Restricted Formula Research Articles

The uniqueness theorem for Gysin coherent characteristic classes of singular spaces

AbstractWe establish a general computational scheme designed for a systematic computation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly geometric and of a recursive nature terminating on genera of explicit characteristic subvarieties that we construct. It enables us, for example, to apply intersection theory of Schubert varieties to obtain a uniqueness result for such characteristic classes in the homology of an ambient Grassmannian. Our framework applies in particular to the Goresky–MacPherson ‐class by virtue of the Gysin restriction formula obtained by the first author in previous work. We illustrate our approach for a systematic computation of the ‐class in terms of normally nonsingular expansions in examples of singular Schubert varieties that do not satisfy Poincaré duality over the rationals.

On the Restriction Formula

On the Restriction Formula

Research on Dynamic Reserve and Energy Arbitrage of Energy Storage System

Replacing the traditional rotating generators with renewable energy will reduce the grid’s inertia and with it the minimum frequency when N-1 contingency occurs triggering an Under-Frequency Load Shedding (UFLS). This study proposes a method for the energy storage system (ESS) to simultaneously provide energy arbitrage, reserve capacity, and assist N-1 contingency, by modifying the restriction formula of economic dispatch (ED) and limiting the SOC range of the ESS. Let the ESS join the Spinning Reserve. Through the PSS®E iterative ESS charging power required at moments when the frequency of contingency is too low in the ED. Let the ESS act as a N-1 contingency extra frequency reserve. This would prevent UFLS and still maintain the demand. The proposed method is applicable to different types of ESS. The method allows energy storages, originally designed for energy arbitrage, to participate in frequency support and spinning reserve.

Flagged Littlewood-Richardson tableaux and branching rule for classical groups

We give a new formula for the branching rule from GLn to On generalizing the Littlewood's restriction formula. The formula is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish in a stable range. As an application, we give a combinatorial formula for the Lusztig t-weight multiplicity Kμ0(t) of type Bn and Dn with highest weight μ and weight 0.

Restriction formula and subadditivity property related to multiplier ideal sheaves

Abstract We give a restriction formula on jumping numbers which is a reformulation of Demailly–Ein–Lazarsfeld’s important restriction formula for multiplier ideal sheaves and a generalization of Demailly–Kollár’s important restriction formula on complex singularity exponents, and then we establish necessary conditions for the extremal case in the reformulated formula; we pose the subadditivity property on the complex singularity exponents of plurisubharmonic functions which is a generalization of Demailly–Kollár’s fundamental subadditivity property, and then we establish necessary conditions for the extremal case in the generalization. We also obtain two sharp relations on jumping numbers, introduce a new invariant of plurisubharmonic singularities and get its decreasing property for consecutive differences.

On the K-theory stable bases of the Springer resolution

Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable bases. We apply it to obtain the restriction formula for partial flag varieties. We also build a relation between the stable basis and the Casselman basis in the principal series representations of the Langlands dual group. As an application, we give a closed formula for the transition matrix between Casselman basis and the characteristic functions.

ON A GENERALISATION OF A RESTRICTED SUM FORMULA FOR MULTIPLE ZETA VALUES AND FINITE MULTIPLE ZETA VALUES

We prove a new linear relation for multiple zeta values. This is a natural generalisation of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.

Amino acid disorders.

Amino acids serve as key building blocks and as an energy source for cell repair, survival, regeneration and growth. Each amino acid has an amino group, a carboxylic acid, and a unique carbon structure. Human utilize 21 different amino acids; most of these can be synthesized endogenously, but 9 are "essential" in that they must be ingested in the diet. In addition to their role as building blocks of protein, amino acids are key energy source (ketogenic, glucogenic or both), are building blocks of Kreb's (aka TCA) cycle intermediates and other metabolites, and recycled as needed. A metabolic defect in the metabolism of tyrosine (homogentisic acid oxidase deficiency) historically defined Archibald Garrod as key architect in linking biochemistry, genetics and medicine and creation of the term 'Inborn Error of Metabolism' (IEM). The key concept of a single gene defect leading to a single enzyme dysfunction, leading to "intoxication" with a precursor in the metabolic pathway was vital to linking genetics and metabolic disorders and developing screening and treatment approaches as described in other chapters in this issue. Amino acid disorders also led to the evolution of the field of metabolic nutrition and offending amino acid restricted formula and foods. This review will discuss the more common disorders caused by inborn errors in amino acid metabolism.

Correcting the predictive validity of a selection test for the effect of indirect range restriction

BackgroundThe validity of selection tests is underestimated if it is determined by simply calculating the predictor-outcome correlation found in the admitted group. This correlation is usually attenuated by two factors: (1) the combination of selection variables which can compensate for each other and (2) range restriction in predictor and outcome due to the absence of outcome measures for rejected applicants.MethodsHere we demonstrate the logic of these artifacts in a situation typical for student selection tests and compare four different methods for their correction: two formulas for the correction of direct and indirect range restriction, expectation maximization algorithm (EM) and multiple imputation by chained equations (MICE). First we show with simulated data how a realistic estimation of predictive validity could be achieved; second we apply the same methods to empirical data from one medical school.ResultsThe results of the four methods are very similar except for the direct range restriction formula which underestimated validity.ConclusionFor practical purposes Thorndike’s case C formula is a relatively straightforward solution to the range restriction problem, provided distributional assumptions are met. With EM and MICE more precision is obtained when distributional requirements are not met, but access to a sophisticated statistical package such as R is needed. The use of true score correlation has its own problems and does not seem to provide a better correction than other methods.

