] said. “We can make them, but it’s not like we can cheaply make a 50-gallon drum full of them.” A fluid situation At this time, the process of manufacturing a nanoparticle typically involves a technician in a chemistry lab mixing up a batch of chemicals by hand in traditional lab flasks and beakers. The new technique used by Brutchey [Richard Brutchey of the USC Dornsife College of Letters, Arts and Sciences] and Malmstadt instead relies on microfluidics — technology that manipulates tiny droplets of fluid in narrow channels. “In order to go large scale, we have to go small,” Brutchey said. Really small. The team 3-D printed tubes about 250 micrometers in diameter, which they believe to be the smallest, fully enclosed 3-D printed tubes anywhere. For reference, your average-sized speck of dust is 50 micrometers wide. They they built a parallel network of four of these tubes, side-by-side, and ran a combination of two nonmixing fluids (like oil and water) through them. As the two fluids fought to get out through the openings, they squeezed off tiny droplets. Each of these droplets acted as a micro-scale chemical reactor in which materials were mixed and nanoparticles were generated. Each microfluidic tube can create millions of identical droplets that perform the same reaction. This sort of system has been envisioned in the past, but it hasn’t been able to be scaled up because the parallel structure meant that if one tube got jammed, it would cause a ripple effect of changing pressures along its neighbors, knocking out the entire system. Think of it like losing a single Christmas light in one of the old-style strands — lose one and you lose them all. Brutchey and Malmstadt bypassed this problem by altering the geometry of the tubes themselves, shaping the junction between the tubes such that the particles come out a uniform size and the system is immune to pressure changes. Here’s a link to and a citation for the paper, Flow invariant droplet formation for stable parallel microreactors by Carson T. Riche, Emily J. Roberts, Malancha Gupta, Richard L. Brutchey & Noah Malmstadt. Nature Communications 7, Article number: 10780 doi:10.1038/ncomms10780 Published 23 February 2016 This is an open access paper. # A nanoparticle ‘printing press’ This research comes from Montréal, Canada via a Jan. 7, 2016 McGill University news release (also on EurekAlert*), Gold nanoparticles have unusual optical, electronic and chemical properties, which scientists are seeking to put to use in a range of new technologies, from nanoelectronics to cancer treatments. Some of the most interesting properties of nanoparticles emerge when they are brought close together – either in clusters of just a few particles or in crystals made up of millions of them. Yet particles that are just millionths of an inch in size are too small to be manipulated by conventional lab tools, so a major challenge has been finding ways to assemble these bits of gold while controlling the three-dimensional shape of their arrangement. One approach that researchers have developed has been to use tiny structures made from synthetic strands of DNA to help organize nanoparticles. Since DNA strands are programmed to pair with other strands in certain patterns, scientists have attached individual strands of DNA to gold particle surfaces to create a variety of assemblies. But these hybrid gold-DNA nanostructures are intricate and expensive to generate, limiting their potential for use in practical materials. The process is similar, in a sense, to producing books by hand. Enter the nanoparticle equivalent of the printing press. It’s efficient, re-usable and carries more information than previously possible. In results reported online in Nature Chemistry, researchers from McGill’s Department of Chemistry outline a procedure for making a DNA [deoxyribonucleic acid] structure with a specific pattern of strands coming out of it; at the end of each strand is a chemical “sticky patch.” When a gold nanoparticle is brought into contact to the DNA nanostructure, it sticks to the patches. The scientists then dissolve the assembly in distilled water, separating the DNA nanostructure into its component strands and leaving behind the DNA imprint on the gold nanoparticle. … The researchers have made an illustration of their concept available, Credit: Thomas Edwardson “These encoded gold nanoparticles are unprecedented in their information content,” says senior author Hanadi Sleiman, who holds the Canada Research Chair in DNA Nanoscience. “The DNA nanostructures, for their part, can be re-used, much like stamps in an old printing press.” The news release includes suggestions for possible future applications, From stained glass to optoelectronics Some of the properties of gold nanoparticles have been recognized for centuries. Medieval artisans added gold chloride to molten glass to create the ruby-red colour in stained-glass windows – the result, as chemists figured out much later, of the light-scattering properties of tiny gold particles. Now, the McGill researchers hope their new production technique will help pave the way for use of DNA-encoded nanoparticles in a range of cutting-edge technologies. First author Thomas Edwardson says the next step for the lab will be to investigate the properties of structures made from these new building blocks. “In much the same way that atoms combine to form complex molecules, patterned DNA gold particles can connect to neighbouring particles to form well-defined nanoparticle assemblies.” These could be put to use in areas including optoelectronic nanodevices and biomedical sciences, the researchers say. The patterns of DNA strands could, for example, be engineered to target specific proteins on cancer cells, and thus serve to detect cancer or to selectively destroy cancer cells. Here’s a link to and a citation for the paper, Transfer of molecular recognition information from DNA nanostructures to gold nanoparticles by Thomas G. W. Edwardson, Kai Lin Lau, Danny Bousmail, Christopher J. Serpell, & Hanadi F. Sleiman. Nature Chemistry (2016) doi:10.1038/nchem.2420 Published online 04 January 2016 This paper is behind a paywall. *’also on EurekAlert’ added on Jan. 8, 2016. # ‘Stained glass nanotechnology’ for color displays From a Dec. 4, 2015 news item on ScienceDaily, A new method for building “drawbridges” between metal nanoparticles may allow electronics makers to build full-color displays using light-scattering nanoparticles that are similar to the gold materials that medieval artisans used to create red stained-glass. “Wouldn’t it be interesting if we could create stained-glass windows that changed colors at the flip of a switch?” said Christy Landes, associate professor of chemistry at Rice and the lead researcher on a new study about the drawbridge method that appears this week in the open-access journal Science Advances. The research by Landes and other experts at Rice University’s Smalley-Curl Institute could allow engineers to use standard electrical switching techniques to construct color displays from pairs of nanoparticles that scatter different colors of light. For centuries, stained-glass makers have tapped the light-scattering properties of tiny gold nanoparticles to produce glass with rich red tones. Similar types of materials could increasingly find use in modern electronics as manufacturers work to make smaller, faster and more energy-efficient components that operate at optical frequencies. A Dec. 4, 2015 Rice University news release (also on EurekAlert), which originated the news item, describes the research in more detail, Though metal nanoparticles scatter bright light, researchers have found it difficult to coax them to produce dramatically different colors, Landes said. Rice’s new drawbridge method for color switching incorporates metal nanoparticles that absorb light energy and convert it into plasmons, waves of electrons that flow like a fluid across a particle’s surface. Each plasmon scatters and absorbs a characteristic frequency of light, and even minor changes in the wave-like sloshing of a plasmon shift that frequency. The greater the change in plasmonic frequency, the greater the difference between the colors observed. “Engineers hoping to make a display from optically active nanoparticles need to be able to switch the color,” Landes said. “That type of switching has proven very difficult to achieve with nanoparticles. People have achieved moderate success using various plasmon-coupling schemes in particle assemblies. What we’ve shown though is variation of the coupling mechanism itself, which can be used to produce huge color changes both rapidly and reversibly.” To demonstrate the method, Landes and study lead author Chad Byers, a graduate student in her lab, anchored pairs of gold nanoparticles to a glass surface covered with indium tin oxide (ITO), the same conductor that’s used in many smartphone screens. By sealing the particles in a chamber filled with a saltwater electrolyte and a silver electrode, Byers and Landes were able form a device with a complete circuit. They then showed they could apply a small voltage to the ITO to electroplate silver onto the surface of the gold particles. In that process, the particles were first coated with a thin layer of silver chloride. By later applying a negative voltage, the researchers caused a conductive silver “drawbridge” to form. Reversing the voltage caused the bridge to withdraw. “The great thing about these chemical bridges is that we can create and eliminate them simply by applying or reversing a voltage,” Landes said. “This is the first method yet demonstrated to produce dramatic, reversible color changes for devices built from light-activated nanoparticles.” This research has its roots in previous work (from the news release), Byers said his research into the plasmonic behavior of gold dimers began about two years ago. “We were pursuing the idea that we could make significant changes in optical properties of individual particles simply by altering charge density,” he said. “Theory predicts that colors can be changed just by adding or removing electrons, and we wanted to see if we could do that reversibly, simply by turning a voltage on or off.” The experiments worked. The color shift was observed and reversible, but the change in the color was minute. “It wasn’t going to get anybody excited about any sort of switchable display applications,” Landes said. But she and Byers also noticed that their results differed from the theoretical predictions. Landes said that was because the predictions were based upon using an inert electrode made of a metal like palladium that isn’t subject to oxidation. But silver is not inert. It reacts easily with oxygen in air or water to form a coat of unsightly silver oxide. This oxidizing layer can also form from silver chloride, and Landes said that is what was occurring when the silver counter electrode was used in