Splitting fields of conics and sums of squares of rational functions

Given a geometrically unirational variety over an infinite base field, we show that every finite separable extension of the base field that splits the variety is the residue field of a closed point. As an application, we obtain a characterization of function fields of smooth conics in which every sum of squares is a sum of two squares.     

On packing squares with equal squares

The following problem arises in connection with certain multidimensional stock cutting problems:How many nonoverlapping open unit squares may be packed into a large square of side α?Of course, if α is a positive integer, it is trivial to see that α² unit squares can be succesfully packed. However, if α is not an integer, the problem becomes much more complicated. Intuitively, one feels that for $\text{α = N + (}\text{1}\text{100}\text{)}$, say (where N is an integer), one should pack N² unit squares in the obvious way and surrender the uncovered border area (which is about $\text{α}\text{50}$) as unusable waste. After all, how could it help to place the unit squares at all sorts of various skew angles?In this note, we show how it helps. In particular, we prove that we can always keep the amount of uncovered area down to at most proportional to $\text{α}{}^{\mathrm{<mtext>7</mtext><mtext>11</mtext>}}$, which for large α is much less than the linear waste produced by the “natural” packing above.     

Splitting necklaces

Let N be an opened necklace with ka_i beads of color i, 1 ⩽ i ⩽ t. We show that it is possible to cut N in (k - 1) · t places and partition the resulting intervals into k collections, each containing precisely a_i beads of color i, 1 ⩽ i ⩽ t. This result is best possible and solves a problem of Goldberg and West. Its proof is topological and uses a generalization, due to Bárány, Shlosman and Szücs, of the Borsuk-Ulam theorem. By similar methods we obtain a generalization of a theorem of Hobby and Rice on L₁-approximation.     

Global unit squares and local unit squares

For units of any Galois number field, we study the relations between the units being global squares and those being local squares.     

Inefficiency in packing squares with unit squares

It is shown that, in packing a square of side $\text{n +}\text{1}\text{2}$ with unit squares, the wasted space always has area $\text{? n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. This answers a question of Erdös and Graham.     

Nonnegative functions as squares or sums of squares

We prove that, for n?4, there are C^∞ nonnegative functions f of n variables (and even flat ones for n?5) which are not a finite sum of squares of C² functions. For n=1, where a decomposition in a sum of two squares is always possible, we investigate the possibility of writing f=g². We prove that, in general, one cannot require a better regularity than g∈C¹. Assuming that f vanishes at all its local minima, we prove that it is possible to get g∈C² but that one cannot require any additional regularity.     

Multi-latin squares

A multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square.In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n≥2m, thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n≥k+2. We also show that for each n≥1, there exists some finite value g(n) such that for all k≥g(n), every k-latin square of order n is separable.We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders.     

Indexed squares

We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1-extendibility is incompatible with square; we prove this is close to optimal by showing that 1-extendibility is compatible with square. First author partially supported by NSF grants DMS-9703945 and DMS-0070549. Second author partially supported by NSF Grants DMS-9305990, DMS-9712580, DMS-9996280 and DMS-0088948.     

A Splitting Inequality

We prove that if A is a finite set, and if is an downwards-closed family of subsets of A, and if fx is the proportion of x-element subsets of A in , then fa · fb fa + b – r, if r < . We connect this result with the Weak Threshold Theorem.     

Splitting Superhereditary Preradicals

Abstract

We study the structure of rings R with the property that, for every right R-module M and an ideal I of R, the annihilator of I in M is a direct summand of M.     

Splitting the shadow

We derive formulae for the theta series of the two translates of the even sublattice L₀ of an odd unimodular lattice L that constitute the shadow of L. The proof rests on special evaluations of the Jacobi theta series attached to L and to a certain vector. We produce an analogous theorem for codes. Additionally, we construct non-linear formally self-dual codes and relate them to lattices.     

Splitting multidimensional necklaces

The well-known “splitting necklace theorem” of Alon [N. Alon, Splitting necklaces, Adv. Math. 63 (1987) 247-253] says that each necklace with k⋅ai beads of color i=1,…,n, can be fairly divided between k thieves by at most n(k−1) cuts. Alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace [0,1] where beads of given color are interpreted as measurable sets Ai⊂[0,1] (or more generally as continuous measures μi). We demonstrate that Alon's result is a special case of a multidimensional consensus division theorem about n continuous probability measures μ₁,…,μn on a d-cube d[0,1]. The dissection is performed by m₁+?+md=n(k−1) hyperplanes parallel to the sides of d[0,1] dividing the cube into m₁⋅?⋅md elementary cuboids (parallelepipeds) where the integers mi are prescribed in advance.     

Monogamous latin squares

We show for all n∉{1,2,4} that there exists a latin square of order n that contains two entries γ₁ and γ₂ such that there are some transversals through γ₁ but they all include γ₂ as well. We use this result to show that if n>6 and n is not of the form 2p for a prime p?11 then there exists a latin square of order n that possesses an orthogonal mate but is not in any triple of MOLS. Such examples provide pairs of 2-maxMOLS.     

Partitions in squares

In this paper we study primarily partitions in different squares. A complete characterization of the least number of terms needed in different cases is given. The asymptotic number of partitions in squares and in different squares is deduced by use of numerical results obtained from extensive computer runs. Some other related problems are also discussed.     

Crisscross Latin squares

Let A be a Latin square of order n. Then the jth right diagonal of A is the set of n cells of A: {(i,j+i):i=0,1…,n?1(modn); and the jth left diagonal of A is the set {(i,j?i):i=0,1…,n?1(modn); A diagonal is said to be complete if every element appears in it exactly once. For n = 2m even, we introduce the concept of a crisscross Latin square which is something in between a diagonal Latin square and a Knut Vik design. A crisscross Latin square is a Latin square such that all the jth right diagonals for even j and all the jth left diagonals for odd j are complete. We show that a necessary and sufficient condition for the existence of a crisscross Latin square of order 2m is that m is even.     

Splitting for Multi-objective Optimization

We introduce a new multi-objective optimization (MOO) methodology based the splitting technique for rare-event simulation. The method generalizes the elite set selection of the traditional splitting framework, and uses both local and global sampling to sample in the decision space. In addition, an ??-dominance method is employed to maintain good solutions. The algorithm was compared with state-of-the art MOO algorithms using a prevailing set of benchmark problems. Numerical experiments demonstrate that the new algorithm is competitive with the well-established MOO algorithms and that it can outperform the best of them in various cases.     

Sumsets being squares

Alon, Angel, Benjamini and Lubetzky [1] recently studied an old problem of Euler on sumsets for which all elements of A+B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B?[0,N] with |B|?(logN)^15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B?[0,N] with |B|?(logN)^9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank.