The Frobenius Formula for A=(a,ha+d,ha+b2d,...,ha+bkd)A=(a,ha+d,ha+b_2d,...,ha+b_kd)

Given relative prime positive integers $A=(a_1, a_2, ..., a_n)$, the Frobenius number $g(A)$ is the largest integer not representable as a linear combination of the $a_i$'s with nonnegative integer coefficients. We find the ``Stable" property introduced for the square sequence $A=(a,a+1,a+2^2,\dots, a+k^2)$ naturally extends for $A(a)=(a,ha+d,ha+b_2d,...,ha+b_kd)$. This gives a parallel characterization of $g(A(a))$ as a ``congruence class function" modulo $b_k$ when $a$ is large enough. For orderly sequence $B=(1,b_2,\dots,b_k)$, we find good bound for $a$. In particular we calculate $g(a,ha+dB)$ for $B=(1,2,b,b+1,2b)$, $B=(1,b,2b-1)$ and $B=(1,2,...,k,K)$

A Generalization of Mersenne and Thabit Numerical Semigroups

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $\langle A\rangle$ generated by $A$. The genus $g(A)$ is the number of positive integer elements that are not in $\langle A\rangle$. The Frobenius problem is to find $F(A)$ and $g(A)$ for a given sequence $A$. In this paper, we study the Frobenius problem of $A=(a,2a+d,2^2a+3d,...,2^ka+(2^k-1)d)$ and obtain formulas for $F(A)$ and $g(A)$ when $a+d\geq k$. Our formulas simplifies further for some special cases, such as Mersenne and Thabit numerical semigroups. We obtain explicit formulas for generalized Mersenne and Thabit numerical semigroups and some more general numerical semigroups

Synthesis and volume phase transition of concanavalin A-based glucose-responsive nanogels

NSFC [21274118, 91227120, 20923004]; FRFCU [2012121016]; NFFTBS [J1210014]; NCETFJGlucose-responsive polymer nanogels that can undergo a reversible and rapid volume phase transition in response to the fluctuation in blood glucose concentration have the potential to regulate the delivery of insulin mimicking pancreatic activity. We report here such a glucose-responsive polymer nanogel, which is made of concanavalin A (ConA) interpenetrated in a chemically crosslinked network of poly(N-isopropylacrylamide) (poly(NIPAM)). The introduction of ConA, a plant lectin protein, into the poly(NIPAM) network makes the newly developed semi-interpenetrating-structured nanogels responsive to glucose over a glucose concentration range of 0-20 mM at a physiological pH of 7.4. While the nanogels can swell and become stable shortly (<1 s) after adding glucose over a concentration range of 50.0 mu M to 20.0 mM, the changes in the average hydrodynamic radius and the size distribution of the nanogels can be fully reversible within the experimental error even after ten cycles of adding/removing glucose. The association rate constant is determined to be ca. 1.8 mM(-1) s(-1), and the dissociation rate constant is ca. 7.5 s(-1), indicating a fast reversible time response to the glucose concentration change of the nanogels. Moreover, in vitro insulin release can be modulated in a pulsatile profile in response to glucose concentrations

Highly sensitive and ultrastable skin sensors for biopressure and bioforce measurements based on hierarchical microstructures

Piezoresistive microsensors are considered to be essential components of the future wearable electronic devices. However, the expensive cost, complex fabrication technology, poor stability, and low yield have limited their developments for practical applications. Here, we present a cost-effective, relatively simple, and high-yield fabrication approach to construct highly sensitive and ultrastable piezoresistive sensors using a bioinspired hierarchically structured graphite/polydimethylsiloxane composite as the active layer. In this fabrication, a commercially available sandpaper is employed as the mold to develop the hierarchical structure. Our devices exhibit fascinating performance including an ultrahigh sensitivity (64.3 kPa^–1), fast response time (<8 ms), low limit of detection of 0.9 Pa, long-term durability (>100 000 cycles), and high ambient stability (>1 year). The applications of these devices in sensing radial artery pulses, acoustic vibrations, and human body motion are demonstrated, exhibiting their enormous potential use in real-time healthcare monitoring and robotic tactile sensing

Graphene@Poly(phenylboronic acid)s microgels with selectively glucose-responsive volume phase transition behavior at a physiological ph

