Root multiplicities and number of nonzero coefficients of a polynomial

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate statement in positive characteristic. Furthermore, we present a new proof of the original result, which produces also the exact number of monic polynomials of a given degree for which the bound is attained. A similar argument allows us to determine the number of monic polynomials of a given degree, multiplicity of a given nonzero root, and number of nonzero coefficients, over a finite field of characteristic larger than the degree.Comment: 6 pages. Minor change from previous version: added Example 6, illustrating the difficulties arising when one tries to relax the hypothesis n<p of Theorem

A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra

A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method.Comment: 10 pages. This version has been revised according to a referee's suggestions. The additions include a discussion of the (lower) density of the set N_p, and the results of more extensive machine computations. Note that the title has also changed. To appear in Israel J. Mat

Laguerre polynomials of derivations

We introduce a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. We take inspiration from a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. Our grading switching is achieved by evaluating certain generalized Laguerre polynomials of degree p − 1, which play the role of generalized exponentials, on a derivation of the algebra. A crucial part of our argument is establishing a congruence for them which is an appropriate analogue of the functional equation exp(x) · exp(y) = exp(x+y) for the classical exponential. Besides having a wider scope, our treatment provides a more transparent explanation of some aspects of the original toral switching, which can be recovered as a special case

A generalized truncated logarithm

We introduce a generalization $G^{(\alpha)}(X)$ of the truncated logarithm $\pounds_1(X)=\sum_{k=1}^{p-1}X^k/k$ in prime characteristic $p$, which depends on a parameter $\alpha$. The main motivation of this study is $G^{(\alpha)}(X)$ being an inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential given by certain Laguerre polynomials. Such Laguerre polynomials play a role in a {\em grading switching} technique for non-associative algebras, previously developed by the authors, because they satisfy a weak analogue of the functional equation $\exp(X)\exp(Y)=\exp(X+Y)$ of the exponential series. We also investigate functional equations satisfied by $G^{(\alpha)}(X)$ motivated by known functional equations for $\pounds_1(X)=-G^{(0)}(X)$

Small-molecule modulators of mitochondrial channels as chemotherapeutic agents

Ion channels residing in the inner (IMM) and outer (OMM) mitochondrial membranes are emerging as noteworthy pharmacological targets in oncology. While these aspects have not been investigated for all of them, a role in cancer growth and/or metastasis and/or drug resistance has been shown at least for the IMM-residing Ca2+ uniporter complex and K+- selective mtKV1.3, mtIKCa, mtSKCa and mtTASK-3, and for the OMM Voltage-Dependent Anion Channel (mitochondrial porin). A special case is that of the Mitochondrial Permeability Transition Pore, a large pore which forms in the IMM of severely stressed cells, and which may be exploited to precipitate the death of cancerous cells. Here we briefly discuss the oncological relevance of mitochondria and their channels, and summarize the methods that can be adopted to selectively target these intracellular organelles. We then present an updated list of known mitochondrial channels, and review the pharmacology of those with proven relevance for cancer

A variable delay integrated receiver for differential phase-shift keying optical transmission systems

An integrated variable delay receiver for DPSK optical transmission systems is presented. The device is realized in silicon-on-insulator technology and can be used to detect DPSK signals at any bit-rates between 10 and 15 Gbit/s

Gradings of non-graded Hamiltonian Lie algebras

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2\colon\n;\omega_2) (of dimension one less than a power of $p$) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2\colon\n;\omega_2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).Comment: 36 pages, to be published in J. Austral. Math. Soc. Ser.

A CXCR4 receptor agonist strongly stimulates axonal regeneration after damage

Objective: To test whether the signaling axis CXCL12\u3b1-CXCR4 is activated upon crush/cut of the sciatic nerve and to test the activity of NUCC-390, a new CXCR4 agonist, in promoting nerve recovery from damage. Methods: The sciatic nerve was either crushed or cut. Expression and localization of CXCL12\u3b1 and CXCR4 were evaluated by imaging with specific antibodies. Their functional involvement in nerve regeneration was determined by antibody-neutralization of CXCL12\u3b1, and by the CXCR4 specific antagonist AMD3100, using as quantitative read-out the compound muscle action potential (CMAP). NUCC-390 activity on nerve regeneration was determined by imaging and CMAP recordings. Results: CXCR4 is expressed at the injury site within the axonal compartment, whilst its ligand CXCL12\u3b1 is expressed in Schwann cells. The CXCL12\u3b1-CXCR4 axis is involved in the recovery of neurotransmission of the injured nerve. More importantly, the small molecule NUCC-390 is a strong promoter of the functional and anatomical recovery of the nerve, by acting very similarly to CXCL12\u3b1. This pharmacological action is due to the capability of NUCC-390 to foster elongation of motor neuron axons both in vitro and in vivo. Interpretation: Imaging and electrophysiological data provide novel and compelling evidence that the CXCL12\u3b1-CXCR4 axis is involved in sciatic nerve repair after crush/cut. This makes NUCC-390 a strong candidate molecule to stimulate nerve repair by promoting axonal elongation. We propose this molecule to be tested in other models of neuronal damage, to lay the basis for clinical trials on the efficacy of NUCC-390 in peripheral nerve repair in humans

