scholarly journals Geometric Realization of $\gamma$-Vectors of Subdivided Cross Polytopes

10.37236/9301 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Natalie Aisbett ◽  
Vadim Volodin
Keyword(s):  
Simplicial Complex ◽  
Geometric Realization ◽  
Simplicial Complexes ◽  
The Face ◽  
The Cross

For any flag simplicial complex $\Theta$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\Delta(\Theta)$ whose $f$-vector is the $\gamma$-vector of $\Theta$. This proves that the $\gamma$-vector of any such simplicial complex is the face vector of a flag simplicial complex, partially solving a conjecture by Nevo and Petersen. As a corollary we obtain that such simplicial complexes satisfy the Frankl-Füredi-Kalai inequalities.

scholarly journals Face numbers of pseudomanifolds with isolated singularities

2012 ◽  
Vol 110 (2) ◽  
pp. 198 ◽  
Author(s):  
Isabella Novik ◽  
Ed Swartz
Keyword(s):  
Simplicial Complex ◽  
Upper Bound ◽  
Hilbert Function ◽  
Simplicial Complexes ◽  
Topological Invariant ◽  
Two Dimensional ◽  
Isolated Singularities ◽  
Face Ring ◽  
The Face

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower and upper bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the $f$-vector can be completely characterized are described. We also show that the Hilbert function of a generic Artinian reduction of the face ring of a simplicial complex $\Delta$ with isolated singularities minus the $h$-vector of $\Delta$ is a PL-topological invariant.

scholarly journals $k$-shellable simplicial complexes and graphs

2018 ◽  
Vol 122 (2) ◽  
pp. 161
Author(s):  
Rahim Rahmati-Asghar
Keyword(s):  
Simplicial Complex ◽  
Large Class ◽  
Simplicial Complexes ◽  
Face Ring ◽  
Stanley Conjecture ◽  
The Face ◽  
Shellable Simplicial Complex

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying an expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture.Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.

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.

“The Last Enemy Is Death”

1989 ◽  
Vol 43 (4) ◽  
pp. 380-392
Author(s):  
Charles E Brown
Keyword(s):  
The Face ◽  
The Cross

Paul's reflections on the universal curse of death and its conquest by the resurrection of God's son who shared that curse in his own death on the cross help define the pastoral approach to those who suffer humanity's common anxiety in the face of death

The Essence of Christianity and the Cross of Christ

1914 ◽  
Vol 7 (4) ◽  
pp. 538-594
Author(s):  
Benjamin B. Warfield
Keyword(s):  
New Haven ◽  
Liberal Theology ◽  
Religious System ◽  
The World ◽  
Cross Of Christ ◽  
The Face ◽  
Divinity School ◽  
The Common ◽  
The Cross ◽  
The Right

In a recent number of The Harvard Theological Review, Professor Douglas Clyde Macintosh of the Yale Divinity School outlines in a very interesting manner the religious system to which he gives his adherence. For “substance of doctrine” (to use a form of speech formerly quite familiar at New Haven) this religious system does not differ markedly from what is usually taught in the circles of the so-called “Liberal Theology.” Professor Macintosh has, however, his own way of construing and phrasing the common “Liberal” teaching; and his own way of construing and phrasing it presents a number of features which invite comment. It is tempting to turn aside to enumerate some of these, and perhaps to offer some remarks upon them. As we must make a selection, however, it seems best to confine ourselves to what appears on the face of it to be the most remarkable thing in Professor Macintosh's representations. This is his disposition to retain for his religious system the historical name of Christianity, although it utterly repudiates the cross of Christ, and in fact feels itself (in case of need) quite able to get along without even the person of Christ. A “new Christianity,” he is willing, to be sure, to allow that it is—a “new Christianity for which the world is waiting”; and as such he is perhaps something more than willing to separate it from what he varyingly speaks of as “the older Christianity,” “actual Christianity,” “historic Christianity,” “actual, historical Christianity.” He strenuously claims for it, nevertheless, the right to call itself by the name of “Christianity.”