Byers’ first experiments. The scientists decided to embrace imperfection (from the news release), “It was an imperfection that was throwing off our results, but rather than run away from it, we decided to use it to our advantage,” Landes said. Rice plasmonics pioneer and study co-author Naomi Halas, director of the Smalley-Curl Institute, said the new research shows how plasmonic components could be used to produce electronically switchable color-displays. “Gold nanoparticles are particularly attractive for display purposes,” said Halas, Rice’s Stanley C. Moore Professor of Electrical and Computer Engineering and professor of chemistry, bioengineering, physics and astronomy, and materials science and nanoengineering. “Depending upon their shape, they can produce a variety of specific colors. They are also extremely stable, and even though gold is expensive, very little is needed to produce an extremely bright color.” In designing, testing and analyzing the follow-up experiments on dimers, Landes and Byers engaged with a brain trust of Rice plasmonics experts that included Halas, physicist and engineer Peter Nordlander, chemist Stephan Link, materials scientist Emilie Ringe and their students, as well as Paul Mulvaney of the University of Melbourne in Australia. Together, the team confirmed the composition and spacing of the dimers and showed how metal drawbridges could be used to induce large color shifts based on voltage inputs. Nordlander and Hui Zhang, the two theorists in the group, examined the device’s “plasmonic coupling,” the interacting dance that plasmons engage in when they are in close contact. For instance, plasmonic dimers are known to act as light-activated capacitors, and prior research has shown that connecting dimers with nanowire bridges brings about a new state of resonance known as a “charge-transfer plasmon,” which has its own distinct optical signature. “The electrochemical bridging of the interparticle gap enables a fully reversible transition between two plasmonic coupling regimes, one capacitive and the other conductive,” Nordlander said. “The shift between these regimes is evident from the dynamic evolution of the charge transfer plasmon.” Halas said the method provides plasmonic researchers with a valuable tool for precisely controlling the gaps between dimers and other multiparticle plasmonic configurations. “In an applied sense, gap control is important for the development of active plasmonic devices like switches and modulators, but it is also an important tool for basic scientists who are conducting curiosity-driven research in the emerging field of quantum plasmonics.” I’m glad the news release writer included the background work leading to this new research and to hint at the level of collaboration needed to achieve the scientists’ new understanding of color switching. Here’s a link to and a citation for the paper, From tunable core-shell nanoparticles to plasmonic drawbridges: Active control of nanoparticle optical properties by Chad P. Byers, Hui Zhang, Dayne F. Swearer, Mustafa Yorulmaz, Benjamin S. Hoener, Da Huang, Anneli Hoggard, Wei-Shun Chang, Paul Mulvaney, Emilie Ringe, Naomi J. Halas, Peter Nordlander, Stephan Link, and Christy F. Landes. Science Advances 04 Dec 2015: Vol. 1, no. 11, e1500988 DOI: 10.1126/sciadv.1500988 In case you missed it in the news release, this is an open access paper. # Combining gold and palladium for catalytic and plasmonic octopods Hopefully I did not the change meaning when I made the title for this piece more succinct. In any event, this research comes from the always prolific Rice University in Texas, US (from a Nov. 30, 2015 news item on Nanotechnology Now), Catalysts are substances that speed up chemical reactions and are essential to many industries, including petroleum, food processing and pharmaceuticals. Common catalysts include palladium and platinum, both found in cars’ catalytic converters. Plasmons are waves of electrons that oscillate in particles, usually metallic, when excited by light. Plasmonic metals like gold and silver can be used as sensors in biological applications and for chemical detection, among others. Plasmonic materials are not the best catalysts, and catalysts are typically very poor for plasmonics. But combining them in the right way shows promise for industrial and scientific applications, said Emilie Ringe, a Rice assistant professor of materials science and nanoengineering and of chemistry who led the study that appears in Scientific Reports. “Plasmonic particles are magnets for light,” said Ringe, who worked on the project with colleagues in the U.S., the United Kingdom and Germany. “They couple with light and create big electric fields that can drive chemical processes. By combining these electric fields with a catalytic surface, we could further push chemical reactions. That’s why we’re studying how palladium and gold can be incorporated together.” The researchers created eight-armed specks of gold and coated them with a gold-palladium alloy. The octopods proved to be efficient catalysts and sensors. A Nov. 30, 2015 Rice University news release (also on EurekAlert), which originated the news item, expands on the theme, “If you simply mix gold and palladium, you may end up with a bad plasmonic material and a pretty bad catalyst, because palladium does not attract light like gold does,” Ringe said. “But our particles have gold cores with palladium at the tips, so they retain their plasmonic properties and the surfaces are catalytic.” Just as important, Ringe said, the team established characterization techniques that will allow scientists to tune application-specific alloys that report on their catalytic activity in real time. The researchers analyzed octopods with a variety of instruments, including Rice’s new Titan Themis microscope, one of the most powerful electron microscopes in the nation. “We confirmed that even though we put palladium on a particle, it’s still capable of doing everything that a similar gold shape would do. That’s really a big deal,” she said. “If you shine a light on these nanoparticles, it creates strong electric fields. Those fields enhance the catalysis, but they also report on the catalysis and the molecules present at the surface of the particles,” Ringe said. The researchers used electron energy loss spectroscopy, cathodoluminescence and energy dispersive X-ray spectroscopy to make 3-D maps of the electric fields produced by exciting the plasmons. They found that strong fields were produced at the palladium-rich tips, where plasmons were the least likely to be excited. Ringe expects further research will produce multifunctional nanoparticles in a variety of shapes that can be greatly refined for applications. Her own Rice lab is working on a metal catalyst to turn inert petroleum derivatives into backbone molecules for novel drugs. Here’s a link to and a citation for the paper, Resonances of nanoparticles with poor plasmonic metal tips by Emilie Ringe, Christopher J. DeSantis, Sean M. Collins, Martial Duchamp, Rafal E. Dunin-Borkowski, Sara E. Skrabalak, & Paul A. Midgley. Scientific Reports 5, Article number: 17431 (2015) doi:10.1038/srep17431 Published online: 30 November 2015 This is an open access paper, # A view to controversies about nanoparticle drug delivery, sticky-flares, and a PNAS surprise Despite all the excitement and claims for nanoparticles as vehicles for drug delivery to ‘sick’ cells there is at least one substantive problem, the drug-laden nanoparticles don’t actually enter the interior of the cell. They are held in a kind of cellular ‘waiting room’. Leonid Schneider in a Nov. 20, 2015 posting on his For Better Science blog describes the process in more detail, A large body of scientific nanotechnology literature is dedicated to the biomedical aspect of nanoparticle delivery into cells and tissues. The functionalization of the nanoparticle surface is designed to insure their specificity at targeting only a certain type of cells, such as cancers cells. Other technological approaches aim at the cargo design, in order to ensure the targeted release of various biologically active agents: small pharmacological substances, peptides or entire enzymes, or nucleotides such as regulatory small RNAs or even genes. There is however a main limitation to this approach: though cells do readily take up nanoparticles through specific membrane-bound receptor interaction (endocytosis) or randomly (pinocytosis), these nanoparticles hardly ever truly reach the inside of the cell, namely its nucleocytoplasmic space. Solid nanoparticles are namely continuously surrounded by the very same membrane barrier they first interacted with when entering the cell. These outer-cell membrane compartments mature into endosomal and then lysosomal vesicles, where their cargo is subjected to low pH and enzymatic digestion. The nanoparticles, though seemingly inside the cell, remain actually outside. … What follows is a stellar piece featuring counterclaims about and including Schneider’s own journalistic research into scientific claims that the problem of gaining entry to a cell’s true interior has been addressed by technologies developed in two different labs. Having featured one of the technologies here in a July 24, 2015 posting titled: Sticky-flares nanotechnology to track and observe RNA (ribonucleic acid) regulation and having been contacted a couple of times by one of the scientists, Raphaël Lévy from the University of Liverpool (UK), challenging the claims made (Lévy’s responses can be found in the comments section of the July 2015 posting), I thought a followup of sorts was in order. Scientific debates (then and now) Scientific debates and controversies are part and parcel of the scientific process and what most outsiders, such as myself, don’t realize is how fraught it is. For a good example from the past, there’s Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life (from its Wikipedia entry), Note: Links have been removed), Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life (published 1985) is a book by Steven Shapin and Simon Schaffer. It examines the debate between Robert Boyle and Thomas Hobbes over Boyle’s air-pump experiments in the 1660s. The style seems more genteel than what a contemporary Canadian or US audience is accustomed to but Hobbes and Boyle (and proponents of both sides) engaged in bruising communication. There was a lot at stake then and now. It’s not just the power, prestige, and money, as powerfully motivating as they are, it’s the research itself. Scientists work for years to achieve breakthroughs or to add more to our common store of knowledge. It’s painstaking and if you work at something for a long time, you tend to be invested in it. Saying you’ve wasted ten years of your life looking at the problem the wrong way or have misunderstood your data is not easy. As for the current debate, Schneider’s description gives no indication