We conjecture formulae of the colored superpolynomials for a class of twist knots K p where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomials for the twist knots with small values of p. The results support the categorified versions of the generalized volume conjecture and the quantum volume conjecture. Furthermore, we obtain the evidence that the Q-deformed Apolynomials can be identified with the augmentation polynomials of knot contact homology in the case of the twist knots.

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In our study, depression occurred in >40% of patients within the first year after stroke. Persistent depression is the first independent determinant of prognosis during the first year after stroke.

Abstract-In wireless sensor networks, some sensor nodes are put in sleep mode while other sensor nodes are in active mode for sensing and communication tasks in order to reduce energy consumption and extend network lifetime. This approach is a special case (k ¼ 2) of a randomized scheduling algorithm, in which k subsets of sensors work alternatively. In this paper, we first study the randomized scheduling algorithm via both analysis and simulations in terms of network coverage intensity, detection delay, and detection probability. We further study asymptotic coverage and other properties. Finally, we analyze a problem of maximizing network lifetime under Quality of Service constraints such as bounded detection delay, detection probability, and network coverage intensity. We prove that the optimal solution exists, and provide conditions of the existence of the optimal solutions.

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots.Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form J(n) = P k c(n, k)Ĵ(k) given a recursion relation for (Ĵ (n)) and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with −15 and 15 crossings.The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

Background: Cognitive dysfunction is considered a core feature among schizophrenia (SZ), bipolar disorder (BD), and major depressive disorder (MDD). Despite abundant literature comparing cognitive dysfunction among these disorders, the relationship between cognitive dysfunction and symptom dimensions remains unclear. The study aims are a) to identify the factor structure of the BPRS-18 and b) to examine the relationship between symptom domains and cognitive function across SZ, BD, and MDD. Methods: A total of 716 participants [262 with SZ, 104 with BD, 101 with MDD, and 249 healthy controls (HC)] were included in the study. One hundred eighty participants (59 with SZ, 23 with BD, 24 with MDD, and 74 HC) completed the MATRICS Consensus Cognitive Battery (MCCB), and 507 participants (85 with SZ, 89 with BD, 90 with MDD, and 243 HC) completed the Wisconsin Card Sorting Test (WCST). All patients completed the Brief Psychiatric Rating Scale (BPRS). Results: We identified five BPRS exploratory factor analysis (EFA) factors (“affective symptoms,” “psychosis,” “negative/disorganized symptoms,” “activation,” and “noncooperation”) and found cognitive dysfunction in all of the participant groups with psychiatric disorders. Negative/disorganized symptoms were the most strongly associated with cognitive dysfunctions across SZ, BD, and MDD. Conclusions: Our findings suggest that cognitive dysfunction severity relates to the negative/disorganized symptom domain across SZ, BD, and MDD, and negative/disorganized symptoms may be an important target for effective cognitive remediation in SZ, BD, and MDD.

In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q -difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative A-polynomial of a knot.In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group U q .sl 2 /) specializes at q D 1 to the better known A-polynomial of a knot, which has to do with genuine SL 2 ‫/ރ.‬ representations of the knot complement.Computing the non-commutative A-polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the C -polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative A-polynomial of twist knots. Finally, we formulate a number of conjectures relating the A, the C -polynomial and the Alexander polynomial, all confirmed for the class of twist knots. 57N10; 57M25