We enumerate all isotopy classes of degree three Morse polynomials R3 * R1 with nonsingular principal homogeneous parts, proving that there are exactly 37 of them. We also count all 2258 isotopy classes of strictly Morse polynomials R3 * R1 of degree three with the maximal possible number (eight) of real critical points. A main tool in this classification is a combinatorial computer program that formalizes Morse surgeries, local monodromy and Picard-Lefschetz theory.
After investigating the geodesic triangles and their angle <^> sums in Nil and SL2 R geometries, we consider the analogous problem in Sol space, which is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sum of geodesic triangles, and we prove that it can be larger than, less than, or equal to pi. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments, and as a special case of this, we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work, we will use the projective model of Sol described by E. Molnar in 1997.
Let G be a locally compact group with the left Haar measure mG. A probability measure & micro; on G is said to be strictly aperiodic if the support of & micro; is not contained in a proper closed left cosets of G. Let & micro; and v be two commuting (with respect to convolution) probability measures on a compact group G and let 1 Mn(& micro;, v) := n X n + 1 i=0 & micro;i & lowast;vn-i be the Birkhoff average of the pair (& micro;, v). Among other results, we show that if one of these measures is strictly aperiodic, then w & lowast;-lim n ->infinity Mn(& micro;, v) = mH, where H is the closed subgroup of G generated by supp & micro; boolean OR supp v and mH is the measure on G defined by mH(B) = mH(B boolean AND H) for every Borel subset B of G. 2020 MATH. SUBJ. CLASS. 28A33, 43A10, 43A77, 47A35.
Minkowski mixed volume of n subpolytopes D1, ... , Dn of a polytope P subset of Rn clearly does not exceed the normalized volume n! Vol(P). Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face F & subne; P intersects at least dim(F) + 1 of the polytopes Di. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems. Motivated by relaxing the bound dim(F) + 1 to dim(F), we prove a combinatorial formula for the mixed volume of a broad class of semi-interlaced polytopes. This class includes, in particular, the off-coordinate polytopes used in computing algebraic degrees such as Maximum Likelihood, Euclidean Distance, and Polar degrees via the Kouchnirenko-Bernshtein theory. We also present applications of our results to the Arnold monotonicity problem (Problem 1982-16), which concerns the dependence of Milnor numbers on the Newton polyhedra.
We construct an irreducible rational curve of degree 10 in CP2 which has 12 triple points, and a union of three rational quartics with 19 triple points. This gives counter-examples to a conjecture by Dimca, Harbourne, and Sticlaru. We also prove that there exists an analytic family C-u of curves of degree 10 with 12 triple points which tends, as u -> 0, to the union of the dual Hesse arrangement of lines (9 lines with 12 triple points) with an additional line. We hope that our approach to the proof of the latter fact could be of independent interest.
It is presented an example of a holomorphic foliation of a non-algebraizable surface which is topologically equivalent to an algebraic foliation.
We study the graded Lie algebra L(RC_K) associated with the lower central series of a right-angled Coxeter group. We construct a surjective homomorphism from the polynomial ring over an explicit Lie algebra N_K to the commutator subalgebra of L(RC_K), and conjecture that it is an isomorphism. The homomorphism is defined in terms of a new operation in Lie algebras associated with groups generated by involutions, which corresponds to the squaring and has an analogue in homotopy theory. We show that the universal enveloping algebra U(N_K) is isomorphic to the mod 2 loop homology algebra of the corresponding moment-angle complex ZK. This allows us to give a presentation of the Lie algebra N_K by generators and relations.
We obtain a complete list of smooth projective threefolds over C for which the dimension of the space of vanishing cycles (in H2(Y, Q) of the smooth hyperplane section Y) equals 2. We also obtain a complete list of rank 2 very ample vector bundles E on smooth projective surfaces with c2(E) = 3.
This article deals with dihedral group actions on compact Riemann surfaces and the interplay between different geometric data associated to them. First, a bijective correspondence between geometric signatures and analytic representations is obtained. Second, a refinement of a result of Bujalance, Cirre, Gamboa and Gromadzki about signature realization is provided. Finally, we apply our results to isogeny decompositions of Jacobians by Prym varieties and by elliptic curves, extending results of Carocca, Recillas and Rodríguez. In particular, we give a complete classification of Jacobians with dihedral action whose group algebra decomposition induces a decomposition into factors of the same dimension.
