Varieties isogenous to a higher product with prescribed numerical invariants

This study incorporated topology Betti number (BN) features into the prediction of primary sites of brain metastases and the construction of magnetic resonance-based imaging biopsy (MRB) models. The significant features of the MRB model were selected from those obtained from gray-scale and three-dimensional wavelet-filtered images, BN and inverted BN (iBN) maps, and clinical variables (age and gender). The primary sites were predicted as either lung cancer or other cancers using MRB models, which were built using seven machine learning methods with significant features chosen by three feature selection methods followed by a combination strategy. Our study dealt with a dataset with relatively smaller brain metastases, which included effective diameters greater than 2mm, with metastases ranging from 2 to 9mm accounting for 17% of the dataset. The MRB models were trained by T1-weighted contrast-enhanced images of 494 metastases chosen from 247 patients and applied to 115 metastases from 62 test patients. The most feasible model attained an area under the receiver operating characteristic curve (AUC) of 0.763 for the test patients when using a signature including features of BN and iBN maps, gray-scale and wavelet-filtered images, and clinical variables. The AUCs of the model were 0.744 for non-small cell lung cancer and 0.861 for small cell lung cancer. The results suggest that the BN signature boosted the performance of MRB for the identification of primary sites of brain metastases including small tumors.

The family of Buchsbaum simplicial posets over a field K provides an algebraic abstraction of the family of (K-homology) manifold triangulations. In 2008, Novik and Swartz established lower bounds on the face numbers of a Buchsbaum simplicial poset as a function of its dimension and its topological Betti numbers over K. They conjectured that these lower bounds are sufficient to classify face numbers of Buchsbaum simplicial posets with prescribed Betti numbers. We prove this conjecture by using methods from the theory of (pseudo)manifold crystallizations to construct simplicial posets with prescribed face numbers and Betti numbers.

We introduce characteristic numbers of a graph and demonstrate that they are a combinatorial analogue of topological Betti numbers. We then use characteristic numbers and related tools to study Hamiltonian GKM manifolds whose moment maps are in general position. We study the connectivity properties of GKM graphs and give an upper bound on the second Betti number of a GKM manifold. When the manifold has dimension at most 10, we use this bound to conclude that the manifold has nondecreasing even Betti numbers up to half the dimension, which is a weak version of the Hard Lefschetz Property.

Topological Characterisation of 3D Digital Image Based on Topological and Betti Numbers

Quantitative methods for correlating dispersion and electrical conductivity in conductor–polymer nanostrand composites

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to determine the $\mathfrak S_n$-module structure of $\mathrm{Tor}_i(I,\Bbbk)$. Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an $\mathfrak S_n$-equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of $\mathfrak S_n$-invariant monomial ideals, and in particular recover formulas for their Castelnuovo--Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or $>n$) we compute the $\mathfrak S_n$-invariant part of $\mathrm{Tor}_i(I,\Bbbk)$ in terms of $\mathrm{Tor}$ groups of the unsymmetrization of $I$.

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a fewnomial hypersurface in R N. In the book Fewnomials (6), A. Khovanskii gives bounds on the Betti numbers of varieties X in R n or in the positive orthant R n . These varieties include algebraic varieties, varieties defined by polynomial functions in the variables and exponentials of the variables, and also by even more general functions. Here, we consider only algebraic varieties. For these, Khovanskii bounds the sum b∗(X) of Betti numbers of X by a function that depends on n, the codimension of X, and the total number of monomials appearing (with non zero coefficients) in polynomial equations definingX. For real algebraic varieties X, the problem of bounding b∗(X) has a long history. O. A. Oleinik (8) and J. Milnor (7) used Morse theory to estimate the number of critical points of a suitable Morse function to obtain a bound. R. Thom (12) used Smith Theory to bound the mod-2 Betti numbers. This Smith-Thom bound has the form b∗(X) ≤ b∗(XC), where XC is the complex variety defined by the same polynomials as X. For instance, if X ⊂ R n is defined by polynomials of degree at most d, then Milnor's bound is b∗(X) ≤ d(2d − 1) n . If X ⊂ (R {0}) n is a smooth hypersurface defined by a polynomial with Newton polytope P , then the Smith-Thom bound is b∗(X) ≤ b∗(XC) = n! ¢ Vol(P ), where Vol(P ) is the usual volume of P. We refer to (10) for an informative history of the subject, and to the book (1) for more details. These bounds are given in term of degree or volume of a Newton polytope which are numerical deformation invariants of the complex variety XC. In contrast, the topology of a real algebraic variety depends on the coefficients of its defini ng equations, and in partic- ular, on the number of monomials involved in these equations. For instance, Descartes's rule of signs implies that the number of positive roots of a real univariate polynomial is less than its number of monomials, but the number of complex roots is equal to its degree. Khovanskii's bound can be seen as a generalization of this Descartes bound. It is smaller than the previous bounds when the defining equations have few monomials compared to their degrees. While it has always been clear that Khovanskii's bounds are unrealistically large, it appears very challenging to sharpen them. Some progress has been made recently for the number of non degenerate positive solutions to a system of n polynomials in n variables (3). Here, we bound b∗(X), when X is a fewnomial hypersurface. Suppose that X ⊂ R n is a smooth hypersurface defined by a Laurent polynomial with

