scholarly journals Volumes of Flow Polytopes Related to Caracol Graphs

10.37236/9187 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Jihyeug Jang ◽  
Jang Soo Kim
Keyword(s):  
Constant Term ◽  
Original Graph ◽  
Dyck Paths ◽  
Flow Vector ◽  
Flow Polytopes ◽  
Flow Polytope

Recently, Benedetti et al. introduced an Ehrhart-like polynomial associated to a graph. This polynomial is defined as the volume of a certain flow polytope related to a graph and has the property that the leading coefficient is the volume of the flow polytope of the original graph with net flow vector $(1,1,\dots,1)$. Benedetti et al. conjectured a formula for the Ehrhart-like polynomial of what they call a caracol graph. In this paper their conjecture is proved using constant term identities, labeled Dyck paths, and a cyclic lemma.

scholarly journals Flow Polytopes and the Space of Diagonal Harmonics

2019 ◽  
Vol 71 (6) ◽  
pp. 1495-1521
Author(s):  
Ricky Ini Liu ◽  
Alejandro H. Morales ◽  
Karola Mészáros
Keyword(s):  
Complete Graph ◽  
Hilbert Series ◽  
Threshold Graphs ◽  
Flow Polytopes ◽  
Diagonal Harmonics ◽  
Flow Polytope

AbstractA result of Haglund implies that the$(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a$(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector$(-n,1,\ldots ,1)$. We study the$(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at$t=1$,$0$, and$q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the$(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.

scholarly journals $h$-Polynomials via Reduced Forms

10.37236/5172 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Karola Meszaros
Keyword(s):  
Reduced Form ◽  
Flow Polytopes ◽  
Flow Polytope ◽  
Reduced Forms

The flow polytope $F_{\widetilde{G}}$ is the set of nonnegative unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope $F_{\widetilde{G}}$ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial $\prod_{(i,j)\in E(G)}x_{ij}$. We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but also regular flag triangulations of flow polytopes. We prove that reduced forms in the subdivision algebra are generalizations of $h$-polynomials of the triangulations of flow polytopes. We deduce several corollaries of the above results, most notably proving certain cases of a conjecture of Kirillov about the nonnegativity of reduced forms in the noncommutative quasi-classical Yang-Baxter algebra.

scholarly journals On Volume Functions of Special Flow Polytopes Associated to the Root System of Type $A$

10.37236/9062 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Takayuki Negishi ◽  
Yuki Sugiyama ◽  
Tatsuru Takakura
Keyword(s):  
Differential Equations ◽  
Root System ◽  
Special Kind ◽  
Type A ◽  
System Of Differential Equations ◽  
Constant Multiple ◽  
Special Flow ◽  
Flow Polytopes ◽  
Flow Polytope ◽  
Volume Functions

In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type $A$.

Studies on Related ms Magnitude Rate Models in Triangular Wave Unloading Section of Calcium Medium-Fine Sandstone

2010 ◽  
Vol 152-153 ◽  
pp. 164-170
Author(s):  
Jie Liu ◽  
Jian Lin Li ◽  
Ying Xia Li ◽  
Shan Shan Yang ◽  
Ji Fang Zhou ◽  
...  
Keyword(s):  
Mechanical Response ◽  
Strain Curve ◽  
Experimental System ◽  
Constant Term ◽  
Lag Time ◽  
Vertical Force ◽  
Time Segment ◽  
Triangular Wave ◽  
Apparent Modulus ◽  
The Impact

Specific to the improvement in the present research of mechanical response under cyclic loading, this paper, taking the calcareous middle- coarse sandstone as the research subject and the RMT-150C experimental system in which data is recoded by ms magnitude as the platform, develops several related models concerning the unloading rate of triangle waves. The unloading process is divided into lag time segment and non-lag time segment, with criterions and related parameters provided as well. The term apparent elastic modulus is defined. The test data analysis shows that there exist a linear relationship between the apparent modulus and instant vertical force before load damage in non-lag time segment. On the preceding basis, a rate-dependent model of triangular wave un-installation section in non-lag time segment is established. Due to the inability of the loading equipment to accurately input the triangle wave, the average loading rate is amended and a constant term is added into it. The model is proved to be reliable, as the predicted value of the deformation rate and the stress strain curve coincides with measured value. At the same time, the impact of the lag time is pointed out quantitatively and a predication model of lag time segment is set up.

scholarly journals Random Assignment Problems on 2d Manifolds

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
D. Benedetto ◽  
E. Caglioti ◽  
S. Caracciolo ◽  
M. D’Achille ◽  
G. Sicuro ◽  
...  
Keyword(s):  
Assignment Problem ◽  
Unit Area ◽  
Constant Term ◽  
Beltrami Operator ◽  
Explicit Calculation ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Random Assignment ◽  
Assignment Problems ◽  
Theoretical Formulation

AbstractWe consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ Ω of unit area. It is known that the average cost scales as $$E_{\Omega }(N)\sim {1}/{2\pi }\ln N$$ E Ω ( N ) ∼ 1 / 2 π ln N with a correction that is at most of order $$\sqrt{\ln N\ln \ln N}$$ ln N ln ln N . In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $$\Omega $$ Ω -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on $$\Omega $$ Ω . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

Intracranial volume-pressure relationship in man

1982 ◽  
Vol 56 (4) ◽  
pp. 524-528 ◽  
Author(s):  
Joseph Th. J. Tans ◽  
Dick C. J. Poortvliet
Keyword(s):  
Best Approximation ◽  
Continuous Monitoring ◽  
Fluid Pressure ◽  
Constant Term ◽  
Volume Index ◽  
Intracranial Volume ◽  
Content Type ◽  
The Best Approximation ◽  
Ventricular Fluid Pressure ◽  
Fine Print

✓ The pressure-volume index (PVI) was determined in 40 patients who underwent continuous monitoring of ventricular fluid pressure. The PVI value was calculated using different mathematical models. From the differences between these values, it is concluded that a monoexponential relationship with a constant term provides the best approximation of the PVI.

Forces on a Circular Hole Located in Functionally Graded Material

2011 ◽  
Vol 704-705 ◽  
pp. 631-635
Author(s):  
Xian Feng Wang ◽  
Feng Xing ◽  
Norio Hasebe
Keyword(s):  
Circular Hole ◽  
Constant Term ◽  
Complex Stress ◽  
Functionally Graded ◽  
Homogeneous Material ◽  
Dimensional Problem ◽  
Stress Components ◽  
Graded Material ◽  
Stress Functions ◽  
Stress Function Method

The complex stress function method is used in this study to formulate the 2-dimensional problem for nonhomogeneous materials. The Young’s modulus E varies linearly with the coordinate x and the Poisson’s ratio of the material is assumed constant and. The stress components and the boundary conditions are expressed in terms of two complex stress functions in explicit forms. It is noted that the constant term in stress functions has an influence on the stress components, which is different from the homogeneous material case. Subsequently, the problem of a nonhomogeneous plane containing a circular hole subjected to a uniform internal pressure is studied.

Sign in / Sign up

Export Citation Format

Share Document