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Let $(M^m, T)$ be a smooth involution on a closed smooth $m$-dimensional manifold and $F = {\bigcup}^n_{j=0} F^j (n \lt m)$ its fixed point set, where $F^j$ denotes the union of those components of $F$ having dimension $j$. The famous Five Halves Theorem of J. Boardman, announced in 1967, establishes that, if $F$ is nonbounding, then $m \leq \frac{5}{2} n$; further, this estimative is best possible. In this paper we obtain improvements of this theorem, taking into account certain natural numbers which we call the decomposability degrees $\ell(F^j)$ of the nonbounding components $F^j$ of $F$ (see the definition in Section 1). Also, these improvements are obtained under assumptions on the set of dimensions occurring in $F$, which we denote $\pi_0(F)$. The main result of this paper is: suppose the involution $(M^m, T)$ has $\pi_0(F) = \{ 0, 1, \dots, j, n \}$, where $2 \leq j \lt n \lt m$ and $F^j$ is nonbounding. Write $\mathcal{M}(n - j)$ for the function of $n - j$ defined in the following way: writing $n - j = 2^p q$, where $q \geq 1$ is odd and $p \geq 0$, $M(n - j) = 2n + p - q + 1$ if $p \leq q$ and $M(n - j) = 2n + 2^{p-q}$ if $p \geq q$. Then $m \leq \mathcal{M}(n - j) + 2j + \ell (F^j)$. In addition, we develop a method to construct involutions $(M^m, T)$ with $\pi_0(F)$ as above, in some special situations, which in some cases will show that the above bound is best possible. This will provide some improvements of the general Five Halves Theorem $(\pi_0(F) = \{ i / 0 \leq i \leq n \} )$, by considering the particular case $j = n - 1$.

Let $2 \leq a \leq b \leq c \in \mathbb{N}$ with $��=1/a+1/b+1/c$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$? (Classically, for $(a,b,c)=(2,3,7)$ and more recently also for general $(a,b,c)$.) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove a conjecture of Marion [21] showing that various finite simple groups are not quotients of $T$, as well as positive results showing that many finite simple groups are quotients of $T$.

In this paper, we investigate the uniqueness problem on entire functionssharing fixed points (ignoring multiplicities). Our main results improve and generalizesome results due to Zhang [13], Qi-Yang [10] and Dou-Qi-Yang [1]. 1. IntroductionIn this paper, a meromorphic function will mean meromorphic in the wholecomplex plane. We assume that the reader is familiar with standard notations andfundamental results of Nevanlinna Theory as explained in [12].We say that two meromorphic functions f and g share a small function a(z) IM(ignoring multiplicities) when f −a and g−a have the same zeros. If f and g havethe same zeros with the same multiplicities, then we say that f and g share a(z)CM (counting multiplicities).Let p be a positive integer and a ∈ C. We denote by N p (r, 1f−a ) the countingfunction of the zeros of f − a where an m-fold zero is counted m times if m ≤ pand p times if m > p. We denote by N L (r, 1f−1 ) the counting function for 1-pointsof both f(z) and g(z) about which f(z) has a larger multiplicity than g(z), withmultiplicity not being counted. We say that a finite value z

This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let \(\left\{ {X_a } \right\}_{a \in Z_ + }\) be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1, 2, ...}. Let T ⊂ S be an arbitrary finite subset of S. Denote by L n the length of the longest run of consecutive i’s for i ∈ T, that occurs in the sequence X 1, ..., X n . In this paper, we obtain a limit law and a week version of an Erdos-Renyi type law for L n . A large deviation result of L n is also discussed.

In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain \({Q = \Omega \times (0, \tau)}\) : Find \({{u \in K}}\) and an \({{\eta \in L^{p'}(Q)}}\) such that $$\eta \in f(\cdot,\cdot,u), \quad \langle u_t + Au, v - u\rangle + \int_Q \eta (v - u)\,{\rm d}x{\rm d}t \ge 0, \quad \forall \, v \in K,$$ where \({{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}\) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function \({{s \mapsto f(\cdot,\cdot,s)}}\) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.

We will show that two functions in the extended Selberg class satisfying the same functional equation must be identically equal if they have sufficiently many common zeros. This paper concerns the question of how L-functions are determined by their zeros. L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑∞ n=1 1 ns as the prototype and are important objects in number theory. The Selberg class S of L-functions is the set of all Dirichlet series L(s) = ∑∞ n=1 a(n) ns of a complex variable s = σ + it with a(1) = 1, satisfying the following axioms (see [7]): (i) (Dirichlet series) For σ > 1, L(s) is an absolutely convergent Dirichlet series. (ii) (Analytic continuation) There is a nonnegative integer k such that (s − 1)L(s) is an entire function of finite order. (iii) (Functional equation) L satisfies a functional equation of type ΛL(s) = ωΛL(1− s), where ΛL(s) = L(s)Q s ∏K j=1 Γ(λjs+μj) with positive real numbers Q, λj and with complex numbers μj , ω with Reμj ≥ 0 and |ω| = 1. (iv) (Ramanujan hypothesis) a(n) n for every e > 0; (v) (Euler product) logL(s) = ∑∞ n=1 b(n) ns , where b(n) = 0 unless n is a positive power of a prime and b(n) n for some θ < 12 . The degree dL of an L-function L is defined to be dL = 2 ∑K j=1 λj , where K, λj are the numbers in axiom (iii). The Selberg class includes the Riemann zeta-function ζ and essentially those Dirichlet series where one might expect the analogue of the Riemann hypothesis. At the same time, there are a whole host of interesting Dirichlet series not possessing a Euler product (see e.g. [3], [8]). Throughout the paper, all L-functions are assumed to be functions from the extended Selberg class of those only satisfying the axioms (i)-(iii) (see [3]). Thus, the results obtained in the present paper particularly apply to L-functions in the Selberg class. Received by the editors October 5, 2011 and, in revised form, February 12, 2012. 2010 Mathematics Subject Classification. Primary 11M36, 30D30.

