The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in the Generalized Complex Plane

In this study, we first calculate the polar moment of inertia of orbit curves under one-parameter planar motion in the generalized complex plane \({{\mathbb{C}_p}}\) and then give the Holditch-type theorem for \({{\mathbb{C}_p}}\): When the fixed points \({X}\) and \({Y}\) on the moving plane \({{\mathbb{K}_p} \subset {\mathbb{C}_p}}\) trace the same curve \({k}\) with the polar moment of inertia \({{T_X}}\), the different point \({Z}\) on this line segment \({XY}\) traces another curve \({{k_Z}}\) with the polar moment of inertia \({{T_Z}}\) during the one-parameter planar motion in the fixed plane \({{\mathbb{K}'_p} \subset {\mathbb{C}_p}}\). Thus, we obtain that the difference between the polar moments of inertia of these curves \({( {{T_Z} - {T_X}} )}\) depends on the only the \({p}\)-distances of this points and \({p}\)-rotation angle of the motion, \({{T_X} - {T_Z} = {\delta _p}ab.}\)     

Generalizations of an Expansion Formula for Top to Random Shuffles

In the top to random shuffle, the first \({a}\) cards are removed from a deck of \({n}\) cards \({12 \cdots n}\) and then inserted back into the deck. This action can be studied by treating the top to random shuffle as an element \({B_a}\), which we define formally in Section 2, of the algebra \({{\mathbb{Q}[S_n]}}\). For \({a = 1}\), Garsia in “On the powers of top to random shuffling” (2002) derived an expansion formula for \({{B^k_1}}\) for \({{k \leq n}}\), though his proof for the formula was non-bijective. We prove, bijectively, an expansion formula for the arbitrary finite product \({B_{a1} B_{a2} \cdots B_{ak}}\) where \({a_{1}, \cdots , a_{k}}\) are positive integers, from which an improved version of Garsia’s aforementioned formula follows. We show some applications of this formula for \({B_{a1} B_{a2} \cdots B_{ak}}\), which include enumeration and calculating probabilities. Then for an arbitrary group \({G}\) we define the group of \({G}\)-permutations \({{S^G_n} := {G \wr S_n}}\) and further generalize the aforementioned expansion formula to the algebra \({{\mathbb{Q} [ S^G_n ]}}\) for the case of finite \({G}\), and we show how other similar expansion formulae in \({{\mathbb{Q} [S_n]}}\) can be generalized to \({{\mathbb{Q} [S^G_n]}}\).     

The exact asymptotic behavior of boundary blow-up solutions to infinity Laplacian equations

In this paper, we study the asymptotic behavior of viscosity solutions to boundary blow-up elliptic problem \({\Delta_{\infty}u=b(x)f(u),\, x\in\Omega,\,u|_{\partial\Omega}=+\infty,}\) where \({\Omega}\) is a bounded domain with C²-boundary in \({\mathbb{R}^{N}}\), \({b\in \rm C(\bar{\Omega})}\) is positive in \({\Omega}\), which may be vanishing on the boundary, \({f\in C^{1}([0, \infty))}\) is regularly varying or is rapidly varying at infinity.     

On complete sequences

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence \({S=\{s_{1}, s_{2}, \dots\}}\) of positive integers and a positive real number α, let S _α denote the sequence \({\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}\), where \({\lfloor x \rfloor}\) denotes the greatest integer not greater than x. Let \({{U_S = \{\alpha \mid S_\alpha} \, is complete\}}\). Hegyvári [6] proved that if \({\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}\), \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma < 2}\), and \({U_{S}\ne\emptyset}\), then \({\mu(U_{S}) > 0}\), where \({\mu(U_{S})}\) is the Lebesgue measure of U _S . Yong-Gao Chen and the first author [4] proved that, if \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma \leqq 7/4=1.75}\), then \({\mu(U_{S}) > 0}\). In this paper, we prove that the conclusion holds for \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.     

Locally Lipschitz composition operators in spaces of functions of bounded variation

We give a necessary and sufficient condition on a function \({f:\mathbb{R}\to\mathbb{R}}\) under which the composition operator (Nemytskij operator) F defined by \({Fh=f\circ h}\) acts in the spaces \({BV_\varphi[a,b], HBV[a,b], {\rm and} \,RV_\varphi[a,b]}\) and satisfies a local Lipschitz condition. While the proof of sufficiency consists in a straightforward calculation, the proof of necessity builds on nontrivial arguments like Helly’s selection principle or the Arzelà–Ascoli compactness criterion.     

Abelian covers of alternating groups

Let \({C={\rm inf} (k/n)\sum_{i=1}^n x_i(x_{i+1}+\cdots+x_{i+k})^{-1}}\), where the infimum is taken over all pairs of integers \({n\geq k\geq 1}\) and all positive \({x_1,\ldots,x_n}\), \({x_{n+i}=x_i}\). We prove that \({\ln 2 \leq C < 0.9305}\). In the definition of the constant C, the operation \({{\rm inf}_{k}\, {\rm inf}_{n}\, {\rm inf}_{x}}\) can be replaced by \({{\rm lim}_{k \to \infty}\, {\rm lim}_{n \to \infty} {\rm inf}_{x}}\).     

Weak law of large numbers for linear processes

We establish sufficient conditions for the Marcinkiewicz–Zygmund type weak law of large numbers for a linear process \({\{X_k:k\in\mathbb Z\}}\) defined by \({X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j}}\) for \({k\in\mathbb Z}\), where \({\{\psi_j:j\in\mathbb Z\}\subset\mathbb R}\) and \({\{\varepsilon_k:k\in\mathbb Z\}}\) are independent and identically distributed random variables such that \({{x^p\Pr\{|\varepsilon_0| > x\}\to 0}}\) as \({{x\to \infty}}\) with \({1 < p < 2}\) and \({E \varepsilon_0=0}\). We use an abstract norming sequence that does not grow faster than \({n^{1/p}}\) if \({\sum|\psi_j| < \infty}\). If \({\sum|\psi_j|=\infty}\), the abstract norming sequence might grow faster than \({n^{1/p}}\) as we illustrate with an example. Also, we investigate the rate of convergence in the Marcinkiewicz–Zygmund type weak law of large numbers for the linear process.     

On Modular Edge-Graceful Graphs

Let G be a connected graph of order \({n\ge 3}\) and size m and \({f:E(G)\to \mathbb{Z}_n}\) an edge labeling of G. Define a vertex labeling \({f': V(G)\to \mathbb{Z}_n}\) by \({f'(v)= \sum_{u\in N(v)}f(uv)}\) where the sum is computed in \({\mathbb{Z}_n}\) . If f′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A graph G is modular edge-graceful if G contains a modular edge-graceful spanning tree. Several classes of modular edge-graceful trees are determined. For a tree T of order n where \({n\not\equiv 2 \pmod 4}\) , it is shown that if T contains at most two even vertices or the set of even vertices of T induces a path, then T is modular edge-graceful. It is also shown that every tree of order n where \({n\not\equiv 2\pmod 4}\) having diameter at most 5 is modular edge-graceful.     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

On stable Baire classes

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps \({f : X\to Y}\) of the class \({\alpha}\) for wide classes of topological spaces. In particular, we prove that for a topological space X and a contractible space Y a map \({f : X \to Y}\) belongs to the nth stable Baire class if and only if there exist a sequence \({(f_k)_{k=1}^\infty}\) of continuous maps \({f_k : {X \to Y}}\) and a sequence \({(F_k)_{k=1}^\infty}\) of functionally ambiguous sets of the nth class in X such that \({f|_{F_k}=f_k|_{F_k}}\) for every k. Moreover, we show that every monotone function \({f : \mathbb{R} \to \mathbb{R}}\) is of the \({\alpha}\) th stable Baire class if and only if it belongs to the first stable Baire class.     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Additive iterative roots of identity and Hamel bases

We describe a class of discontinuous additive functions \({a:X\to X}\) on a real topological vector space X such that \({a^n={\rm id}_X}\) and \({a({\mathcal{H}}){\setminus} {\mathcal{H}}\neq\emptyset}\) for every infinite set \({{\mathcal{H}}\subset X}\) of vectors linearly independent over \({\mathbb{Q}}\). We prove the density of the family of all such functions in the linear topological space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced on \({{\mathcal{A}}_X}\) by the Tychonoff topology of the space X^X. Moreover, we consider additive functions \({a\in{\mathcal{A}}_X}\) satisfying \({a^n={\rm id}_X}\) and \({a({\mathcal{H}})= {\mathcal{H}}}\) for some Hamel basis \({{\mathcal{H}}}\) of X. We show that the class of all such functions is also dense in \({{\mathcal{A}}_X}\). The method is based on decomposition theorems for linear endomorphisms.     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

Lower Bounds on the Graver Complexity of <Emphasis Type="Italic">M</Emphasis>-Fold Matrices