We study the global analytic properties of a space X with a horn type singularity. In particular, we introduce some de Rham complex of square integrable forms and we describe its homology and the spectral properties of the associated Hodge Laplace operator. All this is applied to produce a suitable description of the analytic torsion of X and to prove an extension of the Cheeger Müller theorem.
In this paper, we get a Sobolev-Jawerth embedding for weighted Triebel-Lizorkin-Morrey-Lorentz spaces by the theories of indices of the weighted Lorentz spaces and Littlewood-Paley spaces. Based on this result, the boundedness of the fractional integral operator on weighted Triebel-Lizorkin-Morrey-Lorentz spaces and weighted Hardy-Morrey-Lorentz spaces are established. Specifying the weights we recover the existing results as well as we acquire new results in the new and old settings.
Mixed-norm space is a generalization of classical Lebesgue space L-p(R-d). It considers functions with independent variables under possibly different meanings. In this paper, we give a research on mixed-norm Besov space B-p,q(s)(R-d) defined by modulus of smoothness, which can be decomposed by another method based on the well-known Littlewood-Paley decomposition theory. Then we get two new results of mixed-norm Besov spaces' characterization. The first one is the Littlewood-Paley type characterization, which is received by studying the mutual control between the modulus of smoothness omega(2)(p) and the Littlewood-Paley decomposition operator L(j)f under mixed-norm. Based on this Littlewood-Paley type characterization, we get a characterization theorem for B-p,q(s)(R-d) by wavelets.
In this paper authors will prove generalization of the Lah-Ribaric inequality for sequences of selfadjoint operators in Hilbert space. They will also give further improvement of the same inequality and bounds for difference between its sides. This improvement will likewise result with more accurate bounds for the gap in the Jensen tensorial inequality for sequences of selfadjoint operators.
One of the fundamental results of three-dimensional topology is the Kneser-Milnor unique decomposition theorem. If a 3-manifold admits a Morse-Smale diffeomorphism without hetero clinic curves, the topology of the decomposition summands can be substantially refined. For orientable 3-manifolds this was done by C. Bonatti, V.Z. Grines, V.S. Medvedev and E. Pecou in 2002. In the present paper, we obtain an exhaustive description of the decomposition into a connected sum of non-orientable 3-manifolds admitting Morse-Smale diffeomorphisms without hetero clinic curves.
Let X be a simply connected CW-complex of dimension n. This paper aims to establish a link between the groups E(X) and Gamma n(X ). Here, E (X) represents the group of self-homotopy equivalences of X, while the group Gamma n(X ), which was intruduced by Whitehead, is the image of 7rn of the (n-1)-skeleton of X in 7rn of the n-skeleton of X. This pursuit provides significant insights into discerning the presence of torsion elements within E (X ). 2020 MATH. SUBJ. CLASS. 55P10.
It is well known that the "hit problem" is an important problem in algebraic topology, which involves determining a minimal generating set for a specific module related to the Steenrod algebra. While notable progress has been made for small cases, the general problem remains unsolved, particularly for larger numbers of variables. A related application in this study is to describe the Singer cohomological transfer, which provides insights into the structure of the (mod-2) cohomology groups of the Steenrod algebra. Nonetheless, these cohomology groups remain poorly understood in higher homological degrees. In this work, we strengthen results for the hit problem with five or more variables in certain generic degrees and analyze the behavior of the Singer transfer in the relevant bidegrees. Additionally, we provide a set of efficient, computer-assisted algorithms implementable in SageMath and Maple that effectively address various aspects of the hit problem and the Singer transfer.
We introduce invariants of lattices in real quadratic fields that are constructed from the first derivative at s = 0 of certain L-series. These invariants are able to distinguish the contribution of each of the two embeddings of the base field into R. Our construction makes use of the first cohomology group of PGL2(Q) with coefficients in a module of distributions. This technique allows us to control and sometimes remove the effect of choosing coordinates in the description of such lattices. Furthermore, we explicitly compute the invariants in the simplest cases.
