On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone

In this paper, we study the initial-boundary value problem of the porous medium equation u _t  = Δu ^m  + V(x)u ^p in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|) ^σ . Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l ² + (n − 2)l = ω ₁. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u ₀ ≥ 0.     

A class of optimal linear codes of length one above the Griesmer bound

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g _q (k, d) + 1, k, d] _q code for sq ^k-1 − sq ^k-2 − q ^s  − q ² + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n _q (k, d) = g _q (k, d) + 1 for (k − 2)q ^k-1 − (k − 1)q ^k-2 − q ² + 1 ≤ d ≤ (k − 2)q ^k-1 − (k − 1)q ^k-2, k ≥ 6, q ≥ 2k − 3; and sq ^k-1 − sq ^k-2 − q ^s  − q + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s , s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1. This work was partially supported by the Com²MaC-SRC/ERC program of MOST/KOSEF (grant ^# R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175).     

Wieferich pairs and Barker sequences

We show that if a Barker sequence of length n > 13 exists, then either n = 189 260 468 001 034 441 522 766 781 604, or n > 2 · 10³⁰. This improves the lower bound on the length of a long Barker sequence by a factor of more than 10⁷. We also show that all but fewer than 1600 integers n ≤ 4 · 10²⁶ can be eliminated as the order of a circulant Hadamard matrix. These results are obtained by completing extensive searches for Wieferich prime pairs (q, p), which are defined by the relation q^p-1 o 1{q^{p-1} \equiv1} mod p ², and analyzing their results in combination with a number of arithmetic restrictions on n.     

Exceptional cases of Terai’s conjecture on Diophantine equations

Let a, b, c be relatively prime positive integers such that a ^p  + b ^q  = c ^r for fixed integers p, q, r ≥ 2. Terai conjectured that the equation a ^x  + b ^y  = c ^z in positive integers has only the solution (x, y, z) = (p, q, r) except for specific cases. In this paper, we consider the case q = r = 2 and give some results related to exceptional cases.     

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets S{\mathcal {S}} of the natural numbers such that the Robin inequality holds for all but finitely many n ? S{n \in \mathcal {S}} . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.     

Decremental Dynamic Connectivity

We consider Las Vegas randomized dynamic algorithms for on-line connectivity problems with deletions only. In particular, we show that starting from a graph with m edges and n nodes, we can maintain a spanning forest during m deletions in O(m log(n²/m) + n(log n)³(log log n)²) expected time, which is O(m) if m = Θ(n²) and O(m log n) if m = Ω(n(log n log log n)²). The deletions may be interspersed with connectivity queries, each of which is answered in constant time. The previous best bound was O(m log² n) by Henzinger and Thorup which covered both insertions and deletions. The result is based on a general randomized reduction for edge connectivity problems of many deletions-only queries to a few deletions and insertions queries. For 2-edge connectivity, the complexity is improved from O(m(log n)⁵) to O(m log(n²/m) + n(log n)⁶(log log n)²). For the general decremental k-edge-connectivity problem, we get a total running time of O(k²n² polylog n). Here the previous best bound was O(kmn polylog n). Improved running times are also achieved for the static consensus tree problem, with applications to computational biology and relational data bases.     

Rectifiability and Lipschitz extensions into the Heisenberg group

Denote by \mathbbHⁿ{\mathbb{H}^n} the 2n + 1 dimensional Heisenberg group. We show that the pairs (\mathbbR^k ,\mathbbHⁿ){(\mathbb{R}^k ,\mathbb{H}^n)} and (\mathbbH^k ,\mathbbHⁿ){(\mathbb{H}^k ,\mathbb{H}^n)} do not have the Lipschitz extension property for k  >  n.     

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.      

Contracting endomorphisms and Gorenstein modules

A finite module M over a noetherian local ring R is said to be Gorenstein if Extⁱ(k, M) = 0 for all i ≠ dim R. An endomorphism φ: R → R of rings is called contracting if for some i ≥ 1. Letting ^φR denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if Hom_R(^φR, M) ≅ M and Extⁱ_R(^φR, M) = 0 for 1 ≤ i ≤ depth R. Received: 7 December 2007     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).      

