Bisected theta series,least r-gaps in partitions,and polygonal numbers

The Ramanujan Journal - The least r-gap, $$g_r(\lambda )$$, of a partition $$\lambda $$ is the smallest positive integer that does not appear at least r times as a part of $$\lambda $$. In this...     

Majorana fermions,supersymmetry, and thermofield dynamics

Theoretical and Mathematical Physics - We show the existence of supersymmetry and degeneracy for an arbitrary $$($$ even or odd $$)$$ number of Majorana fermions without invoking any symmetry of...     

Hyperbolic spaces,principal series,and $${\mathrm{O}}(2,\infty )$$O(2,∞)

We prove that there exists no irreducible representation of the identity component of the isometry group $${\mathrm{PO}}(1,n)$$ of the real hyperbolic space of dimension n into the group $${\mathrm{O}}(2,\infty )$$ if $$n\ge 3$$. This is motivated by the existence of irreducible representations (arising from the spherical principal series) of $${\mathrm{PO}}(1,n)^{\circ }$$ into the groups $${\mathrm{O}}(p,\infty )$$ for other values of p.     

Packing chromatic number, $$\mathbf (1, 1, 2, 2) $$ ( 1 , 1 , 2 , 2 ) -colorings,and characterizing the Petersen graph

The packing chromatic number $$\chi _{\rho }(G)$$     

Vandermonde Varieties,Mirrored Spaces,and the Cohomology of Symmetric Semi-algebraic Sets

Foundations of Computational Mathematics - Let $$\mathrm {R}$$ be a real closed field. We prove that for each fixed $$\ell , d \ge 0$$ , there exists an algorithm that takes as input a...     

Multiple expansions of real numbers with digits set $$\left\{ 0,1,q\right\} $$ 0 , 1 , q

Mathematische Zeitschrift - For $$q>1$$ we consider expansions in base q with digits set $$\left\{ 0,1,q\right\} $$ . Let $${{\mathcal {U}}}_q$$ be the set of points which have a unique...     

Functions generating (m,M, $$\varvec{\Psi }$$ )-Schur-convex sums

Aequationes mathematicae - The notion of $$(m,M,\Psi )$$ -Schur-convexity is introduced and functions generating $$(m,M,\Psi )$$ -Schur-convex sums are investigated. An extension of the...     

Certain graphs with exactly one irreducible T-module with endpoint 1, which is thin

Journal of Algebraic Combinatorics - Let $$\Gamma $$ denote a finite, simple and connected graph. Fix a vertex x of $$\Gamma $$ and let $$T=T(x)$$ denote the Terwilliger algebra of $$\Gamma $$ with...     

On the error term $$\Delta _{(k,l)}(x)$$ Δ ( k,l ) ( x )

The Ramanujan Journal - Let k, l be non-negative integers and $$\zeta ^{(k)}(s)$$ denote the kth derivative of the Riemann zeta function $$\zeta (s).$$ Further let $$d_{(k,l)}(n)$$ be the nth...     

A note on polydegree (n, 1) rational inner functions,slice matrices,and singularities

Archiv der Mathematik - We analyze certain compositions of rational inner functions in the unit polydisk $$\mathbb {D}^{d}$$ with polydegree (n, 1), $$n\in \mathbb {N}^{d-1}$$ , and...     

New congruences modulo 2, 4, and 8 for the number of tagged parts over the partitions with designated summands

The Ramanujan Journal - Recently, Lin introduced two new partition functions $$\hbox {PD}_{\mathrm{t}}(n)$$ and $$\hbox {PDO}_{\mathrm{t}}(n)$$, which count the total number of tagged parts over...     

Colouring of successor of regular,again

Acta Mathematica Hungarica - We get a version of the colouring property Pr1 proving $$\Pr_1(\lambda,\lambda,\lambda,\partial)$$ always when $$\lambda= \partial^+,\partial $$ are regular cardinals...     

Surface bundles over surfaces: new inequalities between signature,simplicial volume and Euler characteristic

Geometriae Dedicata - Given a pants decomposition $$\mathcal {PC} = \{\sigma _1, \ldots , \sigma _{\xi }\}$$ on a hyperbolizable surface $$\Sigma $$ and a vector $${{\underline{c}}}= (c_1, \ldots ,...     

A nonstandard characterization of sequential equilibrium,perfect equilibrium,and proper equilibrium

New characterizations of sequential equilibrium, perfect equilibrium, and proper equilibrium are provided that use nonstandard probability. It is shown that there exists a belief system μ such that is a sequential equilibrium in an extensive game with perfect recall iff there exist an infinitesimal and a completely mixed behavioral strategy profile σ′ (so that assigns positive, although possibly infinitesimal, probability to all actions at every information set) that differs only infinitesimally from such that at each information set I for player i, σ _i is an -best response to conditional on having reached I. Note that the characterization of sequential equilibrium does not involve belief systems. There is a similar characterization of perfect equilibrium; the only difference is that σ _i must be a best response to conditional on having reached I. Yet another variant is used to characterize proper equilibrium. This work was supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, and by AFOSR under grant FA9550-05-1-0055.     

Smooth structures,normalized Ricci flows,and finite cyclic groups

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that for any finite cyclic group , where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group , which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of and .     

On Asymptotics, Stirling Numbers, Gamma Function and Polylogs

We apply the Euler–Maclaurin formula to find the asymptotic expansion of the sums , , , in closed form to arbitrary order . The expressions often simplify considerably and the coefficients are recognizable constants. The constant terms of the asymptotics are either (first two sums), 0 (third sum) or yield novel mathematical constants (fourth sum). This allows numerical computation of faster than any current software. One of the constants also appears in the expansion of the function around the singularity at s = 1; this requires the asymptotics of the incomplete gamma function. The manipulations involve polylogs for which we find a representation in terms of Nielsen integrals, as well as mysterious conjectures for Bernoulli numbers. Applications include the determination of the asymptotic growth of the Taylor coefficients of . We also give the asymptotics of Stirling numbers of first kind and their formula in terms of harmonic numbers.     

Coincidences of Centers of Edge-Incentric, or Balloon, Simplices

An edge-incentric d-simplex is defined to be a d-simplex S which admits a (d − 1)-sphere that touches all the edges of S internally. The center of such a sphere is called the edge-incenter of S and is denoted by . Equivalently, S is edge-incentric if and only if its vertices are the centers of d + 1 (d − 1)-spheres in mutual external touch, and for this reason one may call such an S a balloon d-simplex. An orthocentric d-simplex is a d-simplex in which the altitudes are concurrent. The point of concurrence is called the orthocenter and is denoted by . The spaces of edge-incentric and of orthocentric d-simplices have the same dimension d in the sense that a d-simplex in either space can be parametrized, up to shape, by d numbers. Edge-incentric and orthocentric tetrahedra are the first two of the four special classes of tetrahedra studied in [1, Chapter IX.B, pp. 294–333]. The degree of regularity implied by the coincidence of two or more centers of a general d-simplex is investigated in [8], where it is shown that the coincidence of the centroid , the circumcenter , and the incenter does not imply much regularity. For an orthocentric d-simplex S, however, it is proved in [9] that if any two of the centers , and coincide, then S is regular. In this paper, the same question is addressed for edge-incentric d-simplices. Among other things, it is proved that if any three of the centers , and of an edge-incentric d-simplex S coincide, then S is regular, and it is also shown that none of the coincidences , and implies regularity (except when d ≤ 3, d ≤ 4, and d ≤ 6, respectively). In contrast with the afore-mentioned results for orthocentric d-simplices, this emphasizes once more the feeling that, regarding many important properties, orthocentric d-simplices are the true generalizations of triangles. Several open questions are posed. Received: June 19, 2006.     

AUTOMORPHISMS OF SL (2, K) OVER SKEW FIELDS

In this paper, the author proves the following resu: It Let K be a skew field and A be an automorphism of SL(2, K). Then there exists A∈GL(2, K), an automorphism σ or an anti-automorphism τ of K, such that A is of theform AX=AX~σA~(-1) for all X∈SL(2, K)or AX=A(X~τ~2)~(-1)A~(-1) for all X∈SL(2, K),where X~σ, X~τ are the matrices obtained by applying σ, τ on X respee tively and X' is thetranspose of X.     

Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration (q) on the set (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions ₊(q) and ₋(q) of (q) to the set ₊(q) of secant (hyperbolic) lines and to the set ₋(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme ₋(q) is pseudocyclic.We further show that the coherent configurations (q ²) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme ₊(q ²), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes ₊(q ²) and ₋(q ²). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.