Locally Minimal Trees in n-Normed Spaces

The locally minimal trees in normed spaces $({\mathbb{R}}^2, \rho)$ , where the unit circle ${x \in {\mathbb{R}}^2 | \rho (x)=1}$ in the norm ρ coincides with the regular m-gon (m = 2n) inscribed in the Euclidean unit circle S ¹, are completely classified.     

Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality

This paper is a part of our general aim to study properties of elliptic and ordered elliptic geometries and then using some of these properties to introduce new concepts and develop their theories. If ${(P,\mathfrak{G}, \equiv,\tau)}$ denotes an elliptic geometry ordered via a separation ?? then there are polar points o and ?? and on the line ${ \overline{K} := \overline{\infty,o}}$ there can be established an operation ??+?? such that ${(\overline{K},+)}$ becomes a commutative group and the map ${ a^+ :\overline{K}\to \overline{K} ; x \mapsto a + x}$ is a motion on ${\overline{K}}$ . The separation ?? induces on ${\overline{K}}$ a cyclic order ?? with [o, e, ??] = 1 such that ${(\overline{K},+,\omega)}$ becomes a cyclic ordered group. For ${a,b \in K := \overline{K} {\setminus}\{\infty\}}$ we set ${a < b :\Longleftrightarrow [a,b,\infty] =1}$ and for all ${a\in K\,a < \infty}$ . Then (K,?<) is a totally ordered set. We show there is a surjective distance function: $$ \lambda : P \times P \to \overline{K}_+ := \{x \in \overline{K}\,|\,o \leq x\leq\infty\}, $$ with ?? ${\lambda(a,b) = \lambda(c,d) \ \Longleftrightarrow (a,b) \equiv (c,d)}$ ??. We prove in the first part of our project like (cf. Gr?ger in Mitt Math Ges Hamburg 11:441?C457, 1987) the following triangle-inequality: (cf. Theorem 8.2). If (a, b, c) is a triangle consisting of pairwise not polar points with ??(a, c), ??(b, c) < e then ??(a, b) ?? ??(a, c) + ??(b, c) < ??.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

Liouville Type Theorems for Elliptic Equations with Gradient Terms

In this paper we obtain Liouville type theorems for nonnegative supersolutions of the elliptic problem ${-\Delta u + b(x)|\nabla u| = c(x)u}$ in exterior domains of ${\mathbb{R}^N}$ . We show that if lim ${{\rm inf}_{x \longrightarrow \infty} 4c(x) - b(x)^2 > 0}$ then no positive supersolutions can exist, provided the coefficients b and c verify a further restriction related to the fundamental solutions of the homogeneous problem. The weights b and c are allowed to be unbounded. As an application, we also consider supersolutions to the problems ${-\Delta u + b|x|^{\lambda}|{\nabla} u| = c|x|^{\mu} u^p}$ and ${-\Delta u + be^{\lambda |x|}|\nabla u| = ce^{\mu |x|}u^p}$ , where p > 0 and λ, μ ≥ 0, and obtain nonexistence results which are shown to be optimal.     

The formal translation equation for iteration groups of type II

We investigate the translation equation $$F(s+t, x) = F(s, F(t, x)),\quad \quad s,t\in{\mathbb{C}},\qquad\qquad\qquad\qquad({\rm T})$$ in ${\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ , the ring of formal power series over ${\mathbb{C}}$ . Here we restrict ourselves to iteration groups of type II, i.e. to solutions of (T) of the form ${F(s, x) \equiv x + c_k(s)x^k {\rm mod} x^{k + 1}}$ , where k ≥ 2 and c _k ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions c _n (s) of $$F(s, x) = x + \sum_{n \ge q k}c_n(s)x^n$$ are polynomials in c _k (s). It is possible to replace this additive function c _k by an indeterminate. In this way we obtain a formal version of the translation equation in the ring ${(\mathbb{C}[y])\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ . We solve this equation in a completely algebraic way, by deriving formal differential equations or an Aczél–Jabotinsky type equation. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character (depending on certain parameters, the coefficients of the infinitesimal generator H of an iteration group of type II) of these polynomials. Rewriting the solutions G(y, x) of the formal translation equation in the form ${\sum_{n\geq 0}\phi_n(x)y^n}$ as elements of ${(\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right])\left[\kern-0.15em\left[{y}\right]\kern-0.15em\right]}$ , we obtain explicit formulas for ${\phi_n}$ in terms of the derivatives H ^(j)(x) of the generator ${H}$ and also a representation of ${G(y, x)}$ as a Lie–Gröbner series. Eventually, we deduce the canonical form (with respect to conjugation) of the infinitesimal generator ${H}$ as x ^k + hx ^2k-1 and find expansions of the solutions ${G(y, x) = \sum_{r\geq 0} G_r(y, x)h^r}$ of the above mentioned differential equations in powers of the parameter h.     