that there is rancour between any of the parties but it does provide a fascinating view of two scientists challenging one of the US’s nanomedicine rockstars, Chad Mirkin. The following excerpt follows the latest technical breakthroughs to the interior portion of the cell through three phases of the naming conventions (Nano-Flares, also known by its trade name, SmartFlares, which is a precursor technology to Sticky-Flares), Note: Links have been removed, The next family of allegedly nucleocytoplasmic nanoparticles which Lévy turned his attention to, was that of the so called “spherical nucleic acids”, developed in the lab of Chad Mirkin, multiple professor and director of the International Institute for Nanotechnology at the Northwestern University, USA. These so called “Nano-Flares” are gold nanoparticles, functionalized with fluorophore-coupled oligonucleotides matching the messenger RNA (mRNA) of interest (Prigodich et al., ACS Nano 3:2147-2152, 2009; Seferos et al., J Am. Chem.Soc. 129:15477-15479, 2007). The mRNA detection method is such that the fluorescence is initially quenched by the gold nanoparticle proximity. Yet when the oligonucleotide is displaced by the specific binding of the mRNA molecules present inside the cell, the fluorescence becomes detectable and serves thus as quantitative read-out for the intracellular mRNA abundance. Exactly this is where concerns arise. To find and bind mRNA, spherical nucleic acids must leave the endosomal compartments. Is there any evidence that Nano-Flares ever achieve this and reach intact the nucleocytoplasmatic space, where their target mRNA is? Lévy’s lab has focused its research on the commercially available analogue of the Nano-Flares, based on the patent to Mirkin and Northwestern University and sold by Merck Millipore under the trade name of SmartFlares. These were described by Mirkin as “a powerful and prolific tool in biology and medical diagnostics, with ∼ 1,600 unique forms commercially available today”. The work, led by Lévy’s postdoctoral scientist David Mason, now available in post-publication process at ScienceOpen and on Figshare, found no experimental evidence for SmartFlares to be ever found outside the endosomal membrane vesicles. On the contrary, the analysis by several complementary approaches, i.e., electron, fluorescence and photothermal microscopy, revealed that the probes are retained exclusively within the endosomal compartments. In fact, even Merck Millipore was apparently well aware of this problem when the product was developed for the market. As I learned, Merck performed a number of assays to address the specificity issue. Multiple hundred-fold induction of mRNA by biological cell stimulation (confirmed by quantitative RT-PCR) led to no significant changes in the corresponding SmartFlare signal. Similarly, biological gene downregulation or experimental siRNA knock-down had no effect on the corresponding SmartFlare fluorescence. Cell lines confirmed as negative for a certain biomarker proved highly positive in a SmartFlare assay. Live cell imaging showed the SmartFlare signal to be almost entirely mitochondrial, inconsistent with reported patterns of the respective mRNA distributions. Elsewhere however, cyanine dye-labelled oligonucleotides were found to unspecifically localise to mitochondria (Orio et al., J. RNAi Gene Silencing 9:479-485, 2013), which might account to the often observed punctate Smart Flare signal. More recently, Mirkin lab has developed a novel version of spherical nucleic acids, named Sticky-Flares (Briley et al., PNAS 112:9591-9595, 2015), which has also been patented for commercial use. The claim is that “the Sticky-flare is capable of entering live cells without the need for transfection agents and recognizing target RNA transcripts in a sequence-specific manner”. To confirm this, Lévy used the same approach as for the striped nanoparticles [not excerpted here]: he approached Mirkin by email and in person, requesting the original microscopy data from this publication. As Mirkin appeared reluctant, Lévy invoked the rules for data sharing by the journal PNAS, the funder NSF as well as the Northwestern University. After finally receiving Mirkin’s thin-optical microscopy data by air mail, Lévy and Mason re-analyzed it and determined the absence of any evidence for endosomal escape, while all Sticky-Flare particles appeared to be localized exclusively inside vesicular membrane compartments, i.e., endosomes (Mason & Levy, bioRxiv 2015). I encourage you to read Schneider’s Nov. 20, 2015 posting in its entirety as these excerpts can’t do justice to it. The PNAS surprise PNAS (Proceedings of the National Academy of Science) published one of Mirkin’s papers on ‘Sticky-flares’ and is where scientists, Raphaël Lévy and David Mason, submitted a letter outlining their concerns with the ‘Sticky-flares’ research. Here’s the response as reproduced in Lévy’s Nov. 16, 2015 posting on his Rapha-Z-Lab blog Dear Dr. Levy, I regret to inform you that the PNAS Editorial Board has declined to publish your Letter to the Editor. After careful consideration, the Board has decided that your letter does not contribute significantly to the discussion of this paper. Thank you for submitting your comments to PNAS. Sincerely yours, Inder Verma Editor-in-Chief Judge for yourself, Lévy’s and Mason’s letter can be found here (pdf) and here. Conclusions My primary interest in this story is in the view it provides of the scientific process and the importance of and difficulty associated with the debates. I can’t venture an opinion about the research or the counterarguments other than to say that Lévy’s and Mason’s thoughtful challenge bears more examination than PNAS is inclined to accord. If their conclusions or Chad Mirkin’s are wrong, let that be determined in an open process. I’ll leave the very last comment to Schneider who is both writer and cartoonist, from his Nov. 20, 2015 posting, # Attomolar cancer detection: measuring microRNAs in blood The latest research does not lead to a magical disease detector where nanoscale sensors swim through the body continuously monitoring our health and alerting us should something untoward occur (see this Oct. 28, 2014 article on RT.com for more about Google X’s development plans for it and this Nov. 11, 2015 news item on Nanowerk for a measured response from a researcher in the field). Now onto some real research, a Nov. 17, 2015 news item on ScienceDaily announces an ultrasensitive (attoscale) sensor employing gold nanoparticles for detecting cancer, A simple, ultrasensitive microRNA sensor developed and tested by researchers from the schools of science and medicine at Indiana University-Purdue University Indianapolis and the Indiana University Melvin and Bren Simon Cancer Center holds promise for the design of new diagnostic strategies and, potentially, for the prognosis and treatment of pancreatic and other cancers. A Nov. 17, 2015 Indiana University-Purdue University Indianapolis news release on EurekAlert, which originated the news item, provides more detail about research that seems to have focused largely on pancreatic cancer detection (Note: Links have been removed), In a study published in the Nov. [2015] issue of ACS Nano, a peer-reviewed journal of the American Chemical Society focusing on nanoscience and nanotechnology research, the IUPUI researchers describe their design of the novel, low-cost, nanotechnology-enabled reusable sensor. They also report on the promising results of tests of the sensor’s ability to identify pancreatic cancer or indicate the existence of a benign condition by quantifying changes in levels of microRNA signatures linked to pancreatic cancer. MicroRNAs are small molecules of RNA that regulate how larger RNA molecules lead to protein expression. As such, microRNAs are very important in biology and disease states. “We used the fundamental concepts of nanotechnology to design the sensor to detect and quantify biomolecules at very low concentrations,” said Rajesh Sardar, Ph.D., who developed the sensor. “We have designed an ultrasensitive technique so that we can see minute changes in microRNA concentrations in a patient’s blood and confirm the presence of pancreatic cancer.” Sardar is an assistant professor of chemistry and chemical biology in the School of Science at IUPUI and leads an interdisciplinary research program focusing on the intersection of analytical chemistry and the nanoscience of metallic nanoparticles. “If we can establish that there is cancer in the pancreas because the sensor detects high levels of microRNA-10b or one of the other microRNAs associated with that specific cancer, we may be able to treat it sooner,” said Murray Korc, M.D., the Myles Brand Professor of Cancer Research at the IU School of Medicine and a researcher at the IU Simon Cancer Center. Korc, worked with Sardar to improve the sensor’s capabilities and led the testing of the sensor and its clinical uses as well as advancing the understanding of pancreatic cancer biology. “That’s especially significant for pancreatic cancer, because for many patients it is symptom-free for years or even a decade or more, by which time it has spread to other organs, when surgical removal is no longer possible and therapeutic options are limited,” said Korc. “For example, diagnosis of pancreatic cancer at an early stage of the disease followed by surgical removal is associated with a 40 percent five-year survival. Diagnosis of metastatic pancreatic cancer, by contrast, is associated with life expectancy that is often only a year or less. “The beauty of the sensor designed by Dr. Sardar is its ability to accurately detect mild increases in microRNA levels, which could allow for early cancer diagnosis,” Korc added. Over the past decade, studies have shown that microRNAs play important roles in cancer and other diseases, such as diabetes and cardiovascular disorders. The new IUPUI nanotechnology-based sensor can detect changes in any of these microRNAs. The sensor is a small glass chip that contains triangular-shaped gold nanoparticles called ‘nanoprisms.’ After dipping it in a sample of blood or another body fluid, the scientist measures the change in the nanoprism’s optical property to determine the levels of specific microRNAs. For anyone concerned about the cost associated with creating sensors based on gold, about patents, or about current techniques for monitoring microRNAs, there’s more from the news release (Note: A link has been removed), “Using gold nanoprisms may sound expensive, but it isn’t because these particles are so very tiny,” Sardar said. “It’s a rather cheap technique because it uses nanotechnology and needs very little gold.$250 worth of gold makes 4,000 sensors. Four thousand sensors allow you to do at least 4,000 tests. The low cost makes this technique ideal for use anywhere, including in low-resource environments in this country and around the world.”

