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On the least common multiple of several consecutive values of a polynomial St. Petersburg Math. J. (IF 0.8) Pub Date : 2023-03-22 A. Dubickas
Abstract:The periodicity is proved for the arithmetic function defined as the quotient of the product of $k+1$ values (where $k \geq 1$) of a polynomial $f\in {\mathbb Z}[x]$ at $k + 1$ consecutive integers ${f(n) f(n + 1) \cdots f(n + k)}$ and the least common multiple of the corresponding integers $f(n)$, $f(n + 1)$, …, $f(n + k)$. It is shown that this function is periodic if and only if no difference
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Differentiable functions on modules and the equation 𝑔𝑟𝑎𝑑(𝑤)=𝖬𝗀𝗋𝖺𝖽(𝗏) St. Petersburg Math. J. (IF 0.8) Pub Date : 2023-03-22 K. Ciosmak
Abstract:Let $A$ be a finite-dimensional, commutative algebra over $\mathbb {R}$ or $\mathbb {C}$. The notion of $A$-differentiable functions on $A$ is extended to develop a theory of $A$-differentiable functions on finitely generated $A$-modules. Let $U$ be an open, bounded and convex subset of such a module. An explicit formula is given for $A$-differentiable functions on $U$ of prescribed class
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Nevanlinna characteristic and integral inequalities with maximal radial characteristic for meromorphic functions and for differences of subharmonic functions St. Petersburg Math. J. (IF 0.8) Pub Date : 2023-03-22 B. Khabibullin
Abstract:Let $f$ be a meromorphic function on the complex plane with Nevanlinna characteristic $T(r,f)$ and maximal radial characteristic $\ln M(t,f)$, where $M(t,f)$ is the maximum of the modulus $|f|$ on circles centered at zero and of radius $t$. A number of well-known and widely used results make it possible to estimate from above the integrals of $\ln M (t,f)$ over subsets $E$ on segments $[0
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Three dimensions of metric-measure spaces, Sobolev embeddings and optimal sign transport St. Petersburg Math. J. (IF 0.8) Pub Date : 2023-03-22 N. Nikolski
Abstract:A sign interlacing phenomenon for Bessel sequences, frames, and Riesz bases $(u_{k})$ in $L^2$ spaces over the spaces of homogeneous type $\Omega =(\Omega ,\rho ,\mu )$ satisfying the doubling/halving conditions is studied. Under some relations among three basic metric-measure parameters of $\Omega$, asymptotics is obtained for the mass moving norms $\|u_k\|_{KR}$ in the sense of Kantorovich–Rubinstein
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On the local finite separability of finitely generated associative rings St. Petersburg Math. J. (IF 0.8) Pub Date : 2023-03-22 S. Kublanovskii
Abstract:It is proved that analogs of the theorems of M. Hall and N. S. Romanovskii are not true in the class of commutative rings. Necessary and sufficient conditions for the local finite separability of monogenic rings are established. As a corollary, it is proved that a finitely generated torsion-free PI-ring is locally finitely separable if and only if its additive group is finitely generated.
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Twisted forms of classical groups St. Petersburg Math. J. (IF 0.8) Pub Date : 2023-03-22 E. Voronetsky
Abstract:Twisted forms of classical reductive group schemes are described in a unified way. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, called the augmented odd form algebras, consist of $2$-step nilpotent groups with an action of the underlying commutative ring, hence the basic descent theory for them will be developed
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Automorphisms of algebraic varieties and infinite transitivity St. Petersburg Math. J. (IF 0.8) Pub Date : 2023-03-22 I. Arzhantsev
Abstract:This is a survey of recent results on multiple transitivity for automorphism groups of affine algebraic varieties. The property of infinite transitivity of the special automorphism group is treated, which is equivalent to the flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes
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On the algebraic cobordism spectra 𝐌𝐒𝐋 and 𝐌𝐒𝐩 St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-12-16 I. Panin, C. Walter
Abstract:The algebraic cobordism spectra $\mathbf {MSL}$ and $\mathbf {MSp}$ are constructed. They are commutative monoids in the category of symmetric $T^{\wedge 2}$-spectra. The spectrum $\mathbf {MSp}$ comes with a natural symplectic orientation given either by a tautological Thom class $th^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(\mathbf {MSp}_2)$, or a tautological Borel class $b_{1}^{\mathbf {MSp}}
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Arrangements of a plane 𝑀-sextic with respect to a line St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-12-16 S. Orevkov
Abstract:The mutual arrangements of a real algebraic or real pseudoholomorphic plane projective $M$-sextic and a line up to isotopy are studied. A complete list of pseudoholomorphic arrangements is obtained. Four of them are proved to be algebraically unrealizable. All the others with two exceptions are algebraically realized.