Subexponential loss rates in a GI/GI/1 queue with applications

Consider a single server queue with i.i.d. arrival and service processes, $\{ A,A_n ,n \geqslant 0\} $ and $\{ C,\;C_n ,n\;\; \geqslant \;\;0\} $ , respectively, and a finite buffer B. The queue content process $\{ Q_n^B ,n \geqslant 0\} $ is recursively defined as $Q_{n + 1}^B = \min ((Q_n^B + A_{n + 1} - C_{n + 1} )^ + ,B),\;\;q^ + = \max (0,q)$ . When $\mathbb{E}(A - C) < 0$ , and A has a subexponential distribution, we show that the stationary expected loss rate for this queue $E(Q_n^B + A_{n + 1} - C_{n + 1} - B)^ + $ has the following explicit asymptotic characterization: $${\mathbb{E}}\left( {Q_n^B + A_{n + 1} - C_{n + 1} - B} \right)^ + ~{\mathbb{E}}\left( {A - B} \right)^ + {as} B \to \infty ,$$ independently of the server process C _n . For a fluid queue with capacity c, M/G/∞ arrival process A _t , characterized by intermediately regularly varying on periods σ^on, which arrive with Poisson rate Λ, the average loss rate $\lambda _{{loss}}^B $ satisfies λ _loss ^B ～ Λ E(τ^onη — B)⁺ as B → ∞, where $\eta = r + \rho - c,\;\rho \; = \mathbb{E}A_t < \;\;c;r\;\;(c \leqslant r)$ is the rate at which the fluid is arriving during an on period. Accuracy of the above asymptotic relations is verified with extensive numerical and simulation experiments. These explicit formulas have potential application in designing communication networks that will carry traffic with long-tailed characteristics, e.g., Internet data services.     

Harmonic divisors and rationality of zeros of Jacobi polynomials

Let $P_{n}^{ ( \alpha,\beta ) } ( x ) $ be the Jacobi polynomial of degree n with parameters α,β. The main result of the paper states the following: If b≠1,3 and c are non-zero relatively prime natural numbers then $P_{n}^{ ( k+ ( d-3 ) /2,k+ ( d-3 ) /2 ) } ( \sqrt{b/c} ) \neq0$ for all natural numbers d,n and $k\in\mathbb{N}_{0}$ . Moreover, under the above assumption, the polynomial $Q ( x ) = \frac{b}{c} ( x_{1}^{2}+\cdots+x_{d-1}^{2} ) + ( \frac{b}{c}-1 ) x_{d}^{2}$ is not a harmonic divisor, and the Dirichlet problem for the cone {Q(x)<0} has polynomial harmonic solutions for polynomial data functions.     

On the complexity of the closed fragment of Japaridze’s provability logic

We consider the well-known provability logic GLP. We prove that the GLP-provability problem for polymodal formulas without variables is PSPACE-complete. For a number n, let \({L^{n}_0}\) denote the class of all polymodal variable-free formulas without modalities \({\langle n \rangle,\langle n+1\rangle,...}\) . We show that, for every number n, the GLP-provability problem for formulas from \({L^{n}_0}\) is in PTIME.     

The geometric meaning of Zhelobenko operators

Let $ \mathfrak{g} $ be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, $ \mathfrak{b} $ a Borel subalgebra of $ \mathfrak{g} $ , $ \mathfrak{h}\subset \mathfrak{b} $ the Cartan sublagebra, and N ? G the unipotent subgroup corresponding to the nilradical $ \mathfrak{n}\subset \mathfrak{b} $ . We show that the explicit formula for the extremal projection operator for $ \mathfrak{g} $ obtained by Asherova, Smirnov, and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence $ N\times \mathfrak{h}\to \mathfrak{b} $ given by the restriction of the adjoint action. Simple geometric proofs of formulas for the “classical” counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.     

Continuous horizontally rigid functions of two variables are affine

Cain, Clark and Rose defined a function ${{f\colon\mathbb{R}^n \to \mathbb{R}}}$ to be vertically rigid if graph(cf) is isometric to graph(f) for every c ≠?0. It is horizontally rigid if graph ${(f(c \vec{x}))}$ is isometric to graph(f) for every c ≠?0 (see Cain et?al., Real Anal. Exch. 31:515–518, 2005/2006). In Balka and Elekes (J. Math. Anal. Appl. 345:880–888, 2008) the authors of the present paper settled Jankovi?’s conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form a?+?bx or ${{a+be^{kx} (a,b,k \in \mathbb{R})}}$ . Later they proved in Balka and Elekes (Real. Anal. Exch. 35:139–156, 2009) that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form ${a + bx + dy, a +s(y)e^{kx}}$ or ${{a + be^{kx} + dy (a,b,d,k \in \mathbb{R}, k \neq 0, s\colon \mathbb{R}\to \mathbb{R}\,{\rm is\, continuous})}}$ . The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. Richter (Real Anal. Exch. 35:343–354, 2009) proved that a continuous function of one variable is horizontally rigid if and only if it is of the form ${{a+bx (a,b\in \mathbb{R})}}$ . The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form ${{a+ bx + dy (a,b,d \in \mathbb{R})}}$ . This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations.     

