Two-batch liar games on a general bounded channel

We consider an extension of the 2-person R\'enyi-Ulam liar game in which lies are governed by a channel $C$, a set of allowable lie strings of maximum length $k$. Carole selects $x\in[n]$, and Paul makes $t$-ary queries to uniquely determine $x$. In each of $q$ rounds, Paul weakly partitions $[n]=A_0\cup >... \cup A_{t-1}$ and asks for $a$ such that $x\in A_a$. Carole responds with some $b$, and if $a\neq b$, then $x$ accumulates a lie $(a,b)$. Carole's string of lies for $x$ must be in the channel $C$. Paul wins if he determines $x$ within $q$ rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space $[n]$ for which Paul can guarantee finding the distinguished element is $\sim t^{q+k}/(E_k(C)\binom{q}{k})$ as $q\to\infty$, where $E_k(C)$ is the number of lie strings in $C$ of maximum length $k$. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel $C$.Comment: 26 page

Two-player envy-free multi-cake division

We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may not exist for three cakes cut into three pieces each. However, there do exist such divisions for two cakes cut into three pieces each, and for three cakes cut into four pieces each. The resulting allocations of pieces to players are Pareto-optimal with respect to the division. We use a generalization of Sperner's lemma on the polytope of divisions to locate solutions to our generalized cake-cutting problem.Comment: 15 pages, 7 figures, see related work at http://www.math.hmc.edu/~su/papers.htm

A Borsuk-Ulam Equivalent that Directly Implies Sperner\u27s Lemma

We show that Fan’s 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem. Moreover, unlike other Borsuk–Ulam equivalents, we show that this lemma directly implies Sperner’s Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk–Ulam theorem implies the Brouwer fixed-point theorem, or that the Lusternik–Schnirelmann–Borsuk theorem implies the KKM lemma

Two-Player Envy-Free Multi-Cake Division

We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may not exist for three cakes cut into three pieces each. However, there do exist such divisions for two cakes cut into three pieces each, and for three cakes cut into four pieces each. The resulting allocations of pieces to players are Pareto-optimal with respect to the division. We use a generalization of Sperner’s lemma on the polytope of divisions to locate solutions to our generalized cake-cutting problem

Preserving and Restoring Bone with Continuous Insulin Infusion Therapy in a Mouse Model of Type 1 Diabetes

Those with type 1 diabetes (T1D) are more likely to suffer a fracture than age- and sex-matched individuals without diabetes, despite daily insulin therapy. In rodent studies examining the effect of bone- or glucose-targeting therapies on preventing the T1D-related decrease in bone strength, insulin co-therapy is often not included, despite the known importance of insulin signaling to bone mass accrual. Therefore, working toward a relevant pre-clinical model of diabetic bone disease, we assessed the effect of continuous subcutaneous insulin infusion (CSII) therapy at escalating doses on preserving bone and the effect of delayed CSII on rescuing the T1D-related bone deterioration in an established murine model of T1D. Osmotic minipumps were implanted in male DBA/2 J mice 2 weeks (prevention study) and 6 weeks (rescue study) after the first injection of streptozotocin (STZ) to deliver insulin at 0, 0.0625, 0.125, or 0.25 IU/day (prevention study; n = 4–5 per dose) and 0 or 0.25 IU/day (rescue study; n = 10 per group). CSII lasted 4 weeks in both studies, which also included age-matched, non-diabetic DBA/2 J mice (n = 8–12 per study). As the insulin dose increased, blood glucose decreased, body weight increased, a serum maker of bone resorption decreased, and a serum marker of bone formation increased such that each end-point characteristic was linearly correlated with dose. There were insulin dose-dependent relationships (femur diaphysis) with cross-sectional area of cortical bone and cortical thickness (micro-computed tomography) as well as structural strength (peak force endured by the mid-shaft during three-point bending). Likewise, trabecular bone volume fraction (BV/TV), thickness, and number (distal femur metaphysis) increased as the insulin dose increased. Delayed CSII improved glycated hemoglobin (HbA1c), but blood glucose levels remained relatively high (well above non-diabetic levels). Interestingly, it returned the resorption and formation markers to similar levels as those seen in non-T1D control mice. This apparent return after 4 weeks of CSII translated to a partial rescue of the structural strength of the femur mid-shaft. Delayed CSII also increased Tb.Th to levels seen in non-T1D controls but did not fully restore BV/TV. The use of exogenous insulin should be considered in pre-clinical studies investigating the effect of T1D on bone as insulin therapy maintains bone structure without necessarily lowering glucose below diabetic levels

The Impact of SGLT2 Inhibitors, Compared with Insulin, on Diabetic Bone Disease in a Mouse Model of Type 1 Diabetes

