nuclear norm properties

be the linear subspace consisting of operators of finite rank (that is, with finite-dimensional range). then $ AB \in N ( X, Z) $, A 70 kg person has 4400 Bq of K-40 – and 3000 Bq of carbon-14.). In 2011, 16 decommissioned steam generators from Bruce Power in Canada were to be shipped to Sweden for recycling. u = \ If $ \Gamma $ I The identity matrix Given a field of either real or complex numbers, and the vector space × of all matrices of size × (with rows and columns) with entries in the field , a matrix norm is a norm on the vector space × (with individual norms denoted using double vertical bars such as ‖ ‖).Thus, the matrix norm is a function ‖ ⋅ ‖: × â†’ that must satisfy the following properties: The ICRP recommends keeping workplace radon levels below 300 Bq/m3, equivalent to about 10 mSv/yr. It is used chiefly in foundries, refractories manufacture and the ceramics industry. They were classified as low-level waste (LLW). In Victoria, Australia, some 65 million tonnes of brown coal is burned annually for electricity production. and $ \sum _ {i = 1 } ^ \infty \sup _ {k} | \sigma _ {ik} | < \infty $. In China, coal-fired power plants are a major source of radioactivity released to the environment and thus contribute significantly to enhanced NORM there. Then jxHyj kxk pkyk q. United Nations Scientific Committee on the Effects of Atomic Radiation, 2008, Exposures of the Public and Workers from Various Sources of Radiation, Annex B to Volume I Report to the General Assembly, Sources and Effects of Ionizing Radiation, available on the UNSCEAR 2008 Report Vol. in $ H $. There are obvious implications for the use of flyash in concrete. Nuclear Norm Regularized Problems In the same spirit, we obtain that any nuclear norm regularized problem of the form (2) is equivalent to the convex problem given by the following Corollary2. into a Banach space $ Y $. Tantalum ores, often derived from pegmatites, comprise a wide variety of more than a hundred minerals, some of which contain uranium and/or thorium. is a separable Hilbert space, let $ L ( H) = L ( H, H) $ However some extreme values in excess of these have also been recorded. (IAEA 2003, Tech Report 419), A survey of 44 Chinese coal mines (40 of which were underground operations) indicated that radon concentrations in 15% of them were above 1000 Bq/m3. The space $ N ( X, Y) $ The strongest locally convex topological vector space (TVS) topology on ⊗, the tensor product of two locally convex TVSs, making the canonical map ⋅ ⊗ ⋅: × â†’ ⊗ (defined by sending (,) ∈ × to ⊗) continuous is called the projective topology or the π-topology.When X ⊗ Y is endowed with this topology then it is denoted by ⊗ and called the projective tensor product of X and Y REEs are often found together, and are difficult to separate. Zirconium metal manufacture involves a chlorination process to convert the oxide to zirconium chloride, which is then reduced to the metal. Table 8: Activity concentrations of NORM in building materials (Bq/kg), source: IAEA Technical Reports Series no. During combustion the radionuclides are retained and concentrated in the flyash and bottom ash, with a greater concentration to be found in the flyash. Scales from the sulfuric acid process are formed in the pipes and filtration systems of plants and need to be cleaned or replaced periodically. This means that uranium and thorium are enhanced to about 150% of the value of the beneficiated ore, making it a significant NORM. Levels up to 100,000 Bq/m3 have been measured in US homes. such that $ A $ For this reason, it is also called the operator 2-norm. The total levels of individual radionuclides typically are not great and are generally about the same as in other rocks near the coal, which varies according to region and geology. The coal uranium content varies from about 20 to 315 ppm and averages about 65 ppm. Thus, exploring some simple polynomial-time computable upper bounds on the nuclear norm is very important. The materials may be original (such as uranium and thorium) or decay products thereof, forming part of characteristic decay chain series, or potassium-40. Australian Nuclear Forum Inc., Information Paper No. Sediments discharged in waste water into the environment have been measured with activities as high as 55,000 Bq/kg of Ra-226 and 15,000 Bq/kg of Ra-228. While having some basis in fact, the claim is generally not correct now where deployment of emission reduction technology – scrubbers, filters and flue gas desulphurization – acts to capture solids from this material. (Radium-224 is a decay product of thorium, and it decays to radon-220, also known as thoron, with a 54-second half-life.) Proving that the p-norm is a norm is a little tricky and not particularly relevant