Immersed surfaces in the modular orbifold

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface.Comment: 13 pages, 8 figures; version 2 contains minor correction

Stable commutator length in Baumslag-Solitar groups and quasimorphisms for tree actions

This paper has two parts, on Baumslag-Solitar groups and on general G-trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces. In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group BS(2,3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions. Returning to Baumslag-Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval (0, 1/12).Comment: v2: minor changes, incorporates referee suggestions; v1: 36 pages, 10 figure

Rational numbers with odd greedy expansion of fixed length

Given a positive rational number $n/d$ with $d$ odd, its odd greedy expansion starts with the largest odd denominator unit fraction at most $n/d$, adds the largest odd denominator unit fraction so the sum is at most $n/d$, and continues as long as the sum is less than $n/d$. It is an open question whether this expansion always has finitely many terms. Given a fixed positive integer $n$, we find all reduced fractions with numerator $n$ whose odd greedy expansion has length $2$. Given $m-1$ odd positive integers, we find all rational numbers whose odd greedy expansion has length $m$ and begins with these numbers as denominators. Given $m-2$ compatible odd positive integers, we find an infinite family of rational numbers whose odd greedy expansion has length $m$ and begins with these numbers as denominators.Comment: 21 page

Critical groups of arithmetical structures under a generalized star-clique operation

An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group. Keyes and Reiter gave an operation that takes in an arithmetical structure on a finite, connected graph without loops and produces an arithmetical structure on a graph with one fewer vertex. We study how this operation transforms critical groups. We bound the order and the invariant factors of the resulting critical group in terms of the original arithmetical structure and critical group. When the original graph is simple, we determine the resulting critical group exactly.Comment: 19 pages, 3 figures; v2: minor improvements to expositio

Determination of iron in nutrient solutions and its role in plant metabolism

Iron is a very important element in plant nutrition. This wao recognized by the first euooeseful inveetigators in thie fiald. In solution culture methods of research it ie well known that growth in plants may be inhibited by an insufficient supply of available iron. To control this available supply of iron is apparently thr* most difficult of any of the nutrient elements. The quantity lies within relatively narrow limits; both too much, and an undersupply produce physiological disturbances in the plant which is indicated by chlorosis. Among the component salts of the culture solutions are thoss which produce very Insoluble iron compounds. The solutions no* considered best soon lose most of the iron bo that frequent renewal of solution is necessary to keep up the supply of iron

Arithmetical structures on bidents

An arithmetical structure on a finite, connected graph $G$ is a pair of vectors $(\mathbf{d}, \mathbf{r})$ with positive integer entries for which $(\operatorname{diag}(\mathbf{d}) - A)\mathbf{r} = \mathbf{0}$, where $A$ is the adjacency matrix of $G$ and where the entries of $\mathbf{r}$ have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of $(\operatorname{diag}(\mathbf{d}) - A)$. In this paper, we study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two "prongs" at one end. We give a process for determining the number of arithmetical structures on the bident with $n$ vertices and show that this number grows at the same rate as the Catalan numbers as $n$ increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.Comment: 32 page

Synthesis of Macromolecules Containing Phenylalanine and Aliphatic Building Blocks

Aiming at developing efficient interfacial agents for fiber‐reinforced composite materials, macromolecules are designed to have different components able to stick to the fiber and be compatible with the polymer matrix, respectively. Herein, macromolecules are prepared by solid‐phase synthesis considering phenylalanine residues to promote adsorption of the macromolecule on aramid fibers and aliphatic building blocks to interact with a hydrophobic polymer matrix. Using phenylalanine as building block for the preparation of macromolecules by iterative synthesis has been shown to be challenging. Thus, the screening of various parameters for the optimization of the synthesis of these macromolecules is discussed in this communication. A preliminary thermal study by thermal gravimetric analysis is conducted to evaluate their thermal stability

A 1 Volt Switched Transconductor Mixer in 0.18μm CMOS

A new CMOS mixer topology can operate at low supply voltages by using switches connected only to the supplies. Mixing is achieved exploiting two cross-coupled transconductors, which are alternatingly activated by the switches. A down conversion mixer prototype with 12 dB conversion gain was designed and realized in standard 0.18 μm CMOS. It achieves satisfactory mixer performance up to 4 GHz, at a supply voltage of 1 Volt. Moreover, the mixer topology features a fundamental high frequency noise figure benefit