ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

Bounds on the size of PC and URC formulas

In this paper, we investigate CNF encodings, for which unit propagation is strong enough to derive a contradiction if the encoding is not consistent with a partial assignment of the variables (unit refutation complete or URC encoding) or additionally to derive all implied literals if the encoding is consistent with the partial assignment (propagation complete or PC encoding). We prove an exponential separation between the sizes of PC and URC encodings without auxiliary variables and strengthen the known results on their relationship to the PC and URC encodings that can use auxiliary variables. Besides of this, we prove that the sizes of any two irredundant PC formulas representing the same function differ at most by a factor polynomial in the number of the variables and present an example of a function demonstrating that a similar statement is not true for URC formulas. One of the separations above implies that a q-Horn formula may require an exponential number of additional clauses to become a URC formula. On the other hand, for every q-Horn formula, we present a polynomial size URC encoding of the same function using auxiliary variables. This encoding is not q-Horn in general.

al-Uṣūl al-Muhaḏḏabiyya

Ms. St. Petersburg, Russian National Library, Evr Arab I 3951 has 14 leaves, which consist of three fragments: 1) Fols. 1–10, include part of al-Uṣūl al-Muhaḏḏabiyya, the subject of the present paper. 2–3) Fragments of a responsum on forbidden marriages and a theological work. al-Uṣūl al-Muhaḏḏabiyya was written as a concise compendium of Muʿtazili theology, written by a Karaite scholar Sahl b. al-Faḍl al-Tustarī, who was active in Jerusalem (and perhaps later in Egypt) at the end of the 10th century, at the request of al-Qaḍī al-Muhaḏḏab Saniyy al-Dawla, (apparently) a dignitary in the service of the Fāṭimid government. No person with this, or a similar name could be identified in historical or biographic sources as fitting the role of instigator of such an inter-confessional project. On the basis of a comparison between a quotation of a statement on the definition of prophecy by al-Sahl b. al-Faḍl al-Tustarī at an inter-confessional debate, which took place on the Temple Mount in Jerusalem ca. 1095 (quoted in Ibn al-ʿArabī’s Qānūn al-taʾwīl) and a similar statement on prophecy found in the fragment of al-Uṣūl al-Muhaḏḏabiyya, it is quite safe to conclude that the same person is the author of the compendium, and also of the important work Kitāb al-Īmāʾ.

Generalized Cesàro Operators: Geometry of Spectra and Quasi-Nilpotency

For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk, we prove that the spectrum of a generalized Cesàro operator $T_g$ is unchanged if the symbol $g$ is perturbed to $g+h$ by an analytic function $h$ inducing a quasi-nilpotent operator $T_h$, that is, spectrum of $T_h$ equals $\{0\}$. We also show that any $T_g$ operator that can be approximated in the operator norm by an operator $T_h$ with bounded symbol $h$ is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function $g \in \textbf{BMOA}$ to be in the $\textbf{BMOA}$ norm closure of $H^{\infty }$. This condition turns out to be equivalent to quasi-nilpotency of the operator $T_g$ on the Hardy spaces. This raises the question whether similar statement is true in the context of Bergman spaces and the Bloch space. Furthermore, we provide some general geometric properties of the spectrum of $T_{g}$ operators.

A Concordance Analogue of the 4D Light Bulb Theorem

We prove a concordance analogue of Gabai’s $4$D light bulb theorem. That is, we show that when $R$ and $R^{\prime}$ are homotopic, embedded $2$-spheres in a $4$-manifold $X^4,$ where $\pi _1(X^4)$ has no $2$-torsion and one of $R$ or $R^{\prime}$ has a transverse sphere, then $R$ and $R^{\prime}$ are concordant. When $\pi _1(X^4)$ has $2$-torsion, we give a similar statement with extra hypotheses as in the $4$D light bulb theorem. We also give similar statements when $R$ and $R^{\prime}$ are orientable positive-genus surfaces.

Invariant Whitney functions

Theorem 1 of [G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975) 63–68.] says that for a linear action of a compact Lie group [Formula: see text] on a finite dimensional real vector space [Formula: see text], any smooth [Formula: see text]-invariant function on [Formula: see text] can be written as a composite with the Hilbert map. We prove a similar statement for the case of Whitney functions along a subanalytic set [Formula: see text] fulfilling some regularity assumptions. In order to deal with the case when [Formula: see text] is not [Formula: see text]-stable, we use the language of groupoids.

TOWARDS A NON-ARCHIMEDEAN ANALYTIC ANALOG OF THE BASS–QUILLEN CONJECTURE

We suggest an analog of the Bass–Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles $K_{0}$.

An overview of the opioid crisis

The objective of this paper is to highlight the current state of the opioid crisis in Canada, framing it in the context of the global drug problem. The effects of the Opioid Crisis seen in London are part of a larger crisis occurring in North America. The USA has declared the opioid crisis a national emergency and we can expect a similar statement here in Canada as opioid related hospitalizations and deaths continue to rise. Although media attention continues to increase, we continue to see worsening statistics highlighting a flaw in how this complex issue is currently being addressed. Much like how attempts to prevent worsening of the opioid crisis have failed, globally, attempts to reduce the use and misuse of all illicit drugs have failed. Two key documents written by The Global Commission on Drug Policy will be reviewed to provide further insight while highlighting the need to challenge how we currently approach not just the opioid crisis but also the “war on drugs”.

W∗-superrigidity for coinduced actions

We prove W[Formula: see text]-superrigidity for a large class of coinduced actions. We prove that if [Formula: see text] is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group [Formula: see text], the coinduced action [Formula: see text] from an arbitrary probability measure preserving action [Formula: see text] is W[Formula: see text]-superrigid. We also prove a similar statement if [Formula: see text] is an icc non-amenable group which is measure equivalent to a product of two infinite groups. In particular, we obtain that any Bernoulli action of such a group [Formula: see text] is W[Formula: see text]-superrigid.

SUB-RIEMANNIAN RICCI CURVATURES AND UNIVERSAL DIAMETER BOUNDS FOR 3-SASAKIAN MANIFOLDS

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev–Zelenko–Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet–Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\unicode[STIX]{x1D70B}$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf fibrations: $$\begin{eqnarray}\mathbb{S}^{3}{\hookrightarrow}\mathbb{S}^{4d+3}\rightarrow \mathbb{HP}^{d},\end{eqnarray}$$ whose exact sub-Riemannian diameter is $\unicode[STIX]{x1D70B}$, for all $d\geqslant 1$.