Sarajevo Journal of Mathematics

Thanks for Organizing the Conference on the Occasion of My Jubilee

2024-08-01T10:07:30+00:00

My dear colleagues and friends, as well as my ex-students !

It has been such an honour to be a part of the beautiful Conference dedicated to me and my jubilee, that moved me to tears.

I would like to extend my heartfelt gratitude to all of you for this wonderful Conference, to all who presented and dedicated your papers to me.

I am especially thankful to two of my exceptional ex-students now university professors Mirna Džamonja and Duško Pavlović who initiated this event and have lightened up the spark within you and put in a lot of effort to organize this event.

Sincere thanks to the Editorial board of the Sarajevo Journal of Mathematics, too, who with Mirna handled the event throughout.

Finally, I would like to thank all the participants who took the time to attend and be with me at this event and thus contribute to making my jubilee very successful and unforgettable.

Thanks again everyone.

Mirjana Vuković

Happy Birthday Mirjana !

2024-08-01T10:17:05+00:00

This issue of Sarajevo Journal of Mathematics is devoted to our editor in chief Acad. Prof. Dr. Mirjana Vuković on the occasion of her jubilee. Here we will give a brief overview of her life and work.

Further Development on Krasner-Vuković Paragraded Structures and $p$-adic Interpolation of Yubo Jin $L$-values

2024-08-02T07:29:08+00:00

This paper is a joint project with Siegfried Bocherer (Mannheim), developing a recent preprint of Yubo Jin (Durham UK) previous works of Anh Tuan Do (Vietnam) and Dubrovnik, IUC-2016 papers from \textit{Sarajevo Journal of Mathematics} (Vol.12, No.2-Suppl., 2016).

We wish to use paragraded structures on {differential operators and arithmetical automorphic forms on classical groups and show that these structures provide a tool to construct $p$-adic measures and $p$-adic $L$-functions on the corresponding non-archimedean weight spaces.}

An approach to constructions of automorphic $L$-functions on unitary groups and their $p$-adic analogues is presented.

For an algebraic group $G$ over a number field $K$ these $L$ functions

are certain Euler products $L(s,\pi, r, \chi)$.

In particular, our constructions cover the $L$-functions in \cite{Shi00} via the doubling method of Piatetski-Shapiro and Rallis.

A $p$-adic analogue of $L(s,\pi, r, \chi)$ is a $p$-adic analytic function $L_p(s,\pi, r, \chi)$ of $p$-adic arguments $s \in \Z_p$, $\chi \bmod p^r$ which interpolates algebraic numbers defined through the normalized critical values $L^*(s,\pi,r, \chi)$ of the corresponding complex analytic $L$-function. We present a method using arithmetic nearly-holomorphic forms and general quasi-modular forms, related to algebraic automorphic forms. It gives a technique of constructing $p$-adic zeta-functions via general quasi-modular forms and their Fourier coefficients.

Testing Randomness by Matching Pennies

2024-08-02T08:04:59+00:00

In the game of Matching Pennies, Alice and Bob each hold a penny, and at every tick of the clock they simultaneously display the head or the tail sides of their coins. If they both display the same side, then Alice wins Bob's penny; if they display different sides, then Bob wins Alice's penny. To avoid giving the opponent a chance to win, both players seem to have nothing else to do but to randomly play heads and tails with equal frequencies. However, while not losing in this game is easy, not missing an opportunity to win is not. Randomizing your own moves can be made easy. Recognizing when the opponent's moves are not random can be arbitrarily hard.

The notion of randomness is central in game theory, but it is usually taken for granted. The notion of outsmarting is not central in game theory, but it is central in the practice of gaming. We pursue the idea that these two notions can be usefully viewed as two sides of the same coin. The resulting analysis suggests that the methods for strategizing in gaming and security, and for randomizing in computation, can be leveraged against each other.