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Commutators of relative and unrelative elementary unitary groups St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-12-16 N. Vavilov, Z. Zhang
Abstract:In the present paper, which is an outgrowth of the authors’ joint work with Anthony Bak and Roozbeh Hazrat on the unitary commutator calculus [9, 27, 30, 31], generators are found for the mixed commutator subgroups of relative elementary groups and unrelativized versions of commutator formulas are obtained in the setting of Bak’s unitary groups. It is a direct sequel of the papers [71, 76
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Two stars theorems for traces of the Zygmund space St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-12-16 A. Brudnyi
Abstract:For a Banach space $X$ defined in terms of a big-$O$ condition and its subspace x defined by the corresponding little-$o$ condition, the biduality property (generalizing the concept of reflexivity) asserts that the bidual of x is naturally isometrically isomorphic to $X$. The property is known for pairs of many classical function spaces (such as $(\ell _\infty , c_0)$, (BMO, VMO), (Lip, lip)
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Jackson type inequalities for differentiable functions in weighted Orlicz spaces St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-12-16 R. Akgün
Abstract:In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt $A_p$ condition. To obtain a refined version of the Jackson type inequality, an extrapolation theorem, Marcinkiewicz multiplier theorem, and Littlewood–Paley type results are proved. As a consequence, refined inverse Marchaud type
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Geometry of planar curves intersecting many lines at a few points St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 D. Vardakis, A. Volberg
Abstract:The local Lipschitz property is shown for the graphs avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand’s well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is $\sigma$-finite with respect to Hausdorff length and an estimate on the Hausdorff measure
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Boundary quasi-analyticity and a Phragmén–Lindelöf type theorem in classes of functions of bounded type in tubular domains St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 F. Shamoyan
Abstract:A complete description is obtained of the Carleman classes on $\mathbb {R}^n$ such that every function of bounded type in $\mathbb {C}^n_+$ whose boundary values belong to the class under study is in fact a member of the corresponding Carleman class in $\mathbb {C}^n_+\cup \mathbb {R}^n$. Also a refinement of the classical Salinas theorem is obtained, namely: under the conditions of the Salinas
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An effective algorithm for deciding the solvability of a system of polynomial equations over 𝑝-adic integers St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 A. Chistov
Abstract:Consider a system of polynomial equations in $n$ variables of degrees at most $d$ with integer coefficients whose lengths are at most $M$. By using a construction close to smooth stratification of algebraic varieties, it is shown that one can construct a positive integer \begin{equation*} \Delta < 2^{M(nd)^{c\, 2^n n^3}} \end{equation*} (here $c>0$ is a constant) depending on these polynomials
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Isomonodromic quantization of the second Painlevé equation by means of conservative Hamiltonian systems with two degrees of freedom St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 B. Suleimanov
Abstract:For the three nonstationary Schrödinger equations \begin{equation*} i\hbar \Psi _{\tau }=H(x,y,-i\hbar \frac {\partial }{\partial x},-i\hbar \frac {\partial }{\partial y})\Psi , \end{equation*} solutions are constructed that correspond to conservative Hamiltonian systems with two degrees of freedom whose general solutions can be represented by those of the second Painlevé equation. These solutions
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Supercharacters for parabolic contractions of finite groups of 𝐴,𝐵,𝐶,𝐷 Lie types St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 A. Panov
Abstract:Supercharacter theories are constructed for the finite groups obtained by parabolic contraction from simple groups of $A,B,C,D$ Lie types. Supercharacters and superclasses are classified in terms of rook placements in root systems.