The selective response to glucose is possible by using a poly(phenylboronic acid) microgel under a rational design. Such a microgel is made of graphene covalently immobilized in a microgel of poly(4-vinylphenylboronic acid) cross-linked with N,N′-methylenebis(acrylamide). Unlike the microgels reported in previous arts that would undergo volume phase transition in response to both glucose and other monosaccharides, the proposed microgels shrink upon adding glucose, whereas keep unchanged in the size upon adding other monosaccharides (with fructose, galactose, and mannose as models). Although the polysaccharides/glycoproteins (with dextran and Ribonuclease B as models) that contain many glycosyl residues can slightly absorb on the microgel surface and lead to a small impact on glucose-response, it can be addressed by further coating the microgel as a core with a thin nonglucose-responsive poly(N-isopropylacrylamide) gel shell. This selectively glucose-responsive volume phase transition behavior enables "turn-on" photoluminescence detection of glucose in blood serum (a model for complex biosystems)

Graphene@Poly(phenylboronic acid)s Microgels with Selectively Glucose-Responsive Volume Phase Transition Behavior at a Physiological pH

The selective response to glucose is possible by using a poly­(phenylboronic acid) microgel under a rational design. Such a microgel is made of graphene covalently immobilized in a microgel of poly­(4-vinylphenylboronic acid) cross-linked with <i>N</i>,<i>N</i>′-methylenebis­(acrylamide). Unlike the microgels reported in previous arts that would undergo volume phase transition in response to both glucose and other monosaccharides, the proposed microgels shrink upon adding glucose, whereas keep unchanged in the size upon adding other monosaccharides (with fructose, galactose, and mannose as models). Although the polysaccharides/glycoproteins (with dextran and Ribonuclease B as models) that contain many glycosyl residues can slightly absorb on the microgel surface and lead to a small impact on glucose-response, it can be addressed by further coating the microgel as a core with a thin nonglucose-responsive poly­(<i>N</i>-isopropylacrylamide) gel shell. This selectively glucose-responsive volume phase transition behavior enables “turn-on” photoluminescence detection of glucose in blood serum (a model for complex biosystems)

Low-voltage Extended Gate Organic Thin Film Transistors for Ion Sensing Based on Semi-conducting Polymer Electrodes

We report a low-voltage organic field-effect transistor consisting of an extended gate sensory area to detect various ions in a solution. The device distinguishes various ions by the shift in threshold voltage and is sensitive to multiple ions with various concentrations. X-ray photoelectron spectroscopy measurements and the resistance changes at the sensor area prove that the ions are doped into the sensitive film at the sensor area. Because of the effect of doping, the conductivity of the semiconductor polymer film changes thus causing a threshold voltage shift

Deciphering the Role of Fluoroethylene Carbonate towards Highly Reversible Sodium Metal Anodes

Sodium metal anodes (SMAs) suffer from extremely low reversibility (95% with conventional NaPF6 salt at a regular concentration (1.0 M). The peculiar role of FEC is firstly unraveled via its involvement into the solvation structure, where a threshold FEC concentration with a coordination number>1.2 is needed in guaranteeing high Na reversibility over the long-term. Specifically, by incorporating an average number of 1.2 FEC molecules into the primary Na+ solvation sheath, lowest unoccupied molecular orbital (LUMO) levels of such Na+-FEC solvates undergo further decrease, with spin electrons residing either on the O=CO(O) moiety of FEC or sharing between Na+ and its C=O bond, which ensures a prior FEC decomposition in passivating the Na surface against other carbonate molecules. Further, by adopting cryogenic transmission electron microscopy (cryo-TEM), we found that the Na filaments grow into substantially larger diameter from ~400 nm to >1 μm with addition of FEC upon the threshold value. A highly crystalline and much thinner (~40 nm) solid-electrolyte interphase (SEI) is consequently observed to uniformly wrap the Na surface, in contrast to the severely corroded Na as retrieved from the blank electrolyte. The potence of FEC is further demonstrated in a series of “corrosive solvents” such as ethyl acetate (EA), trimethyl phosphate (TMP), and acetonitrile (AN), enabling highly reversible SMAs in the otherwise unusable solvent systems