For a toric pair (X, D), where X is a projective toric variety of dimension d−1 ≥ 1 and D is a very ample T -Cartier divisor, we show that the Hilbert-Kunz density function HKd(X, D)(λ) is the d − 1 dimensional volume of P D ∩ {z = λ}, where P D ⊂ R d is a compact d-dimensional set (which is a finite union of convex polytopes).We also show that, for k ≥ 1, the function HKd(X, kD) can be replaced by another compactly supported continuous function ϕ kD which is 'linear in k'. This gives the formula for the associated coordinate ring (R, m):where ϕ D (see Proposition 1.2) is solely detemined by the shape of the polytope P D , associated to the toric pair (X, D). Moreover ϕ D is a multiplicative function for Segre products. This yields explicit computation of ϕ D (and hence the limit), for smooth Fano toric surfaces with respect to anticanonical divisor. In general, due to this formulation in terms of the polytope P D , one can explicitly compute the limit for two dimensional toric pairs and their Segre products.We further show that (Theorem 6.3) the renormailzed limit takes the minimum value if and only if the polytope P D tiles the space M R = R d−1 (with the lattice M = Z d−1 ). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope.where q = p n , I [q] = n-th Frobenius power of I = the ideal generated by q-th powers of elements of I. This is an ideal of finite colength and (R/I [q] ) denotes the length of the R-module R/I [q] . Existence of the limit was proved by Monsky [Mo1]. This invariant has been extensively studied, over the years (see the survey article [Hu]). As various standard techniques, used for studying multiplicities, are not applicable for the invariant e HK , it has been difficult to compute (there is no general formula even for a hypersurface).In order to study e HK , when R is a standard graded ring (dim R ≥ 2) and I is a homogeneous ideal of finite colength, the second author (in [T2]) has defined the notion of Hilbert-Kunz Density function and its relation with the HK-multiplicity (stated in this paper as Theorem 4.1): the HK density function is a compactly supported continuous function HKd(R, I) : [0, ∞) −→ [0, ∞) such that e HK (R, I) = ∞ 0 HKd(R, I)(x) dx.
We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a β-density function g R,m : [0, ∞) −→ R, where (R, m) is the homogeneous coordinate ring associated to the toric pair (X, D), such thatwhere β(R, m) is the second coefficient of the Hilbert-Kunz function for (R, m), as constructed by Huneke-McDermott-Monsky Moreover we prove, (1) the function g R,m : [0, ∞) −→ R is compactly supported and is continuous except at finitely many points, (2) the function g R,m is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved.Here we also prove and use a result (which is a refined version of a result by Henk-Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes.
Purpose Cleanrooms are highly controlled enclosed rooms where air quality is monitored and ensured to have less contamination according to standard cleanliness level. Air filters are used to optimize indoor air quality and remove air pollutants. Filter media and filtering system are decided as per requirement. Depth filter media are mostly used in cleanroom filtrations. This paper aims to present a comprehensive review of the evolution of cleanroom filter media. It evaluates the advantages and disadvantages of air filter media. It is also studied which air filters have additional properties such as anti-microbial properties, anti-odour properties and chemical absorbent. Development and innovation of air filters and filtration techniques are necessary to improve the performance via the synergistic effect and it can be a possible avenue of future research. Design/methodology/approach This paper aims to drive the future of air filter research and development in achieving high-performance filtration with high filtration efficiency, low operational cost and high durability. Air pollutants are classified into three types: suspended particles, volatile organic pollutants and microorganisms. Technologies involved in purification are filtration, water washing purification, electrostatic precipitation and anion technology. They purify the air by running it through a filter medium that traps dust, hair, pet fur and debris. As air passes through the filter media, they function as a sieve, capturing particles. The fibres in the filter medium provide a winding path for airflow. There are different types of air filters such as the high-efficiency particulate air filter, fibreglass air filter and ultra-low particulate air filter. Findings Emerging filtration technologies and filters such as nanofibres, filters with polytetrafluoroethylene membrane are likely to become prevalent over the coming years globally. The introduction of indoor air filtration with thermal comfort can be a possible avenue of future research along with expanding indoor environment monitoring and improving air quality predictions. New air filters and filtration technologies having better performance with low cost and high durability must be developed which can restrict multiple types of pollutants at the same time. Originality/value The systematic literature review approach used in this paper highlights the emerging trends and issues in cleanroom filtration in a structured and thematic manner, enabling future work to progress as it will continue to develop and evolve.
A. Let be a prime number, k a field of characteristic and a finite -group. Let be a finite-dimensional linear representation of over k. Write = Sym * . For a class of -groups which we call generalised Nakajima groups, we prove the following: (a) e Hilbert ideal is a complete intersection. As a consequence, for the case of generalised Nakajima groups, we prove a conjecture of Shank and Wehlau (reformulated by Broer) that asserts that if the invariant subring is a direct summand of as -modules then is a polynomial ring. (b) e Hilbert ideal has a generating set with elements of degree at most | |. is bound is conjectured by Derksen and Kemper.
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