Homomesy in Products of Three Chains and Multidimensional Recombination

J. Propp and T. Roby isolated a phenomenon in which a statistic on a set has the same average value over any orbit as its global average, naming it homomesy. They proved that the cardinality statistic on order ideals of the product of two chains poset under rowmotion exhibits homomesy. In this paper, we prove an analogous result in the case of the product of three chains where one chain has two elements. In order to prove this result, we generalize from two to n dimensions the recombination technique that D. Einstein and Propp developed to study homomesy. We see that our main homomesy result does not fully generalize to an arbitrary product of three chains, nor to larger products of chains; however, we have a partial generalization to an arbitrary product of three chains. Additional corollaries include refined homomesy results in the product of three chains and a new result on increasing tableaux. We conclude with a generalization of recombination to any ranked poset and a homomesy result for the Type B minuscule poset cross a two element chain.

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Effect of the Short-Segment Internal Fixation with Intermediate Inclined-Angle polyaxial Screw at the Fractured Vertebra on the Treatment of Denis type B Thoracolumbar Fracture

10.21203/rs.2.22037/v3 ◽

2020 ◽

Author(s):

chengjie xiong ◽

Biwang Huang ◽

Tanjun Wei ◽

Hui Kang ◽

Feng Xu

Keyword(s):

Internal Fixation ◽

Vertebral Body ◽

Body Height ◽

Type B ◽

Short Segment ◽

Average Value ◽

Significant Difference ◽

Surgical Options ◽

Inclined Angle ◽

Duration Of Operation

Abstract Background: Short-segment internal fixation with intermediate straight-forward monoaxial screws (SSIF-SFM) and long-segment internal fixation (LSIF) are the tow major surgical options for thoracolumbar (TL) fracture. However, SS IF-SFM might not provide adequate support to the spine and LSIF is un necessarily extensive. SSIF with intermediate inclined-angle polyxial screw (SSIF-IAP) might offer an alternative solution for the treatment of TL fracture. Methods: A retrospective study was conducted. 69 patients (47 males and 22 females; average 34.5 years) with Denis type B TL fracture who met the criteria for inclusion were enrolled. Sagittal Cobb’s angle (SCA), anterior vertebral body height (AVBH), vertebral body index (VBI) and spinal canal encroachment (SCE) were measured and assessed. Visual analogue scale (VAS) and oswestry disability index (ODI) were also evaluated. Results: The average value of incision length, blood loss, duration of operation and hospital stay in the SSIF–IAP group and SSIF-SFM group were significantly decreased compared with those in the LSIF group. The AVBH and VBI in the SSIF-IAP group and LSIF group were significantly improved than those in the SSIF–SFM group at 6-month and the latest follow-ups (p < 0.05). The correction losses of AVBH and VBI (calculated by the reduction of AVBH and VBI) in the SSIF-IAP group and LSIF group were also significantly decreased compared with those in the SSIF–SFM group at 6-month and the latest follow-ups (P < 0.05). There was no significant difference of SCE among the three groups postoperatively. The VAS and ODI in the SSIF-IAP group and SSIF-SFM group were significantly decreased compared with those in the LSIF group at 6-month and the latest follow-ups (P < 0.05). Conclusion: Both SSIF- IAP and LSIF can improve the biomechanical stability as compared with SSIF -SFM . Moreover, SSIF-IAP was an effective and reliable operative technique for patients with Denis type B TL fracture.

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Reply by the author to C. J. Swain

Geophysics ◽

10.1190/1.1486942 ◽

1984 ◽

Vol 49 (9) ◽

pp. 1563-1564

Author(s):

John H. Karl

Keyword(s):

Vertical Gradient ◽

Global Average ◽

Average Value ◽

Mountainous Regions ◽

Elevation Correction

As Dr. Swain observes, my 4πγ[Formula: see text] effect results from a global average, and as such says nothing about the vertical gradient at any given location. That’s my point. If my average value is correct (no one has disagreed with that) and measured values agree with the normal gradient, then there must be anomalies that are even larger than 4πγ[Formula: see text] somewhere. It is likely that these anomalous values occur in mountainous regions—just the place where the elevation correction is most important.

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Iterative Properties of Birational Rowmotion II: Rectangles and Triangles

The Electronic Journal of Combinatorics ◽

10.37236/4335 ◽

2015 ◽

Vol 22 (3) ◽

Cited By ~ 12

Author(s):

Darij Grinberg ◽

Tom Roby

Keyword(s):

Finite Order ◽

P Element ◽

Birational Map ◽

Finite Poset ◽

Finite Set ◽

Element Chain ◽

Order Ideals ◽

Finite Posets

Birational rowmotion — a birational map associated to any finite poset $P$ — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of $P$. Continuing our exploration of this birational rowmotion, we prove that it has order $p+q$ on the $\left( p, q\right) $-rectangle poset (i.e., on the product of a $p$-element chain with a $q$-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the $AA$ case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets.

