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This lecture is concerned primarily with matroids and geometric lattices. Since the intersection lattice of a central arrangement is a geometric lattice, all our results can be applied to arrangements.

Combinatorial Analysis.--The Foundations of a New Theory. MacMahon, P Proceedings of the Royal Society of London (1854-1905). 1899-01-01. 66:336–337

Combinatorial Analysis.--The Foundations of a New Theory. MacMahon, P Proceedings of the Royal Society of London (1854-1905). 1899-01-01. 66:336–337

This project has completed. The PI considered optimization problems with uncertainty in the data. Specifically, algorithms and analysis methods were developed for several optimization scenarios where the random parameters were coming from some unknown probability distribution. One notable result efficiently calculates the average optimal solution for a combinatorial regret problem and provides new bounds for how for this may be from the worst case. For details, see the final report. Further, the project produced 8 published papers and 4 more under review or preparation. The specifics of these references are included in uploaded final report. This work has help to advance the state-of-the-art in robust optimization.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

Shannon's seminal 1948 work gave rise to two distinct areas of research: information theory and mathematical coding theory. While information theory has had a strong influence on theoretical neuroscience, ideas from mathematical coding theory have received considerably less attention. Here we take a new look at combinatorial neural codes from a mathematical coding theory perspective, examining the error correction capabilities of familiar receptive field codes (RF codes). We find, perhaps surprisingly, that the high levels of redundancy present in these codes does not support accurate error correction, although the error-correcting performance of RF codes "catches up" to that of random comparison codes when a small tolerance to error is introduced. On the other hand, RF codes are good at reflecting distances between represented stimuli, while the random comparison codes are not. We suggest that a compromise in error-correcting capability may be a necessary price to pay for a neural code whose structure serves not only error correction, but must also reflect relationships between stimuli.

The $\mathcal{A}$-tracial algebras are algebras endowed with multi-linear forms, compatible with the product, and indexed by partitions. Using the notion of $\mathcal{A}$-cumulants, we define and study the $\mathcal{A}$-freeness property which generalizes the independence and freeness properties, and some invariance properties which model the invariance by conjugation for random matrices. A central limit theorem is given in the setting of $\mathcal{A}$-tracial algebras. A generalization of the normalized moments for random matrices is used to define convergence in $\mathcal{A}$-distribution: this allows us to apply the theory of $\mathcal{A}$-tracial algebras to random matrices. This study is deepened with the use of $\mathcal{A}$-finite dimensional cumulants which are related to some dualities as the Schur-Weyl's duality. This gives a unified and simple framework in order to understand families of random matrices which are invariant by conjugation in law by any group whose associated tensor category is spanned by partitions, this includes for example the unitary groups or the symmetric groups. Among the various by-products, we prove that unitary invariance and convergence in distribution implies convergence in $\mathcal{P}$-distribution. Besides, a new notion of strong asymptotic invariance and independence are shown to imply $\mathcal{A}$-freeness. Finally, we prove general theorems about convergence of matrix-valued additive and multiplicative L{\'e}vy processes which are invariant in law by conjugation by the symmetric group. Using these results, a unified point of view on the study of matricial L{\'e}vy processes is given.

We consider the determination of the number $c_k(\alpha)$ of ordered factorisations of an arbitrary permutation on n symbols, with cycle distribution $\alpha$, into k-cycles such that the factorisations have minimal length and such that the group generated by the factors acts transitively on the n symbols. The case k=2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the alternating group, and this is another case that, in principle, is of considerable interest. We conjecture an explicit form, for arbitrary k, for the generating series for $c_k(\alpha)$, and prove that it holds for factorisations of permutations with one, two and three cycles ($\alpha$ is a partition with at most three parts). The generating series is naturally expressed in terms of the symmetric functions dual to those introduced by Macdonald for the ``top'' connection coefficients in the class algebra of the symmetric group. Our approach is to determine a differential equation for the generating series from a combinatorial analysis of the creation and annihilation of cycles in products under the minimality condition.

We consider the determination of the number $c_k(\alpha)$ of ordered factorisations of an arbitrary permutation on n symbols, with cycle distribution $\alpha$, into k-cycles such that the factorisations have minimal length and such that the group generated by the factors acts transitively on the n symbols. The case k=2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the alternating group, and this is another case that, in principle, is of considerable interest. We conjecture an explicit form, for arbitrary k, for the generating series for $c_k(\alpha)$, and prove that it holds for factorisations of permutations with one, two and three cycles ($\alpha$ is a partition with at most three parts). The generating series is naturally expressed in terms of the symmetric functions dual to those introduced by Macdonald for the ``top'' connection coefficients in the class algebra of the symmetric group. Our approach is to determine a differential equation for the generating series from a combinatorial analysis of the creation and annihilation of cycles in products under the minimality condition.

Combinatorial Analysis. The Foundations of a New Theory. MacMahon, P Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Math. or Phys. Character (1896-1934). 1900-01-01. 194:361–386

In the paper: Fans in the Theory of Real Semigroups. I. Algebraic Theory (submitted) we introduced the notion of fan in the categories of real semigoups and their dual abstract real spectra and developed the algebraic theory of these structures. In this paper we develop the combinatorial theory of ARS-fans, i.e., fans in the dual category of abstract real spectra. Every ARS is a spectral space and hence carries a natural partial order called the {\it specialization partial order}. Our main result shows that the isomorphism type of a finite fan in the category ARS is entirely determined by its order of specialization. The main tools used to prove this result are: (1) Crucial use of the theory of {\it ternary semigroups}, a class of semigroups underlying that of RSs; (2) Every ARS-fan is a disjoint union of abstract order spaces (called {\it levels}); (3) Every level carries a natural involution of abstract order spaces, and (4) The notion of a {\it standard generating system}, a combinatorial tool replacing, in the context of ARSs, the (absent) tools of combinatorial geometry (matroid theory) employed in the cases of fields and of abstract order spaces.