In a paper by Z. Lu and J. Rowlett, it is shown that the eigenvalues of the weighted Laplacian can be approximated by eigenvalues of a naturally associated family of narrow graphs. In this paper, we generalize this result to the p p -Laplacian. Our approach features overcoming the nonlinearity of the p p -Laplacian when p ≠ 2 p\neq 2 , which is different from the Laplacian case.

In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher– Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation. In this sequel, we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent Fisher– Bingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on Stiefel manifolds. In each case, the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables. Both first-order and second-order saddlepoint approximations are considered. Computational algorithms, numerical results and theoretical properties of the approximations are presented. In the challenging high-dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered. Some key words: Directional data; Fisher matrix distribution; Kent distribution; Orientation statistics.

Let Λ be a robustly transitive set of a diffeomorphism f ona closed C ∞ manifold. In this paper, we characterize hyperbolicity of Λin C 1 -generic sense. 1. IntroductionA fundamental problem in differentiable dynamical systems is to understandhow a robust dynamic property (that is, a property that holds for a system andall C 1 nearby ones) on the underlying manifold would influences the behaviorof the tangent map on the tangent bundle. In this paper, we study the robustdynamic property for a transitive set. Let M be a closed C ∞ manifold, and letDiff(M) be the space of diffeomorphisms of M endowed with the C 1 -topology.Denote by d the distance on M induced from a Riemannian metric k·k on thetangent bundle TM. Let f ∈ Diff(M) and Λ be a closed f-invariant set. Theset Λ is transitive if there is a point x ∈ Λ such that ω(x) = Λ. Here ω(x) isthe forward limit set of x. Denote by f| Λ the restriction of f to the set Λ. Amaximal invariant set of f in an open set U, denoted by Λ f (U), is the set ofpoints whose whole orbit contained in U, that is,Λ

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

For a fixed digraph G, the Constraint Satisfaction Problem with the template G, or CSP(G) for short, is the problem of deciding whether a given input digraph H admits a homomorphism to G. The dichotomy conjecture of Feder and Vardi states that CSP(G), for any choice of G, is solvable in polynomial time or NP-complete. This paper confirms the conjecture for a class of oriented trees called special triads. As a corollary we get the smallest known example of an oriented tree (with 33 vertices) defining an NP-complete CSP(G).