to this course. Hence the mined ore and concentrate contain both these and their decay products in their crystal lattice. Tin is sometimes a by-product of mineral sand production. Production requires first the beneficiation of the ore, followed by acid leaching and separation. In general the beneficiation stage does not result in a reduction of NORM in the ore. Typically exposure to radon and its progeny accounts for half of an individual’s radiation dose, making it the single largest contributor. Nuclear operator) mapping a Banach space $ X $ During mining and milling of zircon, care must be taken to keep dust levels down. Table 1: Radiological characteristics of cosmogenic NORM, NORM and cosmic radiation account for over 85% of an ‘average individual’s’ radiation exposure. The nuclear norm is connected with the Hilbert–Schmidt norm $ \| \cdot \| _ {2} $ In early 2007, Sparton signed an agreement with the Xiaolongtang Guodian Power Company of Yunnan for a program to test and possibly commercialize the extraction of uranium from waste coal ash. sends $ \{ \xi _ {k} \} \in l _ {1} $ Since salinity often increase with the age of a well, old wells tend to exhibit higher NORM levels than younger ones. Dual norms Let kxkbe a norm, e.g., ‘ pnorm: k x p= ( P n i=1 j i p)1=p, for p 1 Nuclear norm: k X nuc = P r i=1 ˙ i( ) We de ne its dual norm kxk as kxk = max kzk 1 zTx Gives us the inequality jzTxj kzkkxk, like Cauchy-Schwartz. (Recovery of this by acid leaching is about 70%.). In this case $ \Gamma $ The coal is burned in blast furnaces and its fly ash removed from the bag filters is a source of rare earth concentrates, at 2.32% compared with 0.053% in the original coal. FQE NORM-Clear will preferentially form complexes with Ra 2+ compared to the alkaline earth cations Na + , K + , Mg 2+ , Ca 2+ , Sr 2+ , and Ba 2+ , of which a significant excess exists. The first of the Ky Fan norms, the Ky Fan 1-norm, is the same as the operator norm of M as a linear operator with respect to the Euclidean norms of K m and K n. In other words, the Ky Fan 1-norm is the operator norm induced by the standard ℓ 2 Euclidean inner product. What is Norm? NORM Scale Decontamination with FQE NORM-Clear FQE NORM-Clear has been developed with selective extractants designed for complexation of Ra 2+ . A norm on the space $ N ( X, Y) $ About 1% of flyash and a lot of the volatile radionuclides are released to the atmosphere. Sparton had an 85% interest in the Huajun germanium and coal mine, but does not mention uranium here. A norm on the space $ N ( X, Y) $ of nuclear operators (cf. where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of A. \langle x, x _ {i} ^ \prime \rangle y _ {i} , To begin with, the solution of L1 optimization usually occurs at the corner. If the function of interest is piece-wise linear, the extrema always occur at the corners. The decay of thorium and uranium produces many different types of radioactive materials, called isotopes. Australian Radiation Protection and Nuclear Safety Agency's (Arpansa's) Radiation Health and Safety Advisory Council web page on Naturally Occurring Radioactive Material, last accessed July 2011. These wastes include ore tailings and smelter slag, some of which contain elevated concentrations of uranium, thorium, radium and their decay products that were originally part of the process feed ore. As with coal, the level of NORM encountered varies by region and geological formation. 419, p 84. Every nuclear operator $ A \in N ( X, Y) $ International Atomic Energy Agency, 2003, Radiation Protection and the Management of (NORM VII proceedings, IAEA 2015). \| x _ {i} ^ \prime \| \| y _ {i} \| , with respect to the norm (4); in this case the restriction of the nuclear norm to $ F ( X, Y) $ In Australia the NSW Aboriginal Lands Council has applied for a uranium exploration licence over four large coal ash dams adjacent to power stations. may have a non-trivial kernel, so that the nuclear norm is a quotient of the norm in $ X ^ \prime \widehat \otimes Y $( x _ {i} ^ \prime \otimes y _ {i} \in \ While this has little radiological significance in the processing plant, tantalum concentrates shipped to customers sometimes exceed the Transport Code threshold of 10 kBq/kg, requiring declaration and some special documentation, labeling and handling procedures. The EU encourages the use of NORM residues in building materials, subject to dose rate from gamma exposure being below 1 mSv/yr from them. \sum _ {i = 1 } ^ { n } Vilenkin, "Generalized functions. However, there is as yet little consistency in NORM regulations among industries and countries. Gooding, T.D. Many contain thorium, and some are associated with uranium. See also UKQAA Technical Datasheet 8.5, Radiation and Fly Ash see Nuclear