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Upper estimates of the Morse numbers for the matrix elements of real linear irreducible representations of compact connected simple Lie groups St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 M. Meshcheryakov
Abstract:The Morse numbers of spaces of matrix elements for real irreducible linear representations of compact connected simple Lie groups are estimate from above in a variety of ways, in terms of the dimension, the Dynkin index of the representation, the eigenvalues of the invariant Laplace operator, and the volume of the group.
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Homotopic invariance of dihedral homologies for 𝐴_{∞}-algebras with involution St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 S. Lapin
Abstract:It is established that the dihedral homologies of involutive $A_{\infty }$-algebras are homotopically invariant with respect to the homotopy equivalences of involutive $A_{\infty }$-algebras. As a consequence, it is shown that over any field, the dihedral homologies of a topological space are isomorphic to the dihedral homologies of the involutive $A_{\infty }$-algebra of homologies for the
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Inverse mean value property of solutions to the modified Helmholtz equation St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 N. Kuznetsov
Abstract:A theorem characterizing balls in the Euclidean space $\mathbb {R}^m$ analytically is proved. For this purpose, positive solutions of the modified Helmholtz equation are applied instead of harmonic functions used in previous results. The resulting Kuran type theorem involves the volume mean value property of solutions to this equation. Other plausible inverse mean value properties of these
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Invariant subspaces of the generalized backward shift operator and rational functions St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 O. Ivanova, S. Melikhov, Yu. Melikhov
Abstract:The paper is devoted to a complete characterization of proper closed invariant subspaces of the generalized backward shift operator (Pommiez operator) in the Fréchet space of all holomorphic functions in a simply connected domain $\Omega \ni 0$ in the complex plane. In the case when the function that generates this operator does not have zeros in $\Omega$, all such subspaces are finite-dimensional
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Overgroups of subsystem subgroups in exceptional groups: nonideal levels St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-10-31 P. Gvozdevsky
Abstract:In the present paper, a description of overgroups for the subsystem subgroups $E(\Delta ,R)$ of the Chevalley groups $G(\Phi ,R)$ over the ring $R$, where $\Phi$ is a simply laced root system and $\Delta$ is its sufficiently large subsystem, is almost entirely finished. Namely, objects called levels are defined and it is shown that for any such overgroup $H$ there exists a unique level $\sigma$
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A new characterization of GCD domains of formal power series St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-08-24 A. Hamed
Abstract:By using the $v$-operation, a new characterization of the property for a power series ring to be a GCD domain is discussed. It is shown that if $D$ is a $\operatorname {UFD}$, then $D\lBrack X\rBrack$ is a GCD domain if and only if for any two integral $v$-invertible $v$-ideals $I$ and $J$ of $D\lBrack X\rBrack$ such that $(IJ)_{0}\neq (0),$ we have $((IJ)_{0})_{v}$ $= ((IJ)_{v})_{0},$ where
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The absence of eigenvalues for certain operators with partially periodic coefficients St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-08-24 N. Filonov
Abstract:The absence of eigenvalues is proved for the Schrödinger operator $-\Delta + V(x,y)$ in the Euclidean space whose potential is periodic in some variables and decays in the remaining variables faster than power $1$. A similar result for the Maxwell operator is established.