Comultiplication modules over not necessarily commutative rings are studied. All rings are assumed to be associative and with nonzero identity element; all modules are assumed to be unitary. Expressions of the form “A is an invariant ring” mean that “AA and AA are right and left invariant rings.” If A is a ring and M is a right A-module, then for any subset B in A, we denote by lM (B) the left annihilator {m ∈ M | mB = 0} of B in M . According to [2], a module MA is called a comultiplication module if for any submodule X of M , there exists an ideal B of the ring A such that X = lM (B). Comultiplication modules are studied in many papers (see, e.g., [1–6]), where the focus is mainly on comultiplication modules over commutative rings. Theorem 1 ([1]). Let A be a commutative ring and let M be a comultiplication A-module. (1) M is an essential extension of a direct sum of pairwise nonisomorphic simple modules, and any submodule of the module M that is a finite direct sum of cyclic modules is a cyclic module. (2) If the module M is nonsingular, then M is a projective semisimple module. In connection with Theorem 1, we prove Theorem 2, which is the main result of the present paper. Theorem 2. Let A be a ring and let M be a comultiplication right A-module. (1) Any submodule of the module M that is a finite direct sum of cyclic modules is a cyclic module. (2) If A is an invariant ring with commutative multiplication of ideals, then M is an essential extension of a direct sum of pairwise nonisomorphic simple modules; in addition, if the module M is nonsingular, then M is a projective semisimple module. In connection with Theorem 2, we note that the class of all invariant rings with commutative multiplication of ideals contains all commutative rings, all rings of formal power series in one variable over division rings, all factor rings of any direct products of division rings, and all strongly regular rings. (A ring A is said to be strongly regular if every principal one-sided ideal of A is generated by a central idempotent.) The proof of Theorem 2 is decomposed into a series of assertions; some of the assertions are of independent interest. We present the necessary notation and definitions. A ring A is said to be right invariant if all right ideals of A are ideals. A ring A is said to be dual if B = rA(lA)(B) for any right ideal B of A and C = lA(rA)(C) for any left ideal C of A. We denote by J(M) the Jacobson radical of the module M , i.e., J(M) is the intersection of all maximal submodules in M ; we have J(M) = M if M does not have maximal submodules. We denote by Soc(M) the socle of the module M , i.e., Soc(M) is the sum of all minimal submodules in M , and Soc(M) = 0 if M does not have minimal submodules. A ring A is said to be semiperfect if A/J(A) is an Artinian ring and all idempotents of A/J(A) are lifted to idempotents of the ring A. A cyclic module M is said to be local if the module M/J(M) is simple. A submodule X of the module M is said to be essential if X has the nonzero intersection with every nonzero submodule in M . In this case, we say that M is an essential extension of the module X. A module MA is said to be nonsingular if M does not have nonzero elements whose annihilators are essential right ideals of the ring A. A module M is said to be cocyclic if M is an essential extension of a simple module. A module M is said to be finite-dimensional if M does not contain an infinite direct sum of nonzero submodules. A module M is said to be quotient finite-dimensional if all factor modules Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 4, pp. 217–224, 2011/12. 1072–3374/13/1915–0743 c © 2013 Springer Science+Business Media New York 743 of the module M are finite-dimensional. For a module M , a submodule X in M is said to be superfluous (in M) if X + Y = M for each proper submodule Y in M . Any submodule of a factor module of M is called a subfactor of M . A module MA is said to be faithful if rA(M) = 0. The following two assertions are known; they can be directly verified. Lemma 3 ([1, Lemma 1.2]). Let A be a ring, B be an ideal in A, and let M be a right A-module with MB = 0. Then M is a comultiplication A-module if and only if M is a comultiplication A/B-module. Lemma 4 ([2, Theorem 3.17, Lemma 3.7]). Let A be a ring and let M be a right A-module. The following conditions are equivalent : (1) M is a comultiplication module; (2) X = lM ( rA(X) ) for every submodule X of the module M ; (3) every submodule of the module M is a comultiplication module. Lemma 5. Let A be a ring, M be a comultiplication right A-module, and let X be a nonzero submodule in M . (1) X is a comultiplication A/rA(X)-module and f(X) ⊆ X for any homomorphism f : X → M . In particular, M does not contain a direct sum of two isomorphic nonzero modules. (2) If X = X1⊕· · ·⊕Xn is a submodule in M and there exists a module N such that every module Xi is a homomorphic image of the module N , then X is a homomorphic image of the module N . (3) If X = X1 ⊕ · · · ⊕ Xn is a submodule in M and every module Xi is cyclic, then X is a cyclic module. (4) In M , every finitely generated semisimple submodule is cyclic. (5) If there exists an element m ∈ M with rA(m) = 0, then A is a comultiplication right A-module. (6) If the ring A is right invariant and M contains a faithful cyclic submodule mA, then A is a comultiplication right A-module. (7) If the module M is nonsingular and the multiplication of right ideals of the ring A is commutative, then M is a projective semisimple module. Proof. (1) By Lemma 4(3) and Lemma 3, X is a comultiplication A/rA(X)-module. Since M is a comultiplication module, X = rM (B) for some ideal B in A. Then f(X)B ⊆ f(XB) = f(0) = 0, whence f(X) ⊆ rM (B) = X. (2) By assumption, there exist epimorphisms hi : N → Xi, i = 1, . . . , n. We denote by h the homomorphism h1 + · · · + hn from N into X = X1 ⊕ · · · ⊕Xn. Let πi be the natural projection from the module X onto the module Xi, i = 1, . . . , n. By (1), πi ( h(N) ) ⊆ h(N) for all i. Therefore, h(N) = π1 ( h(N) ⊕ · · · ⊕ πn ( h(N) ) = X1 ⊕ · · · ⊕Xn = X, and h is the required epimorphism from the module N onto the module X. (3) Since every module Xi is cyclic, there exist epimorphisms hi : AA → Xi, i = 1, . . . , n. By (2), there exists an epimorphism from the module AA onto the module X. Therefore, X is a cyclic module. (4) Since every finitely generated semisimple module is a finite direct sum of simple modules, the assertion follows from (3) and the property that every simple module is cyclic. (5) By Lemma 4, mAA is a comultiplication module. In addition, mAA is a free cyclic module with free generator m. Now it is directly verified that AA is a comultiplication module. (6) Since the ring A is right invariant, rA(mA) = rA(m). Therefore, (6) follows from (5). (7) First, we prove that M is a semisimple module. Let Y be a submodule in M . By Zorn’s lemma, there exists a submodule Z in M such that Y ∩Z = 0 and Y ⊕Z is an essential submodule in M . We set N = Y ⊕ Z. It is sufficient to prove that M = N . We assume the contrary. Then there exists a nonzero element m ∈ M \ N . Since M is a comultiplication module, there exists an ideal B of the ring A with N = lM (B). Then mB = 0. Since N is an essential submodule in M , we have that mC ⊆ N for some

The subject of this paper is to give the description, up to topological conjugacy, of possible minimal sets of triangular maps of the square of type 2 ∞ . In [4] , we give a general method allowing to embed any zero-dimensional almost 1–1 extension of the dyadic odometer (in particular any dyadic Toeplitz system) as a minimal set of a triangular map of this type. In this paper we present a method (a combination of that described in [4] with one introduced in [1] ) of similarly embedding a special class of zero-dimensional almost 2–1 extensions of the odometer. We conjecture that these two embedding theorems exhaust all possibilities for nonperiodic minimal sets. The paper was inspired by the last unsolved problem in the Sharkovski classification program of triangular maps: does there exist a triangular map with positive entropy attained on the set of uniformly recurrent points but with entropy zero on the set of regularly recurrent points. The paper answers this question positively, concluding the program.

Based on the concept of the folding, the folding in X-direction and in Y-direction are de ned and denoted by the X-Folding and the Y-Folding respectively. Weconsider a random variable X which follows a rectangular distribution "R(a;b) distribu-tion" with two parameters a;b. This paper aims to apply the folding on the unit areaP(a X b) and also to study the proposed folding in each direction for R(a;b) distri-bution and the generated family of the corresponding constructed rectangular probabilitydistributions. Some main properties of this family are reviewed. According to the pro-posed folding, we derive and discuss some important corresponding functions in closedforms. 1. IntroductionThe main idea of folding on manifolds has been introduced by S. A. Robertsonwho studied the strati cation determined by the folds or singularities [13]. Based onthis paper, some studies on folding in many branches such as manifolds, topology,graphs-theory and algebra have been presented in some literatures [1, 2, 3, 4, 5, 6,7, 8, 9, 12]. The conditional foldings of manifolds have been de ned by M. El-Ghoul[3]. Some applications on the folding of a manifold into itself were introduced byP. Di. Francesco [2]. Also a graph folding has been presented and discussed by E.El-Kholy [7]. Moreover, the theory of isometric foldings has been pushed and alsosome di erent types of foldings have been discussed by E. El-Kholy and others [9].In this article we will discuss the folding of a family of probability distributions intoa generated family of the same type of the probability distributions. Our study will

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL2-factorizations of length 5N, where N is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov, it follows that the elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length 4N. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to 3N. On the other hand, for SL(n, R), over a Bezout ring R, we further improve the bound to 2N = n 2-n. Bibliography: 25 titles.