scholarly journals Simplicial Complexes of Whisker Type

10.37236/4894 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mina Bigdeli ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Antonio Macchia
Keyword(s):  
Simplicial Complex ◽  
Short Proof ◽  
Monomial Ideal ◽  
Simplicial Complexes ◽  
Linear Resolution ◽  
Cw Complex ◽  
Alexander Dual ◽  
Vertex Decomposable ◽  
The Ideal

Let $I\subset K[x_1,\ldots,x_n]$ be  a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that  $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.

scholarly journals Optimal Decision Trees on Simplicial Complexes

10.37236/1900 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Decision Tree ◽  
Decision Trees ◽  
Simplicial Complex ◽  
Elementary Theory ◽  
Simplicial Complexes ◽  
Optimal Decision ◽  
Property A ◽  
Recursive Definition ◽  
Topological Combinatorics ◽  
Definition Of

We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman's discrete Morse theory, the number of evasive faces of a given dimension $i$ with respect to a decision tree on a simplicial complex is greater than or equal to the $i$th reduced Betti number (over any field) of the complex. Under certain favorable circumstances, a simplicial complex admits an "optimal" decision tree such that equality holds for each $i$; we may hence read off the homology directly from the tree. We provide a recursive definition of the class of semi-nonevasive simplicial complexes with this property. A certain generalization turns out to yield the class of semi-collapsible simplicial complexes that admit an optimal discrete Morse function in the analogous sense. In addition, we develop some elementary theory about semi-nonevasive and semi-collapsible complexes. Finally, we provide explicit optimal decision trees for several well-known simplicial complexes.

scholarly journals An explanation for phonological word-final vowel shortening: Evidence from Tokyo Japanese

2013 ◽  
Vol 4 (2) ◽  
Author(s):  
Satsuki Nakai
Keyword(s):  
Short Vowel ◽  
Contributing Factors ◽  
Vowel Length ◽  
Phonological Word ◽  
Distributional Properties ◽  
Tokyo Japanese ◽  
The Face ◽  
The Cross

AbstractThis paper offers an account for the cross-linguistic prevalence of phonological word-final vowel shortening, in the face of phonetic final lengthening, also commonly observed across languages. Two contributing factors are hypothesized: (1) an overlap in the durational distributions of short and long vowel phonemes across positions in the utterance can lead to the misidentification of phonemic vowel length and (2) the direction of bias in such misidentification is determined by the distributional properties of the short and long vowel phonemes in the region of the durational overlap. Because short vowel phonemes are typically more frequent in occurrence and less variable in duration than long vowel phonemes, long vowel phonemes are more likely to be misidentified than short vowel phonemes. Results of production and perception studies in Tokyo Japanese support these hypotheses.

Spectral gap of a weighted 3-simplicial complex

2021 ◽  
pp. 2150084
Author(s):  
Khalid Hatim ◽  
Azeddine Baalal
Keyword(s):  
Lower Bound ◽  
Simplicial Complex ◽  
Upper Bound ◽  
Spectral Gap ◽  
Simplicial Complexes ◽  
Cheeger Constant ◽  
Nonzero Eigenvalue ◽  
New Framework

In this paper, we construct a new framework that’s we call the weighted [Formula: see text]-simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted [Formula: see text]-simplicial complexes.

The Death of God

2011 ◽  
Author(s):  
Krzysztof Michalski
Keyword(s):  
Human Nature ◽  
Gospel Of Matthew ◽  
Human Life ◽  
Death Of God ◽  
New Beginning ◽  
The World ◽  
The Face ◽  
The Cross

This chapter turns to Plato's Phaedo as well as the Gospel of Matthew: two narratives about death, and two visions of human nature. Christ's cry on the cross, as told by Matthew, gives voice to an understanding of human life that is radically different from that of Socrates. For Phaedo's Socrates, the truly important things in life are ideas: the eternal order of the world, the understanding of which leads to unperturbed peace and serenity in the face of death. The Gospel is the complete opposite: it testifies to the incurable presence of the Unknown in every moment of life, a presence that rips apart every human certainty built on what is known, that disturbs all peace, all serenity—that severs the continuity of time, opening every moment of our lives to nothingness, thereby inscribing within them the possibility of an abrupt end and the chance at a new beginning.

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