Maximal and typical topology of real polynomial singularities

Products of knots, branched fibrations and sums of singularities

This paper is focused on the bifurcation of critical periods from a quartic rigidly isochronous center under any small quartic homogeneous perturbations. By studying the number of zeros of the first several terms in the expansion of the period function in ε, it shows that under any small quartic homogeneous perturbations, up to orders 1 and 2 in ε, there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system respectively, and the upper bound can be reached. Up to order 3 in ε, there are at most six critical periods from the periodic orbits of the unperturbed system. Moreover, we consider a family of perturbed systems of this quartic rigidly isochronous center, and obtain that up to any order in ε, there are at most two critical periods bifurcating from the periodic orbits of the unperturbed one, and the upper bound is sharp.

In this paper, we address some flaws in the material allocation function of Materials Requirements Planning (MRP). The problem formulation differs from standard MRP logic in certain important ways: start and finish times for orders are forced to be realistic and material allocations are made to minimize the total tardiness penalty associated with late completion. We show that the resulting MRP material allocation problem is NP-hard in the strong sense. A lower bound and a heuristic are developed from a mixed integer linear formulation and its Lagrangean relaxation. The lower bound and the heuristics are closer to the optimum in cases where there is either abundant material or considerable competition for material; in intermediate cases, small perturbations in material allocation can have a significant effect. A group of heuristics based on the MRP approach and its modifications is examined; they are optimal under certain conditions. An improvement method that preserves priorities inherent in any given starting solution is also presented. The Lagrangean heuristic performs better than the MRP based heuristics for a set of 3900 small problems, yielding solutions that are about 5% to 10% over the optimal. The best MRP based heuristic does about as well as the Lagrangean heuristic on a set of 120 larger problems, and is 25% to 40% better than the standard MRP approach, on the data sets tested.

In this article, by using the Riccati equation method, we investigate the maximal values of the isolated zeros of the Abelian integral for a set of quadratic reversible systems (r11) that belong to genus one, while experiencing varying 3rd, 2nd, and 1st-polynomial perturbations. Specifically, we aimed to find the upper bound for the maximal zeros of the system’s limit cycle (a special dynamic behavior in a stable state, characterized by the existence of specific periodic orbits). We know that the Abelian integral is a function of h, so when studying the maximal zeros of the function related to h, we not only consider the highest degree of the relevant function but also take into account the parity of the function and the range of values of h. Then through variable substitution, a smaller upper bound can be obtained: our findings show that the maximal values of the isolated zeros count under varying 3rd, 2nd, and 1st-polynomial perturbations is 12, improving upon previous results where the upper bound was 34 for the 3rd polynomial perturbation and 22 for the 2nd and 1st polynomial perturbations. This study represents an improvement upon previous research.

For an input graph G, an additive spanner is a sparse subgraph H whose shortest paths match those of G up to small additive error. We prove two new lower bounds in the area of additive spanners:•We construct n-node graphs G for which any spanner on $O(n)$ edges must increase a pairwise distance by $+\Omega(n^{1/7})$. This improves on a recent lower bound of $+\Omega(n^{1/10.5})$ by Lu, Wein, Vassilevska Williams, and Xu [SODA 22].•A classic result by Coppersmith and Elkin [SODA 05] proves that for any n-node graph G and set of $p=O(n^{1/2})$ demand pairs, one can exactly preserve all pairwise distances among demand pairs using a spanner on $O(n)$ edges. They also provided a lower bound construction, establishing that that this range $p=O(n^{1/2})$ cannot be improved. We strengthen this lower bound by proving that, for any constant k, this range of p is still unimprovable even if the spanner is allowed $+k$ additive error among the demand pairs. This negatively resolves an open question asked by Coppersmith and Elkin [SODA 05] and again by Cygan, Grandoni, and Kavitha [STACS 13] and Abboud and Bodwin [SODA 16].At a technical level, our lower bounds are obtained by an improvement to the entire obstacle product framework used to compose “inner” and outer” graphs into lower bound instances. In particular, we develop a new strategy for analysis that allows certain non-layered graphs to be used in the product, and we use this freedom to design better inner and outer graphs that lead to our new lower bounds.