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Birational Rowmotion and Coxeter-Motion on Minuscule Posets

The Electronic Journal of Combinatorics ◽

10.37236/9557 ◽

2021 ◽

Vol 28 (1) ◽

Author(s):

Soichi Okada

Keyword(s):

Dynamical System ◽

Discrete Dynamical System ◽

Order Ideal ◽

Positive Real ◽

Finite Poset ◽

Coxeter Number ◽

Order Ideals ◽

Minuscule Poset

Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions on a finite poset, which is a birational lift of combinatorial rowmotion on order ideals. It is known that combinatorial rowmotion for a minuscule poset has order equal to the Coxeter number, and exhibits the file homomesy phenomenon for refined order ideal cardinality statistics. In this paper we generalize these results to the birational setting. Moreover, as a generalization of birational promotion on a product of two chains, we introduce birational Coxeter-motion on minuscule posets, and prove that it enjoys periodicity and file homomesy.

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Comparison of available measurements of the absolute air-fluorescence yield and determination of its global average value

10.1063/1.3628711 ◽

2011 ◽

Cited By ~ 3

Author(s):

J. Rosado ◽

F. Blanco ◽

F. Arqueros ◽

Hiroyuki Sagawa ◽

Yoshiya Kawasaki ◽

...

Keyword(s):

Fluorescence Yield ◽

Global Average ◽

Average Value ◽

Air Fluorescence ◽

The Absolute

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Biological and Morphological Studies on Low-Leukemia-Strain Mice with Hodgkin's-Like Neoplasia Induced by Serial Cell-Free Transmission of Leukemic Organs of SJL/J Strain Mice

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100100792 ◽

1968 ◽

Vol 26 ◽

pp. 210-211

Author(s):

S. Fujinaga ◽

K. Maruyama ◽

C.W. Williams ◽

K. Sekhri ◽

L. Dmochowski

Keyword(s):

Tissue Culture ◽

Type A ◽

Reticulum Cell ◽

Type B ◽

Viral Origin ◽

Serial Transfer ◽

Strain Mouse ◽

Morphological Studies ◽

Cell Neoplasm ◽

Free Material

Yumoto and Dmochowski (Cancer Res.27, 2098 (1967)) reported the presence of mature and immature type C leukemia virus particles in leukemic organs and tissues such as lymph nodes, spleen, thymus, liver, and kidneys of SJL/J strain mice with Hodgki's-like disease or reticulum cell neoplasm (type B). In an attempt to ascertain the possibility that this neoplasia may be of viral origin, experiments with induction and transmission of this neoplasm were carried out using cell-free extracts of leukemic organs from an SJL/J strain mouse with spontaneous disease.It has been possible to induce the disease in low-leukemia BALB/c and C3HZB strain mice and serially transfer the neoplasia by cell-free extracts of leukemic organs of these mice. Histological examination revealed the neoplasia to be of either reticulum cell-type A or type B. Serial transfer is now in its fifth passage. In addition leukemic spleen from another SJL/J strain mouse with spontaneous reticulum cell neoplasm (type A) was set up in tissue culture and is now in its 141st serial passage in vitro. Preliminary results indicate that cell-free material of 39th tissue culture passage can reproduce neoplasia in BALB/c mice.

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Cryo-electron microscopy of two-dimensional crystals of reduced type B botulinum neurotoxin

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100123052 ◽

1992 ◽

Vol 50 (1) ◽

pp. 530-531

Author(s):

P. F. Flicker ◽

V.S. Kulkarni ◽

J. P. Robinson ◽

G. Stubbs ◽

B. R. DasGupta

Keyword(s):

Disulfide Bonds ◽

Clostridium Botulinum ◽

Structural Changes ◽

Two Dimensional ◽

Type B ◽

Single Chain ◽

Muscle Paralysis ◽

Cryo Electron Microscopy ◽

Disulfide Linkages ◽

Intracellular Action

Botulinum toxin is a potent neurotoxin produced by Clostridium botulinum. The toxin inhibits release of neurotransmitter, causing muscle paralysis. There are several serotypes, A to G, all of molecular weight about 150,000. The protein exists as a single chain or or as two chains, with two disulfide linkages. In a recent investigation on intracellular action of neurotoxins it was reported that type B neurotoxin can inhibit the release of Ca++-activated [3H] norepinephrine only if the disulfide bonds are reduced. In order to investigate possible structural changes in the toxin upon reduction of the disulfide bonds, we have prepared two-dimensional crystals of reduced type B neurotoxin. These two-dimensional crystals will be compared with those of the native (unreduced) type B toxin.

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