A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of ?^{[3]}???^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials ? satisfy that for every two polynomials Q?,Q? ? ? there is a subset ? ? ?, such that Q?,Q? ? ? and whenever Q? and Q? vanish then ?_{Q_i??} Q_i vanishes, then the linear span of the polynomials in ? has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when |?| = 1. An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics

Lower Bounds on Stabilizer Rank

The stabilizer rank of a quantum state ? is the minimal r such that |?? = ?_{j = 1}^r c_j |?_j? for c_j ? ? and stabilizer states ?_j. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n-th tensor power of single-qubit magic states. We prove a lower bound of ?(n) on the stabilizer rank of such states, improving a previous lower bound of ?(?n) of Bravyi, Smith and Smolin [Bravyi et al., 2016]. Further, we prove that for a sufficiently small constant ?, the stabilizer rank of any state which is ?-close to those states is ?(?n/log n). This is the first non-trivial lower bound for approximate stabilizer rank. Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of ???, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function

Genetic Relationship in Cicer Sp. Expose Evidence for Geneflow between the Cultigen and Its Wild Progenitor

There is a debate concerning mono- or poly-phyletic origins of the Near Eastern crops. In parallel, some authors claim that domestication was not possible within the natural range of the wild progenitors due to wild alleles flow into the nascent crops. Here we address both, the mono- or poly-phyletic origins and the domestications within or without the natural range of the progenitor, debates in order to understand the relationship between domesticated chickpea (Cicer arietinum L.) and its wild progenitor (C. reticulatum Ladizinsky) with special emphasis on its domestication centre in southeastern Turkey. A set of 103 chickpea cultivars and landraces from the major growing regions alongside wild accessions (C. reticulatum, C. echinospermum P.H Davis and C. bijugum K.H. Rech) sampled across the natural distribution range in eastern Turkey were genotyped with 194 SNPs markers. The genetic affinities between and within the studied taxa were assessed. The analysis suggests a mono-phyletic origin of the cultigen, with several wild accession as likely members of the wild stock of the cultigen. Clear separation between the wild and domesticated germplasm was apparent, with negligible level of admixture. A single C. reticulatum accession shows morphological and allelic signatures of admixture, a likely result of introgression. No evidence of geneflow from the wild into domesticated germplasm was found. The traditional farming systems of southeaster Turkey are characterized by occurrence of sympatric wild progenitor-domesticated forms of chickpea (and likewise cereals and other grain legumes). Therefore, both the authentic crop landraces and the wild populations native to the area are a unique genetic resource. Our results grant support to the notion of domestication within the natural distribution range of the wild progenitor, suggesting that the Neolithic domesticators were fully capable of selecting the desired phenotypes even when facing rare wild-domesticated introgression events

A Complete Axiom System for Propositional Interval Temporal Logic with Infinite Time

Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike completeness proofs of equally expressive logics with nonelementary computational complexity, our semantic approach does not use tableaux, subformula closures or explicit deductions involving encodings of omega automata and nontrivial techniques for complementing them. We believe that our result also provides evidence of the naturalness of interval-based reasoning

Biallelic ADAM22 pathogenic variants cause progressive encephalopathy and infantile-onset refractory epilepsy

Pathogenic variants in A Disintegrin And Metalloproteinase (ADAM) 22, the postsynaptic cell membrane receptor for the glycoprotein leucine-rich repeat glioma-inactivated protein 1 (LGI1), have been recently associated with recessive developmental and epileptic encephalopathy. However, so far, only two affected individuals have been described and many features of this disorder are unknown. We refine the phenotype and report 19 additional individuals harboring compound heterozygous or homozygous inactivating ADAM22 variants, of whom 18 had clinical data available. Additionally, we provide follow-up data from two previously reported cases. All affected individuals exhibited infantile-onset, treatment-resistant epilepsy. Additional clinical features included moderate to profound global developmental delay/intellectual disability (20/20), hypotonia (12/20), delayed motor development (19/20). Brain MRI findings included cerebral atrophy (13/20), supported by post-mortem histological examination in patient-derived brain tissue, cerebellar vermis atrophy (5/20), and callosal hypoplasia (4/20). Functional studies in transfected cell lines confirmed the deleteriousness of all identified variants and indicated at least three distinct pathological mechanisms: defective cell membrane expression (1), impaired LGI1-binding (2), and/or impaired interaction with the postsynaptic density protein PSD-95 (3). We reveal novel clinical and molecular hallmarks of ADAM22 deficiency and provide knowledge that might inform clinical management and early diagnostics

Biallelic ADAM22 pathogenic variants cause progressive encephalopathy and infantile-onset refractory epilepsy

Pathogenic variants in A Disintegrin And Metalloproteinase (ADAM) 22, the postsynaptic cell membrane receptor for the glycoprotein leucine-rich repeat glioma-inactivated protein 1 (LGI1), have been recently associated with recessive developmental and epileptic encephalopathy. However, so far, only two affected individuals have been described and many features of this disorder are unknown. We refine the phenotype and report 19 additional individuals harbouring compound heterozygous or homozygous inactivating ADAM22 variants, of whom 18 had clinical data available. Additionally, we provide follow-up data from two previously reported cases. All affected individuals exhibited infantile-onset, treatment-resistant epilepsy. Additional clinical features included moderate to profound global developmental delay/intellectual disability (20/20), hypotonia (12/20) and delayed motor development (19/20). Brain MRI findings included cerebral atrophy (13/20), supported by post-mortem histological examination in patient-derived brain tissue, cerebellar vermis atrophy (5/20), and callosal hypoplasia (4/20). Functional studies in transfected cell lines confirmed the deleteriousness of all identified variants and indicated at least three distinct pathological mechanisms: (i) defective cell membrane expression; (ii) impaired LGI1-binding; and/or (iii) impaired interaction with the postsynaptic density protein PSD-95. We reveal novel clinical and molecular hallmarks of ADAM22 deficiency and provide knowledge that might inform clinical management and early diagnostics. Van der Knoop et al. describe the clinical features of 21 individuals with biallelic pathogenic variants in ADAM22 and confirm the deleteriousness of the variants with functional studies. Clinical hallmarks of this rare disorder comprise progressive encephalopathy and infantile-onset refractory epilepsy.Peer reviewe

Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials

In this work, we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if $\mathcal{Q}\subset \mathbb{C}[x_1.\ldots,x_n]$ is a finite set, $|\mathcal{Q}|=m$, of irreducible quadratic polynomials that satisfy the following condition: There is $\delta>0$ such that for every $Q\in\mathcal{Q}$ there are at least $\delta m$ polynomials $P\in \mathcal{Q}$ such that whenever $Q$ and $P$ vanish then so does a third polynomial in $\mathcal{Q}\setminus\{Q,P\}$, then $\dim(\text{span}({\mathcal{Q}}))=\text{poly}(1/\delta)$. The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of $O(1/\delta)$ on the dimension (in the first work an upper bound of $O(1/\delta^2)$ was given, which was improved to $O(1/\delta)$ in the second work).Comment: arXiv admin note: text overlap with arXiv:2006.0826