Let E S ( n ) ES(n) be the smallest integer such that any set of E S ( n ) ES(n) points in the plane in general position contains n n points in convex position. In their seminal 1935 paper, Erdős and Szekeres showed that E S ( n ) ≤ ( 2 n − 4 n − 2 ) + 1 = 4 n − o ( n ) ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)} . In 1960, they showed that E S ( n ) ≥ 2 n − 2 + 1 ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal. In this paper, we nearly settle the Erdős-Szekeres conjecture by showing that E S ( n ) = 2 n + o ( n ) ES(n) =2^{n +o(n)} .

The first three Zagreb indices of a graph G denoted, M1(G),M2(G) and M3(G), are well known. Equally well known is the eccentricity connectivity index denoted, ξ(G). In this paper we derive closed formula for these indices for thorn cycles, thorn star graphs and thorn complete graphs, respectively. Same is repeated for the eccentricity connectivity index. The further aim of the paper is to emphasize the subtle difference between mathematical induction and immediate induction as equally valid techniques of proof. Valid immediate induction requires the rarely utilised property of many graphs namely, well-defineness. AMS Subject Classification: 05C07, 05C38, 05C75, 05C85

Let G be a graph and v be any vertex of G. Then the neighborhood contracted graph Gv of G, with respect to the vertex v, is the graph with vertex set V − N (v), where two vertices u, w ∈ V − N (v) are adjacent in Gv if either w = v and u is adjacent to any vertex of N (v) in G or u, w / ∈ N [v] and u, w are adjacent in G. The properties of the neighborhood contracted graphs are discussed in this paper. The neighborhood contraction in some special class of graphs, the domination in a graph and the neighborhood contracted graphs are discussed in the paper.

This paper is concerned with the asymptotic behavior on $${\partial\Omega}$$ of boundary blow-up solutions to semilinear elliptic equations $$\left\{\begin{array}{ll} \Delta u=b(x)f(u),~~ &x\in \Omega, \\ u(x)=\infty, ~~ &x\in\partial\Omega,\end{array} \right.$$ where b(x) is a nonnegative function on $${\Omega}$$ and may vanish on $${\partial\Omega}$$ at a very degenerate rate; f is nonnegative function on [0,∞) and normalized regularly varying or rapidly varying at infinity. The main feature of this paper is to establish a unified and explicit asymptotic formula when the function f is normalized regularly varying or grows faster than any power function at infinity. The effect of the mean curvature of the nearest point on the boundary in the second-order approximation of the boundary blow-up solution is also discussed. Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.

Inspired by work of McMullen, we show that any orbit of the diagonal group in the space of lattices accumulates on the set of stable lattices. As consequences, we settle a conjecture of Ramharter concerning the asymptotic behaviour of the Mordell constant, and reduce Minkowski's conjecture on products of linear forms to a geometric question, yielding two new proofs of the conjecture in dimensions up to 7., Comment: This paper was part of the paper "On stable lattices and the diagonal group" (arXiv:1309.4025) which was split into three shorter papers (now all on arXiv) following the referee's request

A recent paper of Shekhar compares the ranks of elliptic curves $E_1$ and $E_2$ for which there is an isomorphism $E_1[p] \simeq E_2[p]$ as $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$-modules, where $p$ is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where $p$ is a prime of good supersingular reduction., Comment: fixed a persistent typo in the results of Section 3. this typo led to the original example for p=3 being incorrect; it has been replaced with a correct example

In this paper, for the purpose of measuring the non-self-centrality extent of non-selfcentered graphs, a novel eccentricity-based invariant, named as non-self-centrality number (NSC number for short), of a graph G is defined as follows: \(N\left( G \right) = \sum\nolimits_{{v_i}{v_j} \in V\left( G \right)} {|{e_i} - {e_j}|} \) where the summation goes over all the unordered pairs of vertices in G and ei is the eccentricity of vertex vi in G, whereas the invariant will be called third Zagreb eccentricity index if the summation only goes over the adjacent vertex pairs of graph G. In this paper, we determine the lower and upper bounds on N(G) and characterize the corresponding graphs at which the lower and upper bounds are attained. Finally we propose some attractive research topics for this new invariant of graphs.

The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of [GP1]. The aim of this paper is to show the following result which is essential in proving the main theorem of [GP1]: given an infinite perfect base field $k$, any $k$-smooth scheme $X$ and any $n\geq 1$, the map of simplicial pointed Nisnevich sheaves $(-,\mathbb{A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ induces a Nisnevich local level weak equivalence of $S^1$-spectra $$M_{fr}(X\times (\mathbb{A}^1// \mathbb G_m)^{\wedge n})\to M_{fr}(X\times T^n).$$ Moreover, it is proven that the sequence of $S^1$-spectra $$M_{fr}(X \times T^n \times \mathbb G_m) \to M_{fr}(X \times T^n \times\mathbb A^1) \to M_{fr}(X \times T^{n+1})$$ is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of [GP1]. This computation is crucial for the whole machinery of framed motives [GP1]., Comment: This is the final revised version

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0 or 1 , by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m=5,7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1, m as sum of row terms of corresponding triangle. This version might be out of date, proceed by the link to see actual version., 16 pages, 8 figures, 2 tables, typos revised, added missing references, results generalized and shortened, derivations detailed.

