Nonnegative Degree Research Articles

Analytic solution of the two-star model with correlated degrees.

Exponential random graphs are important to model the structure of real-world complex networks. Here we solve the two-star model with degree-degree correlations in the sparse regime. The model constraints the average correlation between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end-points of two-stars (next nearest neighbors). We compute exactly the network free energy and show that this model undergoes a first-order transition to a condensed phase. For non-negative degree correlations between next nearest neighbors, the degree distribution inside the condensed phase has a single peak at the largest degree, while for negative degree correlations between next nearest neighbors the condensed phase is characterized by a bimodal degree distribution. We calculate the degree assortativities and show they are nonmonotonic functions of the model parameters, with a discontinuous behavior at the first-order transition. The first-order critical line terminates at a second-order critical point, whose location in the phase diagram can be accurately determined. Our results can help to develop more detailed models of complex networks with correlated degrees.

SELF-OSCILLATIONS DESCRIBED BY THE GENERALIZED VAN DER POL EQUATION

Quasilinear self-oscillations, represented by the generalized Van der Pol equation, are considered. The generalization of the equation is carried out by replacing the second degree of the velocity by its arbitrary non-negative degree. An approximate analytical solution, describing the transition of the oscillatory system to the regime of steady-state self-oscillations, is constructed using the energy balance method. A compact formula is obtained for calculating the amplitude of this regime and it is proved that it does not depend on the initial conditions. The calculation of the indicated amplitude involves using the table of gamma functions. It is shown that in some cases the obtained approximate analytical solution generalizes the known results of the theory of oscillations. To identify the errors of this solution, we integrated the generalized differential equation numerically using a computer for specific numerical data. The satisfactory consistency of the numerical results obtained by applying two different methods confirmed the adequacy of the approximate formulas for engineering calculations. The oscillations described by the generalized equation with the dissipative force of opposite sign are also studied. In this case the motion of the oscillatory system depends on the initial conditions. For the deviations of the oscillator from the position of static equilibrium smaller than a threshold value, free damped oscillations occur. In the case of large initial deviations beyond the threshold, free oscillations build up and over time the oscillator sweeps tend to the infinity for a limited period of time. The threshold deviation formula is derived which makes it possible to draw a conclusion about the stability of a dynamic system for various nonlinearity indices in the equation of motion and various initial perturbations.

A Linear-algebraic Proof of Hilbert’s Ternary Quartic Theorem

Hilbert’s ternary quartic theorem states that every nonnegative degree 4 homogeneous polynomial in three variables can be written as a sum of three squares of homogeneous quadratic polynomials. We give a linear-algebraic approach to Hilbert’s theorem by showing that a structured cone of positive semidefinite matrices is generated by rank 1 elements.

Gerstenhaber algebra and Deligne’s conjecture on the Tate–Hochschild cohomology

Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we also obtain that the singular Hochschild cochain complex has a $B$-infinity algebra structure and its cohomology ring is a Gerstenhaber algebra. Inspired by the original definition of Tate cohomology for finite groups, we define a generalized Tate-Hochschild complex with the Hochschild chains in negative degrees and the Hochschild cochains in non-negative degrees. There is a natural embedding of this complex into the singular Hochschild cochain complex. In the case of a self-injective algebra, this embedding becomes a quasi-isomorphism. In particular, for a symmetric algebra, this allows us to show that the Tate-Hochschild cohomology ring, equipped with the Gerstenhaber algebra structure, is a Batalin-Vilkovisky algebra.

Approximating Nonnegative Polynomials via Spectral Sparsification

We study polyhedral approximations to the cone of nonnegative polynomials. We show that any constant ratio polyhedral approximation to the cone of nonnegative degree $2d$ forms in $n$ variables has to have exponentially many facets in terms of $n$. We also show that for any fixed $m \geq 3$, all linear $m$-dimensional sections of the nonnegative cone that include $(x_1^2+x_2^2+\cdots + x_n^2)^d$ have a constant ratio polyhedral approximation with $O(n^{m-2})$ many facets. Our approach is convex geometric, and parts of the argument rely on the recent solution of the Kadison--Singer problem. We also discuss a randomized polyhedral approximation which might be of independent interest.