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Limit behavior of Weyl coefficients St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-08-24 R. Pruckner, H. Woracek
Abstract:The sets of radial or nontangential limit points towards $i\infty$ of a Nevanlinna function $q$ are studied. Given a nonempty, closed, and connected subset ${\mathcal {L}}$ of $\overline {{\mathbb {C}}_+}$, a Hamiltonian $H$ is constructed explicitly such that the radial and outer angular cluster sets towards $i\infty$ of the Weyl coefficient $q_H$ are both equal to ${\mathcal {L}}$. The method
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Approximation by polyanalytic functions in Hölder spaces St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-08-24 M. Mazalov
Abstract:The problem of approximation of functions on plane compact sets by polyanalytic functions of order higher than two in the Hölder spaces $C^m$, $m\in (0,1)$, is significantly more complicated than the well-studied problem of approximation by analytic functions. In particular, the fundamental solutions of the corresponding operators belong to all the indicated Hölder spaces, but this does not
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On a mathematical model of a repressilator St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-08-24 S. Glyzin, A. Kolesov, N. Rozov
Abstract:A mathematical model of the simplest three-link oscillatory gene network, the so-called repressilator, is considered. This model is a nonlinear singularly perturbed system of three ordinary differential equations. The existence and stability of a relaxation periodic solution invariant with respect to cyclic permutations of coordinates are investigated for this system.
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Smooth weight structures and birationality filtrations on motivic categories St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-08-24 M. Bondarko, D. Kumallagov
Abstract:In various triangulated motivic categories, a vast family of aisles (these are certain classes of objects) is introduced. These aisles are defined in terms of the corresponding “motives” (or motivic spectra) of smooth varieties; it is proved that they are expressed in terms of the corresponding homotopy $t$-structures. The aisles in question are described in terms of stalks at function fields
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Problems on the loss of heat: herd instinct versus individual feelings St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-08-24 A. Solynin
Abstract:Several problems are discussed concerning steady-state distribution of heat in domains in $\mathbb {R}^3$ that are complementary to a finite number of balls. The study of these problems was initiated by M. L. Glasser in 1977. Then, in 1978, M. L. Glasser and S. G. Davison presented numerical evidence that the heat flux from two equal balls in $\mathbb {R}^3$ decreases when the balls move closer
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Do some nontrivial closed 𝑧-invariant subspaces have the division property? St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-06-27 J. Esterle
Abstract:Banach spaces $E$ of functions holomorphic on the open unit disk $\mathbb {D}$ are considered such that the unilateral shift $S$ and the backward shift $T$ are bounded on $E$. Under the assumption that the spectra of $S$ and $T$ are equal to the closed unit disk, the existence is discussed of closed $z$-invariant subspaces $N$ of $E$ having the “division property,” which means that the function
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Weighted string equation where the weight is a noncompact multiplier: continuous spectrum and eigenvalues St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-06-27 E. B. Sharov, I. Sheipak
Abstract:The oscillation equation for a singular string with discrete weight generated by a self-similar $n$-link multiplier in the Sobolev space with a negative smoothness index is considered. It is shown that in the case of a noncompact multiplier, the string problem is equivalent to the spectral problem for an $(n-1)$-periodic Jacobi matrix. In the case of $n=3$, a complete description of the spectrum
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Heptagon relation in a direct sum St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-06-27 I. G. Korepanov
Abstract:An Ansatz is proposed for the heptagon relation, that is, an algebraic imitation of the five-dimensional Pachner move 4–3. The formula in question is realized in terms of matrices acting in a direct sum of one-dimensional linear spaces corresponding to 4-faces.
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The set of zeros of the Riemann zeta function as the point spectrum of an operator St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-06-27 V. Kapustin
Abstract:A possible way of proving the Riemann hypothesis consists of constructing a selfadjoint operartor whose spectrum coincides with the set $\{z\,: \, |\operatorname {Im}z|<\frac 12, \ \zeta \big (\frac {1}{2}-iz\big )=0\}$. In the paper we construct a rank-one perturbation of a selfadjoint operator related to a certain canonical system for which a similar property is fulfilled.