Let k be an infinite perfect field and let X = SpecR, where R is the localization of a k-algebra B of finite type with respect to a regular prime ideal p ∈ SpecB. LetD be a divisor onX with a decompositionD = ∪i=1Di into its irreducible components Di such that each Di is nonsingular and they intersect transversally. Consider Y = X − D, a deleted neighbourhood of the closed point x0 of X . Let E be a vector bundle on Y . In this set up, E is trivial in the following cases : (i) dim(X) ≤ 3 (Gabber [Ga1]) ; (ii) t = 1 (Bhatwadekar-Rao [BR]) ; (iii) rankE ≥ min{[dimR 2 ], t} (Rao [Ra1]). The above exposition is borrowed from a paper of Nisnevich [Ni], which contains some beautiful applications of these ‘purity theorems’. In the same set up, now let y ∈ Y be a closed point and let my be the corresponding maximal ideal of OY,y. It is natural to ask whether my is a complete intersection. Observing that the conormal bundle my/my is trivial, we consider a more general situation : Let I be the defining ideal of a closed subscheme Z of Y such that the conormal bundle I/I2 (defined on Z) is trivial. Is I a complete intersection? The questions we are addressing in this paper are a little stronger in nature : Can a given basis of I/I2 be lifted to a minimal set of generators of I? We remark that with the hypothesis on X , each component Di is defined by a single equation fi which vanishes at the closed point x0 of X . Let m be the maximal ideal at x0. The condition of transversality of intersection of Di’s is equivalent to saying that the images of f1, · · · , ft in the A/m-vector space m/m2 are linearly independent. Keeping this discussion in mind, we now give an excerpt of our main results in commutative algebraic terms (see 4.2, 4.3, 4.4).

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., 20 pages

We consider connectivity properties of certain i.i.d. random environments on i¾?d, where at each location some steps may not be available. Site percolation and oriented percolation are examples of such environments. In these models, one of the quantities most often studied is the random set of vertices that can be reached from the origin by following a connected path. More generally, for the models we consider, multiple different types of connectivity are of interest, including: the set of vertices that can be reached from the origin; the set of vertices from which the origin can be reached; the intersection of the two. As with percolation models, many of the models we consider admit, or are expected to admit phase transitions. Among the main results of the paper is a proof of the existence of phase transitions for some two-dimensional models that are non-monotone in their underlying parameter, and an improved bound on the critical value for oriented site percolation on the triangular lattice. The connectivity of the random directed graphs provides a foundation for understanding the asymptotic properties of random walks in these random environments, which we study in a second paper. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 111-137, 2014

The object of this short note is to give some observations on Bernoulli numbers and their function field analogs and point out ‘known’ counter-examples to a conjecture of Chowla. Bernoulli numbers Bn defined (for integer n > 1) by z/(e z − 1) = ∑ Bnz /n!, and their important cousins Bn/n, play interesting roles in many areas of mathematics. (Below we only restrict to these for n even, precisely the case when they are non-zero.) We mention some key words by which the reader can search: Power sums, Zeta special values, Eisenstein series, measures, p-adic L-functions, finite differences, combinatorics, Euler-Maclaurin formula, Todd classes in topology, Grothendieck-Hirzebruch-Riemann-Roch formula, K-theory of integers, Stable homotopy, Bhargava factorial associated to the set of primes, Kummer-HerbrandRibet theorems in cyclotomic theory, Kervaire-Milnor formula for diffeomorphism classes of exotic spheres. Their factorization is of interest, the denominators (which show up explicitly in the third-fourth items from the end) are well understood via theorems of von-Staudt, but the numerators (which show up explicitly in the last two items above) are mysterious and connected to many interesting phenomena. In one of the rare lapses, Ramanujan, in his very first paper [R1911, (14), (18) and Sec. 12], claimed to have proved (editors downgrade it to a conjecture) that the numerator Nn of Bn/n is always a prime, when it was already known since Kummer (in Fermat’s last theorem connection) that ‘irregular’ prime 37 is a proper divisor of N32, and even N20 is composite. In [C1930], Chowla showed that Ramanujan’s claim had infinity of counter-examples. Note that this also follows from one counterexample and the Kummer congruences (recalled below) for that prime! Interestingly, in his last paper [CC1986], Chowla (jointly with his daughter) asks as unsolved problem whether the numerator is always square-free. (This is also mentioned in the nice survey article by Murtys and Williams on Chowla’s work in Vol. 1 of [C1999], where the author learned about it.) Theorem 1. Chowla’s conjecture stated above has infinity of counter-examples. In fact, for any given irregular prime p less than 163 million, and given arbitrarily large k, there is n such that p divides Nn. Proof. Using the tables (or the reader can try to check directly!) giving factorizations of Bn/n, for example the table by Wagstaff at the Bernoulli web page www.bernoulli.org, we see that 37 divides N284. Now recall the well-known Kummer congruences that the value of (1 − p)Bn/n modulo p depends only on (even) n modulo pk−1(p − 1), for n not divisible by p − 1. The first claim follows by taking p = 37. Supported in part by NSA grant H98230-10-1-0200.

Let G =(V,E)beagraphwithchromaticnumber χ(G). Adominating set D of G is called a chromatic transversal dominating set(ctd-set) if D intersects every color class of every χ-partitionof G. Theminimumcardinalityofactd-setofG iscalledthechromatictransversaldomination number of G and is denoted by γ ct (G). In this paper wecharacterizetheclassoftrees,unicyclicgraphsandcubicgraphsforwhichthe chromatic transversal domination number is equal to the connecteddominationnumber. 1. IntroductionAll the graphs considered in this paper unless otherwise specifically statedare finite, connected and simple and are consistent with the terminology usedin Harary [4]. Let G = (V,E) be a simple graph of order p. For a subset Sof V, N(S) denotes the set of all vertices adjacent to some vertex in S andN[S] = N(S) ∪S.A vertex v of G is called a support if it is adjacent to a pendant vertex. Anyvertex of degree greater than one is called an internal vertex. A graph G iscalled a unicyclic graph, if G contains exactly one cycle.A subset D ⊆ V is a dominating set, if every v ∈ V − D is adjacent tosome u ∈ D. The domination number γ = γ(G) is the minimum cardinality ofa dominating set of G. A dominating set D is called a connected dominatingset if the induced subgraph hDi is connected. The minimum cardinality of aconnected dominating set is called the connected domination number and isdenoted by γ