On One Property of Bounded Complexes of Discrete $$\mathbb{F}_p[\pi]$$ -modules

The aim of this paper is to prove the following assertion: let π be a profinite group and K* be a bounded complex of discret $$\mathbb{F}_p[\pi]$$ -modules. Suppose that Hi(K*) are finite Abelian groups. Then, there exists a quasi-isomorphism L* → K*, where L* is a bounded complex of discrete $$\mathbb{F}_p[\pi]$$ -modules such that all Li are finite Abelian groups. This is an analog for discrete $$\mathbb{F}_p[\pi]$$ -modules of the wellknown lemma on bounded complexes of A-modules (e.g., concentrated in nonnegative degrees), where A is a Noetherian ring, which states that any such complex is quasi-isomorphic to a complex of finitely generated A-modules, that are free with a possible exception of the module lying in degree 0. This lemma plays a key role in the proof of the base-change theorem for cohomology of coherent sheaves on Noetherian schemes, which, in turn, can be used to prove the Grothendieck theorem on the behavior of dimensions of cohomology groups of a family of vector bundles over a flat family of varieties.

Homogeneous Lyapunov Functions: From Converse Design to Numerical Implementation

The problem of the synthesis of a homogeneous Lyapunov function for an asymptotically stable homogeneous system is studied. First, for systems with nonnegative degree of homogeneity, several expressions of homogeneous Lyapunov functions are derived, which depend explicitly on the supremum or the integral (over finite or infinite intervals of time) of the system solutions. Second, a numeric procedure is proposed, which ensures the construction of a homogeneous Lyapunov function. The analytical results are illustrated by simulations.

Positivity of the Height Jump Divisor

We study the degeneration of semipositive smooth hermitian line bundles on open complex manifolds, assuming that the metric extends well away from a codimension two analytic subset of the boundary. Using terminology introduced by R. Hain, we show that under these assumptions the so-called height jump divisors are always effective. This result is of particular interest in the context of biextension line bundles on Griffiths intermediate jacobian fibrations of polarized variations of Hodge structure of weight -1, pulled back along normal function sections. In the case of the normal function on M_g associated to the Ceresa cycle, our result proves a conjecture of Hain. As an application of our result we obtain that the Moriwaki divisor on \bar M_g has non-negative degree on all complete curves in \bar M_g not entirely contained in the locus of irreducible singular curves.

NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES

AbstractWe consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

Planar homogeneous systems of difference equations

This paper is concerned to additive and multiplicative systems of homogeneous difference equations of non-negative degree. We apply a reduction in order for both additive and multiplicative systems. Then, we consider convergence and monotony of positive solutions. In fact, using convergence results on factor maps, we obtain convergence results on homogeneous systems. We will conclude that monotonic behaviour on the invariant ray (i.e. for multiplicative systems and for additive systems) may or may not be the representative of other solutions. To illustrate our results, some examples are presented by multiplicative and additive homogeneous systems of rational equations. Copyright © 2014 John Wiley & Sons, Ltd.

The negative side of cohomology for Calabi-Yau categories

We study Z-graded cohomology rings defined over Calabi–Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length 2. In particular, the products of elements of negative degrees are zero. As corollaries, we apply this to Tate–Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.

Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

We use the Thom–Whitney construction to show that infinitesimal deformations of a coherent sheaf {\mathcal F} are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf \mathcal E nd^*(\mathcal E^\cdot) , where \mathcal E^\cdot is any locally free resolution of {\mathcal F} . In particular, one recovers the well known fact that the tangent space to \mathrm{Def}_{\mathcal F} is \mathrm{Ext}^1({\mathcal F},{\mathcal F}) , and obstructions are contained in \mathrm{Ext}^2({\mathcal F},{\mathcal F}) . The main tool is the identification of the deformation functor associated with the Thom–Whitney DGLA of a semicosimplicial DGLA {\mathfrak g}^\Delta , whose cohomology is concentrated in nonnegative degrees, with a noncommutative \v{C}ech cohomology-type functor H^1_{\mathrm{sc}}(\exp {\mathfrak g}^\Delta) .

CUBIC FORMAL POWER SERIES IN CHARACTERISTIC 2 WITH UNBOUNDED PARTIAL QUOTIENTS

There is a theory of continued fractions for formal power series in $x^{−1}$ with coefficients in a field $\mathbb{F}_q$. This theory bears a close analogy with classical continued fractions for real numbers with formal power series playing the role of real numbers and the sum of the terms of non-negative degree in $x$ playing the role of the integral part. We give a family of cubic power series over $\mathbb{F}_2$ with unbounded partial quotients. To be more precise, let $f \in \mathbb{F}_2((x^{−1}))$ such that $f$ is not polynomial but $f^3$ is polynomial with degreed $d \in 3\mathbb{N}$, we prove that the continued fraction expansion of $f$ is unbounded.