Linking and Multiplicity Results for the p-Laplacian on Unbounded Cylinders

We consider the p-Laplacian problem[formula]on unbounded cylinders Ω = Ω̃ × R^N − m  R^N, N − m ≥ 2, where Δ_pu = div(|u|^p − 2u), λ is a constant in a certain range, and a  L^N/p(Ω) ∩ L^∞(Ω) is nonnegative, a  0. Using the principle of symmetric criticality, existence and multiplicity are proved under suitable conditions on a and f.     

Greedily Finding a Dense Subgraph

Given an n-vertex graph with nonnegative edge weights and a positive integer k ≤ n, our goal is to find a k-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1/2 + n/2k)² − O(n ^− 1/3) ≤ R ≤ (1/2 + n/2k)² + O(1/n) for k in the range n/3 ≤ k ≤ n and 2(n/k − 1) − O(1/k) ≤ R ≤ 2(n/k − 1) + O(n/k²) for k < n/3. For k = n/2, for example, these bounds are 9/4 ± O(1/n), improving on naive lower and upper bounds of 2 and 4, respectively. The upper bound for general k compares well with currently the best (and much more complicated) approximation algorithm based on semidefinite programming.     

The sum of squared logarithms inequality in arbitrary dimensions

We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y ∈ ℝⁿ whose elementary symmetric polynomials satisfy e_k(x) ≤ e_k(y) (for 1 ≤ k < n) and e_n(x) = e_n(y) , the inequality ∑_i(log x_i)² ≤ ∑_i(log y_i)² holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f : M ⊆ ℂⁿ → ℝ with f(z) = ∑_i(log z_i)² has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. We conclude by providing applications and wider connections of the SSLI. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semiconcavity estimates for viscous Hamilton–Jacobi equations

We present sharp Hessian estimates of the form D² S^e(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation

Global existence and L ∞ estimates of solutions for a quasilinear parabolic system

In this paper, we study the global existence, L^∞ estimates and decay estimates of solutions for the quasilinear parabolic system u_t = div (|∇ u|^m ∇ u) + f(u, v), v_t = div (|∇ v|^m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ R^N. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation u_t = div (|∇ u|^m ∇ u) + λ |u|^{α - 1} u.     

Unitary Graphs and Their Automorphisms

The (isotropic) unitary graph U (n, q²){U \left(n, q^{2}\right)} is introduced. When n = 2 or 3, U (2, q²){U \left(2, q^{2}\right)} or U (3, q²){U \left(3, q^{2}\right)} are complete graphs with q + 1 or q ³ + 1 vertices, respectively. When n ≥ 4, it is shown that U (n, q²){U \left(n, q^{2}\right)} is strongly regular and its parameters are computed. The group of graph automorphisms of U (n, q²){U \left(n, q^{2}\right)} , when n ≠ 4, 5, is determined.     

A Pair of Families of Non-isomorphic Group Association Schemes with the same Parameters

We show that for the split and non-split extensions ofFq²bySL (2, q) (q =  2^e,e  ≥  3), the group association schemes have the same parameters but are not isomorphic. For the split and non-split extensions ofFq²by the standard Borel subgroup of SL(2,q ) (q =  2^e, e ≥  3), the group association schemes are shown to be isomorphic.     

Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes

Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q ^k/2 of the values of d < q ^k-1. In this article we shall prove that, for q = p prime and roughly \frac38{\frac{3}{8}}-th’s of the values of d < q ^k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis’ theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(δ, q), for arbitrary q and δ ≥ 3. It is known that from a linear code of dimension k with minimum distance d < q ^k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k−1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis’ theorem, that nearly all linear codes over \mathbb F_p{{\mathbb F}_p} of dimension k with minimum distance d < q ^k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.     

Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

For arbitrary [α, β] ⊂ R and p > 0, we solve the extremal problem

Isoperimetric profile comparisons and Yamabe constants

We estimate from below the isoperimetric profile of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} and use this information to obtain lower bounds for the Yamabe constant of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} . This provides a lower bound for the Yamabe invariants of products S ² ×  M ² for any closed Riemann surface M. Explicitly we show that Y (S ² ×  M ²) >  (2/3)Y(S ⁴).