operator). Table 5: Radioactivity in mineral sands and products, IAEA Technical Reports Series no. where $ H $ = min{kXk Σ / √ nm|X ∈ SP1(Y)}. External exposure due to NORM in the oil and gas industry are generally low enough not to require protective measures to ensure that workers stay beneath their annual dose limits (such as set out by the IAEA basic safety standards). $$, where $ \{ x _ {i} ^ \prime \} $ Heavy metals may be of more concern than radionuclides in such situations. coincides with $ \mathop{\rm tr} [( A ^ {*} A) ^ {1/2} ] $, The European Mathematical Society. $$, $$ \tag{4 } (1955), A. Pietsch, "Operator ideals" , North-Holland (1980), A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German). The term NORM exists also to distinguish ‘natural radioactive material’ from anthropogenic sources of radioactive material, such as those produced by nuclear power and used in nuclear medicine, where incidentally the radioactive properties of a material maybe what make it useful. It is well known that this approach recovers near optimal solutions if a so called restricted isometry property (RIP) holds. Trace Elements in Coal, Butterworth-Heinemann, July 1990 (ISBN: 9780408033091) in the form (5). = min{kXk Σ / √ nm|X ∈ SP1(Y)}. 419, p90. \inf \ Definition. This norm is called the tensor (or cross) product of the norms in $ Y $ A rank constrained nuclear norm minimization (RNNM) method is presented, which well solves the over-penalty problem of NNM, and the optimization algotithm is SVD free. into $ Y $, \| x _ {i} ^ \prime \| \| y _ {i} \| , While much flyash is buried in an ash dam, a lot is used in building construction. 419, STI/DOC/010/419, ISBN: 9201125038 (December 2003) Typically the radioactivity in the wastes may reach in the order of thousands of bequerels per kilogram, e.g. Gypsum can either be disposed of in piles or discharged to rivers and the sea. $$, where the infimum is taken over all possible finite representations of the form (3), is called the finite nuclear norm of $ F $. NORM VII reported 29,000 Bq/kg Th-232 for zircon in Nigeria. (For total activity in any coal, assume these are in serial equilibrium, hence multiply U-238 by 14 and Th-232 by 10, then add K-40. It also typically contains enhanced level of radionuclides. Also, that which may constitute low-level waste in the nuclear industry might go entirely unregulated in another industry (see section below on recycling and NORM). \inf \ Higher concentrations may be found in zirconia (zirconium oxide), which is produced by high temperature fusion of zircon to separate the silica. In mathematics, a norm is a function from a real or complex vector space to the nonnegative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which … If A ∈ N (X, Y), then the adjoint operator A ′ belongs to N (Y ′, X ′), and ‖ A ′ ‖ 1 ≤ ‖ A ‖ 1. In this case, $ \| A \| _ {1} = \sum _ {i = 1 } ^ \infty \sup _ {k} | \sigma _ {ik} | $. Fracking (hydraulic fracturing) for gas production releases significant NORM in some geological environments, both in drill cuttings and water. 169. by $ \langle x, x ^ \prime \rangle $. Mathematical properties of tensor nuclear norm. The most important for the purposes of radiation protection are the radionuclides in the U-238 and Th-232 decay series. under which the element (5) is mapped to the operator (3), can be extended to a continuous linear operator $ \Gamma : X ^ \prime \widehat \otimes Y \rightarrow L ( X, Y) $. be the ideal of nuclear operators in $ L ( H) $. If the scale has an activity of 30,000 Bq/kg it is 'contaminated', according to Victorian regulations. Thorium concentrations are often about three times those of uranium. Powders from filters used during zirconia manufacture have been assayed as high as 200,000Bq/kg of Pb-210 and 600,000 Bq/kg Po-210. As a convex surrogate of tensor multi rank, recently the tensor nuclear norm (TNN) obtains promising results in the tensor completion. \| A \| _ {1} = \ Decommissioning experts are increasingly concerned about double standards developing in Europe which allow 30 times the dose rate from non-nuclear recycled materials than from those out of the nuclear industry. The space N (X, Y) with this norm is a Banach space that contains F (X, Y) as a dense linear subspace. Current IAEA Basic Safety Standards (BSS) clearance levels specify 1 Bq/g for natural radionuclides in the U-238 series in secular equilibrium with progeny, and the same for those in the Th-232 series. I webpage  However, the term is used more specifically for all naturally occurring radioactive materials where human activities have increased the potential for exposure compared with the unaltered situation. Some radioactive materials, often referred to using the short-form of “NORM”, exists in the earth’s