Restriction formula for stable basis of the Springer resolution

We give restriction formula for stable basis of the Springer resolution, and generalize it to cotangent bundles of homogeneous spaces. By a limiting process, we get the restriction formula of Schubert varieties.

Translation of first order formulas into ground formulas via a completion theory

A translation technique is presented which transforms a class of First Order Logic formulas, called Restricted formulas, into ground formulas. For the formulas in this class the range of quantified variables is restricted by Domain formulas.If we have a complete knowledge of the predicates involved in the Domain formulas their extensions can be evaluated with the Relational Algebra and these extensions are used to transform universal (respectively existential) quantifiers into finite conjunctions (respectively disjunctions).It is assumed that the complete knowledge is represented by Completion Axioms and Unique Name Axioms à la Reiter. These axioms involve the equality predicate. However, the translation allows to remove the equality in the ground formulas and for a large class of formulas their consequences are the same as the initial First Order formulas. This result open the door for the design of efficient deduction techniques.

Equivariant quantum cohomology of cotangent bundle of G/P

Let G denote a complex semisimple linear algebraic group, P a parabolic subgroup of G and P=G/P. We identify the quantum multiplication by divisors in T⁎P in terms of stable basis, which is introduced in [9]. Using this and the restriction formula for stable basis [17], we show that the G×C⁎-equivariant quantum multiplication formula in T⁎P is conjugate to the formula conjectured by Braverman.

Excited Young diagrams, equivariant $K$-theory, and Schubert varieties

We give combinatorial descriptions of the restrictions to T-fixed points of the classes of structure sheaves of Schubert varieties in the T-equivariant K-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at T-fixed points of the corresponding Schubert varieties. These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction fomulas are positive, in that for a Schubert variety of codimension d, the formula equals (-1)^d times a sum, with nonnegative coefficients, of monomials in the expressions (e^{-\alpha}-1), as \alpha runs over the positive roots. In types A_n and C_n the restriction formulas had been proved earlier by [Kreiman 05], [Kreiman 06] by a different method. In type A_n, the formula for the Hilbert series had been proved earlier by [Li-Yong 12]. The method of this paper, which relies on a restriction formula of [Graham 02] and [Willems 06], is based on the method used by [Ikeda-Naruse 09] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the K-theoretic restriction formulas given by [Ikeda-Naruse 11], which use different versions of excited Young diagrams and set-valued tableaux. We also give Hilbert series and Hilbert polynomial formulas which are valid for Schubert varieties in any cominuscule flag variety, in terms of the 0-Hecke algebra.

Applications of shuffle product to restricted decomposition formulas for multiple zeta values

In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs). The first formula generalizes the decomposition formula of Euler and is similar to the restricted formula of Eie and Wei for MZVs with one strings of 1's. The second formula generalizes the previous results to the product of two MZVs with one and two strings of 1's respectively.

Dirac cohomology, K -characters and branching laws

Inspired by work of Enright and Willenbring, we prove a generalized Littlewood’s restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willenbring in terms of nilpotent Lie algebra cohomology. This follows from the close relationship between the Dirac cohomology and the corresponding nilpotent Lie algebra cohomology for unitary representations of semisimple Lie groups of Hermitian type, which was established by Huang, Pandzic, and Renard.

Some new proof of the sum formula and restricted sum formula

The sum formula is a basic identify of multiple zeta values that expresses a Riemann-zeta value as a homogeneous sum of multiple zeta values of given depth and weight. This formula was already known to Euler in the depth two case. Conjectured in the early 1990s, for higher depth and then proved by Granville and Zagier independently. Restricted sum formula was given in Eie \cite{2}. In this paper, we present some new proofs of those formulas.

The maximum likelihood degree of a very affine variety

AbstractWe show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern–Schwartz–MacPherson class. The strengthened version recovers the geometric deletion–restriction formula of Denhamet al. for arrangement complements, and generalizes Kouchnirenko’s theorem on the Newton polytope for nondegenerate hypersurfaces.

Multiplicity formula of G 2 through restriction formula

In this paper we consider the explicit formulas for the multiplicities of an irreducible representation of a complex simple Lie algebra of class G2.

A geometric deletion–restriction formula

In this paper, we recover the characteristic polynomial of an arrangement of hyperplanes by computing the rational equivalence class of the variety defined by the logarithmic ideal of the arrangement. The logarithmic ideal was introduced in Cohen et al. (2011) [5] in a study of the critical points of the master function. The above result is used to understand the asymptotic behaviour of the Hilbert series of the logarithmic ideal. As an application, we note that a well-known formula due to Solomon and Terao may be expressed as an Euler characteristic and, at least in the case of tame arrangements, deduced from our main theorem.

Towards generalizing Schubert calculus in the symplectic category

The main purpose of this article is to extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. Given a generic component of the moment map, we define a canonical class \alpha_p in the equivariant cohomology of the manifold M for each fixed point p of M. When they exist, canonical classes form a natural basis of the equivariant cohomology of M; in particular, when M is a flag variety, these classes are the equivariant Schubert classes. We show that the restriction of a canonical class \alpha_p to a fixed point q can be calculated by a rational function which depends only on the value of the moment map, and the restriction of other canonical classes to points of index exactly two higher. Therefore, the structure constants can be calculated by a similar rational function. Our restriction formula is manifestly positive in many cases, including when M is a flag manifold. Finally, we prove the existence of integral canonical classes in the case that M is a GKM manifold the moment map component is index increasing. In this case, our restriction formula specializes to an easily computable rational sum which depends only on the GKM graph.