In this paper we introduce the notion of P-strongly regu-lar near-ring. We have shown that a zero-symmetric near-ring N is P-strongly regular if and only if N is P-regular and P is a completelysemiprime ideal. We have also shown that in a P-strongly regular near-ring N, the following holds: (i) Na+ P is an ideal of N for any a∈N.(ii) Every P-prime ideal of N containing P is maximal. (iii) Every idealI of N fulfills I+ P = I 2 + P. 1. IntroductionThroughout this paper, N denotes a zero-symmetric right near-ring. Aright N-subgroup (left N-subgroup) of N is a subgroup I of (N,+) such thatIN ⊆ I(NI ⊆ I). A quasi-ideal of N is a subgroup Q of (N,+) such thatQN ∩ NQ ⊆ Q. Right N-subgroups and left N-subgroups are quasi-ideals.The intersection of a family of quasi-ideals is again a quasi-ideal.Nis called regular, if for every element aof Nthere exists an element x∈ Nsuch that a= axa. Let P be an ideal of N. Then the near-ring N is said tobe a P-regular near-ring if for each a ∈ N, there exists an element x ∈ Nsuch that a= axa+ p for some p∈ P. If P = 0, then a P-regular near-ringis a regular near-ring. Here the notion of P-regularity is a generalization ofregularity. There are near-rings which are P-regular but not regular.V. A. Andrunakievich [1] defined P-regular rings and S. J. Choi [3] extendedthe P-regularity of a ring to the P-regularity of a near-ring. In this paper weintroduce the notion of P-strongly regular near-ring and obtain equivalent con-ditions for a near-ring to be P-strongly regular. We also introduce the notionsof P-prime ideals and P-near-ring in this paper. I. Yakabe [7] characterizedregular zero-symmetricnear-rings without non-zero nilpotent elements in termsof quasi-ideals. In this paper we characterize P-strongly regular near-ring interms of quasi-ideals. For the basic terminology and notation we refer to [6].

The Erdos-Pósa theorem (1965) states that in each graph G which contains at most k disjoint cycles, there is a 'blocking' set B of at most f(k) vertices such that the graph G - B is acyclic. Robertson and Seymour (1986) give an extension concerning any minor-closed class \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} of graphs, as long as \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} does not contain all planar graphs: in each graph G which contains at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, there is a set B of at most g(k) vertices such that G - B is in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}. In an earlier paper (Kurauskas and McDiarmid, Combin, Probab Comput 20 (2011) 763-775), we showed that, amongst all graphs on vertex set [n] = {1,...,n} which contain at most k disjoint cycles, all but an exponentially small proportion contain a blocking set of just k vertices. In the present paper we build on the previous work, and give an extension concerning any minor-closed graph class \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} with 2-connected excluded minors, as long as \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} does not contain all fans (here a 'fan' is a graph consisting of a path together with a vertex joined to each vertex on the path). We show that amongst all graphs G on [n] which contain at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, all but an exponentially small proportion contain a set B of k vertices such that G - B is in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}. (This is not the case when \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} contains all fans.) For a random graph R sampled uniformly from the graphs on [n] with at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, we consider also vertex degrees and the uniqueness of small blockers, the clique number and chromatic number, and the probability of being connected. © 2012 Wiley Periodicals, Inc.

In this paper we present an estimation for the diameter of random subgraph of a hypercube. In the article by A. V. Kostochka (Random Struct Algorithms 4 (1993) 215–229) the authors obtained lower and upper bound for the diameter. According to their work, the inequalities n + mp ≤ D(Gn) ≤ n + mp + 8 almost surely hold as n → ∞, where n is dimension of the hypercube and mp depends only on sampling probabilities. It is not clear from their work, whether the values of the diameter are really distributed on these 9 values, or whether the inequality can be sharpened. In this paper we introduce several new ideas, using which we are able to obtain an exact result: D(Gn) = n + mp (almost surely). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.

A sharp asymptotic formula for the number of strongly connected digraphs on n labelled vertices with m arcs, under the condition m - n ≫ n2/3, m = O(n), is obtained; this provides a partial solution of a problem posed by Wright back in 1977. This formula is a counterpart of a classic asymptotic formula, due to Bender, Canfield and McKay, for the total number of connected undirected graphs on n vertices with m edges. A key ingredient of their proof was a recurrence equation for the connected graphs count due to Wright. No analogue of Wright's recurrence seems to exist for digraphs. In a previous paper with Nick Wormald we rederived the BCM formula by counting first connected graphs among the graphs of minimum degree 2, at least. In this paper, using a similar embedding for directed graphs, we find an asymptotic formula, which includes an explicit error term, for the fraction of strongly-connected digraphs with parameters m and n among all such digraphs with positive in/out-degrees. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x, \mathcal{P})$ is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matom\"aki and their main conjecture is proved in this paper. In particular our main theorem implies that, if not too many large primes are sieved out in the sense that \[ \sum_{\substack{p \in \mathcal{P} \\ x^{1/v} < p \leq x^{1/u}}} \frac{1}{p} \geq \frac{1 + \varepsilon}{u}, \] for some $\varepsilon > 0$ and $v \geq u \geq 1$, then \[ \Psi(x, \mathcal{P}) \gg_{\varepsilon, v} x \prod_{\substack{p \leq x\\ p \notin\mathcal{P}}} \left(1 - \frac{1}{p}\right). \], Comment: 23 pages