The Kodaira dimension of a non-minimal manifold is defined to be that of any of its minimal models. It is shown in [12] that, if ω is a Kahler form on a complex surface (M,J), then κ(M,ω) agrees with the usual holomorphic Kodaira dimension of (M,J). It is also shown in [12] that minimal symplectic 4−manifolds with κ = 0 are exactly those with torsion canonical class, thus can be viewed as symplectic Calabi-Yau surfaces. Known examples of symplectic 4−manifolds with torsion canonical class are either Kahler surfaces with (holomorphic) Kodaira dimension zero or T 2−bundles over T 2 ([10], [12]). They all have small Betti numbers and Euler numbers: b+ ≤ 3, b ≤ 19 and b1 ≤ 4; and the Euler number is between 0 and 24. It is speculated in [12] that these are the only ones. In this paper we prove that it is true up to rational homology.

It is well known that a compact analytic manifold carries, besides the usual topological invariants (its Euler characteristic, the Betti numbers), some numerical invariants like the degree of the canonical divisor, the Hodge numbers, the arithmetic genus, and so on. A common property of all these invariants is that they coincide with the dimension of the cohomology spaces of certain analytic vector bundles or analytic sheaves. In this way, these invariants become accessible to algebraic and homological methods.

We show that if the degree of a nonsingular projective variety is high enough, maximization of any of the most important numerical invariants, such as class, Betti number, and any of the Chern or middle Hodge numbers, leads to the same class of extremal varieties. Moreover, asymptotically (say, for varieties whose total Betti number is big enough) the ratio of any two of these invariants tends to a well-defined constant.

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1(C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.

The properties of the algebraic structure of fermion density matrices are studied. The algebraic structure of a density matrix leads to a more varied and detailed classification scheme than that offered by the usual shell structure approach. Investigation of the algebraic structure of fermion density matrices by the methods of algebraic topology leads to a classification scheme based on Betti numbers.

Over the recent decades, a variety of indices, such as the fractal dimension, Hurst exponent, or Betti numbers, have been used to characterize structural or topological properties of art via a singular parameter, which could then help to classify artworks. A single fractal dimension, in particular, has been commonly interpreted as characteristic of the entire image, such as an abstract painting, whether binary, gray-scale, or in color, and whether self-similar or not. There is now ample evidence, however, that fractal exponents obtained using the standard box-counting are strongly dependent on the details of the method adopted, and on fitting straight lines to the entire scaling plots, which are typically nonlinear. Here, we propose a more discriminating approach with the aim of obtaining robust scaling plots and extracting relevant information encoded in them without any fitting routines. To this goal, we carefully average over all possible grid locations at each scale, rendering scaling plots independent of any particular choice of grids and, crucially, of the orientation of images. We then calculate the derivatives of the scaling plots, so that an image is described by a continuous function, its fractal contour, rather than a single scaling exponent valid over a limited range of scales. We test this method on synthetic examples, ordered and random, then on images of algorithmically defined fractals, and finally, examine selected abstract paintings and prints by acknowledged masters of modern art.

Let $\mathcal{F}$ be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that $\mathcalF$ is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity; therefore the limit sets of the leaves are continuous images of the circle. One important corollary is that if $\mathcal{F}$ is a Reebless, infinite depth foliation in a hyperbolic 3-manifold, then it has the continuous extension property. Such infinite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classes of foliations, including a large class of foliations where all leaves are dense, and infinitely many examples with one sided branching. One extremely useful tool is a detailed understanding of the topological structure and asymptotic properties of the 1-dimensional foliations in the leaves of e $\mathcal{F}$ induced by the stable and unstable foliations of the flow.

Chordal clutters in the sense of Bigdeli et al. (J Comb Theory Ser A 145:129---149, 2017) and Morales et al. (Ann Fac Sci Toulouse Ser 6 23(4):877---891, 2014) are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear resolution appears as the Betti sequence of the circuit ideal of such a chordal clutter. Associated with any simplicial order is a sequence of integers which we call the $$\lambda $$ź-sequence of the chordal clutter. All possible $$\lambda $$ź-sequences are characterized. They are intimately related to the Hilbert function of a suitable standard graded K-algebra attached to the chordal clutter. By the $$\lambda $$ź-sequence of a chordal clutter, we determine other numerical invariants of the circuit ideal, such as the $$\mathbf h $$h-vector and the Betti numbers.