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

On weak solutions of linear coercive integral equations in spacesL 2(X; ℋ k )

Пусть (X,A, μ) - полное про странство с σ-конечно й мерой, и пусть \(\overline {\mu \times \mu } \) . - замык ание меры μ×μ. Пусть далееg: X×X→C - квадратично интегрируемая функц ия по мере \(\overline {\mu \times \mu } \) . Рассматривается лин ейное интегральное у равнение (слабого) типа (1) (1) $$u(t) + A(\mathop \smallint \limits_x g(t,s)u(s)d\mu ) = f(t)\Pi .B.B\,X,$$ гдеА - максимальное р асширение L _k ^′ (в простр анстве ХëрмандераH ₁=B₂к) соотв ествующего линейного (псевдодиф ференциального) опер атораL: S→S; иS обозначает класс Щварца функций Rⁿ→-C. Уст анавливается сущест вование (слабых) решений (1) при н екотором условии коэрпитивно сти на оператор (2) (2) $$(L\Psi )(t) = \Psi (t) + \int\limits_x {g(t,s)L(\Psi (s))d\mu ,} $$ где Ψ принадлежит про странстувуD(Х, S) всех конечно-значных функ ций изX→S. Далее, изучается обобщенна я обратимость максим ального расширения оператора L. Наконец, пр иводится некоторое алгебраическое усло вие, обеспечивающее коэрцитивность L.     

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

From Analogical Proportion to Logical Proportions

Given a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators ( ${a \wedge b}$ and ${\overline{a} \wedge \overline{b}}$ ), or dissimilarity indicators ( ${a \wedge \overline{b}}$ and ${\overline{a} \wedge b}$ ) pertaining to the pair (a, b), to the ones associated with the pair (c, d). There are 120 semantically distinct logical proportions. One of them models the analogical proportion which corresponds to a statement of the form “a is to b as c is to d”. The paper inventories the whole set of logical proportions by dividing it into five subfamilies according to what they express, and then identifies the proportions that satisfy noticeable properties such as full identity (the pair of equivalences defining the proportion hold as true for the 4-tuple (a, a, a, a)), symmetry (if the proportion holds for (a, b, c, d), it also holds for (c, d, a, b)), or code independency (if the proportion holds for (a, b, c, d), it also holds for their negations ${{(\overline{a},\overline{b}, \overline{c}, \overline{d})}}$ ). It appears that only four proportions (including analogical proportion) are homogeneous in the sense that they use only one type of indicator (either similarity or dissimilarity) in their definition. Due to their specific patterns, they have a particular cognitive appeal, and as such are studied in greater details. Finally, the paper provides a discussion of the other existing works on analogical proportions.     

An interstice relationship for flowers with four petals

Given three mutually tangent circles with bends (related to the reciprocal of the radius) a, b and c respectively, an important quantity associated with the triple is the value ${\langle a,b,c \rangle:=ab+ac+bc}$ . In this note we show in the case when a central circle with bend b ₀ is “surrounded” by four circles, i.e., a flower with four petals, with bends b ₁, b ₂, b ₃,b ₄ that either $$\sqrt{\langle b_{0},b_{1},b_{2} \rangle}+\sqrt{\langle b_{0},b_{3},b_{4} \rangle}=\sqrt{\langle b_{0},b_{2},b_{3} \rangle}+\sqrt{\langle b_{0},b_{4},b_{1} \rangle}$$ or $$\sqrt{\langle b_{0},b_{1},b_{2} \rangle}=\sqrt{\langle b_{0},b_{2},b_{3} \rangle}+\sqrt{\langle b_{0},b_{3},b_{4} \rangle}+\sqrt{\langle b_{0},b_{4},b_{1} \rangle}$$ (where ${\langle b_{0},b_{1},b_{2} \rangle}$ is chosen to be maximal). As an application we give a sufficient condition for the alternating sum of the ${\sqrt{\langle a,b,c\rangle}}$ of a packing in standard position to be 0. (A packing is in standard position when we have two circles with bend 0, i.e., parallel lines, and the remaining circles are packed in between.)     

An estimate for the sum of Legendre symbols

For the sum S of the Legendre symbols of a polynomial of odd degree n ≥ 3 modulo primes p ≥ 3, Weil’s estimate |S| ≤ (n ? 1) $ \sqrt p $ and Korobov’s estimate $$ \left| S \right| \leqslant (n - 1)\sqrt {p - \frac{{(n - 3)(n - 4)}} {4}} forp \geqslant \frac{{n^2 + 9}} {2} $$ are well known. In this paper, we prove a stronger estimate, namely, $$ \left| S \right| < (n - 1)\sqrt {p - \frac{{(n - 3)(n + 1)}} {4}} $$ .     