2010 Mathematics Subject Classification. 03D32,91A26,91A26, 68Q32.

Isotypic equivalence of Abelian $p$-groups with separable reduced parts

2024-08-02T08:21:23+00:00

We prove that two Abelian $p$-groups with separable reduced parts are isotypically equivalent if and only if their divisible parts and their basic subgroups are elementarily equivalent. Also as a corollary we prove that any Abelian $p$-group with a separable reduced part is $\omega$-strongly homogeneous

Dessins D’Enfants on Reducible Surfaces

2024-08-02T08:38:46+00:00

In this paper we introduce dessins d'enfants on unions of surfaces, possibly glued. We show why they are natural, discuss their relations with Belyi pairs on reducible curves and provide some examples. In particular, we provide an example of a Fried pair which degenerates to a dessin on a reducible and singular curve.

Linear maps preserving the Cullis determinant of $(n+1)\times n$ matrices

2024-08-02T08:29:02+00:00

In this paper we give an explicit description of linear maps preserving the Cullis determinant of rectangular matrices of the size $(n+1)\times n.$ Unlike the result about the ordinary determinant, it appears that linear preservers of Cullis determinant can be singular. We provide the corresponding examples and characterize the case when these maps are non-singular.

On Invariant Hypercomplex Structures on Homogeneous Spaces

2024-08-02T08:31:42+00:00

An existence of invariant hypercomplex structure on compact homogeneous spaces implies strong restrictions on their root structure and consequently on their characteristic Pontrjagin classes and the corresponding Chern classes. We describe these constraints by making use of Lie theory.

2010 Mathematics Subject Classification. Primary: 22F30, 32Q60, Secondary: 22E60, 57R20.

Applications of the Multisubset Sum Problem Over Finite Abelian Groups

2024-08-02T08:38:13+00:00

In the article, we use the subset sum formula over a finite abelian group on the product of finite groups to derive the number of restricted partitions of elements in the group and to count the number of compositions over finite abelian groups. Later, we apply the formula for the multisubset sum problem on a group $\mathbb{Z}_n$ to produce a new technique for studying restricted partitions of positive integers.

2020 Mathematics Subject Classification. 05A17, 11P81

Almost Diagonalization of $\Psi$DO’s Over Various Generalized Function Spaces

2024-08-02T08:48:12+00:00

Inductive and projective type sequence spaces of sub- and super-exponential growth, and the corresponding inductive and projective limits of modulation spaces are considered as a framework for almost diagonalization of pseudo-differential operators. Moreover, recent results of the first author and B. Prangoski related to the almost diagonalization of pseudo-differential operators in the context of Hormander metrics are reviewed.

2010 Mathematics Subject Classification. 47G30, 46F05, 42C15, 58J40.

The Rate of Convergence of a Certain Mixed Monotone Difference Equation

2024-08-02T08:55:50+00:00

This paper investigates the rate of convergence of a certain mixed monotone rational second-order difference equation with quadratic terms. More precisely we give the precise rate of convergence for all attractors of the difference equation $x_{n+1}=\frac{Ax_{n}^{2}+Ex_{n-1}}{x_{n}^{2}+f}$, where all parameters are positive and initial conditions are non-negative.
The mentioned methods are illustrated in several characteristic examples.

2020 Mathematics Subject Classification. 39A10, 39A20, 65L20.

The Half-Inverse Transmission Problem for a Sturm-Liouville-Type Differential Equation With the Fixed Delay and Non Zero Initial Function

2024-08-02T09:03:36+00:00

In this paper, we consider the boundary value problem for the Sturm-Liouville type equation with the fixed delay $\frac{\pi}{2}$ and a non zero initial function under transmission conditions at the delay point. We study the case when all parameters within the transmission conditions are known and the potential function is known on the interval $\left(0,\frac{\pi}{2}\right)$. We will prove the uniqueness theorem from two spectra, first with Neumann boundary conditions and second with Cauchy boundary condition. Additionally, we will present an algorithm for the construction of the potential function over the interval $\left(\frac{\pi}{2},\pi\right)$

2020 Mathematics Subject Classification. 34K29,34B24

Characterization of Weyl Functions in the Class of Operator-Valued Generalized Nevanlinna Functions

2024-08-02T09:10:52+00:00

We provide the necessary and sufficient conditions for a generalized Nevanlinna function $Q$ ($Q\in N_{\kappa }\left( \mathcal{H} \right)$) to be a Weyl function (also known as a Weyl-Titchmarch function).