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On a Bellman function associated with the Chang–Wilson–Wolff theorem: a case study St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-06-27 F. Nazarov, V. Vasyunin, A. Volberg
Abstract:The tail of distribution (i.e., the measure of the set $\{f\ge x\}$) is estimated for those functions $f$ whose dyadic square function is bounded by a given constant. In particular, an estimate following from the Chang–Wilson–Wolf theorem is slightly improved. The study of the Bellman function corresponding to the problem reveals a curious structure of this function: it has jumps of the first
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Projective free algebras of bounded holomorphic functions on infinitely connected domains St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-06-27 A. Brudnyi
Abstract:The algebra $H^\infty (D)$ of bounded holomorphic functions on $D\subset \mathbb {C}$ is projective free for a wide class of infinitely connected domains. In particular, for such $D$ every rectangular left-invertible matrix with entries in $H^\infty (D)$ can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal
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Banach limits: extreme properties, invariance and the Fubini theorem St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-06-27 N. N. Avdeev, E. Semenov, A. Usachev
Abstract:A Banach limit on the space of all bounded real sequences is a positive normalized linear functional that is invariant with respect to the shift. The paper studies such properties of Banach limits as multiplicativity and the validity of Fubini’s theorem. A subset of Banach limits invariant with respect to dilation operators is also treated.
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Preservation of classes of entire functions defined in terms of growth restrictions along the real axis under perturbations of their zero sets St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-06-27 N. Abuzyarova
Abstract:Four special subsets of the Schwartz algebra are defined (this algebra consists of all entire functions of exponential type and of polynomial growth on the real axis). Perturbations of the zero sets for functions belonging to each of these subsets are studied. It is shown that the boundedness of the real part of the perturbing sequence is a sufficient and, generally speaking, unimprovable
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Compact Hankel operators on compact Abelian groups St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 A. Mirotin
Abstract:The classical theorems of Kronecker, Hartman, Peller, and Adamyan–Arov–Krein are extended to the context of a connected compact Abelian group $G$ with linearly ordered group of characters, on the basis of a description of the structure of compact Hankel operators on $G$. Beurling’s theorem on invariant subspaces is also generalized. Some applications to Hankel operators on discrete groups
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On the number of faces of the Gelfand–Zetlin polytope St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 E. Melikhova
Abstract:The combinatorics of the Gelfand–Zetlin polytope is studied. Geometric properties of a linear projection of this polytope onto a cube are employed to derive a recurrence relation for the $f$-polynomial of the polytope. This recurrence relation is applied to finding the $f$-polynomials and $h$-polynomials for one-parameter families of Gelfand–Zetlin polytopes of simplest types.
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Elliptic solitons and “freak waves” St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 V. Matveev, A. Smirnov
Abstract:It is shown that elliptic solutions to the AKNS hierarchy equations can be obtained by exploring spectral curves that correspond to elliptic solutions of the KdV hierarchy. This also allows one to get the quasirational and trigonometric solutions for AKNS hierarchy equations as a limit case of the elliptic solutions mentioned above.
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Joint weighted universality of the Hurwitz zeta-functions St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 A. Laurinčikas, G. Vadeikis
Abstract:Joint weighted universality theorems are proved concerning simultaneous approximation of a collection of analytic functions by a collection of shifts of Hurwitz zeta-functions with parameters $\alpha _1,\dots ,\alpha _r$. For this, linear independence is required over the field of rational numbers for the set $\{\log (m+\alpha _j)\,:\, m\in \mathbb {N}_0=\mathbb {N}\cup \{0\},\;j=1,\dots
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Dihedral modules with ∞-simplicial faces and dihedral homology for involutive 𝐴_{∞}-algebras over rings St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 S. Lapin
Abstract:On the basis of combinatorial techniques of dihedral modules with $\infty$-simplicial faces, dihedral homology is constructed for involutive $A_{\infty }$-algebras over arbitrary commutative unital rings.
