scholarly journals Playing Nim on a Simplicial Complex

10.37236/1233 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Richard Ehrenborg ◽  
Einar Steingrímsson
Keyword(s):  
Simplicial Complex ◽  
Winning Strategy ◽  
Independent Sets ◽  
Binary Matroid ◽  
Classical Game

We introduce a generalization of the classical game of Nim by placing the piles on the vertices of a simplicial complex and allowing a move to affect the piles on any set of vertices that forms a face of the complex. Under certain conditions on the complex we present a winning strategy. These conditions are satisfied, for instance, when the simplicial complex consists of the independent sets of a binary matroid. Moreover, we study four operations on a simplicial complex under which games on the complex behave nicely. We also consider particular complexes that correspond to natural generalizations of classical Nim.

scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

scholarly journals Linear syzygies of Stanley-Reisner ideals

2001 ◽  
Vol 89 (1) ◽  
pp. 117 ◽  
Author(s):  
V Reiner ◽  
V Welker
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Monomial Ideal ◽  
Independent Sets ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Linear Syzygies ◽  
Alexander Dual ◽  
Syzygy Modules

We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex $\Delta$. The description is in terms of the homology of the canonical Alexander dual complex $\Delta^*$. As applications we are able to prove for monomial ideals and $j=1$ a conjecture of J. Herzog giving lower bounds on the number of $i$-syzygies in the linear strand of $j^{th}$-syzygy modules show that the maps in the linear strand can be written using only $\pm 1$ coefficients if $\Delta^*$ is a pseudomanifold exhibit an example where multigraded maps in the linear strand cannot be written using only $\pm 1$ coefficients compute the entire resolution explicitly when $\Delta^*$ is the complex of independent sets of a matroid

scholarly journals Topological graph persistence

2020 ◽  
Vol 11 (1) ◽  
pp. 72-87
Author(s):  
Mattia G. Bergomi ◽  
Massimo Ferri ◽  
Lorenzo Zuffi
Keyword(s):  
Simplicial Complex ◽  
Independent Sets ◽  
Data Representation ◽  
Basic Tool ◽  
Topological Graph ◽  
Topological Information ◽  
One Dimensional ◽  
Low Dimensional ◽  
Dimensional Simplicial Complex

Abstract Graphs are a basic tool in modern data representation. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do this by extending previous work in homological persistence, and proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

scholarly journals Cooperative Colorings and Independent Systems of Representatives

10.37236/2488 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ron Aharoni ◽  
Ron Holzman ◽  
David Howard ◽  
Philipp Sprüssel
Keyword(s):  
Simplicial Complex ◽  
Independent Sets ◽  
Choice Functions ◽  
Sample Result ◽  
List Colorings ◽  
Vertex Set ◽  
Degree Conditions

We study a generalization of the notion of coloring of graphs, similar in spirit to that of list colorings: a cooperative coloring of a family of graphs $G_1,G_2, \ldots,G_k$ on the same vertex set $V$ is a choice of independent sets $A_i$ in $G_i$ ($1 \le i \le k)$ such that $\bigcup_{i=1}^kA_i=V$. This notion is linked (with translation in both directions) to the notion of ISRs, which are choice functions on given sets, whose range belongs to some simplicial complex. When the complex is that of the independent sets in a graph $G$, an ISR for a partition of the vertex set of a graph $G$ into sets $V_1,\ldots, V_n$ is a choice of a vertex $v_i \in V_i$ for each $i$ such that $\{v_1,\ldots,v_n\}$ is independent in $G$. Using topological tools, we study degree conditions for the existence of cooperative colorings and of ISRs. A sample result: Three cycles on the same vertex set have a cooperative coloring.

scholarly journals Hard Squares with Negative Activity on Cylinders with Odd Circumference

10.37236/71 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Square Grid ◽  
Unit Square ◽  
Hard Squares ◽  
Vertex Set