In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an integer parameter N. Then we emulate the theory of random matrices in a combinatorial framework: for any parameter N, we introduce a family of linear forms on the partition algebras which allows us to define a notion of weak convergence similar to the convergence in moments in random matrices theory. A renormalization of the partition algebras allows us to consider the weak convergence as a simple convergence in a fixed space. This leads us to the definition of a deformed partition algebra for any integer parameter N and to the definition of two transforms: the cumulants transform and the exclusive moments transform. Using an improved triangular inequality for the distance defined on partitions, we prove that the deformed partition algebras, endowed with a deformation of the linear forms converge as N go to infinity. This result allows us to prove combinatorial properties about geodesics and a convergence theorem for semi-groups of functions on partitions. At the end we study a sub-algebra of functions on infinite partitions with finite support : a new addition operation and a notion of R-transform are defined. We introduce the set of multiplicative functions which becomes a Lie group for the new addition and multiplication operations. For each of them, the Lie algebra is studied. The appropriate tools are developed in order to understand the algebraic fluctuations of the moments and cumulants for converging sequences. This allows us to extend all the results we got for the zero order of fluctuations to any order.

We analyse the perturbative series expansion of the vacuum expectation value of a Wilson loop in Chern-Simons gauge theory in the temporal gauge. From the analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel of a Vassiliev invariant of order n is not a knot invariant, since it depends on the regular knot projection chosen, but it differs from a Vassiliev invariant by terms that vanish on knots with n singular crossings. We conjecture that Vassiliev invariants can be reconstructed from their kernels. We present the general form of the kernel of a Vassiliev invariant and we describe the reconstruction of the full primitive Vassiliev invariants at orders two, three and four. At orders two and three we recover known combinatorial expressions for these invariants. At order four we present new combinatorial expressions for the two primitive Vassiliev invariants present at this order.

We analyse the perturbative series expansion of the vacuum expectation value of a Wilson loop in Chern-Simons gauge theory in the temporal gauge. From the analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel of a Vassiliev invariant of order n is not a knot invariant, since it depends on the regular knot projection chosen, but it differs from a Vassiliev invariant by terms that vanish on knots with n singular crossings. We conjecture that Vassiliev invariants can be reconstructed from their kernels. We present the general form of the kernel of a Vassiliev invariant and we describe the reconstruction of the full primitive Vassiliev invariants at orders two, three and four. At orders two and three we recover known combinatorial expressions for these invariants. At order four we present new combinatorial expressions for the two primitive Vassiliev invariants present at this order.

Holding on a thing is essentially to recognize the cause and effect in its evolution. The combinatorial notion implied in words of the sophist in the fable of the blind men with an elephant, is a philosophic notion recognizing things with only the local knowing. Under this notion, this book surveys the quantitative knowing of things, starts with the universal questions raised by children, explains the understanding of things including the reference frame, relativity, system with synchronization, non-harmonious system with non-solvable equations, complex network, etc., and the role of science under the harmonious coexistence of humans with the nature, which is a systematic review on mathematical science and philosophy, suitable for researchers in mathematics, systems, physics and science's philosophy.

The purpose of this project is to find the optimal combination of retention base incentives offered by the Surface Warfare Officer (SWO) community. The SWO community has experienced the lowest junior officer retention rate of all Unrestricted Line Officer (URL) communities since the early 1990's. The community is required to maintain at least 275 junior officers annually, in order to fill operational Department Head billets, in support of sustaining maritime mission requirements. Due to numerous community dissatisfiers, the SWO community has been unable to produce the required force strength of 275 Department Heads. Induction of a monetary bonus was the SWO community's answer to addressing its critical junior officer retention rates. But retention rates are still critically low and are viewed negatively by many junior officers, because the monetary bonus doesn't address the reasons junior officers are dissatisfied with the community. Several non-monetary incentives have been identified by junior SWO as possible tools to increase retention rates. This project will build a Combinatorial Auction Model to include both monetary and non-monetary incentives to determine which combination offers the greatest value to officers, at least cost to the Navy's SWO community.

We show that the well-known Konig's Min-Max Theorem (KMM), a fundamental result in combinatorial matrix theory, can be proven in the first order theory $\LA$ with induction restricted to $\Sigma_1^B$ formulas. This is an improvement over the standard textbook proof of KMM which requires $\Pi_2^B$ induction, and hence does not yield feasible proofs --- while our new approach does. $\LA$ is a weak theory that essentially captures the ring properties of matrices; however, equipped with $\Sigma_1^B$ induction $\LA$ is capable of proving KMM, and a host of other combinatorial properties such as Menger's, Hall's and Dilworth's Theorems. Therefore, our result formalizes Min-Max type of reasoning within a feasible framework.

