On the spectrum of critical sets in latin squares of order 2n

Suppose that L is a latin square of order m and P ? L is a partial latin square. If L is the only latin square of order m which contains P, and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2ⁿ. The back circulant latin square of even order m has a well‐known critical set of size m²/4, and this is the smallest known critical set for a latin square of order m. The abelian 2‐group of order 2ⁿ has a critical set of size 4ⁿ‐3ⁿ, and this is the largest known critical set for a latin square of order 2ⁿ. We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2ⁿ and the abelian 2‐group of order 2ⁿ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 25–43, 2008     

Critical Points Index for Vector Functions and Vector Optimization

In this work, we study the critical points of vector functions from ℝ ⁿ to ℝ ^m with n≥m, following the definition introduced by Smale in the context of vector optimization. The local monotonicity properties of a vector function around a critical point which are invariant with respect to local coordinate changes are considered. We propose a classification of critical points through the introduction of a generalized Morse index for a critical point, consisting of a triplet of nonnegative integers. The proposed index is based on the sign of an appropriate invariant vector-valued second-order differential.     

On a nonlinear elliptic problem with critical potential in ?2

Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the critical potential in ℝ². By establishing a weighted inequality with the best constant, determine the critical potential in ℝ², and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexistence result of solutions with singular points to the nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the mountain pass theorem.     

Critical exponents in a degenerate parabolic equation with weighted source

This article deals with the Fujita-type theorems to the Cauchy problem of degenerate parabolic equation not in divergence form with weighted source u _t ?=?u ^p Δu?+?a(x)u ^q in ? ⁿ ?×?(0,?T), where p?≥?1, q?>?1, and the positive weight function a(x) is of the order |x| ^m with m?>??2. It was known that for the degenerate diffusion equation in divergence form, the weight function affects both of the critical Fujita exponent and the second critical exponent (describing the critical smallness of initial data required by global solutions via the decay rates of the initial data at space-infinity). Contrarily, it is interesting to prove that the weight function in the present model with degenerate diffusion not in divergence form influences the second critical exponent only, without changing the critical Fujita exponent.     

Bond percolation critical probability bounds for three Archimedean lattices

Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices: .7385 < p_c((3, 12²) bond) < .7449, .6430 < p_c((4, 6, 12) bond) < .7376, .6281 < p_c((4, 8²) bond) < .7201. Consequently, the bond percolation critical probability of the (3, 12²) lattice is strictly larger than those of the other ten Archimedean lattices. Thus, the (3, 12²) bond percolation critical probability is possibly the largest of any vertex‐transitive graph with bond percolation critical probability that is strictly less than one. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 507–518, 2002     

Conditional extremals in complete Riemannian manifolds

Conditional extremal curves in a complete Riemannian manifold M are defined as the critical points of the squared L² distance between the tangent vector field of a curve and a so-called prior vector field. We prove that this L² distance satisfies the Palais-Smale condition on the space of absolutely continuous curves joining two submanifolds of M, and thus establish the existence of critical points. We also prove a Morse index theorem in the case where the two submanifolds are single points, and use the Morse inequalities to place lower bounds on the number of critical points of each index.     

Robust rigidity for circle diffeomorphisms with singularities

We prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities, their C ¹-smooth equivalence follows from exponential convergence of those renormalizations. As an easy corollary, any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are C ¹-smoothly conjugate.     

Periodic solutions for a class of superquadratic Hamiltonian systems

We prove the existence of nontrivial critical points for a class of superquadratic nonautonomous second-order Hamiltonian systems by applying condition ^∗(C) to critical point theory, and some new solvability conditions of nontrivial periodic solutions are obtained.     

Restricted Percolation Critical Exponents in High Dimensions

Despite great progress in the study of critical percolation on ℤ^d for d large, properties of critical clusters in high-dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high-dimensional critical exponents; see in particular the conjecture by Kozma-Nachmias that the probability that 0 and (n, n, n, …) are connected within [−n, n]^d scales as n^{−2 − 2d} . In this paper, we study the properties of critical clusters in high-dimensional half-spaces and boxes. In half-spaces, we show that the probability of an open connection (“arm”) from 0 to the boundary of a sidelength n box scales as n⁻³ . We also find the scaling of the half-space two-point function (the probability of an open connection between two vertices) and the tail of the cluster size distribution. In boxes, we obtain the scaling of the two-point function between vertices which are any macroscopic distance away from the boundary. Our argument involves a new application of the “mass transport" principle which we expect will be useful to obtain quantitative estimates for a range of other problems. © 2020 Wiley Periodicals LLC     