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Index of a singular point of a vector field or of a 1-form on an orbifold St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 S. Gusein-Zade
Abstract:Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related to the Euler characteristic through the classical Poincaré–Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related to corresponding analogs of indices of singular points. Earlier, a notion of the universal
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Diagonal complexes for surfaces of finite type and surfaces with involution St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 G. Panina, J. Gordon
Abstract:Two constructions are studied that are inspired by the ideas of a recent paper by the authors. — The diagonal complex $\mathcal {D}$ and its barycentric subdivision $\mathcal {BD}$ related to an oriented surface of finite type $F$ equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called holes
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The order of growth of an exponential series near the boundary of the convergence domain St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 G. Gaisina
Abstract:For a class of analytic functions in a bounded convex domain $G$ that admit an exponential series expansion in $D$, the behavior of the coefficients of this expansion is studied in terms of the growth order near the boundary $\partial G$. In the case where $G$ has a smooth boundary, unimprovable two-sided estimates are established for the order via characteristics depending only on the exponents
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On interpolation and 𝐾-monotonicity for discrete local Morrey spaces St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-05-05 E. Berezhnoi
Abstract:A formula is given that makes it possible to reduce the calculation of an interpolation functor on a pair of local Morrey spaces to the calculation of this functor on pairs of vector function spaces constructed from the ideal spaces involved in the definition of the Morrey spaces in question. It is shown that a pair of local Morrey spaces is $K$-monotone if and only if the pair of vector function
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Complex WKB method for a system of two linear difference equations St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 A. Fedotov
Abstract:Analytic solutions of the difference equation $\Psi (z+h)=M(z)\Psi (z)$ are explored. Here $z$ is a complex variable, $h>0$ is a parameter, and $M$ is a given $SL(2,\mathbb {C})$-valued function. It is assumed that $M$ either is analytic in a bounded domain or is a trigonometric polynomial. A simple method to derive the asymptotics of solutions as $h\to 0$ is described.
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Searchlight asymptotics for high-frequency scattering by boundary inflection St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 V. Smyshlyaev, I. Kamotski
Abstract:The paper is devoted to an inner problem for a whispering gallery high-frequency asymptotic mode’s scattering by a boundary inflection. The related boundary-value problem for a Schrödinger equation on a half-line with a potential linear in both space and time turns out to be fundamental for describing transitions from modal to scattered asymptotic patterns, and despite having been intensively
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Threshold approximations for the resolvent of a polynomial nonnegative operator pencil St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 V. Sloushch, T. Suslina
Abstract:In a Hilbert space $\mathfrak {H}$, a family of operators $A(t)$, $t\in \mathbb {R}$, is treated admitting a factorization of the form $A(t) = X(t)^* X(t)$, where $X(t)=X_0+X_1t+\cdots +X_pt^p$, $p\ge 2$. It is assumed that the point $\lambda _0=0$ is an isolated eigenvalue of finite multiplicity for $A(0)$. Let $F(t)$ be the spectral projection of $A(t)$ for the interval $[0,\delta ]$. For
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Eigenvalue asymptotics for polynomially compact pseudodifferential operators St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 G. Rozenblum
Abstract:The asymptotics is found for eigenvalues of polynomially compact pseudodifferential operators of the zeroth order.
