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Jillian Boustred is an eponymous, Sydney based fashion label founded in 2015. Leading the way in locally made and ethical practices, Jillian Boustred has produced the majority of her collections in Australia since its founding.
The label has grown and evolved over the years but has always retained a sharp focus on designing classic, flattering styles and staple separates - that will live in your wardrobe for years.
Our vision at Jillian Boustred is to be able to celebrate fashion and the expression it brings whilst maintaining a sense of responsibility and authenticity as a socially conscious brand. Thus, the ethos of the brand is formed on three key intentions:
Responsible | Transparent | Inclusive
As the future of fashion unfolds, we believe it is our job to question and deconstruct the norm of accepted fashion processes and establish new ones to help pave a more responsible way forward; a path that values the protection of people and our planet while honoring the inherent creativity and individual expression that comes with the art of fashion.
Jillian Boustred
Founder and Director
Jillian Boustred is a Sydney based fashion designer. Passionate about design, art and making from an early age, Jillian's path into the fashion industry was intuitive and organic. Jillian studied a Bachelor of Design in Fashion and Textiles at the University of Technology Sydney where she majored in Womenswear. During her studies she interned and worked for various Sydney based fashion labels. Following her degree, Jillian went on to do a brief stint in the sampling team at Australian fashion label Ellery. With industry experience under her belt, Jillian launched her first collection in 2015 under the label 'Jillian Boustred', initially working out of her Annandale share house before moving into a creative co-working space in Alexandria. For the last four years the brand has been based in Chippendale. The slow and steady growth of the label has allowed Jillian to perfect her styles, fabrication and fit.
A note from JB:
"Australian made, timeless styles with a contemporary feel and natural fabrication are the core of the label. I have always loved exploring shape and silhouette and designing styles that compliment women's curves. It's so important to me that our dresses are worn over and over again and become "go to" pieces in our customers wardrobes, for me the best way to achieve this is through combining comfort, fit and fabrication" - Jillian
Gemma Geyer
Head of Production and Sustainability
Gemma Geyer is a maker and manager with a compelling passion for people and the planet. Gemma studied a Bachelor of Design (Honours) at the University of New South Wales' College of Fine Arts, majoring in Textiles with a focus on sustainable practices. After working with a number of independent designers and small businesses across the globe (most notably as a seamstress for slow fashion label Elizabeth Suzann in Nashville USA), Gemma feels a conscious responsibility to address the growing environmental and human-centric concerns of the textile and apparel industry. Heading up our Production and Sustainability team, Gemma loves answering your customer care emails, sourcing responsible fibres and fabrics, liaising with our makers across Sydney, researching new innovations in the industry, and occasionally whipping up a tote bag or two from our scrap fabrics!
A note from GG:
"I believe that close relationships between makers, designers and consumers imbue a piece of clothing with value and longevity. I am passionate about making an impactful difference on a local scale and have oriented my career to reflect this. Responsibility for the art of mindful wearing begins in the design and production process and is why I have aligned my values with the ethical management of people and products." - Gemma
Our Australian Made Network
Our team is also made up of a network of Australian makers and suppliers who we have worked with since 2015. Read our Australian Made Journal here.


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Lira, eToro's New Open-source Programming Language
Place/Date: - September 16th, 2019 at 8:38 am UTC · 6 min read
Contact: eToro, Source: eToro
eToro, the global multi-asset investment company, has today released the details of Lira, a new open-source programming language for financial contracts. Lira is the first step in bringing the $500 trillion OTC derivatives market onto the Blockchain by introducing a new formal contract language.(i)
Lira is a domain-specific language that can be used to write OTC financial contracts for assets currently on the Ethereum blockchain. It is both secure and easy to programme whilst guaranteeing self-executing global settlement and automated trade reporting and monitoring. It's easy tracking and compression will enable better collateral requirement efficiencies.
Furthermore, it is the first decentralised finance (DeFi) project that will look to expand the code across different blockchains now that it is open-sourced.
Yoni Assia, CEO and Co-Founder of eToro, commented:
"Bringing the OTC derivatives market onto the blockchain will bring more transparency and capital efficiencies to the industry. Activities in the post-trade cycle, such as settlement and the clearing of derivatives, are both expensive and a source of systemic risk. We believe that blockchain technology can provide a secure execution environment in which settlement is guaranteed by design. That is why today we are introducing a new formal contract language – Lira. This has the potential to open up and transform the derivatives market."
At present, native blockchain scripting languages are subject to broad design requirements. This makes them error-prone (as seen for example by the Dao Hack) and thus unsuitable for writing financial contracts. The flexibility that is offered by the native language is also a potential security threat, presenting an opportunity for attackers.
Professor Omri Ross, eToro's Chief Blockchain Scientist, said:
"45% of smart contracts on the Ethereum blockchain have errors because of the broad nature of their design. Lira is built on formal verification, a mathematical equation that proves the language will deliver the promised outcomes."
"We have decided to make this language open source so that the community can help to move forward what we hope will be an important addition to DeFi. We envision this code to be developed further across different blockchains and will include algorithms to help reduce collateral burdens thereby helping to increase liquidity in the derivatives market. By being able to write financial contracts on the blockchain, Lira could open up the derivatives market and decentralise one of the most important financial industries in the world."
Lira was developed by eToroX Labs. Led by Professor Omri Ross, eToro's Chief Blockchain Scientist, it acts as an internal innovation unit tasked with the research and development of infrastructure for the representation of assets and the execution of trade processes on blockchain infrastructure. Omri will be hosting a two hour workshop session on Lira at the forthcoming Devcon in Osaka in October.
eToro has a history of supporting the development of the crypto community with projects such as Coloured Coins and GoodDollar. Lira is another contribution to the blockchain community who support this latest project.
Greg Dipriso, Global Head of Business Development at MakerDAO, said,
"Lira is a great step in opening the Defi space for new products suited for servicing the OTC market."
Nicolas Arqueros, CTO of Emurgo, added,
"Lira is based on functional language and formal verification, we believe these will be an essential part in the future of blockchain technology and financial protocols."
Yoav Weiss, CTO of Tabookey, commented,
"The full potential of blockchain technology for financial markets has not yet been realised. Lira presents a practical solution for the derivatives market and is underpinned a strong team. We'll be following the project's development closely."
Derivatives are at their most basic, a contract that derives its value from the performance of an underlying entity (asset, index or interest rate).
(i) Source: Bank for International Settlements, 2019: https://www.bis.org/publ/otc_hy1905.htm
About eToro
eToro was founded in 2007 with the vision of opening up the global markets so that everyone can trade and invest in a simple and transparent way. The eToro Group consists of the eToro platform, our multi-asset trading and investment venue, and eToroX, which manages our crypto wallet and exchange.
The eToro platform enables people to invest in the assets they want, from stocks and commodities to cryptoassets. We are a global community of more than ten million registered users who share their investment strategies; and anyone can follow the approaches of those who have been the most successful. Due to the simplicity of the platform users can easily buy, hold and sell assets, monitor their portfolio in real time, and transact whenever they want.
As technology has evolved, so has our business. In 2018, we created eToroX, our tokenized asset subsidiary. eToroX provides the infrastructure, in the form of a crypto wallet and the forthcoming exchange, that supports our commitment to facilitating the evolution of tokenized assets. We believe that leveraging blockchain technology will enable us to become the first truly global service provider allowing everyone to trade, invest and save.
About eToroX
eToroX is a subsidiary of the eToro Group. The business was formed in 2018 to provide the infrastructure to support eToro Group's commitment to facilitating the evolution of tokenized assets. It provides eToro's newly launched wallet and runs the exchange.
The distributed ledger technology (DLT) provider licence was granted by the Gibraltar Financial Services Commission to eToroX in December 2018 (licence number FSC1333B). eToroX is incorporated in Gibraltar with company number 116348 and its registered office is at 57/63 Line Wall Road, Gibraltar.
About eToroX Labs
Created in 2019, eToroX Labs is a hugely talented team at the forefront of blockchain exploration and innovation. Led by Professor Omri Ross, eToro's Chief Blockchain Scientist, it acts as an internal innovation unit tasked with the research and development of infrastructure for the representation of assets and the execution of trade processes on blockchain infrastructure.
eToro is regulated in Europe by the Cyprus Securities and Exchange Commission, regulated by the Financial Conduct Authority in the UK, and by the Australian Securities and Investments Commission in Australia.
eToroX is incorporated in Gibraltar with company number 116348 and its registered office is at 57/63 Line Wall Road, Gibraltar. It's distributed ledger technology (DLT) provider licence was granted by the Gibraltar Financial Services Commission in December 2018 (licence number FSC1333B).

redpajama_set_name: RedPajamaCommonCrawl


A thermoplastic film (100) having a substantially continuous adherent coating (110) on at least one surface, wherein the coating includes a polymeric binder and an additive comprising nano-scale particles (112). When the coating is formed on the surface of the film, the nano-scale particles (112) form a barrier that substantially reduces the diffusion of moisture and oxygen. In preferred embodiments, the coating includes a polymeric binder and silicate particles formed on a base substrate layer (120) that includes a polyolefin selected from the group consisting of homopolymers, copolymers and terpolymers of propylene, ethylene and butene and blends thereof.
This invention relates to coating compositions which can be applied to thermoplastic films to provide improved moisture barrier properties and good thermal stability. More specifically, the invention relates to coating compositions which use a polymeric binder and a nano-scale particle size additive to provide improved moisture barriers for thermoplastic films.
Polymeric films are used in many commercial applications. One particularly important application is the packaging of food products. Films employed in the food packaging industry are chosen and/or designed to provide characteristics necessary for proper food containment. Such characteristics include water vapor barrier properties, oxygen and gas barrier properties and flavor and aroma barrier properties. In order to impart these desirable characteristics in a multi-layer film, the films are often designed with one or more barrier layers. In addition to providing barrier properties, the layers must have acceptable clarity so that the products can be viewed through the film. Prior art films that used non-halogenic layers or coatings to provide barrier characteristics suffered from poor clarity and found limited use in the food packaging industry.
Polypropylene and polyethylene films are widely used in the food packaging industry because of their desirable physical properties, such as transparency and strength. However, polypropylene and polyethylene films do not provide barrier characteristics that are acceptable for many food packaging applications and in the past additional barrier layers have been used to provide polypropylene and polyethylene films with the required barrier properties.
Halogenic barrier materials are commonly used in the food packaging industry to impart moisture and gas barrier properties to films which do not inherently possess these characteristics. High barrier halogenic coatings or layers, such as polyvinyl alcohol (PNOH) and polyvinylidene dichloride (PVDC), have been used to provide moisture and gas barrier properties in thermoplastic films that include layers of biaxially oriented polypropylene (BOPP) and/or polyethylene terephathalate (PET). However, PNOH and PVDC are difficult to recycle because they are thermally unstable and tend to degrade rapidly under typical film reprocessing temperatures (350-550°F), (177-288°C). Furthermore, the halogenic barrier materials are relatively expensive and have processing problems that have limited their effectiveness. In particular, when films containing PVDC are irradiated to promote cross linking, they discolor above certain radiation levels. Thus, there has been a demand for a new barrier coating material that gives improved recyclability and improved processing characteristics.
Aqueous coating compositions containing, in addition to a water soluble or water dispersible film-forming polymer, one or more additives intended to impart a particular property or characteristic to the substrate to which the compositions are applied are known. The most commonly used coating compositions confer clarity, good slip and good anti-block characteristics upon the film. In addition, coatings containing halogenic polymers, such as PVDC, have been used to impart barrier properties in films but non-halogenic coatings have not been successfully used.
It has been discovered that the problems experienced in prior art films can be overcome by the present invention which provides a substantially continuous, adherent coating for at least one surface of a thermoplastic film that includes a polymeric binder and an additive containing nano-scale particles. The nano-scale particles are selected from silica, silicate, clay, organomontmorillonite, calcium carbonate, calcined aluminum silicate, hydrated aluminum silicate, calcium phosphate, alumina, barium sulfate, magnesium sulfate, and diatomacious earth. The nano-scale particles have an average particle size of from 10 nanometers to 100 nanometers and make up 5 to 20 weight percent of the additive.
The polymeric binder is an olefinic copolymer containing a carboxylic acid group or its metal salt-functionalized olefinic copolymer and a thermoplastic additive. The polymeric binder is selected from ethylene acrylic acid, ethylene methyl acrylate and ethylene ethyl acrylate copolymers. The thermoplastic additive makes up 40 to 90 weight percent of the coating and the polymeric binder makes up 10 to 60 weight percent of the coating.
The thermoplastic film can contain a base substrate layer that includes a polyolefin selected from homopolymers, copolymers and terpolymers of propylene, ethylene and butene and blends thereof. The coating can be applied to the surface of a single layer film, or it can be applied to a coextruded layer of a multi-layer film, using techniques well known to those skilled in the art, including by gravure coating, roll coating or spraying. The coatings of the present invention provide non-halogenic barrier materials for thermoplastic films that are environmentally acceptable because they are readily biodegradable. The coatings also provide a cost-effective alternative to the halogenic barrier materials used in prior art films without sacrificing the high performance required when packaging food products. In addition, the coatings can be applied after the film has been fabricated and so the processing problems experienced with coextruded films having halogenic barrier layers are avoided.
Figure 1 shows a side view of a prior art coating containing micron-scale particles.
Figure 2 shows a side view of a coating of the present invention containing nano-scale particles.
It has been discovered that a layered nano-scale particle dispersed in a polymeric binder provide excellent moisture barrier properties and good thermal stability. Dispersions containing particles of silicates and other materials that are presently in use employ micron- scale particles that are too large to form an effective gas or moisture barrier. The spaces between these micron-scale particles form sites that allow O or H O molecules to diffuse through the coatings with relative ease. In contrast, the nano-scale particles in the coatings of the present invention provide relatively few sites between the particles that can be occupied by O2 and H2O molecules and form an effective barrier layer.
While the coatings of the present invention are suitable for application to a wide variety of thermoplastic films, they are especially suitable for application to polypropylene and polyethylene films. Both polypropylene and polyethylene films are well known in the art and widely used in the food packaging industry. Polypropylene films can be the highly stereoregular or highly crystalline type. Preferred polypropylene films are formed by polymerizing polypropylene in the presence of stereospecific catalyst systems. Polypropylene films can also be blends of two or more different polypropylenes. Polyethylene films can be high density polyethylene, low density polyethylene, linear low density polyethylene, very low density polyethylene, or blends thereof. In addition, the films can be multiple layer films having different layers of polypropylene and/or polyethylene or layers that include blends thereof. The present coating composition includes, as a first component, an acrylic copolymer emulsion or solution. The first component includes at least 5 weight percent of a polar, functional comonomer based upon the total weight of the first component. Preferably, the polar functional comonomer can be present in an amount of 5 to 60 weight % and, most preferably, from 10 to 50 weight %. The polar, functional comonomer of the first component can include acrylic acid, methacrylic acid, hydroxyethyl acrylate, hydroxyethyl methacrylate, hydroxypropyl acrylate, hydroxypropyl methacrylate, crotonic acid, fumaric acid, itaconic acid and maleic acid. The principal advantage for having the polar functional comonomer present in this amount is that the carboxylic acid or hydroxyl functional groups are expected to have hydrogen bonding interactions with the carboxylic salt group of the particles, and thus they behave as binding or anchoring agents for the particles.
The second component of the coating composition is the nano-scale particles. Nano- scale particles are particles which have a maximum cross-sectional dimension ranging from 1 to 999 nanometers (i.e., 10"9 meters). The nano-scale particles are selected from silica, silicate, organomontmorillonite, clay, calcium carbonate, calcined aluminum silicate, hydrated aluminum silicate, calcium phosphate, alumina, barium sulfate, magnesium sulfate, and diatomacious earth. The nano-scale particles are combined with the first component, the acrylic copolymer emulsion or solution, to form the coating. In a preferred embodiment, the nano-scale particles are formed into an aqueous solution before being combined with the first component to form the coating. The nano-scale particles of the present invention have an average particle size of from 10 nanometers to 100 nanometers and make up between 1 to 20 weight percent of the coating, and preferably between 3 to 12 weight percent.