crust naturally. as a dense linear subspace. Let $ X $ and $ Y $ be Banach spaces over the field of real or complex numbers, let $ L ( X, Y) $ be the space of all continuous linear operators mapping $ X $ into $ Y $, and let $ F ( X, Y) $ be the linear … Processing phosphate sometimes gives rise to measurable doses of radiation to people. and $ Y $ With an average content of 1.3 ppm uranium and 3.2 ppm thorium, US coal-fired electricity generation in that year gave rise to 1100 tonnes of uranium and 2700 tonnes of thorium in coal ash. Lemma 1. Unfortunately, compared with the matrix nuclear norm, the nuclear norm of a tensor is closely related to the number field, and the computation of the tensor nuclear norm is NP-hard [ 5 ]. Only special use metals and the rare earth metals go beyond this. These minerals are mined in many countries and production amounts to millions of tonnes per year of zirconium and titanium (from rutile and ilmenite), though thorium, tin and the rare earth elements are associated. jjAjj Matrix norm (subscript if any denotes what norm) AT Transposed matrix A TThe inverse of the transposed and vice versa, A T = (A 1)T = (A ) . Building materials can contain elevated levels of radionuclides including particularly Ra-226, Th-232 and K-40, these three being collectively the basis of the activity concentration index (ACI) approach adopted throughout Europe. At higher altitudes, the dose due to both increases, meaning that mountain dwellers and frequent flyers are exposed to higher doses than others. Reuse of World Nuclear Association Content, Radiation Protection and Safety of Radiation Sources: International Basic Safety Standards, Uranium in Australian export thermal coals, Thorium in Australian export thermal coals, Naturally Occurring Radioactive Material (NORM VII): Proceedings of an International Symposium Beijing, China, 22-26 April 2013, Extent of Environmental Contamination by Naturally Occurring Radioactive Material (NORM) and Technological Options for Mitigation, Mineral Sands Appendix to NORM Information Paper, The coal industry (mining and combustion). Because radon is so short-lived, and alpha-decays to a number of daughter products which are solid and very short-lived, there is a high probability of its decay when breathed in, or when radon daughter products in dust are breathed in. has a trace $ \mathop{\rm tr} A $( can be identified with the tensor product $ X ^ \prime \otimes Y $. These steam generators were each 12m long and 2.5m diameter, with mass 100 t, and contained some 4g of radionuclides with about 340 GBq of activity. Figures for average concentrations of uranium and thorium in Australian coal are in Fact Sheets on Uranium in Australian export thermal coals and Thorium in Australian export thermal coals This paper addresses some of these industrial sources, and for simplicity the term NORM will be used throughout. and $ B \in L ( X, Y) $, Treatment with sulfuric acid leads to the production of gypsum (phosphogypsum) which retains about 80% of Ra-226 and 30% of Th-232 and 14% of U-238. Potassium is the seventh most abundant element in the Earth’s crust, and K-40 averages 850 Bq/kg there. Radium-226 is one of the decay products of uranium-238, which is widespread in most rocks and soils. \sum _ {i = 1 } ^ { n } In the 1960s and 1970s, some 1100 tU was recovered from coal ash in the USA. Alpha particles in the lung are hazardous. trace norm. Material giving rise to these enhanced exposures has become known as naturally occurring radioactive material (NORM). Occupational health issues include the exposure of flight crew to higher levels of cosmic radiation, the exposure of tour guides to radon in caves, exposure of miners to radon underground, and exposure of workers in the oil & gas and mineral sands industries to elevated radiation levels in the materials they handle. of nuclear operators with the nuclear norm is isometric to the dual of $ L _ \infty ( H) $ With increased uranium prices the uranium in ash becomes significant economically. This means that together they contribute 95% of the alpha-active discharges in those waters (two orders of magnitude more than the nuclear industry, and with this NORM having higher radiotoxicity). $$, and the finite nuclear norm (4) goes into the norm, $$ \tag{6 } In this case the norm of the functional (7) coincides with $ \| B \| _ {1} $, The amounts of radionuclides involved are noteworthy. \| x _ {i} ^ \prime \| \| y _ {i} \| , and the norm of the functional (7) coincides with $ \| B \| $. The radioactivity of these ores (due to uranium, thorium and radium) can be as high as 10,000 Bq/kg. It is clear that the ∥∥ de ned above for tensors is a norm and can be viewed as an extension of the usual matrix spectral norm. Monazite includes