Indiana University Research and Technology Corporation has filed a patent application on Sardar’s and Korc’s groundbreaking nanotechnology-enabled sensor. The researchers’ ultimate goal is to design ultrasensitive and extremely selective low-cost point-of-care diagnostics enabling individual therapeutic approaches to diseases.

Currently, polymerase chain reaction technology is used to determine microRNA signatures, which requires extraction of the microRNA from blood or other biological fluid and reverse transcription or amplification of the microRNA. PCR provides relative values. By contrast, the process developed at IUPUI is simpler, quantitative, more sensitive and highly specific even when two different microRNAs vary in a single position. The study demonstrated that the IUPUI nanotechnology-enabled sensor is as good as if not better than the most advanced PCR in detection and quantification of microRNA.

Here’s a link to and a citation for the paper,

Label-Free Nanoplasmonic-Based Short Noncoding RNA Sensing at Attomolar Concentrations Allows for Quantitative and Highly Specific Assay of MicroRNA-10b in Biological Fluids and Circulating Exosomes by Gayatri K. Joshi, Samantha Deitz-McElyea, Thakshila Liyanage, Katie Lawrence, Sonali Mali, Rajesh Sardar*, and Murray Korc. ACS Nano, Article ASAP DOI: 10.1021/acsnano.5b04527 Publication Date (Web): October 7, 2015

This is an open access paper.

The researchers have provided this illustration of gold nanoprisms,

Caption: Indiana University-Purdue University Indianapolis researchers have developed a novel, low-cost, nanotechnology-enabled reusable sensor for which a patent application has been filed. Credit: Department of Chemistry and Chemical Biology, School of Science, Indiana University-Purdue University Indianapolis

# Nano-alchemy: silver nanoparticles that look like and act like gold

This work on ‘nano-alchemy’ comes out of the King Abduhllah University of Science and Technology (KAUST) according to a Sept. 22, 2015 article by Lisa Zynga for phys.org (Note: A link has been removed),

In an act of “nano-alchemy,” scientists have synthesized a silver (Ag) nanocluster that is virtually identical to a gold (Au) nanocluster. On the outside, the silver nanocluster has a golden yellow color, and on the inside, its chemical structure and properties also closely mimic those of its gold counterpart. The work shows that it may be possible to create silver nanoparticles that look and behave like gold despite underlying differences between the two elements, and could lead to creating similar analogues between other pairs of elements.

“In some aspects, this is very similar to alchemy, but we call it ‘nano-alchemy,'” Bakr [Osman Bakr, Associate Professor of Materials Science and Engineering at King Abdullah University of Science and Technology (KAUST) in Saudi Arabia] told Phys.org. “When we first encountered the optical spectrum of the silver nanocluster, we thought that we may have inadvertently switched the chemical reagents for silver with gold, and ended up with gold nanoparticles instead. But repeated synthesis and measurements proved that the clusters were indeed silver and yet show properties akin to gold. It was really surprising to us as scientists to find not only similarities in the color and optical properties, but also the X-ray structure.”

In their study, the researchers performed tests demonstrating that the silver and gold nanoclusters have very similar optical properties. Typically, silver nanoclusters are brown or red in color, but this one looks just like gold because it emits light at almost the same wavelength (around 675 nm) as gold. The golden color can be explained by the fact that both nanoclusters have virtually identical crystal structures.

The question naturally arises: why are these silver and gold nanoclusters so similar, when individual atoms of silver and gold are very different, in terms of their optical and structural properties? As Bakr explained, the answer may have to do with the fact that, although larger in size, the nanoclusters behave like “superatoms” in the sense that their electrons orbit the entire nanocluster as if it were a single giant atom. These superatomic orbitals in the silver and gold nanoclusters are very similar, and, in general, an atom’s electron configuration contributes significantly to its properties.

Here’s one of the images used to illustrate Zynga’s article and the paper published by the American Chemical Society,

(Left) Optical properties of the silver and gold nanoclusters, with the inset showing photographs of the actual color of the synthesized nanoclusters. The graph shows the absorption (solid lines) and normalized emission (dotted lines) spectra. (Right) Various representations of the X-ray structure of the silver nanocluster. Credit: Joshi, et al. ©2015 American Chemical Society

I encourage you to read Zynga’s article in its entirety. For the more technically inclined, here’s a link to and a citation for the researchers’ paper,

[Ag25(SR)18]: The “Golden” Silver Nanoparticle by Chakra P. Joshi, Megalamane S. Bootharaju, Mohammad J. Alhilaly, and Osman M. Bakr.J. Am. Chem. Soc., 2015, 137 (36), pp 11578–11581 DOI: 10.1021/jacs.5b07088 Publication Date (Web): August 31, 2015


text: problem 8.3 in introduction to elementary particle physics asks for the ratio between    $$K^{0}_{1} and K^{0}_{2}$$ at the production point and at l=10m from the production point if the momentum is 10GeV and it's from a    $$\pi^{-}$$ striking a tarkget and making K mesons and    $$\Lambda$$

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# Metric Space and Subsets

1. Jun 30, 2009

### seyma

Let (X,d) be a metric space A and B nonempty subsets of X and A is open. Show:
A$$\cap$$B = $$\oslash$$ Iff A$$\cap$$B(closure)= empty
Only B closure
it is easy to show rigth to left but how can i use A's open property I try to solve with contradiction s.t. there exist r>0 Br(p)$$\subseteq$$A$$\cap$$B(closure) but i cannot come conlusion. Can you help me please ? :(

Last edited: Jun 30, 2009
2. Jun 30, 2009

### Office_Shredder

Staff Emeritus
Umm.... I think you're missing part of the iff statement

3. Jun 30, 2009

### seyma

yeah sorry i forget to write empty

4. Jun 30, 2009

### VeeEight

Are you able to show that A $$\cap$$ B' is empty (where B' is the set of limit points for B)? What is the definition of A being open (use the neighborhood definition)? What is the requirement that a point be an element of B' (use the deleted neighborhood definition)?

Using these definitions, you can work a proof by contradiction.


text:
# 19.2 Electric potential in a uniform electric field

 Page 1 / 5
• Describe the relationship between voltage and electric field.
• Derive an expression for the electric potential and electric field.
• Calculate electric field strength given distance and voltage.

In the previous section, we explored the relationship between voltage and energy. In this section, we will explore the relationship between voltage and electric field. For example, a uniform electric field $\mathbf{\text{E}}$ is produced by placing a potential difference (or voltage) $\Delta V$ across two parallel metal plates, labeled A and B. (See [link] .) Examining this will tell us what voltage is needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential and electric field. From a physicist’s point of view, either $\Delta V$ or $\mathbf{\text{E}}$ can be used to describe any charge distribution. $\Delta V$ is most closely tied to energy, whereas $\mathbf{\text{E}}$ is most closely related to force. $\Delta V$ is a scalar    quantity and has no direction, while $\mathbf{\text{E}}$ is a vector    quantity, having both magnitude and direction. (Note that the magnitude of the electric field strength, a scalar quantity, is represented by $E$ below.) The relationship between $\Delta V$ and $\mathbf{\text{E}}$ is revealed by calculating the work done by the force in moving a charge from point A to point B. But, as noted in Electric Potential Energy: Potential Difference , this is complex for arbitrary charge distributions, requiring calculus. We therefore look at a uniform electric field as an interesting special case.

The work done by the electric field in [link] to move a positive charge $q$ from A, the positive plate, higher potential, to B, the negative plate, lower potential, is

$W=\phantom{\rule{0.25em}{0ex}}–\Delta \text{PE}=\phantom{\rule{0.25em}{0ex}}–q\Delta V.$

The potential difference between points A and B is

$–\Delta V=\phantom{\rule{0.25em}{0ex}}–\left({V}_{\text{B}}–{V}_{\text{A}}\right)={V}_{\text{A}}–{V}_{\text{B}}={V}_{\text{AB}}.$

Entering this into the expression for work yields

$W={\text{qV}}_{\text{AB}}.$

Work is $W=\text{Fd}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta$ ; here $\text{cos}\phantom{\rule{0.25em}{0ex}}\theta =1$ , since the path is parallel to the field, and so $W=\text{Fd}$ . Since $F=\text{qE}$ , we see that $W=\text{qEd}$ . Substituting this expression for work into the previous equation gives

$\mathrm{qEd}={\text{qV}}_{\text{AB}}.$

The charge cancels, and so the voltage between points A and B is seen to be

$\begin{array}{c}\left(\begin{array}{c}{V}_{\text{AB}}=\mathrm{Ed}\\ E=\frac{{V}_{\text{AB}}}{d}\end{array}}\text{(uniform}\phantom{\rule{0.25em}{0ex}}E\phantom{\rule{0.25em}{0ex}}\text{- field only),}\end{array}$

where $d$ is the distance from A to B, or the distance between the plates in [link] . Note that the above equation implies the units for electric field are volts per meter. We already know the units for electric field are newtons per coulomb; thus the following relation among units is valid:

$\text{1 N}/C=\text{1 V}/m.$

## Voltage between points a and b

$\begin{array}{c}\left(\begin{array}{c}{V}_{\text{AB}}=\mathrm{Ed}\\ E=\frac{{V}_{\text{AB}}}{d}\end{array}}\text{(uniform}\phantom{\rule{0.25em}{0ex}}E\phantom{\rule{0.25em}{0ex}}\text{- field only),}\end{array}$

where $d$ is the distance from A to B, or the distance between the plates.

