Two examples related to conical energies

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

Plenty of big projections imply big pieces of Lipschitz graphs

I prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$ E ⊂ R d with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.

Three Monuments

Chapter 16 is the last of the music chapters. It focuses on Tallis’s three most monumental works. The first work is the seven-voice Puer natus mass, built around an esoteric number-and-letter code in the tenor part; this piece was rediscovered only in the twentieth century and remains an enigma. The second work is the votive antiphon Gaude gloriosa, the boldest and most ambitious of all Tallis’s works in the traditional pre-Reformation English style. The third work is his forty-part motet Spem in alium, composed on a scale that was never equaled or even attempted by any other Tudor composer. This chapter discusses these three big pieces, their musical and cultural contexts, and their possible origins.

Nonexistence of Almost Moore Digraphs of Diameter Three

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound $M(d,k)=1+d+\cdots +d^k$, where $d>1$ and $k>1$ denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when $d=2,3$ or $k=2$. In this paper, we prove that almost Moore digraphs of diameter $k=3$ do not exist for any degree $d$. The enumeration of almost Moore digraphs of degree $d$ and diameter $k=3$ turns out to be equivalent to the search of binary matrices $A$ fulfilling that $AJ=dJ$ and $I+A+A^2+A^3=J+P$, where $J$ denotes the all-one matrix and $P$ is a permutation matrix. We use spectral techniques in order to show that such equation has no $(0,1)$-matrix solutions. More precisely, we obtain the factorization in ${\Bbb Q}[x]$ of the characteristic polynomial of $A$, in terms of the cycle structure of $P$, we compute the trace of $A$ and we derive a contradiction on some algebraic multiplicities of the eigenvalues of $A$. In order to get the factorization of $\det(xI-A)$ we determine when the polynomials $F_n(x)=\Phi_n(1+x+x^2+x^3)$ are irreducible in ${\Bbb Q}[x]$, where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial, since in such case they become 'big pieces' of $\det(xI-A)$. By using concepts and techniques from algebraic number theory, we prove that $F_n(x)$ is always irreducible in ${\Bbb Q}[x]$, unless $n=1,10$. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.

(428) Seed Threshing and Cleaningin Flowers

Flower seed threshing and cleaning are challenging because many flowers have tiny seed, e.g., the 1000-seed weight of Begonia is 0.01 g, and others have odd-shaped seed, e.g., Tagetes has pappus-bearing seed and Fibigia has winged seed. There is a lack of information on the threshing and cleaning of flower seeds. At the Ornamental Plant Germplasm Center, a small-plot grain belt thresher was modified by disengaging its winnower and a special chute installed to collect the threshed seed and chaff together for cleaning. A custom-made threshing board is used for small samples. The seed with chaff is passed through screen with mesh size that allows all the seed to pass through so that the big pieces of chaff are retained and separated, i.e., scalping. Accurate selection of the next scalping screen (SS) is critical so that the mesh size is just right for at least 95% of the seed to pass through to remove all the chaff larger than the seed. The seed is then sieved on a grading screen (GS) of mesh size that retains at least 95% of the seed to remove all the chaff smaller than the seed. A seed blower is used to further separate the remaining chaff and empty seed based on weight and surface area by adjusting the blowing velocity (BV). A vibratory separator (VS) is used for species with round seed, e.g., Antirrhinum. An X-ray machine is used to monitor the cleaning process. The SC, GS, BV, and VT are given for Agastache, Anisodontea, Antirrhinum, Aquilegia, Aster, Astilboides, Begonia, Belamcanda, Bergenia, Cleome, Coreopsis, Dianthus, Eupatorium, Gaillardia, Geranium, Gypsophila, Iris, Lilium, Lysimachia, Myosotis, Nothoscordom, Oenothera, Passiflora, Penstemon, Petunia, Platycodon, Ranunculus, Rudbeckia, Silene, Stokesia, Synnotia, Tagetes, Talinum, Thalictrum, Verbena, Veronica, and Zinnia.

WILLIAM CARLOS WILLIAMS RIDS THE PEDIATRIC WARDS OF THE NEW YORK NURSERY AND CHILD'S HOSPITAL OF BEDBUGS

William Carlos Williams (1883-1963), one of the most significant American poets of this century, practiced pediatrics for more than 40 years in his birthplace, Rutherford, New Jersey. As an intern in 1908-1909 at the New York Nursery and Child's Hospital, he discovered that the persistent screaming of several of his patients, especially at night, was caused by bedbug bites. In his Autobiography Williams described his tactics as an exterminator. Next day I got permission to buy half a barrel of bar-sulfur, big pieces round as my wrist.