In this paper, we present a construction that turns certain relations on Graver basis elements of an M-fold matrix \({{A^{(M)}}}\) into relations on Graver basis elements of an \({(M+1)}\)-fold matrix \({{A^{(M+1)}}}\). In doing so, we strengthen the bound on the Graver complexity of the M-fold matrix \({{A_{3\times{M}}}}\) from \({{g(A_{3\times{M}}) \geq 17\cdot2^{M-3}-7}}\) (Berstein and Onn) to \({{g(A_{3\times{M}}) \geq 24\cdot2^{M-3}-21}}\), for \({M \geq 4}\). Moreover, we give a lower bound on the Graver complexity \({{g(A^{(M)})}}\) of general \({M}\)-fold matrices \({{A^{(M)}}}\) and we prove that the bound for \({g(A_{3\times{M}})}\) is not tight.     

Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

We consider the nonlinear curl-curl problem \({\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}\) in \({\mathbb{R}^3}\) related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of \({H({\rm curl};\mathbb{R}^3)}\). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and \({\Gamma}\) the variational problem can be posed in a symmetric subspace of \({H({\rm curl};\mathbb{R}^3)}\). For a defocusing case \({{\rm sup} \Gamma < 0}\) with large negative values of \({\Gamma}\) at infinity we obtain ground states by the direct minimization method. For the focusing case \({{\rm inf} \Gamma > 0}\) the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator \({\nabla \times \nabla \times +V(x)}\). Examples of cylindrically symmetric functions V are provided for which this holds.     

On polynomial congruences

We deal with functions which fulfil the condition \({\Delta_h^{n+1} \varphi(x)\in\mathbb{Z}}\) for all x, h taken from some linear space V. We derive necessary and sufficient conditions for such a function to be decent in the following sense: there exist functions \({f\colon V\rightarrow \mathbb{R},\ g\colon V \rightarrow \mathbb{Z}}\) such that \({\varphi = f + g}\) and \({\Delta_h^{n+1}f(x)=0}\) for all \({x, h\in V}\).     

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let \({L_{w}}{:=-w^{-1}{\rm div}(A\nabla)}\) be the degenerate elliptic operator on the Euclidean space \({{\mathbb{R}^{n}}}\), where w is a Muckenhoupt \({A_{2}({\mathbb{R}^{n}})}\) weight. In this article, the authors establish the Riesz transform characterization of the Hardy space \({H^{p}_{L_{w}}({\mathbb{R}}^{n})}\) associated with L_w, for \({w \in A_{q}({\mathbb{R}}^{n}) \cap RH_{\frac{n}{n-2}}({\mathbb{R}^{n}})}\) with \({n \geq 3}\), \({q \in [1,2]}\) and \({p \in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]}\) if, for some \({r \in (1,\,2]}\), \({{\{tL_w e^{-tL_w}\}}_{t\geq 0}}\) satisfies the weighted \({L^{r}-L^{2}}\) full off-diagonal estimates.     

A Schwarz Lemma at the Boundary of Hilbert Balls

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B′in separable complex Hilbert spaces H and H′, respectively. It is found that if the mapping f ∈ C~(1+α)at z_0∈ ?B with f(z_0) = w_0∈ ?B′, then the Fr′echet derivative operator Df(z_0) maps the tangent space Tz_0(?B~n) to Tw_0(?B′), the holomorphic tangent space T_(z_0)~(1,0)(?B~n) to T_(w_0)~(1,0)(?B′),respectively.     

The Eight Cayley–Dickson Doubling Products

The purpose of this paper is to identify all eight of the basic Cayley–Dickson doubling products. A Cayley–Dickson algebra \({\mathbb{A}_{N+1}}\) of dimension \({2^{N+1}}\) consists of all ordered pairs of elements of a Cayley–Dickson algebra \({\mathbb{A}_{N}}\) of dimension \({2^N}\) where the product \({(a, b)(c, d)}\) of elements of \({\mathbb{A}_{N+1}}\) is defined in terms of a pair of second degree binomials \({(f(a, b, c, d), g(a, b, c,d))}\) satisfying certain properties. The polynomial pair\({(f, g)}\) is called a ‘doubling product.’ While \({\mathbb{A}_{0}}\) may denote any ring, here it is taken to be the set \({\mathbb{R}}\) of real numbers. The binomials \({f}\) and \({g}\) should be devised such that \({\mathbb{A}_{1} = \mathbb{C}}\) the complex numbers, \({\mathbb{A}_{2} = \mathbb{H}}\) the quaternions, and \({\mathbb{A}_{3} = \mathbb{O}}\) the octonions. Historically, various researchers have used different yet equivalent doubling products.