This is the study of game comparison in \emph{combinatorial game spaces}, and their sub-spaces, called \emph{universes} of games. The concept of a combinatorial game space allows for a general framework, which includes many standard classes of terminating games. In an \emph{absolute} universe, it is shown that the comparison of two games need only include followers of the two and atomic games instead of all games. This leads to a constructive game comparison for studied absolute universes (e.g. Milnor, Conway, Ettinger, Siegel, Milley and Renault). On the way, we introduce the dicot \emph{kernel} of a game space, and show how to construct infinitely many absolute universes, by extending specifically the mis\`ere kernel.

We begin by reviewing and proving the basic facts of combinatorial game theory. We then consider scoring games (also known as Milnor games or positional games), focusing on the "fixed-length" games for which all sequences of play terminate after the same number of moves. The theory of fixed-length scoring games is shown to have properties similar to the theory of loopy combinatorial games, with operations similar to onsides and offsides. We give a complete description of the structure of fixed-length scoring games in terms of the class of short partizan games. We also consider fixed-length scoring games taking values in the two-element boolean algebra, and classify these games up to indistinguishability. We then apply these results to analyze some positions in the knotting-unknotting game of Pechenik, Townsend, Henrich, MacNaughton, and Silversmith.

A drawing of a graph is said to be a {\em straight-line drawing} if the vertices of $G$ are represented by distinct points in the plane and every edge is represented by a straight-line segment connecting the corresponding pair of vertices and not passing through any other vertex of $G$. The minimum number of slopes in a straight-line drawing of $G$ is called the slope number of $G$. We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes $\{0,\pi/4,\pi/2,-\pi/4\}$. We also prove that four slopes have this property if and only if we can draw $K_4$ with them. Given a graph $G$, an {\em obstacle representation} of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of obstacles (connected polygons) such that two vertices of $G$ are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The {\em obstacle number} of $G$ is the minimum number of obstacles in an obstacle representation of $G$. We show that there are graphs on $n$ vertices with obstacle number $\Omega({n}/{\log n})$. We show that there is an $m=2n+o(n)$, such that, in the Maker-Breaker game played on $\Z^d$ where Maker needs to put at least $m$ of his marks consecutively in one of $n$ given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg who showed that such a pairing strategy exits if $m\ge 3n$. A simple argument shows that $m$ has to be at least $2n+1$ if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a self-contained and elementary proof of the analog for the groups G(r,p,n), with r>1, of Gordon's theorem (previously Haiman's conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a self-contained and elementary proof of the analog for the groups G(r,p,n), with r>1, of Gordon's theorem (previously Haiman's conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the $\mathbb{F}_p$-dimension of an $n$-th graded piece of the Zassenhaus filtration for various finitely generated pro-$p$ groups, including free pro-$p$ groups, Demushkin groups and their free pro-$p$ products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-$p$ Galois groups. We also show that knowledge of small quotients of pro-$p$ Galois groups can provide information regarding the form of relations among the group generators.

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the $\mathbb{F}_p$-dimension of an $n$-th graded piece of the Zassenhaus filtration for various finitely generated pro-$p$ groups, including free pro-$p$ groups, Demushkin groups and their free pro-$p$ products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-$p$ Galois groups. We also show that knowledge of small quotients of pro-$p$ Galois groups can provide information regarding the form of relations among the group generators.

In a previous paper, Sarkar and the third author gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism (modulo torsion). Under this assumption, we present a procedure for finding the rank of the induced Heegaard Floer map combinatorially, in the hat version.

We investigate the exploitation of various combinatorial properties of graphs and set systems to study several issues in quantum information theory. We characterize the combinatorics of distributed EPR pairs for preparing multi-partite entanglement in a real communication network. This combinatorics helps in the study of various problems in multi-party case by just reducing to the two-party case. Particularly, we use this combinatorics to (1) study various possible and impossible transformations of multi-partite states under LOCC, thus presenting an entirely new approach, not based on entropic criterion, to study such state transformations. (2) present a protocol and proof of its unconditional security for quantum key distribution amongst several trusted parties. (3) propose an idea to combine the features of quantum key distribution and quantum secret sharing. We investigate all the above issues in great detail and finally conclude briefly with some open research directions based on our research.

We investigate the exploitation of various combinatorial properties of graphs and set systems to study several issues in quantum information theory. We characterize the combinatorics of distributed EPR pairs for preparing multi-partite entanglement in a real communication network. This combinatorics helps in the study of various problems in multi-party case by just reducing to the two-party case. Particularly, we use this combinatorics to (1) study various possible and impossible transformations of multi-partite states under LOCC, thus presenting an entirely new approach, not based on entropic criterion, to study such state transformations. (2) present a protocol and proof of its unconditional security for quantum key distribution amongst several trusted parties. (3) propose an idea to combine the features of quantum key distribution and quantum secret sharing. We investigate all the above issues in great detail and finally conclude briefly with some open research directions based on our research.

Accompanied with humanity into the 21st century, a highlight trend for developing a science is its overlap and hybrid, and harmoniously with other sciences, which enables one to handle complex systems in the WORLD. This is also for developing mathematics.

Accompanied with humanity into the 21st century, a highlight trend for developing a science is its overlap and hybrid, and harmoniously with other sciences, which enables one to handle complex systems in the WORLD. This is also for developing mathematics.

combinatorial

combinatorial

This is a preliminary version of Lecture 6. Section 6.5 in particular is un�nished.