The layered nano-scale particles dispersed in a polymeric binder can be either a water- based coating or a melt-extrudable coating. For example, an aqueous organomontmorillonite dispersion can be easily blended with an aqueous ethylene acrylic acid copolymer dispersion to form a water-based coating. In addition, a powdery montmorillonite can be blended with ethylene acrylic acid copolymer using conventional melt blending equipment to form an extrusion coating. Either of these coatings can be applied onto thermoplastic films to improve their barrier properties. For a small molecule, such as O2 or H2O, to permeate a plastic film, the molecule must be soluble in the film and must be able to diffuse through the film. In an amorphous film, small molecules can occupy the spaces between the molecules, or bundles of molecules, that are only loosely entangled. The polymer chain, or bundle of chains, that forms the amorphous film is relatively flexible and holes open rather easily. These holes allow the small molecules to move within the film structure. Diffusion occurs when these small molecules move from site to site within the amorphous structure until they eventually pass through the structure. For relatively stiff and bulky polymer molecules, movement is slow and infrequent and, consequently, fewer holes open.
The halogenic barrier materials used in the prior art combine a highly crystalline structure and a specific chemical structure to restrict the movement of small molecules through films. Crystalline structures reduce the mobility of the polymer system and thereby reduce the diffusion rate of small molecules. In addition, the chemical structures of halogenic polymers have intramolecular bonding, efficiency of packing and polymer chain rigidity which not only enhance the degree of crystallinity but also decrease the rate of permeability.
The coatings of the present invention, unlike halogenic barrier materials, do not rely on either a crystalline polymer structure or a specific chemical structure to form an impermeable barrier to moisture and oxygen. Instead, the coatings of the present invention are formed from layered nano-scale particles. These nano-scale particles are so small in size that when they are formed into a layer, there are very few spaces between the particles that are large enough to form sites for the O2 or H2O molecules to occupy. Since diffusion depends on a molecule moving from site to site as it passes through a structure, the tightly packed particles minimize the number of sites in a structure and, thus, limit diffusion through the structure. Especially effective in forming such layers of nano-scale particles are silicate particles.
Nano-scale silicate particles also provide good clarity and gloss because the particles are too small to either reflect or diffuse light. Silicate particles that are known in the art and that have been used in coatings are relatively large when compared to the nano-scale particles of the present invention. The silicate particles known in the art are measured in terms of microns. Thus, the micron-scale particles are of an order of magnitude of 1,000 times greater than nano-scale particles of the present invention. Silicate particles have been used in prior art films to improve certain properties of the film, including antiblocking properties. The silicate particles can vary in size within a range of from 0.5 to 15 microns. When used as antiblocking agents, the silicate particles prevent plastic film surfaces from sticking together when they are tightly rolled up on a mandrel but they do not form a barrier. Figure 1 shows a side view of a coated film 100 that includes a base layer 120 with a coating 110 of micron-size silicate particles 112. The silicate coating 110 has relatively large spaces between the particles 112. These spaces provide sites for the O2 or H2O molecules to occupy and allow diffusion through the coating 110.
The nano-scale particles of the present invention because of their small size have unique characteristics and novel applications when used in plastic films. It has been found that the size of the particles used in the coatings of the present invention affect the barrier properties of the films. Larger particles, that is particles in the 0.5 to 15 micron range (micron- scale particles), do not disperse as well in polymeric binders as do smaller scale particles and have larger spaces between the particles. Consequently, coatings made using micron scale particles do not have good barrier properties.
Silicate coated films 100 that are presently used (as illustrated in Figure 1) have a base film 120 coated with micron-scale silicate particles 112. The coating 110 of micron-scale particles 112 has relatively large interstitial sites between the particles 112 which can be occupied by molecules of O2 and H2O. The large number of sites and their relatively large size provide paths in the structure through which O2 and H2O molecules can diffuse. In contrast, Figure 2 shows the coated film structure 200 of the present invention, having a base film 220 and a coating 210 of nano-scale particles 212 with relatively few interstitial sites that O2 and H2O molecules can occupy.
In addition to having smaller interstitial sites, the nano-scale particles 212 of the present invention form a more effective barrier because there are more layers of particles in the coating 210. As shown in Figure 1 and Figure 2, a coating 210 of nano-scale particles 212 (Figure 2) when compared to a coating 110 of micron-scale particles 112 (Figure 1) of equal thickness has a greater number of layers of particles between the top and bottom surfaces of the coating. The greater number of particles 212 in the coating 210 of the present invention creates a more tortuous path through the coating 210 and makes it more difficult for O2 and H2O molecules to diffuse.
