NKG2D Engagement Alone Is Sufficient to Activate Cytokine-Induced Killer Cells While 2B4 Only Provides Limited Coactivation

Cytokine-induced killer (CIK) cells are an ex vivo expanded heterogeneous cell population with an enriched NK-T phenotype (CD3+CD56+). Due to the convenient and relatively inexpensive expansion capability, together with low incidence of graft versus host disease (GVHD) in allogeneic cancer patients, CIK cells are a promising candidate for immunotherapy. It is well known that natural killer group 2D (NKG2D) plays an important role in CIK cell-mediated antitumor activity; however, it remains unclear whether its engagement alone is sufficient or if it requires additional co-stimulatory signals to activate the CIK cells. Likewise, the role of 2B4 has not yet been identified in CIK cells. Herein, we investigated the individual and cumulative contribution of NKG2D and 2B4 in the activation of CIK cells. Our analysis suggests that (a) NKG2D (not 2B4) is implicated in CIK cell (especially CD3+CD56+ subset)-mediated cytotoxicity, IFN-γ secretion, E/T conjugate formation, and degranulation; (b) NKG2D alone is adequate enough to induce degranulation, IFN-γ secretion, and LFA-1 activation in CIK cells, while 2B4 only provides limited synergy with NKG2D (e.g., in LFA-1 activation); and (c) NKG2D was unable to costimulate CD3. Collectively, we conclude that NKG2D engagement alone suffices to activate CIK cells, thereby strengthening the idea that targeting the NKG2D axis is a promising approach to improve CIK cell therapy for cancer patients. Furthermore, CIK cells exhibit similarities to classical invariant natural killer (iNKT) cells with deficiencies in 2B4 stimulation and in the costimulation of CD3 with NKG2D. In addition, based on the current data, the divergence in receptor function between CIK cells and NK (or T) cells can be assumed, pointing to the possibility that molecular modifications (e.g., using chimeric antigen receptor technology) on CIK cells may need to be customized and optimized to maximize their functional potential.

Symmetry of Dressed Photon

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n≥r≥s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with n=4,r=2,s=1.

A NOTE ON RATIONAL HOMOLOGICAL STABILITY OF AUTOMORPHISMS OF MANIFOLDS

By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp (S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $*\le (g-6)/2$. In this note, we explain how this range can be improved to $*\le g-2$ using cohomological vanishing results due to Borel and the classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.

A diagrammatic approach to the AJ Conjecture

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $$\hat{A}$$ A ^ polynomial), with a classical invariant, namely the defining polynomial A of the $${\mathrm {PSL}_2(\mathbb {C})}$$ PSL 2 ( C ) character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the $$\hat{A}$$ A ^ -polynomial (after we set $$q=1$$ q = 1 , and excluding those of L-degree zero) coincides with those of the A-polynomial. In this paper, we introduce a version of the $$\hat{A}$$ A ^ -polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the $$\hat{A}$$ A ^ -polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the R-matrix state sum formula for the colored Jones polynomial, and its certificate.

Hilbert polynomials of the algebras of $SL_ 2$-invariants

We consider one of the fundamental problems of classical invariant theory, the research of Hilbert polynomials for an algebra of invariants of Lie group $SL_2$. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in algebraic geometry. It is well known that the Hilbert function of the algebra $SL_n$-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary $d$-form $\mathcal{C}_{d}= \mathbb{C}[V_d\oplus \mathbb{C}^2]_{SL_2}$ (here $V_d$ is the $d+1$-dimensional space of binary forms of degree $d$) was obtained by Sylvester. Then it was generalized to the algebra of joint invariants for $n$ binary forms. But the Cayley-Sylvester formula is not expressed in terms of polynomials.In our article we consider the problem of computing the Hilbert polynomials for the algebras of joint invariants and joint covariants of $n$ linear forms and $n$ quadratic forms. We express the Hilbert polynomials $\mathcal{H} \mathcal{I}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i, \mathcal{H}(\mathcal{C}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i,$ $\mathcal{H}(\mathcal{I}^{(n)}_2,i)=\dim(\mathcal{I}^{(n)}_2)_i, \mathcal{H}(\mathcal{C}^{(n)}_2,i)=\dim(\mathcal{C}^{(n)}_2)_i$ of those algebras in terms of quasi-polynomial. We also present them in the form of Narayana numbers and generalized hypergeometric series.

Poincare series for the algebras of joint invariants and covariants of $n$ quadratic forms

We consider one of the fundamental problems of classical invariant theory - the research of Poincare series for an algebra of invariants of Lie group $SL_2$. The first two terms of the Laurent series expansion of Poincare series at the point $z = 1$ give us important information about the structure of the algebra $\mathcal{I}_{d}.$ It was derived by Hilbert for the algebra ${\mathcal{I}_{d}=\mathbb{C}[V_d]^{\,SL_2}}$ of invariants for binary $d-$form (by $V_d$ denote the vector space over $\mathbb{C}$ consisting of all binary forms homogeneous of degree $d$). Springer got this result, using explicit formula for the Poincare series of this algebra. We consider this problem for the algebra of joint invariants $\mathcal{I}_{2n}=\mathbb{C}[\underbrace{V_2 \oplus V_2 \oplus \cdots \oplus V_2}_{\text{n times}}]^{SL_2}$ and the algebra of joint covariants $\mathcal{C}_{2n}=\mathbb{C}[\underbrace{V_2 {\oplus} V_2 {\oplus} \cdots {\oplus} V_2}_{\text{n times}}{\oplus}\mathbb{C}^2 ]^{SL_2}$ of $n$ quadratic forms. We express the Poincare series $\mathcal{P}(\mathcal{C}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{C}_{2n})_{j}\, z^j$ and $\mathcal{P}(\mathcal{I}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{I}_{2n})_{j}\, z^j$ of these algebras in terms of Narayana polynomials. Also, for these algebras we calculate the degrees and asymptotic behavious of the degrees, using their Poincare series.

On the Erdős–Ginzburg–Ziv constant of groups of the form C2r ⊕ Cn

Let [Formula: see text] be a finite abelian group. The Erdős–Ginzburg–Ziv constant [Formula: see text] of [Formula: see text] is defined as the smallest integer [Formula: see text] such that every sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] has a zero-sum subsequence [Formula: see text] of length [Formula: see text]. The value of this classical invariant for groups with rank at most two is known. But the precise value of [Formula: see text] for the groups of rank larger than two is difficult to determine. In this paper, we pay attention to the groups of the form [Formula: see text], where [Formula: see text] and [Formula: see text]. We give a new upper bound of [Formula: see text] for odd integer [Formula: see text]. For [Formula: see text], we obtain that [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text].