Some classical results are generalized to finitely combinatorial manifolds. Euler-Poincare characteristic is discussed and geometrical inclusions in Smarandache geometries for various geometries are also presented by the geometrical theory on finitely combinatorial manifolds in this paper.

Richard Stanley played a crucial role, through his work and his students, in the development of the relatively new area known as combinatorial representation theory. In the early stages, he has the merit to have pointed out to combinatorialists the potential that representation theory has for applications of combinatorial methods. Throughout his distinguished career, he wrote significant articles which touch upon various combinatorial aspects related to representation theory (of Lie algebras, the symmetric group, etc.). I describe some of Richard's contributions involving Lie algebras, as well as recent developments inspired by them (including some open problems), which attest the lasting impact of his work.

For proper subsets U of {1,2,...,31} we define and construct U-regular isotopy invariants of the Carter-Rieger-Saito movies (representing knotted surfaces) and use these to show that there are ambient isotopic knotted surfaces represented by movies M1 and M2 for which movie-moves of type 31 are required to get from M1 to M2.

Keller proposed a combinatorial conjecture on construction of an n-by-infinite matrix, which comes from showing the existence of many orbits of different sizes in certain linear group actions. He proved it for the case n=4, and we show that conjecture is true in the general case. We also propose a combinatorial game version of the conjecture which even further generalizes the problem.

In the paper we obtain some new applications of well--known W. Rudin's theorem concerning lacunary series to problems of combinatorial number theory. We generalize a result of M.-C. Chang on L_2 (L)-norm of Fourier coefficients of a set (here L is a dissociated set), and prove a dual version of the theorem. Our main instrument is computing of eigenvalues of some operators.

In the paper we obtain some new applications of well--known W. Rudin's theorem concerning lacunary series to problems of combinatorial number theory. We generalize a result of M.-C. Chang on L_2 (L)-norm of Fourier coefficients of a set (here L is a dissociated set), and prove a dual version of the theorem. Our main instrument is computing of eigenvalues of some operators.

The talk focuses on expander graphs in conjunction with the combined use of SDPs and eigenvalue techniques for approximating optimal solutions to combinatorial optimization problems. In the first part of the talk I will explain how to construct cost-effective, expanding networks by using 'local' sparsifiers of graphs that emerge as a solution to a semidefinite program. In the second part of the talk I will show that the Unique Games Conjecture is false when the underlying constraint graph is a (spectral) expander. Namely, I will present a polynomial-time algorithm for Unique Games on expanding instances that finds a good assignment when there exists one. ©2009 Microsoft Corporation. All rights reserved.

After some excitement generated by recently suggested public key exchange protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al., it is a prevalent opinion now that the conjugacy search problem is unlikely to provide sufficient level of security if a braid group is used as the platform. In this paper we address the following questions: (1) whether choosing a different group, or a class of groups, can remedy the situation; (2) whether some other "hard" problem from combinatorial group theory can be used, instead of the conjugacy search problem, in a public key exchange protocol. Another question that we address here, although somewhat vague, is likely to become a focus of the future research in public key cryptography based on symbolic computation: (3) whether one can efficiently disguise an element of a given group (or a semigroup) by using defining relations.

We propose a number of techniques for obtaining a global ranking from data that may be incomplete and imbalanced -- characteristics almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in score or rating-based cardinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method uses the graph Helmholtzian, the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is an analogue of the Laplace operator or scalar Laplacian. We study the graph Helmholtzian using combinatorial Hodge theory: we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components, a gradient flow that represents the L2-optimal global ranking and a divergence-free flow (cyclic) that measures the validity of the global ranking obtained -- if this is large, then the data does not have a meaningful global ranking. This divergence-free flow can be further decomposed orthogonally into a curl flow (locally cyclic) and a harmonic flow (locally acyclic but globally cyclic); these provides information on whether inconsistency arises locally or globally. An obvious advantage over the NP-hard Kemeny optimization is that discrete Hodge decomposition may be computed via a linear least squares regression. We also investigated the L1-projection of edge flows, showing that this is dual to correlation maximization over bounded divergence-free flows, and the L1-approximate sparse cyclic ranking, showing that this is dual to correlation maximization over bounded curl-free flows. We discuss relations with Kemeny optimization, Borda count, and Kendall-Smith consistency index from social choice theory and statistics.

The paper is devoted to the study of some well-knonw combinatorial functions on the symmetric group $\sn$ --- the major index $\maj$, the descent number $\des$, and the inversion number $\inv$ --- from the representation-theoretic point of view. We show that each of these functions generates in the group algebra the same ideal, and the restriction of the left regular representation to this ideal is isomorphic to the representation of $\sn$ in the space of $n\times n$ skew-symmetric matrices. This allows us to obtain formulas for the functions $\maj,\des,\inv$ in terms of matrices of an exceptionally simple form. These formulas are applied to find the spectra of the elements under study in the regular representation, as well as to deduce a series of identities relating these functions to one another and to the number of fixed points $\fix$. \smallskip\noindent {\it Keywords:} major index, descent number, inversion number, representations of the symmetric group, skew-symmetric matrices, dual complexity.

We introduce a new model construction for Martin-L\"{o}f intensional type theory, which is sound and complete for the 1-truncated version of the theory. The model formally combines the syntactic model with a notion of realizability; it also encompasses the well-known Hofmann- Streicher groupoid semantics. As our main application, we use the model to analyse the syntactic groupoid associated to the type theory generated by a graph G, showing that it has the same homotopy type as the free groupoid generated by G.

Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are no published results how they perform in practice. With this paper we address this gap. We provide the first implementation of the combinatorial solver by [Kelner et al., STOC 2013], which is particularly appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix corresponds to a graph; solving Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning tree with the low-stretch property. The results of our comprehensive experimental study are ambivalent. They confirm a nearly-linear running time, but for reasonable inputs the constant factors make the solver much slower than methods with higher asymptotic complexity. One other aspect predicted by theory is confirmed by our findings, though: Spanning trees with lower stretch indeed reduce the solver's running time. Yet, simple spanning tree algorithms perform in practice better than those with a guaranteed low stretch.

Let $G$ be a finite abelian group with exponent $n>1$. For $a_1,\ldots,a_{n-1}\in G$, we determine completely when there is a permutation $\sigma$ on $\{1,\ldots,n-1\}$ such that $sa_{\sigma(s)}\not=0$ for all $s=1,\ldots,n-1$. When $G$ is the cyclic group $\mathbb Z/n\mathbb Z$, this confirms a conjecture of Z.-W. Sun.

The interplay between algebro-geometric and combinatorial Brill-Noether theory is studied. The Brill-Noether variety of a graph shown to be non-empty if the Brill-Noether number is non-negative, as a consequence of the analogous fact for smooth projective curves. Similarly, the existence of a graph for which the Brill-Noether variety is empty implies the emptiness of the corresponding Brill-Noether variety for a general curve. The main tool is a refinement of Baker's Specialization Lemma.

The talk focuses on expander graphs in conjunction with the combined use of SDPs and eigenvalue techniques for approximating optimal solutions to combinatorial optimization problems. In the first part of the talk I will explain how to construct cost-effective, expanding networks by using 'local' sparsifiers of graphs that emerge as a solution to a semidefinite program. In the second part of the talk I will show that the Unique Games Conjecture is false when the underlying constraint graph is a (spectral) expander. Namely, I will present a polynomial-time algorithm for Unique Games on expanding instances that finds a good assignment when there exists one. ©2009 Microsoft Corporation. All rights reserved.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we define assigned graphs, that are graphs with appropriate decorations for the multi-scale framework. We then define Hopf algebras on these assigned graphs and on the Gallavotti-Nicol\`o trees, particular class of trees encoding the supplementary informations of the assigned graphs. Several morphisms between these combinatorial Hopf algebras and the Connes-Kreimer algebra are given. Finally, scale dependent couplings are analyzed via this combinatorial algebraic setting.

We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we define assigned graphs, that are graphs with appropriate decorations for the multi-scale framework. We then define Hopf algebras on these assigned graphs and on the Gallavotti-Nicol\`o trees, particular class of trees encoding the supplementary informations of the assigned graphs. Several morphisms between these combinatorial Hopf algebras and the Connes-Kreimer algebra are given. Finally, scale dependent couplings are analyzed via this combinatorial algebraic setting.

Our Island Story was first published in 1905 and became an instant classic. Beginning with the Romans and following the triumphs and foibles of the good, not so good and the downright despicable figures of history; we are treated to a dazzling montage of kings, queens, barons, knights, explorers, movers and shakers that have played a key role in the history of England. Marshall freely mixes folk tale with historical fact and in so doing paints a very vivid picture of the past in a style reminiscent of all that is finest in the children’s story telling tradition.This is the first section of that work and will carry you from the time when Tacitus first sang the praises of Britannica to his Roman readers up to the vicious and bloodthirsty confusion that is the War of the Roses (about 1500 years). (Summary by Jim)<br /><br />Part 2 of this book may be found at <a href="http://librivox.org/our-island-story-part-2-by-he-marshall/" target="_blank">http://librivox.org/our-island-story-part-2-by-he-marshall/</a>.

Our Island Story was first published in 1905 and became an instant classic. Beginning with the Romans and following the triumphs and foibles of the good, not so good and the downright despicable figures of history; we are treated to a dazzling montage of kings, queens, barons, knights, explorers, movers and shakers that have played a key role in the history of England. Marshall freely mixes folk tale with historical fact and in so doing paints a very vivid picture of the past in a style reminiscent of all that is finest in the children’s story-telling tradition. This is the second section of that work and will carry you from the convoluted and bloody ‘Wars of the Roses’ up to the death of Queen Victoria and the resolution of the Boer War. Please be advised that this book was written in the early years of the 20th century and there will be words and phrases used then, in all innocence, that are considered politically incorrect in this age. It’s perfectly fine for children to listen to but please ensure there is someone on hand who can explain these terms in a mature fashion.(Summary by Jim)<br /><br />Part 1 of this book may be found at <a href="https://librivox.org/our-island-story-part-1-by-he-marshall/" target="_blank">https://librivox.org/our-island-story-part-1-by-he-marshall/</a>.