Homogeneous oscillations of the inflaton after inflation can be unstable to small spatial perturbations even without coupling to other fields. We show that for inflaton potentials ∝Φ|<sup>2n</sup> near |Φ|=0 and flatter beyond some |Φ|=M, the inflaton condensate oscillations can lead to self-resonance, followed by its complete fragmentation. We find that for non-quadratic minima (n>1), shortly after backreaction, the equation of state parameter, w→1/3. If M << m<sub>Pl</sub>, radiation domination is established within less than an e-fold of expansion after the end of inflation. In this case self-resonance is efficient and the condensate fragments into transient, localised spherical objects which are unstable and decay, leaving behind them a virialized field with mean kinetic and gradient energies much greater than the potential energy. This end-state yields w=1/3. When M~m<sub>Pl </sub>we observe slow and steady, self-resonace that can last many e-folds before backreaction eventually shuts it off, followed by fragmentation and w→1/3. We provide analytical estimates for the duration to w→1/3 after inflation, which can be used as an upper bound (under certain assumptions) on the duration of the transition between the inflationary and the radiation dominated states of expansion. This upper bound can reduce uncertainties in CMB observables such as the spectral tilt n<sub>s</sub>, and the tensor-to-scalar ratio r. For quadratic minima (n=1), w→0 regardless of the value of M. Finally, this is because when M << m<sub>Pl</sub>, long-lived oscillons form within an e -fold after inflation, and collectively behave as pressureless dust thereafter. For M ~ m <sub>Pl</sub> , the self-resonance is inefficient and the condensate remains intact (ignoring long-term gravitational clustering) and keeps oscillating about the quadratic minimum, again implying w = 0 .

Ranged hash functions generalize hash tables to the setting where hash buckets may come and go over time, a typical case in distributed settings where hash buckets may correspond to unreliable servers or network connections. Monotone ranged hash functions are a particular class of ranged hash functions that minimize item reassignments in response to churn: changes in the set of available buckets. The canonical example of a monotone ranged hash function is the ring-based consistent hashing mechanism of Karger et al. [13]. These hash functions give a maximum load of Θ (n/mlogm) when n is the number of items and m is the number of buckets. The question of whether some better bound could be obtained using a more sophisticated hash function has remained open. We resolve this question by showing two lower bounds. First, the maximum load of any randomized monotone ranged hash function is Ω(√n/mlnm) when n = o(mlogm). This bound covers almost all of the nontrivial case, because when n = Ω(mlogm) simple random assignment matches the trivial lower bound of Ω(n/m). We give a matching (though impractical) upper bound that shows that our lower bound is tight over almost all of its range. Second, for randomized monotone ranged hash functions derived from metric spaces, there is a further trade-off between the expansion factor of the metric and the load balance, which for the special case of growth-restricted metrics gives a bound of Ω(n/mlogm), asymptotically equal to that of consistent hashing. These are the first known non-trivial lower bounds for ranged hash functions. They also explain why in ten years no better ranged hash functions have arisen to replace consistent hashing.

