Universal Polynomial Research Articles

Universal polynomials for tautological integrals on Hilbert schemes

We show that tautological integrals on Hilbert schemes of points can be written in terms of universal polynomials in Chern numbers. The results hold in all dimensions, though they strengthen known results even for surfaces by allowing integrals over arbitrary "geometric" subsets (and their Chern-Schwartz-MacPherson classes). We apply this to enumerative questions, proving a generalised G\"ottsche Conjecture for all singularity types and in all dimensions. So if L is a sufficiently ample line bundle on a smooth variety X, in a general subsystem P^d of |L| of appropriate dimension the number of hypersurfaces with given singularity types is a polynomial in the Chern numbers of (X,L). When X is a surface, we get similar results for the locus of curves with fixed "BPS spectrum" in the sense of stable pairs theory.

7. Normal Form for Infinite Type Hypersurfaces in C^2 with Nonvanishing Levi Form Derivative

In this paper, we study real hypersurfaces $M$ in ${ C}2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR transversal direction. A new phenomenon, compared to known normal forms in other cases, is the presence of resonances as roots of a universal polynomial in the 7-jet of the defining function of $M$. The main result is a complete (formal) normal form at points $p$ with no resonances. Remarkably, our normal form at such infinite type points resembles closely the Chern-Moser normal form at Levi-nondegenerate points. For a fixed hypersurface, its normal forms are parametrized by $S1\times { R}^*$, and as a corollary we find that the automorphisms in the stability group of $M$ at $p$ without resonances are determined by their 1-jets at $p$. In the last section, as a contrast, we also give examples of hypersurfaces with arbitrarily high resonances that possess families of distinct automorphisms whose jets agree up to the resonant order.

Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is,∫-11Pl′m′xt-1/1+t2-2xtPk′m′(x)/(1+t2-2tx)(l′+1)/2dx, wheret∈(0,1). The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument∫0∞Jn(αx2+z2)/(x2+z2)nx2m+1dx.

On Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1 − x2) and Their Byproducts**Supported by the National Natural Science Foundation of China under Grant No. 11275165 and Partially by 20160978-SIP-IPN, Mexico

The associated Legendre polynomials play an important role in the central fields, but in the case of the non-central field we have to introduce the universal associated Legendre polynomials Pl'm'(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential. We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor (1−x2)−p−1 as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction. The calculations are obtained systematically using some properties of the generalized hypergeometric series.

Chern classes of tensor products

We prove explicit formulas for Chern classes of tensor products of virtual vector bundles, whose coefficients are given by certain universal polynomials in the ranks of the two bundles.

Regularity of roots of polynomials

We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset \mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there exists a positive integer $k$ and a rational number $p >1$, both depending only on the degree $n$, such that if $a_j \in C^{k}$ then any continuous choice of roots of $P_a$ is absolutely continuous with derivatives in $L^q$ for all $1 \le q < p$, in a uniform way with respect to $\max_j\|a_j\|_{C^k}$. The uniformity allows us to deduce also a multiparameter version of this result. The proof is based on formulas for the roots of the universal polynomial $P_a$ in terms of its coefficients $a_j$ which we derive using resolution of singularities. For cubic polynomials we compute the formulas as well as bounds for $k$ and $p$ explicitly.

A combinatorial analysis of Severi degrees

Based on results by Brugallé and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees Nd,δ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic version of a special function associated to long-edge graphs which appeared in Fomin–Mikhalkin's formula, and conjectured it to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we consider a special multivariate function associated to long-edge graphs that generalizes their function. The main result of this paper is that the multivariate function we define is always linear. A special case of our result gives an independent proof of Block–Colley–Kennedy's conjecture.The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of Qd,δ and a bound δ for the threshold of polynomiality of Nd,δ. Next, in separate joint work with Osserman, we apply the linearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the Göttsche–Yau–Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series B1(q) and B2(q) appearing in the Göttsche–Yau–Zaslow formula.The proof of our linearity result is completely combinatorial. We define τ-graphs which generalize long-edge graphs, and a closely related family of combinatorial objects we call (τ,n)-words. By introducing height functions and a concept of irreducibility, we describe ways to decompose certain families of (τ,n)-words into irreducible words, which leads to the desired results.

Universal Racah matrices and adjoint knot polynomials: Arborescent knots

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso–Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SUN) and Kauffman (SON) polynomials. For E8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the “eigenvalue conjecture”, which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6×6 case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.

On universal knot polynomials

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel’s plane, respectively and give their exceptional group’s counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel’s plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.

Witt vector rings and the relative de Rham Witt complex

In this paper we develop a novel approach to Witt vector rings and to the (relative) de Rham Witt complex. We do this in the generality of arbitrary commutative algebras and arbitrary truncation sets. In our construction of Witt vector rings the ring structure is obvious and there is no need for universal polynomials. Moreover a natural generalization of the construction easily leads to the relative de Rham Witt complex. Our approach is based on the use of free or at least torsion free presentations of a given commutative ring R and it is an important fact that the resulting objects are independent of all choices. The approach via presentations also sheds new light on our previous description of the ring of p-typical Witt vectors of a perfect Fp-algebra as a completion of a semigroup algebra. We develop this description in different directions. For example, we show that the semigroup algebra can be replaced by any free presentation of R equipped with a linear lift of the Frobenius automorphism. Using the result in Appendix A by Umberto Zannier we also extend the description of the Witt vector ring as a completion to all F‾p-algebras with injective Frobenius map.