3This Country of Ours, Part 1

By Henrietta Elizabeth Marshall

Another fine history book for children! Published in 1917, Marshall&#8217;s book of stories from the history of the United States begins with accounts of exploration and settlement, and ends with the presidency of Woodrow Wilson.<br /><br /> This is Part 1: Stories of Explorers and Pioneers, from Leif Ericson to Sir Walter Raleigh. Read along and view the original illustrations at <a href="http://www.mainlesson.com/display.php?author=marshall&book=country&story=_contents">mainlesson.com</a>. <br /><br /> Part 2: <a href="http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/</a><br /> Part 3: <a href="http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/</a><br /> Part 4: <a href="http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/</a><br /> Part 5: <a href="http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/</a><br /> Part 6: <a href="http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/</a><br /> Part 7: <a href="http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/">http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/</a><br />(Summary by Kara)

4Sinking of the Titanic and Great Sea Disasters

By Logan Marshall

This is a detailed and accurate account of the most awful marine disaster in history, constructed from the real facts as obtained from those on board who survived (gutenberg.org)

Principles of Economics was a leading economics textbook of Alfred Marshall (1842-1924), first published in 1890. Marshall began writing the book in 1881, and he spent much of the next decade at work on it.<br /><br />His plan for the work gradually extended to a two-volume compilation on the whole of economic thought; the first volume was published in 1890 to worldwide acclaim that established him as one of the leading economists of his time. It brought the ideas of supply and demand, of marginal utility and of the costs of production into a coherent whole, and became the dominant economic textbook in England for a long period. The second volume, which was to address foreign trade, money, trade fluctuations, taxation, and collectivism, was never published at all. (Summary from Wikipedia)<br /><br />This reading is based on the eighth edition, published in 1920.

Principles of Economics was a leading economics textbook of Alfred Marshall (1842-1924), first published in 1890. Marshall began writing the Principles of Economics in 1881 and he spent much of the next decade at work on the treatise. His plan for the work gradually extended to a two-volume compilation on the whole of economic thought; the first volume was published in 1890 to worldwide acclaim that established him as one of the leading economists of his time. It brought the ideas of supply and demand, of marginal utility and of the costs of production into a coherent whole, and became the dominant economic textbook in England for a long period. The second volume, which was to address foreign trade, money, trade fluctuations, taxation, and collectivism, was never published at all. (Wikipedia)<br /><br />The recording is based on the eighth edition, published in 1920.

7Myths and Legends of All Nations

By Logan Marshall

This excellent book contains many great stories from the various mythologies of man throughout the ages.

Principles of Economics was a leading economics textbook of Alfred Marshall (1842-1924), first published in 1890. Marshall began writing the book in 1881, and he spent much of the next decade at work on it.<br /><br />His plan for the work gradually extended to a two-volume compilation on the whole of economic thought; the first volume was published in 1890 to worldwide acclaim that established him as one of the leading economists of his time. It brought the ideas of supply and demand, of marginal utility and of the costs of production into a coherent whole, and became the dominant economic textbook in England for a long period. The second volume, which was to address foreign trade, money, trade fluctuations, taxation, and collectivism, was never published at all. (Summary from Wikipedia)<br /><br />This reading is based on the eighth edition, published in 1920.

Principles of Economics was a leading economics textbook of Alfred Marshall (1842-1924), first published in 1890. Marshall began writing the book in 1881, and he spent much of the next decade at work on it.<br /><br />His plan for the work gradually extended to a two-volume compilation on the whole of economic thought; the first volume was published in 1890 to worldwide acclaim that established him as one of the leading economists of his time. It brought the ideas of supply and demand, of marginal utility and of the costs of production into a coherent whole, and became the dominant economic textbook in England for a long period. The second volume, which was to address foreign trade, money, trade fluctuations, taxation, and collectivism, was never published at all. (Summary from Wikipedia)<br /><br />This reading is based on the eighth edition, published in 1920.

Principles of Economics was a leading economics textbook of Alfred Marshall (1842-1924), first published in 1890. Marshall began writing the book in 1881, and he spent much of the next decade at work on it.<br /><br />His plan for the work gradually extended to a two-volume compilation on the whole of economic thought; the first volume was published in 1890 to worldwide acclaim that established him as one of the leading economists of his time. It brought the ideas of supply and demand, of marginal utility and of the costs of production into a coherent whole, and became the dominant economic textbook in England for a long period. The second volume, which was to address foreign trade, money, trade fluctuations, taxation, and collectivism, was never published at all. (Summary from Wikipedia)<br /><br />This reading is based on the eighth edition, published in 1920.

Principles of Economics was a leading economics textbook of Alfred Marshall (1842-1924), first published in 1890. Marshall began writing the Principles of Economics in 1881 and he spent much of the next decade at work on the treatise. His plan for the work gradually extended to a two-volume compilation on the whole of economic thought.<br /><br />The first volume was published in 1890 to worldwide acclaim that established him as one of the leading economists of his time. It brought the ideas of supply and demand, of marginal utility and of the costs of production into a coherent whole, and became the dominant economic textbook in England for a long period. The second volume, which was to address foreign trade, money, trade fluctuations, taxation, and collectivism, was never published at all. (Wikipedia).<br /><br />This reading is based on the eighth edition, published in 1920.

Principles of Economics was a leading economics textbook of Alfred Marshall (1842-1924), first published in 1890. Marshall began writing the book in 1881, and he spent much of the next decade at work on it.<br /><br />His plan for the work gradually extended to a two-volume compilation on the whole of economic thought; the first volume was published in 1890 to worldwide acclaim that established him as one of the leading economists of his time. It brought the ideas of supply and demand, of marginal utility and of the costs of production into a coherent whole, and became the dominant economic textbook in England for a long period. The second volume, which was to address foreign trade, money, trade fluctuations, taxation, and collectivism, was never published at all. (Summary from Wikipedia)<br /><br />This reading is based on the eighth edition, published in 1920.