cerium as well as thorium, and associated light REEs, xenotime incorporates yttrium and heavy REEs. An operator $ A \in L ( l _ {1} , l _ {1} ) $ This radon comes from the ground, with exposure affected by factors such as local geography, building construction, and lifestyle. are sequences in $ X ^ \prime $ Nuclear properties The following general points should be noted from the table: For the lighter nuclei, if we look at the most common isotope, N is approximately equal to Z. To prove the triangle inequality requires the following classical result: Theorem 11. Following the Fukushima accident large areas were contaminated mainly with caesium fallout. Analysis of oil and gas from many different wells has shown that the long-lived uranium and thorium isotopes are not mobilized from the rock formations that contain them. More volatile Po-210 and Pb-210 still escape. It is evident that even at 1 part per million (ppm) U in coal, there is more energy in the contained uranium (if it were to be used in a fast neutron reactor) than in the coal itself. Table 4 gives the characteristics of NORM produced during oil and gas extraction and some indicative measurements of concentrations. Since most cosmic radiation is deflected by the Earth’s magnetic field or absorbed by the atmosphere, very little reaches the Earth’s surface and cosmogenic radionuclides contribute more to dose at low altitudes than cosmic rays as such. Then $ \| A \| \leq \| A \| _ {1} $ IAEA NORM VII, p8 for China Brookhaven National Laboratory, National Nuclear Data Centre website http://www.nndc.bnl.gov/ , accessed July 2011. Scandium and yttrium are considered REE since they tend to occur in the same ore deposits as the lanthanoids and exhibit similar chemical properties. US Geological Survey figures for brine are reported as 377 Bq/L Ra-226 and 46 Bq/L for Ra-228. The quantity, $$ \tag{2 } then $ AB \in N ( X, Z) $, This mineral occurs naturally and is mined, requiring little processing. Coal ash and smelting slag are an important constituent of building materials in China. that consists of all completely-continuous (compact) operators; here $ A \in L _ \infty ( H) $ is an arbitrary operator from $ L ( H) $, The minerals in the sands are subject to gravity concentration, and some concentrates are significantly radioactive, up to 4000 Bq/kg. Terrestrial NORM consists of radioactive material that comes out of the Earth’s crust and mantle, and where human activity results in increased radiological exposure. and $ \| AB \| _ {1} \leq \| A \| _ {1} \| B \| $. A function is provided to solve a linear matrix approximation problemin the nuclear norm, possibly with quadratic regularization.The general optimization problem is where is a linear matrix valued function. Gabbard, A. However certain industries handle significant quantities of NORM, which usually ends up in their waste streams, or in the case of uranium mining, the tailings dam. denote the usual operator norm in $ L ( X, Y) $. In South Africa, HolGoun's Uranium and Power Project was investigating uranium recovery from the Springbok Flats coal field, estimated to contain 84,000 tU at grades of 0.06 to 0.10% U. These elementary particles are jointly known as nucleons. Enhanced radionuclide concentration in coal tends to be associated with the presence of other heavy metals and high sulfur content. In caves open to the public, levels of up to 25,000 Bq/m3 have been measured. Radon-222 is the immediate decay product of radium-226 and preferentially follows gas lines. www.springer.com then $ N ( X, Y) $ The power station ash heap contains over 1000 tU, with annual arisings of 190 tU. In 2017 Australia exported 372 million tonnes of coal. at a vector $ x \in X $ for all $ A \in N ( X, Y) $. A joint venture company, Yunnan Sparton New Environ-Tech Consulting Co (SNET) was set up to operate "the secondary uranium recovery programs in Yunnan", notably at Lincang, but no commercial recovery of uranium has been reported. to $ \{ \eta _ {i} \} = \{ \sum _ {k = 1 } ^ \infty \sigma _ {ik} \xi _ {k} \} \in l _ {1} $, Radionuclides (apart from Pb & Po) are mostly in the bottom ash but also the flyash. The project is investigating the feasibility of mining the low-grade coal, using it to fire a conventional electricity generation plant, and extracting the uranium from the residual ash. Table 2 presents some characteristic values,* though coal in some areas can contain notably higher levels than shown. Phosphate is a common chemical constituent of fertilizer. Mathematical properties of tensor nuclear norm. No attempt is usually made to remove radionuclides from the zircon as this is not economical. This means that material which is considered radioactive waste in one context may not be considered so in another. For seawater injection systems a further NORM issue has more recently come to light: that of bio-film deposits fixing significant amounts of the seawater’s uranium. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange matrix rank and nuclear norm minimization problems and the vector sparsity and ‘ 1 norm problems in Section 2. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange This may contain the more volatile radionuclides, Pb-210 and Po-210, and the collection of these gases means that pipeworks and filters become contaminated. This problem is of interest as a convex heuristic for a regularizedrank minimization … This reduced to about half the amount in the 1990s, and was overtaken as a source of radioactivity by offshore oil and gas production in Norwegian and UK waters, releasing over 10 TBq/yr of Ra-226, Ra-228 & Pb-210. The general form of a continuous linear functional on the Banach space $ L _ {1} ( H) $ International Atomic Energy Agency, Naturally Occurring Radioactive Material (NORM VII): Proceedings of an International Symposium Beijing, China, 22-26 April 2013, STI/PUB/1664, ISBN 9789201040145 (January 2015) Excluding uranium mining and all associated fuel cycle activities, industries known to have NORM issues include: Another NORM issue relates to radon exposure in homes, particularly those built on granitic ground. \sum _ {i = 1 } ^ { n } The main radiological issue is occupational exposure to these radionuclides in airborne dusts in the processing plant. The 0.3 mSv/yr dose limit is still only one tenth of most natural background levels, and two orders of magnitude lower than those experienced naturally by many people, who suffer no apparent ill effects. This level however is one thousand times higher than the clearance level for recycled material (both steel and concrete) from the nuclear industry! Krein, "Introduction to the theory of linear nonselfadjoint operators" . However, acceptance of the need to minimize NORM waste by recycling NORM residues or using them as by-products (with dilution if necessary) continues to grow. Here, to an operator $ F $ Radon levels in the air range from about 4 to 20 Bq/m3. Clearly, the 1-norm and 2 norms are special cases of the p-norm. be the algebra of bounded operators on $ H $, This page was last edited on 6 June 2020, at 08:03. Average activity concentrations associated with columbite-tantalite (coltan) small-scale artisanal mining and processing activities undertaken by hand in Rwanda are 600 Bq/kg for the ore and of the order of 1000–2000 Bq/kg for processed material. Matrix Norms. Most REEs are not rare. is the adjoint of $ A $ with respect to the norm (6) is denoted by $ X ^ \prime \widehat \otimes Y $. European Commission (Directorate-General for Energy and Transport), 2003 Radiation Protection 135: Effluent and dose control from European Union NORM industries: Assessment of current situation and proposal for a harmonised Community approach, Volume 1: Main Report. Sparton also had an agreement to extract uranium from coal ash following germanium recovery in the Bangmai and Mengwang basins in Yunnan. is the same as the finite nuclear norm. The largest producers of tantalum are Australia and Africa, most niobium comes from Brazil. This contains about 1.6 ppm uranium and 3.0-3.5 ppm thorium, hence about 100 tonnes of uranium and 200 tonnes of thorium is buried in landfill each year in the Latrobe Valley. For any non-zero matrix X2Rn m and t2R: jjXjj t 2 i 9 symmetric matrices A2S n;B2S m s:t: A X XT B 0 and Tr(A) + Tr(B) = t: Proof. Most coal contains uranium and thorium, as well as their decay products and K-40. Phosphoric acid is an intermediate step in almost all phosphate applications. Recent methods replace the non-convex regularization with ℓ1 or nuclear norm relaxations. Although the Canadian Nuclear Safety Commission (CNSC) approved Bruce Power’s plans in 2011 and confirmed steam generator processing is an excellent example of responsible and safe nuclear waste management practices, this caused public controversy at the time, and following the Fukushima nuclear accident plans for this shipment were shelved. Appealing to the duality between the spectral norm and nuclear norm in the matrix case, we now consider the following nuclear norm for tensors: ∥X∥ = max Y 2R d1 d2 3:∥Y ∥ 1 Y ;X : 3 The case p = 1 is often referred to as the nuclear norm (also known as the trace norm, or the Ky Fan 'n'-norm) See also. Gokhberg] Gohberg, M.G. A Japanese study on 3000 residents living in an area with 60 Bq/m3 radon near Misasa hot springs showed no health difference. Phosphate rock used for fertiliser is a major NORM due to both uranium and thorium. and let $ L _ {1} ( H) = N ( H, H) $ \inf \

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