## What is the highest voltage possible between two plates?

Dry air will support a maximum electric field strength of about $3.0×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{V/m}$ . Above that value, the field creates enough ionization in the air to make the air a conductor. This allows a discharge or spark that reduces the field. What, then, is the maximum voltage between two parallel conducting plates separated by 2.5 cm of dry air?

Strategy

We are given the maximum electric field $E$ between the plates and the distance $d$ between them. The equation ${V}_{\text{AB}}=\mathrm{Ed}$ can thus be used to calculate the maximum voltage.

Solution

The potential difference or voltage between the plates is

${\text{V}}_{\text{AB}}=\mathrm{Ed}.$

Entering the given values for $E$ and $d$ gives

${V}_{\text{AB}}=\left(3.0×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{V/m}\right)\left(0.025 m\right)=7.5×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}V$

or

${V}_{\text{AB}}=\text{75 kV}.$

(The answer is quoted to only two digits, since the maximum field strength is approximate.)

Discussion

One of the implications of this result is that it takes about 75 kV to make a spark jump across a 2.5 cm (1 in.) gap, or 150 kV for a 5 cm spark. This limits the voltages that can exist between conductors, perhaps on a power transmission line. A smaller voltage will cause a spark if there are points on the surface, since points create greater fields than smooth surfaces. Humid air breaks down at a lower field strength, meaning that a smaller voltage will make a spark jump through humid air. The largest voltages can be built up, say with static electricity, on dry days.

what is normal force?
the force that pushes upward on us. the force that opposes gravity
clifford
upthrust of air
Newton's 3rd law. the force of the ground (earth) that pushes back on gravity, keeping us on the ground instead of sinking into it.
clifford
I really need lots of questions on frictional force
Shii
I can help answering what I can
Shii
does friction also need some force to perform?
Mohit
no friction is a force just like the gravitational force
clifford
yeah but u can't apply friction anywhere else like other forces
Mohit
I don't understand that question. friction does work alongside other forces based on the situation.
clifford
eg. when walking there are two forces acting on us gravitational and frictional force. friction helps us move forward and gravity keeps us on the ground
clifford
friction is a contact force. Two surfaces are necessary for the force to work.
clifford
hope this helped
clifford
the friction force which oppose while it contact with surrounding. there are two kind of friction. slidding and rolling friction.
Neyaz
What is physics?
physics is a branch of science in which we are dealing with the knowledge of our physical things. macroscopic as well as microscopic. we are going look inside the univers with the help of physics. you can learn nature with the help of physics. so many branches of physics you have to learn physics.
vijay
What are quarks?
6 type of quarks
Neyaz
what is candela
Candela is the unit for the measurement of light intensity.
Osei
any one can prove that 1hrpower= 746 watt
Newton second is the unit of ...............?
Neyaz
Impulse and momentum
Fauzia
force×time and mass× velocity
vijay
Good
Neyaz
What is the simple harmonic motion?
oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position
Yuri
Straight out of google, you could do that to, I suppose.
Yuri
*too
Yuri
ok
Fauzia
Oscillatory motion under a regarding force proportional to the amount of displacement from an equilibrium position
Neyaz
examples of work done by load of gravity
What is ehrenfest theorem?
You can look it up, faster and more reliable answer.
Yuri
That isn't a question to ask on a forum and I also have no idea what that is.
Yuri
what is the work done by gravity on the load 87kj,11.684m,mass xkg[g=19m/s
Maureen
What is law of mass action?
rate of chemical reactions is proportional to concentration of reactants ...
ok thanks
Fauzia
what is lenses
lenses are two types
Fauzia
concave and convex
right
Fauzia
speed of light in space
in vacuum speed of light is 3×10^8 m/s
vijay
ok
Vikash
2.99×10^8m/s
Umair
2.8820^8m/s
Muhammed
Vikash
he is correct but we can round up in simple terms
vijay
3×10^8m/s
vijay
is it correct
Fauzia
I mean 3*10^8 m/s ok
vijay
299792458 meter per second
babar
3*10^8m/s
Neyaz
how many Maxwell relations in thermodynamics
vijay
how we can do prove them?
vijay
What is second law of thermodynamics?
Neyaz
please who has a detailed solution to the first two professional application questions under conservation of momentum
I want to know more about pressure
Osei
I can help
Emeh
okay go on
True
I mean on pressure
Emeh
definition of Pressure
John
it is the force per unit area of a substance.S.I unit is Pascal 1pascal is defined as 1N acting on 1m² area i.e 1pa=1N/m²
Emeh
pls explain Doppler effect
Emmex
solve this an inverted differential manometer containing oil specific gravity 0.9 and manometer reading is 400mm find the difference of pressure
Einstine claim that nothing can go with the speed of light even its half (50%) but in to make antimatter they they hit the sub atomic particals 99.9%the speed of light how is it possible
nothing with physical properties. this doesn't include things like particles and gravitational waves
Mustafa
that particles are of very small mass.... near equals to massless
Aritra
but they exist
vijay
yes they exist but mass is too less
Aritra
ok
vijay
greet all
Abayomi
the unit of radioactivity is .....?
Neyaz


text:
Minimal free resolutions of sub-permanents

# On minimal free resolutions of sub-permanents and other ideals arising in complexity theory

## Abstract.

We compute the linear strand of the minimal free resolution of the ideal generated by sub-permanents of an generic matrix and of the ideal generated by square-free monomials of degree . The latter calculation gives the full minimal free resolution by [1]. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.

###### Key words and phrases:
Computational Complexity, Free Resolution, Determinant, Permanent
###### 2001 Mathematics Subject Classification:
68Q17, 13D02, 14L30, 20B30
Landsberg supported by NSF DMS-1405348
Schenck supported by NSF DMS-1312071
Weyman supported by NSF DMS-1400740

## 1. Introduction

We study homological properties of two families of ideals over polynomial rings: the ideals generated by square-free monomials of degree in variables and the ideals generated by sub-permanents of an generic matrix. Recall that the permanent of an matrix is the polynomial

 permm(Y)=∑σ∈Smy1,σ(1)y2,σ(2)⋯ym,σ(m),

where denotes the symmetric group on elements.

We obtain our results via larger ideals (resp. ). The ideal is generated by all monomials of degree in variables. The ideal is generated by permanents of matrices produced from where repetition of rows and columns is allowed. Invariantly, is the ideal generated by and is the ideal generated . The main result in each case says that the linear strand of resolution of (resp. ) is the subcomplex of the linear strand of the resolution of (resp. of ) consisting of elements of regular weights (cf. §2).

Our motivation comes from complexity theory. We seek to find differences between the homological behavior of ideals generated by minors (i.e., subdeterminants) of the generic matrix and the ideals generated by subpermanents. The ideal generated by square-free monomials arises as the -th Jacobian ideal of the monomial .

### Acknowledgments

We thank the anonymous referee for numerous useful suggestions, including simplifications of several proofs. Efremenko, Landsberg and Weyman thank the Simons Institute for the Theory of Computing, UC Berkeley, for providing a wonderful environment during the fall 2014 program Algorithms and Complexity in Algebraic Geometry to work on this article.

## 2. Preliminaries

### 2.1. Representation Theory

For proofs of the statements here, see, e.g., [3] or [9, Ch. 2]. We work exclusively over the complex numbers , although our results hold for an arbitrary field of characteristic . If is a –vector space of dimension , a choice of basis determines a maximal torus of diagonal matrices and a labeling of weights for the torus by -tuples . A weight is dominant if . Irreducible representations of are in one-to-one correspondence with dominant weights . Let denote the irreducible representation associated to . Write for the size of . A weight is regular if each is equal to or . The regular weights will play an important role in stating our results.

When is a dominant weight with , we say that is a partition of , and we write . When dealing with partitions we often omit the trailing zeros. Associated to a partition is its Young diagram which consists of left–justified rows of boxes, with boxes in the –th row: for example, the Young diagram associated to is

 \Yvcentermath1\yng(5,2,1).

The transpose of a partition is obtained by transposing the corresponding Young diagram. For the example above, .

Given finite dimensional –vector spaces , the Cauchy formulas describe the decomposition of the symmetric and exterior powers of into a sum of irreducible –representations, see, e.g., [9, Cor. 2.3.3]:

 (1) Symn(F⊗G)=⨁λ⊢nSλF⊗SλG, n⋀(F⊗G)=⨁λ⊢nSλF⊗Sλ′G.

Let denote the ideal generated by .

### 2.2. GLm and Sm representations

Let equipped with its standard basis. The symmetric group is then contained in as the permutation matrices. Consider the irreducible representation where is a partition of . Inside of we have the -submodule spanned by the elements of regular weight . This submodule is denoted , the Specht module corresponding to . The representations are the distinct irreducible representations of (see, e.g., [5]). Write

 (SλCm)reg=[λ].

Recall that for finite groups , and an -module , is the induced -module. For and ,

 (SλCn)reg≡IndSnSm×Sn−m([λ]⊗[n−m]).

Introduce the notation , and if is a partition of , write for the module that is as an -module and trivial as an -module.

### 2.3. Finite free resolutions

Let be the ring of polynomials in variables equipped with its grading by degree. Let denote its -th graded component. Let denote the maximal ideal . Let be a graded -module. A complex of free graded -modules

 F:0→FNdN→FN−1→⋯→F1d1→F0

is a minimal free resolution of if the only homology of is , and are maps of degree such that for all .

Define the graded Betti numbers of by

 Fi=⊕j≥0S(−i−j)βi,j

where denotes a copy of with generator in degree .

If is an ideal generated in degree , then . The linear strand of is a subcomplex

 Flin:0→FlinNdN→FlinN−1→⋯→Flin1d1→Flin0

where . The graded Betti numbers have the interpretation in terms of functors

 βi,j=dimCTorSi(S/S+,M)i+j

where the subscript denotes the homogeneous component. This means that can be calculated as

 (2) TorSi(S/S+,M)i+j=Hi(C(x1,…,xN;M))i+j

where is the Koszul complex defined by and the differential by

 d(ej1∧⋯∧eji⊗m)=i∑u=1(−1)u+1ej1∧⋯∧^eju∧⋯∧eji⊗xjum.

## 3. The linear strands of the minimal free resolutions of the ideals Ipermn,k

### 3.1. The resolution of I1×k(n,n)

The ideal is the ideal generated by , where . The minimal free resolution of this ideal is known (see, e.g., [7], where it is denoted by ). The linear components of this resolution are generated by

 (3) F––linj=⨁a+b=jS(k+b,1a)E.

So the -th linear term is .

Since the resolution is -equivariant, each module in the complex has a double weight decomposition induced by the restricted action of pairs of diagonal matrices.

### 3.2. The main result

We work over .

Define a sub-complex of the complex given by (3) by setting to be the subspace of spanned by the basis elements of regular content. Note that is indeed a sub-complex of . Let , denote the span of the first basis vectors.

###### Theorem 3.1.

When , the complex is the linear part of the minimal free resolution of the ideal . Moreover, .

As an -module,

 (4) H––linj=IndSn×Sn~Sκ+j×~Sκ+j(⨁a+b=j˜[κ+b,1a]Eκ+j⊗˜[κ+a,1b]Fκ+j).
###### Proof.

Consider the complex giving the linear strand of the ideal . The term consists of the generators of , the space .

Inside is the ideal generated by the sub-permanents which consists of the subspace of regular weights. Note that the set of regular vectors in any (where we assume in order for the set of such vectors to be nonempty) spans a -submodule.

The linear strand of the -the term in the minimal free resolution of the ideal is also a -submodule of . We claim this sub-module is generated by the span of the regular vectors. In what follows denotes the sub-permanent formed from rows and columns .

We work by induction, the case was discussed above. Assume the result has been proven up to homological degree and consider the homological degree and homogeneous degree . The generators of the -th module in the linear strand of the resolution of have to be contained in linear part of , so all its weights are either regular, or such that one of the row indices is , and/or one of the column indices is , and all other are zero or . Call such a weight sub-regular. It remains to show that no linear syzygy with a sub-regular weight can appear. To do this we show that no sub-regular weight vector in maps to zero in .

First consider the case where both the and weights are sub-regular, then (because the space is a -module), the weight must appear in the syzygy. The only way for this to appear is to have a term divisible by . But, since is not a zero-divisor in , such a term cannot map to zero because our syzygy is a syzygy of degree zero multiplied by . But by minimality no such syzygy exists.

Finally consider the case where there is a vector of weight appearing. Here it is more convenient to look at the calculation of the free resolution using the Koszul complex. Such a syzygy would give a Koszul cycle with summands of the form

 (5) z=∑tatea1,t,b1,t∧⋯∧eaj,t,bj,t⊗p(It;Jt)

where are subpermanents formed from distinct rows and columns and . The total weight is and the Koszul differential is zero. Consider this differential. The coefficients of all the basis elements of have to be zero. There are three kinds of basis elements: the indices can contain number twice, once or not contain at all. Consider the basis elements not containing , say the element . The only elements that can appear on the right hand side of the tensor product in are the elements , for .

###### Lemma 3.2.

Let . The elements , for are linearly independent in .

###### Proof.

After re-labeling, the lemma amounts to showing the polynomials are linearly independent, where is the permanent of the matrix obtained by removing the -th column of

 ⎛⎜ ⎜ ⎜ ⎜⎝x1,1x1,2…x1,k+1x2,1x2,2…x2,k+1…………xk,1xk,2…xk,k+1⎞⎟ ⎟ ⎟ ⎟⎠.

Say that

 (6) k+1∑s=1bsx1,sPs=0

for some scalars . We need to show that all are zero. By symmetry it suffices to show that . So set for . Using the Laplace expansion of permanents along the first row, and writing for the permanent obtained by removing row and columns we can rewrite (6) as

 b1x1,1(x1,2P1,2+x1,3P1,3)+b2x1,2(x1,1P1,2+x1,3P1,3)+b3(x1,1P1,3+x1,2P2,3)=0

which gives for , so . ∎

Lemma 3.2 implies that in all the summands in of (5) with , both appearances of the index have to occur among . Now consider the coefficient of the basis element where occurs among , say, in . We obtain a linear combination of the elements for . But these elements are trivially linearly independent (all monomials occurring in them are different) so all coefficients are zero.

The rest of Theorem 3.1 follows because if is a partition of then the weight subspace of , considered as an -module, is (see, e.g., [5]), and the space of regular vectors in is . In the formula for the dimension of , the factor is explained by inducing. The dimensions of hook Specht modules are binomial coefficients, So we need to prove that

 ∑a+b=j(k+j−1a)(k+j−1b)=(2(k+j−1)j)

This has a combinatorial explanation. Given balls, white and black, both sides of the equation calculate number of choices of of them: the left side partitions into how many white (a) and how many black (b) are chosen. ∎

###### Remark 3.3.

For small and , computer computations show no additional first syzygies on the sub-permanents of a generic matrix (besides the linear syzygies) in degree less than the degree of the Koszul relations . For example, for and , there are cubic generators for the ideal and minimal first syzygies of degree six. There can be at most Koszul syzygies, so there must be additional non-Koszul first syzygies.

## 4. The minimal free resolutions of the ideals Isqf;n,k

### 4.1. The resolutions of I1×k(1,n)

Next we consider the case , . In this case and the ideal is just the ideal generated by all monomials of degree . The resolution of this ideal is well-known, see, e.g., [7] or [2].

The whole resolution is linear and -equivariant, and its -th term is

 (7) Fj=S(k,1j)F⊗S(−k−j).

### 4.2. The resolution of Isqf;n,k

We work over .

Define a subcomplex of the complex given by (7) by setting to be the subspace of spanned by the basis elements of regular weight. Note that

 Hlin=⊕jH––linj⊗S(−k−j)

is indeed a subcomplex of .

###### Theorem 4.1.

The complex is the linear part of the minimal free resolution of the ideal . We have the -module decomposition

 (8) H––linj=IndSn~Sκ+j˜[κ,1j],

which has dimension .

###### Proof.

We want to show that is the linear strand of the resolution of . We proceed by induction on . The case is clear because the generators of are precisely the generators of with regular weights. Assume we proved the result for and consider the -th module in the resolution. As with the subpermanent case, it is enough to consider the elements of a subregular weight as linear relations between elements of regular weight are either of regular or subregular weight.

Consider the syzygies in homological dimension and in homogeneous degree in term of cycles in the Koszul complex . These will be cycles of the form

 z=∑tatea1,t∧…∧eaj,t⊗xu1,txu2,t…xuk,t

where the total weight is and all monomials are of regular weights, and are scalars. In each summand with we are forced to have one among and one among . So we can assume that in each summand with we have and . We have . But, looking at the coefficient of with respect to the basis vector we see that its coefficient is just which forces to be zero. The dimension formula follows as in Theorem 3.1. ∎

###### Remark 4.2.

The easiest way to see that the resolution of the ideal is linear and to see the ranks of the modules is to observe that for the matrix

 M(A,X)=⎛⎜⎝a1,1x1…a1,nxn………ak,1x1…ak,nxn⎞⎟⎠

where

 A=⎛⎜⎝a1,1…a1,n………ak,1…ak,n⎞⎟⎠

is a matrix of scalars with all maximal minors non-zero, the ideal of maximal minors of the matrix is just , so the resolution in question is an Eagon-Northcott complex ([2] or [9, 6.1.6]). After this paper was submitted, a characteristic free description of the resolution of , with explicit differentials appeared in [4].