The brave warrior, Beowulf, comes to the aid of King Hrothgar when he hears that Grendel, a horrible monster, is terrorizing the inhabitants of Hart Hall. Beowulf heroically battles Grendel, the Water Witch, and a fierce dragon. (Summary by Laura Caldwell)

14This Country of Ours, Part 2

By Henrietta Elizabeth Marshall

Another fine history book for children! Published in 1917, Marshall&#8217;s book of stories from the history of the United States begins with accounts of exploration and settlement, and ends with the presidency of Woodrow Wilson.<br /><br /> This is Part 2: Stories of Virginia, from chapter 13 (Captain John Smith) to chapter 21 (The Knights of the Golden Horseshoe). Read along and view the original illustrations at <a href="http://www.mainlesson.com/display.php?author=marshall&book=country&story=_contents">mainlesson.com</a>. <br /><br /> Part 1: <a href="http://librivox.org/this-country-of-ours-part-1-by-he-marshall/">http://librivox.org/this-country-of-ours-part-1-by-he-marshall/</a><br /> Part 3: <a href="http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/</a><br /> Part 4: <a href="http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/</a><br /> Part 5: <a href="http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/</a><br /> Part 6: <a href="http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/</a><br /> Part 7: <a href="http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/">http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/</a><br />(Summary by Kara)

15This Country of Ours, Part 3

By Henrietta Elizabeth Marshall

Another fine history book for children! Published in 1917, Marshall's book of stories from the history of the United States begins with accounts of exploration and settlement, and ends with the presidency of Woodrow Wilson.<br /><br /> This is Part 3: Stories of New England, from chapter 22 (The Story of the Pilgrim Fathers) to chapter 34 (The Witches of Salem). Read along and view the original illustrations at <a href="http://www.mainlesson.com/display.php?author=marshall&book=country&story=_contents">mainlesson.com</a>. <br /><br /> Part 1: <a href="http://librivox.org/this-country-of-ours-part-1-by-he-marshall/">http://librivox.org/this-country-of-ours-part-1-by-he-marshall/</a><br /> Part 2: <a href="http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/</a><br /> Part 4: <a href="http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/</a></br> Part 5: <a href="http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/</a></br> Part 6: <a href="http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/</a></br> Part 7: <a href="http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/">http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/</a><br />(Summary by Kara)

16This Country of Ours, Part 4

By Henrietta Elizabeth Marshall

Another fine history book for children! Published in 1917, Marshall&#8217;s book of stories from the history of the United States begins with accounts of exploration and settlement, and ends with the presidency of Woodrow Wilson.<br /><br /> This is Part 4: Stories of the Middle and Southern Colonies, from chapter 35 (The Founding of Maryland) to chapter 44 (The Founding of Georgia). Read along and view the original illustrations at <a href="http://www.mainlesson.com/display.php?author=marshall&book=country&story=_contents">mainlesson.com</a>.<br /><br /> Part 1: <a href="http://librivox.org/this-country-of-ours-part-1-by-he-marshall/">http://librivox.org/this-country-of-ours-part-1-by-he-marshall/</a><br /> Part 2: <a href="http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/</a><br /> Part 3: <a href="http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/</a><br /> Part 5: <a href="http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/</a></br> Part 6: <a href="http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/</a></br> Part 7: <a href="http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/">http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/</a><br />(Summary by Kara)

17This Country of Ours, Part 5

By Henrietta Elizabeth Marshall

Another fine history book for children! Published in 1917, Marshall&#8217;s book of stories from the history of the United States begins with accounts of exploration and settlement, and ends with the presidency of Woodrow Wilson.<br /><br /> This is Part 5: Stories of the French in America, from chapter 45 (How the Mississippi was Discovered) to chapter 50 (The Rebellion of Pontiac). Read along and view the original illustrations at <a href="http://www.mainlesson.com/display.php?author=marshall&book=country&story=_contents">http://mainlesson.com/</a> .<br /><br /> Part 1: <a href="http://librivox.org/this-country-of-ours-part-1-by-he-marshall/">http://librivox.org/this-country-of-ours-part-1-by-he-marshall/</a><br /> Part 2: <a href="http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/</a><br /> Part 3: <a href="http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/</a><br /> Part 4: <a href="http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/</a></br> Part 6: <a href="http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/</a></br> Part 7: <a href="http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/">http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/</a><br />(Summary by Kara)

18This Country of Ours, Part 6

By Henrietta Elizabeth Marshall

Another fine history book for children! Published in 1917, Marshall&#8217;s book of stories from the history of the United States begins with accounts of exploration and settlement, and ends with the presidency of Woodrow Wilson.<br /><br /> This is Part 6: Stories of the Struggle for Liberty, from chapter 51 (The Boston Tea-Party) to chapter 63 (A Turning Point in the World's History). Read along and view the original illustrations at <a href="http://www.mainlesson.com/display.php?author=marshall&book=country&story=_contents">http://mainlesson.com/</a>.<br /><br /> Part 1: <a href="http://librivox.org/this-country-of-ours-part-1-by-he-marshall/">http://librivox.org/this-country-of-ours-part-1-by-he-marshall/</a><br /> Part 2: <a href="http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/</a><br /> Part 3: <a href="http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/</a><br /> Part 4: <a href="http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/</a></br> Part 5: <a href="http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/</a></br> Part 6: <a href="http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/</a><br />(Summary by Kara)