In 1974, Mark Goresky and Robert MacPherson began their development of intersection homology theory (see [24] in these volumes), and their first paper on this topic appeared in 1980; see [12]. At that time, they were missing a fundamental tool which was available for the study of smooth manifolds; they had no Morse Theory for stratified spaces. Goresky and MacPherson wished to have a Stratified Morse Theory to allow them to prove a Lefschetz hyperplane theorem for the intersection homology of complex singular spaces, just as ordinary Morse Theory yields the Lefschetz Hyperplane Theorem for ordinary homology of complex manifolds ([34], §7). The time was ripe for a stratified version of Morse Theory. In 1970, Mather had given a rigorous proof of Thom’s first isotopy lemma [33]; this result says that proper, stratified, submersions are locally-trivial fibrations. In 1973, Morse functions on singular spaces had been defined by Lazzeri in [25], and the density and stability of Morse functions under perturbations had been proved in [37]. We shall recall these definitions and results in Section 2. What was missing was the analog of the fundamental result of Morse Theory, a theorem describing how the topology of a space is related to the critical points of a proper Morse function. In [16], Goresky and MacPherson proved such a theorem for stratified spaces. Suppose that M is a smooth manifold, that X is a Whitney stratified subset ofM , and that f : X → R is a proper function which is the restriction of a smooth function on M . For all v ∈ R, let X≤v := f−1((−∞, v]). Suppose that a, b ∈ R, a < b, and f−1([a, b]) contains a single (stratified) critical point, p, which is non-degenerate (see Definition 2.3) and contained in the open set f−1((a, b)). Let S be the stratum containing p. Then, the Main Theorem of Stratified Morse Theory (see Theorem 2.16) says that the topological space X≤b is obtained from the space X≤a by attaching a space A to X≤a along a subspace B ⊆ A, where the pair (A,B), the Morse data, is the product of the tangential Morse data of f at p and the normal Morse data of f at p. This result is especially powerful in the complex analytic case, where the normal Morse data depends on the stratum S, but not on the point p or on the particular Morse function f . Detailed proofs of these results appeared in the 1988 book Stratified Morse Theory [16]; we present a summary of a number of these results in Section 2. Even before the appearance of [16], Goresky and MacPherson published two papers, Stratified Morse Theory [15] and Morse Theory and Intersection Homology Theory [14], which contained announcements of many of the fundamental definitions and results of Stratified Morse Theory. In addition, these two papers showed that Stratified Morse Theory has a number of important applications to complex analytic spaces, including homotopy results, the desired Lefschetz Hyperplane Theorem for intersection homology, and the first proof that the (shifted) nearby cycles of a perverse sheaf are again perverse. We shall discuss these results and others in Section 3.

This paper deals with the bifurcation of critical periods from a rigidly quartic isochronous center. It shows that under any small homogeneous perturbation of degree four, up to any order in [Formula: see text], there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system, and the upper bound is sharp. In addition, we further prove that under any small polynomial perturbation of degree [Formula: see text], up to the first order in [Formula: see text], there are at most [Formula: see text] critical periods bifurcating from the periodic orbits of the unperturbed quartic system.

A coincidence point of a pair of mappings is an element, at which these mappings take on the same value. Coincidence points of mappings of partially ordered spaces were studied by A.V. Arutyunov, E.S. Zhukovskiy, and S.E. Zhukovskiy (see Topology and its Applications, 2015, V. 179, No. 1, p. 13–33); they proved, in particular, that an orderly covering mapping and a monotone one, both acting from a partially ordered space to a partially ordered one, possess a coincidence point. In this paper, we study the existence of a coincidence point of a pair of mappings that act from a partially ordered space to a set, where no binary relation is defined, and, consequently, it is impossible to define covering and monotonicity properties of mappings. In order to study the mentioned problem, we define the notion of a “quasi-coincidence” point. We understand it as an element, for which there exists another element, which does not exceed the initial one and is such that the value of the first mapping at it equals the value of the second mapping at the initial element. It turns out that the following condition is sufficient for the existence of a coincidence point: any chain of “quasi-coincidence” points is bounded and has a lower bound, which also is a “quasi-coincidence” point. We give an example of mappings that satisfy the above requirements and do not allow the application of results obtained for coincidence points of orderly covering and monotone mappings. In addition, we give an interpretation of the stability notion for coincidence points of mappings that act in partially ordered spaces with respect to their small perturbations and establish the corresponding stability conditions.

We show how one can obtain a lower bound for the electrical, spin, or heat conductivity of correlated quantum systems described by Hamiltonians of the form $H={H}_{0}+g{H}_{1}$. Here, ${H}_{0}$ is an interacting Hamiltonian characterized by conservation laws which lead to an infinite conductivity for $g=0$. The small perturbation $g{H}_{1}$, however, renders the conductivity finite at finite temperatures. For example, ${H}_{0}$ could be a continuum field theory, where momentum is conserved, or an integrable one-dimensional model, while ${H}_{1}$ might describe the effects of weak disorder. In the limit $g\ensuremath{\rightarrow}0$, we derive lower bounds for the relevant conductivities and show how they can be improved systematically using the memory matrix formalism. Furthermore, we discuss various applications and investigate under what conditions our lower bound may become exact.