For , let denote the ideal generated by the partial derivatives of of order .

### 5.1. Size two subpermanents

###### Theorem 5.1.

Let denote the degree component of the ideal generated by the size two sub-permanents of an matrix, so . Then

 dimC[xi,j]/Ipermn,2t= (n2+t−1t)−[(nt)2+n2+(t−1)((n22)−(n2)2)+2(t−12)((n2)2+n(n3)) +2nt−1∑j=3(t−1j)(nj+1)],

Where recall that for , in which case the formula is minus the value of the Hilbert polynomial at .

First, the Hilbert polynomial:

###### Theorem 5.2.

For the ideal of permanents of an matrix, the Hilbert polynomial of is

 (9) n∑i=0fi(t−1i),

where is the entry in the vector

 [n2,(n22)−(n2)2,2(n2)2+2n(n3),2n(n4),2n(n5),…,2n(nn)].
###### Proof.

[6, Thm. 3.2] gives a Gröbner basis for , the radical of , and by [6, Thm. 3.3], has finite length, so vanishes in high degree. The Hilbert polynomial only measures dimension asymptotically, so

By [6], for any diagonal term order, the Gröbner basis for is given by quadrics of the form

 xijxkl+xkjxil with i

and five sets of cubic monomials

 (10) xi1j1xi1j2xi2j3i1>i2j1i2j1j2xi1j1xi2j2xi3j2i1j2xi1j1xi2j2xi3j3i1j2>j3.

The key observation is that all the cubic monomials are square-free, as are the initial terms of the quadrics. Thus the initial ideal of is a square-free monomial ideal and corresponds to the Stanley-Reisner ideal of a simplicial complex . By [8, Lemma 5.2.5], the Hilbert polynomial is as in Equation (9), where is the number of -dimensional faces of . As the vertex set of corresponds to all lattice points with , it is immediate that .

Since is a non-face if , , no edge connects a southwest lattice point to a northeast lattice point. Hence, the edges of consist of all pairs with and , of which there are .

Next, consider the triangles of . Equation (10) says there are no triangles in of the types in Figure 1. Also, there are no triangles which contain an edge connecting vertices at positions and with ,. Thus, the only triangles in are right triangles, but with hypotenuse sloping from northwest to southeast. For a lattice point at position there are exactly right triangles having as their unique north-most vertex. In the rightmost column , there are no such triangles, in the next to last column there are such triangles. Continuing this way yields a total count of

 (n−1)(n2)+(n−2)(n2)+⋯2(n2)+(n2)=(n2)2

such right triangles, and taking into account the right triangles for which is the unique south-most vertex doubles this number.

However, this count neglects thin triangles–those which have all vertices in the same row or column. Since the number of thin triangles is , the final count for the triangles of is

 2(n2)2+2n(n3).

For tetrahedra, the conditions of Equation (10) imply that there can only be thin tetrahedra, and an easy count gives such. The same holds for higher dimensional simplices, and concludes the proof. ∎

###### Corollary 5.3.

For the ideal of permanents of an matrix, the Hilbert function of is, when ,

 (11) HF(Sym(V)/Ipermn,2,t)=(nt)2+HP(Sym(V)/Ipermn,2,t),

and it equals the Hilbert polynomial for .

###### Proof.

The Hilbert function of in degree is by [6, Thm. 3.3]. The result follows by combining Equation 9 with the short exact sequence

 0⟶√Ipermn,2/Ipermn,2⟶Sym(V)/Ipermn,2⟶Sym(V)/√Ipermn,2⟶0,

and additivity of the Hilbert function. ∎

For the purposes of comparing with other ideals, we rephrase this as:

###### Corollary 5.4.

. For :

 dimIpermn,2t= (n2+t−1t)−[(nt)2+n2+(t−1)((n22)−(n2)2)+2(t−12)((n2)2+n(n3)) +2nt−1∑j=3(t−1j)(nj+1)].

### 5.2. Hilbert functions for ideals of square-free monomials

Although these can be deduced from our resolutions, we present the Hilbert functions and polynomials for the ideals generated by square-free monomials.

###### Proposition 5.5.

The Hilbert function of in degree is

 (12) dimI(x1⋯xn),κκ+t=n−κ∑j=0(nκ−j)(κ+t−1κ+j−1)
###### Proof.

The ideal in degree has a basis of the distinct monomials of degree containing at least distinct indices. When we divide such a basis vector by the denominator will have degree at most . For each , the space of possible numerators with a denominator of degree that is fixed, has dimension , and there are possible denominators. Summing over gives the result. ∎

For the Hilbert function of the coordinate ring, we have the following expression:

###### Proposition 5.6.

The Hilbert function of in degree is

 (13) dim(Sym(Cn)/Ix1⋯xn,κ)t=n−κ−2∑j=0(nj+1)(t−1j),

if , and if .

### References

1. Riccardo Biagioli, Sara Faridi, and Mercedes Rosas, Resolutions of De Concini-Procesi ideals of hooks, Comm. Algebra 35 (2007), no. 12, 3875–3891. MR 2371263 (2008i:13018)
2. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995, With a view toward algebraic geometry. MR MR1322960 (97a:13001)
3. William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991, A first course, Readings in Mathematics. MR 1153249 (93a:20069)
4. Federico Galetto, On the ideal generated by all squarefree monomials of a given degree, arXiv:1609.06396
5. David A. Gay, Characters of the Weyl group of on zero weight spaces and centralizers of permutation representations, Rocky Mountain J. Math. 6 (1976), no. 3, 449–455. MR MR0414794 (54 #2886)
6. Reinhard C. Laubenbacher and Irena Swanson, Permanental ideals, J. Symbolic Comput. 30 (2000), 195–205. MR 1777172 (2001i:13039)
7. Claudiu Raicu and Jerzy Weyman, The syzygies of some thickenings of determinantal varieties, Proc. Amer. Math. Soc. 145 (2017), 49–59. MR MR3565359
8. Hal Schenck, Computational algebraic geometry, London Mathematical Society Student Texts, vol. 58, Cambridge University Press, Cambridge, 2003. MR 2011360 (2004k:13001)
9. Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR MR1988690 (2004d:13020)


text:
# How do you find the reciprocal of 6/7?

May 6, 2017

Therefore, the reciprocal of $\frac{6}{7}$ is $\frac{7}{6}$


text:
# The Everything of Theory

## Abstract

In this paper we show that ${\cal N}={\bf R^3_{mnpq}}/4$, where ${\cal N}$ means the number of supersymmetries, {\bf R} means the curvature tensor, and \$ means we don't know how to convert TeX back to ordinary text.

That's not really what this paper is about, but we needed to pad out the abstract, since nowadays abstracts are longer than most introductions. That's because nobody reads papers anymore (unless someone tells them to). Unfortunately, we can't fit anything interesting into the Abstract, because arXiv.org refuses to let us include sound effects, movies, and odors.

In this paper we actually explain the title, authors, institution, and abstract. Next comes some propaganda about how exciting and wonderful some recent developments are. Then we ramble on about some other things bothering us, like why our papers don't get cited more often. (Hey, you know how to use URL's, don't you?) Then there's some physics in there somewhere. Finally, we conclude with an Appendix, Glossary, and References, since that's all that people read anyway (assuming they stayed awake through this Abstract).

## Introduction

Hi. Nice to meet you. Our names are V. Gates, M. Roachcock, E. Kangaroo, and W.C. Gall. How do you like this weather we've been having? Wasn't that a great game last night? No, and we didn't vote for him, either. Well, maybe they'll increase funding next year. Yes, our doctor told us the same thing. That's the court system for you! Ha, ha, yes, we have heard that one, but it's funny every time we hear it. It's down the hall and around the corner.

Well, nice talking to you. Hope to see you again sometime. We'd like to stay and chat some more, but we have a paper to write. Bye for now.

## Acronyms

String theory is often called the "Theory of Everything" (TOE). However, since so far it makes no predictions observable by experiment, a better name might be the "Theory of Everything Not Appearing in Laboratories" (TOENAIL).

This is not string theory's fault; high energy physics now takes a lot more energy than most people have. The real problem with string theory is that there is no alternative. However, the reason there is no alternative is that no one ever bothers to look for one; in fact, there is a strong resistance to even considering looking for one. Consequently, practically all theoretical high energy physics (and even most of phenomenology) is now string theory. Thus, string theory is not so much the Theory of Everything (since it explains nothing), but rather the "Everything of Theory", since it now encompasses all of theory. This era in string research is strongly reminiscent of the Dutch tulip trade just before the Tulip Crash of 1637.

The main reason string theory is incapable of explaining anything is that it has no predictive power. But even if it could predict anything, it would be only at the Planck scale, which is way beyond experiment of the foreseeable future. As a result, string theory has become boring. One solution might be to work on something that has more to do with observable energies. (E.g., some have proposed that the effective Planck scale might be at much larger distances, say, a micron. This would allow for accessible experimental predictions, such as gravitational collapse of microbes.)