On Semistable Principal Bundles over a Complex Projective Manifold

Let G be a simple linear algebraic group defined over the field of complex numbers. Fix a proper parabolic subgroup P of G, and also fix a nontrivial antidominant character χ of P. We prove that a holomorphic principal G-bundle E G over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class vanishes if and only if the line bundle over E G /P defined by χ is numerically effective. Also, a principal G-bundle E G over M is semistable with if and only if for every pair of the form (Y, ψ), where ψ is a holomorphic map to M from a compact connected Riemann surface Y, and for every holomorphic reduction of structure group E P ⊂ ψ*E G to the subgroup P, the line bundle over Y associated with the principal P-bundle E P for χ is of nonnegative degree. Therefore, E G is semistable with if and only if for each pair (Y, ψ) of the above type the G-bundle ψ*E G over Y is semistable. Similar results remain valid for principal bundles over M with a reductive linear algebraic group as the structure group. These generalize an earlier work of Miyaoka [12], where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations, one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.

Bi-relative algebraic K-theory and topological cyclic homology

It is well-known that algebraic K-theory preserves products of rings. However, in general, algebraic K-theory does not preserve fiber-products of rings, and bi-relative algebraic K-theory measures the deviation. It was proved by Cortinas that,rationally, bi-relative algebraic K-theory and bi-relative cyclic homology agree. In this paper, we show that, with finite coefficients, bi-relative algebraic K-theory and bi-relative topological cyclic homology agree. As an application, we show that for a, possibly singular, curve over a perfect field of positive characteristic p, the cyclotomic trace map induces an isomorphism of the p-adic algebraic K-groups and the p-adic topological cyclic homology groups in non-negative degrees. As a further application, we show that the difference between the p-adic K-groups of the integral group ring of a finite group and the p-adic K-groups of a maximal Z-order in the rational group algebra can be expressed entirely in terms of topological cyclic homology.

DIEUDONNÉ DETERMINANTS FOR SKEW POLYNOMIAL RINGS

We observe that the Dieudonné determinant induces a non-negative degree function on the ring of matrices over a skew polynomial ring. We then apply this degree function to two examples. In the first one, we find an expression for the rank of the kernel of an algebraic endomorphism of [Formula: see text] over a field of characteristic p > 0. In the second, we calculate the dimension of the solution space of linear matrix differential equations.

Aggregate Implications of Indivisible Labor

Abstract I suggest that the aggregate implications of indivisible labor are few, and subtle. First, I model behavior in an "indivisible labor" environment like those of Diamond and Mirrlees (1978, 1986), Hansen (1985), Rogerson (1988), Christiano and Eichenbaum (1992) and show how an inverse distribution function describing heterogeneity in the indivisible model is isomorphic with the marginal disutility schedule from the divisible labor model of Lucas and Rapping (1969). It follows that aggregate behavior in such an indivisible model is indistinguishable from aggregates generated by the divisible model; any data on aggregate hours and earnings generated by the divisible (indivisible) model can be generated by a similar parameterization of the indivisible (divisible) model. Second, I generalize the aforementioned models of indivisible labor to allow for labor supply on the "intensive" margin, and to allow for nonlinear taxes. The aggregate implications of doing the former are quite subtle, but doing the latter suggests that the indivisibility of labor may have implications for public finance. My results also imply that backward bending aggregate labor supply, and any nonnegative degree of aggregate intertemporal substitution, are consistent with standard economic theory even when all labor is supplied on the so-called "extensive" margin. Finally, my results suggest that the classic aggregate studies of labor supply by Mincer, Bowen and Finegan, and others have a simple microeconomic interpretation.

Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree

For a projective manifold whose tangent bundle is of nonnegative degree, a vector bundle on it with a holomorphic connection actually admits a compatible flat holomorphic connection, if the manifold satisfies certain conditions. The conditions in question are on the Harder-Narasimhan filtration of the tangent bundle, and on the Neron-Severi group.

Integer-valued polynomials on Krull rings

If R R is a subring of a Krull ring S S such that R Q R_{Q} is a valuation ring for every finite index Q = P ∩ R Q=P\cap R , P P in Spec 1 ( S ) ^{1}(S) , we construct polynomials that map R R into the maximal possible (for a monic polynomial of fixed degree) power of P S P PS_{P} , for all P P in Spec 1 ( S ) ^{1}(S) simultaneously. This gives a direct sum decomposition of Int ( R , S ) (R,S) , the S S -module of polynomials with coefficients in the quotient field of S S that map R R into S S , and a criterion when Int ( R , S ) (R,S) has a regular basis (one consisting of 1 polynomial of each non-negative degree).

Correction: Correction to Algorithm AS 231: The Distribution of a Noncentral χ 2 Variable with Nonnegative Degrees of Freedom

Correction: Correction to Algorithm AS 231: The Distribution of a Noncentral χ 2 Variable with Nonnegative Degrees of Freedom