A Phase 2a Double-Blind Randomized Trial of REL-1017 (Esmethadone) in Patients with MDD: Analysis of Subscales from the Symptoms of Depression Questionnaire

BACKGROUND: Major depressive disorder (MDD) is the second leading cause of disability and chronic disease burden in the United States. The importance of improving functional outcomes in MDD is increasingly recognized. The Symptoms of Depression Questionnaire (SDQ), a patient-reported measure, was developed to capture the heterogeneity of symptoms of MDD. REL-1017 (esmethadone HCl; d-methadone), is a novel N-methyl-d-aspartate receptor (NMDAR) channel blocker and potential rapid antidepressant currently in Phase 3 development. In a Phase 2a trial, REL-1017 showed robust, rapid, and sustained antidepressant efficacy as adjunctive treatment in patients with MDD. The objective of this study was to assess the effects of REL-1017 on SDQ subscales to better characterize the functional implications of its therapeutic effects. METHODS: A double-blind, placebo-controlled, inpatient, two-doses, 25 and 50&nbsp;mg, three-arm, 1:1:1, randomized, phase 2a trial of REL-1017 was conducted at 10 centers in the United States. Least square (LS) mean scores and Cohen's effect sizes of the total score of a 44-item of SDQ and its 5 subscales: lassitude, mood, cognitive/social functioning (SDQ-1); anxiety, agitation, anger, and irritability (SDQ-2); desire to be dead (SDQ-3); disruptions in sleep quality (SDQ-4); changes in appetite and weight (SDQ-5) were compared between REL-1017 and placebo. RESULTS: A total of 62 adult male and female patients (18-65&nbsp;years of age) diagnosed with MDD participated in the trial. On day 14, the last day of efficacy measurement, the difference from placebo of the LS mean (90% CI) for REL-1017 25&nbsp;mg and REL-1017 50&nbsp;mg groups, respectively, showed improvement for both tested doses on SDQ total score (-23.2; P&nbsp;=&nbsp;.0066 [effect size: 0.9]; -26.8 P&nbsp;=&nbsp;.0014 [effect size: 1.1]). Additionally, for SDQ subscales, REL-1017 25&nbsp;mg and REL-1017 50&nbsp;mg groups, respectively, showed significant improvement as compared with placebo: SDQ-1 (-13.9; P&nbsp;=&nbsp;.0025 [effect size: 1.0]; -15.0; P&nbsp;=&nbsp;.0009 [effect size: 1.1]), SDQ-2 (-4.6; P&nbsp;=&nbsp;.0398 [effect size: 0.7]; -7.2; P&nbsp;=&nbsp;.0012 [effect size: 1.1]) and SDQ-4 (-2.7; P&nbsp;=&nbsp;.0055 [effect size: 1.0]; -2.8; P&nbsp;=&nbsp;.0029 [effect size: 1.0]). No significant differences were observed between the treated groups and placebo in the SDQ-3 and SDQ-5 subscales. CONCLUSIONS: In patients with MDD, aside from improving the overall CFB compared to placebo in SDQ total score, REL-1017 resulted in clinically meaningful and statistically significant improvements in cognitive/motivational, anxiety/irritability, and sleep-specific domains. The robust, rapid, and sustained efficacy of REL-1017 for MDD is not limited to improving mood, but potentially extends to cognitive, motivational, sleep, and social functions, with potentially meaningful therapeutic and socioeconomic implications. These results may signal disease-modifying effects of esmethadone for MDD that may offer potential advantages over symptomatic treatment with standard antidepressants. FUNDING: Relmada Therapeutics, Inc

An Agonist of the CXCR4 Receptor Strongly Promotes Regeneration of Degenerated Motor Axon Terminals

The activation of the G-protein coupled receptor CXCR4 by its ligand CXCL12\u3b1 is involved in a large variety of physiological and pathological processes, including the growth of B cells precursors and of motor axons, autoimmune diseases, stem cell migration, inflammation, and several neurodegenerative conditions. Recently, we demonstrated that CXCL12\u3b1 potently stimulates the functional recovery of damaged neuromuscular junctions via interaction with CXCR4. This result prompted us to test the neuroregeneration activity of small molecules acting as CXCR4 agonists, endowed with better pharmacokinetics with respect to the natural ligand. We focused on NUCC-390, recently shown to activate CXCR4 in a cellular system. We designed a novel and convenient chemical synthesis of NUCC-390, which is reported here. NUCC-390 was tested for its capability to induce the regeneration of motor axon terminals completely degenerated by the presynaptic neurotoxin \u3b1-Latrotoxin. NUCC-390 was found to strongly promote the functional recovery of the neuromuscular junction, as assayed by electrophysiology and imaging. This action is CXCR4 dependent, as it is completely prevented by AMD3100, a well-characterized CXCR4 antagonist. These data make NUCC-390 a strong candidate to be tested in human therapy to promote nerve recovery of function after different forms of neurodegeneratio