Let $C_{m,n}$ be the graph on the vertex set $\{1, \ldots, m\} \times \{0, \ldots, n-1\}$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$, where the second index is computed modulo $n$. One may view $C_{m,n}$ as a unit square grid on a cylinder with circumference $n$ units. For odd $n$, we prove that the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $C_{m,n}$ is either $2$ or $-1$, depending on whether or not $\gcd(m-1,n)$ is divisble by $3$. The proof relies heavily on previous work due to Thapper, who reduced the problem of computing the Euler characteristic of $\Sigma_{m,n}$ to that of analyzing a certain subfamily of sets with attractive properties. The situation for even $n$ remains unclear. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

Winning the Argument and Moving the Fight: The Legacy of a Grassroots Humanitarian

Public Voices ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 115
Author(s):  
Mary Coleman
Keyword(s):  
Higher Education ◽  
United States ◽  
Winning Strategy ◽  
Supreme Court Decision ◽  
The United States ◽  
Colleges And Universities ◽  
Lessons Learned ◽  
The Political ◽  
Political Actors ◽  
Trade Offs

The author of this article argues that the two-decades-long litigation struggle was necessary to push the political actors in Mississippi into a more virtuous than vicious legal/political negotiation. The second and related argument, however, is that neither the 1992 United States Supreme Court decision in Fordice nor the negotiation provided an adequate riposte to plaintiffs’ claims. The author shows that their chief counsel for the first phase of the litigation wanted equality of opportunity for historically black colleges and universities (HBCUs), as did the plaintiffs. In the course of explicating the role of a legal grass-roots humanitarian, Coleman suggests lessons learned and trade-offs from that case/negotiation, describing the tradeoffs as part of the political vestiges of legal racism in black public higher education and the need to move HBCUs to a higher level of opportunity at a critical juncture in the life of tuition-dependent colleges and universities in the United States. Throughout the essay the following questions pose themselves: In thinking about the Road to Fordice and to political settlement, would the Justice Department lawyers and the plaintiffs’ lawyers connect at the point of their shared strength? Would the timing of the settlement benefit the plaintiffs and/or the State? Could plaintiffs’ lawyers hold together for the length of the case and move each piece of the case forward in a winning strategy? Who were plaintiffs’ opponents and what was their strategy? With these questions in mind, the author offers an analysis of how the campaign— political/legal arguments and political/legal remedies to remove the vestiges of de jure segregation in higher education—unfolded in Mississippi, with special emphasis on the initiating lawyer in Ayers v. Waller and Fordice, Isaiah Madison

scholarly journals Nanoencapsulated Essential Oils with Enhanced Antifungal Activity for Potential Application on Agri-Food, Material and Environmental Fields

Antibiotics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 31
Author(s):  
Magdaléna Kapustová ◽  
Giuseppe Granata ◽  
Edoardo Napoli ◽  
Andrea Puškárová ◽  
Mária Bučková ◽  
...  
Keyword(s):  
Antifungal Activity ◽  
Essential Oil ◽  
Essential Oils ◽  
Winning Strategy ◽  
Colloidal Suspensions ◽  
Fungal Species ◽  
Natural Environments ◽  
Food Material ◽  
Minimum Fungicidal Concentration ◽  
Fungal Strains

Nanotechnology is a new frontier of this century that finds applications in various fields of science with important effects on our life and on the environment. Nanoencapsulation of bioactive compounds is a promising topic of nanotechnology. The excessive use of synthetic compounds with antifungal activity has led to the selection of resistant fungal species. In this context, the use of plant essential oils (EOs) with antifungal activity encapsulated in ecofriendly nanosystems could be a new and winning strategy to overcome the problem. We prepared nanoencapsules containing the essential oils of Origanum vulgare (OV) and Thymus capitatus (TC) by the nanoprecipitation method. The colloidal suspensions were characterized for size, polydispersity index (PDI), zeta potential, efficiency of encapsulation (EE) and loading capacity (LC). Finally, the essential oil nanosuspensions were assayed against a panel of fourteen fungal strains belonging to the Ascomycota and Basidiomycota phyla. Our results show that the nanosystems containing thyme and oregano essential oils were active against various fungal strains from natural environments and materials. In particular, the minimum inhibitory concentration (MIC) and minimum fungicidal concentration (MFC) values were two to four times lower than the pure essential oils. The aqueous, ecofriendly essential oil nanosuspensions with broad-spectrum antifungal activity could be a valid alternative to synthetic products, finding interesting applications in the agri-food and environmental fields.

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