The Lazard formal group, universal congruences and special values of zeta functions

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist–Meurman–type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group (based on earlier works of the author). Their role in the theory of L L –genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann–Hurwitz–type zeta functions.

A characterization of almost universal ternary inhomogeneous quadratic polynomials with conductor 2

An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we provide a characterization of almost universal ternary quadratic polynomials with conductor 2.

Nonexistence of certain universal polynomials between Banach spaces

A well-known result due to W.B. Johnson (1971) asserts that the formal identity operator from ℓ1 into ℓ∞ is universal for the class of non-compact operators between Banach spaces. We show that there is neither a universal non-compact polynomial nor a universal non-unconditionally converging polynomial between Banach spaces.

Exact solutions to a class of differential equation and some new mathematical properties for the universal associated-Legendre polynomials

The exact solutions to a class of differential equation are studied. Some special cases are discussed for the central potentials, single ring-shaped potential and the angular Teukolsky equation. A new expression to the associated Legendre polynomials is found. Some new properties of the universal associated-Legendre polynomials (UALPs) including the generating function, Rodrigues’ formula, parity, some special values and the recurrence relations are presented. A new different Rodrigues’ formula of associated Legendre polynomials is also obtained.

Universal polynomials for singular curves on surfaces

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system $|L|$ with prescribed singularities is a universal polynomial of Chern numbers of $L$ and $S$, assuming $L$ is sufficiently ample. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche’s conjecture.

TH-C-19A-07: Output Factor Dependence On Range and Modulation for a New Proton Therapy System

Purpose: Proton treatment planning systems are not able to accurately predict output factors and do not calculate monitor units (MU) for proton fields. Output factors (cGy/MU) for patient-specific fields are usually measured in phantoms or modeled empirically. The purpose of this study is to predict the output factors (OFs) for a given proton (R90) and modulation width (Mod) for the first Mevion S250 proton therapy system. Methods: Using water phantoms and a calibrated ionization chamber-electrometer, over 100 OFs were measured for various R90 and Mod combinations for 24 different options. OFs were measured at the center of the Mod, which coincided with the isocenter. The measured OFs were fitted using an analytic model developed by Kooy (Phys.Med.Biol. 50, 2005) for each option and a derived universal empirical-based polynomial as a function of R90 and Mod for all options. Options are devised for ranges of R90 and Mod. The predicted OFs from both models were compared to measurements. Results: Using the empirical-based model, the values could be predicted to within 3% for at least 90% of measurements and within 5% for 98% of the measurements. Using the analytic model to fit each option with the same effective source position, the predictionmore » is much more accurate. The maximal uncertainty between measured and predicted is within 2% and the averaged root-mean-square is 1.5%. Conclusion: Although the measured data was not exhaustive, both models predicted OFs within acceptable uncertainty. Both models are currently used for a sanity check of our continual patient field OF measurements. As we acquire more patient-field OFs, the model will be refined with an ultimate goal of eliminating the time-consuming patient-specific OF measurements.« less

A characterization of almost universal ternary quadratic polynomials with odd prime power conductor

An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an effective characterization of almost universal ternary quadratic polynomials with odd prime power conductor.

Equivariant quantum Schubert polynomials

We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus.

Statistical characteristics of coefficients of a cubic approximation of isotherms of surface active substance films

We explore possibilities for polynomial approximation of the isotherms (the dependence of the pressure, exerted by the film, on the concentration of the substance per unit area) of films of surface active substances in marine environment. The analysis is based on isotherms for samples from the nearshore of the Black Sea and the coasts of the USA. The isotherms are approximated using cubic polynomials. The constant term and the coefficients of this polynomial have a reasonable scatter: their average values are comparable with their standard deviation. The standard deviations of the coefficients at the quadratic and the cubic term exceed the relevant average values by two orders of magnitude. This property signals that it is evidently not possible to approximate the isotherms of surface active films using one universal polynomial function even for one location and that it is necessary to account for the described scatter in estimates of the properties and impact of surface films. We provide cumulative distributions of the coefficients of the cubic approximation for the use in probabilistic express models of properties of surface films in the marine environment.

THE GENERAL SPIN TRIANGLE

We consider the Heisenberg spin triangle with general coupling coefficients and general spin quantum number s. The corresponding classical system is completely integrable. In the quantum case the eigenvalue problem can be reduced to that of tridiagonal matrices in at most 2s+1 dimensions. The corresponding energy spectrum exhibits what we will call spectral symmetries due to the underlying permutational symmetry of the considered class of Hamiltonians. As an application we explicitly calculate six classes of universal polynomials that occur in the high temperature expansion of spin triangles and more general spin systems.