We also investigate an important subclass of $N_{\kappa }(\mathcal{H})$, the functions that have a boundedly invertible derivative at infinity $Q'\left( \infty \right):=\lim \limits_{z \to \infty}{zQ(z)}$. These functions are regular and have the operator representation $Q\left( z \right)=\tilde{\Gamma}^{+}\left( A-z \right)^{-1}\tilde{\Gamma},z\in \rho \left( A \right)$, where $A$ is a bounded self-adjoint operator in a Pontryagin space $\mathcal{K}$. We prove that every such strict function $Q$ is a Weyl function associated with the symmetric operator $S:=A_{\vert (I-P)\mathcal{K}}$, where $P$ is the orthogonal projection, $P:=\tilde{\Gamma} \left( \tilde{\Gamma}^{+} \tilde{\Gamma} \right)^{-1} \tilde{\Gamma}^{+} $.

Additionally, we provide the relation matrices of the adjoint relation $S^{+}$ of $S$, and of $\hat{A}$, where $\hat{A}$ is the representing relation of $\hat{Q}:=-Q^{-1}$. We illustrate our results through examples, wherein we begin with a given function $Q\in N_{\kappa }\left( \mathcal{H} \right)$ and proceed to determine the closed symmetric linear relation $S$ and the boundary triple $\Pi$ so that $Q$ becomes the Weyl function associated with $\Pi$.

2020 Mathematics Subject Classification. 34B20, 47B50, 47A06, 47A56

The First Regularized Trace of the Sturm-Liouville Operator With Robin Boundary Conditions

2024-08-02T09:20:34+00:00

This paper deals with the boundary value problem for the operator Sturm-Liouville type $ D^{2}=D^{2}(h, H, q_{1},q_{2},\tau,\varphi)$ generated by
$$-y''(x)+\sum_{i=1}^{2}q_i(x)y(x-i\tau)=\lambda y(x),\,x\in[0, \pi]$$
$$y'(0)-hy(0)=0,\, y'(\pi)+Hy(\pi)=0$$
where $\ds{\frac{\pi}{3}\leq \tau <\frac{\pi}{2}}$,\,\,$ h, H\in R\setminus\{0\}$ and $\lambda $ is a spectral parameter. We assume that $ q_i$, $i=1,2$ are real-valued potential functions from $ L_2 [0, \pi]$. We establish a formula for the first regularized trace of this operator.

2020 Mathematics Subject Classification. 34B09, 34B24, 34L10

Happy Birthday Alexei !

2024-08-02T08:24:36+00:00

In addition, I would like to send my birthday greetings to Alexei Panchishkin my long-time friend from way back in 1975, when I first was on a study stay as a doctoral "stazhor" at the famous Moscow State University – Lomonosov and first met him. At that time Alexei was one of the best and brilliant students of one of the greatest and famous scientists Yuri Ivanovich Manin from the Moscow's Lomonosov University. It was during the golden age of Russian and Soviet Mathematics.

I met him, for the first time, at the Manin's lectures Algebraic Questions in Differential Equations, or, in other words, Theory of Korteweg – de Vries. Among the listeners, in the large Lomonosov's amphiteather 16-10, were numerous well-known leading Lomonosov's professors: A.I. Kostrikin, L.A. Skornykov, A.V. Mikhalev, E.S. Golod, V.A. Zorich, Yu.A. Bahturin, A.Yu. Olshansky, as well as domestic and foreign visiting professors, doctoral students, and students.

Alexei Panchihkin – Lomonosov’s professor became professor and professor emeritus of the Grenoble Alpes University and a well-known scientist in the amazing worlds of numbers, functions, and varieties – the field of algebraic number theory with a special reference to the arithmetic of automorphic forms and the theory of complex and $p$-adic zeta functions, as well as the founder of new mathematical directions:

Arithmetic of automorphic $p$-adic $L$-functions,
Algorithmic arithmetic of function fields,
Solution of a problem of Coleman-Mazur,

and the author of new mathematical notions named after him:

Panchishkin distributions and
Panchishkin condition for the existence of motivic $p$-adic $L$-functions.

Dear Alexei Panchshkin,

My sincere congratulations on your 70th anniversary, once more !

Best wishes of good health and many new achievements – new books and much enthusiasm in mathematics and life.

Mirjana Vuković