When a molecule diffuses through a structure, the molecule moves from one interstitial site in the structure to another interstitial site. In order for the molecule to pass on to the next site, there must be an adjacent unoccupied site available. If a site is not available, the molecule remains where it is until a site becomes free. The more sites a molecule has to occupy before diffusing through a structure, the slower the rate of diffusion. Therefore, the nano-scale particles 212 of the present invention slow the rate of diffusion in two ways. First, the nano- scale particles 212 decrease the number of available sites that can be occupied by O2 and H2O molecules, and second, they increase the number of interstitial sites that a molecule must occupy in order to diffuse through the coating 210.
The base film 220 of the coated film structure 200 of the present invention can be a single layer film, or it can be a coextruded or laminated layer of a multi-layer film. The base film 220 includes a polyolefin selected from homopolymers, copolymers and terpolymers of propylene, ethylene and butene and blends thereof. Preferred polyolefins include polypropylene (PP), high density polyethylene (HDPE), low density polyethylene (LDPE), linear low density polyethylene (LLDPE), polybutylene (PB), ethylene-vinyl acetate (EVA) and blends thereof. The base film 220 can be made using well known extrusion and laminating methods and, in preferred embodiments, the base film 220 is oriented in at least one direction.
The coating 210 can be applied to the surface of a single layer film, or it can be applied to a coextruded layer of a multi-layer film, using techniques well known to those skilled in the art, including by gravure coating, roll coating or spraying. In a preferred embodiment, the coating 210 of the present invention is applied to the base film 220 using a dispersion coating process. Dispersion coating of plastic films is similar to a printing process. The base film 220 is unwound from a mandrel and the coating 210 applied at the desired thickness. After the coating 210 has dried, the coated film structure 200 is wound up again into a uniform roll. A gravure coating process is the preferred method of applying coatings because it can apply a thin even coating on films. The preferred coatings of the present invention are dispersions of nano-scale silicate particles and a polymeric binder. The polymeric binder is selected from a group of binders including ethylene acrylic acid, ethylene methyl acrylate and ethylene ethyl acrylate copolymers. In a preferred embodiment, the uniformity of the coating is increased by the addition of wetting or dispersing agents (surfactants), such as a fatty acid.
The test results in Table 2 show that films coated with the nano-scale particles of the present invention when compared to the uncoated control film provide improved barrier properties and decrease both the oxygen and moisture transmission rates through the films.
1. A thermoplastic film comprising a substantially continuous, adherent coating on at least one surface, wherein said coating comprises a polymeric binder and an additive comprising nano-scale particles.
2. The thermoplastic film of claim 1, wherein said nano-scale particles are selected from the group consisting of silica, silicate, clay, organomontmorillonite, calcium carbonate, calcined aluminum silicate, hydrated aluminum silicate, calcium phosphate, alumina, barium sulfate, magnesium sulfate, and diatomacious earth.
3. The thermoplastic film of claim 2, wherein said nano-scale particles comprise 5 to 20 weight percent of said additive.
4. The thermoplastic film of claim 3, wherein said nano-scale particles have an average particle size of from 10 nanometers to 100 nanometers.
5. The thermoplastic film of claim 1, wherein said polymeric binder comprises an olefinic copolymer containing carboxylic acid groups or its metal salt-functionalized olefinic copolymer.
6. The thermoplastic film of claim 2, wherein said polymeric binder is selected from the group consisting of ethylene acrylic acid, ethylene methyl acrylate and ethylene ethyl acrylate copolymers.
7. The thermoplastic film of claim 6, wherein said additive comprises 40 to 90 weight percent of said coating and said polymeric binder comprises 10 to 60 weight percent of said coating.
8. The thermoplastic film of claim 2, wherein said film further comprises a base substrate layer comprising a polyolefin selected from the group consisting of homopolymers, copolymers and terpolymers of propylene, ethylene and butene and blends thereof.
9. The thermoplastic film of claim 6, wherein said surface is a coextruded layer of a multi-layer film.
10. The thermoplastic film of claim 6, wherein said coating is applied by gravure coating, roll coating or spraying.