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Steady state non-Newtonian flow in a thin tube structure: equation on the graph St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 G. Panasenko, K. Pileckas, B. Vernescu
Abstract:The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions at the vertices. Nonlinear equations on the graph generated by the non-Newtonian rheology are treated here. The existence and uniqueness of a solution of this problem is proved. This solution describes the leading term of
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Asymptotics of the spectrum of the mixed boundary value problem for the Laplace operator in a thin spindle-shaped domain St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 S. Nazarov, J. Taskinen
Abstract:The asymptotics is examined for solutions to the spectral problem for the Laplace operator in a $d$-dimensional thin, of diameter $O(h)$, spindle-shaped domain $\Omega ^h$ with the Dirichlet condition on small, of size $h\ll 1$, terminal zones $\Gamma ^h_\pm$ and the Neumann condition on the remaining part of the boundary $\partial \Omega ^h$. In the limit as $h\rightarrow +0$, an ordinary
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Scattering of a surface wave in a polygonal domain with impedance boundary St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 M. Lyalinov, N. Zhu
Abstract:The two-dimensional (2D) domain under study is bounded from below by two semi-infinite and, between them, two finite straight lines; on each of the straight lines (segments), a usually individual impedance boundary condition is imposed. An incident surface wave, propagating from infinity along one semi-infinite segment of the polygonal domain, excites outgoing surface waves both on the same
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Metaharmonic functions: Mean flux theorem, its converse and related properties St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 N. Kuznetsov
Abstract:The mean flux theorems are proved for solutions of the Helmholtz equation and its modified version. Also, their converses are considered along with some other properties which generalize those that guarantee harmonicity.
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Solution asymptotics for the system of Landau–Lifshitz equations under a saddle-node dynamical bifurcation St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 L. Kalyakin
Abstract:A system of two nonlinear differential equations with slowly varying coefficients is treated. The asymptotics in the small parameter for the solutions that have a narrow transition layer is studied. Such a layer occurs near the moment where the number of roots of the corresponding algebraic system of equations changes. To construct the asymptotics, the matching method involving three scales
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Short wave diffraction on a contour with a Hölder singularity of the curvature St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 E. Zlobina, A. Kiselev
Abstract:Formulas are constructed for the short-wave asymptotics in the problem of diffraction of a plane wave on a contour with continuous curvature that is smooth everywhere except for one point near which it has a power-like behavior. The wave field is described in the boundary layers surrounding the singular point of the contour and the limit ray. An expression for the diffracted wave is found
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Nonstandard Liouville tori and caustics in asymptotics in the form of Airy and Bessel functions for 2D standing coastal waves St. Petersburg Math. J. (IF 0.8) Pub Date : 2022-03-04 A. Anikin, S. Dobrokhotov, V. Nazaikinskii, A. Tsvetkova
Abstract:The spectral problem $-\langle \nabla ,D(x)\nabla \psi \rangle = \lambda \psi$ in a bounded two-dimensional domain $\Omega$ is considered, where $D(x)$ is a smooth function positive inside the domain and zero on the boundary whose gradient is different from zero on the boundary. This problem arises in the study of long waves trapped by the shore and by bottom irregularities. For its asymptotic
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Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian St. Petersburg Math. J. (IF 0.8) Pub Date : 2021-12-28 N. Ustinov
Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in Ω ∈ C 2 : \Omega \in C^2: ( − Δ ) S p s u ( x ) + u ( x ) = u 2 s ∗ − 1 ( x ) (-\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x) . Here ( − Δ ) S p s (-\Delta )_{Sp}^s stands for the s s th power of the conventional Neumann Laplacian in Ω ⋐ R n \Omega \Subset
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On the sharpness of assumptions in the Federer theorem St. Petersburg Math. J. (IF 0.8) Pub Date : 2021-12-28 B. Makarov,A. Podkorytov
The Federer theorem deals with the “massiveness” of the set of critical values for a t t -smooth map acting from R m \mathbb R^m to R n \mathbb R^n : it claims that the Hausdorff p p -measure of this set is zero for certain p p . If n ≥ m n\ge m , it has long been known that the assumption of that theorem relating the parameters m , n , t , p m,n,t,p is sharp. Here it is shown by an example that this
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Distance difference functions on nonconvex boundaries of Riemannian manifolds St. Petersburg Math. J. (IF 0.8) Pub Date : 2021-12-28 S. Ivanov
It is shown that a complete Riemannian manifold with boundary is uniquely determined, up to isometry, by its distance difference representation on the boundary. Unlike previously known results, no restrictions on the boundary are imposed.