We prove strong lower bounds for the space complexity of (/spl epsi/, /spl delta/)-approximating the number of distinct elements F/sub 0/ in a data stream. Let m be the size of the universe from which the stream elements are drawn. We show that any one-pass streaming algorithm for (/spl epsi/, /spl delta/)-approximating F/sub 0/ must use /spl Omega/(1//spl epsi//sup 2/) space when /spl epsi/ = /spl Omega/(m/sup -1/(9 + k)/), for any k > 0, improving upon the known lower bound of /spl Omega/(1//spl epsi/) for this range of /spl epsi/. This lower bound is tight up to a factor of log log m for small /spl epsi/ and log 1//spl epsi/ for large /spl epsi/. Our lower bound is derived from a reduction from the one-way communication complexity of approximating a Boolean function in Euclidean space. The reduction makes use of a low-distortion embedding from an l/sub 2/ to l/sub 1/ norm.

The design of aircraft for ground maneuvers is an essential part in satisfying the demanding requirements of the aircraft operators. Extensive analysis is done to ensure that a new civil aircraft type will adhere to these requirements, for which the nonlinear nature of the problem generally adds to the complexity of such calculations. Small perturbations in velocity, steering angle, or brake application may lead to significant differences in the final turn widths that can be achieved. Here, the U-turn maneuver is analyzed in detail, with a comparison between the two ways in which this maneuver is conducted. A comparison is also made between existing turn-width prediction methods that consist mainly of geometric methods and simulations and a proposed new method that uses dynamical systems theory. Some assumptions are made with regard to the transient behavior, for which it is shown that these assumptions are conservative when an upper bound is chosen for the transient distance. Furthermore, we demonstrate that the results from the dynamical systems analysis are sufficiently close to the results from simulations to be used as a valuable design tool. Overall, dynamical systems methods provide an order-of-magnitude increase in analysis speed and capability for the prediction of turn widths on the ground when compared with simulations.

We study resonances near the real axis (|Im z | = O ( h N ), N &gt; 1) and the corresponding resonant states for semiclassical long range operators P ( h ). Without a priori assumptions on the distribution or on the multiplicities of the resonances, we show that the truncated resonant states form a family of quasimode states for P ( h ), stable under small perturbations. As a consequence, they also form a family of quasimode states for any suitably defined (self-adjoint) reference operator P #( h ), therefore, those resonances are perturbed eigenvalues of P #( h ). Next we show that the semiclassical wave front set of the resonant states is contained in the set of trapped directions [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. We construct a suitable reference operator from P ( h ) by imposing a microlocal barrier outside [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] to show that the counting function for those resonances admits an upper bound of Weyl's type connected with the measure of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]. We give an example of a system for which this bound is optimal and also prove similar bounds in case of classical scattering by obstacle.

Kulkarni and Kiyavash recently introduced a new method to establish upper bounds on the size of deletion-correcting codes. This method is based upon tools from hypergraph theory. The deletion channel is represented by a hypergraph whose edges are the deletion balls (or spheres), so that a deletion-correcting code becomes a matching in this hypergraph. Consequently, a bound on the size of such a code can be obtained from bounds on the matching number of a hypergraph. Classical results in hypergraph theory are then invoked to compute an upper bound on the matching number as a solution to a linear-programming problem: the problem of finding fractional transversals. The method by Kulkarni and Kiyavash can be applied not only for the deletion channel but also for other error channels. This paper studies this method in its most general setup. First, it is shown that if the error channel is regular and symmetric then the upper bound by this method coincides with the well-known sphere packing bound and thus is called here the generalized sphere packing bound. Even though this bound is explicitly given by a linear programming problem, finding its exact value may still be a challenging task. The art of finding the exact upper bound (or slightly weaker ones) is the assignment of weights to the hypergraph’s vertices in a way that they satisfy the constraints in the linear programming problem. In order to simplify the complexity of the linear programming, we present a technique based upon graph automorphisms that in many cases significantly reduces the number of variables and constraints in the problem. We then apply this method on specific examples of error channels. We start with the $Z$ channel and show how to exactly find the generalized sphere packing bound for this setup. Next studied is the nonbinary limited magnitude channel both for symmetric and asymmetric errors, where we focus on the single-error case. We follow up on the deletion channel, which was the original motivation of the work by Kulkarni and Kiyavash, and show how to improve upon their upper bounds for single-deletion-correcting codes. Since the deletion and grain-error channels have a similar structure for a single error, we also improve upon the existing upper bounds on single-grain error-correcting codes. Finally, we apply this method for projective spaces and find its generalized sphere packing bound for the single-error case.