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Q: Asking for clarification on a step for the proof of the inverse function theorem This is from the book called Calculus and Analysis In Euclidean Space by Shurman et al., pp. 207

Theorem: Let $A$ be an open subset of $\mathbb{R}^n$, and let $f: A \to \mathbb{R}^n$ have continuous partial derivatives at every point of $A$. Let $a$ be a point of $A$, and suppose that $\det f'(a) \neq 0$. Then there exists an open set $V \subset A$ containing $a$ and an open set $W \subset \mathbb{R}^n$ containing $f(a)$ such that $f:V \to W$ has continuously differentiable inverse $f^{-1}:W \to V$. For each $y = f(x) \in W$, the derivative of the inverse is the inverse of the derivative $D(f^{-1})_{y} = (Df_x)^{-1}$.

The author first shows how we can assume without the loss of generality that $f'(a) = I_n$ (the identity matrix). After that the author begins to construct a closed ball $\overline{B}$, in order to apply the difference magnification lemma.
After that the author writes:

As $x$ varies continuously near $a$, the $(i, j)$th entry of $D_jf_i(x)$ of $f'(x)$ varies continuously near $\delta_{ij}$, and so the scalar $\det f'(x)$ varies continuously near $1$.

Question: I'm asking for clarification on the following step made in the proof.

Since $D_jf_i(x) - \delta_{ij} = 0$ and since $\det f'(a) = 1$, applying the persistence of inequality principle $n^2 + 1$ times shows that there exists a closed ball $\overline{B}$ about $a$ small enough that $$\left|D_jf_i(x) - \delta_{ij}\right| < \frac{1}{2n^2} \text{ for all }i, j \in {1,2,\dots,n}, x \in \overline{B}$$
and $\det f'(x) \neq 0, \text{ for all } x \in \overline{B}$

What I don't understand is i.) why does the persistence of the inequality principle show that there exists a closed ball with a radius of at least $\frac{1}{2n^2}$, ii.) why do we apply the principle $n^2 + 1$ times?
I get that the $\det f'(x) \neq 0$ for all $x \in \overline{B}$ results directly from the said principle, but given that the author has stated the principle as

Let $A$ be a subset of $\mathbb{R}^n$ and let $f:A \to \mathbb{R}^n$ be a continuous mapping. Let $p \in A$ and suppose $b \in \mathbb{R}^n \setminus \{f(p)\}$. Then there exists some $\epsilon > 0$ such that $\text{ for all } x \in A \text{ such that } |x - p| < \epsilon, f(x) \neq b$.

I don't immediately see how we construct the closed ball with the said radius.

A: I think one should use the very definition of continuity rather than the property the author stated. Here, we know that for each of the $n^2$ couples $(i,j)\in\{1,\dots,n\}^2$, the following holds: $$\forall\varepsilon>0,\exists\eta_{ij}>0\text{ s.t. }|x-a|<\eta_{ij}\implies|D_jf_i(x)-\delta_{ij}-0|<\varepsilon.$$ Get such $\eta_{ij}$ for all $(i,j)\in\{1,\dots,n\}^2$ for fixed $\varepsilon=\frac{1}{2n^2}$, then set $\eta=\min_{(i,j)\in\{1,\dots,n\}^2}\eta_{ij}$ (note we still have $\eta>0$). Then apply the persistence of inequality principle to $\det f'(a)\neq 0$, then take the minimum between the radius it gives you and $\eta$ to conclude.