Regularity for critical points of a non local energy

We study the regularity of critical points of an energy which stems from micromagnetism theory. First we show that in dimension two critical points are smooth in B ² . In the three dimensional case we prove that the stationary critical points of the energy are smooth except in a subset of one dimensional Hausdorff measure zero. The particularity of this work is the non local character of one term of the energy. Received July 10, 1995 / Accepted April 15, 1996     

Stability of Critical Values and Isolated Critical Continua

We extend the study of critical points in [4]. We show that isolated components of critical points lying on a levelset can be described by an integer which is a lower bound to the “number” of critical points of any function near to the original one in C¹-sup-norm. We also derive a global theorem about continua of critical values similar to that given by Rabinowitz for continua of solutions of certain nonlinear eigenvalue problems. We give a simple application of our abstract results to the problem of bifurcation for gradient systems when the linearization is not completely continuous.     

Exceptional planes of percolation

Consider the standard continuous percolation in ℝ⁴, and choose the parameters so that the induced percolation on a fixed two dimensional linear subspace is critical. Although two dimensional critical percolation dies, we show that there are exceptional two dimensional linear subspaces, in which percolation occurs. Received: 1 April 1997 / Revised version: 20 January 1998     

La dérivée schwarzienne en dynamique unimodale

The first return map of a C³ unimodal map to a neighbourhood of the critical point is conjugated by a real-analytic diffeomorphism to a map with negative Schwarzian derivative. Consequently, we obtain the classification of the metric attractors for smooth unimodal maps with non-degenerate critical point.     

Focal points and support functions in affine differential geometry

The notions of focal point and support function are considered for a nondegenerate hypersurfaceM ⁿ in affine spaceR ⁿ⁺¹ equipped with an equiaffine transversal field. IfM ⁿ is locally strictly convex, these two concepts are related via an Index theorem concerning the critical points of the support functions onM ⁿ . This is used to obtain characterizations of spheres and ellipsoids in terms of the critical point behavior of certain classes of affine support functions.Research supported by NSF Grant No. DMS-9101961.     

Large derivatives,backward contraction and invariant densities for interval maps

In this paper, we study the dynamics of a smooth multimodal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that |Dfⁿ(f(c))|→∞ as n→∞ holds for all critical points c, we show that f satisfies the so-called backward contracting property with an arbitrarily large constant, and that f has an invariant probability μ which is absolutely continuous with respect to Lebesgue measure and the density of μ belongs to L^p for all p<ℓ_max/(ℓ_max-1), where ℓ_max denotes the maximal critical order of f. In the appendix, we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction.     

A note on well-posedness for Camassa-Holm equation

In this note, we investigate the problem of well-posedness for a shallow water equation with data having critical regularity. Our results are based on the use of Besov spaces B_2,r^s (which generalize the Sobolev spaces H^s) with critical index s=3/2.     

Values in preservice mathematics teachers’ discussions of the Body Mass Index - A critical perspective

This article explores the values that come to the fore when preservice mathematics teachers (PTs) ¹ engage in critical discussions about the role of mathematical models in society. The specific model that was discussed was the Body Mass Index (BMI) ². From the analysis of the PTs’ discussions of the BMI from a mathematical and societal point of view several mathematical and mathematics educational values were identified such as openness, rationalism, progress, reasoning, evaluating, and problematizing the instrumental understanding of mathematics. In addition, critical thinking about mathematics in society as emphasized in curricula in the three countries involved in the study, was identified with four categories of complementary pairs. Knowing the mathematical and mathematics educational values underpinning PTs’ discussions and their connection to critical thinking is important for successfully engaging with the role of mathematics in society.     

On non-uniform hyperbolicity assumptions in one-dimensional dynamics

We give an essentially equivalent formulation of the backward contracting property, defined by Juan Rivera-Letelier, in terms of expansion along the orbits of critical values, for complex polynomials of degree at least 2 which are at most finitely renormalizable and have only hyperbolic periodic points, as well as all C ³ interval maps with non-flat critical points.     

Les triplets de la variété de Grassmann G2(R6)

It is shown that a triple of two-dimensional planes in R⁶ is determined up to isometry by nine angular invariants – the six critical angles and the three angles between the critical bases in each plane – and four signs. Conditions of existence of these triples are then analyzed. These results are used to study some noteworthy triples.