Virtual resolutions of points in P1×P1

We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of finite sets of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Specifically, we describe a virtual resolution for a sufficiently general set of points X in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ that only depends on $|X|$. We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$; more precisely, we give an effective bound for their construction that gives a virtual resolution of length two for any set of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$.     

Equivalence of bar induction and bar recursion for continuous functions with continuous moduli

We compare Brouwer's bar theorem and Spector's bar recursion for the lowest type in the context of constructive reverse mathematics. To this end, we reformulate bar recursion as a logical principle stating the existence of a bar recursor for every function which serves as the stopping condition of bar recursion. We then show that the decidable bar induction is equivalent to the existence of a bar recursor for every continuous function from ${\mathbb{N}}^{\mathbb{N}}$ to $\mathbb{N}$ with a continuous modulus. We also introduce fan recursion, the bar recursion for binary trees, and show that the decidable fan theorem is equivalent to the existence of a fan recursor for every continuous function from ${\{0,1\}}^{\mathbb{N}}$ to $\mathbb{N}$ with a continuous modulus. The equivalence for bar induction holds over the extensional version of intuitionistic arithmetic in all finite types augmented with the characteristic principles of Gödel's Dialectica interpretation. On the other hand, we show the equivalence for fan theorem without using such extra principles.     

Solution of the Navier–Stokes problem

A new a priori estimate for solutions to Navier–Stokes equations is derived. Uniqueness and existence of these solutions in ${\mathbb{R}}^3$ for all $t>0$ is proved in a class of solutions locally differentiable in time with values in $H^1\left({\mathbb{R}}^3\right)$, where $H^1\left({\mathbb{R}}^3\right)$ is the Sobolev space. By the solution a solution to an integral equation is understood. No smallness restrictions on the data are imposed.     

Equivalences among Z2s-linear Hadamard codes

The ${\mathbb{Z}}_{2^s}$-additive codes are subgroups of ${\mathbb{Z}}_{2^s}^n$, and can be seen as a generalization of linear codes over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$. A ${\mathbb{Z}}_{2^s}$-linear Hadamard code is a binary Hadamard code which is the Gray map image of a ${\mathbb{Z}}_{2^s}$-additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some ${\mathbb{Z}}_{2^s}$-linear Hadamard codes of length $2^t$ are equivalent, once $t$ is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to $t=11$, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on $s\in \{2,3\}$, the full classification of the ${\mathbb{Z}}_{2^s}$-linear Hadamard codes of length $2^t$ is established by giving the exact number of such codes.     

Proof of the Hénon–Lane–Emden conjecture in R3

We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on $S^{N?1}$, we prove this conjecture is true for space dimension $N=3$; which also implies the single elliptic equation has no positive classical solutions in ${\mathbb{R}}^3$ when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for ${\mathbb{R}}^3$.     

Permutation binomials of the form xr(xq−1 + a) over Fqe

We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\ge 2$ and $a\in {\mathbb{F}}_{q^e}^{⁎}$. As a consequence, we obtain a complete characterization of such permutation binomials over ${\mathbb{F}}_{q^2}$, ${\mathbb{F}}_{q^3}$, ${\mathbb{F}}_{q^4}$, ${\mathbb{F}}_{p^5}$, and ${\mathbb{F}}_{p^6}$, where p is an odd prime.     

An example of the geometry of a 5th-order ODE: The metric on the space of conics in CP2

As an application of the method of [4], we find the metric and connection on the space of conics in ${\mathbb{CP}}^2$ determined as the solution space of the ODE (1). These calculations underpin the twistor construction of the Radon transform on conics in ${\mathbb{CP}}^2$ described in [5]. Two further examples of the method are provided.