19This Country of Ours, Part 7

By Henrietta Elizabeth Marshall

Another fine history book for children! Published in 1917, Marshall&#8217;s book of stories from the history of the United States begins with accounts of exploration and settlement, and ends with the presidency of Woodrow Wilson.<br><br> This is Part 7: Stories of the United States under the Constitution, from chapter 64 (Washington First in War, First in Peace) to chapter 99 (Wilson - The Great War). Read along and view the original illustrations at <a href="http://www.mainlesson.com/display.php?author=marshall&book=country&story=_contents">mainlesson.com</a><br><br> Part 1: <a href="http://librivox.org/this-country-of-ours-part-1-by-he-marshall/">http://librivox.org/this-country-of-ours-part-1-by-he-marshall/</a><br> Part 2: <a href="http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-2-by-h-e-marshall/</a><br> Part 3: <a href="http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-3-by-h-e-marshall/</a><br> Part 4: <a href="http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-4-by-h-e-marshall/</a></br> Part 5: <a href="http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-5-by-h-e-marshall/</a></br> Part 6: <a href="https://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/">http://librivox.org/this-country-of-ours-part-6-by-h-e-marshall/</a><br> Part 7: <a href="http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/">http://librivox.org/this-country-of-ours-part-7-by-henrietta-elizabeth-marshall/</a><br>(Summary by Kara)

20Wonder Book of Bible Stories

By Logan Marshall

It is with the desire of aiding parents and teachers in telling these stories, and aiding children to understand them, also in the hope that they may be read in many schools, that a few among the many interesting stories in the Bible have been chosen, brought together and as far as necessary simplified to meet the minds of the young. - Introduction by Jesse Lyman Hurlbut inside the book itself.

This book gives an exciting account of Napoleon's rise to power. It opens with his humble beginning on the island of Corsica and ultimately emperor of over half of Europe. This book focuses on his victorious military campaigns, his disastrous Russian campaign, his exile to the island of Elba, his final loss at Waterloo, and his last days on the lonely island of St. Helena. Summary by pjcsaville.

22Favorite Fairy Tales

By Logan Marshall

This is a collection of the fairy tales that children love best, told in simple language and lavishly illustrated. They are written by various authors, a selection of the best and most popular fairy stories, culled from many sources and here collected and presented in most attractive form, printed in large clear type, with many pictures, some of them colored. (Summary from the book)<br><br>Read along and view the illustrations at <a href="http://www.gutenberg.org/ebooks/20748"> Project Gutenberg</a>

23Life of Washington, Volume 1

By John James Marshall

If it be true that it takes a great man to interpret the life of a great man then Bushrod Washington made no mistake in the selection of a biographer. For Marshall, under the influence of Washington, came to be nearly as great a man as the character whose life and achievement held his deepest thought for nearly a quarter of a century. . . . Marshall's sympathetic understanding of his subject, his firsthand knowledge of events with his remarkable powers of expression qualified him to produce the masterpiece that has come down to us.

24Cedric the Forester

By Bernard Gay Marshall

A rollicking juvenile adventure tale, this historical fiction book received the Newbery Honor award in 1922. Set in England during the early 1200s, the scion of a Western Marches noble family relates the many encounters and battles that he faces with the Saxon yeoman who becomes his fast friend. This yeoman, great of cross-bow skill and strategy, often saves the day, and after earning knighthood, becomes instrumental in the making of the Magna Carta. - Summary by Lynette Caulkins

25Life of Washington, Volume 2

By John James Marshall

Volume 2 of The Life begins with some early biography, but moves quickly to Washington's military career as a colonel in the battles against the French in Canada until the cessation of his tenure, after which he marries and appears to settle down. But, of course he is called to lead the troops fighting in the revolution against England about which leadership the remaining of Volume 2 is dedicated to the point of the American rejection of England's Plan for Reconciliation.

26Life of Washington, Volume 3

By John James Marshall

Volume 3 of The Life continues the Revolutionary War from the incursion into Jersey in 1778 to its conclusion with the surrender of Lord Cornwallis in 1781 at Yorktown, with emphasis on the role of George Washington, Commander-in-Chief of the United States forces.

27Life of Washington, Volume 4

By John James Marshall

This fourth volume covers the final battles and the peace conditions of the war, Washington at home, Washington as first President, and the internal battles to hammer out a new and heretofore unseen government.

28Life of Washington, Volume 5

By John James Marshall

Volume five of John Marshall's biography follows Washington through his second term ending in the election of John Adams and Washington's retirement to Mount Vernon. This narrative continues the careful and intriguing analysis of the subjects of the previous volume as America navagtes with Washington's wise captaincy the complicated and sometimes violent foreign and domestic conflicts of the young, developing nation.

29English Literature for Boys and Girls

By Henrietta Elizabeth Marshall

"Has there ever been a time when no stories were told? Has there ever been a people who did not care to listen? I think not." Thus begins a journey through the history of English literature, from the early middle ages to the Victorian era. The young person will experience the most well-loved characters of all time, from Beowulf, the warrior who dares to challenge a terrifying and bloodthirsty monster, to virtuous Una of The Faerie Queene, faithful to the end. H. E. Marshall masterfully weaves selections from the stories into her own retelling, leaving the reader with a keen appreciation for the great books that have become the foundation of Western culture. - Summary by cadastra

Book of Bible stories from the Old and New Testaments, in language easy for children to understand. - Summary by Ron Altman