Skeletal co-morbidities in type 1 diabetes include an increased risk for fracture and delayed fracture healing, which are intertwined with disease duration and the presence of other diabetic complications. As such, chronic hyperglycemia is undoubtedly a major contributor to these outcomes, despite standard insulin-replacement therapy. Therefore, using the streptozotocin (STZ)-induced model of hypoinsulinemic hyperglycemia in DBA/2J male mice, we compared the effects of two glucose lowering therapies on the fracture resistance of bone and markers of bone turnover. Twelve week-old diabetic (DM) mice were treated for 9 weeks with: 1) oral canagliflozin (CANA, dose range ~10-16 mg/kg/day), an inhibitor of the renal sodium-dependent glucose co-transporter type 2 (SGLT2); 2) subcutaneous insulin, via minipump (INS, 0.125 units/day); 3) co-therapy (CANA + INS); or 4) no treatment (STZ, without therapy). These groups were also compared to non-diabetic control groups. Untreated diabetic mice experienced increased bone resorption and significant deficits in cortical and trabecular bone that contributed to structural weakness of the femur mid-shaft and the lumbar vertebra, as determined by three-point bending and compression tests, respectively. Treatment with either canagliflozin or insulin alone only partially rectified hyperglycemia and the diabetic bone phenotype. However, when used in combination, normalization of glycemic control was achieved, and a prevention of the DM-related deterioration in bone microarchitecture and bone strength occurred, due to additive effects of canagliflozin and insulin. Nevertheless, CANA-treated mice, whether diabetic or non-diabetic, demonstrated an increase in urinary calcium loss; FGF23 was also increased in CANA-treated DM mice. These findings could herald ongoing bone mineral losses following CANA exposure, suggesting that certain CANA-induced skeletal consequences might detract from therapeutic improvements in glycemic control, as they relate to diabetic bone disease

SGLT2 Inhibitor Therapy Improves Blood Glucose but Does Not Prevent Diabetic Bone Disease in Diabetic DBA/2J Male Mice

Persons with type 1 and type 2 diabetes have increased fracture risk, attributed to deficits in the microarchitecture and strength of diabetic bone, thought to be mediated, in part, by the consequences of chronic hyperglycemia. Therefore, to examine the effects of a glucose-lowering SGLT2 inhibitor on blood glucose (BG) and bone homeostasis in a model of diabetic bone disease, male DBA/2J mice with or without streptozotocin (STZ)-induced hyperglycemia were fed chow containing the SGLT2 inhibitor, canagliflozin (CANA), or chow without drug, for 10 weeks of therapy. Thereafter, serum bone biomarkers were measured, fracture resistance of cortical bone was assessed by μCT analysis and a three-point bending test of the femur, and vertebral bone strength was determined by compression testing. In the femur metaphysis and L6 vertebra, long-term diabetes (DM) induced deficits in trabecular bone microarchitecture. In the femur diaphysis, a decrease in cortical bone area, cortical thickness and minimal moment of inertia occurred in DM (p \u3c 0.0001, for all) while cortical porosity was increased (p \u3c 0.0001). These DM changes were associated with reduced fracture resistance (decreased material strength and toughness; decreased structural strength and rigidity; p \u3c 0.001 for all). Significant increases in PTH (p \u3c 0.0001), RatLAPs (p = 0.0002), and urine calcium concentration (p \u3c 0.0001) were also seen in DM. Canagliflozin treatment improved BG in DM mice by ~35%, but did not improve microarchitectural parameters. Instead, in canagliflozin-treated diabetic mice, a further increase in RatLAPs was evident, possibly suggesting a drug-related intensification of bone resorption. Additionally, detrimental metaphyseal changes were noted in canagliflozin-treated control mice. Hence, diabetic bone disease was not favorably affected by canagliflozin treatment, perhaps due to insufficient glycemic improvement. Instead, in control mice, long-term exposure to SGLT2 inhibition was associated with adverse effects on the trabecular compartment of bone

Mutation update and genotype-phenotype correlations of novel and previously described mutations in TPM2 and TPM3 causing congenital myopathies

Mutations affecting skeletal muscle isoforms of the tropomyosin genes may cause nemaline myopathy, cap myopathy, core-rod myopathy, congenital fiber-type disproportion, distal arthrogryposes, and Escobar syndrome. We correlate the clinical picture of these diseases with novel (19) and previously reported (31) mutations of the TPM2 and TPM3 genes. Included are altogether 93 families: 53 with TPM2 mutations and 40 with TPM3 mutations. Thirty distinct pathogenic variants of TPM2 and 20 of TPM3 have been published or listed in the Leiden Open Variant Database (http://www.dmd.nl/). Most are heterozygous changes associated with autosomal-dominant disease. Patients with TPM2 mutations tended to present with milder symptoms than those with TPM3 mutations, DA being present only in the TPM2 group. Previous studies have shown that five of the mutations in TPM2 and one in TPM3 cause increased Ca2+ sensitivity resulting in a hypercontractile molecular phenotype. Patients with hypercontractile phenotype more often had contractures of the limb joints (18/19) and jaw (6/19) than those with nonhypercontractile ones (2/22 and 1/22), whereas patients with the non-hypercontractile molecular phenotype more often (19/22) had axial contractures than the hypercontractile group (7/19). Our in silico predictions show that most mutations affect tropomyosin–actin association or tropomyosin head-to-tail binding

Research Statement

My research is in the area of algebraic combinatorics. I find the most captivating aspect of combinatorics is the approachability and elegance of its questions. In combinatorics one finds both open problems which are accessible to undergraduate students, and simply stated problems which prove to have deep and sophisticated solutions. In addition, combinatorial problems are integral to many other disciplines; for example my research deals with enumerative questions related to geometry and algebra. Suppose there are n points in the plane. Three natural questions one might ask are: * How many lines do the points determine? * How many triangular shaped regions are formed by the determined lines? * What is the maximum number of lines through a point? All of the information about how an arrangement of points and lines breaks up space is encoded by the flag f- and flag h-vectors, which are combinatorial invariants of a partially ordered set associated to the arrangement. While these vectors are easily defined, they have eluded complete characterization. For example it in not known how to calculate the number of lines which can be determined by n points. My work establishes inequalities which the flag f- and h- vectors satisfy