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-$k$. We treat the other case, that is when Breaker plays randomly, in a separate paper. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkably precise agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias becomes significantly more tilted towards the random player. An important qualitative aspect of the probabilistic intuition carries through nevertheless: the bottleneck for Maker to occupy a connected graph is still the ability to avoid isolated vertices in her graph., This paper has been split in two parts, the second part can be found at http://arxiv.org/abs/1602.04628 . A full version from before the split is submission v2 of this arXiv entry, available at http://arxiv.org/abs/1507.06688v2

Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P}\subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G}/\mathbf{P}$ has been identified with an ind-variety of generalized flags in the paper "Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not. 2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we define a Schubert cell on $\mathbf{G}/\mathbf{P}$ as a $\mathbf{B}$-orbit on $\mathbf{G}/\mathbf{P}$, where $\mathbf{B}$ is any Borel ind-subgroup of $\mathbf{G}$ which intersects $\mathbf{P}$ in a maximal ind-torus. A significant difference with the finite-dimensional case is that in general $\mathbf{B}$ is not conjugate to an ind-subgroup of $\mathbf{P}$, whence $\mathbf{G}/\mathbf{P}$ admits many non-conjugate Schubert decompositions. We study the basic properties of the Schubert cells, proving in particular that they are usual finite-dimensional cells or are isomorphic to affine ind-spaces. We then define Schubert ind-varieties as closures of Schubert cells and study the smoothness of Schubert ind-varieties. Our approach to Schubert ind-varieties differs from an earlier approach by H. Salmasian in "Direct limits of Schubert varieties and global sections of line bundles" (J. Algebra 320 (2008), 3187--3198)., Keywords: Classical ind-group, Bruhat decomposition, Schubert decomposition, generalized flag, homogeneous ind-variety. [26 pages]

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 1969] says that given an integer $n \geq 1$, if the entrywise application of a smooth function $f : (0,\infty) \to \mathbb{R}$ preserves the set of $n \times n$ positive semidefinite matrices with positive entries, then $f$ and its first $n-1$ derivatives are non-negative on $(0,\infty)$. In a recent joint work with Belton-Guillot-Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg-Rudin characterization of dimension-free positivity preservers [Duke Math. J. 1942, 1959]. In recent works with Belton-Guillot-Putinar [Adv. Math. 2016] and with Tao [Amer. J. Math., in press] we used local, real-analytic versions at the origin of the Horn-Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first $n$ nonzero Taylor coefficients at individual points rather than on all of $(0,\infty)$. A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy's (1841) determinant (whose matrix entries are geometric series $1 / (1 - u_j v_k)$), as well as of a determinant of Frobenius [J. reine angew. Math. 1882] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings., Comment: Final version, to appear in Transactions of the American Mathematical Society. 19 pages, no figures

We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty, \] where $k_n$ is the degree excess sequence defined by \[ k_n = \sum_{v \in B_n} (\mbox{deg}\, v - 6) \] for combinatorial balls $B_n$ of radius $n$ and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)+\sum_{j=0}^{n} (k_j +6)} = \infty. \] These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998., Comment: 45 pages, 19 figures; to appear in TAMS

Lately, the following problem attracted a lot of attention in various contexts: find the shortest factorization G = UU - UU - …U ± of a Chevalley group G = G(Φ, R) in terms of the unipotent radical U = U(Φ, R) of the standard Borel subgroup B = B(Φ, R) and the unipotent radical U - = U -(Φ, R) of the opposite Borel subgroup B - = B - (Φ, R). So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits the unitriangular factorization G = UU - UU - U of length 5. Their proof invokes deep analytic and combinatorial tools. In the present paper, we notice that from the work of Bass and Tavgen one immediately gets a much more general results, asserting that over any ring of stable rank 1 one has the unitriangular factorization G = UU - UU - of length 4. Moreover, we give a detailed survey of traingular factorizations, prove some related results, discuss prospects of generalization to other classes of rings, and state several unsolved problems. Another main result of the present paper asserts that, in the assumption of the Generalized Riemann’s Hypothesis, Chevalley groups over the ring $$ \mathbb{Z}\left[ {\frac{1}{p}} \right] $$ admit the unitriangular factorization G = UU - UU - UU - of length 6. Otherwise, the best length estimate for Hasse domains with infinte multiplicative groups that follows from the work of Cooke and Weinberger, gives 9 factors. Bibliography: 67 titles.

In a previous paper, the author proved that all odd cycles, except ve cycles, are highly Ramsey-innite. In this paper, we ll in the missing case, and show that ve cycles are highly Ramsey-innite.

The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian, or Q-matrix. In our previous paper [1] we gave three lower and three upper bounds for the Q-index of nested split graphs, also known as threshold graphs. In this paper, we give another two upper bounds, which are expressed as solutions of cubic equations (in contrast to quadratics from [1]). Some computational results are also included.

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be \(O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)\). This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.

In this paper we investigate the Lipschitz equivalence of dust-like self-similar sets in Rd. One of the fundamental results by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223– 233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. In this paper we extend the study by examining deeper such connections. A key ingredient of our study is the introduction of a new equivalent relation between two dust-like self-similar sets called matchable condition. Thanks to a certain measure-preserving property of bi-Lipschitz maps between dust-like self-similar sets, we show that the matchable condition is a necessary condition for Lipschitz equivalence. Using the matchable condition we prove several conditions on the Lipschitz equivalence of dust-like self-similar sets based on the algebraic properties of the contraction ratios, which include a complete characterization of Lipschitz equivalence when the multiplication groups generated by the contraction ratios have full rank. We also completely characterize the Lipschitz equivalence of dust-like self-similar sets with two branches (i.e. they are generated by IFS with two contractive similarities). Some other results are also presented, including a complete characterization of Lipschitz equivalence when one of the self-similar sets has uniform contraction ratio.