On the properties of the strong Cesàro summability and strong convergence of series

Говорят, что ряд \(\mathop \sum \limits_{k = 0}^\infty a_k \) сумм ируется к s в смысле (С, gа), gа >?1, если $$\sigma _n^{(k)} - s = o(1),n \to \infty ,$$ в смысле [C,α] _λ , α<0, λ>0, если $$\frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n \left| {\sigma _k^{(\alpha - 1)} - s} \right|^\lambda = o(1),n \to \infty ,$$ и в смысле [C,0] _λ , λ>0, если $$\frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n \left| {(k + 1)(s_k - 1) - k(s_{k - 1} - 1)} \right|^\lambda = o(1),n \to \infty ,$$ где σ _n ^(α) обозначаетn-ое ч езаровское среднее р яда. Суммируемость [C,α] _λ , α>?1, λ ≧1 о значает, что $$\mathop \sum \limits_{k = 0}^\infty k^{\lambda - 1} \left| {\sigma _k^{(\alpha )} - \sigma _{k - 1}^{(\alpha )} } \right|^\lambda< \infty .$$ В данной статье содер жится продолжение ис следований свойств [C,α] _λ -суммиру емо сти, которые начали Винн, Х ислоп, Флетт, Танович-М иллер и автор, в частности свя зей между указанными методами суммирования. Наконец, даны некотор ые простые приложени я к вопросам суммируемости ортог ональных рядов.     

Strengthening of strong and approximate convexity?

Let X be a real linear space and $D\subseteq X$ be a nonempty convex subset. Given an error function E:[0,1]×(D?D)?????{+??} and an element $t\in\left]0,1\right[$ , a function f:D??? is called (E,t)-convex if $$f(tx+(1-t)y)\le tf(x)+(1-t)f(y)+E(t,x-y)$$ for all x,y??D. The main result of this paper states that, for all a,b??(???{0})+{0,t,1?t} such that {a,b,a+b}??????, every (E,t)-convex function is also $\big(F,\frac{a}{a+b}\big)$ -convex, where $$F(s,u):=\frac{{(a+b)}^2s(1-s)}{t(1-t)}E\left(t,\frac{u}{a+b}\right),\qquad (u\in (D-D), \, s\in\left]0,1\right[).$$ As a consequence, under further assumptions on E, the strong and approximate convexity properties of (E,t)-convex functions can be strengthened.     

Extremal 1-codes in distance-regular graphs of diameter 3

We study 1-codes in distance-regular graphs of diameter 3 that achieve three different bounds. We show that the intersection array of a distance-regular graph containing such a code has the form $${\{a(p+1), cp, a+1; 1, c, a p\}\quad{\rm or}\quad\{a(p+1), (a+1)p,c; 1, c, a p\}}$$ for c =? c ₂,?a =? a ₃ and ${p = p_{33}^3}$ . These two families contain 10?+?15 known feasible intersection arrays out of which four are uniquely realized by the Sylvester graph, the Hamming graph H(3, 3), the Doro graph and the Johnson graph J(9, 3), but not all members of these two families are feasible. We construct four new one-parameter infinite subfamilies of feasible intersection arrays, two of which have a nontrivial vanishing Krein parameter: $${\{(2r^2-1)(2r+1), 4r(r^2-1), 2r^2; 1, 2(r^2-1), r(4r^2-2)\}}$$ and $${\{2r^2(2r+1), (2r-1)(2r^2+r+1), 2r^2; 1, 2r^2, r(4r^2-1)\}}$$ for r > 1 (the second family actually generalizes to a two-parameter family with the same property). Using this information we calculate some triple intersection numbers for these two families to show that they must contain the desired code. Finally, we use some additional combinatorial arguments to prove nonexistence of distance-regular graphs with such intersection arrays.     

Asymptotics of series of a special form

We have found the asymptotic expansions of the series $$ \sum\limits_{k=0}^{\infty } {{{{\left( {-1} \right)}}^{{\beta k}}}Q\left( {\left( {{\delta^{\alpha }}-{{{\left( {ak+b} \right)}}^{\alpha }}} \right)+} \right)/{{{\left( {ak+b} \right)}}^{r+1 }}} $$ in powers of δ as δ → +∞, where β ∈ $ \mathbb{Z} $ , α, a, b > 0, r ∈ $ \mathbb{C} $ , and Q is an algebraic polynomial satisfying the condition Q(0) = 0. In particular cases, these series arise in the approximation of periodic differentiable functions by the Riesz means and the Cesáro means.