redpajama_set_name: RedPajamaStackExchange


Ah, San Sebastian! The beaches are pristine, the water is clear, and the mountains are green. I spent a very relaxing few days here recovering from my operatic adventure. I cruised on the ocean. I sat on the beach. I read my book. I ate lots of pintxos. The only sightseeing I did (on a rainy day) was visit the Museo San Telmo. It has very cool architecture—a daring, modern add-on to an old convent. It was also hosting an exhibit about the Italian filmmaker Pasolini. Among (many) other accomplishments, he directed Maria Callas is her only non-singing film role!
After getting through a frantic week of opera rehearsals and performances in Burgos (blog post coming soon), I rewarded myself with a trip to San Sebastian. This resort city on the northern coast of Spain is famous not only for its beautiful beaches but also for its excellent food at all levels, from pintxos (Basque tapas) to Michelin-starred restaurants. I splurged last night by getting the tasting menu at Bodegón Alejandro, a place I'd seen recommended in both the Michelin guide and the New York Times when I was planning this trip. It was definitely worth it.
The restaurant lies underground, down an unobtrusive staircase on a busy street of San Sebastian's Old Town. It doesn't look like much, with bare wood tables, plain decor, and a rather kitschy city view mural on the wall. But any doubts about my restaurant choice disappeared with the amuse-bouche—a shot of cold asparagus soup with paper-thin slices of crispy bread. The salty creaminess of the soup went perfectly with the crunchy, fire-blackened toast.
I spent the last week travelling with my fellow Fulbrighter, Vicky. We devoted most of our vacation to Budapest (a blog post is on the way!), but we stopped over in Bratislava for a few days. Bratislava, the capital of Slovakia, feels very small. Our tour guide told us that during the Soviet era, they essentially chose between Prague and Bratislava: one would be thoroughly modernised, while the other would be historically preserved. Bratislava got the short end of the stick (at least in terms of post-Soviet tourist revenue), so most of the old town was destroyed. Everything important in the city proper—the castle, the major museums, the prettiest buildings—can easily be seen in a day.
To start, the castle. We made the mistake of going inside. Don't do it! There's very little of interest there. But the outside is pretty, thanks to a recent rennovation project. And the views from the hill are fabulous. Also, there's a playground!
There's also a playground in front of the castle!
This is part of my trilogy of posts about the Fulbright Berlin Seminar. See the first post for an introduction. This post is about the conference itself, including the socializing that happened around it.
After a welcome dinner with far too much food and wine, we woke up on Monday morning for topic-based discussion panels. I was tasked with moderating the performing arts panel, where scholars who were researching (or simply attending a lot of) theater, opera, music, and dance shared their views. We talked about direction, funding, the experience of being a performer, the past and future of various media, and much more. The conversation was a bit all over the place (my fault, I suppose, given that I was supposed to be moderating), but often intriguing. One Italian scholar lives with an experimental theater company that brings art into public and private spaces as intimate as donated rooms in people's homes!
As a thank-you for moderating, I was presented with the following mug. Look closely—you'll spot a funny Germanism.


Trade at least 10 lots and score a free FC Barcelona Jersey.
Open to both new and existing clients.
Available for the first 200 entrants.
AUD 6.80 for every lot less than 10 lots.
Fully Redeemable. Available for the first 200 entrants.


Buying real estate in Highlands NC can be a smoother process with preparation, education, and experience on your side. The first two steps are up to you, the third comes in the form of a real estate professional. Doing your homework to understand your wants and needs and communicating them to a knowledgeable and experienced Highlands REALTOR® can help get your property search off on the right foot and one step closer to finding the home of your dreams.
Start the buying process by getting pre-approved for a home loan. This will involve some work on your part, making sure you have all of the necessary paperwork in order. If you have this step completed prior to closing your REALTOR® you will be able to start your search knowing exactly what you can afford. When you start your search it is important to be flexible and available, you never know when the right house will get listed or be open for showings.
Contact the team at Meadows Mountain Realty today to learn more about the buying process and homes available in our area. We welcome the opportunity to serve as your guide to real estate in Highlands North Carolina. Put our knowledge and experience to work for you to complete a successful and smooth buying experience.
Click here to read "Five Smart Homebuyer Strategies" from Realty Times.


Develop this Property & Create a Place of Your Own!
This double (side-by-side) home has so much potential! A rarely offered opportunity to own a large lot with two units not yet subdivided. This 54′ x 107′ lot has ample parking for the whole family (PLUS SOME)! Renovate, build a garage or move in and restore this gorgeous home!
This Rarely Offered UNSEVERED side-by-side home with two units boasts old world charm & curb appeal. Located in the heart of Wellington Village, this expansive home gives you 2 kitchens, 2 dining rooms, 2 sun rooms, 2 fireplaces and 2 living rooms and that's just on the main level! Upstairs you'll find 3 bedrooms in each unit for a total of 6 bedrooms of various sizes (plus a lovely sewing room with a large window off the master in unit 30). The home has a total of 3 three piece bathrooms, two independent rear-entry sun rooms & a tall unfinished basement. The large lot accommodates 4-6 parking spots and cedar trees offer privacy along the street and is only steps from from trendy eateries, specialty shops and emporia! Investors & developers might be interested in the potential this large corner lot has to offer. With a little vision and TLC this home can be what you make it: a large single family home or a side-by-side income property!


Projects for Electronics and DIY enthusiasts. Here you can find tested electronics projects with a step-by-step instruction how to build them yourself or make them better. They all have LTspice simulations and EAGLE cad PCB design, so you can test and incorporate them in your own projects.