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

The aim of this paper is to establish the well-known and veryuseful classical Saalschutz’s theorem for the series 3 F 2 (1) by following afft method. In addition to this, two summation formulas closelyrelated to the Saalschutz’s theorem have also been obtained. The resultsestablished in this paper are further utilized to show how one can obtaincertain known and useful hypergeometric identities for the series 3 F 2 (1)and 4 F 3 (1) already available in the literature. 1. Introduction and results requiredWe start with the following well-known and useful classical Saalschutz’stheorem [4, p. 87, Section 51] for the series 3 F 2 (1). If n is a non-negativeinteger and if a , b , c are independent of n ,(1.1) 3 F 2 [ n; a; bc; 1+ a + b c n ; 1]=( c a ) n ( c b ) n ( c ) n ( c a b ) n : As mentioned in almost all the standard books on generalized hypergeomet-ric series that this theorem can be established with the help of the followingEuler’s transformation formula [4, p. 60, Eq.(5)]. If

In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bm\theta\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bm\theta)= 1/n$ for almost every point $\vv x$ on $M$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent $w_0(\vv x)=1/n$ for almost all $\vv x\in \cM$ if and only if for any $\bm\theta\in\R^n$ the inhomogeneous exponent $w_0(\vv x,\bm\theta)=1/n$ for almost all $\vv x\in \cM$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered in \cite{Beresnevich-Velani-new-inhom}. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas of \cite{Beresnevich-Velani-new-inhom} while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory., Comment: Dedicated to A.O. Gelfond on what would have been his 100th birthday 13 pages

We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type. Generalized Harish-Chandra modules are $(\mathfrak{g}, \mathfrak{k})$-modules of finite type where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k} \subset \mathfrak{g}$ is a reductive in $\mathfrak{g}$ subalgebra. A first result of the present paper is that a fundamental series module is a $\mathfrak{g}$-module of finite length. We then define the notions of strongly and weakly reconstructible simple $(\mathfrak{g}, \mathfrak{k})$-modules $M$ which reflect to what extent $M$ can be determined via its appearance in the socle of a fundamental series module. In the second part of the paper we concentrate on the case $\mathfrak{k} \simeq sl(2)$ and prove a sufficient condition for strong reconstructibility. This strengthens our main result from Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type for the case $\mathfrak{k} = sl(2)$. We also compute the $sl(2)$-characters of all simple strongly reconstructible (and some weakly reconstructible) $(\mathfrak{g}, sl(2))$-modules. We conclude the paper by discussing a functor between a generalization of the category $\mathcal{O}$ and a category of $(\mathfrak{g}, sl(2))$-modules, and we conjecture that this functor is an equivalence of categories.

In this paper we shall present discrete Kolmogorov type inequalities for multiply monotone sequences defined on non-positive integers. Moreover, we will provide a more delicate information by obtaining the description of the following modulus of continuity ω j,r p,q (δ ,e) = sup{‖∆ x‖q : x,∆x, . . . ,∆ j x ! 0, ‖x‖p = δ , ‖∆ x‖∞ = e} for δ ! e > 0 and values of j = r−2 or j = r−1 depending on values of other parameters. 1. Notation, definitions, and history Let M := Z− ∪ {0} = {· · · ,−2,−1,0} . We shall define the (forward) difference operator as follows: for the sequence x = {xm}m∈M set ∆x := {xm − xm−1}m∈M, ∆x := x, ∆x := ∆x, and, recursively, ∆ x := ∆(∆ )x for j = 2,3, . . . . Let lp = lp(M) , p ∈ [1,∞] , be the space of all real-valued sequences defined on M such that the norm ‖x‖p :=    ( ∑ m∈M |xm| p )1/p , 1 " p < ∞, sup m∈M |xm|, p = ∞, is finite. Note that in contrast with the derivative operator D = d dt , the difference operator ∆ is a bounded linear operator defined on all lp(M) for any 1 " p " ∞ . In this paper we shall consider sharp discrete inequalities of Kolmogorov type, i.e. inequalities of the form ‖∆x‖q "C‖x‖ α p‖∆ x‖ s , x ∈ lp(U), ∆ x ∈ ls(U) (1) where U = M or U = Z and α ∈ [0,1] is defined by relation (9) below. The exact constants C = C(k,r, p,q,s) in inequalities of this form have been found in the following cases: Mathematics subject classification (2010): 26D10, 47B39, 39A70, 65L12.

In the paper we introduce convexity-like notions based on modification of quasiconvexity. DEFINITION. Let I be a real interval and ω 0 a given number. We say that a function f : I → R is ω -quasiconvex, ω -quasiconcave, respectively, if f (tx+(1− t)y) max( f (x), f (y))−ωmin(t,1− t)|x− y|, f (tx+(1− t)y) max( f (x), f (y))−ωmax(t,1− t)|x− y|, for x,y ∈ I,t ∈ (0,1). If f : I → R is simultaneously ω -quasiconvex and ω -quasiconcave then we say that f is ω -quasiaffine. We characterize these notions, in particular we show that ω -quasiconcave functions coincide with Lipschitz functions with constant ω . We conclude the paper with the following separation type result. THEOREM. Let f : I → R be ω -quasiconvex function and g : I →R ω -quasiconcave such that f g . Then there exists an ω -quasiaffine function h : I → R such that f h g . Mathematics subject classification (2010): 26B25, 39B62.

A selfadjoint involutive matrix J endows C n with the indefinite Krein space struc- ture endowed by the inner product (·,·) given by (x,y) := � Jx,y� , x,y ∈ C n . For a pair of J−selfadjoint matrices A,B with positive eigenvalues, LogA J LogB is called the J−chaotic order or the indefinite chaotic order .S ano (8), proved as an application of Furuta inequality of indefinite type that LogA J LogB if and only if A r J A r 2 B p A r 2 r p+r for all p > 0a ndr > 0. In this paper, we prove Sano's result using a different approach. In the process, some other re- sults due to Bebiano, Lemos, Providencia and Soares (4, 9) are reobtained. The techniques in this paper are inspired by (5).