We propose here a different alternative: Apply string theory to energies beyond the Planck scale, which are not observable even in principle. Then one avoids the objection of the untimeliness of string theory, since it never will be relevant, and thus one may as well work on it now as later.

## D=11 is better than D=10

In the beginning superstrings were assumed to be in 4 dimensions. Then they were found to require 10 dimensions. Originally this was thought to be bad, but then it was decided that it was good. But maximal supergravity liked 11 dimensions, so (non-stringy) supergravity was bad. Finally, superstrings were discovered to be M theory in 11 dimensions. So then it was decided that 11 dimensions was good.

It is now known for certain that both Taipei & Type B(eijing) superstrings are dual to M theory. However, we still do not have the slightest idea what M theory is.

 Strings and M-balls
In fact, the most active research in the area is trying to understand what the "M" stands for. Originally it was thought to stand for "Membrane", but soon it was realized that the Membrane cannot be quantized, so that possibility was rejected. Another possibility was "Matrix", but then it was remembered that Matrix theory works only in D≤2. Finally, it was decided that "M" stands for "Mother", and so far no one has dared to disobey. ("F" theory was briefly considered also, but clearly strings form a matriarchical society.)

Calculations in M theory require making Lorentz boots to the infinite momentum frame. Of course, Lorentz invariance means physics is the same in all frames, but at this speed we are too dizzy to reject hand-waving arguments [we got tired of renumbering references, so guess].

## Branes and GUTs

Various methods have been used in an attempt to apply stringbrane theory to f(enomenol)udgy. One recent approach is compactification on AdS5⊗S5. (According to the Einstein summation convention, sum over 5=0,1,2,3,4). This produces supersymmetric QCD on a pročektive superspace. The results for masses of glueball states totally agree with results from lattice QCD, thus obtaining confirmation of both theories without recourse to experiment.

The details of the calculations have been (/are being/will be) presented elsewhere; here we give cartoons. Since lattice QCD works in Euclidean space, we Wick rotate

The usual OscarWilder-Schraeder conditions are then applied to preserve seepy tea invariance. Holography then follows from the mysterious relation

tea:four::tee:fore

The continuum limit is ambiguous, as always, but we assume all variations of the theory are members of the same Univers(al)ity class. In any case, we hide such ambiguities by use of the subtle-point approximation. Curvature is flattened by coupling it to the Frankensteinlin term, which scares away the ghost current.

Another advantage of branes is that the extra ones can account for the duck matter needed to explain the missing mass, as well as the duck tape (cosmic strings) that hold the compactified dimensions together. Of course, to account for the recent discovery of the nonvanishing cosmological constant, one needs the usual fine tuning to order yottayottayotta=1072.

## Fractalons

Recent cosmological observations have indicated that α=e2/ħc varies with time. The obvious interpretation is that 2 is not a constant. This may be related to the classical limit

[q,p] = iħ → 0     in   lim i → 0

 Every year asymptotic freedom is celebrated with the running of the couplings.
One theoretical justification is extra dimensions: It is known that gauge couplings are dimensionless only in D=4. However, experimentally they are seen to have logarithmic corrections in terms of energy (and thus time), indicating nonvanishing dimension: In D dimensions, gauge couplings have energy dependence

α∼ ED-4

and thus in fractal dimension D=4-ε we have roughly

α ∼ E ≈ 1 -ε ln E

(in the case of asymptotic freedom). Thus, the Standard Model is dual to a model in 4-ε dimensions, where the quantum running of the coupling in the Standard Model appears already classically in the latter model. Note that ε itself does not run, since

dimension(dimension)=0

Of course, we introduce nonabelian fields for gluons because we want to gauge their reaction.

Similarly, in this new model there are "ε-scalars" that can take the place of Higgs fields, just as in monopole solutions of pure Yang-Mills theory, where the time component of the 4-vector potential acts as an adjoint Higgs. Since there are only ε of these Higgs's, the probability of their production is correspondingly reduced, explaining the Higgs's's apparent experimental absence.

This model is similar to the usual braneworld scenarios, with 2 important differences:

1. The number of extra "bulk" dimensions is much smaller than 1.
2. The number of extra bulk dimensions is negative.

## Conclusions

That's it. It's the end. We have to stop now. You can stay and read the Appendices if you want. Thanks for coming. Don't forget your coats. Have a κν year!

## Appendix: How to get a job in physics

### I. Be born a genius

If you are born absolutely brilliant, then everyone will want you. This will require some advanced planning, starting before birth, and maybe before conception. However, few people plan their careers till high school, which is 15 years too late. Unfortunately, it's not clear that matchmaking or even genetic engineering will accomplish this goal: How many of Einstein's relatives won Nobel prizes?

### II. Be in the right place at the right time

If you go to an "in" university, you can get letters of recommendation from an "in" professor, and maybe even get one as an advisor. Of course, just because the university is "in" when you enter grad school doesn't mean it still will be when you graduate. Also, "in" graduates tend to get hired by other (or even the same) "in" universities, since they are victims of their own hype, and we all know how in-breeding affects the gene pool.

### III. Work on popular topics

If you work on just the topics that are most popular at the time, you might also become popular. This requires quick reflexes, since topics often lose popularity as soon as people read the first paper. If you write a paper on the topic before it becomes popular, no one will care, and they won't bother referencing you even when it does become popular. On the other hand, if you write a paper more than a week after it first becomes popular, at the very best your paper will become buried at the end of a long list of references. Furthermore, all that is relevant is that the paper be popular while you are being evaluated for a position, or at least while your letters of recommendation are being written, so if you are really clever you can work on anything you want until your last year in graduate school.

### IV. Give a lot of seminars

If you speak everywhere you are applying at the same time you are being considered, your name has a better chance of being remembered. You might also include your photo in your application, for people who remember faces better than names.

## Glossary

algoraphobia: fear of wide-open races
Cosmo0: Big Bang
electra: plural of "electron"
Hips mechanism: eating destroys symmetry
open-sores software: programs by acne-ridden geeks
well-known: a conjecture that we have no idea how to justify
yymmnnnn: horses in Gulliver's Travels

## References

[1] L. Lorenz, Spell my name right, Philos.Mag. 34 (1867) 287.
[2] Lawrence Sklar, I am not spinless, U of M (-theory) preprint.
[∞] E. Soccerchief, D=10 is D=11, but N=3 is N=4, ν.φ. B52 (2002) 123.
[0,1] Eager Claponoff, d(brane)/dk, Plasma Physics Lab preprint.
[q,p] Enrico ("4") Fermi, A better length than Planck's, my-lab preprint.
Bare Einstein, Moldy Cheeses, and Horseshoe Nasty, pp waves and poopoo waves, disgusting preprint.
[1']
 L.N. Con, Fuzzy Wuzzy was a sphere, FUZZY FUZZY preprinttnirperp.


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# Partial Orders

• September 6th 2010, 09:01 AM
usagi_killer
Partial Orders
Hi there, I have a question relating to partial orders, I can prove only part of the question but can't seem to finish it off, any help would be greatly appreciated!

Let T denote the set of real-valued functions that are differentiable at the point x=0 we define a relation R on T by 'f R g iff $f'(0) \le g'(0)$'. Does R give rise to a partial ordering of T? If not, then which of reflexive, antisymmetry, transitive fails?

So for R to be a partial order on T then R must be reflexive, antisymmetric and transitive.

I can show R is reflexive, for example, let h be any real valued function that is differentiable at the point x=0, ie, $h \in T$

Thus $h'(0) \le h'(0)$ so $(h,h) \in R$ and thus R is reflexive.

I am unsure of how to go about showing R is antisymmetric and transitive (or how to go about showing if one of these fails)

Thanks again!

-----------------------------------------------------------------------------------------

Actually after thinking more, I can show R is a partial order but the answer says it's not?

I showed R is antisymmetric like this:

If $(f, g) \in R$ and $(g,f) \in R$ then it is clear that f'(0)=g'(0)

I also showed R is transitive like this:

If $f'(0) \le g'(0)$ and $g'(0) \le z'(0)$ then $f'(0) \le z'(0)$ and thus $(f, z) \in R$

So where in my working have I gone wrong...? :S
• September 6th 2010, 09:17 AM
Plato
Consider the functions $f(x)=x^2~\&~g(x)=cos(x)$.
Is the relation antisymmetric?
i.e. Does $f=g?$
• September 6th 2010, 09:20 AM
emakarov
Why don't you start by writing what it means for any relation R on T to be antisymmetric and then substituting the actual definition for the letter R in that expression? You will get a proposition of the form, "For every two functions f and g in T, ...". Then we can decide whether it is true or false.
• September 6th 2010, 09:28 AM
usagi_killer
Oh right, sigh, what a careless mistake by me, thank you Plato (you're always helping me ^_^) and emakarov.

I assume my reasoning behind reflexivity and transitivity are valid...?
• September 6th 2010, 09:33 AM
emakarov
Quote:

I assume my reasoning behind reflexivity and transitivity are valid...?
Yes.